src/HOL/List.thy
author nipkow
Fri Dec 16 16:59:32 2005 +0100 (2005-12-16)
changeset 18423 d7859164447f
parent 18336 1a2e30b37ed3
child 18447 da548623916a
permissions -rw-r--r--
new lemmas
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(*  Title:      HOL/List.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow
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*)
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header {* The datatype of finite lists *}
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theory List
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imports PreList
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begin
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datatype 'a list =
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    Nil    ("[]")
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  | Cons 'a  "'a list"    (infixr "#" 65)
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subsection{*Basic list processing functions*}
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consts
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  "@" :: "'a list => 'a list => 'a list"    (infixr 65)
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  filter:: "('a => bool) => 'a list => 'a list"
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  concat:: "'a list list => 'a list"
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  foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"
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  foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"
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  hd:: "'a list => 'a"
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  tl:: "'a list => 'a list"
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  last:: "'a list => 'a"
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  butlast :: "'a list => 'a list"
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  set :: "'a list => 'a set"
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  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool"
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  map :: "('a=>'b) => ('a list => 'b list)"
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  nth :: "'a list => nat => 'a"    (infixl "!" 100)
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  list_update :: "'a list => nat => 'a => 'a list"
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  take:: "nat => 'a list => 'a list"
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  drop:: "nat => 'a list => 'a list"
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  takeWhile :: "('a => bool) => 'a list => 'a list"
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  dropWhile :: "('a => bool) => 'a list => 'a list"
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  rev :: "'a list => 'a list"
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  zip :: "'a list => 'b list => ('a * 'b) list"
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  upt :: "nat => nat => nat list" ("(1[_..</_'])")
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  remdups :: "'a list => 'a list"
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  remove1 :: "'a => 'a list => 'a list"
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  null:: "'a list => bool"
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  "distinct":: "'a list => bool"
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  replicate :: "nat => 'a => 'a list"
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  rotate1 :: "'a list \<Rightarrow> 'a list"
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  rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
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  sublist :: "'a list => nat set => 'a list"
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(* For efficiency *)
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  mem :: "'a => 'a list => bool"    (infixl 55)
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  list_inter :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
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  list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
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  list_all:: "('a => bool) => ('a list => bool)"
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  itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
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  filtermap :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list"
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  map_filter :: "('a => 'b) => ('a => bool) => 'a list => 'b list"
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nonterminals lupdbinds lupdbind
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syntax
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  -- {* list Enumeration *}
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  "@list" :: "args => 'a list"    ("[(_)]")
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  -- {* Special syntax for filter *}
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_:_./ _])")
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  -- {* list update *}
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  "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
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  "" :: "lupdbind => lupdbinds"    ("_")
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  "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
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  "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
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  upto:: "nat => nat => nat list"    ("(1[_../_])")
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translations
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  "[x, xs]" == "x#[xs]"
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  "[x]" == "x#[]"
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  "[x:xs . P]"== "filter (%x. P) xs"
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  "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"
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  "xs[i:=x]" == "list_update xs i x"
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  "[i..j]" == "[i..<(Suc j)]"
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syntax (xsymbols)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
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syntax (HTML output)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
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text {*
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  Function @{text size} is overloaded for all datatypes. Users may
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  refer to the list version as @{text length}. *}
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syntax length :: "'a list => nat"
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translations "length" => "size :: _ list => nat"
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typed_print_translation {*
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  let
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    fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] =
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          Syntax.const "length" $ t
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      | size_tr' _ _ _ = raise Match;
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  in [("size", size_tr')] end
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*}
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primrec
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  "hd(x#xs) = x"
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primrec
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  "tl([]) = []"
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  "tl(x#xs) = xs"
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primrec
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  "null([]) = True"
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  "null(x#xs) = False"
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primrec
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  "last(x#xs) = (if xs=[] then x else last xs)"
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primrec
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  "butlast []= []"
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  "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
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primrec
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  "set [] = {}"
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  "set (x#xs) = insert x (set xs)"
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primrec
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  "map f [] = []"
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  "map f (x#xs) = f(x)#map f xs"
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primrec
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  append_Nil:"[]@ys = ys"
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  append_Cons: "(x#xs)@ys = x#(xs@ys)"
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primrec
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  "rev([]) = []"
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  "rev(x#xs) = rev(xs) @ [x]"
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primrec
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  "filter P [] = []"
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  "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
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primrec
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  foldl_Nil:"foldl f a [] = a"
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  foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
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primrec
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  "foldr f [] a = a"
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  "foldr f (x#xs) a = f x (foldr f xs a)"
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primrec
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  "concat([]) = []"
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  "concat(x#xs) = x @ concat(xs)"
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primrec
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  drop_Nil:"drop n [] = []"
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  drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  take_Nil:"take n [] = []"
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  take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  "[][i:=v] = []"
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  "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])"
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primrec
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  "takeWhile P [] = []"
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  "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
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primrec
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  "dropWhile P [] = []"
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  "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
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primrec
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  "zip xs [] = []"
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  zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
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primrec
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  upt_0: "[i..<0] = []"
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  upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
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primrec
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  "distinct [] = True"
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  "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
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primrec
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  "remdups [] = []"
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  "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
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primrec
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  "remove1 x [] = []"
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  "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)"
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primrec
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  replicate_0: "replicate 0 x = []"
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  replicate_Suc: "replicate (Suc n) x = x # replicate n x"
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defs
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rotate1_def: "rotate1 xs == (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"
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rotate_def:  "rotate n == rotate1 ^ n"
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list_all2_def:
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 "list_all2 P xs ys ==
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  length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)"
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sublist_def:
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 "sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..<size xs]))"
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primrec
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  "x mem [] = False"
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  "x mem (y#ys) = (if y=x then True else x mem ys)"
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primrec
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 "list_inter [] bs = []"
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 "list_inter (a#as) bs =
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  (if a \<in> set bs then a#(list_inter as bs) else list_inter as bs)"
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primrec
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  "list_all P [] = True"
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  "list_all P (x#xs) = (P(x) \<and> list_all P xs)"
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primrec
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"list_ex P [] = False"
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"list_ex P (x#xs) = (P x \<or> list_ex P xs)"
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primrec
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 "filtermap f [] = []"
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 "filtermap f (x#xs) =
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    (case f x of None \<Rightarrow> filtermap f xs
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     | Some y \<Rightarrow> y # (filtermap f xs))"
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primrec
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  "map_filter f P [] = []"
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  "map_filter f P (x#xs) = (if P x then f x # map_filter f P xs else 
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               map_filter f P xs)"
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primrec
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"itrev [] ys = ys"
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"itrev (x#xs) ys = itrev xs (x#ys)"
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lemma not_Cons_self [simp]: "xs \<noteq> x # xs"
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by (induct xs) auto
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lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
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lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
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by (induct xs) auto
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lemma length_induct:
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"(!!xs. \<forall>ys. length ys < length xs --> P ys ==> P xs) ==> P xs"
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by (rule measure_induct [of length]) iprover
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subsubsection {* @{text length} *}
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text {*
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Needs to come before @{text "@"} because of theorem @{text
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append_eq_append_conv}.
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*}
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lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
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by (induct xs) auto
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lemma length_map [simp]: "length (map f xs) = length xs"
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by (induct xs) auto
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lemma length_rev [simp]: "length (rev xs) = length xs"
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by (induct xs) auto
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lemma length_tl [simp]: "length (tl xs) = length xs - 1"
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by (cases xs) auto
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lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
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by (induct xs) auto
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lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
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by (induct xs) auto
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lemma length_Suc_conv:
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"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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by (induct xs) auto
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lemma Suc_length_conv:
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"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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apply (induct xs, simp, simp)
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apply blast
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done
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lemma impossible_Cons [rule_format]: 
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  "length xs <= length ys --> xs = x # ys = False"
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apply (induct xs, auto)
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done
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lemma list_induct2[consumes 1]: "\<And>ys.
