author  haftmann 
Tue, 31 Oct 2006 14:58:14 +0100  
changeset 21126  4dbc3ccbaab0 
parent 21113  5b76e541cc0a 
child 21131  a447addc14af 
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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*) 
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header {* The datatype of finite lists *} 
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theory List 
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imports PreList FunDef 
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begin 
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datatype 'a list = 
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Nil ("[]") 
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 Cons 'a "'a list" (infixr "#" 65) 

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subsection{*Basic list processing functions*} 
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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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map :: "('a=>'b) => ('a list => 'b list)" 

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

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

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

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

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

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

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

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

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upt :: "nat => nat => nat list" ("(1[_..</_'])") 
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remdups :: "'a list => 'a list" 
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remove1 :: "'a => 'a list => 'a list" 
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"distinct":: "'a list => bool" 
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replicate :: "nat => 'a => 'a list" 

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splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" 
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abbreviation 
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upto:: "nat => nat => nat list" ("(1[_../_])") 

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"[i..j] == [i..<(Suc j)]" 

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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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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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syntax (xsymbols) 
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"@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])") 
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syntax (HTML output) 
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"@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])") 

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New operator "lists" for formalizing sets of lists
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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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abbreviation 
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length :: "'a list => nat" 

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"length == size" 

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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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"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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"set [] = {}" 
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"set (x#xs) = insert x (set 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, but separate 

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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, but separate 

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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, but separate 

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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] = (case i of 0 => v # xs  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, but separate 

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

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

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

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

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primrec 
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"remove1 x [] = []" 
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"remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)" 

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primrec 
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replicate_0: "replicate 0 x = []" 
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replicate_Suc: "replicate (Suc n) x = x # replicate n x" 

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definition 
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rotate1 :: "'a list \<Rightarrow> 'a list" 
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rotate1_def: "rotate1 xs = (case xs of [] \<Rightarrow> []  x#xs \<Rightarrow> xs @ [x])" 
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rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" 
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rotate_def: "rotate n = rotate1 ^ n" 
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list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" 
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list_all2_def: "list_all2 P xs ys = 
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(length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))" 
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sublist :: "'a list => nat set => 'a list" 
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sublist_def: "sublist xs A = 
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map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))" 
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primrec 

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"splice [] ys = ys" 
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"splice (x#xs) ys = (if ys=[] then x#xs else x # hd ys # splice xs (tl ys))" 
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 {*Warning: simpset does not contain the second eqn but a derived one. *} 
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subsubsection {* @{const Nil} and @{const Cons} *} 
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lemma not_Cons_self [simp]: 
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"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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"(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs" 
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by (rule measure_induct [of length]) iprover 
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subsubsection {* @{const 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) 
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apply 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; 

266 
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> 

268 
\<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) 

274 
done 

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subsubsection {* @{text "@"}  append *} 
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lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)" 
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by (induct xs) auto 
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lemma append_Nil2 [simp]: "xs @ [] = xs" 
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by (induct xs) auto 
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lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])" 
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by (induct xs) auto 
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lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])" 
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by (induct xs) auto 
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lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])" 
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by (induct xs) auto 
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lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])" 
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by (induct xs) auto 
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lemma append_eq_append_conv [simp]: 
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"!!ys. length xs = length ys \<or> length us = length vs 
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==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)" 
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apply (induct xs) 
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apply (case_tac ys, simp, force) 
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apply (case_tac ys, force, simp) 

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done 
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lemma append_eq_append_conv2: "!!ys zs ts. 
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(xs @ ys = zs @ ts) = 

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(EX us. xs = zs @ us & us @ ys = ts  xs @ us = zs & ys = us@ ts)" 

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apply (induct xs) 

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apply fastsimp 

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apply(case_tac zs) 

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apply simp 

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apply fastsimp 

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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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lemma Cons_eq_append_conv: "x#xs = ys@zs = 
346 
(ys = [] & x#xs = zs  (EX ys'. x#ys' = ys & xs = ys'@zs))" 

347 
by(cases ys) auto 

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

351 
by(cases ys) auto 

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text {* Trivial rules for solving @{text "@"}equations automatically. *} 
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356 
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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text {* 
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Simplification procedure for all list equalities. 
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Currently only tries to rearrange @{text "@"} to see if 

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 both lists end in a singleton list, 

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 or both lists end in the same list. 

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*} 
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ML_setup {* 

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local 
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val append_assoc = thm "append_assoc"; 
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val append_Nil = thm "append_Nil"; 

380 
val append_Cons = thm "append_Cons"; 

381 
val append1_eq_conv = thm "append1_eq_conv"; 

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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) 
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 last (Const("List.op @",_) $ _ $ ys) = last ys 

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 last t = t; 

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389 
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true 

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 list1 _ = false; 
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392 
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) 
394 
 butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys 

395 
 butlast xs = Const("List.list.Nil",fastype_of xs); 

13114  396 

16973  397 
val rearr_ss = HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]; 
398 

20044
92cc2f4c7335
simprocs: no theory argument  use simpset context instead;
wenzelm
parents:
19890
diff
changeset

399 
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) = 
13462  400 
let 
401 
val lastl = last lhs and lastr = last rhs; 

402 
fun rearr conv = 

403 
let 

404 
val lhs1 = butlast lhs and rhs1 = butlast rhs; 

405 
val Type(_,listT::_) = eqT 

406 
val appT = [listT,listT] > listT 

407 
val app = Const("List.op @",appT) 

