src/HOL/List.thy
author hoelzl
Thu Nov 12 17:21:48 2009 +0100 (2009-11-12)
changeset 33639 603320b93668
parent 33593 ef54e2108b74
child 33640 0d82107dc07a
permissions -rw-r--r--
New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
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(*  Title:      HOL/List.thy
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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 Presburger ATP_Linkup Recdef
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uses ("Tools/list_code.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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  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 nth :: "'a list => nat => 'a" (infixl "!" 100) where
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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 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 characteristic 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_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
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"insort_key f x [] = [x]" |
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"insort_key f x (y#ys) = (if f x \<le> f y then (x#y#ys) else y#(insort_key f x ys))"
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primrec sort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b list \<Rightarrow> 'b list" where
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"sort_key f [] = []" |
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"sort_key f (x#xs) = insort_key f x (sort_key f xs)"
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abbreviation "sort \<equiv> sort_key (\<lambda>x. x)"
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abbreviation "insort \<equiv> insort_key (\<lambda>x. x)"
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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
nipkow@24476
   309
optmized to @{term"map (%x. e) xs"}.
nipkow@23240
   310
nipkow@24349
   311
It is easy to write short list comprehensions which stand for complex
nipkow@24349
   312
expressions. During proofs, they may become unreadable (and
nipkow@24349
   313
mangled). In such cases it can be advisable to introduce separate
nipkow@24349
   314
definitions for the list comprehensions in question.  *}
nipkow@24349
   315
nipkow@23209
   316
(*
nipkow@23240
   317
Proper theorem proving support would be nice. For example, if
nipkow@23192
   318
@{text"set[f x y. x \<leftarrow> xs, y \<leftarrow> ys, P x y]"}
nipkow@23192
   319
produced something like
nipkow@23209
   320
@{term"{z. EX x: set xs. EX y:set ys. P x y \<and> z = f x y}"}.
nipkow@23209
   321
*)
nipkow@23209
   322
nipkow@23240
   323
nonterminals lc_qual lc_quals
nipkow@23192
   324
nipkow@23192
   325
syntax
nipkow@23240
   326
"_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
nipkow@24349
   327
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ <- _")
nipkow@23240
   328
"_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
nipkow@24476
   329
(*"_lc_let" :: "letbinds => lc_qual"  ("let _")*)
nipkow@23240
   330
"_lc_end" :: "lc_quals" ("]")
nipkow@23240
   331
"_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals" (", __")
nipkow@24349
   332
"_lc_abs" :: "'a => 'b list => 'b list"
nipkow@23192
   333
nipkow@24476
   334
(* These are easier than ML code but cannot express the optimized
nipkow@24476
   335
   translation of [e. p<-xs]
nipkow@23192
   336
translations
nipkow@24349
   337
"[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
nipkow@23240
   338
"_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
nipkow@24349
   339
 => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
nipkow@23240
   340
"[e. P]" => "if P then [e] else []"
nipkow@23240
   341
"_listcompr e (_lc_test P) (_lc_quals Q Qs)"
nipkow@23240
   342
 => "if P then (_listcompr e Q Qs) else []"
nipkow@24349
   343
"_listcompr e (_lc_let b) (_lc_quals Q Qs)"
nipkow@24349
   344
 => "_Let b (_listcompr e Q Qs)"
nipkow@24476
   345
*)
nipkow@23240
   346
nipkow@23279
   347
syntax (xsymbols)
nipkow@24349
   348
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
nipkow@23279
   349
syntax (HTML output)
nipkow@24349
   350
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
nipkow@24349
   351
nipkow@24349
   352
parse_translation (advanced) {*
nipkow@24349
   353
let
nipkow@24476
   354
  val NilC = Syntax.const @{const_name Nil};
nipkow@24476
   355
  val ConsC = Syntax.const @{const_name Cons};
nipkow@24476
   356
  val mapC = Syntax.const @{const_name map};
nipkow@24476
   357
  val concatC = Syntax.const @{const_name concat};
nipkow@24476
   358
  val IfC = Syntax.const @{const_name If};
nipkow@24476
   359
  fun singl x = ConsC $ x $ NilC;
nipkow@24476
   360
nipkow@24476
   361
   fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
nipkow@24349
   362
    let
wenzelm@29281
   363
      val x = Free (Name.variant (fold Term.add_free_names [p, e] []) "x", dummyT);
nipkow@24476
   364
      val e = if opti then singl e else e;
nipkow@24476
   365
      val case1 = Syntax.const "_case1" $ p $ e;
nipkow@24349
   366
      val case2 = Syntax.const "_case1" $ Syntax.const Term.dummy_patternN
nipkow@24476
   367
                                        $ NilC;
nipkow@24349
   368
      val cs = Syntax.const "_case2" $ case1 $ case2
haftmann@31784
   369
      val ft = DatatypeCase.case_tr false Datatype.info_of_constr
nipkow@24349
   370
                 ctxt [x, cs]
nipkow@24349
   371
    in lambda x ft end;
nipkow@24349
   372
nipkow@24476
   373
  fun abs_tr ctxt (p as Free(s,T)) e opti =
nipkow@24349
   374
        let val thy = ProofContext.theory_of ctxt;
nipkow@24349
   375
            val s' = Sign.intern_const thy s
nipkow@24476
   376
        in if Sign.declared_const thy s'
nipkow@24476
   377
           then (pat_tr ctxt p e opti, false)
nipkow@24476
   378
           else (lambda p e, true)
nipkow@24349
   379
        end
nipkow@24476
   380
    | abs_tr ctxt p e opti = (pat_tr ctxt p e opti, false);
nipkow@24476
   381
nipkow@24476
   382
  fun lc_tr ctxt [e, Const("_lc_test",_)$b, qs] =
nipkow@24476
   383
        let val res = case qs of Const("_lc_end",_) => singl e
nipkow@24476
   384
                      | Const("_lc_quals",_)$q$qs => lc_tr ctxt [e,q,qs];
nipkow@24476
   385
        in IfC $ b $ res $ NilC end
nipkow@24476
   386
    | lc_tr ctxt [e, Const("_lc_gen",_) $ p $ es, Const("_lc_end",_)] =
nipkow@24476
   387
        (case abs_tr ctxt p e true of
nipkow@24476
   388
           (f,true) => mapC $ f $ es
nipkow@24476
   389
         | (f, false) => concatC $ (mapC $ f $ es))
nipkow@24476
   390
    | lc_tr ctxt [e, Const("_lc_gen",_) $ p $ es, Const("_lc_quals",_)$q$qs] =
nipkow@24476
   391
        let val e' = lc_tr ctxt [e,q,qs];
nipkow@24476
   392
        in concatC $ (mapC $ (fst(abs_tr ctxt p e' false)) $ es) end
nipkow@24476
   393
nipkow@24476
   394
in [("_listcompr", lc_tr)] end
nipkow@24349
   395
*}
nipkow@23279
   396
nipkow@23240
   397
(*
nipkow@23240
   398
term "[(x,y,z). b]"
nipkow@24476
   399
term "[(x,y,z). x\<leftarrow>xs]"
nipkow@24476
   400
term "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]"
nipkow@24476
   401
term "[(x,y,z). x<a, x>b]"
nipkow@24476
   402
term "[(x,y,z). x\<leftarrow>xs, x>b]"
nipkow@24476
   403
term "[(x,y,z). x<a, x\<leftarrow>xs]"
nipkow@24349
   404
term "[(x,y). Cons True x \<leftarrow> xs]"
nipkow@24349
   405
term "[(x,y,z). Cons x [] \<leftarrow> xs]"
nipkow@23240
   406
term "[(x,y,z). x<a, x>b, x=d]"
nipkow@23240
   407
term "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
nipkow@23240
   408
term "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
nipkow@23240
   409
term "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
nipkow@23240
   410
term "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
nipkow@23240
   411
term "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
nipkow@23240
   412
term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
nipkow@23240
   413
term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
nipkow@24349
   414
term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
nipkow@23192
   415
*)
nipkow@23192
   416
haftmann@21061
   417
subsubsection {* @{const Nil} and @{const Cons} *}
haftmann@21061
   418
haftmann@21061
   419
lemma not_Cons_self [simp]:
haftmann@21061
   420
  "xs \<noteq> x # xs"
nipkow@13145
   421
by (induct xs) auto
wenzelm@13114
   422
wenzelm@13142
   423
lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
wenzelm@13114
   424
wenzelm@13142
   425
lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
nipkow@13145
   426
by (induct xs) auto
wenzelm@13114
   427
wenzelm@13142
   428
lemma length_induct:
haftmann@21061
   429
  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
nipkow@17589
   430
by (rule measure_induct [of length]) iprover
wenzelm@13114
   431
wenzelm@13114
   432
haftmann@21061
   433
subsubsection {* @{const length} *}
wenzelm@13114
   434
wenzelm@13142
   435
text {*
haftmann@21061
   436
  Needs to come before @{text "@"} because of theorem @{text
haftmann@21061
   437
  append_eq_append_conv}.
wenzelm@13142
   438
*}
wenzelm@13114
   439
wenzelm@13142
   440
lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
nipkow@13145
   441
by (induct xs) auto
wenzelm@13114
   442
wenzelm@13142
   443
lemma length_map [simp]: "length (map f xs) = length xs"
nipkow@13145
   444
by (induct xs) auto
wenzelm@13114
   445
wenzelm@13142
   446
lemma length_rev [simp]: "length (rev xs) = length xs"
nipkow@13145
   447
by (induct xs) auto
wenzelm@13114
   448
wenzelm@13142
   449
lemma length_tl [simp]: "length (tl xs) = length xs - 1"
nipkow@13145
   450
by (cases xs) auto
wenzelm@13114
   451
wenzelm@13142
   452
lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
nipkow@13145
   453
by (induct xs) auto
wenzelm@13114
   454
wenzelm@13142
   455
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
nipkow@13145
   456
by (induct xs) auto
wenzelm@13114
   457
nipkow@23479
   458
lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
nipkow@23479
   459
by auto
nipkow@23479
   460
wenzelm@13114
   461
lemma length_Suc_conv:
nipkow@13145
   462
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
nipkow@13145
   463
by (induct xs) auto
wenzelm@13142
   464
nipkow@14025
   465
lemma Suc_length_conv:
nipkow@14025
   466
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
paulson@14208
   467
apply (induct xs, simp, simp)
nipkow@14025
   468
apply blast
nipkow@14025
   469
done
nipkow@14025
   470
wenzelm@25221
   471
lemma impossible_Cons: "length xs <= length ys ==> xs = x # ys = False"
wenzelm@25221
   472
  by (induct xs) auto
wenzelm@25221
   473
haftmann@26442
   474
lemma list_induct2 [consumes 1, case_names Nil Cons]:
haftmann@26442
   475
  "length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
haftmann@26442
   476
   (\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
haftmann@26442
   477
   \<Longrightarrow> P xs ys"
haftmann@26442
   478
proof (induct xs arbitrary: ys)
haftmann@26442
   479
  case Nil then show ?case by simp
haftmann@26442
   480
next
haftmann@26442
   481
  case (Cons x xs ys) then show ?case by (cases ys) simp_all
haftmann@26442
   482
qed
haftmann@26442
   483
haftmann@26442
   484
lemma list_induct3 [consumes 2, case_names Nil Cons]:
haftmann@26442
   485
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
haftmann@26442
   486
   (\<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
   487
   \<Longrightarrow> P xs ys zs"
haftmann@26442
   488
proof (induct xs arbitrary: ys zs)
haftmann@26442
   489
  case Nil then show ?case by simp
haftmann@26442
   490
next
haftmann@26442
   491
  case (Cons x xs ys zs) then show ?case by (cases ys, simp_all)
haftmann@26442
   492
    (cases zs, simp_all)
haftmann@26442
   493
qed
wenzelm@13114
   494
krauss@22493
   495
lemma list_induct2': 
krauss@22493
   496
  "\<lbrakk> P [] [];
krauss@22493
   497
  \<And>x xs. P (x#xs) [];
krauss@22493
   498
  \<And>y ys. P [] (y#ys);
krauss@22493
   499
   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
krauss@22493
   500
 \<Longrightarrow> P xs ys"
krauss@22493
   501
by (induct xs arbitrary: ys) (case_tac x, auto)+
krauss@22493
   502
nipkow@22143
   503
lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
nipkow@24349
   504
by (rule Eq_FalseI) auto
wenzelm@24037
   505
wenzelm@24037
   506
simproc_setup list_neq ("(xs::'a list) = ys") = {*
nipkow@22143
   507
(*
nipkow@22143
   508
Reduces xs=ys to False if xs and ys cannot be of the same length.
nipkow@22143
   509
This is the case if the atomic sublists of one are a submultiset
nipkow@22143
   510
of those of the other list and there are fewer Cons's in one than the other.
