src/HOL/List.thy
author wenzelm
Mon Mar 22 20:58:52 2010 +0100 (2010-03-22)
changeset 35898 c890a3835d15
parent 35828 46cfc4b8112e
child 36154 11c6106d7787
child 36198 ead2db2be11a
permissions -rw-r--r--
recovered header;
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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 Sledgehammer 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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syntax
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  -- {* list Enumeration *}
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  "_list" :: "args => 'a list"    ("[(_)]")
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translations
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  "[x, xs]" == "x#[xs]"
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  "[x]" == "x#[]"
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subsection {* Basic list processing functions *}
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primrec
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  hd :: "'a list \<Rightarrow> 'a" where
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  "hd (x # xs) = x"
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primrec
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  tl :: "'a list \<Rightarrow> 'a list" where
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    "tl [] = []"
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  | "tl (x # xs) = xs"
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primrec
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  last :: "'a list \<Rightarrow> 'a" where
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  "last (x # xs) = (if xs = [] then x else last xs)"
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primrec
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  butlast :: "'a list \<Rightarrow> 'a list" where
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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 :: "'a list \<Rightarrow> 'a set" where
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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 :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
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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) 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 :: "'a list \<Rightarrow> 'a list" where
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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:: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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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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syntax
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  -- {* Special syntax for filter *}
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  "_filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_<-_./ _])")
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translations
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  "[x<-xs . P]"== "CONST filter (%x. P) xs"
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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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primrec
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  foldl :: "('b \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b" where
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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 :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where
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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:: "'a list list \<Rightarrow> 'a list" where
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    "concat [] = []"
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  | "concat (x # xs) = x @ concat xs"
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primrec (in monoid_add)
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  listsum :: "'a list \<Rightarrow> 'a" where
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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:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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    drop_Nil: "drop n [] = []"
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  | drop_Cons: "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> 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:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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    take_Nil:"take n [] = []"
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  | take_Cons: "take n (x # xs) = (case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # take m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  nth :: "'a list => nat => 'a" (infixl "!" 100) where
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  nth_Cons: "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> 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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  list_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
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    "list_update [] i v = []"
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  | "list_update (x # xs) i v = (case i of 0 \<Rightarrow> v # xs | Suc j \<Rightarrow> x # list_update xs j v)"
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nonterminals lupdbinds lupdbind
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syntax
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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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  "_LUpdate xs (_lupdbinds b bs)" == "_LUpdate (_LUpdate xs b) bs"
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  "xs[i:=x]" == "CONST list_update xs i x"
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primrec
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  takeWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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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 :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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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 :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
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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 :: "nat \<Rightarrow> nat \<Rightarrow> nat list" ("(1[_..</_'])") where
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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 :: "'a list \<Rightarrow> bool" where
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    "distinct [] \<longleftrightarrow> True"
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  | "distinct (x # xs) \<longleftrightarrow> x \<notin> set xs \<and> distinct xs"
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primrec
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  remdups :: "'a list \<Rightarrow> 'a list" where
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    "remdups [] = []"
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  | "remdups (x # xs) = (if x \<in> set xs then remdups xs else x # remdups xs)"
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definition
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  insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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  "insert x xs = (if x \<in> set xs then xs else x # xs)"
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hide (open) const insert hide (open) fact insert_def
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primrec
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  remove1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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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 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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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 :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
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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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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 \<Rightarrow> nat" where
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  "length \<equiv> size"
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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 :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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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 "List.insert 2 [0::nat,1,2] = [0,1,2]" by (simp add: List.insert_def)}\\
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@{lemma "List.insert 3 [0::nat,1,2] = [3,0,1,2]" by (simp add: List.insert_def)}\\
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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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definition sort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b list \<Rightarrow> 'b list" where
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"sort_key f xs = foldr (insort_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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definition insort_insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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  "insort_insert x xs = (if x \<in> set xs then xs else insort x xs)"
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end
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subsubsection {* List comprehension *}
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nipkow@24349
   301
text{* Input syntax for Haskell-like list comprehension notation.
nipkow@24349
   302
Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
nipkow@24349
   303
the list of all pairs of distinct elements from @{text xs} and @{text ys}.
nipkow@24349
   304
The syntax is as in Haskell, except that @{text"|"} becomes a dot
nipkow@24349
   305
(like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
nipkow@24349
   306
\verb![e| x <- xs, ...]!.
nipkow@24349
   307
nipkow@24349
   308
The qualifiers after the dot are
nipkow@24349
   309
\begin{description}
nipkow@24349
   310
\item[generators] @{text"p \<leftarrow> xs"},
nipkow@24476
   311
 where @{text p} is a pattern and @{text xs} an expression of list type, or
nipkow@24476
   312
\item[guards] @{text"b"}, where @{text b} is a boolean expression.
nipkow@24476
   313
%\item[local bindings] @ {text"let x = e"}.
nipkow@24349
   314
\end{description}
nipkow@23240
   315
nipkow@24476
   316
Just like in Haskell, list comprehension is just a shorthand. To avoid
nipkow@24476
   317
misunderstandings, the translation into desugared form is not reversed
nipkow@24476
   318
upon output. Note that the translation of @{text"[e. x \<leftarrow> xs]"} is
nipkow@24476
   319
optmized to @{term"map (%x. e) xs"}.
nipkow@23240
   320
nipkow@24349
   321
It is easy to write short list comprehensions which stand for complex
nipkow@24349
   322
expressions. During proofs, they may become unreadable (and
nipkow@24349
   323
mangled). In such cases it can be advisable to introduce separate
nipkow@24349
   324
definitions for the list comprehensions in question.  *}
nipkow@24349
   325
nipkow@23209
   326
(*
nipkow@23240
   327
Proper theorem proving support would be nice. For example, if
nipkow@23192
   328
@{text"set[f x y. x \<leftarrow> xs, y \<leftarrow> ys, P x y]"}
nipkow@23192
   329
produced something like
nipkow@23209
   330
@{term"{z. EX x: set xs. EX y:set ys. P x y \<and> z = f x y}"}.
nipkow@23209
   331
*)
nipkow@23209
   332
nipkow@23240
   333
nonterminals lc_qual lc_quals
nipkow@23192
   334
nipkow@23192
   335
syntax
nipkow@23240
   336
"_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
nipkow@24349
   337
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ <- _")
nipkow@23240
   338
"_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
nipkow@24476
   339
(*"_lc_let" :: "letbinds => lc_qual"  ("let _")*)
nipkow@23240
   340
"_lc_end" :: "lc_quals" ("]")
nipkow@23240
   341
"_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals" (", __")
nipkow@24349
   342
"_lc_abs" :: "'a => 'b list => 'b list"
nipkow@23192
   343
nipkow@24476
   344
(* These are easier than ML code but cannot express the optimized
nipkow@24476
   345
   translation of [e. p<-xs]
nipkow@23192
   346
translations
nipkow@24349
   347
"[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
nipkow@23240
   348
"_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
nipkow@24349
   349
 => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
nipkow@23240
   350
"[e. P]" => "if P then [e] else []"
nipkow@23240
   351
"_listcompr e (_lc_test P) (_lc_quals Q Qs)"
nipkow@23240
   352
 => "if P then (_listcompr e Q Qs) else []"
nipkow@24349
   353
"_listcompr e (_lc_let b) (_lc_quals Q Qs)"
nipkow@24349
   354
 => "_Let b (_listcompr e Q Qs)"
nipkow@24476
   355
*)
nipkow@23240
   356
nipkow@23279
   357
syntax (xsymbols)
nipkow@24349
   358
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
nipkow@23279
   359
syntax (HTML output)
nipkow@24349
   360
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
nipkow@24349
   361
nipkow@24349
   362
parse_translation (advanced) {*
nipkow@24349
   363
let
wenzelm@35256
   364
  val NilC = Syntax.const @{const_syntax Nil};
wenzelm@35256
   365
  val ConsC = Syntax.const @{const_syntax Cons};
wenzelm@35256
   366
  val mapC = Syntax.const @{const_syntax map};
wenzelm@35256
   367
  val concatC = Syntax.const @{const_syntax concat};
wenzelm@35256
   368
  val IfC = Syntax.const @{const_syntax If};
wenzelm@35115
   369
nipkow@24476
   370
  fun singl x = ConsC $ x $ NilC;
nipkow@24476
   371
wenzelm@35115
   372
  fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
nipkow@24349
   373
    let
wenzelm@29281
   374
      val x = Free (Name.variant (fold Term.add_free_names [p, e] []) "x", dummyT);
nipkow@24476
   375
      val e = if opti then singl e else e;
wenzelm@35115
   376
      val case1 = Syntax.const @{syntax_const "_case1"} $ p $ e;
wenzelm@35256
   377
      val case2 =
wenzelm@35256
   378
        Syntax.const @{syntax_const "_case1"} $
wenzelm@35256
   379
          Syntax.const @{const_syntax dummy_pattern} $ NilC;
wenzelm@35115
   380
      val cs = Syntax.const @{syntax_const "_case2"} $ case1 $ case2;
wenzelm@35115
   381
      val ft = Datatype_Case.case_tr false Datatype.info_of_constr ctxt [x, cs];
nipkow@24349
   382
    in lambda x ft end;
nipkow@24349
   383
wenzelm@35256
   384
  fun abs_tr ctxt (p as Free (s, T)) e opti =
wenzelm@35115
   385
        let
wenzelm@35115
   386
          val thy = ProofContext.theory_of ctxt;
wenzelm@35115
   387
          val s' = Sign.intern_const thy s;
wenzelm@35115
   388
        in
wenzelm@35115
   389
          if Sign.declared_const thy s'
wenzelm@35115
   390
          then (pat_tr ctxt p e opti, false)
wenzelm@35115
   391
          else (lambda p e, true)
nipkow@24349
   392
        end
nipkow@24476
   393
    | abs_tr ctxt p e opti = (pat_tr ctxt p e opti, false);
nipkow@24476
   394
wenzelm@35115
   395
  fun lc_tr ctxt [e, Const (@{syntax_const "_lc_test"}, _) $ b, qs] =
wenzelm@35115
   396
        let
wenzelm@35115
   397
          val res =
wenzelm@35115
   398
            (case qs of
wenzelm@35115
   399
              Const (@{syntax_const "_lc_end"}, _) => singl e
wenzelm@35115
   400
            | Const (@{syntax_const "_lc_quals"}, _) $ q $ qs => lc_tr ctxt [e, q, qs]);
nipkow@24476
   401
        in IfC $ b $ res $ NilC end
wenzelm@35115
   402
    | lc_tr ctxt
wenzelm@35115
   403
          [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
wenzelm@35115
   404
            Const(@{syntax_const "_lc_end"}, _)] =
nipkow@24476
   405
        (case abs_tr ctxt p e true of
wenzelm@35115
   406
          (f, true) => mapC $ f $ es
wenzelm@35115
   407
        | (f, false) => concatC $ (mapC $ f $ es))
wenzelm@35115
   408
    | lc_tr ctxt
wenzelm@35115
   409
          [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
wenzelm@35115
   410
            Const (@{syntax_const "_lc_quals"}, _) $ q $ qs] =
wenzelm@35115
   411
        let val e' = lc_tr ctxt [e, q, qs];
wenzelm@35115
   412
        in concatC $ (mapC $ (fst (abs_tr ctxt p e' false)) $ es) end;
wenzelm@35115
   413
wenzelm@35115
   414
in [(@{syntax_const "_listcompr"}, lc_tr)] end
nipkow@24349
   415
*}
nipkow@23279
   416
nipkow@23240
   417
term "[(x,y,z). b]"
nipkow@24476
   418
term "[(x,y,z). x\<leftarrow>xs]"
nipkow@24476
   419
term "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]"
nipkow@24476
   420
term "[(x,y,z). x<a, x>b]"
nipkow@24476
   421
term "[(x,y,z). x\<leftarrow>xs, x>b]"
nipkow@24476
   422
term "[(x,y,z). x<a, x\<leftarrow>xs]"
nipkow@24349
   423
term "[(x,y). Cons True x \<leftarrow> xs]"
nipkow@24349
   424
term "[(x,y,z). Cons x [] \<leftarrow> xs]"
nipkow@23240
   425
term "[(x,y,z). x<a, x>b, x=d]"
nipkow@23240
   426
term "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
nipkow@23240
   427
term "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
nipkow@23240
   428
term "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
nipkow@23240
   429
term "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
nipkow@23240
   430
term "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
nipkow@23240
   431
term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
nipkow@23240
   432
term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
wenzelm@35115
   433
(*
nipkow@24349
   434
term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
nipkow@23192
   435
*)
nipkow@23192
   436
wenzelm@35115
   437
haftmann@21061
   438
subsubsection {* @{const Nil} and @{const Cons} *}
haftmann@21061
   439
haftmann@21061
   440
lemma not_Cons_self [simp]:
haftmann@21061
   441
  "xs \<noteq> x # xs"
nipkow@13145
   442
by (induct xs) auto
wenzelm@13114
   443
wenzelm@13142
   444
lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
wenzelm@13114
   445
wenzelm@13142
   446
lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
nipkow@13145
   447
by (induct xs) auto
wenzelm@13114
   448
wenzelm@13142
   449
lemma length_induct:
haftmann@21061
   450
  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
nipkow@17589
   451
by (rule measure_induct [of length]) iprover
wenzelm@13114
   452
wenzelm@13114
   453
haftmann@21061
   454
subsubsection {* @{const length} *}
wenzelm@13114
   455
wenzelm@13142
   456
text {*
haftmann@21061
   457
  Needs to come before @{text "@"} because of theorem @{text
haftmann@21061
   458
  append_eq_append_conv}.
wenzelm@13142
   459
*}
wenzelm@13114
   460
wenzelm@13142
   461
lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
nipkow@13145
   462
by (induct xs) auto
wenzelm@13114
   463
wenzelm@13142
   464
lemma length_map [simp]: "length (map f xs) = length xs"
nipkow@13145
   465
by (induct xs) auto
wenzelm@13114
   466
wenzelm@13142
   467
lemma length_rev [simp]: "length (rev xs) = length xs"
nipkow@13145
   468
by (induct xs) auto
wenzelm@13114
   469
wenzelm@13142
   470
lemma length_tl [simp]: "length (tl xs) = length xs - 1"
nipkow@13145
   471
by (cases xs) auto
wenzelm@13114
   472
wenzelm@13142
   473
lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
nipkow@13145
   474
by (induct xs) auto
wenzelm@13114
   475
wenzelm@13142
   476
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
nipkow@13145
   477
by (induct xs) auto
wenzelm@13114
   478
nipkow@23479
   479
lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
nipkow@23479
   480
by auto
nipkow@23479
   481
wenzelm@13114
   482
lemma length_Suc_conv:
nipkow@13145
   483
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
nipkow@13145
   484
by (induct xs) auto
wenzelm@13142
   485
nipkow@14025
   486
lemma Suc_length_conv:
nipkow@14025
   487
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
paulson@14208
   488
apply (induct xs, simp, simp)
nipkow@14025
   489
apply blast
nipkow@14025
   490
done
nipkow@14025
   491
wenzelm@25221
   492
lemma impossible_Cons: "length xs <= length ys ==> xs = x # ys = False"
wenzelm@25221
   493
  by (induct xs) auto
wenzelm@25221
   494
haftmann@26442
   495
lemma list_induct2 [consumes 1, case_names Nil Cons]:
haftmann@26442
   496
  "length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
haftmann@26442
   497
   (\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
haftmann@26442
   498
   \<Longrightarrow> P xs ys"
haftmann@26442
   499
proof (induct xs arbitrary: ys)
haftmann@26442
   500
  case Nil then show ?case by simp
haftmann@26442
   501
next
haftmann@26442
   502
  case (Cons x xs ys) then show ?case by (cases ys) simp_all
haftmann@26442
   503
qed
haftmann@26442
   504
haftmann@26442
   505
lemma list_induct3 [consumes 2, case_names Nil Cons]:
haftmann@26442
   506
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
haftmann@26442
   507
   (\<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
   508
   \<Longrightarrow> P xs ys zs"
haftmann@26442
   509
proof (induct xs arbitrary: ys zs)
haftmann@26442
   510
  case Nil then show ?case by simp
haftmann@26442
   511
next
haftmann@26442
   512
  case (Cons x xs ys zs) then show ?case by (cases ys, simp_all)
haftmann@26442
   513
    (cases zs, simp_all)
haftmann@26442
   514
qed
wenzelm@13114
   515
krauss@22493
   516
lemma list_induct2': 
krauss@22493
   517
  "\<lbrakk> P [] [];
krauss@22493
   518
  \<And>x xs. P (x#xs) [];
krauss@22493
   519
  \<And>y ys. P [] (y#ys);
krauss@22493
   520
   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
krauss@22493
   521
 \<Longrightarrow> P xs ys"
krauss@22493
   522
by (induct xs arbitrary: ys) (case_tac x, auto)+
krauss@22493
   523
nipkow@22143
   524
lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
nipkow@24349
   525
by (rule Eq_FalseI) auto
wenzelm@24037
   526
wenzelm@24037
   527
simproc_setup list_neq ("(xs::'a list) = ys") = {*
nipkow@22143
   528
(*
nipkow@22143
   529
Reduces xs=ys to False if xs and ys cannot be of the same length.