408 
val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr) 

13480
bb72bd43c6c3
use Tactic.prove instead of prove_goalw_cterm in internal proofs!
wenzelm
parents:
13462
diff
changeset

409 
val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2)); 
20044
92cc2f4c7335
simprocs: no theory argument  use simpset context instead;
wenzelm
parents:
19890
diff
changeset

410 
val thm = Goal.prove (Simplifier.the_context ss) [] [] eq 
17877
67d5ab1cb0d8
Simplifier.inherit_context instead of Simplifier.inherit_bounds;
wenzelm
parents:
17830
diff
changeset

411 
(K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1)); 
15531  412 
in SOME ((conv RS (thm RS trans)) RS eq_reflection) end; 
13114  413 

13462  414 
in 
415 
if list1 lastl andalso list1 lastr then rearr append1_eq_conv 

416 
else if lastl aconv lastr then rearr append_same_eq 

15531  417 
else NONE 
13462  418 
end; 
419 

13114  420 
in 
13462  421 

422 
val list_eq_simproc = 

21061
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added normal post setup; cleaned up "execution" constants
haftmann
parents:
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diff
changeset

423 
Simplifier.simproc (the_context ()) "list_eq" ["(xs::'a list) = ys"] (K list_eq); 
13462  424 

13114  425 
end; 
426 

427 
Addsimprocs [list_eq_simproc]; 

428 
*} 

429 

430 

15392  431 
subsubsection {* @{text map} *} 
13114  432 

13142  433 
lemma map_ext: "(!!x. x : set xs > f x = g x) ==> map f xs = map g xs" 
13145  434 
by (induct xs) simp_all 
13114  435 

13142  436 
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)" 
13145  437 
by (rule ext, induct_tac xs) auto 
13114  438 

13142  439 
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys" 
13145  440 
by (induct xs) auto 
13114  441 

13142  442 
lemma map_compose: "map (f o g) xs = map f (map g xs)" 
13145  443 
by (induct xs) (auto simp add: o_def) 
13114  444 

13142  445 
lemma rev_map: "rev (map f xs) = map f (rev xs)" 
13145  446 
by (induct xs) auto 
13114  447 

13737  448 
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)" 
449 
by (induct xs) auto 

450 

19770
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HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents:
19623
diff
changeset

451 
lemma map_cong [fundef_cong, recdef_cong]: 
13145  452 
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys" 
453 
 {* a congruence rule for @{text map} *} 

13737  454 
by simp 
13114  455 

13142  456 
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])" 
13145  457 
by (cases xs) auto 
13114  458 

13142  459 
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])" 
13145  460 
by (cases xs) auto 
13114  461 

18447  462 
lemma map_eq_Cons_conv: 
14025  463 
"(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)" 
13145  464 
by (cases xs) auto 
13114  465 

18447  466 
lemma Cons_eq_map_conv: 
14025  467 
"(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)" 
468 
by (cases ys) auto 

469 

18447  470 
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1] 
471 
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1] 

472 
declare map_eq_Cons_D [dest!] Cons_eq_map_D [dest!] 

473 

14111  474 
lemma ex_map_conv: 
475 
"(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)" 

18447  476 
by(induct ys, auto simp add: Cons_eq_map_conv) 
14111  477 

15110
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Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

478 
lemma map_eq_imp_length_eq: 
78b5636eabc7
Added a number of new thms and the new function remove1
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parents:
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diff
changeset

479 
"!!xs. map f xs = map f ys ==> length xs = length ys" 
78b5636eabc7
Added a number of new thms and the new function remove1
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parents:
15072
diff
changeset

480 
apply (induct ys) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
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diff
changeset

481 
apply simp 
78b5636eabc7
Added a number of new thms and the new function remove1
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parents:
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diff
changeset

482 
apply(simp (no_asm_use)) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

483 
apply clarify 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

484 
apply(simp (no_asm_use)) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

485 
apply fast 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

486 
done 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

487 

78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

488 
lemma map_inj_on: 
78b5636eabc7
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nipkow
parents:
15072
diff
changeset

489 
"[ map f xs = map f ys; inj_on f (set xs Un set ys) ] 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

490 
==> xs = ys" 
78b5636eabc7
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nipkow
parents:
15072
diff
changeset

491 
apply(frule map_eq_imp_length_eq) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

492 
apply(rotate_tac 1) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
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diff
changeset

493 
apply(induct rule:list_induct2) 
78b5636eabc7
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nipkow
parents:
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diff
changeset

494 
apply simp 
78b5636eabc7
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parents:
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diff
changeset

495 
apply(simp) 
78b5636eabc7
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nipkow
parents:
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diff
changeset

496 
apply (blast intro:sym) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

497 
done 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

498 

78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
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diff
changeset

499 
lemma inj_on_map_eq_map: 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

500 
"inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)" 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

501 
by(blast dest:map_inj_on) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
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diff
changeset

502 

13114  503 
lemma map_injective: 
14338  504 
"!!xs. map f xs = map f ys ==> inj f ==> xs = ys" 
505 
by (induct ys) (auto dest!:injD) 

13114  506 

14339  507 
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)" 
508 
by(blast dest:map_injective) 

509 

13114  510 
lemma inj_mapI: "inj f ==> inj (map f)" 
17589  511 
by (iprover dest: map_injective injD intro: inj_onI) 
13114  512 

513 
lemma inj_mapD: "inj (map f) ==> inj f" 