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 \<lbrakk> length xs = length ys;
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   P [] [];
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   \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
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 \<Longrightarrow> P xs ys"
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apply(induct xs)
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 apply simp
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apply(case_tac ys)
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 apply simp
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apply(simp)
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done
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subsubsection {* @{text "@"} -- append *}
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lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
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by (induct xs) auto
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lemma append_Nil2 [simp]: "xs @ [] = xs"
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by (induct xs) auto
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lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
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by (induct xs) auto
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lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
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by (induct xs) auto
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lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
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by (induct xs) auto
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lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
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by (induct xs) auto
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lemma append_eq_append_conv [simp]:
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 "!!ys. length xs = length ys \<or> length us = length vs
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 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
berghofe@13883
   345
apply (induct xs)
paulson@14208
   346
 apply (case_tac ys, simp, force)
paulson@14208
   347
apply (case_tac ys, force, simp)
nipkow@13145
   348
done
wenzelm@13142
   349
nipkow@14495
   350
lemma append_eq_append_conv2: "!!ys zs ts.
nipkow@14495
   351
 (xs @ ys = zs @ ts) =
nipkow@14495
   352
 (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
nipkow@14495
   353
apply (induct xs)
nipkow@14495
   354
 apply fastsimp
nipkow@14495
   355
apply(case_tac zs)
nipkow@14495
   356
 apply simp
nipkow@14495
   357
apply fastsimp
nipkow@14495
   358
done
nipkow@14495
   359
wenzelm@13142
   360
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145
   361
by simp
wenzelm@13142
   362
wenzelm@13142
   363
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145
   364
by simp
wenzelm@13114
   365
wenzelm@13142
   366
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   367
by simp
wenzelm@13114
   368
wenzelm@13142
   369
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   370
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   371
wenzelm@13142
   372
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   373
using append_same_eq [of "[]"] by auto
wenzelm@13114
   374
wenzelm@13142
   375
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   376
by (induct xs) auto
wenzelm@13114
   377
wenzelm@13142
   378
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   379
by (induct xs) auto
wenzelm@13114
   380
wenzelm@13142
   381
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   382
by (simp add: hd_append split: list.split)
wenzelm@13114
   383
wenzelm@13142
   384
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   385
by (simp split: list.split)
wenzelm@13114
   386
wenzelm@13142
   387
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   388
by (simp add: tl_append split: list.split)
wenzelm@13114
   389
wenzelm@13114
   390
nipkow@14300
   391
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   392
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   393
by(cases ys) auto
nipkow@14300
   394
nipkow@15281
   395
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   396
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   397
by(cases ys) auto
nipkow@15281
   398
nipkow@14300
   399
wenzelm@13142
   400
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   401
wenzelm@13114
   402
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   403
by simp
wenzelm@13114
   404
wenzelm@13142
   405
lemma Cons_eq_appendI:
nipkow@13145
   406
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   407
by (drule sym) simp
wenzelm@13114
   408
wenzelm@13142
   409
lemma append_eq_appendI:
nipkow@13145
   410
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   411
by (drule sym) simp
wenzelm@13114
   412
wenzelm@13114
   413
wenzelm@13142
   414
text {*
nipkow@13145
   415
Simplification procedure for all list equalities.
nipkow@13145
   416
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   417
- both lists end in a singleton list,
nipkow@13145
   418
- or both lists end in the same list.
wenzelm@13142
   419
*}
wenzelm@13142
   420
wenzelm@13142
   421
ML_setup {*
nipkow@3507
   422
local
nipkow@3507
   423
wenzelm@13122
   424
val append_assoc = thm "append_assoc";
wenzelm@13122
   425
val append_Nil = thm "append_Nil";
wenzelm@13122
   426
val append_Cons = thm "append_Cons";
wenzelm@13122
   427
val append1_eq_conv = thm "append1_eq_conv";
wenzelm@13122
   428
val append_same_eq = thm "append_same_eq";
wenzelm@13122
   429
wenzelm@13114
   430
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462
   431
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
wenzelm@13462
   432
  | last (Const("List.op @",_) $ _ $ ys) = last ys
wenzelm@13462
   433
  | last t = t;
wenzelm@13114
   434
wenzelm@13114
   435
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462
   436
  | list1 _ = false;
wenzelm@13114
   437
wenzelm@13114
   438
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462
   439
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
wenzelm@13462
   440
  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462
   441
  | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114
   442
wenzelm@16973
   443
val rearr_ss = HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons];
wenzelm@16973
   444
wenzelm@16973
   445
fun list_eq sg ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   446
  let
wenzelm@13462
   447
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   448
    fun rearr conv =
wenzelm@13462
   449
      let
wenzelm@13462
   450
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   451
        val Type(_,listT::_) = eqT
wenzelm@13462
   452
        val appT = [listT,listT] ---> listT
wenzelm@13462
   453
        val app = Const("List.op @",appT)
wenzelm@13462
   454
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   455
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@17956
   456
        val thm = Goal.prove sg [] [] eq
wenzelm@17877
   457
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
skalberg@15531
   458
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   459
wenzelm@13462
   460
  in
wenzelm@13462
   461
    if list1 lastl andalso list1 lastr then rearr append1_eq_conv
wenzelm@13462
   462
    else if lastl aconv lastr then rearr append_same_eq
skalberg@15531
   463
    else NONE
wenzelm@13462
   464
  end;
wenzelm@13462
   465
wenzelm@13114
   466
in
wenzelm@13462
   467
wenzelm@13462
   468
val list_eq_simproc =
wenzelm@13462
   469
  Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq;
wenzelm@13462
   470
wenzelm@13114
   471
end;
wenzelm@13114
   472
wenzelm@13114
   473
Addsimprocs [list_eq_simproc];
wenzelm@13114
   474
*}
wenzelm@13114
   475
wenzelm@13114
   476
nipkow@15392
   477
subsubsection {* @{text map} *}
wenzelm@13114
   478
wenzelm@13142
   479
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   480
by (induct xs) simp_all
wenzelm@13114
   481
wenzelm@13142
   482
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   483
by (rule ext, induct_tac xs) auto
wenzelm@13114
   484
wenzelm@13142
   485
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   486
by (induct xs) auto
wenzelm@13114
   487
wenzelm@13142
   488
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   489
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   490
wenzelm@13142
   491
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   492
by (induct xs) auto
wenzelm@13114
   493
nipkow@13737
   494
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   495
by (induct xs) auto
nipkow@13737
   496
wenzelm@13366
   497
lemma map_cong [recdef_cong]:
nipkow@13145
   498
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   499
-- {* a congruence rule for @{text map} *}
nipkow@13737
   500
by simp
wenzelm@13114
   501
wenzelm@13142
   502
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   503
by (cases xs) auto
wenzelm@13114
   504
wenzelm@13142
   505
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   506
by (cases xs) auto
wenzelm@13114
   507
nipkow@14025
   508
lemma map_eq_Cons_conv[iff]:
nipkow@14025
   509
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   510
by (cases xs) auto
wenzelm@13114
   511
nipkow@14025
   512
lemma Cons_eq_map_conv[iff]:
nipkow@14025
   513
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   514
by (cases ys) auto
nipkow@14025
   515
nipkow@14111
   516
lemma ex_map_conv:
nipkow@14111
   517
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
nipkow@14111
   518
by(induct ys, auto)
nipkow@14111
   519
nipkow@15110
   520
lemma map_eq_imp_length_eq:
nipkow@15110
   521
  "!!xs. map f xs = map f ys ==> length xs = length ys"
nipkow@15110
   522
apply (induct ys)
nipkow@15110
   523
 apply simp
nipkow@15110
   524
apply(simp (no_asm_use))
nipkow@15110
   525
apply clarify
nipkow@15110
   526
apply(simp (no_asm_use))
nipkow@15110
   527
apply fast
nipkow@15110
   528
done
nipkow@15110
   529
nipkow@15110
   530
lemma map_inj_on:
nipkow@15110
   531
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   532
  ==> xs = ys"
nipkow@15110
   533
apply(frule map_eq_imp_length_eq)
nipkow@15110
   534
apply(rotate_tac -1)
nipkow@15110
   535
apply(induct rule:list_induct2)
nipkow@15110
   536
 apply simp
nipkow@15110
   537
apply(simp)
nipkow@15110
   538
apply (blast intro:sym)
nipkow@15110
   539
done
nipkow@15110
   540
nipkow@15110
   541
lemma inj_on_map_eq_map:
nipkow@15110
   542
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   543
by(blast dest:map_inj_on)
nipkow@15110
   544
wenzelm@13114
   545
lemma map_injective:
nipkow@14338
   546
 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@14338
   547
by (induct ys) (auto dest!:injD)
wenzelm@13114
   548
nipkow@14339
   549
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   550
