src/HOL/List.thy
author nipkow
Tue Jul 29 13:16:54 2008 +0200 (2008-07-29)
changeset 27693 73253a4e3ee2
parent 27381 19ae7064f00f
child 27715 087db5aacac3
permissions -rw-r--r--
added removeAll
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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 Plain Relation_Power Presburger Recdef ATP_Linkup
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uses "Tools/string_syntax.ML"
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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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  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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  listsum ::  "'a list => 'a::monoid_add"
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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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  removeAll :: "'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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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[_\<leftarrow>_ ./ _])")
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syntax (HTML output)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
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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" where
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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 :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65)
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where
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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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"listsum [] = 0"
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"listsum (x # xs) = x + listsum 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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  "removeAll x [] = []"
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  "removeAll x (y#xs) = (if x=y then removeAll x xs else y # removeAll 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" where
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  "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"
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definition
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  rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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  "rotate n = rotate1 ^ n"
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definition
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  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
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  [code func del]: "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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definition
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  sublist :: "'a list => nat set => 'a list" where
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  "sublist xs A = 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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text{*
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\begin{figure}[htbp]
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\fbox{
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\begin{tabular}{l}
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@{lemma "[a,b]@[c,d] = [a,b,c,d]" by simp}\\
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@{lemma "length [a,b,c] = 3" by simp}\\
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@{lemma "set [a,b,c] = {a,b,c}" by simp}\\
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@{lemma "map f [a,b,c] = [f a, f b, f c]" by simp}\\
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@{lemma "rev [a,b,c] = [c,b,a]" by simp}\\
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@{lemma "hd [a,b,c,d] = a" by simp}\\
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@{lemma "tl [a,b,c,d] = [b,c,d]" by simp}\\
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@{lemma "last [a,b,c,d] = d" by simp}\\
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@{lemma "butlast [a,b,c,d] = [a,b,c]" by simp}\\
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@{lemma[source] "filter (\<lambda>n::nat. n<2) [0,2,1] = [0,1]" by simp}\\
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@{lemma "concat [[a,b],[c,d,e],[],[f]] = [a,b,c,d,e,f]" by simp}\\
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@{lemma "foldl f x [a,b,c] = f (f (f x a) b) c" by simp}\\
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@{lemma "foldr f [a,b,c] x = f a (f b (f c x))" by simp}\\
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@{lemma "zip [a,b,c] [x,y,z] = [(a,x),(b,y),(c,z)]" by simp}\\
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@{lemma "zip [a,b] [x,y,z] = [(a,x),(b,y)]" by simp}\\
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@{lemma "splice [a,b,c] [x,y,z] = [a,x,b,y,c,z]" by simp}\\
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@{lemma "splice [a,b,c,d] [x,y] = [a,x,b,y,c,d]" by simp}\\
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@{lemma "take 2 [a,b,c,d] = [a,b]" by simp}\\
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@{lemma "take 6 [a,b,c,d] = [a,b,c,d]" by simp}\\
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@{lemma "drop 2 [a,b,c,d] = [c,d]" by simp}\\
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@{lemma "drop 6 [a,b,c,d] = []" by simp}\\
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@{lemma "takeWhile (%n::nat. n<3) [1,2,3,0] = [1,2]" by simp}\\
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@{lemma "dropWhile (%n::nat. n<3) [1,2,3,0] = [3,0]" by simp}\\
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@{lemma "distinct [2,0,1::nat]" by simp}\\
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@{lemma "remdups [2,0,2,1::nat,2] = [0,1,2]" by simp}\\
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@{lemma "remove1 2 [2,0,2,1::nat,2] = [0,2,1,2]" by simp}\\
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@{lemma "removeAll 2 [2,0,2,1::nat,2] = [0,1]" by simp}\\
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@{lemma "nth [a,b,c,d] 2 = c" by simp}\\
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@{lemma "[a,b,c,d][2 := x] = [a,b,x,d]" by simp}\\
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@{lemma "sublist [a,b,c,d,e] {0,2,3} = [a,c,d]" by (simp add:sublist_def)}\\
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@{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by (simp add:rotate1_def)}\\
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@{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate1_def rotate_def nat_number)}\\
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@{lemma "replicate 4 a = [a,a,a,a]" by (simp add:nat_number)}\\
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@{lemma "[2..<5] = [2,3,4]" by (simp add:nat_number)}\\
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@{lemma "listsum [1,2,3::nat] = 6" by simp}
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\end{tabular}}
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\caption{Characteristic examples}
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\label{fig:Characteristic}
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\end{figure}
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Figure~\ref{fig:Characteristic} shows charachteristic examples
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that should give an intuitive understanding of the above functions.
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*}
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text{* The following simple sort functions are intended for proofs,
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not for efficient implementations. *}
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context linorder
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begin
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fun sorted :: "'a list \<Rightarrow> bool" where
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"sorted [] \<longleftrightarrow> True" |
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"sorted [x] \<longleftrightarrow> True" |
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"sorted (x#y#zs) \<longleftrightarrow> x <= y \<and> sorted (y#zs)"
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primrec insort :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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"insort x [] = [x]" |
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"insort x (y#ys) = (if x <= y then (x#y#ys) else y#(insort x ys))"
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primrec sort :: "'a list \<Rightarrow> 'a list" where
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"sort [] = []" |
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"sort (x#xs) = insort x (sort xs)"
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end
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subsubsection {* List comprehension *}
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text{* Input syntax for Haskell-like list comprehension notation.
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Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
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the list of all pairs of distinct elements from @{text xs} and @{text ys}.
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The syntax is as in Haskell, except that @{text"|"} becomes a dot
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(like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
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\verb![e| x <- xs, ...]!.
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The qualifiers after the dot are
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\begin{description}
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\item[generators] @{text"p \<leftarrow> xs"},
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 where @{text p} is a pattern and @{text xs} an expression of list type, or
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\item[guards] @{text"b"}, where @{text b} is a boolean expression.
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%\item[local bindings] @ {text"let x = e"}.
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\end{description}
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Just like in Haskell, list comprehension is just a shorthand. To avoid
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misunderstandings, the translation into desugared form is not reversed
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upon output. Note that the translation of @{text"[e. x \<leftarrow> xs]"} is
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optmized to @{term"map (%x. e) xs"}.
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It is easy to write short list comprehensions which stand for complex
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expressions. During proofs, they may become unreadable (and
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mangled). In such cases it can be advisable to introduce separate
nipkow@24349
   313
definitions for the list comprehensions in question.  *}
nipkow@24349
   314
nipkow@23209
   315
(*
nipkow@23240
   316
Proper theorem proving support would be nice. For example, if
nipkow@23192
   317
@{text"set[f x y. x \<leftarrow> xs, y \<leftarrow> ys, P x y]"}
nipkow@23192
   318
produced something like
nipkow@23209
   319
@{term"{z. EX x: set xs. EX y:set ys. P x y \<and> z = f x y}"}.
nipkow@23209
   320
*)
nipkow@23209
   321
nipkow@23240
   322
nonterminals lc_qual lc_quals
nipkow@23192
   323
nipkow@23192
   324
syntax
nipkow@23240
   325
"_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
nipkow@24349
   326
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ <- _")
nipkow@23240
   327
"_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
nipkow@24476
   328
(*"_lc_let" :: "letbinds => lc_qual"  ("let _")*)
nipkow@23240
   329
"_lc_end" :: "lc_quals" ("]")
nipkow@23240
   330
"_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals" (", __")
nipkow@24349
   331
"_lc_abs" :: "'a => 'b list => 'b list"
nipkow@23192
   332
nipkow@24476
   333
(* These are easier than ML code but cannot express the optimized
nipkow@24476
   334
   translation of [e. p<-xs]
nipkow@23192
   335
translations
nipkow@24349
   336
"[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
nipkow@23240
   337
"_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
nipkow@24349
   338
 => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
nipkow@23240
   339
"[e. P]" => "if P then [e] else []"
nipkow@23240
   340
"_listcompr e (_lc_test P) (_lc_quals Q Qs)"
nipkow@23240
   341
 => "if P then (_listcompr e Q Qs) else []"
nipkow@24349
   342
"_listcompr e (_lc_let b) (_lc_quals Q Qs)"
nipkow@24349
   343
 => "_Let b (_listcompr e Q Qs)"
nipkow@24476
   344
*)
nipkow@23240
   345
nipkow@23279
   346
syntax (xsymbols)
nipkow@24349
   347
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
nipkow@23279
   348
syntax (HTML output)
nipkow@24349
   349
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
nipkow@24349
   350
nipkow@24349
   351
parse_translation (advanced) {*
nipkow@24349
   352
let
nipkow@24476
   353
  val NilC = Syntax.const @{const_name Nil};
nipkow@24476
   354
  val ConsC = Syntax.const @{const_name Cons};
nipkow@24476
   355
  val mapC = Syntax.const @{const_name map};
nipkow@24476
   356
  val concatC = Syntax.const @{const_name concat};
nipkow@24476
   357
  val IfC = Syntax.const @{const_name If};
nipkow@24476
   358
  fun singl x = ConsC $ x $ NilC;
nipkow@24476
   359
nipkow@24476
   360
   fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
nipkow@24349
   361
    let
nipkow@24476
   362
      val x = Free (Name.variant (add_term_free_names (p$e, [])) "x", dummyT);
nipkow@24476
   363
      val e = if opti then singl e else e;
nipkow@24476
   364
      val case1 = Syntax.const "_case1" $ p $ e;
nipkow@24349
   365
      val case2 = Syntax.const "_case1" $ Syntax.const Term.dummy_patternN
nipkow@24476
   366
                                        $ NilC;
nipkow@24349
   367
      val cs = Syntax.const "_case2" $ case1 $ case2
nipkow@24349
   368
      val ft = DatatypeCase.case_tr false DatatypePackage.datatype_of_constr
nipkow@24349
   369
                 ctxt [x, cs]
nipkow@24349
   370
    in lambda x ft end;
nipkow@24349
   371
nipkow@24476
   372
  fun abs_tr ctxt (p as Free(s,T)) e opti =
nipkow@24349
   373
        let val thy = ProofContext.theory_of ctxt;
nipkow@24349
   374
            val s' = Sign.intern_const thy s
nipkow@24476
   375
        in if Sign.declared_const thy s'
nipkow@24476
   376
           then (pat_tr ctxt p e opti, false)
nipkow@24476
   377
           else (lambda p e, true)
nipkow@24349
   378
        end
nipkow@24476
   379
    | abs_tr ctxt p e opti = (pat_tr ctxt p e opti, false);
nipkow@24476
   380
nipkow@24476
   381
  fun lc_tr ctxt [e, Const("_lc_test",_)$b, qs] =
nipkow@24476
   382
        let val res = case qs of Const("_lc_end",_) => singl e
nipkow@24476
   383
                      | Const("_lc_quals",_)$q$qs => lc_tr ctxt [e,q,qs];
nipkow@24476
   384
        in IfC $ b $ res $ NilC end
nipkow@24476
   385
    | lc_tr ctxt [e, Const("_lc_gen",_) $ p $ es, Const("_lc_end",_)] =
nipkow@24476
   386
        (case abs_tr ctxt p e true of
nipkow@24476
   387
           (f,true) => mapC $ f $ es
nipkow@24476
   388
         | (f, false) => concatC $ (mapC $ f $ es))
nipkow@24476
   389
    | lc_tr ctxt [e, Const("_lc_gen",_) $ p $ es, Const("_lc_quals",_)$q$qs] =
nipkow@24476
   390
        let val e' = lc_tr ctxt [e,q,qs];
nipkow@24476
   391
        in concatC $ (mapC $ (fst(abs_tr ctxt p e' false)) $ es) end
nipkow@24476
   392
nipkow@24476
   393
in [("_listcompr", lc_tr)] end
nipkow@24349
   394
*}
nipkow@23279
   395
nipkow@23240
   396
(*
nipkow@23240
   397
term "[(x,y,z). b]"
nipkow@24476
   398
term "[(x,y,z). x\<leftarrow>xs]"
nipkow@24476
   399
term "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]"
nipkow@24476
   400
term "[(x,y,z). x<a, x>b]"
nipkow@24476
   401
term "[(x,y,z). x\<leftarrow>xs, x>b]"
nipkow@24476
   402
term "[(x,y,z). x<a, x\<leftarrow>xs]"
nipkow@24349
   403
term "[(x,y). Cons True x \<leftarrow> xs]"
nipkow@24349
   404
term "[(x,y,z). Cons x [] \<leftarrow> xs]"
nipkow@23240
   405
term "[(x,y,z). x<a, x>b, x=d]"
nipkow@23240
   406
term "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
nipkow@23240
   407
term "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
nipkow@23240
   408
term "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
nipkow@23240
   409
term "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
nipkow@23240
   410
term "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
nipkow@23240
   411
term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
nipkow@23240
   412
term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
nipkow@24349
   413
term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
nipkow@23192
   414
*)
nipkow@23192
   415
haftmann@21061
   416
subsubsection {* @{const Nil} and @{const Cons} *}
haftmann@21061
   417
haftmann@21061
   418
lemma not_Cons_self [simp]:
haftmann@21061
   419
  "xs \<noteq> x # xs"
nipkow@13145
   420
by (induct xs) auto
wenzelm@13114
   421
wenzelm@13142
   422
lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
wenzelm@13114
   423
wenzelm@13142
   424
lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
nipkow@13145
   425
by (induct xs) auto
wenzelm@13114
   426
wenzelm@13142
   427
lemma length_induct:
haftmann@21061
   428
  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
nipkow@17589
   429
by (rule measure_induct [of length]) iprover
wenzelm@13114
   430
wenzelm@13114
   431
haftmann@21061
   432
subsubsection {* @{const length} *}
wenzelm@13114
   433
wenzelm@13142
   434
text {*
haftmann@21061
   435
  Needs to come before @{text "@"} because of theorem @{text
haftmann@21061
   436
  append_eq_append_conv}.