14208  514 
apply (unfold inj_on_def, clarify) 
13145  515 
apply (erule_tac x = "[x]" in ballE) 
14208  516 
apply (erule_tac x = "[y]" in ballE, simp, blast) 
13145  517 
apply blast 
518 
done 

13114  519 

14339  520 
lemma inj_map[iff]: "inj (map f) = inj f" 
13145  521 
by (blast dest: inj_mapD intro: inj_mapI) 
13114  522 

15303  523 
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A" 
524 
apply(rule inj_onI) 

525 
apply(erule map_inj_on) 

526 
apply(blast intro:inj_onI dest:inj_onD) 

527 
done 

528 

14343  529 
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs" 
530 
by (induct xs, auto) 

13114  531 

14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset

532 
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs" 
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset

533 
by (induct xs) auto 
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset

534 

15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

535 
lemma map_fst_zip[simp]: 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

536 
"length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs" 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

537 
by (induct rule:list_induct2, simp_all) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

538 

78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

539 
lemma map_snd_zip[simp]: 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

540 
"length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys" 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

541 
by (induct rule:list_induct2, simp_all) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

542 

78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

543 

15392  544 
subsubsection {* @{text rev} *} 
13114  545 

13142  546 
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs" 
13145  547 
by (induct xs) auto 
13114  548 

13142  549 
lemma rev_rev_ident [simp]: "rev (rev xs) = xs" 
13145  550 
by (induct xs) auto 
13114  551 

15870  552 
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)" 
553 
by auto 

554 

13142  555 
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])" 
13145  556 
by (induct xs) auto 
13114  557 

13142  558 
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])" 
13145  559 
by (induct xs) auto 
13114  560 

15870  561 
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])" 
562 
by (cases xs) auto 

563 

564 
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])" 

565 
by (cases xs) auto 

566 

21061
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset

567 
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)" 
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset

568 
apply (induct xs arbitrary: ys, force) 
14208  569 
apply (case_tac ys, simp, force) 
13145  570 
done 
13114  571 

15439  572 
lemma inj_on_rev[iff]: "inj_on rev A" 
573 
by(simp add:inj_on_def) 

574 

13366  575 
lemma rev_induct [case_names Nil snoc]: 
576 
"[ P []; !!x xs. P xs ==> P (xs @ [x]) ] ==> P xs" 

15489
d136af442665
Replaced application of subst by simplesubst in proof of rev_induct
berghofe
parents:
15439
diff
changeset

577 
apply(simplesubst rev_rev_ident[symmetric]) 
13145  578 
apply(rule_tac list = "rev xs" in list.induct, simp_all) 
579 
done 

13114  580 

13145  581 
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *} "compatibility" 
13114  582 

13366  583 
lemma rev_exhaust [case_names Nil snoc]: 
584 
"(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P" 

13145  585 
by (induct xs rule: rev_induct) auto 
13114  586 

13366  587 
lemmas rev_cases = rev_exhaust 
588 

18423  589 
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])" 
590 
by(rule rev_cases[of xs]) auto 

591 

13114  592 

15392  593 
subsubsection {* @{text set} *} 
13114  594 

13142  595 
lemma finite_set [iff]: "finite (set xs)" 
13145  596 
by (induct xs) auto 
13114  597 

13142  598 
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)" 
13145  599 
by (induct xs) auto 
13114  600 

17830  601 
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs" 
602 
by(cases xs) auto 

14099  603 

13142  604 
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)" 
13145  605 
by auto 
13114  606 

14099  607 
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
608 
by auto 

609 

13142  610 
lemma set_empty [iff]: "(set xs = {}) = (xs = [])" 
13145  611 
by (induct xs) auto 
13114  612 

15245  613 
lemma set_empty2[iff]: "({} = set xs) = (xs = [])" 
614 
by(induct xs) auto 

615 

13142  616 
lemma set_rev [simp]: "set (rev xs) = set xs" 
13145  617 
by (induct xs) auto 
13114  618 

13142  619 
lemma set_map [simp]: "set (map f xs) = f`(set xs)" 
13145  620 
by (induct xs) auto 
13114  621 

13142  622 
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}" 
13145  623 
by (induct xs) auto 
13114  624 

15425  625 
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}" 
14208  626 
apply (induct j, simp_all) 
627 
apply (erule ssubst, auto) 

13145  628 
done 
13114  629 

13142  630 
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)" 
15113  631 
proof (induct xs) 
632 
case Nil show ?case by simp 

633 
case (Cons a xs) 

634 
show ?case 

635 
proof 

636 
assume "x \<in> set (a # xs)" 

637 
with prems show "\<exists>ys zs. a # xs = ys @ x # zs" 

638 
by (simp, blast intro: Cons_eq_appendI) 

639 
next 

640 
assume "\<exists>ys zs. a # xs = ys @ x # zs" 

641 
then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast 

642 
show "x \<in> set (a # xs)" 

643 
by (cases ys, auto simp add: eq) 

644 
qed 

645 
qed 

13142  646 

18049  647 
lemma in_set_conv_decomp_first: 
648 
"(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)" 

649 
proof (induct xs) 

650 
case Nil show ?case by simp 

651 
next 

652 
case (Cons a xs) 

653 
show ?case 

654 
proof cases 

655 
assume "x = a" thus ?case using Cons by force 

656 
next 

657 
assume "x \<noteq> a" 

658 
show ?case 

659 
proof 

660 
assume "x \<in> set (a # xs)" 