nipkow@22143
   511
*)
wenzelm@24037
   512
wenzelm@24037
   513
let
nipkow@22143
   514
huffman@29856
   515
fun len (Const(@{const_name Nil},_)) acc = acc
huffman@29856
   516
  | len (Const(@{const_name Cons},_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
huffman@29856
   517
  | len (Const(@{const_name append},_) $ xs $ ys) acc = len xs (len ys acc)
huffman@29856
   518
  | len (Const(@{const_name rev},_) $ xs) acc = len xs acc
huffman@29856
   519
  | len (Const(@{const_name map},_) $ _ $ xs) acc = len xs acc
nipkow@22143
   520
  | len t (ts,n) = (t::ts,n);
nipkow@22143
   521
wenzelm@24037
   522
fun list_neq _ ss ct =
nipkow@22143
   523
  let
wenzelm@24037
   524
    val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
nipkow@22143
   525
    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
nipkow@22143
   526
    fun prove_neq() =
nipkow@22143
   527
      let
nipkow@22143
   528
        val Type(_,listT::_) = eqT;
haftmann@22994
   529
        val size = HOLogic.size_const listT;
nipkow@22143
   530
        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
nipkow@22143
   531
        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
nipkow@22143
   532
        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
haftmann@22633
   533
          (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
haftmann@22633
   534
      in SOME (thm RS @{thm neq_if_length_neq}) end
nipkow@22143
   535
  in
wenzelm@23214
   536
    if m < n andalso submultiset (op aconv) (ls,rs) orelse
wenzelm@23214
   537
       n < m andalso submultiset (op aconv) (rs,ls)
nipkow@22143
   538
    then prove_neq() else NONE
nipkow@22143
   539
  end;
wenzelm@24037
   540
in list_neq end;
nipkow@22143
   541
*}
nipkow@22143
   542
nipkow@22143
   543
nipkow@15392
   544
subsubsection {* @{text "@"} -- append *}
wenzelm@13114
   545
wenzelm@13142
   546
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
nipkow@13145
   547
by (induct xs) auto
wenzelm@13114
   548
wenzelm@13142
   549
lemma append_Nil2 [simp]: "xs @ [] = xs"
nipkow@13145
   550
by (induct xs) auto
nipkow@3507
   551
wenzelm@13142
   552
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
nipkow@13145
   553
by (induct xs) auto
wenzelm@13114
   554
wenzelm@13142
   555
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
nipkow@13145
   556
by (induct xs) auto
wenzelm@13114
   557
wenzelm@13142
   558
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
nipkow@13145
   559
by (induct xs) auto
wenzelm@13114
   560
wenzelm@13142
   561
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
nipkow@13145
   562
by (induct xs) auto
wenzelm@13114
   563
wenzelm@25221
   564
lemma append_eq_append_conv [simp, noatp]:
nipkow@24526
   565
 "length xs = length ys \<or> length us = length vs
berghofe@13883
   566
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
nipkow@24526
   567
apply (induct xs arbitrary: ys)
paulson@14208
   568
 apply (case_tac ys, simp, force)
paulson@14208
   569
apply (case_tac ys, force, simp)
nipkow@13145
   570
done
wenzelm@13142
   571
nipkow@24526
   572
lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
nipkow@24526
   573
  (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
nipkow@24526
   574
apply (induct xs arbitrary: ys zs ts)
nipkow@14495
   575
 apply fastsimp
nipkow@14495
   576
apply(case_tac zs)
nipkow@14495
   577
 apply simp
nipkow@14495
   578
apply fastsimp
nipkow@14495
   579
done
nipkow@14495
   580
wenzelm@13142
   581
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145
   582
by simp
wenzelm@13142
   583
wenzelm@13142
   584
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145
   585
by simp
wenzelm@13114
   586
wenzelm@13142
   587
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   588
by simp
wenzelm@13114
   589
wenzelm@13142
   590
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   591
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   592
wenzelm@13142
   593
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   594
using append_same_eq [of "[]"] by auto
wenzelm@13114
   595
paulson@24286
   596
lemma hd_Cons_tl [simp,noatp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   597
by (induct xs) auto
wenzelm@13114
   598
wenzelm@13142
   599
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   600
by (induct xs) auto
wenzelm@13114
   601
wenzelm@13142
   602
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   603
by (simp add: hd_append split: list.split)
wenzelm@13114
   604
wenzelm@13142
   605
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   606
by (simp split: list.split)
wenzelm@13114
   607
wenzelm@13142
   608
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   609
by (simp add: tl_append split: list.split)
wenzelm@13114
   610
wenzelm@13114
   611
nipkow@14300
   612
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   613
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   614
by(cases ys) auto
nipkow@14300
   615
nipkow@15281
   616
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   617
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   618
by(cases ys) auto
nipkow@15281
   619
nipkow@14300
   620
wenzelm@13142
   621
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   622
wenzelm@13114
   623
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   624
by simp
wenzelm@13114
   625
wenzelm@13142
   626
lemma Cons_eq_appendI:
nipkow@13145
   627
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   628
by (drule sym) simp
wenzelm@13114
   629
wenzelm@13142
   630
lemma append_eq_appendI:
nipkow@13145
   631
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   632
by (drule sym) simp
wenzelm@13114
   633
wenzelm@13114
   634
wenzelm@13142
   635
text {*
nipkow@13145
   636
Simplification procedure for all list equalities.
nipkow@13145
   637
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   638
- both lists end in a singleton list,
nipkow@13145
   639
- or both lists end in the same list.
wenzelm@13142
   640
*}
wenzelm@13142
   641
wenzelm@26480
   642
ML {*
nipkow@3507
   643
local
nipkow@3507
   644
huffman@29856
   645
fun last (cons as Const(@{const_name Cons},_) $ _ $ xs) =
huffman@29856
   646
  (case xs of Const(@{const_name Nil},_) => cons | _ => last xs)
huffman@29856
   647
  | last (Const(@{const_name append},_) $ _ $ ys) = last ys
wenzelm@13462
   648
  | last t = t;
wenzelm@13114
   649
huffman@29856
   650
fun list1 (Const(@{const_name Cons},_) $ _ $ Const(@{const_name Nil},_)) = true
wenzelm@13462
   651
  | list1 _ = false;
wenzelm@13114
   652
huffman@29856
   653
fun butlast ((cons as Const(@{const_name Cons},_) $ x) $ xs) =
huffman@29856
   654
  (case xs of Const(@{const_name Nil},_) => xs | _ => cons $ butlast xs)
huffman@29856
   655
  | butlast ((app as Const(@{const_name append},_) $ xs) $ ys) = app $ butlast ys
huffman@29856
   656
  | butlast xs = Const(@{const_name Nil},fastype_of xs);
wenzelm@13114
   657
haftmann@22633
   658
val rearr_ss = HOL_basic_ss addsimps [@{thm append_assoc},
haftmann@22633
   659
  @{thm append_Nil}, @{thm append_Cons}];
wenzelm@16973
   660
wenzelm@20044
   661
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   662
  let
wenzelm@13462
   663
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   664
    fun rearr conv =
wenzelm@13462
   665
      let
wenzelm@13462
   666
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   667
        val Type(_,listT::_) = eqT
wenzelm@13462
   668
        val appT = [listT,listT] ---> listT
huffman@29856
   669
        val app = Const(@{const_name append},appT)
wenzelm@13462
   670
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   671
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@20044
   672
        val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
wenzelm@17877
   673
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
skalberg@15531
   674
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   675
wenzelm@13462
   676
  in
haftmann@22633
   677
    if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
haftmann@22633
   678
    else if lastl aconv lastr then rearr @{thm append_same_eq}
skalberg@15531
   679
    else NONE
wenzelm@13462
   680
  end;
wenzelm@13462
   681
wenzelm@13114
   682
in
wenzelm@13462
   683
wenzelm@13462
   684
val list_eq_simproc =
wenzelm@32010
   685
  Simplifier.simproc @{theory} "list_eq" ["(xs::'a list) = ys"] (K list_eq);
wenzelm@13462
   686
wenzelm@13114
   687
end;
wenzelm@13114
   688
wenzelm@13114
   689
Addsimprocs [list_eq_simproc];
wenzelm@13114
   690
*}
wenzelm@13114
   691
wenzelm@13114
   692
nipkow@15392
   693
subsubsection {* @{text map} *}
wenzelm@13114
   694
wenzelm@13142
   695
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   696
by (induct xs) simp_all
wenzelm@13114
   697
wenzelm@13142
   698
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   699
by (rule ext, induct_tac xs) auto
wenzelm@13114
   700
wenzelm@13142
   701
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   702
by (induct xs) auto
wenzelm@13114
   703
hoelzl@33639
   704
lemma map_map [simp]: "map f (map g xs) = map (f \<circ> g) xs"
hoelzl@33639
   705
by (induct xs) auto
hoelzl@33639
   706
hoelzl@33639
   707
lemma map_compose: "map (f \<circ> g) xs = map f (map g xs)"
hoelzl@33639
   708
by simp
wenzelm@13114
   709
wenzelm@13142
   710
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   711
by (induct xs) auto
wenzelm@13114
   712
nipkow@13737
   713
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   714
by (induct xs) auto
nipkow@13737
   715
krauss@19770
   716
lemma map_cong [fundef_cong, recdef_cong]:
nipkow@13145
   717
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   718
-- {* a congruence rule for @{text map} *}
nipkow@13737
   719
by simp
wenzelm@13114
   720
wenzelm@13142
   721
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   722
by (cases xs) auto
wenzelm@13114
   723
wenzelm@13142
   724
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   725
by (cases xs) auto
wenzelm@13114
   726
paulson@18447
   727
lemma map_eq_Cons_conv:
nipkow@14025
   728
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   729
by (cases xs) auto
wenzelm@13114
   730
paulson@18447
   731
lemma Cons_eq_map_conv:
nipkow@14025
   732
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   733
by (cases ys) auto
nipkow@14025
   734
paulson@18447
   735
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
paulson@18447
   736
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
paulson@18447
   737
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
paulson@18447
   738
nipkow@14111
   739
lemma ex_map_conv:
nipkow@14111
   740
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
paulson@18447
   741
by(induct ys, auto simp add: Cons_eq_map_conv)
nipkow@14111
   742
nipkow@15110
   743
lemma map_eq_imp_length_eq:
haftmann@26734
   744
  assumes "map f xs = map f ys"
haftmann@26734
   745
  shows "length xs = length ys"
haftmann@26734
   746
using assms proof (induct ys arbitrary: xs)
haftmann@26734
   747
  case Nil then show ?case by simp
haftmann@26734
   748
next
haftmann@26734
   749
  case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto
haftmann@26734
   750
  from Cons xs have "map f zs = map f ys" by simp
haftmann@26734
   751
  moreover with Cons have "length zs = length ys" by blast
haftmann@26734
   752
  with xs show ?case by simp
haftmann@26734
   753
qed
haftmann@26734
   754
  
nipkow@15110
   755
lemma map_inj_on:
nipkow@15110
   756
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   757
  ==> xs = ys"
nipkow@15110
   758
apply(frule map_eq_imp_length_eq)
nipkow@15110
   759
apply(rotate_tac -1)
nipkow@15110
   760
apply(induct rule:list_induct2)
nipkow@15110
   761
 apply simp
nipkow@15110
   762
apply(simp)
nipkow@15110
   763
apply (blast intro:sym)
nipkow@15110
   764
done
nipkow@15110
   765
nipkow@15110
   766
lemma inj_on_map_eq_map:
nipkow@15110
   767
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   768
by(blast dest:map_inj_on)
nipkow@15110
   769
wenzelm@13114
   770
lemma map_injective:
nipkow@24526
   771
 "map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@24526
   772
by (induct ys arbitrary: xs) (auto dest!:injD)
wenzelm@13114
   773
nipkow@14339
   774
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   775
by(blast dest:map_injective)
nipkow@14339
   776
wenzelm@13114
   777
lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@17589
   778
by (iprover dest: map_injective injD intro: inj_onI)
wenzelm@13114
   779
wenzelm@13114
   780
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   781
apply (unfold inj_on_def, clarify)
nipkow@13145
   782
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   783
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   784
apply blast
nipkow@13145
   785
done
wenzelm@13114
   786
nipkow@14339
   787
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   788
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   789
nipkow@15303
   790
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
   791
apply(rule inj_onI)
nipkow@15303
   792
apply(erule map_inj_on)
nipkow@15303
   793
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
   794
done
nipkow@15303
   795
kleing@14343
   796
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   797
by (induct xs, auto)
wenzelm@13114
   798
nipkow@14402
   799
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   800
by (induct xs) auto
nipkow@14402
   801
nipkow@15110
   802
lemma map_fst_zip[simp]:
nipkow@15110
   803
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   804
by (induct rule:list_induct2, simp_all)
nipkow@15110
   805
nipkow@15110
   806
lemma map_snd_zip[simp]:
nipkow@15110
   807
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   808
by (induct rule:list_induct2, simp_all)
nipkow@15110
   809
nipkow@15110
   810
nipkow@15392
   811
subsubsection {* @{text rev} *}
wenzelm@13114
   812
wenzelm@13142
   813
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   814
by (induct xs) auto
wenzelm@13114
   815
wenzelm@13142
   816
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   817
by (induct xs) auto
wenzelm@13114
   818
kleing@15870
   819
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870
   820
by auto
kleing@15870
   821
wenzelm@13142
   822
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   823
by (induct xs) auto
wenzelm@13114
   824
wenzelm@13142
   825
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   826
by (induct xs) auto
wenzelm@13114
   827
kleing@15870
   828
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870
   829
by (cases xs) auto
kleing@15870
   830
kleing@15870
   831
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870
   832
by (cases xs) auto
kleing@15870
   833
haftmann@21061
   834
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
haftmann@21061
   835
apply (induct xs arbitrary: ys, force)
paulson@14208
   836
apply (case_tac ys, simp, force)
nipkow@13145
   837
done
wenzelm@13114
   838
nipkow@15439
   839
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
   840
by(simp add:inj_on_def)
nipkow@15439
   841
wenzelm@13366
   842
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   843
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489