nipkow@22143
   530
This is the case if the atomic sublists of one are a submultiset
nipkow@22143
   531
of those of the other list and there are fewer Cons's in one than the other.
nipkow@22143
   532
*)
wenzelm@24037
   533
wenzelm@24037
   534
let
nipkow@22143
   535
huffman@29856
   536
fun len (Const(@{const_name Nil},_)) acc = acc
huffman@29856
   537
  | len (Const(@{const_name Cons},_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
huffman@29856
   538
  | len (Const(@{const_name append},_) $ xs $ ys) acc = len xs (len ys acc)
huffman@29856
   539
  | len (Const(@{const_name rev},_) $ xs) acc = len xs acc
huffman@29856
   540
  | len (Const(@{const_name map},_) $ _ $ xs) acc = len xs acc
nipkow@22143
   541
  | len t (ts,n) = (t::ts,n);
nipkow@22143
   542
wenzelm@24037
   543
fun list_neq _ ss ct =
nipkow@22143
   544
  let
wenzelm@24037
   545
    val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
nipkow@22143
   546
    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
nipkow@22143
   547
    fun prove_neq() =
nipkow@22143
   548
      let
nipkow@22143
   549
        val Type(_,listT::_) = eqT;
haftmann@22994
   550
        val size = HOLogic.size_const listT;
nipkow@22143
   551
        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
nipkow@22143
   552
        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
nipkow@22143
   553
        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
haftmann@22633
   554
          (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
haftmann@22633
   555
      in SOME (thm RS @{thm neq_if_length_neq}) end
nipkow@22143
   556
  in
wenzelm@23214
   557
    if m < n andalso submultiset (op aconv) (ls,rs) orelse
wenzelm@23214
   558
       n < m andalso submultiset (op aconv) (rs,ls)
nipkow@22143
   559
    then prove_neq() else NONE
nipkow@22143
   560
  end;
wenzelm@24037
   561
in list_neq end;
nipkow@22143
   562
*}
nipkow@22143
   563
nipkow@22143
   564
nipkow@15392
   565
subsubsection {* @{text "@"} -- append *}
wenzelm@13114
   566
wenzelm@13142
   567
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
nipkow@13145
   568
by (induct xs) auto
wenzelm@13114
   569
wenzelm@13142
   570
lemma append_Nil2 [simp]: "xs @ [] = xs"
nipkow@13145
   571
by (induct xs) auto
nipkow@3507
   572
wenzelm@13142
   573
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
nipkow@13145
   574
by (induct xs) auto
wenzelm@13114
   575
wenzelm@13142
   576
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
nipkow@13145
   577
by (induct xs) auto
wenzelm@13114
   578
wenzelm@13142
   579
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
nipkow@13145
   580
by (induct xs) auto
wenzelm@13114
   581
wenzelm@13142
   582
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
nipkow@13145
   583
by (induct xs) auto
wenzelm@13114
   584
blanchet@35828
   585
lemma append_eq_append_conv [simp, no_atp]:
nipkow@24526
   586
 "length xs = length ys \<or> length us = length vs
berghofe@13883
   587
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
nipkow@24526
   588
apply (induct xs arbitrary: ys)
paulson@14208
   589
 apply (case_tac ys, simp, force)
paulson@14208
   590
apply (case_tac ys, force, simp)
nipkow@13145
   591
done
wenzelm@13142
   592
nipkow@24526
   593
lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
nipkow@24526
   594
  (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
nipkow@24526
   595
apply (induct xs arbitrary: ys zs ts)
nipkow@14495
   596
 apply fastsimp
nipkow@14495
   597
apply(case_tac zs)
nipkow@14495
   598
 apply simp
nipkow@14495
   599
apply fastsimp
nipkow@14495
   600
done
nipkow@14495
   601
berghofe@34910
   602
lemma same_append_eq [iff, induct_simp]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145
   603
by simp
wenzelm@13142
   604
wenzelm@13142
   605
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145
   606
by simp
wenzelm@13114
   607
berghofe@34910
   608
lemma append_same_eq [iff, induct_simp]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   609
by simp
wenzelm@13114
   610
wenzelm@13142
   611
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   612
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   613
wenzelm@13142
   614
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   615
using append_same_eq [of "[]"] by auto
wenzelm@13114
   616
blanchet@35828
   617
lemma hd_Cons_tl [simp,no_atp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   618
by (induct xs) auto
wenzelm@13114
   619
wenzelm@13142
   620
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   621
by (induct xs) auto
wenzelm@13114
   622
wenzelm@13142
   623
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   624
by (simp add: hd_append split: list.split)
wenzelm@13114
   625
wenzelm@13142
   626
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   627
by (simp split: list.split)
wenzelm@13114
   628
wenzelm@13142
   629
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   630
by (simp add: tl_append split: list.split)
wenzelm@13114
   631
wenzelm@13114
   632
nipkow@14300
   633
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   634
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   635
by(cases ys) auto
nipkow@14300
   636
nipkow@15281
   637
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   638
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   639
by(cases ys) auto
nipkow@15281
   640
nipkow@14300
   641
wenzelm@13142
   642
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   643
wenzelm@13114
   644
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   645
by simp
wenzelm@13114
   646
wenzelm@13142
   647
lemma Cons_eq_appendI:
nipkow@13145
   648
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   649
by (drule sym) simp
wenzelm@13114
   650
wenzelm@13142
   651
lemma append_eq_appendI:
nipkow@13145
   652
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   653
by (drule sym) simp
wenzelm@13114
   654
wenzelm@13114
   655
wenzelm@13142
   656
text {*
nipkow@13145
   657
Simplification procedure for all list equalities.
nipkow@13145
   658
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   659
- both lists end in a singleton list,
nipkow@13145
   660
- or both lists end in the same list.
wenzelm@13142
   661
*}
wenzelm@13142
   662
wenzelm@26480
   663
ML {*
nipkow@3507
   664
local
nipkow@3507
   665
huffman@29856
   666
fun last (cons as Const(@{const_name Cons},_) $ _ $ xs) =
huffman@29856
   667
  (case xs of Const(@{const_name Nil},_) => cons | _ => last xs)
huffman@29856
   668
  | last (Const(@{const_name append},_) $ _ $ ys) = last ys
wenzelm@13462
   669
  | last t = t;
wenzelm@13114
   670
huffman@29856
   671
fun list1 (Const(@{const_name Cons},_) $ _ $ Const(@{const_name Nil},_)) = true
wenzelm@13462
   672
  | list1 _ = false;
wenzelm@13114
   673
huffman@29856
   674
fun butlast ((cons as Const(@{const_name Cons},_) $ x) $ xs) =
huffman@29856
   675
  (case xs of Const(@{const_name Nil},_) => xs | _ => cons $ butlast xs)
huffman@29856
   676
  | butlast ((app as Const(@{const_name append},_) $ xs) $ ys) = app $ butlast ys
huffman@29856
   677
  | butlast xs = Const(@{const_name Nil},fastype_of xs);
wenzelm@13114
   678
haftmann@22633
   679
val rearr_ss = HOL_basic_ss addsimps [@{thm append_assoc},
haftmann@22633
   680
  @{thm append_Nil}, @{thm append_Cons}];
wenzelm@16973
   681
wenzelm@20044
   682
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   683
  let
wenzelm@13462
   684
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   685
    fun rearr conv =
wenzelm@13462
   686
      let
wenzelm@13462
   687
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   688
        val Type(_,listT::_) = eqT
wenzelm@13462
   689
        val appT = [listT,listT] ---> listT
huffman@29856
   690
        val app = Const(@{const_name append},appT)
wenzelm@13462
   691
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   692
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@20044
   693
        val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
wenzelm@17877
   694
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
skalberg@15531
   695
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   696
wenzelm@13462
   697
  in
haftmann@22633
   698
    if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
haftmann@22633
   699
    else if lastl aconv lastr then rearr @{thm append_same_eq}
skalberg@15531
   700
    else NONE
wenzelm@13462
   701
  end;
wenzelm@13462
   702
wenzelm@13114
   703
in
wenzelm@13462
   704
wenzelm@13462
   705
val list_eq_simproc =
wenzelm@32010
   706
  Simplifier.simproc @{theory} "list_eq" ["(xs::'a list) = ys"] (K list_eq);
wenzelm@13462
   707
wenzelm@13114
   708
end;
wenzelm@13114
   709
wenzelm@13114
   710
Addsimprocs [list_eq_simproc];
wenzelm@13114
   711
*}
wenzelm@13114
   712
wenzelm@13114
   713
nipkow@15392
   714
subsubsection {* @{text map} *}
wenzelm@13114
   715
wenzelm@13142
   716
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   717
by (induct xs) simp_all
wenzelm@13114
   718
wenzelm@13142
   719
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   720
by (rule ext, induct_tac xs) auto
wenzelm@13114
   721
wenzelm@13142
   722
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   723
by (induct xs) auto
wenzelm@13114
   724
hoelzl@33639
   725
lemma map_map [simp]: "map f (map g xs) = map (f \<circ> g) xs"
hoelzl@33639
   726
by (induct xs) auto
hoelzl@33639
   727
nipkow@35208
   728
lemma map_comp_map[simp]: "((map f) o (map g)) = map(f o g)"
nipkow@35208
   729
apply(rule ext)
nipkow@35208
   730
apply(simp)
nipkow@35208
   731
done
nipkow@35208
   732
wenzelm@13142
   733
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   734
by (induct xs) auto
wenzelm@13114
   735
nipkow@13737
   736
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   737
by (induct xs) auto
nipkow@13737
   738
krauss@19770
   739
lemma map_cong [fundef_cong, recdef_cong]:
nipkow@13145
   740
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   741
-- {* a congruence rule for @{text map} *}
nipkow@13737
   742
by simp
wenzelm@13114
   743
wenzelm@13142
   744
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   745
by (cases xs) auto
wenzelm@13114
   746
wenzelm@13142
   747
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   748
by (cases xs) auto
wenzelm@13114
   749
paulson@18447
   750
lemma map_eq_Cons_conv:
nipkow@14025
   751
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   752
by (cases xs) auto
wenzelm@13114
   753
paulson@18447
   754
lemma Cons_eq_map_conv:
nipkow@14025
   755
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   756
by (cases ys) auto
nipkow@14025
   757
paulson@18447
   758
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
paulson@18447
   759
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
paulson@18447
   760
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
paulson@18447
   761
nipkow@14111
   762
lemma ex_map_conv:
nipkow@14111
   763
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
paulson@18447
   764
by(induct ys, auto simp add: Cons_eq_map_conv)
nipkow@14111
   765
nipkow@15110
   766
lemma map_eq_imp_length_eq:
paulson@35510
   767
  assumes "map f xs = map g ys"
haftmann@26734
   768
  shows "length xs = length ys"
haftmann@26734
   769
using assms proof (induct ys arbitrary: xs)
haftmann@26734
   770
  case Nil then show ?case by simp
haftmann@26734
   771
next
haftmann@26734
   772
  case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto
paulson@35510
   773
  from Cons xs have "map f zs = map g ys" by simp
haftmann@26734
   774
  moreover with Cons have "length zs = length ys" by blast
haftmann@26734
   775
  with xs show ?case by simp
haftmann@26734
   776
qed
haftmann@26734
   777
  
nipkow@15110
   778
lemma map_inj_on:
nipkow@15110
   779
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   780
  ==> xs = ys"
nipkow@15110
   781
apply(frule map_eq_imp_length_eq)
nipkow@15110
   782
apply(rotate_tac -1)
nipkow@15110
   783
apply(induct rule:list_induct2)
nipkow@15110
   784
 apply simp
nipkow@15110
   785
apply(simp)
nipkow@15110
   786
apply (blast intro:sym)
nipkow@15110
   787
done
nipkow@15110
   788
nipkow@15110
   789
lemma inj_on_map_eq_map:
nipkow@15110
   790
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   791
by(blast dest:map_inj_on)
nipkow@15110
   792
wenzelm@13114
   793
lemma map_injective:
nipkow@24526
   794
 "map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@24526
   795
by (induct ys arbitrary: xs) (auto dest!:injD)
wenzelm@13114
   796
nipkow@14339
   797
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   798
by(blast dest:map_injective)
nipkow@14339
   799
wenzelm@13114
   800
lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@17589
   801
by (iprover dest: map_injective injD intro: inj_onI)
wenzelm@13114
   802
wenzelm@13114
   803
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   804
apply (unfold inj_on_def, clarify)
nipkow@13145
   805
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   806
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   807
apply blast
nipkow@13145
   808
done
wenzelm@13114
   809
nipkow@14339
   810
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   811
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   812
nipkow@15303
   813
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
   814
apply(rule inj_onI)
nipkow@15303
   815
apply(erule map_inj_on)
nipkow@15303
   816
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
   817
done
nipkow@15303
   818
kleing@14343
   819
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   820
by (induct xs, auto)
wenzelm@13114
   821
nipkow@14402
   822
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   823
by (induct xs) auto
nipkow@14402
   824
nipkow@15110
   825
lemma map_fst_zip[simp]:
nipkow@15110
   826
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   827
by (induct rule:list_induct2, simp_all)
nipkow@15110
   828
nipkow@15110
   829
lemma map_snd_zip[simp]:
nipkow@15110
   830
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   831
by (induct rule:list_induct2, simp_all)
nipkow@15110
   832
nipkow@15110
   833
nipkow@15392
   834
subsubsection {* @{text rev} *}
wenzelm@13114
   835
wenzelm@13142
   836
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   837
by (induct xs) auto
wenzelm@13114
   838
wenzelm@13142
   839
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   840
by (induct xs) auto
wenzelm@13114
   841
kleing@15870
   842