by(blast dest:map_injective)
nipkow@14339
   551
wenzelm@13114
   552
lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@17589
   553
by (iprover dest: map_injective injD intro: inj_onI)
wenzelm@13114
   554
wenzelm@13114
   555
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   556
apply (unfold inj_on_def, clarify)
nipkow@13145
   557
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   558
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   559
apply blast
nipkow@13145
   560
done
wenzelm@13114
   561
nipkow@14339
   562
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   563
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   564
nipkow@15303
   565
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
   566
apply(rule inj_onI)
nipkow@15303
   567
apply(erule map_inj_on)
nipkow@15303
   568
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
   569
done
nipkow@15303
   570
kleing@14343
   571
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   572
by (induct xs, auto)
wenzelm@13114
   573
nipkow@14402
   574
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   575
by (induct xs) auto
nipkow@14402
   576
nipkow@15110
   577
lemma map_fst_zip[simp]:
nipkow@15110
   578
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   579
by (induct rule:list_induct2, simp_all)
nipkow@15110
   580
nipkow@15110
   581
lemma map_snd_zip[simp]:
nipkow@15110
   582
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   583
by (induct rule:list_induct2, simp_all)
nipkow@15110
   584
nipkow@15110
   585
nipkow@15392
   586
subsubsection {* @{text rev} *}
wenzelm@13114
   587
wenzelm@13142
   588
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   589
by (induct xs) auto
wenzelm@13114
   590
wenzelm@13142
   591
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   592
by (induct xs) auto
wenzelm@13114
   593
kleing@15870
   594
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870
   595
by auto
kleing@15870
   596
wenzelm@13142
   597
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   598
by (induct xs) auto
wenzelm@13114
   599
wenzelm@13142
   600
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   601
by (induct xs) auto
wenzelm@13114
   602
kleing@15870
   603
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870
   604
by (cases xs) auto
kleing@15870
   605
kleing@15870
   606
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870
   607
by (cases xs) auto
kleing@15870
   608
wenzelm@13142
   609
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"
paulson@14208
   610
apply (induct xs, force)
paulson@14208
   611
apply (case_tac ys, simp, force)
nipkow@13145
   612
done
wenzelm@13114
   613
nipkow@15439
   614
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
   615
by(simp add:inj_on_def)
nipkow@15439
   616
wenzelm@13366
   617
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   618
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489
   619
apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145
   620
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   621
done
wenzelm@13114
   622
nipkow@13145
   623
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114
   624
wenzelm@13366
   625
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   626
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   627
by (induct xs rule: rev_induct) auto
wenzelm@13114
   628
wenzelm@13366
   629
lemmas rev_cases = rev_exhaust
wenzelm@13366
   630
nipkow@18423
   631
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
nipkow@18423
   632
by(rule rev_cases[of xs]) auto
nipkow@18423
   633
wenzelm@13114
   634
nipkow@15392
   635
subsubsection {* @{text set} *}
wenzelm@13114
   636
wenzelm@13142
   637
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   638
by (induct xs) auto
wenzelm@13114
   639
wenzelm@13142
   640
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   641
by (induct xs) auto
wenzelm@13114
   642
nipkow@17830
   643
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
nipkow@17830
   644
by(cases xs) auto
oheimb@14099
   645
wenzelm@13142
   646
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   647
by auto
wenzelm@13114
   648
oheimb@14099
   649
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   650
by auto
oheimb@14099
   651
wenzelm@13142
   652
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   653
by (induct xs) auto
wenzelm@13114
   654
nipkow@15245
   655
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   656
by(induct xs) auto
nipkow@15245
   657
wenzelm@13142
   658
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   659
by (induct xs) auto
wenzelm@13114
   660
wenzelm@13142
   661
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   662
by (induct xs) auto
wenzelm@13114
   663
wenzelm@13142
   664
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   665
by (induct xs) auto
wenzelm@13114
   666
nipkow@15425
   667
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
paulson@14208
   668
apply (induct j, simp_all)
paulson@14208
   669
apply (erule ssubst, auto)
nipkow@13145
   670
done
wenzelm@13114
   671
wenzelm@13142
   672
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
paulson@15113
   673
proof (induct xs)
paulson@15113
   674
  case Nil show ?case by simp
paulson@15113
   675
  case (Cons a xs)
paulson@15113
   676
  show ?case
paulson@15113
   677
  proof 
paulson@15113
   678
    assume "x \<in> set (a # xs)"
paulson@15113
   679
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   680
      by (simp, blast intro: Cons_eq_appendI)
paulson@15113
   681
  next
paulson@15113
   682
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   683
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
paulson@15113
   684
    show "x \<in> set (a # xs)" 
paulson@15113
   685
      by (cases ys, auto simp add: eq)
paulson@15113
   686
  qed
paulson@15113
   687
qed
wenzelm@13142
   688
nipkow@18049
   689
lemma in_set_conv_decomp_first:
nipkow@18049
   690
 "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
nipkow@18049
   691
proof (induct xs)
nipkow@18049
   692
  case Nil show ?case by simp
nipkow@18049
   693
next
nipkow@18049
   694
  case (Cons a xs)
nipkow@18049
   695
  show ?case
nipkow@18049
   696
  proof cases
nipkow@18049
   697
    assume "x = a" thus ?case using Cons by force
nipkow@18049
   698
  next
nipkow@18049
   699
    assume "x \<noteq> a"
nipkow@18049
   700
    show ?case
nipkow@18049
   701
    proof
nipkow@18049
   702
      assume "x \<in> set (a # xs)"
nipkow@18049
   703
      from prems show "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@18049
   704
	by(fastsimp intro!: Cons_eq_appendI)
nipkow@18049
   705
    next
nipkow@18049
   706
      assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@18049
   707
      then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
nipkow@18049
   708
      show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq)
nipkow@18049
   709
    qed
nipkow@18049
   710
  qed
nipkow@18049
   711
qed
nipkow@18049
   712
nipkow@18049
   713
lemmas split_list       = in_set_conv_decomp[THEN iffD1, standard]
nipkow@18049
   714
lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard]
nipkow@18049
   715
nipkow@18049
   716
paulson@13508
   717
lemma finite_list: "finite A ==> EX l. set l = A"
paulson@13508
   718
apply (erule finite_induct, auto)
paulson@13508
   719
apply (rule_tac x="x#l" in exI, auto)
paulson@13508
   720
done
paulson@13508
   721
kleing@14388
   722
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
   723
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
   724
paulson@15168
   725
nipkow@15392
   726
subsubsection {* @{text filter} *}
wenzelm@13114
   727
wenzelm@13142
   728
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   729
by (induct xs) auto
wenzelm@13114
   730
nipkow@15305
   731
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
   732
by (induct xs) simp_all
nipkow@15305
   733
wenzelm@13142
   734
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   735
by (induct xs) auto
wenzelm@13114
   736
nipkow@16998
   737
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998
   738
by (induct xs) (auto simp add: le_SucI)
nipkow@16998
   739
nipkow@18423
   740
lemma sum_length_filter_compl:
nipkow@18423
   741
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
nipkow@18423
   742
by(induct xs) simp_all
nipkow@18423
   743
wenzelm@13142
   744
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   745
by (induct xs) auto
wenzelm@13114
   746
wenzelm@13142
   747
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   748
by (induct xs) auto
wenzelm@13114
   749
nipkow@16998
   750
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@16998
   751
  by (induct xs) simp_all
nipkow@16998
   752
nipkow@16998
   753
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998
   754
apply (induct xs)
nipkow@16998
   755
 apply auto
nipkow@16998
   756
apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998
   757
apply simp
nipkow@16998
   758
done
wenzelm@13114
   759
nipkow@16965
   760
lemma filter_map:
nipkow@16965
   761
  "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965
   762
by (induct xs) simp_all
nipkow@16965
   763
nipkow@16965
   764
lemma length_filter_map[simp]:
nipkow@16965
   765
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965
   766
by (simp add:filter_map)
nipkow@16965
   767
wenzelm@13142
   768
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   769
by auto
wenzelm@13114
   770
nipkow@15246
   771
lemma length_filter_less:
nipkow@15246
   772
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
   773
proof (induct xs)
nipkow@15246
   774
  case Nil thus ?case by simp
nipkow@15246
   775
next
nipkow@15246
   776
  case (Cons x xs) thus ?case
nipkow@15246
   777
    apply (auto split:split_if_asm)
nipkow@15246
   778
    using length_filter_le[of P xs] apply arith
nipkow@15246
   779
  done
nipkow@15246
   780
qed
wenzelm@13114
   781
nipkow@15281
   782
lemma length_filter_conv_card:
nipkow@15281
   783
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
   784
proof (induct xs)
nipkow@15281
   785
  case Nil thus ?case by simp
nipkow@15281
   786
next
nipkow@15281
   787
  case (Cons x xs)
nipkow@15281
   788
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
   789
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
   790
  show ?case (is "?l = card ?S'")
nipkow@15281
   791
  proof (cases)
nipkow@15281
   792
    assume "p x"
nipkow@15281
   793
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@15281
   794
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   795
    have "length (filter p (x # xs)) = Suc(card ?S)"
nipkow@15281
   796
      using Cons by simp
nipkow@15281
   797
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
nipkow@15281
   798
      by (simp add: card_image inj_Suc)
nipkow@15281
   799
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   800
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
   801
    finally show ?thesis .