wenzelm@13142
   437
*}
wenzelm@13114
   438
wenzelm@13142
   439
lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
nipkow@13145
   440
by (induct xs) auto
wenzelm@13114
   441
wenzelm@13142
   442
lemma length_map [simp]: "length (map f xs) = length xs"
nipkow@13145
   443
by (induct xs) auto
wenzelm@13114
   444
wenzelm@13142
   445
lemma length_rev [simp]: "length (rev xs) = length xs"
nipkow@13145
   446
by (induct xs) auto
wenzelm@13114
   447
wenzelm@13142
   448
lemma length_tl [simp]: "length (tl xs) = length xs - 1"
nipkow@13145
   449
by (cases xs) auto
wenzelm@13114
   450
wenzelm@13142
   451
lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
nipkow@13145
   452
by (induct xs) auto
wenzelm@13114
   453
wenzelm@13142
   454
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
nipkow@13145
   455
by (induct xs) auto
wenzelm@13114
   456
nipkow@23479
   457
lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
nipkow@23479
   458
by auto
nipkow@23479
   459
wenzelm@13114
   460
lemma length_Suc_conv:
nipkow@13145
   461
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
nipkow@13145
   462
by (induct xs) auto
wenzelm@13142
   463
nipkow@14025
   464
lemma Suc_length_conv:
nipkow@14025
   465
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
paulson@14208
   466
apply (induct xs, simp, simp)
nipkow@14025
   467
apply blast
nipkow@14025
   468
done
nipkow@14025
   469
wenzelm@25221
   470
lemma impossible_Cons: "length xs <= length ys ==> xs = x # ys = False"
wenzelm@25221
   471
  by (induct xs) auto
wenzelm@25221
   472
haftmann@26442
   473
lemma list_induct2 [consumes 1, case_names Nil Cons]:
haftmann@26442
   474
  "length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
haftmann@26442
   475
   (\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
haftmann@26442
   476
   \<Longrightarrow> P xs ys"
haftmann@26442
   477
proof (induct xs arbitrary: ys)
haftmann@26442
   478
  case Nil then show ?case by simp
haftmann@26442
   479
next
haftmann@26442
   480
  case (Cons x xs ys) then show ?case by (cases ys) simp_all
haftmann@26442
   481
qed
haftmann@26442
   482
haftmann@26442
   483
lemma list_induct3 [consumes 2, case_names Nil Cons]:
haftmann@26442
   484
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
haftmann@26442
   485
   (\<And>x xs y ys z zs. length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P xs ys zs \<Longrightarrow> P (x#xs) (y#ys) (z#zs))
haftmann@26442
   486
   \<Longrightarrow> P xs ys zs"
haftmann@26442
   487
proof (induct xs arbitrary: ys zs)
haftmann@26442
   488
  case Nil then show ?case by simp
haftmann@26442
   489
next
haftmann@26442
   490
  case (Cons x xs ys zs) then show ?case by (cases ys, simp_all)
haftmann@26442
   491
    (cases zs, simp_all)
haftmann@26442
   492
qed
wenzelm@13114
   493
krauss@22493
   494
lemma list_induct2': 
krauss@22493
   495
  "\<lbrakk> P [] [];
krauss@22493
   496
  \<And>x xs. P (x#xs) [];
krauss@22493
   497
  \<And>y ys. P [] (y#ys);
krauss@22493
   498
   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
krauss@22493
   499
 \<Longrightarrow> P xs ys"
krauss@22493
   500
by (induct xs arbitrary: ys) (case_tac x, auto)+
krauss@22493
   501
nipkow@22143
   502
lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
nipkow@24349
   503
by (rule Eq_FalseI) auto
wenzelm@24037
   504
wenzelm@24037
   505
simproc_setup list_neq ("(xs::'a list) = ys") = {*
nipkow@22143
   506
(*
nipkow@22143
   507
Reduces xs=ys to False if xs and ys cannot be of the same length.
nipkow@22143
   508
This is the case if the atomic sublists of one are a submultiset
nipkow@22143
   509
of those of the other list and there are fewer Cons's in one than the other.
nipkow@22143
   510
*)
wenzelm@24037
   511
wenzelm@24037
   512
let
nipkow@22143
   513
nipkow@22143
   514
fun len (Const("List.list.Nil",_)) acc = acc
nipkow@22143
   515
  | len (Const("List.list.Cons",_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
haftmann@23029
   516
  | len (Const("List.append",_) $ xs $ ys) acc = len xs (len ys acc)
nipkow@22143
   517
  | len (Const("List.rev",_) $ xs) acc = len xs acc
nipkow@22143
   518
  | len (Const("List.map",_) $ _ $ xs) acc = len xs acc
nipkow@22143
   519
  | len t (ts,n) = (t::ts,n);
nipkow@22143
   520
wenzelm@24037
   521
fun list_neq _ ss ct =
nipkow@22143
   522
  let
wenzelm@24037
   523
    val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
nipkow@22143
   524
    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
nipkow@22143
   525
    fun prove_neq() =
nipkow@22143
   526
      let
nipkow@22143
   527
        val Type(_,listT::_) = eqT;
haftmann@22994
   528
        val size = HOLogic.size_const listT;
nipkow@22143
   529
        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
nipkow@22143
   530
        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
nipkow@22143
   531
        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
haftmann@22633
   532
          (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
haftmann@22633
   533
      in SOME (thm RS @{thm neq_if_length_neq}) end
nipkow@22143
   534
  in
wenzelm@23214
   535
    if m < n andalso submultiset (op aconv) (ls,rs) orelse
wenzelm@23214
   536
       n < m andalso submultiset (op aconv) (rs,ls)
nipkow@22143
   537
    then prove_neq() else NONE
nipkow@22143
   538
  end;
wenzelm@24037
   539
in list_neq end;
nipkow@22143
   540
*}
nipkow@22143
   541
nipkow@22143
   542
nipkow@15392
   543
subsubsection {* @{text "@"} -- append *}
wenzelm@13114
   544
wenzelm@13142
   545
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
nipkow@13145
   546
by (induct xs) auto
wenzelm@13114
   547
wenzelm@13142
   548
lemma append_Nil2 [simp]: "xs @ [] = xs"
nipkow@13145
   549
by (induct xs) auto
nipkow@3507
   550
nipkow@24449
   551
interpretation semigroup_append: semigroup_add ["op @"]
nipkow@24449
   552
by unfold_locales simp
nipkow@24449
   553
interpretation monoid_append: monoid_add ["[]" "op @"]
nipkow@24449
   554
by unfold_locales (simp+)
nipkow@24449
   555
wenzelm@13142
   556
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
nipkow@13145
   557
by (induct xs) auto
wenzelm@13114
   558
wenzelm@13142
   559
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
nipkow@13145
   560
by (induct xs) auto
wenzelm@13114
   561
wenzelm@13142
   562
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
nipkow@13145
   563
by (induct xs) auto
wenzelm@13114
   564
wenzelm@13142
   565
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
nipkow@13145
   566
by (induct xs) auto
wenzelm@13114
   567
wenzelm@25221
   568
lemma append_eq_append_conv [simp, noatp]:
nipkow@24526
   569
 "length xs = length ys \<or> length us = length vs
berghofe@13883
   570
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
nipkow@24526
   571
apply (induct xs arbitrary: ys)
paulson@14208
   572
 apply (case_tac ys, simp, force)
paulson@14208
   573
apply (case_tac ys, force, simp)
nipkow@13145
   574
done
wenzelm@13142
   575
nipkow@24526
   576
lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
nipkow@24526
   577
  (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
nipkow@24526
   578
apply (induct xs arbitrary: ys zs ts)
nipkow@14495
   579
 apply fastsimp
nipkow@14495
   580
apply(case_tac zs)
nipkow@14495
   581
 apply simp
nipkow@14495
   582
apply fastsimp
nipkow@14495
   583
done
nipkow@14495
   584
wenzelm@13142
   585
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145
   586
by simp
wenzelm@13142
   587
wenzelm@13142
   588
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145
   589
by simp
wenzelm@13114
   590
wenzelm@13142
   591
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   592
by simp
wenzelm@13114
   593
wenzelm@13142
   594
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   595
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   596
wenzelm@13142
   597
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   598
using append_same_eq [of "[]"] by auto
wenzelm@13114
   599
paulson@24286
   600
lemma hd_Cons_tl [simp,noatp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   601
by (induct xs) auto
wenzelm@13114
   602
wenzelm@13142
   603
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   604
by (induct xs) auto
wenzelm@13114
   605
wenzelm@13142
   606
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   607
by (simp add: hd_append split: list.split)
wenzelm@13114
   608
wenzelm@13142
   609
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   610
by (simp split: list.split)
wenzelm@13114
   611
wenzelm@13142
   612
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   613
by (simp add: tl_append split: list.split)
wenzelm@13114
   614
wenzelm@13114
   615
nipkow@14300
   616
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   617
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   618
by(cases ys) auto
nipkow@14300
   619
nipkow@15281
   620
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   621
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   622
by(cases ys) auto
nipkow@15281
   623
nipkow@14300
   624
wenzelm@13142
   625
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   626
wenzelm@13114
   627
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   628
by simp
wenzelm@13114
   629
wenzelm@13142
   630
lemma Cons_eq_appendI:
nipkow@13145
   631
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   632
by (drule sym) simp
wenzelm@13114
   633
wenzelm@13142
   634
lemma append_eq_appendI:
nipkow@13145
   635
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   636
by (drule sym) simp
wenzelm@13114
   637
wenzelm@13114
   638
wenzelm@13142
   639
text {*
nipkow@13145
   640
Simplification procedure for all list equalities.
nipkow@13145
   641
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   642
- both lists end in a singleton list,
nipkow@13145
   643
- or both lists end in the same list.
wenzelm@13142
   644
*}
wenzelm@13142
   645
wenzelm@26480
   646
ML {*
nipkow@3507
   647
local
nipkow@3507
   648
wenzelm@13114
   649
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462
   650
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
haftmann@23029
   651
  | last (Const("List.append",_) $ _ $ ys) = last ys
wenzelm@13462
   652
  | last t = t;
wenzelm@13114
   653
wenzelm@13114
   654
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462
   655
  | list1 _ = false;
wenzelm@13114
   656
wenzelm@13114
   657
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462
   658
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
haftmann@23029
   659
  | butlast ((app as Const("List.append",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462
   660
  | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114
   661
haftmann@22633
   662
val rearr_ss = HOL_basic_ss addsimps [@{thm append_assoc},
haftmann@22633
   663
  @{thm append_Nil}, @{thm append_Cons}];
wenzelm@16973
   664
wenzelm@20044
   665
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   666
  let
wenzelm@13462
   667
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   668
    fun rearr conv =
wenzelm@13462
   669
      let
wenzelm@13462
   670
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   671
        val Type(_,listT::_) = eqT
wenzelm@13462
   672
        val appT = [listT,listT] ---> listT
haftmann@23029
   673
        val app = Const("List.append",appT)
wenzelm@13462
   674
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   675
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@20044
   676
        val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
wenzelm@17877
   677
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
skalberg@15531
   678
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   679
wenzelm@13462
   680
  in
haftmann@22633
   681
    if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
haftmann@22633
   682
    else if lastl aconv lastr then rearr @{thm append_same_eq}
skalberg@15531
   683
    else NONE
wenzelm@13462
   684
  end;
wenzelm@13462
   685
wenzelm@13114
   686
in
wenzelm@13462
   687
wenzelm@13462
   688
val list_eq_simproc =
haftmann@22633
   689
  Simplifier.simproc @{theory} "list_eq" ["(xs::'a list) = ys"] (K list_eq);
wenzelm@13462
   690
wenzelm@13114
   691
end;
wenzelm@13114
   692
wenzelm@13114
   693
Addsimprocs [list_eq_simproc];
wenzelm@13114
   694
*}
wenzelm@13114
   695
wenzelm@13114
   696
nipkow@15392
   697
subsubsection {* @{text map} *}
wenzelm@13114
   698
wenzelm@13142
   699
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   700
by (induct xs) simp_all
wenzelm@13114
   701
wenzelm@13142
   702
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   703
by (rule ext, induct_tac xs) auto
wenzelm@13114
   704
wenzelm@13142
   705
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   706
by (induct xs) auto
wenzelm@13114
   707
wenzelm@13142
   708
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   709
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   710
wenzelm@13142
   711
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   712
by (induct xs) auto
wenzelm@13114
   713
nipkow@13737
   714
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   715
by (induct xs) auto
nipkow@13737
   716
krauss@19770
   717
lemma map_cong [fundef_cong, recdef_cong]:
nipkow@13145
   718
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   719
-- {* a congruence rule for @{text map} *}
nipkow@13737
   720
by simp
wenzelm@13114
   721
wenzelm@13142
   722
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   723
by (cases xs) auto
wenzelm@13114
   724
wenzelm@13142
   725
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   726
by (cases xs) auto
wenzelm@13114
   727
paulson@18447
   728
lemma map_eq_Cons_conv:
nipkow@14025
   729
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   730
by (cases xs) auto
wenzelm@13114
   731
paulson@18447
   732
lemma Cons_eq_map_conv:
nipkow@14025
   733
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   734
by (cases ys) auto
nipkow@14025
   735
paulson@18447
   736
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
paulson@18447
   737
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
paulson@18447
   738
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
paulson@18447
   739
nipkow@14111
   740
lemma ex_map_conv:
nipkow@14111
   741
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
paulson@18447
   742
by(induct ys, auto simp add: Cons_eq_map_conv)
nipkow@14111
   743
nipkow@15110
   744
lemma map_eq_imp_length_eq:
haftmann@26734
   745
  assumes "map f xs = map f ys"
haftmann@26734
   746
  shows "length xs = length ys"
haftmann@26734
   747
using assms proof (induct ys arbitrary: xs)
haftmann@26734
   748
  case Nil then show ?case by simp
haftmann@26734
   749
next
haftmann@26734
   750
  case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto
haftmann@26734
   751
  from Cons xs have "map f zs = map f ys" by simp
haftmann@26734
   752
  moreover with Cons have "length zs = length ys" by blast
haftmann@26734
   753
  with xs show ?case by simp
haftmann@26734
   754
qed
haftmann@26734
   755
  
nipkow@15110
   756
lemma map_inj_on:
nipkow@15110
   757
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   758
  ==> xs = ys"
nipkow@15110
   759
apply(frule map_eq_imp_length_eq)
nipkow@15110
   760
apply(rotate_tac -1)
nipkow@15110
   761
apply(induct rule:list_induct2)
nipkow@15110
   762
 apply simp
nipkow@15110
   763
apply(simp)
nipkow@15110
   764
apply (blast intro:sym)
nipkow@15110
   765
done
nipkow@15110
   766
nipkow@15110
   767
lemma inj_on_map_eq_map:
nipkow@15110
   768
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   769
by(blast dest:map_inj_on)
nipkow@15110
   770
wenzelm@13114
   771
lemma map_injective:
nipkow@24526
   772
 "map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@24526
   773
by (induct ys arbitrary: xs) (auto dest!:injD)
wenzelm@13114
   774
nipkow@14339
   775
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   776
by(blast dest:map_injective)
nipkow@14339
   777
wenzelm@13114
   778
lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@17589
   779
by (iprover dest: map_injective injD intro: inj_onI)
wenzelm@13114
   780
wenzelm@13114
   781