   844
apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145
   845
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   846
done
wenzelm@13114
   847
wenzelm@13366
   848
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   849
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   850
by (induct xs rule: rev_induct) auto
wenzelm@13114
   851
wenzelm@13366
   852
lemmas rev_cases = rev_exhaust
wenzelm@13366
   853
nipkow@18423
   854
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
nipkow@18423
   855
by(rule rev_cases[of xs]) auto
nipkow@18423
   856
wenzelm@13114
   857
nipkow@15392
   858
subsubsection {* @{text set} *}
wenzelm@13114
   859
wenzelm@13142
   860
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   861
by (induct xs) auto
wenzelm@13114
   862
wenzelm@13142
   863
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   864
by (induct xs) auto
wenzelm@13114
   865
nipkow@17830
   866
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
nipkow@17830
   867
by(cases xs) auto
oheimb@14099
   868
wenzelm@13142
   869
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   870
by auto
wenzelm@13114
   871
oheimb@14099
   872
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   873
by auto
oheimb@14099
   874
wenzelm@13142
   875
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   876
by (induct xs) auto
wenzelm@13114
   877
nipkow@15245
   878
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   879
by(induct xs) auto
nipkow@15245
   880
wenzelm@13142
   881
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   882
by (induct xs) auto
wenzelm@13114
   883
wenzelm@13142
   884
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   885
by (induct xs) auto
wenzelm@13114
   886
wenzelm@13142
   887
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   888
by (induct xs) auto
wenzelm@13114
   889
nipkow@32417
   890
lemma set_upt [simp]: "set[i..<j] = {i..<j}"
nipkow@32417
   891
by (induct j) (simp_all add: atLeastLessThanSuc)
wenzelm@13114
   892
wenzelm@13142
   893
wenzelm@25221
   894
lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs"
nipkow@18049
   895
proof (induct xs)
nipkow@26073
   896
  case Nil thus ?case by simp
nipkow@26073
   897
next
nipkow@26073
   898
  case Cons thus ?case by (auto intro: Cons_eq_appendI)
nipkow@26073
   899
qed
nipkow@26073
   900
haftmann@26734
   901
lemma in_set_conv_decomp: "x \<in> set xs \<longleftrightarrow> (\<exists>ys zs. xs = ys @ x # zs)"
haftmann@26734
   902
  by (auto elim: split_list)
nipkow@26073
   903
nipkow@26073
   904
lemma split_list_first: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@26073
   905
proof (induct xs)
nipkow@26073
   906
  case Nil thus ?case by simp
nipkow@18049
   907
next
nipkow@18049
   908
  case (Cons a xs)
nipkow@18049
   909
  show ?case
nipkow@18049
   910
  proof cases
wenzelm@25221
   911
    assume "x = a" thus ?case using Cons by fastsimp
nipkow@18049
   912
  next
nipkow@26073
   913
    assume "x \<noteq> a" thus ?case using Cons by(fastsimp intro!: Cons_eq_appendI)
nipkow@26073
   914
  qed
nipkow@26073
   915
qed
nipkow@26073
   916
nipkow@26073
   917
lemma in_set_conv_decomp_first:
nipkow@26073
   918
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
haftmann@26734
   919
  by (auto dest!: split_list_first)
nipkow@26073
   920
nipkow@26073
   921
lemma split_list_last: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
nipkow@26073
   922
proof (induct xs rule:rev_induct)
nipkow@26073
   923
  case Nil thus ?case by simp
nipkow@26073
   924
next
nipkow@26073
   925
  case (snoc a xs)
nipkow@26073
   926
  show ?case
nipkow@26073
   927
  proof cases
nipkow@26073
   928
    assume "x = a" thus ?case using snoc by simp (metis ex_in_conv set_empty2)
nipkow@26073
   929
  next
nipkow@26073
   930
    assume "x \<noteq> a" thus ?case using snoc by fastsimp
nipkow@18049
   931
  qed
nipkow@18049
   932
qed
nipkow@18049
   933
nipkow@26073
   934
lemma in_set_conv_decomp_last:
nipkow@26073
   935
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
haftmann@26734
   936
  by (auto dest!: split_list_last)
nipkow@26073
   937
nipkow@26073
   938
lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs & P x"
nipkow@26073
   939
proof (induct xs)
nipkow@26073
   940
  case Nil thus ?case by simp
nipkow@26073
   941
next
nipkow@26073
   942
  case Cons thus ?case
nipkow@26073
   943
    by(simp add:Bex_def)(metis append_Cons append.simps(1))
nipkow@26073
   944
qed
nipkow@26073
   945
nipkow@26073
   946
lemma split_list_propE:
haftmann@26734
   947
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
   948
  obtains ys x zs where "xs = ys @ x # zs" and "P x"
haftmann@26734
   949
using split_list_prop [OF assms] by blast
nipkow@26073
   950
nipkow@26073
   951
lemma split_list_first_prop:
nipkow@26073
   952
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
nipkow@26073
   953
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)"
haftmann@26734
   954
proof (induct xs)
nipkow@26073
   955
  case Nil thus ?case by simp
nipkow@26073
   956
next
nipkow@26073
   957
  case (Cons x xs)
nipkow@26073
   958
  show ?case
nipkow@26073
   959
  proof cases
nipkow@26073
   960
    assume "P x"
haftmann@26734
   961
    thus ?thesis by simp
haftmann@26734
   962
      (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
nipkow@26073
   963
  next
nipkow@26073
   964
    assume "\<not> P x"
nipkow@26073
   965
    hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
nipkow@26073
   966
    thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
nipkow@26073
   967
  qed
nipkow@26073
   968
qed
nipkow@26073
   969
nipkow@26073
   970
lemma split_list_first_propE:
haftmann@26734
   971
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
   972
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y"
haftmann@26734
   973
using split_list_first_prop [OF assms] by blast
nipkow@26073
   974
nipkow@26073
   975
lemma split_list_first_prop_iff:
nipkow@26073
   976
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
nipkow@26073
   977
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))"
haftmann@26734
   978
by (rule, erule split_list_first_prop) auto
nipkow@26073
   979
nipkow@26073
   980
lemma split_list_last_prop:
nipkow@26073
   981
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
nipkow@26073
   982
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)"
nipkow@26073
   983
proof(induct xs rule:rev_induct)
nipkow@26073
   984
  case Nil thus ?case by simp
nipkow@26073
   985
next
nipkow@26073
   986
  case (snoc x xs)
nipkow@26073
   987
  show ?case
nipkow@26073
   988
  proof cases
nipkow@26073
   989
    assume "P x" thus ?thesis by (metis emptyE set_empty)
nipkow@26073
   990
  next
nipkow@26073
   991
    assume "\<not> P x"
nipkow@26073
   992
    hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
nipkow@26073
   993
    thus ?thesis using `\<not> P x` snoc(1) by fastsimp
nipkow@26073
   994
  qed
nipkow@26073
   995
qed
nipkow@26073
   996
nipkow@26073
   997
lemma split_list_last_propE:
haftmann@26734
   998
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
   999
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z"
haftmann@26734
  1000
using split_list_last_prop [OF assms] by blast
nipkow@26073
  1001
nipkow@26073
  1002
lemma split_list_last_prop_iff:
nipkow@26073
  1003
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
nipkow@26073
  1004
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
haftmann@26734
  1005
by (metis split_list_last_prop [where P=P] in_set_conv_decomp)
nipkow@26073
  1006
nipkow@26073
  1007
lemma finite_list: "finite A ==> EX xs. set xs = A"
haftmann@26734
  1008
  by (erule finite_induct)
haftmann@26734
  1009
    (auto simp add: set.simps(2) [symmetric] simp del: set.simps(2))
paulson@13508
  1010
kleing@14388
  1011
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
  1012
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
  1013
haftmann@26442
  1014
lemma set_minus_filter_out:
haftmann@26442
  1015
  "set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)"
haftmann@26442
  1016
  by (induct xs) auto
paulson@15168
  1017
nipkow@15392
  1018
subsubsection {* @{text filter} *}
wenzelm@13114
  1019
wenzelm@13142
  1020
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
  1021
by (induct xs) auto
wenzelm@13114
  1022
nipkow@15305
  1023
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
  1024
by (induct xs) simp_all
nipkow@15305
  1025
wenzelm@13142
  1026
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
  1027
by (induct xs) auto
wenzelm@13114
  1028
nipkow@16998
  1029
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998
  1030
by (induct xs) (auto simp add: le_SucI)
nipkow@16998
  1031
nipkow@18423
  1032
lemma sum_length_filter_compl:
nipkow@18423
  1033
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
nipkow@18423
  1034
by(induct xs) simp_all
nipkow@18423
  1035
wenzelm@13142
  1036
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
  1037
by (induct xs) auto
wenzelm@13114
  1038
wenzelm@13142
  1039
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
  1040
by (induct xs) auto
wenzelm@13114
  1041
nipkow@16998
  1042
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@24349
  1043
by (induct xs) simp_all
nipkow@16998
  1044
nipkow@16998
  1045
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998
  1046
apply (induct xs)
nipkow@16998
  1047
 apply auto
nipkow@16998
  1048
apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998
  1049
apply simp
nipkow@16998
  1050
done
wenzelm@13114
  1051
nipkow@16965
  1052
lemma filter_map:
nipkow@16965
  1053
  "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965
  1054
by (induct xs) simp_all
nipkow@16965
  1055
nipkow@16965
  1056
lemma length_filter_map[simp]:
nipkow@16965
  1057
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965
  1058
by (simp add:filter_map)
nipkow@16965
  1059
wenzelm@13142
  1060
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
  1061
by auto
wenzelm@13114
  1062
nipkow@15246
  1063
lemma length_filter_less:
nipkow@15246
  1064
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
  1065
proof (induct xs)
nipkow@15246
  1066
  case Nil thus ?case by simp
nipkow@15246
  1067
next
nipkow@15246
  1068
  case (Cons x xs) thus ?case
nipkow@15246
  1069
    apply (auto split:split_if_asm)
nipkow@15246
  1070
    using length_filter_le[of P xs] apply arith
nipkow@15246
  1071
  done
nipkow@15246
  1072
qed
wenzelm@13114
  1073
nipkow@15281
  1074
lemma length_filter_conv_card:
nipkow@15281
  1075
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
  1076
proof (induct xs)
nipkow@15281
  1077
  case Nil thus ?case by simp
nipkow@15281
  1078
next
nipkow@15281
  1079
  case (Cons x xs)
nipkow@15281
  1080
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
  1081
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
  1082
  show ?case (is "?l = card ?S'")
nipkow@15281
  1083
  proof (cases)
nipkow@15281
  1084
    assume "p x"
nipkow@15281
  1085
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@25162
  1086
      by(auto simp: image_def split:nat.split dest:gr0_implies_Suc)
nipkow@15281
  1087
    have "length (filter p (x # xs)) = Suc(card ?S)"
wenzelm@23388
  1088
      using Cons `p x` by simp
nipkow@15281
  1089
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
nipkow@15281
  1090
      by (simp add: card_image inj_Suc)
nipkow@15281
  1091
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
  1092
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
  1093
    finally show ?thesis .
nipkow@15281
  1094
  next
nipkow@15281
  1095
    assume "\<not> p x"
nipkow@15281
  1096
    hence eq: "?S' = Suc ` ?S"
nipkow@25162
  1097
      by(auto simp add: image_def split:nat.split elim:lessE)
nipkow@15281
  1098
    have "length (filter p (x # xs)) = card ?S"
wenzelm@23388
  1099
      using Cons `\<not> p x` by simp
nipkow@15281
  1100
    also have "\<dots> = card(Suc ` ?S)" using fin
nipkow@15281
  1101
      by (simp add: card_image inj_Suc)
nipkow@15281
  1102
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
  1103
      by (simp add:card_insert_if)
nipkow@15281
  1104
    finally show ?thesis .
nipkow@15281
  1105
  qed
nipkow@15281
  1106
qed
nipkow@15281
  1107
nipkow@17629
  1108
lemma Cons_eq_filterD:
nipkow@17629
  1109
 "x#xs = filter P ys \<Longrightarrow>
nipkow@17629
  1110
  \<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
  1111
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
nipkow@17629
  1112
proof(induct ys)
nipkow@17629
  1113
  case Nil thus ?case by simp
nipkow@17629
  1114
next
nipkow@17629
  1115
  case (Cons y ys)
nipkow@17629
  1116
  show ?case (is "\<exists>x. ?Q x")
nipkow@17629
  1117
  proof cases
nipkow@17629
  1118
    assume Py: "P y"
nipkow@17629
  1119
    show ?thesis
nipkow@17629
  1120
    proof cases
wenzelm@25221
  1121
      assume "x = y"
wenzelm@25221
  1122
      with Py Cons.prems have "?Q []" by simp
wenzelm@25221
  1123
      then show ?thesis ..
nipkow@17629
  1124
    next
wenzelm@25221
  1125
      assume "x \<noteq> y"
wenzelm@25221
  1126
      with Py Cons.prems show ?thesis by simp
nipkow@17629
  1127
    qed
nipkow@17629
  1128
  next
wenzelm@25221
  1129
    assume "\<not> P y"
wenzelm@25221
  1130
    with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastsimp
wenzelm@25221
  1131
    then have "?Q (y#us)" by simp
wenzelm@25221
  1132
    then show ?thesis ..
nipkow@17629
  1133
  qed
nipkow@17629
  1134
qed
nipkow@17629
  1135
nipkow@17629
  1136
lemma filter_eq_ConsD:
nipkow@17629
  1137
 "filter P ys = x#xs \<Longrightarrow>
nipkow@17629
  1138
  \<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
  1139
by(rule Cons_eq_filterD) simp
nipkow@17629
  1140
nipkow@17629
  1141
lemma filter_eq_Cons_iff:
nipkow@17629
  1142
 "(filter P ys = x#xs) =
nipkow@17629
  1143
  (\<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
  1144
by(auto dest:filter_eq_ConsD)
nipkow@17629
  1145
nipkow@17629
  1146
lemma Cons_eq_filter_iff:
nipkow@17629
  1147
 "(x#xs = filter P ys) =
nipkow@17629
  1148
  (\<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
  1149
by(auto dest:Cons_eq_filterD)
nipkow@17629
  1150
krauss@19770
  1151
lemma filter_cong[fundef_cong, recdef_cong]:
nipkow@17501
  1152
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
nipkow@17501
  1153
apply simp
nipkow@17501
  1154
apply(erule thin_rl)
nipkow@17501
  1155
by (induct ys) simp_all
nipkow@17501
  1156
nipkow@15281
  1157
haftmann@26442
  1158
subsubsection {* List partitioning *}
haftmann@26442
  1159
haftmann@26442
  1160
primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where
haftmann@26442
  1161
  "partition P [] = ([], [])"
haftmann@26442
  1162
  | "partition P (x # xs) = 
haftmann@26442
  1163
      (let (yes, no) = partition P xs
haftmann@26442
  1164
      in if P x then (x # yes, no) else (yes, x # no))"
haftmann@26442
  1165
haftmann@26442
  1166
lemma partition_filter1:
haftmann@26442
  1167
    "fst (partition P xs) = filter P xs"
haftmann@26442
  1168
by (induct xs) (auto simp add: Let_def split_def)
haftmann@26442
  1169
haftmann@26442
  1170
lemma partition_filter2:
haftmann@26442
  1171
    "snd (partition P xs) = filter (Not o P) xs"
haftmann@26442
  1172