nipkow@15281
   802
  next
nipkow@15281
   803
    assume "\<not> p x"
nipkow@15281
   804
    hence eq: "?S' = Suc ` ?S"
nipkow@15281
   805
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   806
    have "length (filter p (x # xs)) = card ?S"
nipkow@15281
   807
      using Cons by simp
nipkow@15281
   808
    also have "\<dots> = card(Suc ` ?S)" using fin
nipkow@15281
   809
      by (simp add: card_image inj_Suc)
nipkow@15281
   810
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   811
      by (simp add:card_insert_if)
nipkow@15281
   812
    finally show ?thesis .
nipkow@15281
   813
  qed
nipkow@15281
   814
qed
nipkow@15281
   815
nipkow@17629
   816
lemma Cons_eq_filterD:
nipkow@17629
   817
 "x#xs = filter P ys \<Longrightarrow>
nipkow@17629
   818
  \<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
   819
  (concl is "\<exists>us vs. ?P ys us vs")
nipkow@17629
   820
proof(induct ys)
nipkow@17629
   821
  case Nil thus ?case by simp
nipkow@17629
   822
next
nipkow@17629
   823
  case (Cons y ys)
nipkow@17629
   824
  show ?case (is "\<exists>x. ?Q x")
nipkow@17629
   825
  proof cases
nipkow@17629
   826
    assume Py: "P y"
nipkow@17629
   827
    show ?thesis
nipkow@17629
   828
    proof cases
nipkow@17629
   829
      assume xy: "x = y"
nipkow@17629
   830
      show ?thesis
nipkow@17629
   831
      proof from Py xy Cons(2) show "?Q []" by simp qed
nipkow@17629
   832
    next
nipkow@17629
   833
      assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp
nipkow@17629
   834
    qed
nipkow@17629
   835
  next
nipkow@17629
   836
    assume Py: "\<not> P y"
nipkow@17629
   837
    with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp
nipkow@17629
   838
    show ?thesis (is "? us. ?Q us")
nipkow@17629
   839
    proof show "?Q (y#us)" using 1 by simp qed
nipkow@17629
   840
  qed
nipkow@17629
   841
qed
nipkow@17629
   842
nipkow@17629
   843
lemma filter_eq_ConsD:
nipkow@17629
   844
 "filter P ys = x#xs \<Longrightarrow>
nipkow@17629
   845
  \<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
   846
by(rule Cons_eq_filterD) simp
nipkow@17629
   847
nipkow@17629
   848
lemma filter_eq_Cons_iff:
nipkow@17629
   849
 "(filter P ys = x#xs) =
nipkow@17629
   850
  (\<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
   851
by(auto dest:filter_eq_ConsD)
nipkow@17629
   852
nipkow@17629
   853
lemma Cons_eq_filter_iff:
nipkow@17629
   854
 "(x#xs = filter P ys) =
nipkow@17629
   855
  (\<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
   856
by(auto dest:Cons_eq_filterD)
nipkow@17629
   857
krauss@18336
   858
lemma filter_cong[recdef_cong]:
nipkow@17501
   859
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
nipkow@17501
   860
apply simp
nipkow@17501
   861
apply(erule thin_rl)
nipkow@17501
   862
by (induct ys) simp_all
nipkow@17501
   863
nipkow@15281
   864
nipkow@15392
   865
subsubsection {* @{text concat} *}
wenzelm@13114
   866
wenzelm@13142
   867
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
   868
by (induct xs) auto
wenzelm@13114
   869
wenzelm@13142
   870
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   871
by (induct xss) auto
wenzelm@13114
   872
wenzelm@13142
   873
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   874
by (induct xss) auto
wenzelm@13114
   875
wenzelm@13142
   876
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
nipkow@13145
   877
by (induct xs) auto
wenzelm@13114
   878
wenzelm@13142
   879
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
   880
by (induct xs) auto
wenzelm@13114
   881
wenzelm@13142
   882
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
   883
by (induct xs) auto
wenzelm@13114
   884
wenzelm@13142
   885
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
   886
by (induct xs) auto
wenzelm@13114
   887
wenzelm@13114
   888
nipkow@15392
   889
subsubsection {* @{text nth} *}
wenzelm@13114
   890
wenzelm@13142
   891
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
   892
by auto
wenzelm@13114
   893
wenzelm@13142
   894
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
   895
by auto
wenzelm@13114
   896
wenzelm@13142
   897
declare nth.simps [simp del]
wenzelm@13114
   898
wenzelm@13114
   899
lemma nth_append:
nipkow@13145
   900
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
paulson@14208
   901
apply (induct "xs", simp)
paulson@14208
   902
apply (case_tac n, auto)
nipkow@13145
   903
done
wenzelm@13114
   904
nipkow@14402
   905
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
nipkow@14402
   906
by (induct "xs") auto
nipkow@14402
   907
nipkow@14402
   908
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
nipkow@14402
   909
by (induct "xs") auto
nipkow@14402
   910
wenzelm@13142
   911
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
paulson@14208
   912
apply (induct xs, simp)
paulson@14208
   913
apply (case_tac n, auto)
nipkow@13145
   914
done
wenzelm@13114
   915
nipkow@18423
   916
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
nipkow@18423
   917
by(cases xs) simp_all
nipkow@18423
   918
nipkow@18049
   919
nipkow@18049
   920
lemma list_eq_iff_nth_eq:
nipkow@18049
   921
 "!!ys. (xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
nipkow@18049
   922
apply(induct xs)
nipkow@18049
   923
 apply simp apply blast
nipkow@18049
   924
apply(case_tac ys)
nipkow@18049
   925
 apply simp
nipkow@18049
   926
apply(simp add:nth_Cons split:nat.split)apply blast
nipkow@18049
   927
done
nipkow@18049
   928
wenzelm@13142
   929
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
   930
apply (induct xs, simp, simp)
nipkow@13145
   931
apply safe
paulson@14208
   932
apply (rule_tac x = 0 in exI, simp)
paulson@14208
   933
 apply (rule_tac x = "Suc i" in exI, simp)
paulson@14208
   934
apply (case_tac i, simp)
nipkow@13145
   935
apply (rename_tac j)
paulson@14208
   936
apply (rule_tac x = j in exI, simp)
nipkow@13145
   937
done
wenzelm@13114
   938
nipkow@17501
   939
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
nipkow@17501
   940
by(auto simp:set_conv_nth)
nipkow@17501
   941
nipkow@13145
   942
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
   943
by (auto simp add: set_conv_nth)
wenzelm@13114
   944
wenzelm@13142
   945
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
   946
by (auto simp add: set_conv_nth)
wenzelm@13114
   947
wenzelm@13114
   948
lemma all_nth_imp_all_set:
nipkow@13145
   949
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
   950
by (auto simp add: set_conv_nth)
wenzelm@13114
   951
wenzelm@13114
   952
lemma all_set_conv_all_nth:
nipkow@13145
   953
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
   954
by (auto simp add: set_conv_nth)
wenzelm@13114
   955
wenzelm@13114
   956
nipkow@15392
   957
subsubsection {* @{text list_update} *}
wenzelm@13114
   958
wenzelm@13142
   959
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
nipkow@13145
   960
by (induct xs) (auto split: nat.split)
wenzelm@13114
   961
wenzelm@13114
   962
lemma nth_list_update:
nipkow@13145
   963
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@13145