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   782
apply (unfold inj_on_def, clarify)
nipkow@13145
   783
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   784
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   785
apply blast
nipkow@13145
   786
done
wenzelm@13114
   787
nipkow@14339
   788
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   789
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   790
nipkow@15303
   791
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
   792
apply(rule inj_onI)
nipkow@15303
   793
apply(erule map_inj_on)
nipkow@15303
   794
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
   795
done
nipkow@15303
   796
kleing@14343
   797
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   798
by (induct xs, auto)
wenzelm@13114
   799
nipkow@14402
   800
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   801
by (induct xs) auto
nipkow@14402
   802
nipkow@15110
   803
lemma map_fst_zip[simp]:
nipkow@15110
   804
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   805
by (induct rule:list_induct2, simp_all)
nipkow@15110
   806
nipkow@15110
   807
lemma map_snd_zip[simp]:
nipkow@15110
   808
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   809
by (induct rule:list_induct2, simp_all)
nipkow@15110
   810
nipkow@15110
   811
nipkow@15392
   812
subsubsection {* @{text rev} *}
wenzelm@13114
   813
wenzelm@13142
   814
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   815
by (induct xs) auto
wenzelm@13114
   816
wenzelm@13142
   817
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   818
by (induct xs) auto
wenzelm@13114
   819
kleing@15870
   820
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870
   821
by auto
kleing@15870
   822
wenzelm@13142
   823
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   824
by (induct xs) auto
wenzelm@13114
   825
wenzelm@13142
   826
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   827
by (induct xs) auto
wenzelm@13114
   828
kleing@15870
   829
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870
   830
by (cases xs) auto
kleing@15870
   831
kleing@15870
   832
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870
   833
by (cases xs) auto
kleing@15870
   834
haftmann@21061
   835
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
haftmann@21061
   836
apply (induct xs arbitrary: ys, force)
paulson@14208
   837
apply (case_tac ys, simp, force)
nipkow@13145
   838
done
wenzelm@13114
   839
nipkow@15439
   840
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
   841
by(simp add:inj_on_def)
nipkow@15439
   842
wenzelm@13366
   843
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   844
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489
   845
apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145
   846
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   847
done
wenzelm@13114
   848
wenzelm@13366
   849
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   850
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   851
by (induct xs rule: rev_induct) auto
wenzelm@13114
   852
wenzelm@13366
   853
lemmas rev_cases = rev_exhaust
wenzelm@13366
   854
nipkow@18423
   855
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
nipkow@18423
   856
by(rule rev_cases[of xs]) auto
nipkow@18423
   857
wenzelm@13114
   858
nipkow@15392
   859
subsubsection {* @{text set} *}
wenzelm@13114
   860
wenzelm@13142
   861
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   862
by (induct xs) auto
wenzelm@13114
   863
wenzelm@13142
   864
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   865
by (induct xs) auto
wenzelm@13114
   866
nipkow@17830
   867
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
nipkow@17830
   868
by(cases xs) auto
oheimb@14099
   869
wenzelm@13142
   870
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   871
by auto
wenzelm@13114
   872
oheimb@14099
   873
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   874
by auto
oheimb@14099
   875
wenzelm@13142
   876
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   877
by (induct xs) auto
wenzelm@13114
   878
nipkow@15245
   879
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   880
by(induct xs) auto
nipkow@15245
   881
wenzelm@13142
   882
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   883
by (induct xs) auto
wenzelm@13114
   884
wenzelm@13142
   885
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   886
by (induct xs) auto
wenzelm@13114
   887
wenzelm@13142
   888
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   889
by (induct xs) auto
wenzelm@13114
   890
nipkow@15425
   891
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
paulson@14208
   892
apply (induct j, simp_all)
paulson@14208
   893
apply (erule ssubst, auto)
nipkow@13145
   894
done
wenzelm@13114
   895
wenzelm@13142
   896
wenzelm@25221
   897
lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs"
nipkow@18049
   898
proof (induct xs)
nipkow@26073
   899
  case Nil thus ?case by simp
nipkow@26073
   900
next
nipkow@26073
   901
  case Cons thus ?case by (auto intro: Cons_eq_appendI)
nipkow@26073
   902
qed
nipkow@26073
   903
haftmann@26734
   904
lemma in_set_conv_decomp: "x \<in> set xs \<longleftrightarrow> (\<exists>ys zs. xs = ys @ x # zs)"
haftmann@26734
   905
  by (auto elim: split_list)
nipkow@26073
   906
nipkow@26073
   907
lemma split_list_first: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@26073
   908
proof (induct xs)
nipkow@26073
   909
  case Nil thus ?case by simp
nipkow@18049
   910
next
nipkow@18049
   911
  case (Cons a xs)
nipkow@18049
   912
  show ?case
nipkow@18049
   913
  proof cases
wenzelm@25221
   914
    assume "x = a" thus ?case using Cons by fastsimp
nipkow@18049
   915
  next
nipkow@26073
   916
    assume "x \<noteq> a" thus ?case using Cons by(fastsimp intro!: Cons_eq_appendI)
nipkow@26073
   917
  qed
nipkow@26073
   918
qed
nipkow@26073
   919
nipkow@26073
   920
lemma in_set_conv_decomp_first:
nipkow@26073
   921
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
haftmann@26734
   922
  by (auto dest!: split_list_first)
nipkow@26073
   923
nipkow@26073
   924
lemma split_list_last: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
nipkow@26073
   925
proof (induct xs rule:rev_induct)
nipkow@26073
   926
  case Nil thus ?case by simp
nipkow@26073
   927
next
nipkow@26073
   928
  case (snoc a xs)
nipkow@26073
   929
  show ?case
nipkow@26073
   930
  proof cases
nipkow@26073
   931
    assume "x = a" thus ?case using snoc by simp (metis ex_in_conv set_empty2)
nipkow@26073
   932
  next
nipkow@26073
   933
    assume "x \<noteq> a" thus ?case using snoc by fastsimp
nipkow@18049
   934
  qed
nipkow@18049
   935
qed
nipkow@18049
   936
nipkow@26073
   937
lemma in_set_conv_decomp_last:
nipkow@26073
   938
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
haftmann@26734
   939
  by (auto dest!: split_list_last)
nipkow@26073
   940
nipkow@26073
   941
lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs & P x"
nipkow@26073
   942
proof (induct xs)
nipkow@26073
   943
  case Nil thus ?case by simp
nipkow@26073
   944
next
nipkow@26073
   945
  case Cons thus ?case
nipkow@26073
   946
    by(simp add:Bex_def)(metis append_Cons append.simps(1))
nipkow@26073
   947
qed
nipkow@26073
   948
nipkow@26073
   949
lemma split_list_propE:
haftmann@26734
   950
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
   951
  obtains ys x zs where "xs = ys @ x # zs" and "P x"
haftmann@26734
   952
using split_list_prop [OF assms] by blast
nipkow@26073
   953
nipkow@26073
   954
lemma split_list_first_prop:
nipkow@26073
   955
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
nipkow@26073
   956
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)"
haftmann@26734
   957
proof (induct xs)
nipkow@26073
   958
  case Nil thus ?case by simp
nipkow@26073
   959
next
nipkow@26073
   960
  case (Cons x xs)
nipkow@26073
   961
  show ?case
nipkow@26073
   962
  proof cases
nipkow@26073
   963
    assume "P x"
haftmann@26734
   964
    thus ?thesis by simp
haftmann@26734
   965
      (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
nipkow@26073
   966
  next
nipkow@26073
   967
    assume "\<not> P x"
nipkow@26073
   968
    hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
nipkow@26073
   969
    thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
nipkow@26073
   970
  qed
nipkow@26073
   971
qed
nipkow@26073
   972
nipkow@26073
   973
lemma split_list_first_propE:
haftmann@26734
   974
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
   975
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y"
haftmann@26734
   976
using split_list_first_prop [OF assms] by blast
nipkow@26073
   977
nipkow@26073
   978
lemma split_list_first_prop_iff:
nipkow@26073
   979
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
nipkow@26073
   980
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))"
haftmann@26734
   981
by (rule, erule split_list_first_prop) auto
nipkow@26073
   982
nipkow@26073
   983
lemma split_list_last_prop:
nipkow@26073
   984
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
nipkow@26073
   985
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)"
nipkow@26073
   986
proof(induct xs rule:rev_induct)
nipkow@26073
   987
  case Nil thus ?case by simp
nipkow@26073
   988
next
nipkow@26073
   989
  case (snoc x xs)
nipkow@26073
   990
  show ?case
nipkow@26073
   991
  proof cases
nipkow@26073
   992
    assume "P x" thus ?thesis by (metis emptyE set_empty)
nipkow@26073
   993
  next
nipkow@26073
   994
    assume "\<not> P x"
nipkow@26073
   995
    hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
nipkow@26073
   996
    thus ?thesis using `\<not> P x` snoc(1) by fastsimp
nipkow@26073
   997
  qed
nipkow@26073
   998
qed
nipkow@26073
   999
nipkow@26073
  1000
lemma split_list_last_propE:
haftmann@26734
  1001
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
  1002
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z"
haftmann@26734
  1003
using split_list_last_prop [OF assms] by blast
nipkow@26073
  1004
nipkow@26073
  1005
lemma split_list_last_prop_iff:
nipkow@26073
  1006
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
nipkow@26073
  1007
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
haftmann@26734
  1008
by (metis split_list_last_prop [where P=P] in_set_conv_decomp)
nipkow@26073
  1009
nipkow@26073
  1010
lemma finite_list: "finite A ==> EX xs. set xs = A"
haftmann@26734
  1011
  by (erule finite_induct)
haftmann@26734
  1012
    (auto simp add: set.simps(2) [symmetric] simp del: set.simps(2))
paulson@13508
  1013
kleing@14388
  1014
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
  1015
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
  1016
haftmann@26442
  1017
lemma set_minus_filter_out:
haftmann@26442
  1018
  "set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)"
haftmann@26442
  1019
  by (induct xs) auto
paulson@15168
  1020
nipkow@15392
  1021
subsubsection {* @{text filter} *}
wenzelm@13114
  1022
wenzelm@13142
  1023
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
  1024
by (induct xs) auto
wenzelm@13114
  1025
nipkow@15305
  1026
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
  1027
by (induct xs) simp_all
nipkow@15305
  1028
wenzelm@13142
  1029
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
  1030
by (induct xs) auto
wenzelm@13114
  1031
nipkow@16998
  1032
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998
  1033
by (induct xs) (auto simp add: le_SucI)
nipkow@16998
  1034
nipkow@18423
  1035
lemma sum_length_filter_compl:
nipkow@18423
  1036
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
nipkow@18423
  1037
by(induct xs) simp_all
nipkow@18423
  1038
wenzelm@13142
  1039
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
  1040
by (induct xs) auto
wenzelm@13114
  1041
wenzelm@13142
  1042
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
  1043
by (induct xs) auto
wenzelm@13114
  1044
nipkow@16998
  1045
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@24349
  1046
by (induct xs) simp_all
nipkow@16998
  1047
nipkow@16998
  1048
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998
  1049
apply (induct xs)
nipkow@16998
  1050
 apply auto
nipkow@16998
  1051
apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998
  1052
apply simp
nipkow@16998
  1053
done
wenzelm@13114
  1054
nipkow@16965
  1055
lemma filter_map:
nipkow@16965
  1056
  "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965
  1057
by (induct xs) simp_all
nipkow@16965
  1058
nipkow@16965
  1059
lemma length_filter_map[simp]:
nipkow@16965
  1060
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965
  1061
by (simp add:filter_map)
nipkow@16965
  1062
wenzelm@13142
  1063
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
  1064
by auto
wenzelm@13114
  1065
nipkow@15246
  1066
lemma length_filter_less:
nipkow@15246
  1067
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
  1068
proof (induct xs)
nipkow@15246
  1069
  case Nil thus ?case by simp
nipkow@15246
  1070
next
nipkow@15246
  1071
  case (Cons x xs) thus ?case
nipkow@15246
  1072
    apply (auto split:split_if_asm)
nipkow@15246
  1073
    using length_filter_le[of P xs] apply arith
nipkow@15246
  1074
  done
nipkow@15246
  1075
qed
wenzelm@13114
  1076
nipkow@15281
  1077
lemma length_filter_conv_card:
nipkow@15281
  1078
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
  1079
proof (induct xs)
nipkow@15281
  1080
  case Nil thus ?case by simp
nipkow@15281
  1081
next
nipkow@15281
  1082
  case (Cons x xs)
nipkow@15281
  1083
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
  1084
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
  1085
  show ?case (is "?l = card ?S'")
nipkow@15281
  1086
  proof (cases)
nipkow@15281
  1087
    assume "p x"
nipkow@15281
  1088
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@25162
  1089
      by(auto simp: image_def split:nat.split dest:gr0_implies_Suc)
nipkow@15281
  1090
    have "length (filter p (x # xs)) = Suc(card ?S)"
wenzelm@23388
  1091
      using Cons `p x` by simp
nipkow@15281
  1092
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
nipkow@15281
  1093
      by (simp add: card_image inj_Suc)
nipkow@15281
  1094
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
  1095
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
  1096
    finally show ?thesis .
nipkow@15281
  1097
  next
nipkow@15281
  1098
    assume "\<not> p x"
nipkow@15281
  1099
    hence eq: "?S' = Suc ` ?S"
nipkow@25162
  1100
      by(auto simp add: image_def split:nat.split elim:lessE)
nipkow@15281
  1101
    have "length (filter p (x # xs)) = card ?S"
wenzelm@23388
  1102
      using Cons `\<not> p x` by simp
nipkow@15281
  1103
    also have "\<dots> = card(Suc ` ?S)" using fin
nipkow@15281
  1104
      by (simp add: card_image inj_Suc)
nipkow@15281
  1105
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
  1106
      by (simp add:card_insert_if)
nipkow@15281
  1107
    finally show ?thesis .
nipkow@15281
  1108
  qed
nipkow@15281
  1109
qed
nipkow@15281
  1110
nipkow@17629
  1111
lemma Cons_eq_filterD:
nipkow@17629
  1112
 "x#xs = filter P ys \<Longrightarrow>
nipkow@17629
  1113
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
wenzelm@19585
  1114
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
nipkow@17629
  1115
proof(induct ys)
nipkow@17629
  1116
  case Nil thus ?case by simp
nipkow@17629
  1117
next
nipkow@17629
  1118
  case (Cons y ys)
nipkow@17629
  1119
  show ?case (is "\<exists>x. ?Q x")
nipkow@17629
  1120
  proof cases
nipkow@17629
  1121
    assume Py: "P y"
nipkow@17629
  1122
    show ?thesis
nipkow@17629
  1123
    proof cases
wenzelm@25221
  1124
      assume "x = y"
wenzelm@25221
  1125
      with Py Cons.prems have "?Q []" by simp
wenzelm@25221
  1126
      then show ?thesis ..
nipkow@17629
  1127
    next
wenzelm@25221
  1128
      assume "x \<noteq> y"
wenzelm@25221
  1129
      with Py Cons.prems show ?thesis by simp
nipkow@17629
  1130
    qed
nipkow@17629
  1131
  next
wenzelm@25221
  1132
    assume "\<not> P y"
wenzelm@25221
  1133
    with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastsimp
wenzelm@25221
  1134
    then have "?Q (y#us)" by simp
wenzelm@25221
  1135
    then show ?thesis ..