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870
   843
by auto
kleing@15870
   844
wenzelm@13142
   845
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   846
by (induct xs) auto
wenzelm@13114
   847
wenzelm@13142
   848
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   849
by (induct xs) auto
wenzelm@13114
   850
kleing@15870
   851
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870
   852
by (cases xs) auto
kleing@15870
   853
kleing@15870
   854
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870
   855
by (cases xs) auto
kleing@15870
   856
haftmann@21061
   857
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
haftmann@21061
   858
apply (induct xs arbitrary: ys, force)
paulson@14208
   859
apply (case_tac ys, simp, force)
nipkow@13145
   860
done
wenzelm@13114
   861
nipkow@15439
   862
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
   863
by(simp add:inj_on_def)
nipkow@15439
   864
wenzelm@13366
   865
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   866
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489
   867
apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145
   868
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   869
done
wenzelm@13114
   870
wenzelm@13366
   871
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   872
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   873
by (induct xs rule: rev_induct) auto
wenzelm@13114
   874
wenzelm@13366
   875
lemmas rev_cases = rev_exhaust
wenzelm@13366
   876
nipkow@18423
   877
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
nipkow@18423
   878
by(rule rev_cases[of xs]) auto
nipkow@18423
   879
wenzelm@13114
   880
nipkow@15392
   881
subsubsection {* @{text set} *}
wenzelm@13114
   882
wenzelm@13142
   883
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   884
by (induct xs) auto
wenzelm@13114
   885
wenzelm@13142
   886
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   887
by (induct xs) auto
wenzelm@13114
   888
nipkow@17830
   889
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
nipkow@17830
   890
by(cases xs) auto
oheimb@14099
   891
wenzelm@13142
   892
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   893
by auto
wenzelm@13114
   894
oheimb@14099
   895
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   896
by auto
oheimb@14099
   897
wenzelm@13142
   898
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   899
by (induct xs) auto
wenzelm@13114
   900
nipkow@15245
   901
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   902
by(induct xs) auto
nipkow@15245
   903
wenzelm@13142
   904
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   905
by (induct xs) auto
wenzelm@13114
   906
wenzelm@13142
   907
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   908
by (induct xs) auto
wenzelm@13114
   909
wenzelm@13142
   910
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   911
by (induct xs) auto
wenzelm@13114
   912
nipkow@32417
   913
lemma set_upt [simp]: "set[i..<j] = {i..<j}"
nipkow@32417
   914
by (induct j) (simp_all add: atLeastLessThanSuc)
wenzelm@13114
   915
wenzelm@13142
   916
wenzelm@25221
   917
lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs"
nipkow@18049
   918
proof (induct xs)
nipkow@26073
   919
  case Nil thus ?case by simp
nipkow@26073
   920
next
nipkow@26073
   921
  case Cons thus ?case by (auto intro: Cons_eq_appendI)
nipkow@26073
   922
qed
nipkow@26073
   923
haftmann@26734
   924
lemma in_set_conv_decomp: "x \<in> set xs \<longleftrightarrow> (\<exists>ys zs. xs = ys @ x # zs)"
haftmann@26734
   925
  by (auto elim: split_list)
nipkow@26073
   926
nipkow@26073
   927
lemma split_list_first: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@26073
   928
proof (induct xs)
nipkow@26073
   929
  case Nil thus ?case by simp
nipkow@18049
   930
next
nipkow@18049
   931
  case (Cons a xs)
nipkow@18049
   932
  show ?case
nipkow@18049
   933
  proof cases
wenzelm@25221
   934
    assume "x = a" thus ?case using Cons by fastsimp
nipkow@18049
   935
  next
nipkow@26073
   936
    assume "x \<noteq> a" thus ?case using Cons by(fastsimp intro!: Cons_eq_appendI)
nipkow@26073
   937
  qed
nipkow@26073
   938
qed
nipkow@26073
   939
nipkow@26073
   940
lemma in_set_conv_decomp_first:
nipkow@26073
   941
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
haftmann@26734
   942
  by (auto dest!: split_list_first)
nipkow@26073
   943
nipkow@26073
   944
lemma split_list_last: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
nipkow@26073
   945
proof (induct xs rule:rev_induct)
nipkow@26073
   946
  case Nil thus ?case by simp
nipkow@26073
   947
next
nipkow@26073
   948
  case (snoc a xs)
nipkow@26073
   949
  show ?case
nipkow@26073
   950
  proof cases
nipkow@26073
   951
    assume "x = a" thus ?case using snoc by simp (metis ex_in_conv set_empty2)
nipkow@26073
   952
  next
nipkow@26073
   953
    assume "x \<noteq> a" thus ?case using snoc by fastsimp
nipkow@18049
   954
  qed
nipkow@18049
   955
qed
nipkow@18049
   956
nipkow@26073
   957
lemma in_set_conv_decomp_last:
nipkow@26073
   958
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
haftmann@26734
   959
  by (auto dest!: split_list_last)
nipkow@26073
   960
nipkow@26073
   961
lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs & P x"
nipkow@26073
   962
proof (induct xs)
nipkow@26073
   963
  case Nil thus ?case by simp
nipkow@26073
   964
next
nipkow@26073
   965
  case Cons thus ?case
nipkow@26073
   966
    by(simp add:Bex_def)(metis append_Cons append.simps(1))
nipkow@26073
   967
qed
nipkow@26073
   968
nipkow@26073
   969
lemma split_list_propE:
haftmann@26734
   970
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
   971
  obtains ys x zs where "xs = ys @ x # zs" and "P x"
haftmann@26734
   972
using split_list_prop [OF assms] by blast
nipkow@26073
   973
nipkow@26073
   974
lemma split_list_first_prop:
nipkow@26073
   975
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
nipkow@26073
   976
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)"
haftmann@26734
   977
proof (induct xs)
nipkow@26073
   978
  case Nil thus ?case by simp
nipkow@26073
   979
next
nipkow@26073
   980
  case (Cons x xs)
nipkow@26073
   981
  show ?case
nipkow@26073
   982
  proof cases
nipkow@26073
   983
    assume "P x"
haftmann@26734
   984
    thus ?thesis by simp
haftmann@26734
   985
      (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
nipkow@26073
   986
  next
nipkow@26073
   987
    assume "\<not> P x"
nipkow@26073
   988
    hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
nipkow@26073
   989
    thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
nipkow@26073
   990
  qed
nipkow@26073
   991
qed
nipkow@26073
   992
nipkow@26073
   993
lemma split_list_first_propE:
haftmann@26734
   994
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
   995
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y"
haftmann@26734
   996
using split_list_first_prop [OF assms] by blast
nipkow@26073
   997
nipkow@26073
   998
lemma split_list_first_prop_iff:
nipkow@26073
   999
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
nipkow@26073
  1000
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))"
haftmann@26734
  1001
by (rule, erule split_list_first_prop) auto
nipkow@26073
  1002
nipkow@26073
  1003
lemma split_list_last_prop:
nipkow@26073
  1004
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
nipkow@26073
  1005
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)"
nipkow@26073
  1006
proof(induct xs rule:rev_induct)
nipkow@26073
  1007
  case Nil thus ?case by simp
nipkow@26073
  1008
next
nipkow@26073
  1009
  case (snoc x xs)
nipkow@26073
  1010
  show ?case
nipkow@26073
  1011
  proof cases
nipkow@26073
  1012
    assume "P x" thus ?thesis by (metis emptyE set_empty)
nipkow@26073
  1013
  next
nipkow@26073
  1014
    assume "\<not> P x"
nipkow@26073
  1015
    hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
nipkow@26073
  1016
    thus ?thesis using `\<not> P x` snoc(1) by fastsimp
nipkow@26073
  1017
  qed
nipkow@26073
  1018
qed
nipkow@26073
  1019
nipkow@26073
  1020
lemma split_list_last_propE:
haftmann@26734
  1021
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
  1022
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z"
haftmann@26734
  1023
using split_list_last_prop [OF assms] by blast
nipkow@26073
  1024
nipkow@26073
  1025
lemma split_list_last_prop_iff:
nipkow@26073
  1026
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
nipkow@26073
  1027
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
haftmann@26734
  1028
by (metis split_list_last_prop [where P=P] in_set_conv_decomp)
nipkow@26073
  1029
nipkow@26073
  1030
lemma finite_list: "finite A ==> EX xs. set xs = A"
haftmann@26734
  1031
  by (erule finite_induct)
haftmann@26734
  1032
    (auto simp add: set.simps(2) [symmetric] simp del: set.simps(2))
paulson@13508
  1033
kleing@14388
  1034
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
  1035
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
  1036
haftmann@26442
  1037
lemma set_minus_filter_out:
haftmann@26442
  1038
  "set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)"
haftmann@26442
  1039
  by (induct xs) auto
paulson@15168
  1040
wenzelm@35115
  1041
nipkow@15392
  1042
subsubsection {* @{text filter} *}
wenzelm@13114
  1043
wenzelm@13142
  1044
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
  1045
by (induct xs) auto
wenzelm@13114
  1046
nipkow@15305
  1047
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
  1048
by (induct xs) simp_all
nipkow@15305
  1049
wenzelm@13142
  1050
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
  1051
by (induct xs) auto
wenzelm@13114
  1052
nipkow@16998
  1053
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998
  1054
by (induct xs) (auto simp add: le_SucI)
nipkow@16998
  1055
nipkow@18423
  1056
lemma sum_length_filter_compl:
nipkow@18423
  1057
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
nipkow@18423
  1058
by(induct xs) simp_all
nipkow@18423
  1059
wenzelm@13142
  1060
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
  1061
by (induct xs) auto
wenzelm@13114
  1062
wenzelm@13142
  1063
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
  1064
by (induct xs) auto
wenzelm@13114
  1065
nipkow@16998
  1066
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@24349
  1067
by (induct xs) simp_all
nipkow@16998
  1068
nipkow@16998
  1069
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998
  1070
apply (induct xs)
nipkow@16998
  1071
 apply auto
nipkow@16998
  1072
apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998
  1073
apply simp
nipkow@16998
  1074
done
wenzelm@13114
  1075
nipkow@16965
  1076
lemma filter_map:
nipkow@16965
  1077
  "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965
  1078
by (induct xs) simp_all
nipkow@16965
  1079
nipkow@16965
  1080
lemma length_filter_map[simp]:
nipkow@16965
  1081
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965
  1082
by (simp add:filter_map)
nipkow@16965
  1083
wenzelm@13142
  1084
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
  1085
by auto
wenzelm@13114
  1086
nipkow@15246
  1087
lemma length_filter_less:
nipkow@15246
  1088
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
  1089
proof (induct xs)
nipkow@15246
  1090
  case Nil thus ?case by simp
nipkow@15246
  1091
next
nipkow@15246
  1092
  case (Cons x xs) thus ?case
nipkow@15246
  1093
    apply (auto split:split_if_asm)
nipkow@15246
  1094
    using length_filter_le[of P xs] apply arith
nipkow@15246
  1095
  done
nipkow@15246
  1096
qed
wenzelm@13114
  1097
nipkow@15281
  1098
lemma length_filter_conv_card:
nipkow@15281
  1099
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
  1100
proof (induct xs)
nipkow@15281
  1101
  case Nil thus ?case by simp
nipkow@15281
  1102
next
nipkow@15281
  1103
  case (Cons x xs)
nipkow@15281
  1104
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
  1105
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
  1106
  show ?case (is "?l = card ?S'")
nipkow@15281
  1107
  proof (cases)
nipkow@15281
  1108
    assume "p x"
nipkow@15281
  1109
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@25162
  1110
      by(auto simp: image_def split:nat.split dest:gr0_implies_Suc)
nipkow@15281
  1111
    have "length (filter p (x # xs)) = Suc(card ?S)"
wenzelm@23388
  1112
      using Cons `p x` by simp
nipkow@15281
  1113
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
nipkow@15281
  1114
      by (simp add: card_image inj_Suc)
nipkow@15281
  1115
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
  1116
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
  1117
    finally show ?thesis .
nipkow@15281
  1118
  next
nipkow@15281
  1119
    assume "\<not> p x"
nipkow@15281
  1120
    hence eq: "?S' = Suc ` ?S"
nipkow@25162
  1121
      by(auto simp add: image_def split:nat.split elim:lessE)
nipkow@15281
  1122
    have "length (filter p (x # xs)) = card ?S"
wenzelm@23388
  1123
      using Cons `\<not> p x` by simp
nipkow@15281
  1124
    also have "\<dots> = card(Suc ` ?S)" using fin
nipkow@15281
  1125
      by (simp add: card_image inj_Suc)
nipkow@15281
  1126
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
  1127
      by (simp add:card_insert_if)
nipkow@15281
  1128
    finally show ?thesis .
nipkow@15281
  1129
  qed
nipkow@15281
  1130
qed
nipkow@15281
  1131
nipkow@17629
  1132
lemma Cons_eq_filterD:
nipkow@17629
  1133
 "x#xs = filter P ys \<Longrightarrow>
nipkow@17629
  1134
  \<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
  1135
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
nipkow@17629
  1136
proof(induct ys)
nipkow@17629
  1137
  case Nil thus ?case by simp
nipkow@17629
  1138
next
nipkow@17629
  1139
  case (Cons y ys)
nipkow@17629
  1140
  show ?case (is "\<exists>x. ?Q x")
nipkow@17629
  1141
  proof cases
nipkow@17629
  1142
    assume Py: "P y"
nipkow@17629
  1143
    show ?thesis
nipkow@17629
  1144
    proof cases
wenzelm@25221
  1145
      assume "x = y"
wenzelm@25221
  1146
      with Py Cons.prems have "?Q []" by simp
wenzelm@25221
  1147
      then show ?thesis ..
nipkow@17629
  1148
    next
wenzelm@25221
  1149
      assume "x \<noteq> y"
wenzelm@25221
  1150
      with Py Cons.prems show ?thesis by simp
nipkow@17629
  1151
    qed
nipkow@17629
  1152
  next
wenzelm@25221
  1153
    assume "\<not> P y"
wenzelm@25221
  1154
    with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastsimp
wenzelm@25221
  1155
    then have "?Q (y#us)" by simp
wenzelm@25221
  1156
    then show ?thesis ..