by (induct xs) (auto simp add: Let_def split_def)
haftmann@26442
  1173
haftmann@26442
  1174
lemma partition_P:
haftmann@26442
  1175
  assumes "partition P xs = (yes, no)"
haftmann@26442
  1176
  shows "(\<forall>p \<in> set yes.  P p) \<and> (\<forall>p  \<in> set no. \<not> P p)"
haftmann@26442
  1177
proof -
haftmann@26442
  1178
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
haftmann@26442
  1179
    by simp_all
haftmann@26442
  1180
  then show ?thesis by (simp_all add: partition_filter1 partition_filter2)
haftmann@26442
  1181
qed
haftmann@26442
  1182
haftmann@26442
  1183
lemma partition_set:
haftmann@26442
  1184
  assumes "partition P xs = (yes, no)"
haftmann@26442
  1185
  shows "set yes \<union> set no = set xs"
haftmann@26442
  1186
proof -
haftmann@26442
  1187
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
haftmann@26442
  1188
    by simp_all
haftmann@26442
  1189
  then show ?thesis by (auto simp add: partition_filter1 partition_filter2) 
haftmann@26442
  1190
qed
haftmann@26442
  1191
hoelzl@33639
  1192
lemma partition_filter_conv[simp]:
hoelzl@33639
  1193
  "partition f xs = (filter f xs,filter (Not o f) xs)"
hoelzl@33639
  1194
unfolding partition_filter2[symmetric]
hoelzl@33639
  1195
unfolding partition_filter1[symmetric] by simp
hoelzl@33639
  1196
hoelzl@33639
  1197
declare partition.simps[simp del]
haftmann@26442
  1198
nipkow@15392
  1199
subsubsection {* @{text concat} *}
wenzelm@13114
  1200
wenzelm@13142
  1201
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
  1202
by (induct xs) auto
wenzelm@13114
  1203
paulson@18447
  1204
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
  1205
by (induct xss) auto
wenzelm@13114
  1206
paulson@18447
  1207
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
  1208
by (induct xss) auto
wenzelm@13114
  1209
nipkow@24308
  1210
lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
nipkow@13145
  1211
by (induct xs) auto
wenzelm@13114
  1212
nipkow@24476
  1213
lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs"
nipkow@24349
  1214
by (induct xs) auto
nipkow@24349
  1215
wenzelm@13142
  1216
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
  1217
by (induct xs) auto
wenzelm@13114
  1218
wenzelm@13142
  1219
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
  1220
by (induct xs) auto
wenzelm@13114
  1221
wenzelm@13142
  1222
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
  1223
by (induct xs) auto
wenzelm@13114
  1224
wenzelm@13114
  1225
nipkow@15392
  1226
subsubsection {* @{text nth} *}
wenzelm@13114
  1227
haftmann@29827
  1228
lemma nth_Cons_0 [simp, code]: "(x # xs)!0 = x"
nipkow@13145
  1229
by auto
wenzelm@13114
  1230
haftmann@29827
  1231
lemma nth_Cons_Suc [simp, code]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
  1232
by auto
wenzelm@13114
  1233
wenzelm@13142
  1234
declare nth.simps [simp del]
wenzelm@13114
  1235
wenzelm@13114
  1236
lemma nth_append:
nipkow@24526
  1237
  "(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
nipkow@24526
  1238
apply (induct xs arbitrary: n, simp)
paulson@14208
  1239
apply (case_tac n, auto)
nipkow@13145
  1240
done
wenzelm@13114
  1241
nipkow@14402
  1242
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
wenzelm@25221
  1243
by (induct xs) auto
nipkow@14402
  1244
nipkow@14402
  1245
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
wenzelm@25221
  1246
by (induct xs) auto
nipkow@14402
  1247
nipkow@24526
  1248
lemma nth_map [simp]: "n < length xs ==> (map f xs)!n = f(xs!n)"
nipkow@24526
  1249
apply (induct xs arbitrary: n, simp)
paulson@14208
  1250
apply (case_tac n, auto)
nipkow@13145
  1251
done
wenzelm@13114
  1252
nipkow@18423
  1253
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
nipkow@18423
  1254
by(cases xs) simp_all
nipkow@18423
  1255
nipkow@18049
  1256
nipkow@18049
  1257
lemma list_eq_iff_nth_eq:
nipkow@24526
  1258
 "(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
nipkow@24526
  1259
apply(induct xs arbitrary: ys)
paulson@24632
  1260
 apply force
nipkow@18049
  1261
apply(case_tac ys)
nipkow@18049
  1262
 apply simp
nipkow@18049
  1263
apply(simp add:nth_Cons split:nat.split)apply blast
nipkow@18049
  1264
done
nipkow@18049
  1265
wenzelm@13142
  1266
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
  1267
apply (induct xs, simp, simp)
nipkow@13145
  1268
apply safe
paulson@24632
  1269
apply (metis nat_case_0 nth.simps zero_less_Suc)
paulson@24632
  1270
apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
paulson@14208
  1271
apply (case_tac i, simp)
paulson@24632
  1272
apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff)
nipkow@13145
  1273
done
wenzelm@13114
  1274
nipkow@17501
  1275
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
nipkow@17501
  1276
by(auto simp:set_conv_nth)
nipkow@17501
  1277
nipkow@13145
  1278
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
  1279
by (auto simp add: set_conv_nth)
wenzelm@13114
  1280
wenzelm@13142
  1281
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
  1282
by (auto simp add: set_conv_nth)
wenzelm@13114
  1283
wenzelm@13114
  1284
lemma all_nth_imp_all_set:
nipkow@13145
  1285
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
  1286
by (auto simp add: set_conv_nth)
wenzelm@13114
  1287
wenzelm@13114
  1288
lemma all_set_conv_all_nth:
nipkow@13145
  1289
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
  1290
by (auto simp add: set_conv_nth)
wenzelm@13114
  1291
kleing@25296
  1292
lemma rev_nth:
kleing@25296
  1293
  "n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)"
kleing@25296
  1294
proof (induct xs arbitrary: n)
kleing@25296
  1295
  case Nil thus ?case by simp
kleing@25296
  1296
next
kleing@25296
  1297
  case (Cons x xs)
kleing@25296
  1298
  hence n: "n < Suc (length xs)" by simp
kleing@25296
  1299
  moreover
kleing@25296
  1300
  { assume "n < length xs"
kleing@25296
  1301
    with n obtain n' where "length xs - n = Suc n'"
kleing@25296
  1302
      by (cases "length xs - n", auto)
kleing@25296
  1303
    moreover
kleing@25296
  1304
    then have "length xs - Suc n = n'" by simp
kleing@25296
  1305
    ultimately
kleing@25296
  1306
    have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp
kleing@25296
  1307
  }
kleing@25296
  1308
  ultimately
kleing@25296
  1309
  show ?case by (clarsimp simp add: Cons nth_append)
kleing@25296
  1310
qed
wenzelm@13114
  1311
nipkow@31159
  1312
lemma Skolem_list_nth:
nipkow@31159
  1313
  "(ALL i<k. EX x. P i x) = (EX xs. size xs = k & (ALL i<k. P i (xs!i)))"
nipkow@31159
  1314
  (is "_ = (EX xs. ?P k xs)")
nipkow@31159
  1315
proof(induct k)
nipkow@31159
  1316
  case 0 show ?case by simp
nipkow@31159
  1317
next
nipkow@31159
  1318
  case (Suc k)
nipkow@31159
  1319
  show ?case (is "?L = ?R" is "_ = (EX xs. ?P' xs)")
nipkow@31159
  1320
  proof
nipkow@31159
  1321
    assume "?R" thus "?L" using Suc by auto
nipkow@31159
  1322
  next
nipkow@31159
  1323
    assume "?L"
nipkow@31159
  1324
    with Suc obtain x xs where "?P k xs & P k x" by (metis less_Suc_eq)
nipkow@31159
  1325
    hence "?P'(xs@[x])" by(simp add:nth_append less_Suc_eq)
nipkow@31159
  1326
    thus "?R" ..
nipkow@31159
  1327
  qed
nipkow@31159
  1328
qed
nipkow@31159
  1329
nipkow@31159
  1330
nipkow@15392
  1331
subsubsection {* @{text list_update} *}
wenzelm@13114
  1332
nipkow@24526
  1333
lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
nipkow@24526
  1334
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1335
wenzelm@13114
  1336
lemma nth_list_update:
nipkow@24526
  1337
"i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@24526
  1338
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1339
wenzelm@13142
  1340
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
  1341
by (simp add: nth_list_update)
wenzelm@13114
  1342
nipkow@24526
  1343
lemma nth_list_update_neq [simp]: "i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@24526
  1344
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1345
nipkow@24526
  1346
lemma list_update_id[simp]: "xs[i := xs!i] = xs"
nipkow@24526
  1347
by (induct xs arbitrary: i) (simp_all split:nat.splits)
nipkow@24526
  1348
nipkow@24526
  1349
lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
nipkow@24526
  1350
apply (induct xs arbitrary: i)
nipkow@17501
  1351
 apply simp
nipkow@17501
  1352
apply (case_tac i)
nipkow@17501
  1353
apply simp_all
nipkow@17501
  1354
done
nipkow@17501
  1355
nipkow@31077
  1356
lemma list_update_nonempty[simp]: "xs[k:=x] = [] \<longleftrightarrow> xs=[]"
nipkow@31077
  1357
by(metis length_0_conv length_list_update)
nipkow@31077
  1358
wenzelm@13114
  1359
lemma list_update_same_conv:
nipkow@24526
  1360
"i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@24526
  1361
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1362
nipkow@14187
  1363
lemma list_update_append1:
nipkow@24526
  1364
 "i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
nipkow@24526
  1365
apply (induct xs arbitrary: i, simp)
nipkow@14187
  1366
apply(simp split:nat.split)
nipkow@14187
  1367
done
nipkow@14187
  1368
kleing@15868
  1369
lemma list_update_append:
nipkow@24526
  1370
  "(xs @ ys) [n:= x] = 
kleing@15868
  1371
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
nipkow@24526
  1372
by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868
  1373
nipkow@14402
  1374
lemma list_update_length [simp]:
nipkow@14402
  1375
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
  1376
by (induct xs, auto)
nipkow@14402
  1377
nipkow@31264
  1378
lemma map_update: "map f (xs[k:= y]) = (map f xs)[k := f y]"
nipkow@31264
  1379
by(induct xs arbitrary: k)(auto split:nat.splits)
nipkow@31264
  1380
nipkow@31264
  1381
lemma rev_update:
nipkow@31264
  1382
  "k < length xs \<Longrightarrow> rev (xs[k:= y]) = (rev xs)[length xs - k - 1 := y]"
nipkow@31264
  1383
by (induct xs arbitrary: k) (auto simp: list_update_append split:nat.splits)
nipkow@31264
  1384
wenzelm@13114
  1385
lemma update_zip:
nipkow@31080
  1386
  "(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@24526
  1387
by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split)
nipkow@24526
  1388
nipkow@24526
  1389
lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)"
nipkow@24526
  1390
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1391
wenzelm@13114
  1392
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
  1393
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
  1394
nipkow@24526
  1395
lemma set_update_memI: "n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
nipkow@24526
  1396
by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868
  1397
nipkow@31077
  1398
lemma list_update_overwrite[simp]:
haftmann@24796
  1399
  "xs [i := x, i := y] = xs [i := y]"
nipkow@31077
  1400
apply (induct xs arbitrary: i) apply simp
nipkow@31077
  1401
apply (case_tac i, simp_all)
haftmann@24796
  1402
done
haftmann@24796
  1403
haftmann@24796
  1404
lemma list_update_swap:
haftmann@24796
  1405
  "i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]"
haftmann@24796
  1406
apply (induct xs arbitrary: i i')
haftmann@24796
  1407
apply simp
haftmann@24796
  1408
apply (case_tac i, case_tac i')
haftmann@24796
  1409
apply auto
haftmann@24796
  1410
apply (case_tac i')
haftmann@24796
  1411
apply auto
haftmann@24796
  1412
done
haftmann@24796
  1413
haftmann@29827
  1414
lemma list_update_code [code]:
haftmann@29827
  1415
  "[][i := y] = []"
haftmann@29827
  1416
  "(x # xs)[0 := y] = y # xs"
haftmann@29827
  1417
  "(x # xs)[Suc i := y] = x # xs[i := y]"
haftmann@29827
  1418
  by simp_all
haftmann@29827
  1419
wenzelm@13114
  1420
nipkow@15392
  1421
subsubsection {* @{text last} and @{text butlast} *}
wenzelm@13114
  1422
wenzelm@13142
  1423
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
  1424
by (induct xs) auto
wenzelm@13114
  1425
wenzelm@13142
  1426
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
  1427
by (induct xs) auto
wenzelm@13114
  1428
nipkow@14302
  1429
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
  1430
by(simp add:last.simps)
nipkow@14302
  1431
nipkow@14302
  1432
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
  1433
by(simp add:last.simps)
nipkow@14302
  1434
nipkow@14302
  1435
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
  1436
by (induct xs) (auto)
nipkow@14302
  1437
nipkow@14302
  1438
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
  1439
by(simp add:last_append)
nipkow@14302
  1440
nipkow@14302
  1441
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
  1442
by(simp add:last_append)
nipkow@14302
  1443
nipkow@17762
  1444
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
nipkow@17762
  1445
by(rule rev_exhaust[of xs]) simp_all
nipkow@17762
  1446
nipkow@17762
  1447
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
nipkow@17762
  1448
by(cases xs) simp_all
nipkow@17762
  1449
nipkow@17765
  1450
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
nipkow@17765
  1451
by (induct as) auto
nipkow@17762
  1452
wenzelm@13142
  1453
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
  1454
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1455
wenzelm@13114
  1456
lemma butlast_append:
nipkow@24526
  1457
  "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@24526
  1458
by (induct xs arbitrary: ys) auto
wenzelm@13114
  1459
wenzelm@13142
  1460
lemma append_butlast_last_id [simp]:
nipkow@13145
  1461
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
  1462
by (induct xs) auto
wenzelm@13114
  1463
wenzelm@13142
  1464
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
  1465
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1466
wenzelm@13114
  1467
lemma in_set_butlast_appendI:
nipkow@13145
  1468
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
  1469
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
  1470
nipkow@24526
  1471
lemma last_drop[simp]: "n < length xs \<Longrightarrow> last (drop n xs) = last xs"
nipkow@24526
  1472
apply (induct xs arbitrary: n)
nipkow@17501
  1473
 apply simp
nipkow@17501
  1474
apply (auto split:nat.split)
nipkow@17501
  1475
done
nipkow@17501
  1476
huffman@30128
  1477
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
nipkow@17589
  1478
by(induct xs)(auto simp:neq_Nil_conv)
nipkow@17589
  1479
huffman@30128
  1480
lemma butlast_conv_take: "butlast xs = take (length xs - 1) xs"
huffman@26584
  1481
by (induct xs, simp, case_tac xs, simp_all)
huffman@26584
  1482
nipkow@31077
  1483
lemma last_list_update:
nipkow@31077
  1484
  "xs \<noteq> [] \<Longrightarrow> last(xs[k:=x]) = (if k = size xs - 1 then x else last xs)"
nipkow@31077
  1485
by (auto simp: last_conv_nth)
nipkow@31077
  1486
nipkow@31077
  1487
lemma butlast_list_update:
nipkow@31077
  1488
  "butlast(xs[k:=x]) =
nipkow@31077
  1489
 (if k = size xs - 1 then butlast xs else (butlast xs)[k:=x])"
nipkow@31077
  1490
apply(cases xs rule:rev_cases)
nipkow@31077
  1491
apply simp
nipkow@31077
  1492
apply(simp add:list_update_append split:nat.splits)
nipkow@31077
  1493
done
nipkow@31077
  1494
haftmann@24796
  1495
nipkow@15392
  1496
subsubsection {* @{text take} and @{text drop} *}
wenzelm@13114
  1497
wenzelm@13142
  1498
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
  1499
by (induct xs) auto
wenzelm@13114
  1500
wenzelm@13142
  1501
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