nipkow@17629
  1136
  qed
nipkow@17629
  1137
qed
nipkow@17629
  1138
nipkow@17629
  1139
lemma filter_eq_ConsD:
nipkow@17629
  1140
 "filter P ys = x#xs \<Longrightarrow>
nipkow@17629
  1141
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
nipkow@17629
  1142
by(rule Cons_eq_filterD) simp
nipkow@17629
  1143
nipkow@17629
  1144
lemma filter_eq_Cons_iff:
nipkow@17629
  1145
 "(filter P ys = x#xs) =
nipkow@17629
  1146
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
nipkow@17629
  1147
by(auto dest:filter_eq_ConsD)
nipkow@17629
  1148
nipkow@17629
  1149
lemma Cons_eq_filter_iff:
nipkow@17629
  1150
 "(x#xs = filter P ys) =
nipkow@17629
  1151
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
nipkow@17629
  1152
by(auto dest:Cons_eq_filterD)
nipkow@17629
  1153
krauss@19770
  1154
lemma filter_cong[fundef_cong, recdef_cong]:
nipkow@17501
  1155
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
nipkow@17501
  1156
apply simp
nipkow@17501
  1157
apply(erule thin_rl)
nipkow@17501
  1158
by (induct ys) simp_all
nipkow@17501
  1159
nipkow@15281
  1160
haftmann@26442
  1161
subsubsection {* List partitioning *}
haftmann@26442
  1162
haftmann@26442
  1163
primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where
haftmann@26442
  1164
  "partition P [] = ([], [])"
haftmann@26442
  1165
  | "partition P (x # xs) = 
haftmann@26442
  1166
      (let (yes, no) = partition P xs
haftmann@26442
  1167
      in if P x then (x # yes, no) else (yes, x # no))"
haftmann@26442
  1168
haftmann@26442
  1169
lemma partition_filter1:
haftmann@26442
  1170
    "fst (partition P xs) = filter P xs"
haftmann@26442
  1171
by (induct xs) (auto simp add: Let_def split_def)
haftmann@26442
  1172
haftmann@26442
  1173
lemma partition_filter2:
haftmann@26442
  1174
    "snd (partition P xs) = filter (Not o P) xs"
haftmann@26442
  1175
by (induct xs) (auto simp add: Let_def split_def)
haftmann@26442
  1176
haftmann@26442
  1177
lemma partition_P:
haftmann@26442
  1178
  assumes "partition P xs = (yes, no)"
haftmann@26442
  1179
  shows "(\<forall>p \<in> set yes.  P p) \<and> (\<forall>p  \<in> set no. \<not> P p)"
haftmann@26442
  1180
proof -
haftmann@26442
  1181
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
haftmann@26442
  1182
    by simp_all
haftmann@26442
  1183
  then show ?thesis by (simp_all add: partition_filter1 partition_filter2)
haftmann@26442
  1184
qed
haftmann@26442
  1185
haftmann@26442
  1186
lemma partition_set:
haftmann@26442
  1187
  assumes "partition P xs = (yes, no)"
haftmann@26442
  1188
  shows "set yes \<union> set no = set xs"
haftmann@26442
  1189
proof -
haftmann@26442
  1190
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
haftmann@26442
  1191
    by simp_all
haftmann@26442
  1192
  then show ?thesis by (auto simp add: partition_filter1 partition_filter2) 
haftmann@26442
  1193
qed
haftmann@26442
  1194
haftmann@26442
  1195
nipkow@15392
  1196
subsubsection {* @{text concat} *}
wenzelm@13114
  1197
wenzelm@13142
  1198
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
  1199
by (induct xs) auto
wenzelm@13114
  1200
paulson@18447
  1201
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
  1202
by (induct xss) auto
wenzelm@13114
  1203
paulson@18447
  1204
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
  1205
by (induct xss) auto
wenzelm@13114
  1206
nipkow@24308
  1207
lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
nipkow@13145
  1208
by (induct xs) auto
wenzelm@13114
  1209
nipkow@24476
  1210
lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs"
nipkow@24349
  1211
by (induct xs) auto
nipkow@24349
  1212
wenzelm@13142
  1213
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
  1214
by (induct xs) auto
wenzelm@13114
  1215
wenzelm@13142
  1216
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
  1217
by (induct xs) auto
wenzelm@13114
  1218
wenzelm@13142
  1219
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
  1220
by (induct xs) auto
wenzelm@13114
  1221
wenzelm@13114
  1222
nipkow@15392
  1223
subsubsection {* @{text nth} *}
wenzelm@13114
  1224
wenzelm@13142
  1225
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
  1226
by auto
wenzelm@13114
  1227
wenzelm@13142
  1228
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
  1229
by auto
wenzelm@13114
  1230
wenzelm@13142
  1231
declare nth.simps [simp del]
wenzelm@13114
  1232
wenzelm@13114
  1233
lemma nth_append:
nipkow@24526
  1234
  "(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
nipkow@24526
  1235
apply (induct xs arbitrary: n, simp)
paulson@14208
  1236
apply (case_tac n, auto)
nipkow@13145
  1237
done
wenzelm@13114
  1238
nipkow@14402
  1239
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
wenzelm@25221
  1240
by (induct xs) auto
nipkow@14402
  1241
nipkow@14402
  1242
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
wenzelm@25221
  1243
by (induct xs) auto
nipkow@14402
  1244
nipkow@24526
  1245
lemma nth_map [simp]: "n < length xs ==> (map f xs)!n = f(xs!n)"
nipkow@24526
  1246
apply (induct xs arbitrary: n, simp)
paulson@14208
  1247
apply (case_tac n, auto)
nipkow@13145
  1248
done
wenzelm@13114
  1249
nipkow@18423
  1250
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
nipkow@18423
  1251
by(cases xs) simp_all
nipkow@18423
  1252
nipkow@18049
  1253
nipkow@18049
  1254
lemma list_eq_iff_nth_eq:
nipkow@24526
  1255
 "(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
nipkow@24526
  1256
apply(induct xs arbitrary: ys)
paulson@24632
  1257
 apply force
nipkow@18049
  1258
apply(case_tac ys)
nipkow@18049
  1259
 apply simp
nipkow@18049
  1260
apply(simp add:nth_Cons split:nat.split)apply blast
nipkow@18049
  1261
done
nipkow@18049
  1262
wenzelm@13142
  1263
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
  1264
apply (induct xs, simp, simp)
nipkow@13145
  1265
apply safe
paulson@24632
  1266
apply (metis nat_case_0 nth.simps zero_less_Suc)
paulson@24632
  1267
apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
paulson@14208
  1268
apply (case_tac i, simp)
paulson@24632
  1269
apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff)
nipkow@13145
  1270
done
wenzelm@13114
  1271
nipkow@17501
  1272
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
nipkow@17501
  1273
by(auto simp:set_conv_nth)
nipkow@17501
  1274
nipkow@13145
  1275
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
  1276
by (auto simp add: set_conv_nth)
wenzelm@13114
  1277
wenzelm@13142
  1278
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
  1279
by (auto simp add: set_conv_nth)
wenzelm@13114
  1280
wenzelm@13114
  1281
lemma all_nth_imp_all_set:
nipkow@13145
  1282
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
  1283
by (auto simp add: set_conv_nth)
wenzelm@13114
  1284
wenzelm@13114
  1285
lemma all_set_conv_all_nth:
nipkow@13145
  1286
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
  1287
by (auto simp add: set_conv_nth)
wenzelm@13114
  1288
kleing@25296
  1289
lemma rev_nth:
kleing@25296
  1290
  "n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)"
kleing@25296
  1291
proof (induct xs arbitrary: n)
kleing@25296
  1292
  case Nil thus ?case by simp
kleing@25296
  1293
next
kleing@25296
  1294
  case (Cons x xs)
kleing@25296
  1295
  hence n: "n < Suc (length xs)" by simp
kleing@25296
  1296
  moreover
kleing@25296
  1297
  { assume "n < length xs"
kleing@25296
  1298
    with n obtain n' where "length xs - n = Suc n'"
kleing@25296
  1299
      by (cases "length xs - n", auto)
kleing@25296
  1300
    moreover
kleing@25296
  1301
    then have "length xs - Suc n = n'" by simp
kleing@25296
  1302
    ultimately
kleing@25296
  1303
    have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp
kleing@25296
  1304
  }
kleing@25296
  1305
  ultimately
kleing@25296
  1306
  show ?case by (clarsimp simp add: Cons nth_append)
kleing@25296
  1307
qed
wenzelm@13114
  1308
nipkow@15392
  1309
subsubsection {* @{text list_update} *}
wenzelm@13114
  1310
nipkow@24526
  1311
lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
nipkow@24526
  1312
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1313
wenzelm@13114
  1314
lemma nth_list_update:
nipkow@24526
  1315
"i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@24526
  1316
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1317
wenzelm@13142
  1318
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
  1319
by (simp add: nth_list_update)
wenzelm@13114
  1320
nipkow@24526
  1321
lemma nth_list_update_neq [simp]: "i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@24526
  1322
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1323
nipkow@24526
  1324
lemma list_update_id[simp]: "xs[i := xs!i] = xs"
nipkow@24526
  1325
by (induct xs arbitrary: i) (simp_all split:nat.splits)
nipkow@24526
  1326
nipkow@24526
  1327
lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
nipkow@24526
  1328
apply (induct xs arbitrary: i)
nipkow@17501
  1329
 apply simp
nipkow@17501
  1330
apply (case_tac i)
nipkow@17501
  1331
apply simp_all
nipkow@17501
  1332
done
nipkow@17501
  1333
wenzelm@13114
  1334
lemma list_update_same_conv:
nipkow@24526
  1335
"i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@24526
  1336
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1337
nipkow@14187
  1338
lemma list_update_append1:
nipkow@24526
  1339
 "i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
nipkow@24526
  1340
apply (induct xs arbitrary: i, simp)
nipkow@14187
  1341
apply(simp split:nat.split)
nipkow@14187
  1342
done
nipkow@14187
  1343
kleing@15868
  1344
lemma list_update_append:
nipkow@24526
  1345
  "(xs @ ys) [n:= x] = 
kleing@15868
  1346
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
nipkow@24526
  1347
by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868
  1348
nipkow@14402
  1349
lemma list_update_length [simp]:
nipkow@14402
  1350
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
  1351
by (induct xs, auto)
nipkow@14402
  1352
wenzelm@13114
  1353
lemma update_zip:
nipkow@24526
  1354
  "length xs = length ys ==>
nipkow@24526
  1355
  (zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@24526
  1356
by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split)
nipkow@24526
  1357
nipkow@24526
  1358
lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)"
nipkow@24526
  1359
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1360
wenzelm@13114
  1361
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
  1362
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
  1363
nipkow@24526
  1364
lemma set_update_memI: "n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
nipkow@24526
  1365
by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868
  1366
haftmann@24796
  1367
lemma list_update_overwrite:
haftmann@24796
  1368
  "xs [i := x, i := y] = xs [i := y]"
haftmann@24796
  1369
apply (induct xs arbitrary: i)
haftmann@24796
  1370
apply simp
haftmann@24796
  1371
apply (case_tac i)
haftmann@24796
  1372
apply simp_all
haftmann@24796
  1373
done
haftmann@24796
  1374
haftmann@24796
  1375
lemma list_update_swap:
haftmann@24796
  1376
  "i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]"
haftmann@24796
  1377
apply (induct xs arbitrary: i i')
haftmann@24796
  1378
apply simp
haftmann@24796
  1379
apply (case_tac i, case_tac i')
haftmann@24796
  1380
apply auto
haftmann@24796
  1381
apply (case_tac i')
haftmann@24796
  1382
apply auto
haftmann@24796
  1383
done
haftmann@24796
  1384
wenzelm@13114
  1385
nipkow@15392
  1386
subsubsection {* @{text last} and @{text butlast} *}
wenzelm@13114
  1387
wenzelm@13142
  1388
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
  1389
by (induct xs) auto
wenzelm@13114
  1390
wenzelm@13142
  1391
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
  1392
by (induct xs) auto
wenzelm@13114
  1393
nipkow@14302
  1394
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
  1395
by(simp add:last.simps)
nipkow@14302
  1396
nipkow@14302
  1397
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
  1398
by(simp add:last.simps)
nipkow@14302
  1399
nipkow@14302
  1400
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
  1401
by (induct xs) (auto)
nipkow@14302
  1402
nipkow@14302
  1403
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
  1404
by(simp add:last_append)
nipkow@14302
  1405
nipkow@14302
  1406
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
  1407
by(simp add:last_append)
nipkow@14302
  1408
nipkow@17762
  1409
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
nipkow@17762
  1410
by(rule rev_exhaust[of xs]) simp_all
nipkow@17762
  1411
nipkow@17762
  1412
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
nipkow@17762
  1413
by(cases xs) simp_all
nipkow@17762
  1414
nipkow@17765
  1415
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
nipkow@17765
  1416
by (induct as) auto
nipkow@17762
  1417
wenzelm@13142
  1418
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
  1419
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1420
wenzelm@13114
  1421
lemma butlast_append:
nipkow@24526
  1422
  "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@24526
  1423
by (induct xs arbitrary: ys) auto
wenzelm@13114
  1424
wenzelm@13142
  1425
lemma append_butlast_last_id [simp]:
nipkow@13145
  1426
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
  1427
by (induct xs) auto
wenzelm@13114
  1428
wenzelm@13142
  1429
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
  1430
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1431
wenzelm@13114
  1432
lemma in_set_butlast_appendI:
nipkow@13145
  1433
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
  1434
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
  1435
nipkow@24526
  1436
lemma last_drop[simp]: "n < length xs \<Longrightarrow> last (drop n xs) = last xs"
nipkow@24526
  1437
apply (induct xs arbitrary: n)
nipkow@17501
  1438
 apply simp
nipkow@17501
  1439
apply (auto split:nat.split)
nipkow@17501
  1440
done
nipkow@17501
  1441
nipkow@17589
  1442
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
nipkow@17589
  1443
by(induct xs)(auto simp:neq_Nil_conv)
nipkow@17589
  1444
huffman@26584
  1445
lemma butlast_conv_take: "butlast xs = take (length xs - 1) xs"
huffman@26584
  1446
by (induct xs, simp, case_tac xs, simp_all)
huffman@26584
  1447
haftmann@24796
  1448
nipkow@15392
  1449
subsubsection {* @{text take} and @{text drop} *}
wenzelm@13114
  1450
wenzelm@13142
  1451
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
  1452
by (induct xs) auto
wenzelm@13114
  1453
wenzelm@13142
  1454
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
  1455
by (induct xs) auto
wenzelm@13114
  1456
wenzelm@13142
  1457
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
  1458
by simp
wenzelm@13114
  1459
wenzelm@13142
  1460
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
  1461
by simp
wenzelm@13114
  1462
wenzelm@13142
  1463
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