nipkow@17629
  1157
  qed
nipkow@17629
  1158
qed
nipkow@17629
  1159
nipkow@17629
  1160
lemma filter_eq_ConsD:
nipkow@17629
  1161
 "filter P ys = x#xs \<Longrightarrow>
nipkow@17629
  1162
  \<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
  1163
by(rule Cons_eq_filterD) simp
nipkow@17629
  1164
nipkow@17629
  1165
lemma filter_eq_Cons_iff:
nipkow@17629
  1166
 "(filter P ys = x#xs) =
nipkow@17629
  1167
  (\<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
  1168
by(auto dest:filter_eq_ConsD)
nipkow@17629
  1169
nipkow@17629
  1170
lemma Cons_eq_filter_iff:
nipkow@17629
  1171
 "(x#xs = filter P ys) =
nipkow@17629
  1172
  (\<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
  1173
by(auto dest:Cons_eq_filterD)
nipkow@17629
  1174
krauss@19770
  1175
lemma filter_cong[fundef_cong, recdef_cong]:
nipkow@17501
  1176
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
nipkow@17501
  1177
apply simp
nipkow@17501
  1178
apply(erule thin_rl)
nipkow@17501
  1179
by (induct ys) simp_all
nipkow@17501
  1180
nipkow@15281
  1181
haftmann@26442
  1182
subsubsection {* List partitioning *}
haftmann@26442
  1183
haftmann@26442
  1184
primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where
haftmann@26442
  1185
  "partition P [] = ([], [])"
haftmann@26442
  1186
  | "partition P (x # xs) = 
haftmann@26442
  1187
      (let (yes, no) = partition P xs
haftmann@26442
  1188
      in if P x then (x # yes, no) else (yes, x # no))"
haftmann@26442
  1189
haftmann@26442
  1190
lemma partition_filter1:
haftmann@26442
  1191
    "fst (partition P xs) = filter P xs"
haftmann@26442
  1192
by (induct xs) (auto simp add: Let_def split_def)
haftmann@26442
  1193
haftmann@26442
  1194
lemma partition_filter2:
haftmann@26442
  1195
    "snd (partition P xs) = filter (Not o P) xs"
haftmann@26442
  1196
by (induct xs) (auto simp add: Let_def split_def)
haftmann@26442
  1197
haftmann@26442
  1198
lemma partition_P:
haftmann@26442
  1199
  assumes "partition P xs = (yes, no)"
haftmann@26442
  1200
  shows "(\<forall>p \<in> set yes.  P p) \<and> (\<forall>p  \<in> set no. \<not> P p)"
haftmann@26442
  1201
proof -
haftmann@26442
  1202
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
haftmann@26442
  1203
    by simp_all
haftmann@26442
  1204
  then show ?thesis by (simp_all add: partition_filter1 partition_filter2)
haftmann@26442
  1205
qed
haftmann@26442
  1206
haftmann@26442
  1207
lemma partition_set:
haftmann@26442
  1208
  assumes "partition P xs = (yes, no)"
haftmann@26442
  1209
  shows "set yes \<union> set no = set xs"
haftmann@26442
  1210
proof -
haftmann@26442
  1211
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
haftmann@26442
  1212
    by simp_all
haftmann@26442
  1213
  then show ?thesis by (auto simp add: partition_filter1 partition_filter2) 
haftmann@26442
  1214
qed
haftmann@26442
  1215
hoelzl@33639
  1216
lemma partition_filter_conv[simp]:
hoelzl@33639
  1217
  "partition f xs = (filter f xs,filter (Not o f) xs)"
hoelzl@33639
  1218
unfolding partition_filter2[symmetric]
hoelzl@33639
  1219
unfolding partition_filter1[symmetric] by simp
hoelzl@33639
  1220
hoelzl@33639
  1221
declare partition.simps[simp del]
haftmann@26442
  1222
wenzelm@35115
  1223
nipkow@15392
  1224
subsubsection {* @{text concat} *}
wenzelm@13114
  1225
wenzelm@13142
  1226
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
  1227
by (induct xs) auto
wenzelm@13114
  1228
paulson@18447
  1229
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
  1230
by (induct xss) auto
wenzelm@13114
  1231
paulson@18447
  1232
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
  1233
by (induct xss) auto
wenzelm@13114
  1234
nipkow@24308
  1235
lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
nipkow@13145
  1236
by (induct xs) auto
wenzelm@13114
  1237
nipkow@24476
  1238
lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs"
nipkow@24349
  1239
by (induct xs) auto
nipkow@24349
  1240
wenzelm@13142
  1241
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
  1242
by (induct xs) auto
wenzelm@13114
  1243
wenzelm@13142
  1244
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
  1245
by (induct xs) auto
wenzelm@13114
  1246
wenzelm@13142
  1247
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
  1248
by (induct xs) auto
wenzelm@13114
  1249
wenzelm@13114
  1250
nipkow@15392
  1251
subsubsection {* @{text nth} *}
wenzelm@13114
  1252
haftmann@29827
  1253
lemma nth_Cons_0 [simp, code]: "(x # xs)!0 = x"
nipkow@13145
  1254
by auto
wenzelm@13114
  1255
haftmann@29827
  1256
lemma nth_Cons_Suc [simp, code]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
  1257
by auto
wenzelm@13114
  1258
wenzelm@13142
  1259
declare nth.simps [simp del]
wenzelm@13114
  1260
wenzelm@13114
  1261
lemma nth_append:
nipkow@24526
  1262
  "(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
nipkow@24526
  1263
apply (induct xs arbitrary: n, simp)
paulson@14208
  1264
apply (case_tac n, auto)
nipkow@13145
  1265
done
wenzelm@13114
  1266
nipkow@14402
  1267
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
wenzelm@25221
  1268
by (induct xs) auto
nipkow@14402
  1269
nipkow@14402
  1270
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
wenzelm@25221
  1271
by (induct xs) auto
nipkow@14402
  1272
nipkow@24526
  1273
lemma nth_map [simp]: "n < length xs ==> (map f xs)!n = f(xs!n)"
nipkow@24526
  1274
apply (induct xs arbitrary: n, simp)
paulson@14208
  1275
apply (case_tac n, auto)
nipkow@13145
  1276
done
wenzelm@13114
  1277
nipkow@18423
  1278
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
nipkow@18423
  1279
by(cases xs) simp_all
nipkow@18423
  1280
nipkow@18049
  1281
nipkow@18049
  1282
lemma list_eq_iff_nth_eq:
nipkow@24526
  1283
 "(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
nipkow@24526
  1284
apply(induct xs arbitrary: ys)
paulson@24632
  1285
 apply force
nipkow@18049
  1286
apply(case_tac ys)
nipkow@18049
  1287
 apply simp
nipkow@18049
  1288
apply(simp add:nth_Cons split:nat.split)apply blast
nipkow@18049
  1289
done
nipkow@18049
  1290
wenzelm@13142
  1291
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
  1292
apply (induct xs, simp, simp)
nipkow@13145
  1293
apply safe
paulson@24632
  1294
apply (metis nat_case_0 nth.simps zero_less_Suc)
paulson@24632
  1295
apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
paulson@14208
  1296
apply (case_tac i, simp)
paulson@24632
  1297
apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff)
nipkow@13145
  1298
done
wenzelm@13114
  1299
nipkow@17501
  1300
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
nipkow@17501
  1301
by(auto simp:set_conv_nth)
nipkow@17501
  1302
nipkow@13145
  1303
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
  1304
by (auto simp add: set_conv_nth)
wenzelm@13114
  1305
wenzelm@13142
  1306
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
  1307
by (auto simp add: set_conv_nth)
wenzelm@13114
  1308
wenzelm@13114
  1309
lemma all_nth_imp_all_set:
nipkow@13145
  1310
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
  1311
by (auto simp add: set_conv_nth)
wenzelm@13114
  1312
wenzelm@13114
  1313
lemma all_set_conv_all_nth:
nipkow@13145
  1314
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
  1315
by (auto simp add: set_conv_nth)
wenzelm@13114
  1316
kleing@25296
  1317
lemma rev_nth:
kleing@25296
  1318
  "n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)"
kleing@25296
  1319
proof (induct xs arbitrary: n)
kleing@25296
  1320
  case Nil thus ?case by simp
kleing@25296
  1321
next
kleing@25296
  1322
  case (Cons x xs)
kleing@25296
  1323
  hence n: "n < Suc (length xs)" by simp
kleing@25296
  1324
  moreover
kleing@25296
  1325
  { assume "n < length xs"
kleing@25296
  1326
    with n obtain n' where "length xs - n = Suc n'"
kleing@25296
  1327
      by (cases "length xs - n", auto)
kleing@25296
  1328
    moreover
kleing@25296
  1329
    then have "length xs - Suc n = n'" by simp
kleing@25296
  1330
    ultimately
kleing@25296
  1331
    have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp
kleing@25296
  1332
  }
kleing@25296
  1333
  ultimately
kleing@25296
  1334
  show ?case by (clarsimp simp add: Cons nth_append)
kleing@25296
  1335
qed
wenzelm@13114
  1336
nipkow@31159
  1337
lemma Skolem_list_nth:
nipkow@31159
  1338
  "(ALL i<k. EX x. P i x) = (EX xs. size xs = k & (ALL i<k. P i (xs!i)))"
nipkow@31159
  1339
  (is "_ = (EX xs. ?P k xs)")
nipkow@31159
  1340
proof(induct k)
nipkow@31159
  1341
  case 0 show ?case by simp
nipkow@31159
  1342
next
nipkow@31159
  1343
  case (Suc k)
nipkow@31159
  1344
  show ?case (is "?L = ?R" is "_ = (EX xs. ?P' xs)")
nipkow@31159
  1345
  proof
nipkow@31159
  1346
    assume "?R" thus "?L" using Suc by auto
nipkow@31159
  1347
  next
nipkow@31159
  1348
    assume "?L"
nipkow@31159
  1349
    with Suc obtain x xs where "?P k xs & P k x" by (metis less_Suc_eq)
nipkow@31159
  1350
    hence "?P'(xs@[x])" by(simp add:nth_append less_Suc_eq)
nipkow@31159
  1351
    thus "?R" ..
nipkow@31159
  1352
  qed
nipkow@31159
  1353
qed
nipkow@31159
  1354
nipkow@31159
  1355
nipkow@15392
  1356
subsubsection {* @{text list_update} *}
wenzelm@13114
  1357
nipkow@24526
  1358
lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
nipkow@24526
  1359
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1360
wenzelm@13114
  1361
lemma nth_list_update:
nipkow@24526
  1362
"i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@24526
  1363
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1364
wenzelm@13142
  1365
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
  1366
by (simp add: nth_list_update)
wenzelm@13114
  1367
nipkow@24526
  1368
lemma nth_list_update_neq [simp]: "i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@24526
  1369
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1370
nipkow@24526
  1371
lemma list_update_id[simp]: "xs[i := xs!i] = xs"
nipkow@24526
  1372
by (induct xs arbitrary: i) (simp_all split:nat.splits)
nipkow@24526
  1373
nipkow@24526
  1374
lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
nipkow@24526
  1375
apply (induct xs arbitrary: i)
nipkow@17501
  1376
 apply simp
nipkow@17501
  1377
apply (case_tac i)
nipkow@17501
  1378
apply simp_all
nipkow@17501
  1379
done
nipkow@17501
  1380
nipkow@31077
  1381
lemma list_update_nonempty[simp]: "xs[k:=x] = [] \<longleftrightarrow> xs=[]"
nipkow@31077
  1382
by(metis length_0_conv length_list_update)
nipkow@31077
  1383
wenzelm@13114
  1384
lemma list_update_same_conv:
nipkow@24526
  1385
"i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@24526
  1386
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1387
nipkow@14187
  1388
lemma list_update_append1:
nipkow@24526
  1389
 "i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
nipkow@24526
  1390
apply (induct xs arbitrary: i, simp)
nipkow@14187
  1391
apply(simp split:nat.split)
nipkow@14187
  1392
done
nipkow@14187
  1393
kleing@15868
  1394
lemma list_update_append:
nipkow@24526
  1395
  "(xs @ ys) [n:= x] = 
kleing@15868
  1396
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
nipkow@24526
  1397
by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868
  1398
nipkow@14402
  1399
lemma list_update_length [simp]:
nipkow@14402
  1400
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
  1401
by (induct xs, auto)
nipkow@14402
  1402
nipkow@31264
  1403
lemma map_update: "map f (xs[k:= y]) = (map f xs)[k := f y]"
nipkow@31264
  1404
by(induct xs arbitrary: k)(auto split:nat.splits)
nipkow@31264
  1405
nipkow@31264
  1406
lemma rev_update:
nipkow@31264
  1407
  "k < length xs \<Longrightarrow> rev (xs[k:= y]) = (rev xs)[length xs - k - 1 := y]"
nipkow@31264
  1408
by (induct xs arbitrary: k) (auto simp: list_update_append split:nat.splits)
nipkow@31264
  1409
wenzelm@13114
  1410
lemma update_zip:
nipkow@31080
  1411
  "(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@24526
  1412
by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split)
nipkow@24526
  1413
nipkow@24526
  1414
lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)"
nipkow@24526
  1415
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1416
wenzelm@13114
  1417
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
  1418
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
  1419
nipkow@24526
  1420
lemma set_update_memI: "n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
nipkow@24526
  1421
by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868
  1422
nipkow@31077
  1423
lemma list_update_overwrite[simp]:
haftmann@24796
  1424
  "xs [i := x, i := y] = xs [i := y]"
nipkow@31077
  1425
apply (induct xs arbitrary: i) apply simp
nipkow@31077
  1426
apply (case_tac i, simp_all)
haftmann@24796
  1427
done
haftmann@24796
  1428
haftmann@24796
  1429
lemma list_update_swap:
haftmann@24796
  1430
  "i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]"
haftmann@24796
  1431
apply (induct xs arbitrary: i i')
haftmann@24796
  1432
apply simp
haftmann@24796
  1433
apply (case_tac i, case_tac i')
haftmann@24796
  1434
apply auto
haftmann@24796
  1435
apply (case_tac i')
haftmann@24796
  1436
apply auto
haftmann@24796
  1437
done
haftmann@24796
  1438
haftmann@29827
  1439
lemma list_update_code [code]:
haftmann@29827
  1440
  "[][i := y] = []"
haftmann@29827
  1441
  "(x # xs)[0 := y] = y # xs"
haftmann@29827
  1442
  "(x # xs)[Suc i := y] = x # xs[i := y]"
haftmann@29827
  1443
  by simp_all
haftmann@29827
  1444
wenzelm@13114
  1445
nipkow@15392
  1446
subsubsection {* @{text last} and @{text butlast} *}
wenzelm@13114
  1447
wenzelm@13142
  1448
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
  1449
by (induct xs) auto
wenzelm@13114
  1450
wenzelm@13142
  1451
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
  1452
by (induct xs) auto
wenzelm@13114
  1453
nipkow@14302
  1454
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
  1455
by(simp add:last.simps)
nipkow@14302
  1456
nipkow@14302
  1457
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
  1458
by(simp add:last.simps)
nipkow@14302
  1459
nipkow@14302
  1460
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
  1461
by (induct xs) (auto)
nipkow@14302
  1462
nipkow@14302
  1463
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
  1464
by(simp add:last_append)
nipkow@14302
  1465
nipkow@14302
  1466
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
  1467
by(simp add:last_append)
nipkow@14302
  1468
nipkow@17762
  1469
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
nipkow@17762
  1470
by(rule rev_exhaust[of xs]) simp_all
nipkow@17762
  1471
nipkow@17762
  1472
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
nipkow@17762
  1473
by(cases xs) simp_all
nipkow@17762
  1474
nipkow@17765
  1475
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
nipkow@17765
  1476
by (induct as) auto
nipkow@17762
  1477
wenzelm@13142
  1478
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
  1479
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1480
wenzelm@13114
  1481
lemma butlast_append:
nipkow@24526
  1482
  "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@24526
  1483
by (induct xs arbitrary: ys) auto
wenzelm@13114
  1484
wenzelm@13142
  1485
lemma append_butlast_last_id [simp]:
nipkow@13145
  1486
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
  1487
by (induct xs) auto
wenzelm@13114
  1488
wenzelm@13142
  1489
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
  1490
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1491
wenzelm@13114
  1492
lemma in_set_butlast_appendI:
nipkow@13145
  1493
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
  1494
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
  1495
nipkow@24526
  1496
lemma last_drop[simp]: "n < length xs \<Longrightarrow> last (drop n xs) = last xs"
nipkow@24526
  1497
apply (induct xs arbitrary: n)
nipkow@17501
  1498
 apply simp
nipkow@17501
  1499
apply (auto split:nat.split)
nipkow@17501
  1500
done
nipkow@17501
  1501
huffman@30128
  1502
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
nipkow@17589
  1503
by(induct xs)(auto simp:neq_Nil_conv)
nipkow@17589
  1504
huffman@30128
  1505
lemma butlast_conv_take: "butlast xs = take (length xs - 1) xs"
huffman@26584
  1506
by (induct xs, simp, case_tac xs, simp_all)
huffman@26584
  1507
nipkow@31077
  1508
lemma last_list_update:
nipkow@31077
  1509
  "xs \<noteq> [] \<Longrightarrow> last(xs[k:=x]) = (if k = size xs - 1 then x else last xs)"
nipkow@31077
  1510
by (auto simp: last_conv_nth)
nipkow@31077
  1511
nipkow@31077
  1512