   964
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   965
wenzelm@13142
   966
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
   967
by (simp add: nth_list_update)
wenzelm@13114
   968
wenzelm@13142
   969
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145
   970
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   971
wenzelm@13142
   972
lemma list_update_overwrite [simp]:
nipkow@13145
   973
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145
   974
by (induct xs) (auto split: nat.split)
wenzelm@13114
   975
nipkow@14402
   976
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
paulson@14208
   977
apply (induct xs, simp)
nipkow@14187
   978
apply(simp split:nat.splits)
nipkow@14187
   979
done
nipkow@14187
   980
nipkow@17501
   981
lemma list_update_beyond[simp]: "\<And>i. length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
nipkow@17501
   982
apply (induct xs)
nipkow@17501
   983
 apply simp
nipkow@17501
   984
apply (case_tac i)
nipkow@17501
   985
apply simp_all
nipkow@17501
   986
done
nipkow@17501
   987
wenzelm@13114
   988
lemma list_update_same_conv:
nipkow@13145
   989
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145
   990
by (induct xs) (auto split: nat.split)
wenzelm@13114
   991
nipkow@14187
   992
lemma list_update_append1:
nipkow@14187
   993
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
paulson@14208
   994
apply (induct xs, simp)
nipkow@14187
   995
apply(simp split:nat.split)
nipkow@14187
   996
done
nipkow@14187
   997
kleing@15868
   998
lemma list_update_append:
kleing@15868
   999
  "!!n. (xs @ ys) [n:= x] = 
kleing@15868
  1000
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
kleing@15868
  1001
by (induct xs) (auto split:nat.splits)
kleing@15868
  1002
nipkow@14402
  1003
lemma list_update_length [simp]:
nipkow@14402
  1004
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
  1005
by (induct xs, auto)
nipkow@14402
  1006
wenzelm@13114
  1007
lemma update_zip:
nipkow@13145
  1008
"!!i xy xs. length xs = length ys ==>
nipkow@13145
  1009
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145
  1010
by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114
  1011
wenzelm@13114
  1012
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145
  1013
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1014
wenzelm@13114
  1015
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
  1016
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
  1017
kleing@15868
  1018
lemma set_update_memI: "!!n. n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
kleing@15868
  1019
by (induct xs) (auto split:nat.splits)
kleing@15868
  1020
wenzelm@13114
  1021
nipkow@15392
  1022
subsubsection {* @{text last} and @{text butlast} *}
wenzelm@13114
  1023
wenzelm@13142
  1024
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
  1025
by (induct xs) auto
wenzelm@13114
  1026
wenzelm@13142
  1027
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
  1028
by (induct xs) auto
wenzelm@13114
  1029
nipkow@14302
  1030
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
  1031
by(simp add:last.simps)
nipkow@14302
  1032
nipkow@14302
  1033
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
  1034
by(simp add:last.simps)
nipkow@14302
  1035
nipkow@14302
  1036
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
  1037
by (induct xs) (auto)
nipkow@14302
  1038
nipkow@14302
  1039
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
  1040
by(simp add:last_append)
nipkow@14302
  1041
nipkow@14302
  1042
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
  1043
by(simp add:last_append)
nipkow@14302
  1044
nipkow@17762
  1045
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
nipkow@17762
  1046
by(rule rev_exhaust[of xs]) simp_all
nipkow@17762
  1047
nipkow@17762
  1048
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
nipkow@17762
  1049
by(cases xs) simp_all
nipkow@17762
  1050
nipkow@17765
  1051
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
nipkow@17765
  1052
by (induct as) auto
nipkow@17762
  1053
wenzelm@13142
  1054
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
  1055
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1056
wenzelm@13114
  1057
lemma butlast_append:
nipkow@13145
  1058
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145
  1059
by (induct xs) auto
wenzelm@13114
  1060
wenzelm@13142
  1061
lemma append_butlast_last_id [simp]:
nipkow@13145
  1062
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
  1063
by (induct xs) auto
wenzelm@13114
  1064
wenzelm@13142
  1065
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
  1066
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1067
wenzelm@13114
  1068
lemma in_set_butlast_appendI:
nipkow@13145
  1069
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
  1070
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
  1071
nipkow@17501
  1072
lemma last_drop[simp]: "!!n. n < length xs \<Longrightarrow> last (drop n xs) = last xs"
nipkow@17501
  1073
apply (induct xs)
nipkow@17501
  1074
 apply simp
nipkow@17501
  1075
apply (auto split:nat.split)
nipkow@17501
  1076
done
nipkow@17501
  1077
nipkow@17589
  1078
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
nipkow@17589
  1079
by(induct xs)(auto simp:neq_Nil_conv)
nipkow@17589
  1080
wenzelm@13142
  1081
nipkow@15392
  1082
subsubsection {* @{text take} and @{text drop} *}
wenzelm@13114
  1083
wenzelm@13142
  1084
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
  1085
by (induct xs) auto
wenzelm@13114
  1086
wenzelm@13142
  1087
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
  1088
by (induct xs) auto
wenzelm@13114
  1089
wenzelm@13142
  1090
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
  1091
by simp
wenzelm@13114
  1092
wenzelm@13142
  1093
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
  1094
by simp
wenzelm@13114
  1095
wenzelm@13142
  1096
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
  1097
nipkow@15110
  1098
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
  1099
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
  1100
nipkow@14187
  1101
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
  1102
by(cases xs, simp_all)
nipkow@14187
  1103
nipkow@14187
  1104
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
nipkow@14187
  1105
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@14187
  1106
nipkow@14187
  1107
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
paulson@14208
  1108
apply (induct xs, simp)
nipkow@14187
  1109
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
  1110
done
nipkow@14187
  1111
nipkow@13913
  1112
lemma take_Suc_conv_app_nth:
nipkow@13913
  1113
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
paulson@14208
  1114
apply (induct xs, simp)
paulson@14208
  1115
apply (case_tac i, auto)
nipkow@13913
  1116
done
nipkow@13913
  1117
mehta@14591
  1118
lemma drop_Suc_conv_tl:
mehta@14591
  1119
  "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
mehta@14591
  1120
apply (induct xs, simp)
mehta@14591
  1121
apply (case_tac i, auto)
mehta@14591
  1122