  1464
nipkow@15110
  1465
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
  1466
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
  1467
nipkow@14187
  1468
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
  1469
by(cases xs, simp_all)
nipkow@14187
  1470
huffman@26584
  1471
lemma take_tl: "take n (tl xs) = tl (take (Suc n) xs)"
huffman@26584
  1472
by (induct xs arbitrary: n) simp_all
huffman@26584
  1473
nipkow@24526
  1474
lemma drop_tl: "drop n (tl xs) = tl(drop n xs)"
nipkow@24526
  1475
by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@24526
  1476
huffman@26584
  1477
lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)"
huffman@26584
  1478
by (cases n, simp, cases xs, auto)
huffman@26584
  1479
huffman@26584
  1480
lemma tl_drop: "tl (drop n xs) = drop n (tl xs)"
huffman@26584
  1481
by (simp only: drop_tl)
huffman@26584
  1482
nipkow@24526
  1483
lemma nth_via_drop: "drop n xs = y#ys \<Longrightarrow> xs!n = y"
nipkow@24526
  1484
apply (induct xs arbitrary: n, simp)
nipkow@14187
  1485
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
  1486
done
nipkow@14187
  1487
nipkow@13913
  1488
lemma take_Suc_conv_app_nth:
nipkow@24526
  1489
  "i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
nipkow@24526
  1490
apply (induct xs arbitrary: i, simp)
paulson@14208
  1491
apply (case_tac i, auto)
nipkow@13913
  1492
done
nipkow@13913
  1493
mehta@14591
  1494
lemma drop_Suc_conv_tl:
nipkow@24526
  1495
  "i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
nipkow@24526
  1496
apply (induct xs arbitrary: i, simp)
mehta@14591
  1497
apply (case_tac i, auto)
mehta@14591
  1498
done
mehta@14591
  1499
nipkow@24526
  1500
lemma length_take [simp]: "length (take n xs) = min (length xs) n"
nipkow@24526
  1501
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1502
nipkow@24526
  1503
lemma length_drop [simp]: "length (drop n xs) = (length xs - n)"
nipkow@24526
  1504
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1505
nipkow@24526
  1506
lemma take_all [simp]: "length xs <= n ==> take n xs = xs"
nipkow@24526
  1507
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1508
nipkow@24526
  1509
lemma drop_all [simp]: "length xs <= n ==> drop n xs = []"
nipkow@24526
  1510
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114
  1511
wenzelm@13142
  1512
lemma take_append [simp]:
nipkow@24526
  1513
  "take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@24526
  1514
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114
  1515
wenzelm@13142
  1516
lemma drop_append [simp]:
nipkow@24526
  1517
  "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@24526
  1518
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1519
nipkow@24526
  1520
lemma take_take [simp]: "take n (take m xs) = take (min n m) xs"
nipkow@24526
  1521
apply (induct m arbitrary: xs n, auto)
paulson@14208
  1522
apply (case_tac xs, auto)
nipkow@15236
  1523
apply (case_tac n, auto)
nipkow@13145
  1524
done
wenzelm@13114
  1525
nipkow@24526
  1526
lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs"
nipkow@24526
  1527
apply (induct m arbitrary: xs, auto)
paulson@14208
  1528
apply (case_tac xs, auto)
nipkow@13145
  1529
done
wenzelm@13114
  1530
nipkow@24526
  1531
lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)"
nipkow@24526
  1532
apply (induct m arbitrary: xs n, auto)
paulson@14208
  1533
apply (case_tac xs, auto)
nipkow@13145
  1534
done
wenzelm@13114
  1535
nipkow@24526
  1536
lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@24526
  1537
apply(induct xs arbitrary: m n)
nipkow@14802
  1538
 apply simp
nipkow@14802
  1539
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  1540
done
nipkow@14802
  1541
nipkow@24526
  1542
lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs"
nipkow@24526
  1543
apply (induct n arbitrary: xs, auto)
paulson@14208
  1544
apply (case_tac xs, auto)
nipkow@13145
  1545
done
wenzelm@13114
  1546
nipkow@24526
  1547
lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@24526
  1548
apply(induct xs arbitrary: n)
nipkow@15110
  1549
 apply simp
nipkow@15110
  1550
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  1551
done
nipkow@15110
  1552
nipkow@24526
  1553
lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)"
nipkow@24526
  1554
apply(induct xs arbitrary: n)
nipkow@15110
  1555
apply simp
nipkow@15110
  1556
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1557
done
nipkow@15110
  1558
nipkow@24526
  1559
lemma take_map: "take n (map f xs) = map f (take n xs)"
nipkow@24526
  1560
apply (induct n arbitrary: xs, auto)
paulson@14208
  1561
apply (case_tac xs, auto)
nipkow@13145
  1562
done
wenzelm@13114
  1563
nipkow@24526
  1564
lemma drop_map: "drop n (map f xs) = map f (drop n xs)"
nipkow@24526
  1565
apply (induct n arbitrary: xs, auto)
paulson@14208
  1566
apply (case_tac xs, auto)
nipkow@13145
  1567
done
wenzelm@13114
  1568
nipkow@24526
  1569
lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)"
nipkow@24526
  1570
apply (induct xs arbitrary: i, auto)
paulson@14208
  1571
apply (case_tac i, auto)
nipkow@13145
  1572
done
wenzelm@13114
  1573
nipkow@24526
  1574
lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)"
nipkow@24526
  1575
apply (induct xs arbitrary: i, auto)
paulson@14208
  1576
apply (case_tac i, auto)
nipkow@13145
  1577
done
wenzelm@13114
  1578
nipkow@24526
  1579
lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i"
nipkow@24526
  1580
apply (induct xs arbitrary: i n, auto)
paulson@14208
  1581
apply (case_tac n, blast)
paulson@14208
  1582
apply (case_tac i, auto)
nipkow@13145
  1583
done
wenzelm@13114
  1584
wenzelm@13142
  1585
lemma nth_drop [simp]:
nipkow@24526
  1586
  "n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
nipkow@24526
  1587
apply (induct n arbitrary: xs i, auto)
paulson@14208
  1588
apply (case_tac xs, auto)
nipkow@13145
  1589
done
nipkow@3507
  1590
huffman@26584
  1591
lemma butlast_take:
huffman@26584
  1592
  "n <= length xs ==> butlast (take n xs) = take (n - 1) xs"
huffman@26584
  1593
by (simp add: butlast_conv_take min_max.inf_absorb1 min_max.inf_absorb2)
huffman@26584
  1594
huffman@26584
  1595
lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)"
huffman@26584
  1596
by (simp add: butlast_conv_take drop_take)
huffman@26584
  1597
huffman@26584
  1598
lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs"
huffman@26584
  1599
by (simp add: butlast_conv_take min_max.inf_absorb1)
huffman@26584
  1600
huffman@26584
  1601
lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)"
huffman@26584
  1602
by (simp add: butlast_conv_take drop_take)
huffman@26584
  1603
nipkow@18423
  1604
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
nipkow@18423
  1605
by(simp add: hd_conv_nth)
nipkow@18423
  1606
nipkow@24526
  1607
lemma set_take_subset: "set(take n xs) \<subseteq> set xs"
nipkow@24526
  1608
by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split)
nipkow@24526
  1609
nipkow@24526
  1610
lemma set_drop_subset: "set(drop n xs) \<subseteq> set xs"
nipkow@24526
  1611
by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1612
nipkow@14187
  1613
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1614
using set_take_subset by fast
nipkow@14187
  1615
nipkow@14187
  1616
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1617
using set_drop_subset by fast
nipkow@14187
  1618
wenzelm@13114
  1619
lemma append_eq_conv_conj:
nipkow@24526
  1620
  "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
nipkow@24526
  1621
apply (induct xs arbitrary: zs, simp, clarsimp)
paulson@14208
  1622
apply (case_tac zs, auto)
nipkow@13145
  1623
done
wenzelm@13142
  1624
nipkow@24526
  1625
lemma take_add: 
nipkow@24526
  1626
  "i+j \<le> length(xs) \<Longrightarrow> take (i+j) xs = take i xs @ take j (drop i xs)"
nipkow@24526
  1627
apply (induct xs arbitrary: i, auto) 
nipkow@24526
  1628
apply (case_tac i, simp_all)
paulson@14050
  1629
done
paulson@14050
  1630
nipkow@14300
  1631
lemma append_eq_append_conv_if:
nipkow@24526
  1632
 "(xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1633
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1634
   then xs\<^isub>1 = take (size xs\<^isub>1) ys\<^isub>1 \<and> xs\<^isub>2 = drop (size xs\<^isub>1) ys\<^isub>1 @ ys\<^isub>2
nipkow@14300
  1635
   else take (size ys\<^isub>1) xs\<^isub>1 = ys\<^isub>1 \<and> drop (size ys\<^isub>1) xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>2)"
nipkow@24526
  1636
apply(induct xs\<^isub>1 arbitrary: ys\<^isub>1)
nipkow@14300
  1637
 apply simp
nipkow@14300
  1638
apply(case_tac ys\<^isub>1)
nipkow@14300
  1639
apply simp_all
nipkow@14300
  1640
done
nipkow@14300
  1641
nipkow@15110
  1642
lemma take_hd_drop:
nipkow@24526
  1643
  "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
nipkow@24526
  1644
apply(induct xs arbitrary: n)
nipkow@15110
  1645
apply simp
nipkow@15110
  1646
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1647
done
nipkow@15110
  1648
nipkow@17501
  1649
lemma id_take_nth_drop:
nipkow@17501
  1650
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
nipkow@17501
  1651
proof -
nipkow@17501
  1652
  assume si: "i < length xs"
nipkow@17501
  1653
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
nipkow@17501
  1654
  moreover
nipkow@17501
  1655
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
nipkow@17501
  1656
    apply (rule_tac take_Suc_conv_app_nth) by arith
nipkow@17501
  1657
  ultimately show ?thesis by auto
nipkow@17501
  1658
qed
nipkow@17501
  1659
  
nipkow@17501
  1660
lemma upd_conv_take_nth_drop:
nipkow@17501
  1661
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1662
proof -
nipkow@17501
  1663
  assume i: "i < length xs"
nipkow@17501
  1664
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
nipkow@17501
  1665
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
nipkow@17501
  1666
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1667
    using i by (simp add: list_update_append)
nipkow@17501
  1668
  finally show ?thesis .
nipkow@17501
  1669
qed
nipkow@17501
  1670
haftmann@24796
  1671
lemma nth_drop':
haftmann@24796
  1672
  "i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs"
haftmann@24796
  1673
apply (induct i arbitrary: xs)
haftmann@24796
  1674
apply (simp add: neq_Nil_conv)
haftmann@24796
  1675
apply (erule exE)+
haftmann@24796
  1676
apply simp
haftmann@24796
  1677
apply (case_tac xs)
haftmann@24796
  1678
apply simp_all
haftmann@24796
  1679
done
haftmann@24796
  1680
wenzelm@13114
  1681
nipkow@15392
  1682
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1683
wenzelm@13142
  1684
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1685
by (induct xs) auto
wenzelm@13114
  1686
wenzelm@13142
  1687
lemma takeWhile_append1 [simp]:
nipkow@13145
  1688
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1689
by (induct xs) auto
wenzelm@13114
  1690
wenzelm@13142
  1691
lemma takeWhile_append2 [simp]:
nipkow@13145
  1692
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1693
by (induct xs) auto
wenzelm@13114
  1694
wenzelm@13142
  1695
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1696
by (induct xs) auto
wenzelm@13114
  1697
wenzelm@13142
  1698
lemma dropWhile_append1 [simp]:
nipkow@13145
  1699
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1700
by (induct xs) auto
wenzelm@13114
  1701
wenzelm@13142
  1702
lemma dropWhile_append2 [simp]:
nipkow@13145
  1703
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1704
by (induct xs) auto
wenzelm@13114
  1705
krauss@23971
  1706
lemma set_takeWhileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1707
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1708
nipkow@13913
  1709
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1710
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1711
by(induct xs, auto)
nipkow@13913
  1712
nipkow@13913
  1713
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1714
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1715
by(induct xs, auto)
nipkow@13913
  1716
nipkow@13913
  1717
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1718
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1719
by(induct xs, auto)
nipkow@13913
  1720
nipkow@17501
  1721
text{* The following two lemmmas could be generalized to an arbitrary
nipkow@17501
  1722
property. *}
nipkow@17501
  1723
nipkow@17501
  1724
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1725
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
nipkow@17501
  1726
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
nipkow@17501
  1727
nipkow@17501
  1728
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1729
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
nipkow@17501
  1730
apply(induct xs)
nipkow@17501
  1731
 apply simp
nipkow@17501
  1732
apply auto
nipkow@17501
  1733
apply(subst dropWhile_append2)
nipkow@17501
  1734
apply auto
nipkow@17501
  1735
done
nipkow@17501
  1736
nipkow@18423
  1737
lemma takeWhile_not_last:
nipkow@18423
  1738
 "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
nipkow@18423
  1739
apply(induct xs)
nipkow@18423
  1740
 apply simp
nipkow@18423
  1741
apply(case_tac xs)
nipkow@18423
  1742
apply(auto)
nipkow@18423
  1743
done
nipkow@18423
  1744
krauss@19770
  1745
lemma takeWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1746
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1747
  ==> takeWhile P l = takeWhile Q k"
nipkow@24349
  1748
by (induct k arbitrary: l) (simp_all)
krauss@18336
  1749
krauss@19770
  1750
lemma dropWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1751
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1752
  ==> dropWhile P l = dropWhile Q k"
nipkow@24349
  1753
by (induct k arbitrary: l, simp_all)
krauss@18336
  1754
wenzelm@13114
  1755
nipkow@15392
  1756
subsubsection {* @{text zip} *}
wenzelm@13114
  1757
wenzelm@13142
  1758
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1759
by (induct ys) auto
wenzelm@13114
  1760
wenzelm@13142
  1761
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1762
by simp
wenzelm@13114
  1763
wenzelm@13142
  1764
declare zip_Cons [simp del]
wenzelm@13114
  1765
nipkow@15281
  1766
lemma zip_Cons1:
nipkow@15281
  1767
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  1768
by(auto split:list.split)
nipkow@15281
  1769
wenzelm@13142
  1770
lemma length_zip [simp]:
krauss@22493
  1771
"length (zip xs ys) = min (length xs) (length ys)"
krauss@22493
  1772
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1773
wenzelm@13114
  1774
lemma zip_append1:
krauss@22493
  1775
"zip (xs @ ys) zs =
nipkow@13145
  1776
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
krauss@22493
  1777
by (induct xs zs rule:list_induct2') auto
wenzelm@13114
  1778
wenzelm@13114
  1779
lemma zip_append2:
krauss@22493
  1780
"zip xs (ys @ zs) =
nipkow@13145
  1781
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
krauss@22493
  1782
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1783