661 
from prems show "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys" 

662 
by(fastsimp intro!: Cons_eq_appendI) 

663 
next 

664 
assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys" 

665 
then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast 

666 
show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq) 

667 
qed 

668 
qed 

669 
qed 

670 

671 
lemmas split_list = in_set_conv_decomp[THEN iffD1, standard] 

672 
lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard] 

673 

674 

13508  675 
lemma finite_list: "finite A ==> EX l. set l = A" 
676 
apply (erule finite_induct, auto) 

677 
apply (rule_tac x="x#l" in exI, auto) 

678 
done 

679 

14388  680 
lemma card_length: "card (set xs) \<le> length xs" 
681 
by (induct xs) (auto simp add: card_insert_if) 

13114  682 

15168  683 

15392  684 
subsubsection {* @{text filter} *} 
13114  685 

13142  686 
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys" 
13145  687 
by (induct xs) auto 
13114  688 

15305  689 
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)" 
690 
by (induct xs) simp_all 

691 

13142  692 
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs" 
13145  693 
by (induct xs) auto 
13114  694 

16998  695 
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs" 
696 
by (induct xs) (auto simp add: le_SucI) 

697 

18423  698 
lemma sum_length_filter_compl: 
699 
"length(filter P xs) + length(filter (%x. ~P x) xs) = length xs" 

700 
by(induct xs) simp_all 

701 

13142  702 
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs" 
13145  703 
by (induct xs) auto 
13114  704 

13142  705 
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []" 
13145  706 
by (induct xs) auto 
13114  707 

16998  708 
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
709 
by (induct xs) simp_all 

710 

711 
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)" 

712 
apply (induct xs) 

713 
apply auto 

714 
apply(cut_tac P=P and xs=xs in length_filter_le) 

715 
apply simp 

716 
done 

13114  717 

16965  718 
lemma filter_map: 
719 
"filter P (map f xs) = map f (filter (P o f) xs)" 

720 
by (induct xs) simp_all 

721 

722 
lemma length_filter_map[simp]: 

723 
"length (filter P (map f xs)) = length(filter (P o f) xs)" 

724 
by (simp add:filter_map) 

725 

13142  726 
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs" 
13145  727 
by auto 
13114  728 

15246  729 
lemma length_filter_less: 
730 
"\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs" 

731 
proof (induct xs) 

732 
case Nil thus ?case by simp 

733 
next 

734 
case (Cons x xs) thus ?case 

735 
apply (auto split:split_if_asm) 

736 
using length_filter_le[of P xs] apply arith 

737 
done 

738 
qed 

13114  739 

15281  740 
lemma length_filter_conv_card: 
741 
"length(filter p xs) = card{i. i < length xs & p(xs!i)}" 

742 
proof (induct xs) 

743 
case Nil thus ?case by simp 

744 
next 

745 
case (Cons x xs) 

746 
let ?S = "{i. i < length xs & p(xs!i)}" 

747 
have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite) 

748 
show ?case (is "?l = card ?S'") 

749 
proof (cases) 

750 
assume "p x" 

751 
hence eq: "?S' = insert 0 (Suc ` ?S)" 

752 
by(auto simp add: nth_Cons image_def split:nat.split elim:lessE) 

753 
have "length (filter p (x # xs)) = Suc(card ?S)" 

754 
using Cons by simp 

755 
also have "\<dots> = Suc(card(Suc ` ?S))" using fin 

756 
by (simp add: card_image inj_Suc) 

757 
also have "\<dots> = card ?S'" using eq fin 

758 
by (simp add:card_insert_if) (simp add:image_def) 

759 
finally show ?thesis . 

760 
next 

761 
assume "\<not> p x" 

762 
hence eq: "?S' = Suc ` ?S" 

763 
by(auto simp add: nth_Cons image_def split:nat.split elim:lessE) 

764 
have "length (filter p (x # xs)) = card ?S" 

765 
using Cons by simp 

766 
also have "\<dots> = card(Suc ` ?S)" using fin 

767 
by (simp add: card_image inj_Suc) 

768 
also have "\<dots> = card ?S'" using eq fin 

769 
by (simp add:card_insert_if) 

770 
finally show ?thesis . 

771 
qed 

772 
qed 

773 

17629  774 
lemma Cons_eq_filterD: 
775 
"x#xs = filter P ys \<Longrightarrow> 

776 
\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs" 

19585  777 
(is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs") 
17629  778 
proof(induct ys) 
779 
case Nil thus ?case by simp 

780 
next 

781 
case (Cons y ys) 

782 
show ?case (is "\<exists>x. ?Q x") 

783 
proof cases 

784 
assume Py: "P y" 

785 
show ?thesis 

786 
proof cases 

787 
assume xy: "x = y" 

788 
show ?thesis 

789 
proof from Py xy Cons(2) show "?Q []" by simp qed 

790 
next 

791 
assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp 

792 
qed 

793 
next 

794 
assume Py: "\<not> P y" 

795 
with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp 

796 
show ?thesis (is "? us. ?Q us") 

797 
proof show "?Q (y#us)" using 1 by simp qed 

798 
qed 

799 
qed 

800 

801 
lemma filter_eq_ConsD: 

802 
"filter P ys = x#xs \<Longrightarrow> 

803 
\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs" 

804 
by(rule Cons_eq_filterD) simp 

805 

806 
lemma filter_eq_Cons_iff: 