done
mehta@14591
  1123
wenzelm@13142
  1124
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
nipkow@13145
  1125
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1126
wenzelm@13142
  1127
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
nipkow@13145
  1128
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1129
wenzelm@13142
  1130
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
nipkow@13145
  1131
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1132
wenzelm@13142
  1133
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
nipkow@13145
  1134
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1135
wenzelm@13142
  1136
lemma take_append [simp]:
nipkow@13145
  1137
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@13145
  1138
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1139
wenzelm@13142
  1140
lemma drop_append [simp]:
nipkow@13145
  1141
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@13145
  1142
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1143
wenzelm@13142
  1144
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
paulson@14208
  1145
apply (induct m, auto)
paulson@14208
  1146
apply (case_tac xs, auto)
nipkow@15236
  1147
apply (case_tac n, auto)
nipkow@13145
  1148
done
wenzelm@13114
  1149
wenzelm@13142
  1150
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
paulson@14208
  1151
apply (induct m, auto)
paulson@14208
  1152
apply (case_tac xs, auto)
nipkow@13145
  1153
done
wenzelm@13114
  1154
wenzelm@13114
  1155
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
paulson@14208
  1156
apply (induct m, auto)
paulson@14208
  1157
apply (case_tac xs, auto)
nipkow@13145
  1158
done
wenzelm@13114
  1159
nipkow@14802
  1160
lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@14802
  1161
apply(induct xs)
nipkow@14802
  1162
 apply simp
nipkow@14802
  1163
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  1164
done
nipkow@14802
  1165
wenzelm@13142
  1166
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
paulson@14208
  1167
apply (induct n, auto)
paulson@14208
  1168
apply (case_tac xs, auto)
nipkow@13145
  1169
done
wenzelm@13114
  1170
nipkow@15110
  1171
lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@15110
  1172
apply(induct xs)
nipkow@15110
  1173
 apply simp
nipkow@15110
  1174
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  1175
done
nipkow@15110
  1176
nipkow@15110
  1177
lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"
nipkow@15110
  1178
apply(induct xs)
nipkow@15110
  1179
apply simp
nipkow@15110
  1180
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1181
done
nipkow@15110
  1182
wenzelm@13114
  1183
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
paulson@14208
  1184
apply (induct n, auto)
paulson@14208
  1185
apply (case_tac xs, auto)
nipkow@13145
  1186
done
wenzelm@13114
  1187
wenzelm@13142
  1188
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
paulson@14208
  1189
apply (induct n, auto)
paulson@14208
  1190
apply (case_tac xs, auto)
nipkow@13145
  1191
done
wenzelm@13114
  1192
wenzelm@13114
  1193
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
paulson@14208
  1194
apply (induct xs, auto)
paulson@14208
  1195
apply (case_tac i, auto)
nipkow@13145
  1196
done
wenzelm@13114
  1197
wenzelm@13114
  1198
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
paulson@14208
  1199
apply (induct xs, auto)
paulson@14208
  1200
apply (case_tac i, auto)
nipkow@13145
  1201
done
wenzelm@13114
  1202
wenzelm@13142
  1203
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
paulson@14208
  1204
apply (induct xs, auto)
paulson@14208
  1205
apply (case_tac n, blast)
paulson@14208
  1206
apply (case_tac i, auto)
nipkow@13145
  1207
done
wenzelm@13114
  1208
wenzelm@13142
  1209
lemma nth_drop [simp]:
nipkow@13145
  1210
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
paulson@14208
  1211
apply (induct n, auto)
paulson@14208
  1212
apply (case_tac xs, auto)
nipkow@13145
  1213
done
nipkow@3507
  1214
nipkow@18423
  1215
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
nipkow@18423
  1216
by(simp add: hd_conv_nth)
nipkow@18423
  1217
nipkow@14025
  1218
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
nipkow@14025
  1219
by(induct xs)(auto simp:take_Cons split:nat.split)
nipkow@14025
  1220
nipkow@14025
  1221
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
nipkow@14025
  1222
by(induct xs)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1223
nipkow@14187
  1224
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1225
using set_take_subset by fast
nipkow@14187
  1226
nipkow@14187
  1227
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1228
using set_drop_subset by fast
nipkow@14187
  1229
wenzelm@13114
  1230
lemma append_eq_conv_conj:
nipkow@13145
  1231
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
paulson@14208
  1232
apply (induct xs, simp, clarsimp)
paulson@14208
  1233
apply (case_tac zs, auto)
nipkow@13145
  1234
done
wenzelm@13142
  1235
paulson@14050
  1236
lemma take_add [rule_format]: 
paulson@14050
  1237
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
paulson@14050
  1238
apply (induct xs, auto) 
paulson@14050
  1239
apply (case_tac i, simp_all) 
paulson@14050
  1240
done
paulson@14050
  1241
nipkow@14300
  1242
lemma append_eq_append_conv_if:
nipkow@14300
  1243
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1244
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1245
   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
  1246
   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@14300
  1247
apply(induct xs\<^isub>1)
nipkow@14300
  1248
 apply simp
nipkow@14300
  1249
apply(case_tac ys\<^isub>1)
nipkow@14300
  1250
apply simp_all
nipkow@14300
  1251
done
nipkow@14300
  1252
nipkow@15110
  1253
lemma take_hd_drop:
nipkow@15110
  1254
  "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
nipkow@15110
  1255
apply(induct xs)
nipkow@15110
  1256
apply simp
nipkow@15110
  1257
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1258
done
nipkow@15110
  1259
nipkow@17501
  1260
lemma id_take_nth_drop:
nipkow@17501
  1261
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
nipkow@17501
  1262
proof -
nipkow@17501
  1263
  assume si: "i < length xs"
nipkow@17501
  1264
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
nipkow@17501
  1265
  moreover
nipkow@17501
  1266
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
nipkow@17501
  1267
    apply (rule_tac take_Suc_conv_app_nth) by arith
nipkow@17501
  1268
  ultimately show ?thesis by auto
nipkow@17501
  1269
qed
nipkow@17501
  1270
  
nipkow@17501
  1271
lemma upd_conv_take_nth_drop:
nipkow@17501
  1272
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1273
proof -
nipkow@17501
  1274
  assume i: "i < length xs"
nipkow@17501
  1275
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
nipkow@17501
  1276
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
nipkow@17501
  1277
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1278
    using i by (simp add: list_update_append)
nipkow@17501
  1279
  finally show ?thesis .