wenzelm@13142
  1784
lemma zip_append [simp]:
wenzelm@13142
  1785
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1786
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1787
by (simp add: zip_append1)
wenzelm@13114
  1788
wenzelm@13114
  1789
lemma zip_rev:
nipkow@14247
  1790
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1791
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1792
nipkow@23096
  1793
lemma map_zip_map:
nipkow@23096
  1794
 "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
nipkow@23096
  1795
apply(induct xs arbitrary:ys) apply simp
nipkow@23096
  1796
apply(case_tac ys)
nipkow@23096
  1797
apply simp_all
nipkow@23096
  1798
done
nipkow@23096
  1799
nipkow@23096
  1800
lemma map_zip_map2:
nipkow@23096
  1801
 "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
nipkow@23096
  1802
apply(induct xs arbitrary:ys) apply simp
nipkow@23096
  1803
apply(case_tac ys)
nipkow@23096
  1804
apply simp_all
nipkow@23096
  1805
done
nipkow@23096
  1806
wenzelm@13142
  1807
lemma nth_zip [simp]:
nipkow@24526
  1808
"[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
nipkow@24526
  1809
apply (induct ys arbitrary: i xs, simp)
nipkow@13145
  1810
apply (case_tac xs)
nipkow@13145
  1811
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1812
done
wenzelm@13114
  1813
wenzelm@13114
  1814
lemma set_zip:
nipkow@13145
  1815
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
  1816
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1817
wenzelm@13114
  1818
lemma zip_update:
nipkow@13145
  1819
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
  1820
by (rule sym, simp add: update_zip)
wenzelm@13114
  1821
wenzelm@13142
  1822
lemma zip_replicate [simp]:
nipkow@24526
  1823
  "zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
nipkow@24526
  1824
apply (induct i arbitrary: j, auto)
paulson@14208
  1825
apply (case_tac j, auto)
nipkow@13145
  1826
done
wenzelm@13114
  1827
nipkow@19487
  1828
lemma take_zip:
nipkow@24526
  1829
  "take n (zip xs ys) = zip (take n xs) (take n ys)"
nipkow@24526
  1830
apply (induct n arbitrary: xs ys)
nipkow@19487
  1831
 apply simp
nipkow@19487
  1832
apply (case_tac xs, simp)
nipkow@19487
  1833
apply (case_tac ys, simp_all)
nipkow@19487
  1834
done
nipkow@19487
  1835
nipkow@19487
  1836
lemma drop_zip:
nipkow@24526
  1837
  "drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
nipkow@24526
  1838
apply (induct n arbitrary: xs ys)
nipkow@19487
  1839
 apply simp
nipkow@19487
  1840
apply (case_tac xs, simp)
nipkow@19487
  1841
apply (case_tac ys, simp_all)
nipkow@19487
  1842
done
nipkow@19487
  1843
krauss@22493
  1844
lemma set_zip_leftD:
krauss@22493
  1845
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
krauss@22493
  1846
by (induct xs ys rule:list_induct2') auto
krauss@22493
  1847
krauss@22493
  1848
lemma set_zip_rightD:
krauss@22493
  1849
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
krauss@22493
  1850
by (induct xs ys rule:list_induct2') auto
wenzelm@13142
  1851
nipkow@23983
  1852
lemma in_set_zipE:
nipkow@23983
  1853
  "(x,y) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
nipkow@23983
  1854
by(blast dest: set_zip_leftD set_zip_rightD)
nipkow@23983
  1855
nipkow@15392
  1856
subsubsection {* @{text list_all2} *}
wenzelm@13114
  1857
kleing@14316
  1858
lemma list_all2_lengthD [intro?]: 
kleing@14316
  1859
  "list_all2 P xs ys ==> length xs = length ys"
nipkow@24349
  1860
by (simp add: list_all2_def)
haftmann@19607
  1861
haftmann@19787
  1862
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
nipkow@24349
  1863
by (simp add: list_all2_def)
haftmann@19607
  1864
haftmann@19787
  1865
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
nipkow@24349
  1866
by (simp add: list_all2_def)
haftmann@19607
  1867
haftmann@19607
  1868
lemma list_all2_Cons [iff, code]:
haftmann@19607
  1869
  "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@24349
  1870
by (auto simp add: list_all2_def)
wenzelm@13114
  1871
wenzelm@13114
  1872
lemma list_all2_Cons1:
nipkow@13145
  1873
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1874
by (cases ys) auto
wenzelm@13114
  1875
wenzelm@13114
  1876
lemma list_all2_Cons2:
nipkow@13145
  1877
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1878
by (cases xs) auto
wenzelm@13114
  1879
wenzelm@13142
  1880
lemma list_all2_rev [iff]:
nipkow@13145
  1881
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1882
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1883
kleing@13863
  1884
lemma list_all2_rev1:
kleing@13863
  1885
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1886
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1887
wenzelm@13114
  1888
lemma list_all2_append1:
nipkow@13145
  1889
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1890
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1891
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1892
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1893
apply (rule iffI)
nipkow@13145
  1894
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1895
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  1896
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1897
apply (simp add: ball_Un)
nipkow@13145
  1898
done
wenzelm@13114
  1899
wenzelm@13114
  1900
lemma list_all2_append2:
nipkow@13145
  1901
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1902
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1903
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1904
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1905
apply (rule iffI)
nipkow@13145
  1906
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1907
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  1908
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1909
apply (simp add: ball_Un)
nipkow@13145
  1910
done
wenzelm@13114
  1911
kleing@13863
  1912
lemma list_all2_append:
nipkow@14247
  1913
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  1914
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  1915
by (induct rule:list_induct2, simp_all)
kleing@13863
  1916
kleing@13863
  1917
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1918
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
nipkow@24349
  1919
by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1920
wenzelm@13114
  1921
lemma list_all2_conv_all_nth:
nipkow@13145
  1922
"list_all2 P xs ys =
nipkow@13145
  1923
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1924
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1925
berghofe@13883
  1926
lemma list_all2_trans:
berghofe@13883
  1927
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1928
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1929
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1930
proof (induct as)
berghofe@13883
  1931
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1932
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1933
  proof (induct bs)
berghofe@13883
  1934
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1935
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1936
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1937
  qed simp
berghofe@13883
  1938
qed simp
berghofe@13883
  1939
kleing@13863
  1940
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1941
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
nipkow@24349
  1942
by (simp add: list_all2_conv_all_nth)
kleing@13863
  1943
paulson@14395
  1944
lemma list_all2I:
paulson@14395
  1945
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
nipkow@24349
  1946
by (simp add: list_all2_def)
paulson@14395
  1947
kleing@14328
  1948
lemma list_all2_nthD:
kleing@13863
  1949
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@24349
  1950
by (simp add: list_all2_conv_all_nth)
kleing@13863
  1951
nipkow@14302
  1952
lemma list_all2_nthD2:
nipkow@14302
  1953
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@24349
  1954
by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  1955
kleing@13863
  1956
lemma list_all2_map1: 
kleing@13863
  1957
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
nipkow@24349
  1958
by (simp add: list_all2_conv_all_nth)
kleing@13863
  1959
kleing@13863
  1960
lemma list_all2_map2: 
kleing@13863
  1961
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
nipkow@24349
  1962
by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1963
kleing@14316
  1964
lemma list_all2_refl [intro?]:
kleing@13863
  1965
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
nipkow@24349
  1966
by (simp add: list_all2_conv_all_nth)
kleing@13863
  1967
kleing@13863
  1968
lemma list_all2_update_cong:
kleing@13863
  1969
  "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
nipkow@24349
  1970
by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1971
kleing@13863
  1972
lemma list_all2_update_cong2:
kleing@13863
  1973
  "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
nipkow@24349
  1974
by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1975
nipkow@14302
  1976
lemma list_all2_takeI [simp,intro?]:
nipkow@24526
  1977
  "list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@24526
  1978
apply (induct xs arbitrary: n ys)
nipkow@24526
  1979
 apply simp
nipkow@24526
  1980
apply (clarsimp simp add: list_all2_Cons1)
nipkow@24526
  1981
apply (case_tac n)
nipkow@24526
  1982
apply auto
nipkow@24526
  1983
done
nipkow@14302
  1984
nipkow@14302
  1985
lemma list_all2_dropI [simp,intro?]:
nipkow@24526
  1986
  "list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
nipkow@24526
  1987
apply (induct as arbitrary: n bs, simp)
nipkow@24526
  1988
apply (clarsimp simp add: list_all2_Cons1)
nipkow@24526
  1989
apply (case_tac n, simp, simp)
nipkow@24526
  1990
done
kleing@13863
  1991
kleing@14327
  1992
lemma list_all2_mono [intro?]:
nipkow@24526
  1993
  "list_all2 P xs ys \<Longrightarrow> (\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys) \<Longrightarrow> list_all2 Q xs ys"
nipkow@24526
  1994
apply (induct xs arbitrary: ys, simp)
nipkow@24526
  1995
apply (case_tac ys, auto)
nipkow@24526
  1996
done
kleing@13863
  1997
haftmann@22551
  1998
lemma list_all2_eq:
haftmann@22551
  1999
  "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
nipkow@24349
  2000
by (induct xs ys rule: list_induct2') auto
haftmann@22551
  2001
wenzelm@13142
  2002
nipkow@15392
  2003
subsubsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  2004
wenzelm@13142
  2005
lemma foldl_append [simp]:
nipkow@24526
  2006
  "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@24526
  2007
by (induct xs arbitrary: a) auto
wenzelm@13142
  2008
nipkow@14402
  2009
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  2010
by (induct xs) auto
nipkow@14402
  2011
nipkow@23096
  2012
lemma foldr_map: "foldr g (map f xs) a = foldr (g o f) xs a"
nipkow@23096
  2013
by(induct xs) simp_all
nipkow@23096
  2014
nipkow@24449
  2015
text{* For efficient code generation: avoid intermediate list. *}
nipkow@24449
  2016
lemma foldl_map[code unfold]:
nipkow@24449
  2017
  "foldl g a (map f xs) = foldl (%a x. g a (f x)) a xs"
nipkow@23096
  2018
by(induct xs arbitrary:a) simp_all
nipkow@23096
  2019
krauss@19770
  2020
lemma foldl_cong [fundef_cong, recdef_cong]:
krauss@18336
  2021
  "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] 
krauss@18336
  2022
  ==> foldl f a l = foldl g b k"
nipkow@24349
  2023
by (induct k arbitrary: a b l) simp_all
krauss@18336
  2024
krauss@19770
  2025
lemma foldr_cong [fundef_cong, recdef_cong]:
krauss@18336
  2026
  "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] 
krauss@18336
  2027
  ==> foldr f l a = foldr g k b"
nipkow@24349
  2028
by (induct k arbitrary: a b l) simp_all
krauss@18336
  2029
nipkow@24449
  2030
lemma (in semigroup_add) foldl_assoc:
haftmann@25062
  2031
shows "foldl op+ (x+y) zs = x + (foldl op+ y zs)"
nipkow@24449
  2032
by (induct zs arbitrary: y) (simp_all add:add_assoc)
nipkow@24449
  2033
nipkow@24449
  2034
lemma (in monoid_add) foldl_absorb0:
haftmann@25062
  2035
shows "x + (foldl op+ 0 zs) = foldl op+ x zs"
nipkow@24449
  2036
by (induct zs) (simp_all add:foldl_assoc)
nipkow@24449
  2037
nipkow@24449
  2038
nipkow@23096
  2039
text{* The ``First Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096
  2040
nipkow@23096
  2041
lemma foldl_foldr1_lemma:
nipkow@23096
  2042
 "foldl op + a xs = a + foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096
  2043
by (induct xs arbitrary: a) (auto simp:add_assoc)
nipkow@23096
  2044
nipkow@23096
  2045
corollary foldl_foldr1:
nipkow@23096
  2046
 "foldl op + 0 xs = foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096
  2047
by (simp add:foldl_foldr1_lemma)
nipkow@23096
  2048
nipkow@23096
  2049
nipkow@23096
  2050
text{* The ``Third Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096
  2051
nipkow@14402
  2052
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  2053
by (induct xs) auto
nipkow@14402
  2054
nipkow@14402
  2055
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  2056
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  2057
haftmann@25062
  2058
lemma (in ab_semigroup_add) foldr_conv_foldl: "foldr op + xs a = foldl op + a xs"
chaieb@24471
  2059
  by (induct xs, auto simp add: foldl_assoc add_commute)
chaieb@24471
  2060
wenzelm@13142
  2061
text {*
nipkow@13145
  2062
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  2063
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  2064
*}
wenzelm@13142
  2065
nipkow@24526
  2066
lemma start_le_sum: "(m::nat) <= n ==> m <= foldl (op +) n ns"
nipkow@24526
  2067
by (induct ns arbitrary: n) auto
nipkow@24526
  2068
nipkow@24526
  2069
lemma elem_le_sum: "(n::nat) : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  2070
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  2071
wenzelm@13142
  2072
lemma sum_eq_0_conv [iff]:
nipkow@24526
  2073
  "(foldl (op +) (m::nat) ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@24526
  2074
by (induct ns arbitrary: m) auto
wenzelm@13114
  2075
chaieb@24471
  2076
lemma foldr_invariant: 
chaieb@24471
  2077
  "\<lbrakk>Q x ; \<forall> x\<in> set xs. P x; \<forall> x y. P x \<and> Q y \<longrightarrow> Q (f x y) \<rbrakk> \<Longrightarrow> Q (foldr f xs x)"
chaieb@24471
  2078
  by (induct xs, simp_all)
chaieb@24471
  2079
chaieb@24471
  2080
lemma foldl_invariant: 
chaieb@24471
  2081
  "\<lbrakk>Q x ; \<forall> x\<in> set xs. P x; \<forall> x y. P x \<and> Q y \<longrightarrow> Q (f y x) \<rbrakk> \<Longrightarrow> Q (foldl f x xs)"
chaieb@24471
  2082
  by (induct xs arbitrary: x, simp_all)
chaieb@24471
  2083
nipkow@24449
  2084
text{* @{const foldl} and @{text concat} *}
nipkow@24449
  2085
nipkow@24449
  2086
lemma concat_conv_foldl: "concat xss = foldl op@ [] xss"
nipkow@24449
  2087
by (induct xss) (simp_all add:monoid_append.foldl_absorb0)
nipkow@24449
  2088
nipkow@24449
  2089
lemma foldl_conv_concat:
nipkow@24449
  2090
  "foldl (op @) xs xxs = xs @ (concat xxs)"
nipkow@24449