lemma butlast_list_update:
nipkow@31077
  1513
  "butlast(xs[k:=x]) =
nipkow@31077
  1514
 (if k = size xs - 1 then butlast xs else (butlast xs)[k:=x])"
nipkow@31077
  1515
apply(cases xs rule:rev_cases)
nipkow@31077
  1516
apply simp
nipkow@31077
  1517
apply(simp add:list_update_append split:nat.splits)
nipkow@31077
  1518
done
nipkow@31077
  1519
haftmann@24796
  1520
nipkow@15392
  1521
subsubsection {* @{text take} and @{text drop} *}
wenzelm@13114
  1522
wenzelm@13142
  1523
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
  1524
by (induct xs) auto
wenzelm@13114
  1525
wenzelm@13142
  1526
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
  1527
by (induct xs) auto
wenzelm@13114
  1528
wenzelm@13142
  1529
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
  1530
by simp
wenzelm@13114
  1531
wenzelm@13142
  1532
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
  1533
by simp
wenzelm@13114
  1534
wenzelm@13142
  1535
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
  1536
huffman@30128
  1537
lemma take_1_Cons [simp]: "take 1 (x # xs) = [x]"
huffman@30128
  1538
  unfolding One_nat_def by simp
huffman@30128
  1539
huffman@30128
  1540
lemma drop_1_Cons [simp]: "drop 1 (x # xs) = xs"
huffman@30128
  1541
  unfolding One_nat_def by simp
huffman@30128
  1542
nipkow@15110
  1543
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
  1544
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
  1545
nipkow@14187
  1546
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
  1547
by(cases xs, simp_all)
nipkow@14187
  1548
huffman@26584
  1549
lemma take_tl: "take n (tl xs) = tl (take (Suc n) xs)"
huffman@26584
  1550
by (induct xs arbitrary: n) simp_all
huffman@26584
  1551
nipkow@24526
  1552
lemma drop_tl: "drop n (tl xs) = tl(drop n xs)"
nipkow@24526
  1553
by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@24526
  1554
huffman@26584
  1555
lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)"
huffman@26584
  1556
by (cases n, simp, cases xs, auto)
huffman@26584
  1557
huffman@26584
  1558
lemma tl_drop: "tl (drop n xs) = drop n (tl xs)"
huffman@26584
  1559
by (simp only: drop_tl)
huffman@26584
  1560
nipkow@24526
  1561
lemma nth_via_drop: "drop n xs = y#ys \<Longrightarrow> xs!n = y"
nipkow@24526
  1562
apply (induct xs arbitrary: n, simp)
nipkow@14187
  1563
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
  1564
done
nipkow@14187
  1565
nipkow@13913
  1566
lemma take_Suc_conv_app_nth:
nipkow@24526
  1567
  "i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
nipkow@24526
  1568
apply (induct xs arbitrary: i, simp)
paulson@14208
  1569
apply (case_tac i, auto)
nipkow@13913
  1570
done
nipkow@13913
  1571
mehta@14591
  1572
lemma drop_Suc_conv_tl:
nipkow@24526
  1573
  "i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
nipkow@24526
  1574
apply (induct xs arbitrary: i, simp)
mehta@14591
  1575
apply (case_tac i, auto)
mehta@14591
  1576
done
mehta@14591
  1577
nipkow@24526
  1578
lemma length_take [simp]: "length (take n xs) = min (length xs) n"
nipkow@24526
  1579
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1580
nipkow@24526
  1581
lemma length_drop [simp]: "length (drop n xs) = (length xs - n)"
nipkow@24526
  1582
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1583
nipkow@24526
  1584
lemma take_all [simp]: "length xs <= n ==> take n xs = xs"
nipkow@24526
  1585
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1586
nipkow@24526
  1587
lemma drop_all [simp]: "length xs <= n ==> drop n xs = []"
nipkow@24526
  1588
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114
  1589
wenzelm@13142
  1590
lemma take_append [simp]:
nipkow@24526
  1591
  "take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@24526
  1592
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114
  1593
wenzelm@13142
  1594
lemma drop_append [simp]:
nipkow@24526
  1595
  "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@24526
  1596
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1597
nipkow@24526
  1598
lemma take_take [simp]: "take n (take m xs) = take (min n m) xs"
nipkow@24526
  1599
apply (induct m arbitrary: xs n, auto)
paulson@14208
  1600
apply (case_tac xs, auto)
nipkow@15236
  1601
apply (case_tac n, auto)
nipkow@13145
  1602
done
wenzelm@13114
  1603
nipkow@24526
  1604
lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs"
nipkow@24526
  1605
apply (induct m arbitrary: xs, auto)
paulson@14208
  1606
apply (case_tac xs, auto)
nipkow@13145
  1607
done
wenzelm@13114
  1608
nipkow@24526
  1609
lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)"
nipkow@24526
  1610
apply (induct m arbitrary: xs n, auto)
paulson@14208
  1611
apply (case_tac xs, auto)
nipkow@13145
  1612
done
wenzelm@13114
  1613
nipkow@24526
  1614
lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@24526
  1615
apply(induct xs arbitrary: m n)
nipkow@14802
  1616
 apply simp
nipkow@14802
  1617
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  1618
done
nipkow@14802
  1619
nipkow@24526
  1620
lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs"
nipkow@24526
  1621
apply (induct n arbitrary: xs, auto)
paulson@14208
  1622
apply (case_tac xs, auto)
nipkow@13145
  1623
done
wenzelm@13114
  1624
nipkow@24526
  1625
lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@24526
  1626
apply(induct xs arbitrary: n)
nipkow@15110
  1627
 apply simp
nipkow@15110
  1628
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  1629
done
nipkow@15110
  1630
nipkow@24526
  1631
lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)"
nipkow@24526
  1632
apply(induct xs arbitrary: n)
nipkow@15110
  1633
apply simp
nipkow@15110
  1634
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1635
done
nipkow@15110
  1636
nipkow@24526
  1637
lemma take_map: "take n (map f xs) = map f (take n xs)"
nipkow@24526
  1638
apply (induct n arbitrary: xs, auto)
paulson@14208
  1639
apply (case_tac xs, auto)
nipkow@13145
  1640
done
wenzelm@13114
  1641
nipkow@24526
  1642
lemma drop_map: "drop n (map f xs) = map f (drop n xs)"
nipkow@24526
  1643
apply (induct n arbitrary: xs, auto)
paulson@14208
  1644
apply (case_tac xs, auto)
nipkow@13145
  1645
done
wenzelm@13114
  1646
nipkow@24526
  1647
lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)"
nipkow@24526
  1648
apply (induct xs arbitrary: i, auto)
paulson@14208
  1649
apply (case_tac i, auto)
nipkow@13145
  1650
done
wenzelm@13114
  1651
nipkow@24526
  1652
lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)"
nipkow@24526
  1653
apply (induct xs arbitrary: i, auto)
paulson@14208
  1654
apply (case_tac i, auto)
nipkow@13145
  1655
done
wenzelm@13114
  1656
nipkow@24526
  1657
lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i"
nipkow@24526
  1658
apply (induct xs arbitrary: i n, auto)
paulson@14208
  1659
apply (case_tac n, blast)
paulson@14208
  1660
apply (case_tac i, auto)
nipkow@13145
  1661
done
wenzelm@13114
  1662
wenzelm@13142
  1663
lemma nth_drop [simp]:
nipkow@24526
  1664
  "n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
nipkow@24526
  1665
apply (induct n arbitrary: xs i, auto)
paulson@14208
  1666
apply (case_tac xs, auto)
nipkow@13145
  1667
done
nipkow@3507
  1668
huffman@26584
  1669
lemma butlast_take:
huffman@30128
  1670
  "n <= length xs ==> butlast (take n xs) = take (n - 1) xs"
huffman@26584
  1671
by (simp add: butlast_conv_take min_max.inf_absorb1 min_max.inf_absorb2)
huffman@26584
  1672
huffman@26584
  1673
lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)"
huffman@30128
  1674
by (simp add: butlast_conv_take drop_take add_ac)
huffman@26584
  1675
huffman@26584
  1676
lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs"
huffman@26584
  1677
by (simp add: butlast_conv_take min_max.inf_absorb1)
huffman@26584
  1678
huffman@26584
  1679
lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)"
huffman@30128
  1680
by (simp add: butlast_conv_take drop_take add_ac)
huffman@26584
  1681
nipkow@18423
  1682
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
nipkow@18423
  1683
by(simp add: hd_conv_nth)
nipkow@18423
  1684
nipkow@35248
  1685
lemma set_take_subset_set_take:
nipkow@35248
  1686
  "m <= n \<Longrightarrow> set(take m xs) <= set(take n xs)"
nipkow@35248
  1687
by(induct xs arbitrary: m n)(auto simp:take_Cons split:nat.split)
nipkow@35248
  1688
nipkow@24526
  1689
lemma set_take_subset: "set(take n xs) \<subseteq> set xs"
nipkow@24526
  1690
by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split)
nipkow@24526
  1691
nipkow@24526
  1692
lemma set_drop_subset: "set(drop n xs) \<subseteq> set xs"
nipkow@24526
  1693
by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1694
nipkow@35248
  1695
lemma set_drop_subset_set_drop:
nipkow@35248
  1696
  "m >= n \<Longrightarrow> set(drop m xs) <= set(drop n xs)"
nipkow@35248
  1697
apply(induct xs arbitrary: m n)
nipkow@35248
  1698
apply(auto simp:drop_Cons split:nat.split)
nipkow@35248
  1699
apply (metis set_drop_subset subset_iff)
nipkow@35248
  1700
done
nipkow@35248
  1701
nipkow@14187
  1702
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1703
using set_take_subset by fast
nipkow@14187
  1704
nipkow@14187
  1705
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1706
using set_drop_subset by fast
nipkow@14187
  1707
wenzelm@13114
  1708
lemma append_eq_conv_conj:
nipkow@24526
  1709
  "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
nipkow@24526
  1710
apply (induct xs arbitrary: zs, simp, clarsimp)
paulson@14208
  1711
apply (case_tac zs, auto)
nipkow@13145
  1712
done
wenzelm@13142
  1713
nipkow@24526
  1714
lemma take_add: 
nipkow@24526
  1715
  "i+j \<le> length(xs) \<Longrightarrow> take (i+j) xs = take i xs @ take j (drop i xs)"
nipkow@24526
  1716
apply (induct xs arbitrary: i, auto) 
nipkow@24526
  1717
apply (case_tac i, simp_all)
paulson@14050
  1718
done
paulson@14050
  1719
nipkow@14300
  1720
lemma append_eq_append_conv_if:
nipkow@24526
  1721
 "(xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1722
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1723
   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
  1724
   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
  1725
apply(induct xs\<^isub>1 arbitrary: ys\<^isub>1)
nipkow@14300
  1726
 apply simp
nipkow@14300
  1727
apply(case_tac ys\<^isub>1)
nipkow@14300
  1728
apply simp_all
nipkow@14300
  1729
done
nipkow@14300
  1730
nipkow@15110
  1731
lemma take_hd_drop:
huffman@30079
  1732
  "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (Suc n) xs"
nipkow@24526
  1733
apply(induct xs arbitrary: n)
nipkow@15110
  1734
apply simp
nipkow@15110
  1735
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1736
done
nipkow@15110
  1737
nipkow@17501
  1738
lemma id_take_nth_drop:
nipkow@17501
  1739
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
nipkow@17501
  1740
proof -
nipkow@17501
  1741
  assume si: "i < length xs"
nipkow@17501
  1742
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
nipkow@17501
  1743
  moreover
nipkow@17501
  1744
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
nipkow@17501
  1745
    apply (rule_tac take_Suc_conv_app_nth) by arith
nipkow@17501
  1746
  ultimately show ?thesis by auto
nipkow@17501
  1747
qed
nipkow@17501
  1748
  
nipkow@17501
  1749
lemma upd_conv_take_nth_drop:
nipkow@17501
  1750
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1751
proof -
nipkow@17501
  1752
  assume i: "i < length xs"
nipkow@17501
  1753
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
nipkow@17501
  1754
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
nipkow@17501
  1755
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1756
    using i by (simp add: list_update_append)
nipkow@17501
  1757
  finally show ?thesis .
nipkow@17501
  1758
qed
nipkow@17501
  1759
haftmann@24796
  1760
lemma nth_drop':
haftmann@24796
  1761
  "i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs"
haftmann@24796
  1762
apply (induct i arbitrary: xs)
haftmann@24796
  1763
apply (simp add: neq_Nil_conv)
haftmann@24796
  1764
apply (erule exE)+
haftmann@24796
  1765
apply simp
haftmann@24796
  1766
apply (case_tac xs)
haftmann@24796
  1767
apply simp_all
haftmann@24796
  1768
done
haftmann@24796
  1769
wenzelm@13114
  1770
nipkow@15392
  1771
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1772
hoelzl@33639
  1773
lemma length_takeWhile_le: "length (takeWhile P xs) \<le> length xs"
hoelzl@33639
  1774
  by (induct xs) auto
hoelzl@33639
  1775
wenzelm@13142
  1776
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1777
by (induct xs) auto
wenzelm@13114
  1778
wenzelm@13142
  1779
lemma takeWhile_append1 [simp]:
nipkow@13145
  1780
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1781
by (induct xs) auto
wenzelm@13114
  1782
wenzelm@13142
  1783
lemma takeWhile_append2 [simp]:
nipkow@13145
  1784
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1785
by (induct xs) auto
wenzelm@13114
  1786
wenzelm@13142
  1787
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1788
by (induct xs) auto
wenzelm@13114
  1789
hoelzl@33639
  1790
lemma takeWhile_nth: "j < length (takeWhile P xs) \<Longrightarrow> takeWhile P xs ! j = xs ! j"
hoelzl@33639
  1791
apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
hoelzl@33639
  1792
hoelzl@33639
  1793
lemma dropWhile_nth: "j < length (dropWhile P xs) \<Longrightarrow> dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))"
hoelzl@33639
  1794
apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
hoelzl@33639
  1795
hoelzl@33639
  1796
lemma length_dropWhile_le: "length (dropWhile P xs) \<le> length xs"
hoelzl@33639
  1797
by (induct xs) auto
hoelzl@33639
  1798
wenzelm@13142
  1799
lemma dropWhile_append1 [simp]:
nipkow@13145
  1800
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1801
by (induct xs) auto
wenzelm@13114
  1802
wenzelm@13142
  1803
lemma dropWhile_append2 [simp]:
nipkow@13145
  1804
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1805
by (induct xs) auto
wenzelm@13114
  1806
krauss@23971
  1807
lemma set_takeWhileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1808
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1809
nipkow@13913
  1810
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1811
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1812
by(induct xs, auto)
nipkow@13913
  1813
nipkow@13913
  1814
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1815
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1816
by(induct xs, auto)
nipkow@13913
  1817
nipkow@13913
  1818
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1819
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1820
by(induct xs, auto)
nipkow@13913
  1821
nipkow@31077
  1822
lemma distinct_takeWhile[simp]: "distinct xs ==> distinct (takeWhile P xs)"
nipkow@31077
  1823
by (induct xs) (auto dest: set_takeWhileD)
nipkow@31077
  1824
nipkow@31077
  1825
lemma distinct_dropWhile[simp]: "distinct xs ==> distinct (dropWhile P xs)"
nipkow@31077
  1826
by (induct xs) auto
nipkow@31077
  1827
hoelzl@33639
  1828
lemma takeWhile_map: "takeWhile P (map f xs) = map f (takeWhile (P \<circ> f) xs)"
hoelzl@33639
  1829
by (induct xs) auto
hoelzl@33639
  1830
hoelzl@33639
  1831
lemma dropWhile_map: "dropWhile P (map f xs) = map f (dropWhile (P \<circ> f) xs)"
hoelzl@33639
  1832
by (induct xs) auto
hoelzl@33639
  1833
hoelzl@33639
  1834
lemma takeWhile_eq_take: "takeWhile P xs = take (length (takeWhile P xs)) xs"
hoelzl@33639
  1835
by (induct xs) auto
hoelzl@33639
  1836
hoelzl@33639
  1837
lemma dropWhile_eq_drop: "dropWhile P xs = drop (length (takeWhile P xs)) xs"
hoelzl@33639
  1838
by (induct xs) auto
hoelzl@33639
  1839
hoelzl@33639
  1840
lemma hd_dropWhile:
hoelzl@33639
  1841
  "dropWhile P xs \<noteq> [] \<Longrightarrow> \<not> P (hd (dropWhile P xs))"
hoelzl@33639
  1842
using assms by (induct xs) auto
hoelzl@33639
  1843
hoelzl@33639
  1844
lemma takeWhile_eq_filter:
hoelzl@33639
  1845
  assumes "\<And> x. x \<in> set (dropWhile P xs) \<Longrightarrow> \<not> P x"
hoelzl@33639
  1846
  shows "takeWhile P xs = filter P xs"
hoelzl@33639
  1847
proof -
hoelzl@33639
  1848
  have A: "filter P xs = filter P (takeWhile P xs @ dropWhile P xs)"
hoelzl@33639
  1849
    by simp
hoelzl@33639
  1850
  have B: "filter P (dropWhile P xs) = []"
hoelzl@33639
  1851
    unfolding filter_empty_conv using assms by blast
hoelzl@33639
  1852
  have "filter P xs = takeWhile P xs"
hoelzl@33639
  1853
    unfolding A filter_append B
hoelzl@33639
  1854
    by (auto simp add: filter_id_conv dest: set_takeWhileD)
hoelzl@33639
  1855
  thus ?thesis ..