nipkow@17501
  1280
qed
nipkow@17501
  1281
wenzelm@13114
  1282
nipkow@15392
  1283
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1284
wenzelm@13142
  1285
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1286
by (induct xs) auto
wenzelm@13114
  1287
wenzelm@13142
  1288
lemma takeWhile_append1 [simp]:
nipkow@13145
  1289
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1290
by (induct xs) auto
wenzelm@13114
  1291
wenzelm@13142
  1292
lemma takeWhile_append2 [simp]:
nipkow@13145
  1293
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1294
by (induct xs) auto
wenzelm@13114
  1295
wenzelm@13142
  1296
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1297
by (induct xs) auto
wenzelm@13114
  1298
wenzelm@13142
  1299
lemma dropWhile_append1 [simp]:
nipkow@13145
  1300
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1301
by (induct xs) auto
wenzelm@13114
  1302
wenzelm@13142
  1303
lemma dropWhile_append2 [simp]:
nipkow@13145
  1304
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1305
by (induct xs) auto
wenzelm@13114
  1306
wenzelm@13142
  1307
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1308
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1309
nipkow@13913
  1310
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1311
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1312
by(induct xs, auto)
nipkow@13913
  1313
nipkow@13913
  1314
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1315
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1316
by(induct xs, auto)
nipkow@13913
  1317
nipkow@13913
  1318
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1319
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1320
by(induct xs, auto)
nipkow@13913
  1321
nipkow@17501
  1322
text{* The following two lemmmas could be generalized to an arbitrary
nipkow@17501
  1323
property. *}
nipkow@17501
  1324
nipkow@17501
  1325
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1326
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
nipkow@17501
  1327
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
nipkow@17501
  1328
nipkow@17501
  1329
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1330
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
nipkow@17501
  1331
apply(induct xs)
nipkow@17501
  1332
 apply simp
nipkow@17501
  1333
apply auto
nipkow@17501
  1334
apply(subst dropWhile_append2)
nipkow@17501
  1335
apply auto
nipkow@17501
  1336
done
nipkow@17501
  1337
nipkow@18423
  1338
lemma takeWhile_not_last:
nipkow@18423
  1339
 "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
nipkow@18423
  1340
apply(induct xs)
nipkow@18423
  1341
 apply simp
nipkow@18423
  1342
apply(case_tac xs)
nipkow@18423
  1343
apply(auto)
nipkow@18423
  1344
done
nipkow@18423
  1345
krauss@18336
  1346
lemma takeWhile_cong [recdef_cong]:
krauss@18336
  1347
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1348
  ==> takeWhile P l = takeWhile Q k"
krauss@18336
  1349
  by (induct k fixing: l, simp_all)
krauss@18336
  1350
krauss@18336
  1351
lemma dropWhile_cong [recdef_cong]:
krauss@18336
  1352
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1353
  ==> dropWhile P l = dropWhile Q k"
krauss@18336
  1354
  by (induct k fixing: l, simp_all)
krauss@18336
  1355
wenzelm@13114
  1356
nipkow@15392
  1357
subsubsection {* @{text zip} *}
wenzelm@13114
  1358
wenzelm@13142
  1359
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1360
by (induct ys) auto
wenzelm@13114
  1361
wenzelm@13142
  1362
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1363
by simp
wenzelm@13114
  1364
wenzelm@13142
  1365
declare zip_Cons [simp del]
wenzelm@13114
  1366
nipkow@15281
  1367
lemma zip_Cons1:
nipkow@15281
  1368
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  1369
by(auto split:list.split)
nipkow@15281
  1370
wenzelm@13142
  1371
lemma length_zip [simp]:
nipkow@13145
  1372
"!!xs. length (zip xs ys) = min (length xs) (length ys)"
paulson@14208
  1373
apply (induct ys, simp)
paulson@14208
  1374
apply (case_tac xs, auto)
nipkow@13145
  1375
done
wenzelm@13114
  1376
wenzelm@13114
  1377
lemma zip_append1:
nipkow@13145
  1378
"!!xs. zip (xs @ ys) zs =
nipkow@13145
  1379
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
paulson@14208
  1380
apply (induct zs, simp)
paulson@14208
  1381
apply (case_tac xs, simp_all)
nipkow@13145
  1382
done
wenzelm@13114
  1383
wenzelm@13114
  1384
lemma zip_append2:
nipkow@13145
  1385
"!!ys. zip xs (ys @ zs) =
nipkow@13145
  1386
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
paulson@14208
  1387
apply (induct xs, simp)
paulson@14208
  1388
apply (case_tac ys, simp_all)
nipkow@13145
  1389
done
wenzelm@13114
  1390
wenzelm@13142
  1391
lemma zip_append [simp]:
wenzelm@13142
  1392
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1393
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1394
by (simp add: zip_append1)
wenzelm@13114
  1395
wenzelm@13114
  1396
lemma zip_rev:
nipkow@14247
  1397
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1398
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1399
wenzelm@13142
  1400
lemma nth_zip [simp]:
nipkow@13145
  1401
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
paulson@14208
  1402
apply (induct ys, simp)
nipkow@13145
  1403
apply (case_tac xs)
nipkow@13145
  1404
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1405
done
wenzelm@13114
  1406
wenzelm@13114
  1407
lemma set_zip:
nipkow@13145
  1408
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
  1409
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1410
wenzelm@13114
  1411
lemma zip_update:
nipkow@13145
  1412
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
  1413
by (rule sym, simp add: update_zip)
wenzelm@13114
  1414
wenzelm@13142
  1415
lemma zip_replicate [simp]:
nipkow@13145
  1416
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
paulson@14208
  1417
apply (induct i, auto)
paulson@14208
  1418
apply (case_tac j, auto)
nipkow@13145
  1419
done
wenzelm@13114
  1420
wenzelm@13142
  1421
nipkow@15392
  1422
subsubsection {* @{text list_all2} *}
wenzelm@13114
  1423
kleing@14316
  1424
lemma list_all2_lengthD [intro?]: 
kleing@14316
  1425
  "list_all2 P xs ys ==> length xs = length ys"
nipkow@13145
  1426
by (simp add: list_all2_def)
wenzelm@13114
  1427
nipkow@17090
  1428
lemma list_all2_Nil [iff,code]: "list_all2 P [] ys = (ys = [])"
nipkow@13145
  1429
by (simp add: list_all2_def)
wenzelm@13114
  1430
wenzelm@13142
  1431
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"
nipkow@13145
  1432
by (simp add: list_all2_def)
wenzelm@13114
  1433
nipkow@17090
  1434
lemma list_all2_Cons [iff,code]:
nipkow@13145
  1435
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@13145
  1436
by (auto simp add: list_all2_def)
wenzelm@13114
  1437
wenzelm@13114
  1438
lemma list_all2_Cons1:
nipkow@13145
  1439
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1440
by (cases ys) auto
wenzelm@13114
  1441
wenzelm@13114
  1442
lemma list_all2_Cons2:
nipkow@13145
  1443
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1444
by (cases xs) auto
wenzelm@13114
  1445
wenzelm@13142
  1446
lemma list_all2_rev [iff]:
nipkow@13145
  1447
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1448
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1449
kleing@13863
  1450
lemma list_all2_rev1:
kleing@13863
  1451
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1452
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1453
wenzelm@13114
  1454
lemma list_all2_append1:
nipkow@13145
  1455
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1456
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1457
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1458
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1459
apply (rule iffI)
nipkow@13145
  1460
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1461
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  1462
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1463
apply (simp add: ball_Un)
nipkow@13145
  1464
done
wenzelm@13114
  1465
wenzelm@13114
  1466
lemma list_all2_append2:
nipkow@13145
  1467
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1468
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1469
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1470
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1471
apply (rule iffI)
nipkow@13145
  1472
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1473
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  1474
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1475
apply (simp add: ball_Un)
nipkow@13145
  1476
done
wenzelm@13114
  1477
kleing@13863
  1478
lemma list_all2_append:
nipkow@14247
  1479
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  1480
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  1481
by (induct rule:list_induct2, simp_all)
kleing@13863
  1482
kleing@13863
  1483
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1484
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
kleing@13863
  1485
  by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1486
wenzelm@13114
  1487
lemma list_all2_conv_all_nth:
nipkow@13145
  1488
"list_all2 P xs ys =
nipkow@13145
  1489
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1490
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1491
berghofe@13883
  1492
lemma list_all2_trans:
berghofe@13883
  1493
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1494
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1495
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1496
proof (induct as)
berghofe@13883
  1497
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1498
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1499
  proof (induct bs)
berghofe@13883
  1500
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1501
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1502