  1502
by (induct xs) auto
wenzelm@13114
  1503
wenzelm@13142
  1504
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
  1505
by simp
wenzelm@13114
  1506
wenzelm@13142
  1507
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
  1508
by simp
wenzelm@13114
  1509
wenzelm@13142
  1510
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
  1511
huffman@30128
  1512
lemma take_1_Cons [simp]: "take 1 (x # xs) = [x]"
huffman@30128
  1513
  unfolding One_nat_def by simp
huffman@30128
  1514
huffman@30128
  1515
lemma drop_1_Cons [simp]: "drop 1 (x # xs) = xs"
huffman@30128
  1516
  unfolding One_nat_def by simp
huffman@30128
  1517
nipkow@15110
  1518
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
  1519
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
  1520
nipkow@14187
  1521
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
  1522
by(cases xs, simp_all)
nipkow@14187
  1523
huffman@26584
  1524
lemma take_tl: "take n (tl xs) = tl (take (Suc n) xs)"
huffman@26584
  1525
by (induct xs arbitrary: n) simp_all
huffman@26584
  1526
nipkow@24526
  1527
lemma drop_tl: "drop n (tl xs) = tl(drop n xs)"
nipkow@24526
  1528
by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@24526
  1529
huffman@26584
  1530
lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)"
huffman@26584
  1531
by (cases n, simp, cases xs, auto)
huffman@26584
  1532
huffman@26584
  1533
lemma tl_drop: "tl (drop n xs) = drop n (tl xs)"
huffman@26584
  1534
by (simp only: drop_tl)
huffman@26584
  1535
nipkow@24526
  1536
lemma nth_via_drop: "drop n xs = y#ys \<Longrightarrow> xs!n = y"
nipkow@24526
  1537
apply (induct xs arbitrary: n, simp)
nipkow@14187
  1538
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
  1539
done
nipkow@14187
  1540
nipkow@13913
  1541
lemma take_Suc_conv_app_nth:
nipkow@24526
  1542
  "i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
nipkow@24526
  1543
apply (induct xs arbitrary: i, simp)
paulson@14208
  1544
apply (case_tac i, auto)
nipkow@13913
  1545
done
nipkow@13913
  1546
mehta@14591
  1547
lemma drop_Suc_conv_tl:
nipkow@24526
  1548
  "i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
nipkow@24526
  1549
apply (induct xs arbitrary: i, simp)
mehta@14591
  1550
apply (case_tac i, auto)
mehta@14591
  1551
done
mehta@14591
  1552
nipkow@24526
  1553
lemma length_take [simp]: "length (take n xs) = min (length xs) n"
nipkow@24526
  1554
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1555
nipkow@24526
  1556
lemma length_drop [simp]: "length (drop n xs) = (length xs - n)"
nipkow@24526
  1557
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1558
nipkow@24526
  1559
lemma take_all [simp]: "length xs <= n ==> take n xs = xs"
nipkow@24526
  1560
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1561
nipkow@24526
  1562
lemma drop_all [simp]: "length xs <= n ==> drop n xs = []"
nipkow@24526
  1563
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114
  1564
wenzelm@13142
  1565
lemma take_append [simp]:
nipkow@24526
  1566
  "take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@24526
  1567
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114
  1568
wenzelm@13142
  1569
lemma drop_append [simp]:
nipkow@24526
  1570
  "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@24526
  1571
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1572
nipkow@24526
  1573
lemma take_take [simp]: "take n (take m xs) = take (min n m) xs"
nipkow@24526
  1574
apply (induct m arbitrary: xs n, auto)
paulson@14208
  1575
apply (case_tac xs, auto)
nipkow@15236
  1576
apply (case_tac n, auto)
nipkow@13145
  1577
done
wenzelm@13114
  1578
nipkow@24526
  1579
lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs"
nipkow@24526
  1580
apply (induct m arbitrary: xs, auto)
paulson@14208
  1581
apply (case_tac xs, auto)
nipkow@13145
  1582
done
wenzelm@13114
  1583
nipkow@24526
  1584
lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)"
nipkow@24526
  1585
apply (induct m arbitrary: xs n, auto)
paulson@14208
  1586
apply (case_tac xs, auto)
nipkow@13145
  1587
done
wenzelm@13114
  1588
nipkow@24526
  1589
lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@24526
  1590
apply(induct xs arbitrary: m n)
nipkow@14802
  1591
 apply simp
nipkow@14802
  1592
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  1593
done
nipkow@14802
  1594
nipkow@24526
  1595
lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs"
nipkow@24526
  1596
apply (induct n arbitrary: xs, auto)
paulson@14208
  1597
apply (case_tac xs, auto)
nipkow@13145
  1598
done
wenzelm@13114
  1599
nipkow@24526
  1600
lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@24526
  1601
apply(induct xs arbitrary: n)
nipkow@15110
  1602
 apply simp
nipkow@15110
  1603
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  1604
done
nipkow@15110
  1605
nipkow@24526
  1606
lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)"
nipkow@24526
  1607
apply(induct xs arbitrary: n)
nipkow@15110
  1608
apply simp
nipkow@15110
  1609
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1610
done
nipkow@15110
  1611
nipkow@24526
  1612
lemma take_map: "take n (map f xs) = map f (take n xs)"
nipkow@24526
  1613
apply (induct n arbitrary: xs, auto)
paulson@14208
  1614
apply (case_tac xs, auto)
nipkow@13145
  1615
done
wenzelm@13114
  1616
nipkow@24526
  1617
lemma drop_map: "drop n (map f xs) = map f (drop n xs)"
nipkow@24526
  1618
apply (induct n arbitrary: xs, auto)
paulson@14208
  1619
apply (case_tac xs, auto)
nipkow@13145
  1620
done
wenzelm@13114
  1621
nipkow@24526
  1622
lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)"
nipkow@24526
  1623
apply (induct xs arbitrary: i, auto)
paulson@14208
  1624
apply (case_tac i, auto)
nipkow@13145
  1625
done
wenzelm@13114
  1626
nipkow@24526
  1627
lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)"
nipkow@24526
  1628
apply (induct xs arbitrary: i, auto)
paulson@14208
  1629
apply (case_tac i, auto)
nipkow@13145
  1630
done
wenzelm@13114
  1631
nipkow@24526
  1632
lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i"
nipkow@24526
  1633
apply (induct xs arbitrary: i n, auto)
paulson@14208
  1634
apply (case_tac n, blast)
paulson@14208
  1635
apply (case_tac i, auto)
nipkow@13145
  1636
done
wenzelm@13114
  1637
wenzelm@13142
  1638
lemma nth_drop [simp]:
nipkow@24526
  1639
  "n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
nipkow@24526
  1640
apply (induct n arbitrary: xs i, auto)
paulson@14208
  1641
apply (case_tac xs, auto)
nipkow@13145
  1642
done
nipkow@3507
  1643
huffman@26584
  1644
lemma butlast_take:
huffman@30128
  1645
  "n <= length xs ==> butlast (take n xs) = take (n - 1) xs"
huffman@26584
  1646
by (simp add: butlast_conv_take min_max.inf_absorb1 min_max.inf_absorb2)
huffman@26584
  1647
huffman@26584
  1648
lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)"
huffman@30128
  1649
by (simp add: butlast_conv_take drop_take add_ac)
huffman@26584
  1650
huffman@26584
  1651
lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs"
huffman@26584
  1652
by (simp add: butlast_conv_take min_max.inf_absorb1)
huffman@26584
  1653
huffman@26584
  1654
lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)"
huffman@30128
  1655
by (simp add: butlast_conv_take drop_take add_ac)
huffman@26584
  1656
nipkow@18423
  1657
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
nipkow@18423
  1658
by(simp add: hd_conv_nth)
nipkow@18423
  1659
nipkow@24526
  1660
lemma set_take_subset: "set(take n xs) \<subseteq> set xs"
nipkow@24526
  1661
by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split)
nipkow@24526
  1662
nipkow@24526
  1663
lemma set_drop_subset: "set(drop n xs) \<subseteq> set xs"
nipkow@24526
  1664
by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1665
nipkow@14187
  1666
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1667
using set_take_subset by fast
nipkow@14187
  1668
nipkow@14187
  1669
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1670
using set_drop_subset by fast
nipkow@14187
  1671
wenzelm@13114
  1672
lemma append_eq_conv_conj:
nipkow@24526
  1673
  "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
nipkow@24526
  1674
apply (induct xs arbitrary: zs, simp, clarsimp)
paulson@14208
  1675
apply (case_tac zs, auto)
nipkow@13145
  1676
done
wenzelm@13142
  1677
nipkow@24526
  1678
lemma take_add: 
nipkow@24526
  1679
  "i+j \<le> length(xs) \<Longrightarrow> take (i+j) xs = take i xs @ take j (drop i xs)"
nipkow@24526
  1680
apply (induct xs arbitrary: i, auto) 
nipkow@24526
  1681
apply (case_tac i, simp_all)
paulson@14050
  1682
done
paulson@14050
  1683
nipkow@14300
  1684
lemma append_eq_append_conv_if:
nipkow@24526
  1685
 "(xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1686
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1687
   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
  1688
   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
  1689
apply(induct xs\<^isub>1 arbitrary: ys\<^isub>1)
nipkow@14300
  1690
 apply simp
nipkow@14300
  1691
apply(case_tac ys\<^isub>1)
nipkow@14300
  1692
apply simp_all
nipkow@14300
  1693
done
nipkow@14300
  1694
nipkow@15110
  1695
lemma take_hd_drop:
huffman@30079
  1696
  "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (Suc n) xs"
nipkow@24526
  1697
apply(induct xs arbitrary: n)
nipkow@15110
  1698
apply simp
nipkow@15110
  1699
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1700
done
nipkow@15110
  1701
nipkow@17501
  1702
lemma id_take_nth_drop:
nipkow@17501
  1703
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
nipkow@17501
  1704
proof -
nipkow@17501
  1705
  assume si: "i < length xs"
nipkow@17501
  1706
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
nipkow@17501
  1707
  moreover
nipkow@17501
  1708
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
nipkow@17501
  1709
    apply (rule_tac take_Suc_conv_app_nth) by arith
nipkow@17501
  1710
  ultimately show ?thesis by auto
nipkow@17501
  1711
qed
nipkow@17501
  1712
  
nipkow@17501
  1713
lemma upd_conv_take_nth_drop:
nipkow@17501
  1714
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1715
proof -
nipkow@17501
  1716
  assume i: "i < length xs"
nipkow@17501
  1717
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
nipkow@17501
  1718
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
nipkow@17501
  1719
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1720
    using i by (simp add: list_update_append)
nipkow@17501
  1721
  finally show ?thesis .
nipkow@17501
  1722
qed
nipkow@17501
  1723
haftmann@24796
  1724
lemma nth_drop':
haftmann@24796
  1725
  "i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs"
haftmann@24796
  1726
apply (induct i arbitrary: xs)
haftmann@24796
  1727
apply (simp add: neq_Nil_conv)
haftmann@24796
  1728
apply (erule exE)+
haftmann@24796
  1729
apply simp
haftmann@24796
  1730
apply (case_tac xs)
haftmann@24796
  1731
apply simp_all
haftmann@24796
  1732
done
haftmann@24796
  1733
wenzelm@13114
  1734
nipkow@15392
  1735
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1736
hoelzl@33639
  1737
lemma length_takeWhile_le: "length (takeWhile P xs) \<le> length xs"
hoelzl@33639
  1738
  by (induct xs) auto
hoelzl@33639
  1739
wenzelm@13142
  1740
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1741
by (induct xs) auto
wenzelm@13114
  1742
wenzelm@13142
  1743
lemma takeWhile_append1 [simp]:
nipkow@13145
  1744
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1745
by (induct xs) auto
wenzelm@13114
  1746
wenzelm@13142
  1747
lemma takeWhile_append2 [simp]:
nipkow@13145
  1748
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1749
by (induct xs) auto
wenzelm@13114
  1750
wenzelm@13142
  1751
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1752
by (induct xs) auto
wenzelm@13114
  1753
hoelzl@33639
  1754
lemma takeWhile_nth: "j < length (takeWhile P xs) \<Longrightarrow> takeWhile P xs ! j = xs ! j"
hoelzl@33639
  1755
apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
hoelzl@33639
  1756
hoelzl@33639
  1757
lemma dropWhile_nth: "j < length (dropWhile P xs) \<Longrightarrow> dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))"
hoelzl@33639
  1758
apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
hoelzl@33639
  1759
hoelzl@33639
  1760
lemma length_dropWhile_le: "length (dropWhile P xs) \<le> length xs"
hoelzl@33639
  1761
by (induct xs) auto
hoelzl@33639
  1762
wenzelm@13142
  1763
lemma dropWhile_append1 [simp]:
nipkow@13145
  1764
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1765
by (induct xs) auto
wenzelm@13114
  1766
wenzelm@13142
  1767
lemma dropWhile_append2 [simp]:
nipkow@13145
  1768
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1769
by (induct xs) auto
wenzelm@13114
  1770
krauss@23971
  1771
lemma set_takeWhileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1772
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1773
nipkow@13913
  1774
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1775
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1776
by(induct xs, auto)
nipkow@13913
  1777
nipkow@13913
  1778
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1779
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1780
by(induct xs, auto)
nipkow@13913
  1781
nipkow@13913
  1782
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1783
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1784
by(induct xs, auto)
nipkow@13913
  1785
nipkow@31077
  1786
lemma distinct_takeWhile[simp]: "distinct xs ==> distinct (takeWhile P xs)"
nipkow@31077
  1787
by (induct xs) (auto dest: set_takeWhileD)
nipkow@31077
  1788
nipkow@31077
  1789
lemma distinct_dropWhile[simp]: "distinct xs ==> distinct (dropWhile P xs)"
nipkow@31077
  1790
by (induct xs) auto
nipkow@31077
  1791
hoelzl@33639
  1792
lemma takeWhile_map: "takeWhile P (map f xs) = map f (takeWhile (P \<circ> f) xs)"
hoelzl@33639
  1793
by (induct xs) auto
hoelzl@33639
  1794
hoelzl@33639
  1795
lemma dropWhile_map: "dropWhile P (map f xs) = map f (dropWhile (P \<circ> f) xs)"
hoelzl@33639
  1796
by (induct xs) auto
hoelzl@33639
  1797
hoelzl@33639
  1798
lemma takeWhile_eq_take: "takeWhile P xs = take (length (takeWhile P xs)) xs"
hoelzl@33639
  1799
by (induct xs) auto
hoelzl@33639
  1800
hoelzl@33639
  1801
lemma dropWhile_eq_drop: "dropWhile P xs = drop (length (takeWhile P xs)) xs"
hoelzl@33639
  1802
by (induct xs) auto
hoelzl@33639
  1803
hoelzl@33639
  1804
lemma hd_dropWhile:
hoelzl@33639
  1805
  "dropWhile P xs \<noteq> [] \<Longrightarrow> \<not> P (hd (dropWhile P xs))"
hoelzl@33639
  1806
using assms by (induct xs) auto
hoelzl@33639
  1807
hoelzl@33639
  1808
lemma takeWhile_eq_filter:
hoelzl@33639
  1809
  assumes "\<And> x. x \<in> set (dropWhile P xs) \<Longrightarrow> \<not> P x"
hoelzl@33639
  1810
  shows "takeWhile P xs = filter P xs"
hoelzl@33639
  1811
proof -
hoelzl@33639
  1812
  have A: "filter P xs = filter P (takeWhile P xs @ dropWhile P xs)"
hoelzl@33639
  1813
    by simp
hoelzl@33639
  1814
  have B: "filter P (dropWhile P xs) = []"
hoelzl@33639
  1815
    unfolding filter_empty_conv using assms by blast
hoelzl@33639
  1816
  have "filter P xs = takeWhile P xs"
hoelzl@33639
  1817
    unfolding A filter_append B
hoelzl@33639
  1818
    by (auto simp add: filter_id_conv dest: set_takeWhileD)
hoelzl@33639
  1819
  thus ?thesis ..