807 
"(filter P ys = x#xs) = 

808 
(\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)" 

809 
by(auto dest:filter_eq_ConsD) 

810 

811 
lemma Cons_eq_filter_iff: 

812 
"(x#xs = filter P ys) = 

813 
(\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)" 

814 
by(auto dest:Cons_eq_filterD) 

815 

19770
be5c23ebe1eb
HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents:
19623
diff
changeset

816 
lemma filter_cong[fundef_cong, recdef_cong]: 
17501  817 
"xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys" 
818 
apply simp 

819 
apply(erule thin_rl) 

820 
by (induct ys) simp_all 

821 

15281  822 

15392  823 
subsubsection {* @{text concat} *} 
13114  824 

13142  825 
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys" 
13145  826 
by (induct xs) auto 
13114  827 

18447  828 
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])" 
13145  829 
by (induct xss) auto 
13114  830 

18447  831 
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])" 
13145  832 
by (induct xss) auto 
13114  833 

13142  834 
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)" 
13145  835 
by (induct xs) auto 
13114  836 

13142  837 
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)" 
13145  838 
by (induct xs) auto 
13114  839 

13142  840 
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)" 
13145  841 
by (induct xs) auto 
13114  842 

13142  843 
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))" 
13145  844 
by (induct xs) auto 
13114  845 

846 

15392  847 
subsubsection {* @{text nth} *} 
13114  848 

13142  849 
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x" 
13145  850 
by auto 
13114  851 

13142  852 
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n" 
13145  853 
by auto 
13114  854 

13142  855 
declare nth.simps [simp del] 
13114  856 

857 
lemma nth_append: 

13145  858 
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n  length xs))" 
14208  859 
apply (induct "xs", simp) 
860 
apply (case_tac n, auto) 

13145  861 
done 
13114  862 

14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset

863 
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x" 
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset

864 
by (induct "xs") auto 
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset

865 

4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset

866 
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n" 
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset

867 
by (induct "xs") auto 
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset

868 

13142  869 
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)" 
14208  870 
apply (induct xs, simp) 
871 
apply (case_tac n, auto) 

13145  872 
done 
13114  873 

18423  874 
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0" 
875 
by(cases xs) simp_all 

876 

18049  877 

878 
lemma list_eq_iff_nth_eq: 

879 
"!!ys. (xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))" 

880 
apply(induct xs) 

881 
apply simp apply blast 

882 
apply(case_tac ys) 

883 
apply simp 

884 
apply(simp add:nth_Cons split:nat.split)apply blast 

885 
done 

886 

13142  887 
lemma set_conv_nth: "set xs = {xs!i  i. i < length xs}" 
15251  888 
apply (induct xs, simp, simp) 
13145  889 
apply safe 
14208  890 
apply (rule_tac x = 0 in exI, simp) 
891 
apply (rule_tac x = "Suc i" in exI, simp) 

892 
apply (case_tac i, simp) 

13145  893 
apply (rename_tac j) 
14208  894 
apply (rule_tac x = j in exI, simp) 
13145  895 
done 
13114  896 

17501  897 
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)" 
898 
by(auto simp:set_conv_nth) 

899 

13145  900 
lemma list_ball_nth: "[ n < length xs; !x : set xs. P x] ==> P(xs!n)" 
901 
by (auto simp add: set_conv_nth) 

13114  902 

13142  903 
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs" 
13145  904 
by (auto simp add: set_conv_nth) 
13114  905 

906 
lemma all_nth_imp_all_set: 

13145  907 
"[ !i < length xs. P(xs!i); x : set xs] ==> P x" 
908 
by (auto simp add: set_conv_nth) 

13114  909 

910 
lemma all_set_conv_all_nth: 

13145  911 
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs > P (xs ! i))" 
912 
by (auto simp add: set_conv_nth) 

13114  913 

914 

15392  915 
subsubsection {* @{text list_update} *} 
13114  916 

13142  917 
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs" 
13145  918 
by (induct xs) (auto split: nat.split) 
13114  919 

920 
lemma nth_list_update: 

13145  921 
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)" 
922 
by (induct xs) (auto simp add: nth_Cons split: nat.split) 

13114  923 

13142  924 
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x" 
13145  925 
by (simp add: nth_list_update) 
13114  926 

13142  927 
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j" 
13145  928 
by (induct xs) (auto simp add: nth_Cons split: nat.split) 
13114  929 

13142  930 
lemma list_update_overwrite [simp]: 
13145  931 
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]" 
932 
by (induct xs) (auto split: nat.split) 

13114  933 

14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset

934 
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs" 
14208  935 
apply (induct xs, simp) 
14187  936 
apply(simp split:nat.splits) 
937 
done 

938 

17501  939 
lemma list_update_beyond[simp]: "\<And>i. length xs \<le> i \<Longrightarrow> xs[i:=x] = xs" 
940 
apply (induct xs) 

941 
apply simp 

942 
apply (case_tac i) 

943 
apply simp_all 

944 
done 

945 

13114  946 
lemma list_update_same_conv: 
13145  947 
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)" 
948 
by (induct xs) (auto split: nat.split) 

13114  949 

14187  950 
lemma list_update_append1: 
951 
"!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys" 

14208  952 
apply (induct xs, simp) 
14187  953 
apply(simp split:nat.split) 
954 
done 

955 

15868  956 
lemma list_update_append: 
957 
"!!n. (xs @ ys) [n:= x] = 

958 
(if n < length xs then xs[n:= x] @ ys else xs @ (ys [nlength xs:= x]))" 