hoelzl@33639
  1856
qed
hoelzl@33639
  1857
hoelzl@33639
  1858
lemma takeWhile_eq_take_P_nth:
hoelzl@33639
  1859
  "\<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
  1860
  takeWhile P xs = take n xs"
hoelzl@33639
  1861
proof (induct xs arbitrary: n)
hoelzl@33639
  1862
  case (Cons x xs)
hoelzl@33639
  1863
  thus ?case
hoelzl@33639
  1864
  proof (cases n)
hoelzl@33639
  1865
    case (Suc n') note this[simp]
hoelzl@33639
  1866
    have "P x" using Cons.prems(1)[of 0] by simp
hoelzl@33639
  1867
    moreover have "takeWhile P xs = take n' xs"
hoelzl@33639
  1868
    proof (rule Cons.hyps)
hoelzl@33639
  1869
      case goal1 thus "P (xs ! i)" using Cons.prems(1)[of "Suc i"] by simp
hoelzl@33639
  1870
    next case goal2 thus ?case using Cons by auto
hoelzl@33639
  1871
    qed
hoelzl@33639
  1872
    ultimately show ?thesis by simp
hoelzl@33639
  1873
   qed simp
hoelzl@33639
  1874
qed simp
hoelzl@33639
  1875
hoelzl@33639
  1876
lemma nth_length_takeWhile:
hoelzl@33639
  1877
  "length (takeWhile P xs) < length xs \<Longrightarrow> \<not> P (xs ! length (takeWhile P xs))"
hoelzl@33639
  1878
by (induct xs) auto
hoelzl@33639
  1879
hoelzl@33639
  1880
lemma length_takeWhile_less_P_nth:
hoelzl@33639
  1881
  assumes all: "\<And> i. i < j \<Longrightarrow> P (xs ! i)" and "j \<le> length xs"
hoelzl@33639
  1882
  shows "j \<le> length (takeWhile P xs)"
hoelzl@33639
  1883
proof (rule classical)
hoelzl@33639
  1884
  assume "\<not> ?thesis"
hoelzl@33639
  1885
  hence "length (takeWhile P xs) < length xs" using assms by simp
hoelzl@33639
  1886
  thus ?thesis using all `\<not> ?thesis` nth_length_takeWhile[of P xs] by auto
hoelzl@33639
  1887
qed
nipkow@31077
  1888
nipkow@17501
  1889
text{* The following two lemmmas could be generalized to an arbitrary
nipkow@17501
  1890
property. *}
nipkow@17501
  1891
nipkow@17501
  1892
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1893
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
nipkow@17501
  1894
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
nipkow@17501
  1895
nipkow@17501
  1896
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1897
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
nipkow@17501
  1898
apply(induct xs)
nipkow@17501
  1899
 apply simp
nipkow@17501
  1900
apply auto
nipkow@17501
  1901
apply(subst dropWhile_append2)
nipkow@17501
  1902
apply auto
nipkow@17501
  1903
done
nipkow@17501
  1904
nipkow@18423
  1905
lemma takeWhile_not_last:
nipkow@18423
  1906
 "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
nipkow@18423
  1907
apply(induct xs)
nipkow@18423
  1908
 apply simp
nipkow@18423
  1909
apply(case_tac xs)
nipkow@18423
  1910
apply(auto)
nipkow@18423
  1911
done
nipkow@18423
  1912
krauss@19770
  1913
lemma takeWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1914
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1915
  ==> takeWhile P l = takeWhile Q k"
nipkow@24349
  1916
by (induct k arbitrary: l) (simp_all)
krauss@18336
  1917
krauss@19770
  1918
lemma dropWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1919
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1920
  ==> dropWhile P l = dropWhile Q k"
nipkow@24349
  1921
by (induct k arbitrary: l, simp_all)
krauss@18336
  1922
wenzelm@13114
  1923
nipkow@15392
  1924
subsubsection {* @{text zip} *}
wenzelm@13114
  1925
wenzelm@13142
  1926
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1927
by (induct ys) auto
wenzelm@13114
  1928
wenzelm@13142
  1929
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1930
by simp
wenzelm@13114
  1931
wenzelm@13142
  1932
declare zip_Cons [simp del]
wenzelm@13114
  1933
nipkow@15281
  1934
lemma zip_Cons1:
nipkow@15281
  1935
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  1936
by(auto split:list.split)
nipkow@15281
  1937
wenzelm@13142
  1938
lemma length_zip [simp]:
krauss@22493
  1939
"length (zip xs ys) = min (length xs) (length ys)"
krauss@22493
  1940
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1941
haftmann@34978
  1942
lemma zip_obtain_same_length:
haftmann@34978
  1943
  assumes "\<And>zs ws n. length zs = length ws \<Longrightarrow> n = min (length xs) (length ys)
haftmann@34978
  1944
    \<Longrightarrow> zs = take n xs \<Longrightarrow> ws = take n ys \<Longrightarrow> P (zip zs ws)"
haftmann@34978
  1945
  shows "P (zip xs ys)"
haftmann@34978
  1946
proof -
haftmann@34978
  1947
  let ?n = "min (length xs) (length ys)"
haftmann@34978
  1948
  have "P (zip (take ?n xs) (take ?n ys))"
haftmann@34978
  1949
    by (rule assms) simp_all
haftmann@34978
  1950
  moreover have "zip xs ys = zip (take ?n xs) (take ?n ys)"
haftmann@34978
  1951
  proof (induct xs arbitrary: ys)
haftmann@34978
  1952
    case Nil then show ?case by simp
haftmann@34978
  1953
  next
haftmann@34978
  1954
    case (Cons x xs) then show ?case by (cases ys) simp_all
haftmann@34978
  1955
  qed
haftmann@34978
  1956
  ultimately show ?thesis by simp
haftmann@34978
  1957
qed
haftmann@34978
  1958
wenzelm@13114
  1959
lemma zip_append1:
krauss@22493
  1960
"zip (xs @ ys) zs =
nipkow@13145
  1961
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
krauss@22493
  1962
by (induct xs zs rule:list_induct2') auto
wenzelm@13114
  1963
wenzelm@13114
  1964
lemma zip_append2:
krauss@22493
  1965
"zip xs (ys @ zs) =
nipkow@13145
  1966
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
krauss@22493
  1967
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1968
wenzelm@13142
  1969
lemma zip_append [simp]:
wenzelm@13142
  1970
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1971
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1972
by (simp add: zip_append1)
wenzelm@13114
  1973
wenzelm@13114
  1974
lemma zip_rev:
nipkow@14247
  1975
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1976
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1977
hoelzl@33639
  1978
lemma zip_map_map:
hoelzl@33639
  1979
  "zip (map f xs) (map g ys) = map (\<lambda> (x, y). (f x, g y)) (zip xs ys)"
hoelzl@33639
  1980
proof (induct xs arbitrary: ys)
hoelzl@33639
  1981
  case (Cons x xs) note Cons_x_xs = Cons.hyps
hoelzl@33639
  1982
  show ?case
hoelzl@33639
  1983
  proof (cases ys)
hoelzl@33639
  1984
    case (Cons y ys')
hoelzl@33639
  1985
    show ?thesis unfolding Cons using Cons_x_xs by simp
hoelzl@33639
  1986
  qed simp
hoelzl@33639
  1987
qed simp
hoelzl@33639
  1988
hoelzl@33639
  1989
lemma zip_map1:
hoelzl@33639
  1990
  "zip (map f xs) ys = map (\<lambda>(x, y). (f x, y)) (zip xs ys)"
hoelzl@33639
  1991
using zip_map_map[of f xs "\<lambda>x. x" ys] by simp
hoelzl@33639
  1992
hoelzl@33639
  1993
lemma zip_map2:
hoelzl@33639
  1994
  "zip xs (map f ys) = map (\<lambda>(x, y). (x, f y)) (zip xs ys)"
hoelzl@33639
  1995
using zip_map_map[of "\<lambda>x. x" xs f ys] by simp
hoelzl@33639
  1996
nipkow@23096
  1997
lemma map_zip_map:
hoelzl@33639
  1998
  "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
hoelzl@33639
  1999
unfolding zip_map1 by auto
nipkow@23096
  2000
nipkow@23096
  2001
lemma map_zip_map2:
hoelzl@33639
  2002
  "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
hoelzl@33639
  2003
unfolding zip_map2 by auto
nipkow@23096
  2004
nipkow@31080
  2005
text{* Courtesy of Andreas Lochbihler: *}
nipkow@31080
  2006
lemma zip_same_conv_map: "zip xs xs = map (\<lambda>x. (x, x)) xs"
nipkow@31080
  2007
by(induct xs) auto
nipkow@31080
  2008
wenzelm@13142
  2009
lemma nth_zip [simp]:
nipkow@24526
  2010
"[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
nipkow@24526
  2011
apply (induct ys arbitrary: i xs, simp)
nipkow@13145
  2012
apply (case_tac xs)
nipkow@13145
  2013
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  2014
done
wenzelm@13114
  2015
wenzelm@13114
  2016
lemma set_zip:
nipkow@13145
  2017
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@31080
  2018
by(simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  2019
hoelzl@33639
  2020
lemma zip_same: "((a,b) \<in> set (zip xs xs)) = (a \<in> set xs \<and> a = b)"
hoelzl@33639
  2021
by(induct xs) auto
hoelzl@33639
  2022
wenzelm@13114
  2023
lemma zip_update:
nipkow@31080
  2024
  "zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@31080
  2025
by(rule sym, simp add: update_zip)
wenzelm@13114
  2026
wenzelm@13142
  2027
lemma zip_replicate [simp]:
nipkow@24526
  2028
  "zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
nipkow@24526
  2029
apply (induct i arbitrary: j, auto)
paulson@14208
  2030
apply (case_tac j, auto)
nipkow@13145
  2031
done
wenzelm@13114
  2032
nipkow@19487
  2033
lemma take_zip:
nipkow@24526
  2034
  "take n (zip xs ys) = zip (take n xs) (take n ys)"
nipkow@24526
  2035
apply (induct n arbitrary: xs ys)
nipkow@19487
  2036
 apply simp
nipkow@19487
  2037
apply (case_tac xs, simp)
nipkow@19487
  2038
apply (case_tac ys, simp_all)
nipkow@19487
  2039
done
nipkow@19487
  2040
nipkow@19487
  2041
lemma drop_zip:
nipkow@24526
  2042
  "drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
nipkow@24526
  2043
apply (induct n arbitrary: xs ys)
nipkow@19487
  2044
 apply simp
nipkow@19487
  2045
apply (case_tac xs, simp)
nipkow@19487
  2046
apply (case_tac ys, simp_all)
nipkow@19487
  2047
done
nipkow@19487
  2048
hoelzl@33639
  2049
lemma zip_takeWhile_fst: "zip (takeWhile P xs) ys = takeWhile (P \<circ> fst) (zip xs ys)"
hoelzl@33639
  2050
proof (induct xs arbitrary: ys)
hoelzl@33639
  2051
  case (Cons x xs) thus ?case by (cases ys) auto
hoelzl@33639
  2052
qed simp
hoelzl@33639
  2053
hoelzl@33639
  2054
lemma zip_takeWhile_snd: "zip xs (takeWhile P ys) = takeWhile (P \<circ> snd) (zip xs ys)"
hoelzl@33639
  2055
proof (induct xs arbitrary: ys)
hoelzl@33639
  2056
  case (Cons x xs) thus ?case by (cases ys) auto
hoelzl@33639
  2057
qed simp
hoelzl@33639
  2058
krauss@22493
  2059
lemma set_zip_leftD:
krauss@22493
  2060
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
krauss@22493
  2061
by (induct xs ys rule:list_induct2') auto
krauss@22493
  2062
krauss@22493
  2063
lemma set_zip_rightD:
krauss@22493
  2064
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
krauss@22493
  2065
by (induct xs ys rule:list_induct2') auto
wenzelm@13142
  2066
nipkow@23983
  2067
lemma in_set_zipE:
nipkow@23983
  2068
  "(x,y) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
nipkow@23983
  2069
by(blast dest: set_zip_leftD set_zip_rightD)
nipkow@23983
  2070
haftmann@29829
  2071
lemma zip_map_fst_snd:
haftmann@29829
  2072
  "zip (map fst zs) (map snd zs) = zs"
haftmann@29829
  2073
  by (induct zs) simp_all
haftmann@29829
  2074
haftmann@29829
  2075
lemma zip_eq_conv:
haftmann@29829
  2076
  "length xs = length ys \<Longrightarrow> zip xs ys = zs \<longleftrightarrow> map fst zs = xs \<and> map snd zs = ys"
haftmann@29829
  2077
  by (auto simp add: zip_map_fst_snd)
haftmann@29829
  2078
hoelzl@33639
  2079
lemma distinct_zipI1:
hoelzl@33639
  2080
  "distinct xs \<Longrightarrow> distinct (zip xs ys)"
hoelzl@33639
  2081
proof (induct xs arbitrary: ys)
hoelzl@33639
  2082
  case (Cons x xs)
hoelzl@33639
  2083
  show ?case
hoelzl@33639
  2084
  proof (cases ys)
hoelzl@33639
  2085
    case (Cons y ys')
hoelzl@33639
  2086
    have "(x, y) \<notin> set (zip xs ys')"
hoelzl@33639
  2087
      using Cons.prems by (auto simp: set_zip)
hoelzl@33639
  2088
    thus ?thesis
hoelzl@33639
  2089
      unfolding Cons zip_Cons_Cons distinct.simps
hoelzl@33639
  2090
      using Cons.hyps Cons.prems by simp
hoelzl@33639
  2091
  qed simp
hoelzl@33639
  2092
qed simp
hoelzl@33639
  2093
hoelzl@33639
  2094
lemma distinct_zipI2:
hoelzl@33639
  2095
  "distinct xs \<Longrightarrow> distinct (zip xs ys)"
hoelzl@33639
  2096
proof (induct xs arbitrary: ys)
hoelzl@33639
  2097
  case (Cons x xs)
hoelzl@33639
  2098
  show ?case
hoelzl@33639
  2099
  proof (cases ys)
hoelzl@33639
  2100
    case (Cons y ys')
hoelzl@33639
  2101
     have "(x, y) \<notin> set (zip xs ys')"
hoelzl@33639
  2102
      using Cons.prems by (auto simp: set_zip)
hoelzl@33639
  2103
    thus ?thesis
hoelzl@33639
  2104
      unfolding Cons zip_Cons_Cons distinct.simps
hoelzl@33639
  2105
      using Cons.hyps Cons.prems by simp
hoelzl@33639
  2106
  qed simp
hoelzl@33639
  2107
qed simp
haftmann@29829
  2108
wenzelm@35115
  2109
nipkow@15392
  2110
subsubsection {* @{text list_all2} *}
wenzelm@13114
  2111
kleing@14316
  2112
lemma list_all2_lengthD [intro?]: 
kleing@14316
  2113
  "list_all2 P xs ys ==> length xs = length ys"
nipkow@24349
  2114
by (simp add: list_all2_def)
haftmann@19607
  2115
haftmann@19787
  2116
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
nipkow@24349
  2117
by (simp add: list_all2_def)
haftmann@19607
  2118
haftmann@19787
  2119
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
nipkow@24349
  2120
by (simp add: list_all2_def)
haftmann@19607
  2121
haftmann@19607
  2122
lemma list_all2_Cons [iff, code]:
haftmann@19607
  2123
  "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@24349
  2124
by (auto simp add: list_all2_def)
wenzelm@13114
  2125
wenzelm@13114
  2126
lemma list_all2_Cons1:
nipkow@13145
  2127
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  2128
by (cases ys) auto
wenzelm@13114
  2129
wenzelm@13114
  2130
lemma list_all2_Cons2:
nipkow@13145
  2131
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  2132
by (cases xs) auto
wenzelm@13114
  2133
wenzelm@13142
  2134
lemma list_all2_rev [iff]:
nipkow@13145
  2135
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  2136
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  2137
kleing@13863
  2138
lemma list_all2_rev1:
kleing@13863
  2139
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  2140
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  2141
wenzelm@13114
  2142
lemma list_all2_append1:
nipkow@13145
  2143