hoelzl@33639
  1820
qed
hoelzl@33639
  1821
hoelzl@33639
  1822
lemma takeWhile_eq_take_P_nth:
hoelzl@33639
  1823
  "\<lbrakk> \<And> i. \<lbrakk> i < n ; i < length xs \<rbrakk> \<Longrightarrow> P (xs ! i) ; n < length xs \<Longrightarrow> \<not> P (xs ! n) \<rbrakk> \<Longrightarrow>
hoelzl@33639
  1824
  takeWhile P xs = take n xs"
hoelzl@33639
  1825
proof (induct xs arbitrary: n)
hoelzl@33639
  1826
  case (Cons x xs)
hoelzl@33639
  1827
  thus ?case
hoelzl@33639
  1828
  proof (cases n)
hoelzl@33639
  1829
    case (Suc n') note this[simp]
hoelzl@33639
  1830
    have "P x" using Cons.prems(1)[of 0] by simp
hoelzl@33639
  1831
    moreover have "takeWhile P xs = take n' xs"
hoelzl@33639
  1832
    proof (rule Cons.hyps)
hoelzl@33639
  1833
      case goal1 thus "P (xs ! i)" using Cons.prems(1)[of "Suc i"] by simp
hoelzl@33639
  1834
    next case goal2 thus ?case using Cons by auto
hoelzl@33639
  1835
    qed
hoelzl@33639
  1836
    ultimately show ?thesis by simp
hoelzl@33639
  1837
   qed simp
hoelzl@33639
  1838
qed simp
hoelzl@33639
  1839
hoelzl@33639
  1840
lemma nth_length_takeWhile:
hoelzl@33639
  1841
  "length (takeWhile P xs) < length xs \<Longrightarrow> \<not> P (xs ! length (takeWhile P xs))"
hoelzl@33639
  1842
by (induct xs) auto
hoelzl@33639
  1843
hoelzl@33639
  1844
lemma length_takeWhile_less_P_nth:
hoelzl@33639
  1845
  assumes all: "\<And> i. i < j \<Longrightarrow> P (xs ! i)" and "j \<le> length xs"
hoelzl@33639
  1846
  shows "j \<le> length (takeWhile P xs)"
hoelzl@33639
  1847
proof (rule classical)
hoelzl@33639
  1848
  assume "\<not> ?thesis"
hoelzl@33639
  1849
  hence "length (takeWhile P xs) < length xs" using assms by simp
hoelzl@33639
  1850
  thus ?thesis using all `\<not> ?thesis` nth_length_takeWhile[of P xs] by auto
hoelzl@33639
  1851
qed
nipkow@31077
  1852
nipkow@17501
  1853
text{* The following two lemmmas could be generalized to an arbitrary
nipkow@17501
  1854
property. *}
nipkow@17501
  1855
nipkow@17501
  1856
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1857
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
nipkow@17501
  1858
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
nipkow@17501
  1859
nipkow@17501
  1860
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1861
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
nipkow@17501
  1862
apply(induct xs)
nipkow@17501
  1863
 apply simp
nipkow@17501
  1864
apply auto
nipkow@17501
  1865
apply(subst dropWhile_append2)
nipkow@17501
  1866
apply auto
nipkow@17501
  1867
done
nipkow@17501
  1868
nipkow@18423
  1869
lemma takeWhile_not_last:
nipkow@18423
  1870
 "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
nipkow@18423
  1871
apply(induct xs)
nipkow@18423
  1872
 apply simp
nipkow@18423
  1873
apply(case_tac xs)
nipkow@18423
  1874
apply(auto)
nipkow@18423
  1875
done
nipkow@18423
  1876
krauss@19770
  1877
lemma takeWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1878
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1879
  ==> takeWhile P l = takeWhile Q k"
nipkow@24349
  1880
by (induct k arbitrary: l) (simp_all)
krauss@18336
  1881
krauss@19770
  1882
lemma dropWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1883
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1884
  ==> dropWhile P l = dropWhile Q k"
nipkow@24349
  1885
by (induct k arbitrary: l, simp_all)
krauss@18336
  1886
wenzelm@13114
  1887
nipkow@15392
  1888
subsubsection {* @{text zip} *}
wenzelm@13114
  1889
wenzelm@13142
  1890
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1891
by (induct ys) auto
wenzelm@13114
  1892
wenzelm@13142
  1893
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1894
by simp
wenzelm@13114
  1895
wenzelm@13142
  1896
declare zip_Cons [simp del]
wenzelm@13114
  1897
nipkow@15281
  1898
lemma zip_Cons1:
nipkow@15281
  1899
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  1900
by(auto split:list.split)
nipkow@15281
  1901
wenzelm@13142
  1902
lemma length_zip [simp]:
krauss@22493
  1903
"length (zip xs ys) = min (length xs) (length ys)"
krauss@22493
  1904
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1905
wenzelm@13114
  1906
lemma zip_append1:
krauss@22493
  1907
"zip (xs @ ys) zs =
nipkow@13145
  1908
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
krauss@22493
  1909
by (induct xs zs rule:list_induct2') auto
wenzelm@13114
  1910
wenzelm@13114
  1911
lemma zip_append2:
krauss@22493
  1912
"zip xs (ys @ zs) =
nipkow@13145
  1913
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
krauss@22493
  1914
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1915
wenzelm@13142
  1916
lemma zip_append [simp]:
wenzelm@13142
  1917
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1918
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1919
by (simp add: zip_append1)
wenzelm@13114
  1920
wenzelm@13114
  1921
lemma zip_rev:
nipkow@14247
  1922
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1923
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1924
hoelzl@33639
  1925
lemma zip_map_map:
hoelzl@33639
  1926
  "zip (map f xs) (map g ys) = map (\<lambda> (x, y). (f x, g y)) (zip xs ys)"
hoelzl@33639
  1927
proof (induct xs arbitrary: ys)
hoelzl@33639
  1928
  case (Cons x xs) note Cons_x_xs = Cons.hyps
hoelzl@33639
  1929
  show ?case
hoelzl@33639
  1930
  proof (cases ys)
hoelzl@33639
  1931
    case (Cons y ys')
hoelzl@33639
  1932
    show ?thesis unfolding Cons using Cons_x_xs by simp
hoelzl@33639
  1933
  qed simp
hoelzl@33639
  1934
qed simp
hoelzl@33639
  1935
hoelzl@33639
  1936
lemma zip_map1:
hoelzl@33639
  1937
  "zip (map f xs) ys = map (\<lambda>(x, y). (f x, y)) (zip xs ys)"
hoelzl@33639
  1938
using zip_map_map[of f xs "\<lambda>x. x" ys] by simp
hoelzl@33639
  1939
hoelzl@33639
  1940
lemma zip_map2:
hoelzl@33639
  1941
  "zip xs (map f ys) = map (\<lambda>(x, y). (x, f y)) (zip xs ys)"
hoelzl@33639
  1942
using zip_map_map[of "\<lambda>x. x" xs f ys] by simp
hoelzl@33639
  1943
nipkow@23096
  1944
lemma map_zip_map:
hoelzl@33639
  1945
  "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
hoelzl@33639
  1946
unfolding zip_map1 by auto
nipkow@23096
  1947
nipkow@23096
  1948
lemma map_zip_map2:
hoelzl@33639
  1949
  "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
hoelzl@33639
  1950
unfolding zip_map2 by auto
nipkow@23096
  1951
nipkow@31080
  1952
text{* Courtesy of Andreas Lochbihler: *}
nipkow@31080
  1953
lemma zip_same_conv_map: "zip xs xs = map (\<lambda>x. (x, x)) xs"
nipkow@31080
  1954
by(induct xs) auto
nipkow@31080
  1955
wenzelm@13142
  1956
lemma nth_zip [simp]:
nipkow@24526
  1957
"[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
nipkow@24526
  1958
apply (induct ys arbitrary: i xs, simp)
nipkow@13145
  1959
apply (case_tac xs)
nipkow@13145
  1960
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1961
done
wenzelm@13114
  1962
wenzelm@13114
  1963
lemma set_zip:
nipkow@13145
  1964
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@31080
  1965
by(simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1966
hoelzl@33639
  1967
lemma zip_same: "((a,b) \<in> set (zip xs xs)) = (a \<in> set xs \<and> a = b)"
hoelzl@33639
  1968
by(induct xs) auto
hoelzl@33639
  1969
wenzelm@13114
  1970
lemma zip_update:
nipkow@31080
  1971
  "zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@31080
  1972
by(rule sym, simp add: update_zip)
wenzelm@13114
  1973
wenzelm@13142
  1974
lemma zip_replicate [simp]:
nipkow@24526
  1975
  "zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
nipkow@24526
  1976
apply (induct i arbitrary: j, auto)
paulson@14208
  1977
apply (case_tac j, auto)
nipkow@13145
  1978
done
wenzelm@13114
  1979
nipkow@19487
  1980
lemma take_zip:
nipkow@24526
  1981
  "take n (zip xs ys) = zip (take n xs) (take n ys)"
nipkow@24526
  1982
apply (induct n arbitrary: xs ys)
nipkow@19487
  1983
 apply simp
nipkow@19487
  1984
apply (case_tac xs, simp)
nipkow@19487
  1985
apply (case_tac ys, simp_all)
nipkow@19487
  1986
done
nipkow@19487
  1987
nipkow@19487
  1988
lemma drop_zip:
nipkow@24526
  1989
  "drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
nipkow@24526
  1990
apply (induct n arbitrary: xs ys)
nipkow@19487
  1991
 apply simp
nipkow@19487
  1992
apply (case_tac xs, simp)
nipkow@19487
  1993
apply (case_tac ys, simp_all)
nipkow@19487
  1994
done
nipkow@19487
  1995
hoelzl@33639
  1996
lemma zip_takeWhile_fst: "zip (takeWhile P xs) ys = takeWhile (P \<circ> fst) (zip xs ys)"
hoelzl@33639
  1997
proof (induct xs arbitrary: ys)
hoelzl@33639
  1998
  case (Cons x xs) thus ?case by (cases ys) auto
hoelzl@33639
  1999
qed simp
hoelzl@33639
  2000
hoelzl@33639
  2001
lemma zip_takeWhile_snd: "zip xs (takeWhile P ys) = takeWhile (P \<circ> snd) (zip xs ys)"
hoelzl@33639
  2002
proof (induct xs arbitrary: ys)
hoelzl@33639
  2003
  case (Cons x xs) thus ?case by (cases ys) auto
hoelzl@33639
  2004
qed simp
hoelzl@33639
  2005
krauss@22493
  2006
lemma set_zip_leftD:
krauss@22493
  2007
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
krauss@22493
  2008
by (induct xs ys rule:list_induct2') auto
krauss@22493
  2009
krauss@22493
  2010
lemma set_zip_rightD:
krauss@22493
  2011
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
krauss@22493
  2012
by (induct xs ys rule:list_induct2') auto
wenzelm@13142
  2013
nipkow@23983
  2014
lemma in_set_zipE:
nipkow@23983
  2015
  "(x,y) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
nipkow@23983
  2016
by(blast dest: set_zip_leftD set_zip_rightD)
nipkow@23983
  2017
haftmann@29829
  2018
lemma zip_map_fst_snd:
haftmann@29829
  2019
  "zip (map fst zs) (map snd zs) = zs"
haftmann@29829
  2020
  by (induct zs) simp_all
haftmann@29829
  2021
haftmann@29829
  2022
lemma zip_eq_conv:
haftmann@29829
  2023
  "length xs = length ys \<Longrightarrow> zip xs ys = zs \<longleftrightarrow> map fst zs = xs \<and> map snd zs = ys"
haftmann@29829
  2024
  by (auto simp add: zip_map_fst_snd)
haftmann@29829
  2025
hoelzl@33639
  2026
lemma distinct_zipI1:
hoelzl@33639
  2027
  "distinct xs \<Longrightarrow> distinct (zip xs ys)"
hoelzl@33639
  2028
proof (induct xs arbitrary: ys)
hoelzl@33639
  2029
  case (Cons x xs)
hoelzl@33639
  2030
  show ?case
hoelzl@33639
  2031
  proof (cases ys)
hoelzl@33639
  2032
    case (Cons y ys')
hoelzl@33639
  2033
    have "(x, y) \<notin> set (zip xs ys')"
hoelzl@33639
  2034
      using Cons.prems by (auto simp: set_zip)
hoelzl@33639
  2035
    thus ?thesis
hoelzl@33639
  2036
      unfolding Cons zip_Cons_Cons distinct.simps
hoelzl@33639
  2037
      using Cons.hyps Cons.prems by simp
hoelzl@33639
  2038
  qed simp
hoelzl@33639
  2039
qed simp
hoelzl@33639
  2040
hoelzl@33639
  2041
lemma distinct_zipI2:
hoelzl@33639
  2042
  "distinct xs \<Longrightarrow> distinct (zip xs ys)"
hoelzl@33639
  2043
proof (induct xs arbitrary: ys)
hoelzl@33639
  2044
  case (Cons x xs)
hoelzl@33639
  2045
  show ?case
hoelzl@33639
  2046
  proof (cases ys)
hoelzl@33639
  2047
    case (Cons y ys')
hoelzl@33639
  2048
     have "(x, y) \<notin> set (zip xs ys')"
hoelzl@33639
  2049
      using Cons.prems by (auto simp: set_zip)
hoelzl@33639
  2050
    thus ?thesis
hoelzl@33639
  2051
      unfolding Cons zip_Cons_Cons distinct.simps
hoelzl@33639
  2052
      using Cons.hyps Cons.prems by simp
hoelzl@33639
  2053
  qed simp
hoelzl@33639
  2054
qed simp
haftmann@29829
  2055
nipkow@15392
  2056
subsubsection {* @{text list_all2} *}
wenzelm@13114
  2057
kleing@14316
  2058
lemma list_all2_lengthD [intro?]: 
kleing@14316
  2059
  "list_all2 P xs ys ==> length xs = length ys"
nipkow@24349
  2060
by (simp add: list_all2_def)
haftmann@19607
  2061
haftmann@19787
  2062
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
nipkow@24349
  2063
by (simp add: list_all2_def)
haftmann@19607
  2064
haftmann@19787
  2065
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
nipkow@24349
  2066
by (simp add: list_all2_def)
haftmann@19607
  2067
haftmann@19607
  2068
lemma list_all2_Cons [iff, code]:
haftmann@19607
  2069
  "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@24349
  2070
by (auto simp add: list_all2_def)
wenzelm@13114
  2071
wenzelm@13114
  2072
lemma list_all2_Cons1:
nipkow@13145
  2073
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  2074
by (cases ys) auto
wenzelm@13114
  2075
wenzelm@13114
  2076
lemma list_all2_Cons2:
nipkow@13145
  2077
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  2078
by (cases xs) auto
wenzelm@13114
  2079
wenzelm@13142
  2080
lemma list_all2_rev [iff]:
nipkow@13145
  2081
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  2082
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  2083
kleing@13863
  2084
lemma list_all2_rev1:
kleing@13863
  2085
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  2086
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  2087
wenzelm@13114
  2088
lemma list_all2_append1:
nipkow@13145
  2089
"list_all2 P (xs @ ys) zs =
nipkow@13145
  2090
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  2091
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  2092
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  2093
apply (rule iffI)
nipkow@13145
  2094
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  2095
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  2096
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  2097
apply (simp add: ball_Un)
nipkow@13145
  2098
done
wenzelm@13114
  2099
wenzelm@13114
  2100
lemma list_all2_append2:
nipkow@13145
  2101
"list_all2 P xs (ys @ zs) =
nipkow@13145
  2102
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  2103
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  2104
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  2105
apply (rule iffI)
nipkow@13145
  2106
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  2107
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  2108
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  2109
apply (simp add: ball_Un)
nipkow@13145
  2110
done
wenzelm@13114
  2111
kleing@13863
  2112
lemma list_all2_append:
nipkow@14247
  2113
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  2114
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  2115
by (induct rule:list_induct2, simp_all)
kleing@13863
  2116
kleing@13863
  2117
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  2118
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
nipkow@24349
  2119
by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  2120
wenzelm@13114