  2091
by(simp add:concat_conv_foldl monoid_append.foldl_absorb0)
nipkow@24449
  2092
nipkow@23096
  2093
subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
nipkow@23096
  2094
haftmann@26442
  2095
lemma listsum_append [simp]: "listsum (xs @ ys) = listsum xs + listsum ys"
nipkow@24449
  2096
by (induct xs) (simp_all add:add_assoc)
nipkow@24449
  2097
haftmann@26442
  2098
lemma listsum_rev [simp]:
haftmann@26442
  2099
  fixes xs :: "'a\<Colon>comm_monoid_add list"
haftmann@26442
  2100
  shows "listsum (rev xs) = listsum xs"
nipkow@24449
  2101
by (induct xs) (simp_all add:add_ac)
nipkow@24449
  2102
haftmann@26442
  2103
lemma listsum_foldr: "listsum xs = foldr (op +) xs 0"
haftmann@26442
  2104
by (induct xs) auto
haftmann@26442
  2105
haftmann@26442
  2106
lemma length_concat: "length (concat xss) = listsum (map length xss)"
haftmann@26442
  2107
by (induct xss) simp_all
nipkow@23096
  2108
nipkow@24449
  2109
text{* For efficient code generation ---
nipkow@24449
  2110
       @{const listsum} is not tail recursive but @{const foldl} is. *}
nipkow@24449
  2111
lemma listsum[code unfold]: "listsum xs = foldl (op +) 0 xs"
nipkow@23096
  2112
by(simp add:listsum_foldr foldl_foldr1)
nipkow@23096
  2113
nipkow@24449
  2114
nipkow@23096
  2115
text{* Some syntactic sugar for summing a function over a list: *}
nipkow@23096
  2116
nipkow@23096
  2117
syntax
nipkow@23096
  2118
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
nipkow@23096
  2119
syntax (xsymbols)
nipkow@23096
  2120
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096
  2121
syntax (HTML output)
nipkow@23096
  2122
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096
  2123
nipkow@23096
  2124
translations -- {* Beware of argument permutation! *}
nipkow@23096
  2125
  "SUM x<-xs. b" == "CONST listsum (map (%x. b) xs)"
nipkow@23096
  2126
  "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (map (%x. b) xs)"
nipkow@23096
  2127
haftmann@26442
  2128
lemma listsum_triv: "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
haftmann@26442
  2129
  by (induct xs) (simp_all add: left_distrib)
haftmann@26442
  2130
nipkow@23096
  2131
lemma listsum_0 [simp]: "(\<Sum>x\<leftarrow>xs. 0) = 0"
haftmann@26442
  2132
  by (induct xs) (simp_all add: left_distrib)
nipkow@23096
  2133
nipkow@23096
  2134
text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
nipkow@23096
  2135
lemma uminus_listsum_map:
haftmann@26442
  2136
  fixes f :: "'a \<Rightarrow> 'b\<Colon>ab_group_add"
haftmann@26442
  2137
  shows "- listsum (map f xs) = (listsum (map (uminus o f) xs))"
haftmann@26442
  2138
by (induct xs) simp_all
nipkow@23096
  2139
wenzelm@13114
  2140
nipkow@24645
  2141
subsubsection {* @{text upt} *}
wenzelm@13114
  2142
nipkow@17090
  2143
lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
nipkow@17090
  2144
-- {* simp does not terminate! *}
nipkow@13145
  2145
by (induct j) auto
wenzelm@13142
  2146
nipkow@15425
  2147
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
nipkow@13145
  2148
by (subst upt_rec) simp
wenzelm@13114
  2149
nipkow@15425
  2150
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
nipkow@15281
  2151
by(induct j)simp_all
nipkow@15281
  2152
nipkow@15281
  2153
lemma upt_eq_Cons_conv:
nipkow@24526
  2154
 "([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
nipkow@24526
  2155
apply(induct j arbitrary: x xs)
nipkow@15281
  2156
 apply simp
nipkow@15281
  2157
apply(clarsimp simp add: append_eq_Cons_conv)
nipkow@15281
  2158
apply arith
nipkow@15281
  2159
done
nipkow@15281
  2160
nipkow@15425
  2161
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
nipkow@13145
  2162
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  2163
by simp
wenzelm@13114
  2164
nipkow@15425
  2165
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
haftmann@26734
  2166
  by (simp add: upt_rec)
wenzelm@13114
  2167
nipkow@15425
  2168
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
nipkow@13145
  2169
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  2170
by (induct k) auto
wenzelm@13114
  2171
nipkow@15425
  2172
lemma length_upt [simp]: "length [i..<j] = j - i"
nipkow@13145
  2173
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  2174
nipkow@15425
  2175
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
nipkow@13145
  2176
apply (induct j)
nipkow@13145
  2177
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  2178
done
wenzelm@13114
  2179
nipkow@17906
  2180
nipkow@17906
  2181
lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
nipkow@17906
  2182
by(simp add:upt_conv_Cons)
nipkow@17906
  2183
nipkow@17906
  2184
lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
nipkow@17906
  2185
apply(cases j)
nipkow@17906
  2186
 apply simp
nipkow@17906
  2187
by(simp add:upt_Suc_append)
nipkow@17906
  2188
nipkow@24526
  2189
lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]"
nipkow@24526
  2190
apply (induct m arbitrary: i, simp)
nipkow@13145
  2191
apply (subst upt_rec)
nipkow@13145
  2192
apply (rule sym)
nipkow@13145
  2193
apply (subst upt_rec)
nipkow@13145
  2194
apply (simp del: upt.simps)
nipkow@13145
  2195
done
nipkow@3507
  2196
nipkow@17501
  2197
lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
nipkow@17501
  2198
apply(induct j)
nipkow@17501
  2199
apply auto
nipkow@17501
  2200
done
nipkow@17501
  2201
nipkow@24645
  2202
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]"
nipkow@13145
  2203
by (induct n) auto
wenzelm@13114
  2204
nipkow@24526
  2205
lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
nipkow@24526
  2206
apply (induct n m  arbitrary: i rule: diff_induct)
nipkow@13145
  2207
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  2208
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  2209
done
wenzelm@13114
  2210
berghofe@13883
  2211
lemma nth_take_lemma:
nipkow@24526
  2212
  "k <= length xs ==> k <= length ys ==>
berghofe@13883
  2213
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
nipkow@24526
  2214
apply (atomize, induct k arbitrary: xs ys)
paulson@14208
  2215
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  2216
txt {* Both lists must be non-empty *}
paulson@14208
  2217
apply (case_tac xs, simp)
paulson@14208
  2218
apply (case_tac ys, clarify)
nipkow@13145
  2219
 apply (simp (no_asm_use))
nipkow@13145
  2220
apply clarify
nipkow@13145
  2221
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  2222
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  2223
apply blast
nipkow@13145
  2224
done
wenzelm@13114
  2225
wenzelm@13114
  2226
lemma nth_equalityI:
wenzelm@13114
  2227
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  2228
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  2229
apply (simp_all add: take_all)
nipkow@13145
  2230
done
wenzelm@13142
  2231
haftmann@24796
  2232
lemma map_nth:
haftmann@24796
  2233
  "map (\<lambda>i. xs ! i) [0..<length xs] = xs"
haftmann@24796
  2234
  by (rule nth_equalityI, auto)
haftmann@24796
  2235
kleing@13863
  2236
(* needs nth_equalityI *)
kleing@13863
  2237
lemma list_all2_antisym:
kleing@13863
  2238
  "\<lbrakk> (\<And>x y. \<lbrakk>P x y; Q y x\<rbrakk> \<Longrightarrow> x = y); list_all2 P xs ys; list_all2 Q ys xs \<rbrakk> 
kleing@13863
  2239
  \<Longrightarrow> xs = ys"
kleing@13863
  2240
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  2241
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  2242
  done
kleing@13863
  2243
wenzelm@13142
  2244
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  2245
-- {* The famous take-lemma. *}
nipkow@13145
  2246
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  2247
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  2248
done
wenzelm@13142
  2249
wenzelm@13142
  2250
nipkow@15302
  2251
lemma take_Cons':
nipkow@15302
  2252
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@15302
  2253
by (cases n) simp_all
nipkow@15302
  2254
nipkow@15302
  2255
lemma drop_Cons':
nipkow@15302
  2256
     "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@15302
  2257
by (cases n) simp_all
nipkow@15302
  2258
nipkow@15302
  2259
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@15302
  2260
by (cases n) simp_all
nipkow@15302
  2261
paulson@18622
  2262
lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard]
paulson@18622
  2263
lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard]
paulson@18622
  2264
lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard]
paulson@18622
  2265
paulson@18622
  2266
declare take_Cons_number_of [simp] 
paulson@18622
  2267
        drop_Cons_number_of [simp] 
paulson@18622
  2268
        nth_Cons_number_of [simp] 
nipkow@15302
  2269
nipkow@15302
  2270
nipkow@15392
  2271
subsubsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  2272
wenzelm@13142
  2273
lemma distinct_append [simp]:
nipkow@13145
  2274
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  2275
by (induct xs) auto
wenzelm@13142
  2276
nipkow@15305
  2277
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
nipkow@15305
  2278
by(induct xs) auto
nipkow@15305
  2279
wenzelm@13142
  2280
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  2281
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  2282
wenzelm@13142
  2283
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  2284
by (induct xs) auto
wenzelm@13142
  2285
nipkow@25287
  2286
lemma distinct_remdups_id: "distinct xs ==> remdups xs = xs"
nipkow@25287
  2287
by (induct xs, auto)
nipkow@25287
  2288
haftmann@26734
  2289
lemma remdups_id_iff_distinct [simp]: "remdups xs = xs \<longleftrightarrow> distinct xs"
haftmann@26734
  2290
by (metis distinct_remdups distinct_remdups_id)
nipkow@25287
  2291
nipkow@24566
  2292
lemma finite_distinct_list: "finite A \<Longrightarrow> EX xs. set xs = A & distinct xs"
paulson@24632
  2293
by (metis distinct_remdups finite_list set_remdups)
nipkow@24566
  2294
paulson@15072
  2295
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
nipkow@24349
  2296
by (induct x, auto) 
paulson@15072
  2297
paulson@15072
  2298
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
nipkow@24349
  2299
by (induct x, auto)
paulson@15072
  2300
nipkow@15245
  2301
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245
  2302
by (induct xs) auto
nipkow@15245
  2303
nipkow@15245
  2304
lemma length_remdups_eq[iff]:
nipkow@15245
  2305
  "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245
  2306
apply(induct xs)
nipkow@15245
  2307
 apply auto
nipkow@15245
  2308
apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245
  2309
 apply arith
nipkow@15245
  2310
apply(rule length_remdups_leq)
nipkow@15245
  2311
done
nipkow@15245
  2312
nipkow@18490
  2313
nipkow@18490
  2314
lemma distinct_map:
nipkow@18490
  2315
  "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
nipkow@18490
  2316
by (induct xs) auto
nipkow@18490
  2317
nipkow@18490
  2318
wenzelm@13142
  2319
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  2320
by (induct xs) auto
wenzelm@13114
  2321
nipkow@17501
  2322
lemma distinct_upt[simp]: "distinct[i..<j]"
nipkow@17501
  2323
by (induct j) auto
nipkow@17501
  2324
nipkow@24526
  2325
lemma distinct_take[simp]: "distinct xs \<Longrightarrow> distinct (take i xs)"
nipkow@24526
  2326
apply(induct xs arbitrary: i)
nipkow@17501
  2327
 apply simp
nipkow@17501
  2328
apply (case_tac i)
nipkow@17501
  2329
 apply simp_all
nipkow@17501
  2330
apply(blast dest:in_set_takeD)
nipkow@17501
  2331
done
nipkow@17501
  2332
nipkow@24526
  2333
lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)"
nipkow@24526
  2334
apply(induct xs arbitrary: i)
nipkow@17501
  2335
 apply simp
nipkow@17501
  2336
apply (case_tac i)
nipkow@17501
  2337
 apply simp_all
nipkow@17501
  2338
done
nipkow@17501
  2339
nipkow@17501
  2340
lemma distinct_list_update:
nipkow@17501
  2341
assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
nipkow@17501
  2342
shows "distinct (xs[i:=a])"
nipkow@17501
  2343
proof (cases "i < length xs")
nipkow@17501
  2344
  case True
nipkow@17501
  2345
  with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
nipkow@17501
  2346
    apply (drule_tac id_take_nth_drop) by simp
nipkow@17501
  2347
  with d True show ?thesis
nipkow@17501
  2348
    apply (simp add: upd_conv_take_nth_drop)
nipkow@17501
  2349
    apply (drule subst [OF id_take_nth_drop]) apply assumption
nipkow@17501
  2350
    apply simp apply (cases "a = xs!i") apply simp by blast
nipkow@17501
  2351
next
nipkow@17501
  2352
  case False with d show ?thesis by auto
nipkow@17501
  2353
qed
nipkow@17501
  2354
nipkow@17501
  2355
nipkow@17501
  2356
text {* It is best to avoid this indexed version of distinct, but
nipkow@17501
  2357
sometimes it is useful. *}
nipkow@17501
  2358
wenzelm@13142
  2359
lemma distinct_conv_nth:
nipkow@17501
  2360
"distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@15251
  2361
apply (induct xs, simp, simp)
paulson@14208
  2362
apply (rule iffI, clarsimp)
nipkow@13145
  2363
 apply (case_tac i)
paulson@14208
  2364
apply (case_tac j, simp)
nipkow@13145
  2365
apply (simp add: set_conv_nth)
nipkow@13145
  2366
 apply (case_tac j)
paulson@24648
  2367
apply (clarsimp simp add: set_conv_nth, simp) 
nipkow@13145
  2368
apply (rule conjI)
paulson@24648
  2369
(*TOO SLOW
paulson@24632
  2370
apply (metis Zero_neq_Suc gr0_conv_Suc in_set_conv_nth lessI less_trans_Suc nth_Cons' nth_Cons_Suc)
paulson@24648
  2371
*)
paulson@24648
  2372
 apply (clarsimp simp add: set_conv_nth)
paulson@24648
  2373
 apply (erule_tac x = 0 in allE, simp)
paulson@24648
  2374
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
wenzelm@25130
  2375
(*TOO SLOW
paulson@24632
  2376
apply (metis Suc_Suc_eq lessI less_trans_Suc nth_Cons_Suc)
wenzelm@25130
  2377
*)
wenzelm@25130
  2378
apply (erule_tac x = "Suc i" in allE, simp)
wenzelm@25130
  2379
apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145
  2380
done
wenzelm@13114
  2381
nipkow@18490
  2382
lemma nth_eq_iff_index_eq:
nipkow@18490
  2383
 "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
nipkow@18490
  2384
by(auto simp: distinct_conv_nth)
nipkow@18490
  2385
nipkow@15110
  2386
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
nipkow@24349
  2387
by (induct xs) auto
kleing@14388
  2388
nipkow@15110
  2389
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388
  2390
proof (induct xs)
kleing@14388
  2391
  case Nil thus ?case by simp
kleing@14388
  2392
next
kleing@14388
  2393
  case (Cons x xs)
kleing@14388
  2394
  show ?case
kleing@14388
  2395
  proof (cases "x \<in> set xs")
kleing@14388
  2396
    case False with Cons show ?thesis by simp
kleing@14388
  2397
  next
kleing@14388
  2398
    case True with Cons.prems
kleing@14388
  2399
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  2400
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  2401
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  2402
    ultimately have False by simp
kleing@14388
  2403
    thus ?thesis ..