"list_all2 P (xs @ ys) zs =
nipkow@13145
  2144
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  2145
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  2146
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  2147
apply (rule iffI)
nipkow@13145
  2148
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  2149
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  2150
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  2151
apply (simp add: ball_Un)
nipkow@13145
  2152
done
wenzelm@13114
  2153
wenzelm@13114
  2154
lemma list_all2_append2:
nipkow@13145
  2155
"list_all2 P xs (ys @ zs) =
nipkow@13145
  2156
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  2157
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  2158
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  2159
apply (rule iffI)
nipkow@13145
  2160
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  2161
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  2162
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  2163
apply (simp add: ball_Un)
nipkow@13145
  2164
done
wenzelm@13114
  2165
kleing@13863
  2166
lemma list_all2_append:
nipkow@14247
  2167
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  2168
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  2169
by (induct rule:list_induct2, simp_all)
kleing@13863
  2170
kleing@13863
  2171
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  2172
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
nipkow@24349
  2173
by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  2174
wenzelm@13114
  2175
lemma list_all2_conv_all_nth:
nipkow@13145
  2176
"list_all2 P xs ys =
nipkow@13145
  2177
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  2178
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  2179
berghofe@13883
  2180
lemma list_all2_trans:
berghofe@13883
  2181
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  2182
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  2183
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  2184
proof (induct as)
berghofe@13883
  2185
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  2186
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  2187
  proof (induct bs)
berghofe@13883
  2188
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  2189
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  2190
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  2191
  qed simp
berghofe@13883
  2192
qed simp
berghofe@13883
  2193
kleing@13863
  2194
lemma list_all2_all_nthI [intro?]:
kleing@13863
  2195
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
nipkow@24349
  2196
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2197
paulson@14395
  2198
lemma list_all2I:
paulson@14395
  2199
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
nipkow@24349
  2200
by (simp add: list_all2_def)
paulson@14395
  2201
kleing@14328
  2202
lemma list_all2_nthD:
kleing@13863
  2203
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@24349
  2204
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2205
nipkow@14302
  2206
lemma list_all2_nthD2:
nipkow@14302
  2207
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@24349
  2208
by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  2209
kleing@13863
  2210
lemma list_all2_map1: 
kleing@13863
  2211
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
nipkow@24349
  2212
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2213
kleing@13863
  2214
lemma list_all2_map2: 
kleing@13863
  2215
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
nipkow@24349
  2216
by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  2217
kleing@14316
  2218
lemma list_all2_refl [intro?]:
kleing@13863
  2219
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
nipkow@24349
  2220
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2221
kleing@13863
  2222
lemma list_all2_update_cong:
kleing@13863
  2223
  "\<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
  2224
by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  2225
kleing@13863
  2226
lemma list_all2_update_cong2:
kleing@13863
  2227
  "\<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
  2228
by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  2229
nipkow@14302
  2230
lemma list_all2_takeI [simp,intro?]:
nipkow@24526
  2231
  "list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@24526
  2232
apply (induct xs arbitrary: n ys)
nipkow@24526
  2233
 apply simp
nipkow@24526
  2234
apply (clarsimp simp add: list_all2_Cons1)
nipkow@24526
  2235
apply (case_tac n)
nipkow@24526
  2236
apply auto
nipkow@24526
  2237
done
nipkow@14302
  2238
nipkow@14302
  2239
lemma list_all2_dropI [simp,intro?]:
nipkow@24526
  2240
  "list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
nipkow@24526
  2241
apply (induct as arbitrary: n bs, simp)
nipkow@24526
  2242
apply (clarsimp simp add: list_all2_Cons1)
nipkow@24526
  2243
apply (case_tac n, simp, simp)
nipkow@24526
  2244
done
kleing@13863
  2245
kleing@14327
  2246
lemma list_all2_mono [intro?]:
nipkow@24526
  2247
  "list_all2 P xs ys \<Longrightarrow> (\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys) \<Longrightarrow> list_all2 Q xs ys"
nipkow@24526
  2248
apply (induct xs arbitrary: ys, simp)
nipkow@24526
  2249
apply (case_tac ys, auto)
nipkow@24526
  2250
done
kleing@13863
  2251
haftmann@22551
  2252
lemma list_all2_eq:
haftmann@22551
  2253
  "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
nipkow@24349
  2254
by (induct xs ys rule: list_induct2') auto
haftmann@22551
  2255
wenzelm@13142
  2256
nipkow@15392
  2257
subsubsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  2258
wenzelm@13142
  2259
lemma foldl_append [simp]:
nipkow@24526
  2260
  "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@24526
  2261
by (induct xs arbitrary: a) auto
wenzelm@13142
  2262
nipkow@14402
  2263
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  2264
by (induct xs) auto
nipkow@14402
  2265
nipkow@23096
  2266
lemma foldr_map: "foldr g (map f xs) a = foldr (g o f) xs a"
nipkow@23096
  2267
by(induct xs) simp_all
nipkow@23096
  2268
nipkow@24449
  2269
text{* For efficient code generation: avoid intermediate list. *}
haftmann@31998
  2270
lemma foldl_map[code_unfold]:
nipkow@24449
  2271
  "foldl g a (map f xs) = foldl (%a x. g a (f x)) a xs"
nipkow@23096
  2272
by(induct xs arbitrary:a) simp_all
nipkow@23096
  2273
haftmann@34978
  2274
lemma foldl_apply:
haftmann@34978
  2275
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x \<circ> h = h \<circ> g x"
haftmann@34978
  2276
  shows "foldl (\<lambda>s x. f x s) (h s) xs = h (foldl (\<lambda>s x. g x s) s xs)"
haftmann@34978
  2277
  by (rule sym, insert assms, induct xs arbitrary: s) (simp_all add: expand_fun_eq)
haftmann@31930
  2278
krauss@19770
  2279
lemma foldl_cong [fundef_cong, recdef_cong]:
krauss@18336
  2280
  "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] 
krauss@18336
  2281
  ==> foldl f a l = foldl g b k"
nipkow@24349
  2282
by (induct k arbitrary: a b l) simp_all
krauss@18336
  2283
krauss@19770
  2284
lemma foldr_cong [fundef_cong, recdef_cong]:
krauss@18336
  2285
  "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] 
krauss@18336
  2286
  ==> foldr f l a = foldr g k b"
nipkow@24349
  2287
by (induct k arbitrary: a b l) simp_all
krauss@18336
  2288
haftmann@35195
  2289
lemma foldl_fun_comm:
haftmann@35195
  2290
  assumes "\<And>x y s. f (f s x) y = f (f s y) x"
haftmann@35195
  2291
  shows "f (foldl f s xs) x = foldl f (f s x) xs"
haftmann@35195
  2292
  by (induct xs arbitrary: s)
haftmann@35195
  2293
    (simp_all add: assms)
haftmann@35195
  2294
nipkow@24449
  2295
lemma (in semigroup_add) foldl_assoc:
haftmann@25062
  2296
shows "foldl op+ (x+y) zs = x + (foldl op+ y zs)"
nipkow@24449
  2297
by (induct zs arbitrary: y) (simp_all add:add_assoc)
nipkow@24449
  2298
nipkow@24449
  2299
lemma (in monoid_add) foldl_absorb0:
haftmann@25062
  2300
shows "x + (foldl op+ 0 zs) = foldl op+ x zs"
nipkow@24449
  2301
by (induct zs) (simp_all add:foldl_assoc)
nipkow@24449
  2302
haftmann@35195
  2303
lemma foldl_rev:
haftmann@35195
  2304
  assumes "\<And>x y s. f (f s x) y = f (f s y) x"
haftmann@35195
  2305
  shows "foldl f s (rev xs) = foldl f s xs"
haftmann@35195
  2306
proof (induct xs arbitrary: s)
haftmann@35195
  2307
  case Nil then show ?case by simp
haftmann@35195
  2308
next
haftmann@35195
  2309
  case (Cons x xs) with assms show ?case by (simp add: foldl_fun_comm)
haftmann@35195
  2310
qed
haftmann@35195
  2311
nipkow@24449
  2312
nipkow@23096
  2313
text{* The ``First Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096
  2314
nipkow@23096
  2315
lemma foldl_foldr1_lemma:
nipkow@23096
  2316
 "foldl op + a xs = a + foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096
  2317
by (induct xs arbitrary: a) (auto simp:add_assoc)
nipkow@23096
  2318
nipkow@23096
  2319
corollary foldl_foldr1:
nipkow@23096
  2320
 "foldl op + 0 xs = foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096
  2321
by (simp add:foldl_foldr1_lemma)
nipkow@23096
  2322
nipkow@23096
  2323
nipkow@23096
  2324
text{* The ``Third Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096
  2325
nipkow@14402
  2326
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  2327
by (induct xs) auto
nipkow@14402
  2328
nipkow@14402
  2329
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  2330
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  2331
haftmann@25062
  2332
lemma (in ab_semigroup_add) foldr_conv_foldl: "foldr op + xs a = foldl op + a xs"
chaieb@24471
  2333
  by (induct xs, auto simp add: foldl_assoc add_commute)
chaieb@24471
  2334
wenzelm@13142
  2335
text {*
nipkow@13145
  2336
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  2337
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  2338
*}
wenzelm@13142
  2339
nipkow@24526
  2340
lemma start_le_sum: "(m::nat) <= n ==> m <= foldl (op +) n ns"
nipkow@24526
  2341
by (induct ns arbitrary: n) auto
nipkow@24526
  2342
nipkow@24526
  2343
lemma elem_le_sum: "(n::nat) : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  2344
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  2345
wenzelm@13142
  2346
lemma sum_eq_0_conv [iff]:
nipkow@24526
  2347
  "(foldl (op +) (m::nat) ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@24526
  2348
by (induct ns arbitrary: m) auto
wenzelm@13114
  2349
chaieb@24471
  2350
lemma foldr_invariant: 
chaieb@24471
  2351
  "\<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
  2352
  by (induct xs, simp_all)
chaieb@24471
  2353
chaieb@24471
  2354
lemma foldl_invariant: 
chaieb@24471
  2355
  "\<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
  2356
  by (induct xs arbitrary: x, simp_all)
chaieb@24471
  2357
haftmann@34978
  2358
lemma foldl_weak_invariant:
haftmann@34978
  2359
  assumes "P s"
haftmann@34978
  2360
    and "\<And>s x. x \<in> set xs \<Longrightarrow> P s \<Longrightarrow> P (f s x)"
haftmann@34978
  2361
  shows "P (foldl f s xs)"
haftmann@34978
  2362
  using assms by (induct xs arbitrary: s) simp_all
haftmann@34978
  2363
haftmann@31455
  2364
text {* @{const foldl} and @{const concat} *}
nipkow@24449
  2365
nipkow@24449
  2366
lemma foldl_conv_concat:
haftmann@29782
  2367
  "foldl (op @) xs xss = xs @ concat xss"
haftmann@29782
  2368
proof (induct xss arbitrary: xs)
haftmann@29782
  2369
  case Nil show ?case by simp
haftmann@29782
  2370
next
haftmann@35267
  2371
  interpret monoid_add "op @" "[]" proof qed simp_all
haftmann@29782
  2372
  case Cons then show ?case by (simp add: foldl_absorb0)
haftmann@29782
  2373
qed
haftmann@29782
  2374
haftmann@29782
  2375
lemma concat_conv_foldl: "concat xss = foldl (op @) [] xss"
haftmann@29782
  2376
  by (simp add: foldl_conv_concat)
haftmann@29782
  2377
haftmann@31455
  2378
text {* @{const Finite_Set.fold} and @{const foldl} *}
haftmann@31455
  2379
haftmann@35195
  2380
lemma (in fun_left_comm) fold_set_remdups:
haftmann@35195
  2381
  "fold f y (set xs) = foldl (\<lambda>y x. f x y) y (remdups xs)"
haftmann@35195
  2382
  by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm insert_absorb)
haftmann@35195
  2383
haftmann@31455
  2384
lemma (in fun_left_comm_idem) fold_set:
haftmann@31455
  2385
  "fold f y (set xs) = foldl (\<lambda>y x. f x y) y xs"
haftmann@31455
  2386
  by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm)
haftmann@31455
  2387
haftmann@32681
  2388
lemma (in ab_semigroup_idem_mult) fold1_set:
haftmann@32681
  2389
  assumes "xs \<noteq> []"
haftmann@32681
  2390
  shows "fold1 times (set xs) = foldl times (hd xs) (tl xs)"
haftmann@32681
  2391
proof -
haftmann@32681
  2392
  interpret fun_left_comm_idem times by (fact fun_left_comm_idem)
haftmann@32681
  2393
  from assms obtain y ys where xs: "xs = y # ys"
haftmann@32681
  2394
    by (cases xs) auto
haftmann@32681
  2395
  show ?thesis
haftmann@32681
  2396
  proof (cases "set ys = {}")
haftmann@32681
  2397
    case True with xs show ?thesis by simp
haftmann@32681
  2398
  next
haftmann@32681
  2399
    case False
haftmann@32681
  2400
    then have "fold1 times (insert y (set ys)) = fold times y (set ys)"
haftmann@32681
  2401
      by (simp only: finite_set fold1_eq_fold_idem)
haftmann@32681
  2402
    with xs show ?thesis by (simp add: fold_set mult_commute)
haftmann@32681
  2403
  qed
haftmann@32681
  2404
qed
haftmann@32681
  2405
haftmann@32681
  2406
lemma (in lattice) Inf_fin_set_fold [code_unfold]:
haftmann@32681
  2407
  "Inf_fin (set (x # xs)) = foldl inf x xs"
haftmann@32681
  2408
proof -
haftmann@32681
  2409
  interpret ab_semigroup_idem_mult "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@32681
  2410
    by (fact ab_semigroup_idem_mult_inf)
haftmann@32681
  2411
  show ?thesis
haftmann@32681
  2412