  2121
lemma list_all2_conv_all_nth:
nipkow@13145
  2122
"list_all2 P xs ys =
nipkow@13145
  2123
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  2124
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  2125
berghofe@13883
  2126
lemma list_all2_trans:
berghofe@13883
  2127
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  2128
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  2129
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  2130
proof (induct as)
berghofe@13883
  2131
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  2132
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  2133
  proof (induct bs)
berghofe@13883
  2134
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  2135
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  2136
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  2137
  qed simp
berghofe@13883
  2138
qed simp
berghofe@13883
  2139
kleing@13863
  2140
lemma list_all2_all_nthI [intro?]:
kleing@13863
  2141
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
nipkow@24349
  2142
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2143
paulson@14395
  2144
lemma list_all2I:
paulson@14395
  2145
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
nipkow@24349
  2146
by (simp add: list_all2_def)
paulson@14395
  2147
kleing@14328
  2148
lemma list_all2_nthD:
kleing@13863
  2149
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@24349
  2150
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2151
nipkow@14302
  2152
lemma list_all2_nthD2:
nipkow@14302
  2153
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@24349
  2154
by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  2155
kleing@13863
  2156
lemma list_all2_map1: 
kleing@13863
  2157
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
nipkow@24349
  2158
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2159
kleing@13863
  2160
lemma list_all2_map2: 
kleing@13863
  2161
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
nipkow@24349
  2162
by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  2163
kleing@14316
  2164
lemma list_all2_refl [intro?]:
kleing@13863
  2165
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
nipkow@24349
  2166
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2167
kleing@13863
  2168
lemma list_all2_update_cong:
kleing@13863
  2169
  "\<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
  2170
by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  2171
kleing@13863
  2172
lemma list_all2_update_cong2:
kleing@13863
  2173
  "\<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
  2174
by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  2175
nipkow@14302
  2176
lemma list_all2_takeI [simp,intro?]:
nipkow@24526
  2177
  "list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@24526
  2178
apply (induct xs arbitrary: n ys)
nipkow@24526
  2179
 apply simp
nipkow@24526
  2180
apply (clarsimp simp add: list_all2_Cons1)
nipkow@24526
  2181
apply (case_tac n)
nipkow@24526
  2182
apply auto
nipkow@24526
  2183
done
nipkow@14302
  2184
nipkow@14302
  2185
lemma list_all2_dropI [simp,intro?]:
nipkow@24526
  2186
  "list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
nipkow@24526
  2187
apply (induct as arbitrary: n bs, simp)
nipkow@24526
  2188
apply (clarsimp simp add: list_all2_Cons1)
nipkow@24526
  2189
apply (case_tac n, simp, simp)
nipkow@24526
  2190
done
kleing@13863
  2191
kleing@14327
  2192
lemma list_all2_mono [intro?]:
nipkow@24526
  2193
  "list_all2 P xs ys \<Longrightarrow> (\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys) \<Longrightarrow> list_all2 Q xs ys"
nipkow@24526
  2194
apply (induct xs arbitrary: ys, simp)
nipkow@24526
  2195
apply (case_tac ys, auto)
nipkow@24526
  2196
done
kleing@13863
  2197
haftmann@22551
  2198
lemma list_all2_eq:
haftmann@22551
  2199
  "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
nipkow@24349
  2200
by (induct xs ys rule: list_induct2') auto
haftmann@22551
  2201
wenzelm@13142
  2202
nipkow@15392
  2203
subsubsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  2204
wenzelm@13142
  2205
lemma foldl_append [simp]:
nipkow@24526
  2206
  "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@24526
  2207
by (induct xs arbitrary: a) auto
wenzelm@13142
  2208
nipkow@14402
  2209
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  2210
by (induct xs) auto
nipkow@14402
  2211
nipkow@23096
  2212
lemma foldr_map: "foldr g (map f xs) a = foldr (g o f) xs a"
nipkow@23096
  2213
by(induct xs) simp_all
nipkow@23096
  2214
nipkow@24449
  2215
text{* For efficient code generation: avoid intermediate list. *}
haftmann@31998
  2216
lemma foldl_map[code_unfold]:
nipkow@24449
  2217
  "foldl g a (map f xs) = foldl (%a x. g a (f x)) a xs"
nipkow@23096
  2218
by(induct xs arbitrary:a) simp_all
nipkow@23096
  2219
haftmann@31930
  2220
lemma foldl_apply_inv:
haftmann@31930
  2221
  assumes "\<And>x. g (h x) = x"
haftmann@31930
  2222
  shows "foldl f (g s) xs = g (foldl (\<lambda>s x. h (f (g s) x)) s xs)"
haftmann@31930
  2223
  by (rule sym, induct xs arbitrary: s) (simp_all add: assms)
haftmann@31930
  2224
krauss@19770
  2225
lemma foldl_cong [fundef_cong, recdef_cong]:
krauss@18336
  2226
  "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] 
krauss@18336
  2227
  ==> foldl f a l = foldl g b k"
nipkow@24349
  2228
by (induct k arbitrary: a b l) simp_all
krauss@18336
  2229
krauss@19770
  2230
lemma foldr_cong [fundef_cong, recdef_cong]:
krauss@18336
  2231
  "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] 
krauss@18336
  2232
  ==> foldr f l a = foldr g k b"
nipkow@24349
  2233
by (induct k arbitrary: a b l) simp_all
krauss@18336
  2234
nipkow@24449
  2235
lemma (in semigroup_add) foldl_assoc:
haftmann@25062
  2236
shows "foldl op+ (x+y) zs = x + (foldl op+ y zs)"
nipkow@24449
  2237
by (induct zs arbitrary: y) (simp_all add:add_assoc)
nipkow@24449
  2238
nipkow@24449
  2239
lemma (in monoid_add) foldl_absorb0:
haftmann@25062
  2240
shows "x + (foldl op+ 0 zs) = foldl op+ x zs"
nipkow@24449
  2241
by (induct zs) (simp_all add:foldl_assoc)
nipkow@24449
  2242
nipkow@24449
  2243
nipkow@23096
  2244
text{* The ``First Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096
  2245
nipkow@23096
  2246
lemma foldl_foldr1_lemma:
nipkow@23096
  2247
 "foldl op + a xs = a + foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096
  2248
by (induct xs arbitrary: a) (auto simp:add_assoc)
nipkow@23096
  2249
nipkow@23096
  2250
corollary foldl_foldr1:
nipkow@23096
  2251
 "foldl op + 0 xs = foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096
  2252
by (simp add:foldl_foldr1_lemma)
nipkow@23096
  2253
nipkow@23096
  2254
nipkow@23096
  2255
text{* The ``Third Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096
  2256
nipkow@14402
  2257
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  2258
by (induct xs) auto
nipkow@14402
  2259
nipkow@14402
  2260
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  2261
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  2262
haftmann@25062
  2263
lemma (in ab_semigroup_add) foldr_conv_foldl: "foldr op + xs a = foldl op + a xs"
chaieb@24471
  2264
  by (induct xs, auto simp add: foldl_assoc add_commute)
chaieb@24471
  2265
wenzelm@13142
  2266
text {*
nipkow@13145
  2267
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  2268
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  2269
*}
wenzelm@13142
  2270
nipkow@24526
  2271
lemma start_le_sum: "(m::nat) <= n ==> m <= foldl (op +) n ns"
nipkow@24526
  2272
by (induct ns arbitrary: n) auto
nipkow@24526
  2273
nipkow@24526
  2274
lemma elem_le_sum: "(n::nat) : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  2275
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  2276
wenzelm@13142
  2277
lemma sum_eq_0_conv [iff]:
nipkow@24526
  2278
  "(foldl (op +) (m::nat) ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@24526
  2279
by (induct ns arbitrary: m) auto
wenzelm@13114
  2280
chaieb@24471
  2281
lemma foldr_invariant: 
chaieb@24471
  2282
  "\<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
  2283
  by (induct xs, simp_all)
chaieb@24471
  2284
chaieb@24471
  2285
lemma foldl_invariant: 
chaieb@24471
  2286
  "\<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
  2287
  by (induct xs arbitrary: x, simp_all)
chaieb@24471
  2288
haftmann@31455
  2289
text {* @{const foldl} and @{const concat} *}
nipkow@24449
  2290
nipkow@24449
  2291
lemma foldl_conv_concat:
haftmann@29782
  2292
  "foldl (op @) xs xss = xs @ concat xss"
haftmann@29782
  2293
proof (induct xss arbitrary: xs)
haftmann@29782
  2294
  case Nil show ?case by simp
haftmann@29782
  2295
next
haftmann@29782
  2296
  interpret monoid_add "[]" "op @" proof qed simp_all
haftmann@29782
  2297
  case Cons then show ?case by (simp add: foldl_absorb0)
haftmann@29782
  2298
qed
haftmann@29782
  2299
haftmann@29782
  2300
lemma concat_conv_foldl: "concat xss = foldl (op @) [] xss"
haftmann@29782
  2301
  by (simp add: foldl_conv_concat)
haftmann@29782
  2302
haftmann@31455
  2303
text {* @{const Finite_Set.fold} and @{const foldl} *}
haftmann@31455
  2304
haftmann@31455
  2305
lemma (in fun_left_comm_idem) fold_set:
haftmann@31455
  2306
  "fold f y (set xs) = foldl (\<lambda>y x. f x y) y xs"
haftmann@31455
  2307
  by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm)
haftmann@31455
  2308
haftmann@32681
  2309
lemma (in ab_semigroup_idem_mult) fold1_set:
haftmann@32681
  2310
  assumes "xs \<noteq> []"
haftmann@32681
  2311
  shows "fold1 times (set xs) = foldl times (hd xs) (tl xs)"
haftmann@32681
  2312
proof -
haftmann@32681
  2313
  interpret fun_left_comm_idem times by (fact fun_left_comm_idem)
haftmann@32681
  2314
  from assms obtain y ys where xs: "xs = y # ys"
haftmann@32681
  2315
    by (cases xs) auto
haftmann@32681
  2316
  show ?thesis
haftmann@32681
  2317
  proof (cases "set ys = {}")
haftmann@32681
  2318
    case True with xs show ?thesis by simp
haftmann@32681
  2319
  next
haftmann@32681
  2320
    case False
haftmann@32681
  2321
    then have "fold1 times (insert y (set ys)) = fold times y (set ys)"
haftmann@32681
  2322
      by (simp only: finite_set fold1_eq_fold_idem)
haftmann@32681
  2323
    with xs show ?thesis by (simp add: fold_set mult_commute)
haftmann@32681
  2324
  qed
haftmann@32681
  2325
qed
haftmann@32681
  2326
haftmann@32681
  2327
lemma (in lattice) Inf_fin_set_fold [code_unfold]:
haftmann@32681
  2328
  "Inf_fin (set (x # xs)) = foldl inf x xs"
haftmann@32681
  2329
proof -
haftmann@32681
  2330
  interpret ab_semigroup_idem_mult "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@32681
  2331
    by (fact ab_semigroup_idem_mult_inf)
haftmann@32681
  2332
  show ?thesis
haftmann@32681
  2333
    by (simp add: Inf_fin_def fold1_set del: set.simps)
haftmann@32681
  2334
qed
haftmann@32681
  2335
haftmann@32681
  2336
lemma (in lattice) Sup_fin_set_fold [code_unfold]:
haftmann@32681
  2337
  "Sup_fin (set (x # xs)) = foldl sup x xs"
haftmann@32681
  2338
proof -
haftmann@32681
  2339
  interpret ab_semigroup_idem_mult "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@32681
  2340
    by (fact ab_semigroup_idem_mult_sup)
haftmann@32681
  2341
  show ?thesis
haftmann@32681
  2342
    by (simp add: Sup_fin_def fold1_set del: set.simps)
haftmann@32681
  2343
qed
haftmann@32681
  2344
haftmann@32681
  2345
lemma (in linorder) Min_fin_set_fold [code_unfold]:
haftmann@32681
  2346
  "Min (set (x # xs)) = foldl min x xs"
haftmann@32681
  2347
proof -
haftmann@32681
  2348
  interpret ab_semigroup_idem_mult "min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@32681
  2349
    by (fact ab_semigroup_idem_mult_min)
haftmann@32681
  2350
  show ?thesis
haftmann@32681
  2351
    by (simp add: Min_def fold1_set del: set.simps)
haftmann@32681
  2352
qed
haftmann@32681
  2353
haftmann@32681
  2354
lemma (in linorder) Max_fin_set_fold [code_unfold]:
haftmann@32681
  2355
  "Max (set (x # xs)) = foldl max x xs"
haftmann@32681
  2356
proof -
haftmann@32681
  2357
  interpret ab_semigroup_idem_mult "max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@32681
  2358
    by (fact ab_semigroup_idem_mult_max)
haftmann@32681
  2359
  show ?thesis
haftmann@32681
  2360
    by (simp add: Max_def fold1_set del: set.simps)
haftmann@32681
  2361
qed
haftmann@32681
  2362
haftmann@32681
  2363
lemma (in complete_lattice) Inf_set_fold [code_unfold]:
haftmann@32681
  2364
  "Inf (set xs) = foldl inf top xs"
haftmann@32681
  2365
  by (cases xs)
haftmann@32681
  2366
    (simp_all add: Inf_fin_Inf [symmetric] Inf_fin_set_fold
haftmann@32681
  2367
      inf_commute del: set.simps, simp add: top_def)
haftmann@32681
  2368
haftmann@32681
  2369
lemma (in complete_lattice) Sup_set_fold [code_unfold]:
haftmann@32681
  2370
  "Sup (set xs) = foldl sup bot xs"
haftmann@32681
  2371
  by (cases xs)
haftmann@32681
  2372
    (simp_all add: Sup_fin_Sup [symmetric] Sup_fin_set_fold
haftmann@32681
  2373
      sup_commute del: set.simps, simp add: bot_def)
haftmann@31455
  2374
nipkow@24449
  2375
nipkow@23096
  2376
subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
nipkow@23096
  2377
haftmann@26442
  2378
lemma listsum_append [simp]: "listsum (xs @ ys) = listsum xs + listsum ys"
nipkow@24449
  2379
by (induct xs) (simp_all add:add_assoc)
nipkow@24449
  2380
haftmann@26442
  2381
lemma listsum_rev [simp]:
haftmann@26442
  2382
  fixes xs :: "'a\<Colon>comm_monoid_add list"
haftmann@26442
  2383
  shows "listsum (rev xs) = listsum xs"
nipkow@24449
  2384
by (induct xs) (simp_all add:add_ac)
nipkow@24449
  2385
nipkow@31022
  2386
lemma listsum_map_remove1:
nipkow@31022
  2387
fixes f :: "'a \<Rightarrow> ('b::comm_monoid_add)"
nipkow@31022
  2388
shows "x : set xs \<Longrightarrow> listsum(map f xs) = f x + listsum(map f (remove1 x xs))"
nipkow@31022
  2389
by (induct xs)(auto simp add:add_ac)
nipkow@31022
  2390
nipkow@31022
  2391
lemma list_size_conv_listsum:
nipkow@31022
  2392
  "list_size f xs = listsum (map f xs) + size xs"
nipkow@31022
  2393
by(induct xs) auto
nipkow@31022
  2394
haftmann@26442
  2395
lemma listsum_foldr: "listsum xs = foldr (op +) xs 0"
haftmann@26442
  2396
by (induct xs) auto
haftmann@26442
  2397
haftmann@26442
  2398
lemma length_concat: "length (concat xss) = listsum (map length xss)"
haftmann@26442
  2399
by (induct xss) simp_all
nipkow@23096
  2400
hoelzl@33639
  2401
lemma listsum_map_filter:
hoelzl@33639