959 
by (induct xs) (auto split:nat.splits) 

960 

14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset

961 
lemma list_update_length [simp]: 
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset

962 
"(xs @ x # ys)[length xs := y] = (xs @ y # ys)" 
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset

963 
by (induct xs, auto) 
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset

964 

13114  965 
lemma update_zip: 
13145  966 
"!!i xy xs. length xs = length ys ==> 
967 
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])" 

968 
by (induct ys) (auto, case_tac xs, auto split: nat.split) 

13114  969 

970 
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)" 

13145  971 
by (induct xs) (auto split: nat.split) 
13114  972 

973 
lemma set_update_subsetI: "[ set xs <= A; x:A ] ==> set(xs[i := x]) <= A" 

13145  974 
by (blast dest!: set_update_subset_insert [THEN subsetD]) 
13114  975 

15868  976 
lemma set_update_memI: "!!n. n < length xs \<Longrightarrow> x \<in> set (xs[n := x])" 
977 
by (induct xs) (auto split:nat.splits) 

978 

13114  979 

15392  980 
subsubsection {* @{text last} and @{text butlast} *} 
13114  981 

13142  982 
lemma last_snoc [simp]: "last (xs @ [x]) = x" 
13145  983 
by (induct xs) auto 
13114  984 

13142  985 
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs" 
13145  986 
by (induct xs) auto 
13114  987 

14302  988 
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x" 
989 
by(simp add:last.simps) 

990 

991 
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs" 

992 
by(simp add:last.simps) 

993 

994 
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)" 

995 
by (induct xs) (auto) 

996 

997 
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs" 

998 
by(simp add:last_append) 

999 

1000 
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys" 

1001 
by(simp add:last_append) 

1002 

17762  1003 
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs" 
1004 
by(rule rev_exhaust[of xs]) simp_all 

1005 

1006 
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs" 

1007 
by(cases xs) simp_all 

1008 

17765  1009 
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as" 
1010 
by (induct as) auto 

17762  1011 

13142  1012 
lemma length_butlast [simp]: "length (butlast xs) = length xs  1" 
13145  1013 
by (induct xs rule: rev_induct) auto 
13114  1014 

1015 
lemma butlast_append: 

13145  1016 
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)" 
1017 
by (induct xs) auto 

13114  1018 

13142  1019 
lemma append_butlast_last_id [simp]: 
13145  1020 
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs" 
1021 
by (induct xs) auto 

13114  1022 

13142  1023 
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs" 
13145  1024 
by (induct xs) (auto split: split_if_asm) 
13114  1025 

1026 
lemma in_set_butlast_appendI: 

13145  1027 
"x : set (butlast xs)  x : set (butlast ys) ==> x : set (butlast (xs @ ys))" 
1028 
by (auto dest: in_set_butlastD simp add: butlast_append) 

13114  1029 

17501  1030 
lemma last_drop[simp]: "!!n. n < length xs \<Longrightarrow> last (drop n xs) = last xs" 
1031 
apply (induct xs) 

1032 
apply simp 

1033 
apply (auto split:nat.split) 

1034 
done 

1035 

17589  1036 
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs  1)" 
1037 
by(induct xs)(auto simp:neq_Nil_conv) 

1038 

15392  1039 
subsubsection {* @{text take} and @{text drop} *} 
13114  1040 

13142  1041 
lemma take_0 [simp]: "take 0 xs = []" 
13145  1042 
by (induct xs) auto 
13114  1043 

13142  1044 
lemma drop_0 [simp]: "drop 0 xs = xs" 
13145  1045 
by (induct xs) auto 
13114  1046 

13142  1047 
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs" 
13145  1048 
by simp 
13114  1049 

13142  1050 
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs" 
13145  1051 
by simp 
13114  1052 

13142  1053 
declare take_Cons [simp del] and drop_Cons [simp del] 
13114  1054 

15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1055 
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)" 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1056 
by(clarsimp simp add:neq_Nil_conv) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1057 

14187  1058 
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)" 
1059 
by(cases xs, simp_all) 

1060 

1061 
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)" 

1062 
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split) 

1063 

1064 
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y" 

14208  1065 
apply (induct xs, simp) 
14187  1066 
apply(simp add:drop_Cons nth_Cons split:nat.splits) 
1067 
done 

1068 

13913  1069 
lemma take_Suc_conv_app_nth: 
1070 
"!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]" 

14208  1071 
apply (induct xs, simp) 
1072 
apply (case_tac i, auto) 

13913  1073 
done 
1074 

14591  1075 
lemma drop_Suc_conv_tl: 
1076 
"!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs" 

1077 
apply (induct xs, simp) 

1078 
apply (case_tac i, auto) 

1079 
done 

1080 

13142  1081 
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n" 
13145  1082 
by (induct n) (auto, case_tac xs, auto) 
13114  1083 

13142  1084 
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs  n)" 
13145  1085 
by (induct n) (auto, case_tac xs, auto) 
13114  1086 

13142  1087 
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs" 
13145  1088 
by (induct n) (auto, case_tac xs, auto) 
13114  1089 

13142  1090 
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []" 
13145  1091 
by (induct n) (auto, case_tac xs, auto) 
13114  1092 

13142  1093 
lemma take_append [simp]: 
13145  1094 
"!!xs. take n (xs @ ys) = (take n xs @ take (n  length xs) ys)" 
1095 
by (induct n) (auto, case_tac xs, auto) 