kleing@14388
  2404
  qed
kleing@14388
  2405
qed
kleing@14388
  2406
nipkow@25287
  2407
lemma not_distinct_decomp: "~ distinct ws ==> EX xs ys zs y. ws = xs@[y]@ys@[y]@zs"
nipkow@25287
  2408
apply (induct n == "length ws" arbitrary:ws) apply simp
nipkow@25287
  2409
apply(case_tac ws) apply simp
nipkow@25287
  2410
apply (simp split:split_if_asm)
nipkow@25287
  2411
apply (metis Cons_eq_appendI eq_Nil_appendI split_list)
nipkow@25287
  2412
done
nipkow@18490
  2413
nipkow@18490
  2414
lemma length_remdups_concat:
nipkow@18490
  2415
 "length(remdups(concat xss)) = card(\<Union>xs \<in> set xss. set xs)"
nipkow@24308
  2416
by(simp add: set_concat distinct_card[symmetric])
nipkow@17906
  2417
nipkow@17906
  2418
nipkow@15392
  2419
subsubsection {* @{text remove1} *}
nipkow@15110
  2420
nipkow@18049
  2421
lemma remove1_append:
nipkow@18049
  2422
  "remove1 x (xs @ ys) =
nipkow@18049
  2423
  (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
nipkow@18049
  2424
by (induct xs) auto
nipkow@18049
  2425
nipkow@23479
  2426
lemma in_set_remove1[simp]:
nipkow@23479
  2427
  "a \<noteq> b \<Longrightarrow> a : set(remove1 b xs) = (a : set xs)"
nipkow@23479
  2428
apply (induct xs)
nipkow@23479
  2429
apply auto
nipkow@23479
  2430
done
nipkow@23479
  2431
nipkow@15110
  2432
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110
  2433
apply(induct xs)
nipkow@15110
  2434
 apply simp
nipkow@15110
  2435
apply simp
nipkow@15110
  2436
apply blast
nipkow@15110
  2437
done
nipkow@15110
  2438
paulson@17724
  2439
lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110
  2440
apply(induct xs)
nipkow@15110
  2441
 apply simp
nipkow@15110
  2442
apply simp
nipkow@15110
  2443
apply blast
nipkow@15110
  2444
done
nipkow@15110
  2445
nipkow@23479
  2446
lemma length_remove1:
nipkow@23479
  2447
  "length(remove1 x xs) = (if x : set xs then length xs - 1 else length xs)"
nipkow@23479
  2448
apply (induct xs)
nipkow@23479
  2449
 apply (auto dest!:length_pos_if_in_set)
nipkow@23479
  2450
done
nipkow@23479
  2451
nipkow@18049
  2452
lemma remove1_filter_not[simp]:
nipkow@18049
  2453
  "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
nipkow@18049
  2454
by(induct xs) auto
nipkow@18049
  2455
nipkow@15110
  2456
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110
  2457
apply(insert set_remove1_subset)
nipkow@15110
  2458
apply fast
nipkow@15110
  2459
done
nipkow@15110
  2460
nipkow@15110
  2461
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110
  2462
by (induct xs) simp_all
nipkow@15110
  2463
wenzelm@13114
  2464
nipkow@27693
  2465
subsubsection {* @{text removeAll} *}
nipkow@27693
  2466
nipkow@27693
  2467
lemma removeAll_append[simp]:
nipkow@27693
  2468
  "removeAll x (xs @ ys) = removeAll x xs @ removeAll x ys"
nipkow@27693
  2469
by (induct xs) auto
nipkow@27693
  2470
nipkow@27693
  2471
lemma set_removeAll[simp]: "set(removeAll x xs) = set xs - {x}"
nipkow@27693
  2472
by (induct xs) auto
nipkow@27693
  2473
nipkow@27693
  2474
lemma removeAll_id[simp]: "x \<notin> set xs \<Longrightarrow> removeAll x xs = xs"
nipkow@27693
  2475
by (induct xs) auto
nipkow@27693
  2476
nipkow@27693
  2477
(* Needs count:: 'a \<Rightarrow> a' list \<Rightarrow> nat
nipkow@27693
  2478
lemma length_removeAll:
nipkow@27693
  2479
  "length(removeAll x xs) = length xs - count x xs"
nipkow@27693
  2480
*)
nipkow@27693
  2481
nipkow@27693
  2482
lemma removeAll_filter_not[simp]:
nipkow@27693
  2483
  "\<not> P x \<Longrightarrow> removeAll x (filter P xs) = filter P xs"
nipkow@27693
  2484
by(induct xs) auto
nipkow@27693
  2485
nipkow@27693
  2486
nipkow@27693
  2487
lemma distinct_remove1_removeAll:
nipkow@27693
  2488
  "distinct xs ==> remove1 x xs = removeAll x xs"
nipkow@27693
  2489
by (induct xs) simp_all
nipkow@27693
  2490
nipkow@27693
  2491
lemma map_removeAll_inj_on: "inj_on f (insert x (set xs)) \<Longrightarrow>
nipkow@27693
  2492
  map f (removeAll x xs) = removeAll (f x) (map f xs)"
nipkow@27693
  2493
by (induct xs) (simp_all add:inj_on_def)
nipkow@27693
  2494
nipkow@27693
  2495
lemma map_removeAll_inj: "inj f \<Longrightarrow>
nipkow@27693
  2496
  map f (removeAll x xs) = removeAll (f x) (map f xs)"
nipkow@27693
  2497
by(metis map_removeAll_inj_on subset_inj_on subset_UNIV)
nipkow@27693
  2498
nipkow@27693
  2499
nipkow@15392
  2500
subsubsection {* @{text replicate} *}
wenzelm@13114
  2501
wenzelm@13142
  2502
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  2503
by (induct n) auto
nipkow@13124
  2504
wenzelm@13142
  2505
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  2506
by (induct n) auto
wenzelm@13114
  2507
wenzelm@13114
  2508
lemma replicate_app_Cons_same:
nipkow@13145
  2509
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  2510
by (induct n) auto
wenzelm@13114
  2511
wenzelm@13142
  2512
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208
  2513
apply (induct n, simp)
nipkow@13145
  2514
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  2515
done
wenzelm@13114
  2516
wenzelm@13142
  2517
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  2518
by (induct n) auto
wenzelm@13114
  2519
nipkow@16397
  2520
text{* Courtesy of Matthias Daum: *}
nipkow@16397
  2521
lemma append_replicate_commute:
nipkow@16397
  2522
  "replicate n x @ replicate k x = replicate k x @ replicate n x"
nipkow@16397
  2523
apply (simp add: replicate_add [THEN sym])
nipkow@16397
  2524
apply (simp add: add_commute)
nipkow@16397
  2525
done
nipkow@16397
  2526
wenzelm@13142
  2527
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  2528
by (induct n) auto
wenzelm@13114
  2529
wenzelm@13142
  2530
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  2531
by (induct n) auto
wenzelm@13114
  2532
wenzelm@13142
  2533
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  2534
by (atomize (full), induct n) auto
wenzelm@13114
  2535
nipkow@24526
  2536
lemma nth_replicate[simp]: "i < n ==> (replicate n x)!i = x"
nipkow@24526
  2537
apply (induct n arbitrary: i, simp)
nipkow@13145
  2538
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  2539
done
wenzelm@13114
  2540
nipkow@16397
  2541
text{* Courtesy of Matthias Daum (2 lemmas): *}
nipkow@16397
  2542
lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
nipkow@16397
  2543
apply (case_tac "k \<le> i")
nipkow@16397
  2544
 apply  (simp add: min_def)
nipkow@16397
  2545
apply (drule not_leE)
nipkow@16397
  2546
apply (simp add: min_def)
nipkow@16397
  2547
apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
nipkow@16397
  2548
 apply  simp
nipkow@16397
  2549
apply (simp add: replicate_add [symmetric])
nipkow@16397
  2550
done
nipkow@16397
  2551
nipkow@24526
  2552
lemma drop_replicate[simp]: "drop i (replicate k x) = replicate (k-i) x"
nipkow@24526
  2553
apply (induct k arbitrary: i)
nipkow@16397
  2554
 apply simp
nipkow@16397
  2555
apply clarsimp
nipkow@16397
  2556
apply (case_tac i)
nipkow@16397
  2557
 apply simp
nipkow@16397
  2558
apply clarsimp
nipkow@16397
  2559
done
nipkow@16397
  2560
nipkow@16397
  2561
wenzelm@13142
  2562
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  2563
by (induct n) auto
wenzelm@13114
  2564
wenzelm@13142
  2565
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  2566
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  2567
wenzelm@13142
  2568
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  2569
by auto
wenzelm@13114
  2570
wenzelm@13142
  2571
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  2572
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  2573
haftmann@24796
  2574
lemma replicate_append_same:
haftmann@24796
  2575
  "replicate i x @ [x] = x # replicate i x"
haftmann@24796
  2576
  by (induct i) simp_all
haftmann@24796
  2577
haftmann@24796
  2578
lemma map_replicate_trivial:
haftmann@24796
  2579
  "map (\<lambda>i. x) [0..<i] = replicate i x"
haftmann@24796
  2580
  by (induct i) (simp_all add: replicate_append_same)
haftmann@24796
  2581
wenzelm@13114
  2582
nipkow@15392
  2583
subsubsection{*@{text rotate1} and @{text rotate}*}
nipkow@15302
  2584
nipkow@15302
  2585
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
nipkow@15302
  2586
by(simp add:rotate1_def)
nipkow@15302
  2587
nipkow@15302
  2588
lemma rotate0[simp]: "rotate 0 = id"
nipkow@15302
  2589
by(simp add:rotate_def)
nipkow@15302
  2590
nipkow@15302
  2591
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
nipkow@15302
  2592
by(simp add:rotate_def)
nipkow@15302
  2593
nipkow@15302
  2594
lemma rotate_add:
nipkow@15302
  2595
  "rotate (m+n) = rotate m o rotate n"
nipkow@15302
  2596
by(simp add:rotate_def funpow_add)
nipkow@15302
  2597
nipkow@15302
  2598
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
nipkow@15302
  2599
by(simp add:rotate_add)
nipkow@15302
  2600
nipkow@18049
  2601
lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
nipkow@18049
  2602
by(simp add:rotate_def funpow_swap1)
nipkow@18049
  2603
nipkow@15302
  2604
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
nipkow@15302
  2605
by(cases xs) simp_all
nipkow@15302
  2606
nipkow@15302
  2607
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2608
apply(induct n)
nipkow@15302
  2609
 apply simp
nipkow@15302
  2610
apply (simp add:rotate_def)
nipkow@13145
  2611
done
wenzelm@13114
  2612
nipkow@15302
  2613
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
nipkow@15302
  2614
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2615
nipkow@15302
  2616
lemma rotate_drop_take:
nipkow@15302
  2617
  "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
nipkow@15302
  2618
apply(induct n)
nipkow@15302
  2619
 apply simp
nipkow@15302
  2620
apply(simp add:rotate_def)
nipkow@15302
  2621
apply(cases "xs = []")
nipkow@15302
  2622
 apply (simp)
nipkow@15302
  2623
apply(case_tac "n mod length xs = 0")
nipkow@15302
  2624
 apply(simp add:mod_Suc)
nipkow@15302
  2625
 apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
nipkow@15302
  2626
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
nipkow@15302
  2627
                take_hd_drop linorder_not_le)
nipkow@13145
  2628
done
wenzelm@13114
  2629
nipkow@15302
  2630
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
nipkow@15302
  2631
by(simp add:rotate_drop_take)
nipkow@15302
  2632
nipkow@15302
  2633
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2634
by(simp add:rotate_drop_take)
nipkow@15302
  2635
nipkow@15302
  2636
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
nipkow@15302
  2637
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2638
nipkow@24526
  2639
lemma length_rotate[simp]: "length(rotate n xs) = length xs"
nipkow@24526
  2640
by (induct n arbitrary: xs) (simp_all add:rotate_def)
nipkow@15302
  2641
nipkow@15302
  2642
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
nipkow@15302
  2643
by(simp add:rotate1_def split:list.split) blast
nipkow@15302
  2644
nipkow@15302
  2645
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
nipkow@15302
  2646
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2647
nipkow@15302
  2648
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
nipkow@15302
  2649
by(simp add:rotate_drop_take take_map drop_map)
nipkow@15302
  2650
nipkow@15302
  2651
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
nipkow@15302
  2652
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2653
nipkow@15302
  2654
lemma set_rotate[simp]: "set(rotate n xs) = set xs"
nipkow@15302
  2655
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2656
nipkow@15302
  2657
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
nipkow@15302
  2658
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2659
nipkow@15302
  2660
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
nipkow@15302
  2661
by (induct n) (simp_all add:rotate_def)
wenzelm@13114
  2662
nipkow@15439
  2663
lemma rotate_rev:
nipkow@15439
  2664
  "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
nipkow@15439
  2665
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2666
apply(cases "length xs = 0")
nipkow@15439
  2667
 apply simp
nipkow@15439
  2668
apply(cases "n mod length xs = 0")
nipkow@15439
  2669
 apply simp
nipkow@15439
  2670
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2671
done
nipkow@15439
  2672
nipkow@18423
  2673
lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
nipkow@18423
  2674
apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
nipkow@18423
  2675
apply(subgoal_tac "length xs \<noteq> 0")
nipkow@18423
  2676
 prefer 2 apply simp
nipkow@18423
  2677
using mod_less_divisor[of "length xs" n] by arith
nipkow@18423
  2678
wenzelm@13114
  2679
nipkow@15392
  2680
subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  2681
wenzelm@13142
  2682
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  2683
by (auto simp add: sublist_def)
wenzelm@13114
  2684
wenzelm@13142
  2685
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  2686
by (auto simp add: sublist_def)
wenzelm@13114
  2687
nipkow@15281
  2688
lemma length_sublist:
nipkow@15281
  2689
  "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
nipkow@15281
  2690
by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
nipkow@15281
  2691
nipkow@15281
  2692
lemma sublist_shift_lemma_Suc:
nipkow@24526
  2693
  "map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
nipkow@24526
  2694
   map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
nipkow@24526
  2695
apply(induct xs arbitrary: "is")
nipkow@15281
  2696
 apply simp
nipkow@15281
  2697
apply (case_tac "is")
nipkow@15281
  2698
 apply simp
nipkow@15281
  2699
apply simp
nipkow@15281
  2700
done
nipkow@15281
  2701
wenzelm@13114
  2702
lemma sublist_shift_lemma:
nipkow@23279
  2703
     "map fst [p<-zip xs [i..<i + length xs] . snd p : A] =
nipkow@23279
  2704
      map fst [p<-zip xs [0..<length xs] . snd p + i : A]"
nipkow@13145
  2705
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  2706
wenzelm@13114
  2707
lemma sublist_append:
paulson@15168
  2708
     "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  2709
apply (unfold sublist_def)
paulson@14208
  2710
apply (induct l' rule: rev_induct, simp)
nipkow@13145
  2711
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  2712
apply (simp add: add_commute)
nipkow@13145
  2713
done
wenzelm@13114
  2714
wenzelm@13114
  2715
lemma sublist_Cons:
nipkow@13145
  2716
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  2717
apply (induct l rule: rev_induct)
nipkow@13145
  2718
 apply (simp add: sublist_def)
nipkow@13145
  2719
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  2720
done
wenzelm@13114
  2721
nipkow@24526
  2722
lemma set_sublist: "set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
nipkow@24526
  2723
apply(induct xs arbitrary: I)
nipkow@25162
  2724
apply(auto simp: sublist_Cons nth_Cons split:nat.split dest!: gr0_implies_Suc)
nipkow@15281
  2725
done
nipkow@15281
  2726
nipkow@15281
  2727
lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
nipkow@15281
  2728
by(auto simp add:set_sublist)
nipkow@15281
  2729
nipkow@15281
  2730
lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
nipkow@15281
  2731
by(auto simp add:set_sublist)
nipkow@15281
  2732
nipkow@15281
  2733
lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
nipkow@15281
  2734
by(auto simp add:set_sublist)
nipkow@15281
  2735
wenzelm@13142
  2736
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  2737
by (simp add: sublist_Cons)
wenzelm@13114
  2738
nipkow@15281
  2739
nipkow@24526
  2740
lemma distinct_sublistI[simp]: "distinct xs \<Longrightarrow> distinct(sublist xs I)"
nipkow@24526
  2741
apply(induct xs arbitrary: I)
nipkow@15281
  2742
 apply simp
nipkow@15281
  2743
apply(auto simp add:sublist_Cons)
nipkow@15281
  2744
done
nipkow@15281
  2745
nipkow@15281
  2746
nipkow@15045
  2747
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
paulson@14208
  2748
apply (induct l rule: rev_induct, simp)
nipkow@13145
  2749
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  2750
done
wenzelm@13114
  2751
nipkow@24526
  2752
lemma filter_in_sublist:
nipkow@24526
  2753
 "distinct xs \<Longrightarrow> filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
nipkow@24526
  2754
proof (induct xs arbitrary: s)
nipkow@17501
  2755
  case Nil thus ?case by simp
nipkow@17501