  2402
  fixes f :: "'a \<Rightarrow> 'b \<Colon> comm_monoid_add"
hoelzl@33639
  2403
  assumes "\<And> x. \<lbrakk> x \<in> set xs ; \<not> P x \<rbrakk> \<Longrightarrow> f x = 0"
hoelzl@33639
  2404
  shows "listsum (map f (filter P xs)) = listsum (map f xs)"
hoelzl@33639
  2405
using assms by (induct xs) auto
hoelzl@33639
  2406
nipkow@24449
  2407
text{* For efficient code generation ---
nipkow@24449
  2408
       @{const listsum} is not tail recursive but @{const foldl} is. *}
haftmann@31998
  2409
lemma listsum[code_unfold]: "listsum xs = foldl (op +) 0 xs"
nipkow@23096
  2410
by(simp add:listsum_foldr foldl_foldr1)
nipkow@23096
  2411
nipkow@31077
  2412
lemma distinct_listsum_conv_Setsum:
nipkow@31077
  2413
  "distinct xs \<Longrightarrow> listsum xs = Setsum(set xs)"
nipkow@31077
  2414
by (induct xs) simp_all
nipkow@31077
  2415
nipkow@24449
  2416
nipkow@23096
  2417
text{* Some syntactic sugar for summing a function over a list: *}
nipkow@23096
  2418
nipkow@23096
  2419
syntax
nipkow@23096
  2420
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
nipkow@23096
  2421
syntax (xsymbols)
nipkow@23096
  2422
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096
  2423
syntax (HTML output)
nipkow@23096
  2424
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096
  2425
nipkow@23096
  2426
translations -- {* Beware of argument permutation! *}
nipkow@23096
  2427
  "SUM x<-xs. b" == "CONST listsum (map (%x. b) xs)"
nipkow@23096
  2428
  "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (map (%x. b) xs)"
nipkow@23096
  2429
haftmann@26442
  2430
lemma listsum_triv: "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
haftmann@26442
  2431
  by (induct xs) (simp_all add: left_distrib)
haftmann@26442
  2432
nipkow@23096
  2433
lemma listsum_0 [simp]: "(\<Sum>x\<leftarrow>xs. 0) = 0"
haftmann@26442
  2434
  by (induct xs) (simp_all add: left_distrib)
nipkow@23096
  2435
nipkow@23096
  2436
text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
nipkow@23096
  2437
lemma uminus_listsum_map:
haftmann@26442
  2438
  fixes f :: "'a \<Rightarrow> 'b\<Colon>ab_group_add"
haftmann@26442
  2439
  shows "- listsum (map f xs) = (listsum (map (uminus o f) xs))"
haftmann@26442
  2440
by (induct xs) simp_all
nipkow@23096
  2441
huffman@31258
  2442
lemma listsum_addf:
huffman@31258
  2443
  fixes f g :: "'a \<Rightarrow> 'b::comm_monoid_add"
huffman@31258
  2444
  shows "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
huffman@31258
  2445
by (induct xs) (simp_all add: algebra_simps)
huffman@31258
  2446
huffman@31258
  2447
lemma listsum_subtractf:
huffman@31258
  2448
  fixes f g :: "'a \<Rightarrow> 'b::ab_group_add"
huffman@31258
  2449
  shows "(\<Sum>x\<leftarrow>xs. f x - g x) = listsum (map f xs) - listsum (map g xs)"
huffman@31258
  2450
by (induct xs) simp_all
huffman@31258
  2451
huffman@31258
  2452
lemma listsum_const_mult:
huffman@31258
  2453
  fixes f :: "'a \<Rightarrow> 'b::semiring_0"
huffman@31258
  2454
  shows "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
huffman@31258
  2455
by (induct xs, simp_all add: algebra_simps)
huffman@31258
  2456
huffman@31258
  2457
lemma listsum_mult_const:
huffman@31258
  2458
  fixes f :: "'a \<Rightarrow> 'b::semiring_0"
huffman@31258
  2459
  shows "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
huffman@31258
  2460
by (induct xs, simp_all add: algebra_simps)
huffman@31258
  2461
huffman@31258
  2462
lemma listsum_abs:
huffman@31258
  2463
  fixes xs :: "'a::pordered_ab_group_add_abs list"
huffman@31258
  2464
  shows "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
huffman@31258
  2465
by (induct xs, simp, simp add: order_trans [OF abs_triangle_ineq])
huffman@31258
  2466
huffman@31258
  2467
lemma listsum_mono:
huffman@31258
  2468
  fixes f g :: "'a \<Rightarrow> 'b::{comm_monoid_add, pordered_ab_semigroup_add}"
huffman@31258
  2469
  shows "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (\<Sum>x\<leftarrow>xs. f x) \<le> (\<Sum>x\<leftarrow>xs. g x)"
huffman@31258
  2470
by (induct xs, simp, simp add: add_mono)
huffman@31258
  2471
wenzelm@13114
  2472
nipkow@24645
  2473
subsubsection {* @{text upt} *}
wenzelm@13114
  2474
nipkow@17090
  2475
lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
nipkow@17090
  2476
-- {* simp does not terminate! *}
nipkow@13145
  2477
by (induct j) auto
wenzelm@13142
  2478
nipkow@32005
  2479
lemmas upt_rec_number_of[simp] = upt_rec[of "number_of m" "number_of n", standard]
nipkow@32005
  2480
nipkow@15425
  2481
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
nipkow@13145
  2482
by (subst upt_rec) simp
wenzelm@13114
  2483
nipkow@15425
  2484
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
nipkow@15281
  2485
by(induct j)simp_all
nipkow@15281
  2486
nipkow@15281
  2487
lemma upt_eq_Cons_conv:
nipkow@24526
  2488
 "([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
nipkow@24526
  2489
apply(induct j arbitrary: x xs)
nipkow@15281
  2490
 apply simp
nipkow@15281
  2491
apply(clarsimp simp add: append_eq_Cons_conv)
nipkow@15281
  2492
apply arith
nipkow@15281
  2493
done
nipkow@15281
  2494
nipkow@15425
  2495
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
nipkow@13145
  2496
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  2497
by simp
wenzelm@13114
  2498
nipkow@15425
  2499
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
haftmann@26734
  2500
  by (simp add: upt_rec)
wenzelm@13114
  2501
nipkow@15425
  2502
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
nipkow@13145
  2503
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  2504
by (induct k) auto
wenzelm@13114
  2505
nipkow@15425
  2506
lemma length_upt [simp]: "length [i..<j] = j - i"
nipkow@13145
  2507
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  2508
nipkow@15425
  2509
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
nipkow@13145
  2510
apply (induct j)
nipkow@13145
  2511
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  2512
done
wenzelm@13114
  2513
nipkow@17906
  2514
nipkow@17906
  2515
lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
nipkow@17906
  2516
by(simp add:upt_conv_Cons)
nipkow@17906
  2517
nipkow@17906
  2518
lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
nipkow@17906
  2519
apply(cases j)
nipkow@17906
  2520
 apply simp
nipkow@17906
  2521
by(simp add:upt_Suc_append)
nipkow@17906
  2522
nipkow@24526
  2523
lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]"
nipkow@24526
  2524
apply (induct m arbitrary: i, simp)
nipkow@13145
  2525
apply (subst upt_rec)
nipkow@13145
  2526
apply (rule sym)
nipkow@13145
  2527
apply (subst upt_rec)
nipkow@13145
  2528
apply (simp del: upt.simps)
nipkow@13145
  2529
done
nipkow@3507
  2530
nipkow@17501
  2531
lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
nipkow@17501
  2532
apply(induct j)
nipkow@17501
  2533
apply auto
nipkow@17501
  2534
done
nipkow@17501
  2535
nipkow@24645
  2536
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]"
nipkow@13145
  2537
by (induct n) auto
wenzelm@13114
  2538
nipkow@24526
  2539
lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
nipkow@24526
  2540
apply (induct n m  arbitrary: i rule: diff_induct)
nipkow@13145
  2541
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  2542
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  2543
done
wenzelm@13114
  2544
berghofe@13883
  2545
lemma nth_take_lemma:
nipkow@24526
  2546
  "k <= length xs ==> k <= length ys ==>
berghofe@13883
  2547
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
nipkow@24526
  2548
apply (atomize, induct k arbitrary: xs ys)
paulson@14208
  2549
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  2550
txt {* Both lists must be non-empty *}
paulson@14208
  2551
apply (case_tac xs, simp)
paulson@14208
  2552
apply (case_tac ys, clarify)
nipkow@13145
  2553
 apply (simp (no_asm_use))
nipkow@13145
  2554
apply clarify
nipkow@13145
  2555
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  2556
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  2557
apply blast
nipkow@13145
  2558
done
wenzelm@13114
  2559
wenzelm@13114
  2560
lemma nth_equalityI:
wenzelm@13114
  2561
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  2562
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  2563
apply (simp_all add: take_all)
nipkow@13145
  2564
done
wenzelm@13142
  2565
haftmann@24796
  2566
lemma map_nth:
haftmann@24796
  2567
  "map (\<lambda>i. xs ! i) [0..<length xs] = xs"
haftmann@24796
  2568
  by (rule nth_equalityI, auto)
haftmann@24796
  2569
kleing@13863
  2570
(* needs nth_equalityI *)
kleing@13863
  2571
lemma list_all2_antisym:
kleing@13863
  2572
  "\<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
  2573
  \<Longrightarrow> xs = ys"
kleing@13863
  2574
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  2575
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  2576
  done
kleing@13863
  2577
wenzelm@13142
  2578
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  2579
-- {* The famous take-lemma. *}
nipkow@13145
  2580
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  2581
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  2582
done
wenzelm@13142
  2583
wenzelm@13142
  2584
nipkow@15302
  2585
lemma take_Cons':
nipkow@15302
  2586
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@15302
  2587
by (cases n) simp_all
nipkow@15302
  2588
nipkow@15302
  2589
lemma drop_Cons':
nipkow@15302
  2590
     "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@15302
  2591
by (cases n) simp_all
nipkow@15302
  2592
nipkow@15302
  2593
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@15302
  2594
by (cases n) simp_all
nipkow@15302
  2595
paulson@18622
  2596
lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard]
paulson@18622
  2597
lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard]
paulson@18622
  2598
lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard]
paulson@18622
  2599
paulson@18622
  2600
declare take_Cons_number_of [simp] 
paulson@18622
  2601
        drop_Cons_number_of [simp] 
paulson@18622
  2602
        nth_Cons_number_of [simp] 
nipkow@15302
  2603
nipkow@15302
  2604
nipkow@32415
  2605
subsubsection {* @{text upto}: interval-list on @{typ int} *}
nipkow@32415
  2606
nipkow@32415
  2607
(* FIXME make upto tail recursive? *)
nipkow@32415
  2608
nipkow@32415
  2609
function upto :: "int \<Rightarrow> int \<Rightarrow> int list" ("(1[_../_])") where
nipkow@32415
  2610
"upto i j = (if i \<le> j then i # [i+1..j] else [])"
nipkow@32415
  2611
by auto
nipkow@32415
  2612
termination
nipkow@32415
  2613
by(relation "measure(%(i::int,j). nat(j - i + 1))") auto
nipkow@32415
  2614
nipkow@32415
  2615
declare upto.simps[code, simp del]
nipkow@32415
  2616
nipkow@32415
  2617
lemmas upto_rec_number_of[simp] =
nipkow@32415
  2618
  upto.simps[of "number_of m" "number_of n", standard]
nipkow@32415
  2619
nipkow@32415
  2620
lemma upto_empty[simp]: "j < i \<Longrightarrow> [i..j] = []"
nipkow@32415
  2621
by(simp add: upto.simps)
nipkow@32415
  2622
nipkow@32415
  2623
lemma set_upto[simp]: "set[i..j] = {i..j}"
nipkow@32415
  2624
apply(induct i j rule:upto.induct)
nipkow@32415
  2625
apply(simp add: upto.simps simp_from_to)
nipkow@32415
  2626
done
nipkow@32415
  2627
nipkow@32415
  2628
nipkow@15392
  2629
subsubsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  2630
wenzelm@13142
  2631
lemma distinct_append [simp]:
nipkow@13145
  2632
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  2633
by (induct xs) auto
wenzelm@13142
  2634
nipkow@15305
  2635
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
nipkow@15305
  2636
by(induct xs) auto
nipkow@15305
  2637
wenzelm@13142
  2638
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  2639
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  2640
wenzelm@13142
  2641
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  2642
by (induct xs) auto
wenzelm@13142
  2643
nipkow@25287
  2644
lemma distinct_remdups_id: "distinct xs ==> remdups xs = xs"
nipkow@25287
  2645
by (induct xs, auto)
nipkow@25287
  2646
haftmann@26734
  2647
lemma remdups_id_iff_distinct [simp]: "remdups xs = xs \<longleftrightarrow> distinct xs"
haftmann@26734
  2648
by (metis distinct_remdups distinct_remdups_id)
nipkow@25287
  2649
nipkow@24566
  2650
lemma finite_distinct_list: "finite A \<Longrightarrow> EX xs. set xs = A & distinct xs"
paulson@24632
  2651
by (metis distinct_remdups finite_list set_remdups)
nipkow@24566
  2652
paulson@15072
  2653
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
nipkow@24349
  2654
by (induct x, auto) 
paulson@15072
  2655
paulson@15072
  2656
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
nipkow@24349
  2657
by (induct x, auto)
paulson@15072
  2658
nipkow@15245
  2659
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245
  2660
by (induct xs) auto
nipkow@15245
  2661
nipkow@15245
  2662
lemma length_remdups_eq[iff]:
nipkow@15245
  2663
  "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245
  2664
apply(induct xs)
nipkow@15245
  2665
 apply auto
nipkow@15245
  2666
apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245
  2667
 apply arith
nipkow@15245
  2668
apply(rule length_remdups_leq)
nipkow@15245
  2669
done
nipkow@15245
  2670
nipkow@18490
  2671
nipkow@18490
  2672
lemma distinct_map:
nipkow@18490
  2673
  "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
nipkow@18490
  2674
by (induct xs) auto
nipkow@18490
  2675
nipkow@18490
  2676
wenzelm@13142
  2677
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  2678
by (induct xs) auto
wenzelm@13114
  2679
nipkow@17501
  2680
lemma distinct_upt[simp]: "distinct[i..<j]"
nipkow@17501
  2681
by (induct j) auto
nipkow@17501
  2682
nipkow@32415
  2683
lemma distinct_upto[simp]: "distinct[i..j]"
nipkow@32415
  2684
apply(induct i j rule:upto.induct)
nipkow@32415
  2685
apply(subst upto.simps)
nipkow@32415
  2686
apply(simp)
nipkow@32415
  2687
done
nipkow@32415
  2688
nipkow@24526
  2689
lemma distinct_take[simp]: "distinct xs \<Longrightarrow> distinct (take i xs)"
nipkow@24526
  2690
apply(induct xs arbitrary: i)
nipkow@17501
  2691
 apply simp
nipkow@17501
  2692
apply (case_tac i)
nipkow@17501
  2693
 apply simp_all
nipkow@17501
  2694
apply(blast dest:in_set_takeD)
nipkow@17501
  2695
done
nipkow@17501
  2696
nipkow@24526
  2697
lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)"
nipkow@24526
  2698
apply(induct xs arbitrary: i)
nipkow@17501
<