13114  1096 

13142  1097 
lemma drop_append [simp]: 
13145  1098 
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n  length xs) ys" 
1099 
by (induct n) (auto, case_tac xs, auto) 

13114  1100 

13142  1101 
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs" 
14208  1102 
apply (induct m, auto) 
1103 
apply (case_tac xs, auto) 

15236
f289e8ba2bb3
Proofs needed to be updated because induction now preserves name of
nipkow
parents:
15176
diff
changeset

1104 
apply (case_tac n, auto) 
13145  1105 
done 
13114  1106 

13142  1107 
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs" 
14208  1108 
apply (induct m, auto) 
1109 
apply (case_tac xs, auto) 

13145  1110 
done 
13114  1111 

1112 
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)" 

14208  1113 
apply (induct m, auto) 
1114 
apply (case_tac xs, auto) 

13145  1115 
done 
13114  1116 

14802  1117 
lemma drop_take: "!!m n. drop n (take m xs) = take (mn) (drop n xs)" 
1118 
apply(induct xs) 

1119 
apply simp 

1120 
apply(simp add: take_Cons drop_Cons split:nat.split) 

1121 
done 

1122 

13142  1123 
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs" 
14208  1124 
apply (induct n, auto) 
1125 
apply (case_tac xs, auto) 

13145  1126 
done 
13114  1127 

15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1128 
lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])" 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1129 
apply(induct xs) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1130 
apply simp 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1131 
apply(simp add:take_Cons split:nat.split) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1132 
done 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1133 

78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1134 
lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)" 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1135 
apply(induct xs) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1136 
apply simp 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1137 
apply(simp add:drop_Cons split:nat.split) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1138 
done 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1139 

13114  1140 
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)" 
14208  1141 
apply (induct n, auto) 
1142 
apply (case_tac xs, auto) 

13145  1143 
done 
13114  1144 

13142  1145 
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)" 
14208  1146 
apply (induct n, auto) 
1147 
apply (case_tac xs, auto) 

13145  1148 
done 
13114  1149 

1150 
lemma rev_take: "!!i. rev (take i xs) = drop (length xs  i) (rev xs)" 

14208  1151 
apply (induct xs, auto) 
1152 
apply (case_tac i, auto) 

13145  1153 
done 
13114  1154 

1155 
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs  i) (rev xs)" 

14208  1156 
apply (induct xs, auto) 
1157 
apply (case_tac i, auto) 

13145  1158 
done 
13114  1159 

13142  1160 
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i" 
14208  1161 
apply (induct xs, auto) 
1162 
apply (case_tac n, blast) 

1163 
apply (case_tac i, auto) 

13145  1164 
done 
13114  1165 

13142  1166 
lemma nth_drop [simp]: 
13145  1167 
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)" 
14208  1168 
apply (induct n, auto) 
1169 
apply (case_tac xs, auto) 

13145  1170 
done 
3507  1171 

18423  1172 
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n" 
1173 
by(simp add: hd_conv_nth) 

1174 

14025  1175 
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs" 
1176 
by(induct xs)(auto simp:take_Cons split:nat.split) 

1177 

1178 
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs" 

1179 
by(induct xs)(auto simp:drop_Cons split:nat.split) 

1180 

14187  1181 
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs" 
1182 
using set_take_subset by fast 

1183 

1184 
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs" 

1185 
using set_drop_subset by fast 

1186 

13114  1187 
lemma append_eq_conv_conj: 
13145  1188 
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)" 
14208  1189 
apply (induct xs, simp, clarsimp) 
1190 
apply (case_tac zs, auto) 

13145  1191 
done 
13142  1192 

14050  1193 
lemma take_add [rule_format]: 
1194 
"\<forall>i. i+j \<le> length(xs) > take (i+j) xs = take i xs @ take j (drop i xs)" 

1195 
apply (induct xs, auto) 

1196 
apply (case_tac i, simp_all) 

1197 
done 

1198 

14300  1199 
lemma append_eq_append_conv_if: 
1200 
"!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) = 

1201 
(if size xs\<^isub>1 \<le> size ys\<^isub>1 

1202 
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 

1203 
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)" 

1204 
apply(induct xs\<^isub>1) 

1205 
apply simp 

1206 
apply(case_tac ys\<^isub>1) 

1207 
apply simp_all 

1208 
done 

1209 

15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1210 
lemma take_hd_drop: 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1211 
"!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs" 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1212 
apply(induct xs) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1213 
apply simp 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1214 
apply(simp add:drop_Cons split:nat.split) 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1215 
done 
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset

1216 

17501  1217 
lemma id_take_nth_drop: 
1218 
"i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 

1219 
proof  

1220 
assume si: "i < length xs" 

1221 
hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto 

1222 
moreover 

1223 
from si have "take (Suc i) xs = take i xs @ [xs!i]" 

1224 
apply (rule_tac take_Suc_conv_app_nth) by arith 

1225 
ultimately show ?thesis by auto 

1226 
qed 

1227 

1228 
lemma upd_conv_take_nth_drop: 

1229 
"i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs" 

1230 
proof  

1231 
assume i: "i < length xs" 

1232 
have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]" 

1233 
by(rule arg_cong[OF id_take_nth_drop[OF i]]) 

1234 
also have "\<dots> = take i xs @ a # drop (Suc i) xs" 

1235 
using i by (simp add: list_update_append) 

1236 
finally show ?thesis . 