    by (simp add: Inf_fin_def fold1_set del: set.simps)
haftmann@32681
  2413
qed
haftmann@32681
  2414
haftmann@32681
  2415
lemma (in lattice) Sup_fin_set_fold [code_unfold]:
haftmann@32681
  2416
  "Sup_fin (set (x # xs)) = foldl sup x xs"
haftmann@32681
  2417
proof -
haftmann@32681
  2418
  interpret ab_semigroup_idem_mult "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@32681
  2419
    by (fact ab_semigroup_idem_mult_sup)
haftmann@32681
  2420
  show ?thesis
haftmann@32681
  2421
    by (simp add: Sup_fin_def fold1_set del: set.simps)
haftmann@32681
  2422
qed
haftmann@32681
  2423
haftmann@32681
  2424
lemma (in linorder) Min_fin_set_fold [code_unfold]:
haftmann@32681
  2425
  "Min (set (x # xs)) = foldl min x xs"
haftmann@32681
  2426
proof -
haftmann@32681
  2427
  interpret ab_semigroup_idem_mult "min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@32681
  2428
    by (fact ab_semigroup_idem_mult_min)
haftmann@32681
  2429
  show ?thesis
haftmann@32681
  2430
    by (simp add: Min_def fold1_set del: set.simps)
haftmann@32681
  2431
qed
haftmann@32681
  2432
haftmann@32681
  2433
lemma (in linorder) Max_fin_set_fold [code_unfold]:
haftmann@32681
  2434
  "Max (set (x # xs)) = foldl max x xs"
haftmann@32681
  2435
proof -
haftmann@32681
  2436
  interpret ab_semigroup_idem_mult "max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@32681
  2437
    by (fact ab_semigroup_idem_mult_max)
haftmann@32681
  2438
  show ?thesis
haftmann@32681
  2439
    by (simp add: Max_def fold1_set del: set.simps)
haftmann@32681
  2440
qed
haftmann@32681
  2441
haftmann@32681
  2442
lemma (in complete_lattice) Inf_set_fold [code_unfold]:
haftmann@32681
  2443
  "Inf (set xs) = foldl inf top xs"
haftmann@34007
  2444
proof -
haftmann@34007
  2445
  interpret fun_left_comm_idem "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@34007
  2446
    by (fact fun_left_comm_idem_inf)
haftmann@34007
  2447
  show ?thesis by (simp add: Inf_fold_inf fold_set inf_commute)
haftmann@34007
  2448
qed
haftmann@32681
  2449
haftmann@32681
  2450
lemma (in complete_lattice) Sup_set_fold [code_unfold]:
haftmann@32681
  2451
  "Sup (set xs) = foldl sup bot xs"
haftmann@34007
  2452
proof -
haftmann@34007
  2453
  interpret fun_left_comm_idem "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@34007
  2454
    by (fact fun_left_comm_idem_sup)
haftmann@34007
  2455
  show ?thesis by (simp add: Sup_fold_sup fold_set sup_commute)
haftmann@34007
  2456
qed
haftmann@34007
  2457
haftmann@34007
  2458
lemma (in complete_lattice) INFI_set_fold:
haftmann@34007
  2459
  "INFI (set xs) f = foldl (\<lambda>y x. inf (f x) y) top xs"
haftmann@34007
  2460
  unfolding INFI_def set_map [symmetric] Inf_set_fold foldl_map
haftmann@34007
  2461
    by (simp add: inf_commute)
haftmann@34007
  2462
haftmann@34007
  2463
lemma (in complete_lattice) SUPR_set_fold:
haftmann@34007
  2464
  "SUPR (set xs) f = foldl (\<lambda>y x. sup (f x) y) bot xs"
haftmann@34007
  2465
  unfolding SUPR_def set_map [symmetric] Sup_set_fold foldl_map
haftmann@34007
  2466
    by (simp add: sup_commute)
haftmann@31455
  2467
wenzelm@35115
  2468
nipkow@23096
  2469
subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
nipkow@23096
  2470
haftmann@26442
  2471
lemma listsum_append [simp]: "listsum (xs @ ys) = listsum xs + listsum ys"
nipkow@24449
  2472
by (induct xs) (simp_all add:add_assoc)
nipkow@24449
  2473
haftmann@26442
  2474
lemma listsum_rev [simp]:
haftmann@26442
  2475
  fixes xs :: "'a\<Colon>comm_monoid_add list"
haftmann@26442
  2476
  shows "listsum (rev xs) = listsum xs"
nipkow@24449
  2477
by (induct xs) (simp_all add:add_ac)
nipkow@24449
  2478
nipkow@31022
  2479
lemma listsum_map_remove1:
nipkow@31022
  2480
fixes f :: "'a \<Rightarrow> ('b::comm_monoid_add)"
nipkow@31022
  2481
shows "x : set xs \<Longrightarrow> listsum(map f xs) = f x + listsum(map f (remove1 x xs))"
nipkow@31022
  2482
by (induct xs)(auto simp add:add_ac)
nipkow@31022
  2483
nipkow@31022
  2484
lemma list_size_conv_listsum:
nipkow@31022
  2485
  "list_size f xs = listsum (map f xs) + size xs"
nipkow@31022
  2486
by(induct xs) auto
nipkow@31022
  2487
haftmann@26442
  2488
lemma listsum_foldr: "listsum xs = foldr (op +) xs 0"
haftmann@26442
  2489
by (induct xs) auto
haftmann@26442
  2490
haftmann@26442
  2491
lemma length_concat: "length (concat xss) = listsum (map length xss)"
haftmann@26442
  2492
by (induct xss) simp_all
nipkow@23096
  2493
hoelzl@33639
  2494
lemma listsum_map_filter:
hoelzl@33639
  2495
  fixes f :: "'a \<Rightarrow> 'b \<Colon> comm_monoid_add"
hoelzl@33639
  2496
  assumes "\<And> x. \<lbrakk> x \<in> set xs ; \<not> P x \<rbrakk> \<Longrightarrow> f x = 0"
hoelzl@33639
  2497
  shows "listsum (map f (filter P xs)) = listsum (map f xs)"
hoelzl@33639
  2498
using assms by (induct xs) auto
hoelzl@33639
  2499
nipkow@24449
  2500
text{* For efficient code generation ---
nipkow@24449
  2501
       @{const listsum} is not tail recursive but @{const foldl} is. *}
haftmann@31998
  2502
lemma listsum[code_unfold]: "listsum xs = foldl (op +) 0 xs"
nipkow@23096
  2503
by(simp add:listsum_foldr foldl_foldr1)
nipkow@23096
  2504
nipkow@31077
  2505
lemma distinct_listsum_conv_Setsum:
nipkow@31077
  2506
  "distinct xs \<Longrightarrow> listsum xs = Setsum(set xs)"
nipkow@31077
  2507
by (induct xs) simp_all
nipkow@31077
  2508
nipkow@24449
  2509
nipkow@23096
  2510
text{* Some syntactic sugar for summing a function over a list: *}
nipkow@23096
  2511
nipkow@23096
  2512
syntax
nipkow@23096
  2513
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
nipkow@23096
  2514
syntax (xsymbols)
nipkow@23096
  2515
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096
  2516
syntax (HTML output)
nipkow@23096
  2517
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096
  2518
nipkow@23096
  2519
translations -- {* Beware of argument permutation! *}
haftmann@34941
  2520
  "SUM x<-xs. b" == "CONST listsum (CONST map (%x. b) xs)"
haftmann@34941
  2521
  "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"
nipkow@23096
  2522
haftmann@26442
  2523
lemma listsum_triv: "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
haftmann@26442
  2524
  by (induct xs) (simp_all add: left_distrib)
haftmann@26442
  2525
nipkow@23096
  2526
lemma listsum_0 [simp]: "(\<Sum>x\<leftarrow>xs. 0) = 0"
haftmann@26442
  2527
  by (induct xs) (simp_all add: left_distrib)
nipkow@23096
  2528
nipkow@23096
  2529
text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
nipkow@23096
  2530
lemma uminus_listsum_map:
haftmann@26442
  2531
  fixes f :: "'a \<Rightarrow> 'b\<Colon>ab_group_add"
haftmann@26442
  2532
  shows "- listsum (map f xs) = (listsum (map (uminus o f) xs))"
haftmann@26442
  2533
by (induct xs) simp_all
nipkow@23096
  2534
huffman@31258
  2535
lemma listsum_addf:
huffman@31258
  2536
  fixes f g :: "'a \<Rightarrow> 'b::comm_monoid_add"
huffman@31258
  2537
  shows "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
huffman@31258
  2538
by (induct xs) (simp_all add: algebra_simps)
huffman@31258
  2539
huffman@31258
  2540
lemma listsum_subtractf:
huffman@31258
  2541
  fixes f g :: "'a \<Rightarrow> 'b::ab_group_add"
huffman@31258
  2542
  shows "(\<Sum>x\<leftarrow>xs. f x - g x) = listsum (map f xs) - listsum (map g xs)"
huffman@31258
  2543
by (induct xs) simp_all
huffman@31258
  2544
huffman@31258
  2545
lemma listsum_const_mult:
huffman@31258
  2546
  fixes f :: "'a \<Rightarrow> 'b::semiring_0"
huffman@31258
  2547
  shows "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
huffman@31258
  2548
by (induct xs, simp_all add: algebra_simps)
huffman@31258
  2549
huffman@31258
  2550
lemma listsum_mult_const:
huffman@31258
  2551
  fixes f :: "'a \<Rightarrow> 'b::semiring_0"
huffman@31258
  2552
  shows "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
huffman@31258
  2553
by (induct xs, simp_all add: algebra_simps)
huffman@31258
  2554
huffman@31258
  2555
lemma listsum_abs:
haftmann@35028
  2556
  fixes xs :: "'a::ordered_ab_group_add_abs list"
huffman@31258
  2557
  shows "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
huffman@31258
  2558
by (induct xs, simp, simp add: order_trans [OF abs_triangle_ineq])
huffman@31258
  2559
huffman@31258
  2560
lemma listsum_mono:
haftmann@35028
  2561
  fixes f g :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_ab_semigroup_add}"
huffman@31258
  2562
  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
  2563
by (induct xs, simp, simp add: add_mono)
huffman@31258
  2564
wenzelm@13114
  2565
nipkow@24645
  2566
subsubsection {* @{text upt} *}
wenzelm@13114
  2567
nipkow@17090
  2568
lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
nipkow@17090
  2569
-- {* simp does not terminate! *}
nipkow@13145
  2570
by (induct j) auto
wenzelm@13142
  2571
nipkow@32005
  2572
lemmas upt_rec_number_of[simp] = upt_rec[of "number_of m" "number_of n", standard]
nipkow@32005
  2573
nipkow@15425
  2574
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
nipkow@13145
  2575
by (subst upt_rec) simp
wenzelm@13114
  2576
nipkow@15425
  2577
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
nipkow@15281
  2578
by(induct j)simp_all
nipkow@15281
  2579
nipkow@15281
  2580
lemma upt_eq_Cons_conv:
nipkow@24526
  2581
 "([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
nipkow@24526
  2582
apply(induct j arbitrary: x xs)
nipkow@15281
  2583
 apply simp
nipkow@15281
  2584
apply(clarsimp simp add: append_eq_Cons_conv)
nipkow@15281
  2585
apply arith
nipkow@15281
  2586
done
nipkow@15281
  2587
nipkow@15425
  2588
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
nipkow@13145
  2589
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  2590
by simp
wenzelm@13114
  2591
nipkow@15425
  2592
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
haftmann@26734
  2593
  by (simp add: upt_rec)
wenzelm@13114
  2594
nipkow@15425
  2595
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
nipkow@13145
  2596
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  2597
by (induct k) auto
wenzelm@13114
  2598
nipkow@15425
  2599
lemma length_upt [simp]: "length [i..<j] = j - i"
nipkow@13145
  2600
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  2601
nipkow@15425
  2602
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
nipkow@13145
  2603
apply (induct j)
nipkow@13145
  2604
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  2605
done
wenzelm@13114
  2606
nipkow@17906
  2607
nipkow@17906
  2608
lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
nipkow@17906
  2609
by(simp add:upt_conv_Cons)
nipkow@17906
  2610
nipkow@17906
  2611
lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
nipkow@17906
  2612
apply(cases j)
nipkow@17906
  2613
 apply simp
nipkow@17906
  2614
by(simp add:upt_Suc_append)
nipkow@17906
  2615
nipkow@24526
  2616
lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]"
nipkow@24526
  2617
apply (induct m arbitrary: i, simp)
nipkow@13145
  2618
apply (subst upt_rec)
nipkow@13145
  2619
apply (rule sym)
nipkow@13145
  2620
apply (subst upt_rec)
nipkow@13145
  2621
apply (simp del: upt.simps)
nipkow@13145
  2622
done
nipkow@3507
  2623
nipkow@17501
  2624
lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
nipkow@17501
  2625
apply(induct j)
nipkow@17501
  2626
apply auto
nipkow@17501
  2627
done
nipkow@17501
  2628
nipkow@24645
  2629
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]"
nipkow@13145
  2630
by (induct n) auto
wenzelm@13114
  2631
nipkow@24526
  2632
lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
nipkow@24526
  2633
apply (induct n m  arbitrary: i rule: diff_induct)
nipkow@13145
  2634
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  2635
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  2636
done
wenzelm@13114
  2637
berghofe@13883
  2638
lemma nth_take_lemma:
nipkow@24526
  2639
  "k <= length xs ==> k <= length ys ==>
berghofe@13883
  2640
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
nipkow@24526
  2641
apply (atomize, induct k arbitrary: xs ys)
paulson@14208
  2642
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  2643
txt {* Both lists must be non-empty *}
paulson@14208
  2644
apply (case_tac xs, simp)
paulson@14208
  2645
apply (case_tac ys, clarify)
nipkow@13145
  2646
 apply (simp (no_asm_use))
nipkow@13145
  2647
apply clarify
nipkow@13145
  2648
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  2649
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  2650
apply blast
nipkow@13145
  2651
done
wenzelm@13114
  2652
wenzelm@13114
  2653
lemma nth_equalityI:
wenzelm@13114
  2654
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  2655
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  2656
apply (simp_all add: take_all)
nipkow@13145
  2657
done
wenzelm@13142
  2658
haftmann@24796
  2659
lemma map_nth:
haftmann@24796
  2660
  "map (\<lambda>i. xs ! i) [0..<length xs] = xs"
haftmann@24796
  2661
  by (rule nth_equalityI, auto)
haftmann@24796
  2662
kleing@13863
  2663
(* needs nth_equalityI *)
kleing@13863
  2664
lemma list_all2_antisym:
kleing@13863
  2665
  "\<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
  2666
  \<Longrightarrow> xs = ys"
kleing@13863
  2667
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  2668
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  2669
  done
kleing@13863
  2670
wenzelm@13142
  2671
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  2672
-- {* The famous take-lemma. *}
nipkow@13145
  2673
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  2674
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  2675
done
wenzelm@13142
  2676
wenzelm@13142
  2677
nipkow@15302
  2678
lemma take_Cons':
nipkow@15302
  2679
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@15302
  2680
by (cases n) simp_all
ni