1237 
qed 

1238 

13114  1239 

15392  1240 
subsubsection {* @{text takeWhile} and @{text dropWhile} *} 
13114  1241 

13142  1242 
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs" 
13145  1243 
by (induct xs) auto 
13114  1244 

13142  1245 
lemma takeWhile_append1 [simp]: 
13145  1246 
"[ x:set xs; ~P(x)] ==> takeWhile P (xs @ ys) = takeWhile P xs" 
1247 
by (induct xs) auto 

13114  1248 

13142  1249 
lemma takeWhile_append2 [simp]: 
13145  1250 
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys" 
1251 
by (induct xs) auto 

13114  1252 

13142  1253 
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs" 
13145  1254 
by (induct xs) auto 
13114  1255 

13142  1256 
lemma dropWhile_append1 [simp]: 
13145  1257 
"[ x : set xs; ~P(x)] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys" 
1258 
by (induct xs) auto 

13114  1259 

13142  1260 
lemma dropWhile_append2 [simp]: 
13145  1261 
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys" 
1262 
by (induct xs) auto 

13114  1263 

13142  1264 
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x" 
13145  1265 
by (induct xs) (auto split: split_if_asm) 
13114  1266 

13913  1267 
lemma takeWhile_eq_all_conv[simp]: 
1268 
"(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)" 

1269 
by(induct xs, auto) 

1270 

1271 
lemma dropWhile_eq_Nil_conv[simp]: 

1272 
"(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)" 

1273 
by(induct xs, auto) 

1274 

1275 
lemma dropWhile_eq_Cons_conv: 

1276 
"(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)" 

1277 
by(induct xs, auto) 

1278 

17501  1279 
text{* The following two lemmmas could be generalized to an arbitrary 
1280 
property. *} 

1281 

1282 
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow> 

1283 
takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))" 

1284 
by(induct xs) (auto simp: takeWhile_tail[where l="[]"]) 

1285 

1286 
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow> 

1287 
dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)" 

1288 
apply(induct xs) 

1289 
apply simp 

1290 
apply auto 

1291 
apply(subst dropWhile_append2) 

1292 
apply auto 

1293 
done 

1294 

18423  1295 
lemma takeWhile_not_last: 
1296 
"\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs" 

1297 
apply(induct xs) 

1298 
apply simp 

1299 
apply(case_tac xs) 

1300 
apply(auto) 

1301 
done 

1302 

19770
be5c23ebe1eb
HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents:
19623
diff
changeset

1303 
lemma takeWhile_cong [fundef_cong, recdef_cong]: 
18336
1a2e30b37ed3
Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents:
18049
diff
changeset

1304 
"[ l = k; !!x. x : set l ==> P x = Q x ] 
1a2e30b37ed3
Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents:
18049
diff
changeset

1305 
==> takeWhile P l = takeWhile Q k" 
20503  1306 
by (induct k arbitrary: l) (simp_all) 
18336
1a2e30b37ed3
Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents:
18049
diff
changeset

1307 

19770
be5c23ebe1eb
HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents:
19623
diff
changeset

1308 
lemma dropWhile_cong [fundef_cong, recdef_cong]: 
18336
1a2e30b37ed3
Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents:
18049
diff
changeset

1309 
"[ l = k; !!x. x : set l ==> P x = Q x ] 
1a2e30b37ed3
Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents:
18049
diff
changeset

1310 
==> dropWhile P l = dropWhile Q k" 
20503  1311 
by (induct k arbitrary: l, simp_all) 
18336
1a2e30b37ed3
Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents:
18049
diff
changeset

1312 

13114  1313 

15392  1314 
subsubsection {* @{text zip} *} 
13114  1315 

13142  1316 
lemma zip_Nil [simp]: "zip [] ys = []" 
13145  1317 
by (induct ys) auto 
13114  1318 

13142  1319 
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys" 
13145  1320 
by simp 
13114  1321 

13142  1322 
declare zip_Cons [simp del] 
13114  1323 

15281  1324 
lemma zip_Cons1: 
1325 
"zip (x#xs) ys = (case ys of [] \<Rightarrow> []  y#ys \<Rightarrow> (x,y)#zip xs ys)" 

1326 
by(auto split:list.split) 

1327 

13142  1328 
lemma length_zip [simp]: 
13145  1329 
"!!xs. length (zip xs ys) = min (length xs) (length ys)" 
14208  1330 
apply (induct ys, simp) 
1331 
apply (case_tac xs, auto) 

13145  1332 
done 
13114  1333 

1334 
lemma zip_append1: 

13145  1335 
"!!xs. zip (xs @ ys) zs = 
1336 
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)" 

14208  1337 
apply (induct zs, simp) 
1338 
apply (case_tac xs, simp_all) 

13145  1339 
done 
13114  1340 

1341 
lemma zip_append2: 

13145  1342 
"!!ys. zip xs (ys @ zs) = 
1343 
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs" 

14208  1344 
apply (induct xs, simp) 
1345 
apply (case_tac ys, simp_all) 

13145  1346 
done 
13114  1347 

13142  1348 
lemma zip_append [simp]: 
1349 
"[ length xs = length us; length ys = length vs ] ==> 

13145  1350 
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs" 
1351 
by (simp add: zip_append1) 

13114  1352 

1353 
lemma zip_rev: 

14247  1354 
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)" 
1355 
by (induct rule:list_induct2, simp_all) 

13114  1356 

13142 