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1503
  qed simp
berghofe@13883
  1504
qed simp
berghofe@13883
  1505
kleing@13863
  1506
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1507
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
kleing@13863
  1508
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1509
paulson@14395
  1510
lemma list_all2I:
paulson@14395
  1511
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
paulson@14395
  1512
  by (simp add: list_all2_def)
paulson@14395
  1513
kleing@14328
  1514
lemma list_all2_nthD:
kleing@13863
  1515
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
kleing@13863
  1516
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1517
nipkow@14302
  1518
lemma list_all2_nthD2:
nipkow@14302
  1519
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@14302
  1520
  by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  1521
kleing@13863
  1522
lemma list_all2_map1: 
kleing@13863
  1523
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
kleing@13863
  1524
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1525
kleing@13863
  1526
lemma list_all2_map2: 
kleing@13863
  1527
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
kleing@13863
  1528
  by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1529
kleing@14316
  1530
lemma list_all2_refl [intro?]:
kleing@13863
  1531
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
kleing@13863
  1532
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1533
kleing@13863
  1534
lemma list_all2_update_cong:
kleing@13863
  1535
  "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
kleing@13863
  1536
  by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1537
kleing@13863
  1538
lemma list_all2_update_cong2:
kleing@13863
  1539
  "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
kleing@13863
  1540
  by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1541
nipkow@14302
  1542
lemma list_all2_takeI [simp,intro?]:
nipkow@14302
  1543
  "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@14302
  1544
  apply (induct xs)
nipkow@14302
  1545
   apply simp
nipkow@14302
  1546
  apply (clarsimp simp add: list_all2_Cons1)
nipkow@14302
  1547
  apply (case_tac n)
nipkow@14302
  1548
  apply auto
nipkow@14302
  1549
  done
nipkow@14302
  1550
nipkow@14302
  1551
lemma list_all2_dropI [simp,intro?]:
kleing@13863
  1552
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
paulson@14208
  1553
  apply (induct as, simp)
kleing@13863
  1554
  apply (clarsimp simp add: list_all2_Cons1)
paulson@14208
  1555
  apply (case_tac n, simp, simp)
kleing@13863
  1556
  done
kleing@13863
  1557
kleing@14327
  1558
lemma list_all2_mono [intro?]:
kleing@13863
  1559
  "\<And>y. list_all2 P x y \<Longrightarrow> (\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> list_all2 Q x y"
paulson@14208
  1560
  apply (induct x, simp)
paulson@14208
  1561
  apply (case_tac y, auto)
kleing@13863
  1562
  done
kleing@13863
  1563
wenzelm@13142
  1564
nipkow@15392
  1565
subsubsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  1566
wenzelm@13142
  1567
lemma foldl_append [simp]:
nipkow@13145
  1568
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145
  1569
by (induct xs) auto
wenzelm@13142
  1570
nipkow@14402
  1571
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  1572
by (induct xs) auto
nipkow@14402
  1573
krauss@18336
  1574
lemma foldl_cong [recdef_cong]:
krauss@18336
  1575
  "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] 
krauss@18336
  1576
  ==> foldl f a l = foldl g b k"
krauss@18336
  1577
  by (induct k fixing: a b l, simp_all)
krauss@18336
  1578
krauss@18336
  1579
lemma foldr_cong [recdef_cong]:
krauss@18336
  1580
  "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] 
krauss@18336
  1581
  ==> foldr f l a = foldr g k b"
krauss@18336
  1582
  by (induct k fixing: a b l, simp_all)
krauss@18336
  1583
nipkow@14402
  1584
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  1585
by (induct xs) auto
nipkow@14402
  1586
nipkow@14402
  1587
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  1588
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  1589
wenzelm@13142
  1590
text {*
nipkow@13145
  1591
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1592
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1593
*}
wenzelm@13142
  1594
wenzelm@13142
  1595
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
nipkow@13145
  1596
by (induct ns) auto
wenzelm@13142
  1597
wenzelm@13142
  1598
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1599
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1600
wenzelm@13142
  1601
lemma sum_eq_0_conv [iff]:
nipkow@13145
  1602
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145
  1603
by (induct ns) auto
wenzelm@13114
  1604
wenzelm@13114
  1605
nipkow@15392
  1606
subsubsection {* @{text upto} *}
wenzelm@13114
  1607
nipkow@17090
  1608
lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
nipkow@17090
  1609
-- {* simp does not terminate! *}
nipkow@13145
  1610
by (induct j) auto
wenzelm@13142
  1611
nipkow@15425
  1612
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
nipkow@13145
  1613
by (subst upt_rec) simp
wenzelm@13114
  1614
nipkow@15425
  1615
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
nipkow@15281
  1616
by(induct j)simp_all
nipkow@15281
  1617
nipkow@15281
  1618
lemma upt_eq_Cons_conv:
nipkow@15425
  1619
 "!!x xs. ([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
nipkow@15281
  1620
apply(induct j)
nipkow@15281
  1621
 apply simp
nipkow@15281
  1622
apply(clarsimp simp add: append_eq_Cons_conv)
nipkow@15281
  1623
apply arith
nipkow@15281
  1624
done
nipkow@15281
  1625
nipkow@15425
  1626
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
nipkow@13145
  1627
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1628
by simp
wenzelm@13114
  1629
nipkow@15425
  1630
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
nipkow@13145
  1631
apply(rule trans)
nipkow@13145
  1632
apply(subst upt_rec)
paulson@14208
  1633
 prefer 2 apply (rule refl, simp)
nipkow@13145
  1634
done
wenzelm@13114
  1635
nipkow@15425
  1636
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
nipkow@13145
  1637
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1638
by (induct k) auto
wenzelm@13114
  1639
nipkow@15425
  1640
lemma length_upt [simp]: "length [i..<j] = j - i"
nipkow@13145
  1641
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1642
nipkow@15425
  1643
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
nipkow@13145
  1644
apply (induct j)
nipkow@13145
  1645
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1646
done
wenzelm@13114
  1647
nipkow@17906
  1648
nipkow@17906
  1649
lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
nipkow@17906
  1650
by(simp add:upt_conv_Cons)
nipkow@17906
  1651
nipkow@17906
  1652
lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
nipkow@17906
  1653
apply(cases j)
nipkow@17906
  1654
 apply simp
nipkow@17906
  1655
by(simp add:upt_Suc_append)
nipkow@17906
  1656
nipkow@15425
  1657
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..<n] = [i..<i+m]"
paulson@14208
  1658
apply (induct m, simp)
nipkow@13145
  1659
apply (subst upt_rec)
nipkow@13145
  1660
apply (rule sym)
nipkow@13145
  1661
apply (subst upt_rec)
nipkow@13145
  1662
apply (simp del: upt.simps)
nipkow@13145
  1663
done
nipkow@3507
  1664
nipkow@17501
  1665
lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
nipkow@17501
  1666
apply(induct j)
nipkow@17501
  1667
apply auto
nipkow@17501
  1668
apply arith
nipkow@17501
  1669
done
nipkow@17501
  1670
nipkow@15425
  1671
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..n]"
nipkow@13145
  1672
by (induct n) auto
wenzelm@13114
  1673
nipkow@15425
  1674
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
nipkow@13145
  1675
apply (induct n m rule: diff_induct)
nipkow@13145
  1676
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1677
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1678
done
wenzelm@13114
  1679
berghofe@13883
  1680
lemma nth_take_lemma:
berghofe@13883
  1681
  "!!xs ys. k <= length xs ==> k <= length ys ==>
berghofe@13883
  1682
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
berghofe@13883
  1683
apply (atomize, induct k)
paulson@14208
  1684
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  1685
txt {* Both lists must be non-empty *}
paulson@14208
  1686
apply (case_tac xs, simp)
paulson@14208
  1687
apply (case_tac ys, clarify)
nipkow@13145
  1688
 apply (simp (no_asm_use))
nipkow@13145
  1689
apply clarify
nipkow@13145
  1690
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  1691
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  1692
apply blast
nipkow@13145
  1693
done
wenzelm@13114
  1694
wenzelm@13114
  1695
lemma nth_equalityI:
wenzelm@13114
  1696
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  1697
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  1698
apply (simp_all add: take_all)
nipkow@13145
  1699
done
wenzelm@13142
  1700
kleing@13863
  1701
(* needs nth_equalityI *)
kleing@13863
  1702
lemma list_all2_antisym:
kleing@13863
  1703
  "\<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
  1704
  \<Longrightarrow> xs = ys"
kleing@13863
  1705
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  1706
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  1707
  done
kleing@13863
  1708
wenzelm@13142
  1709
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  1710
-- {* The famous take-lemma. *}
nipkow@13145
  1711
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  1712
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  1713
done
wenzelm@13142
  1714
wenzelm@13142
  1715
nipkow@15302
  1716
lemma take_Cons':
nipkow@15302
  1717
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@15302
  1718
by (cases n) simp_all
nipkow@15302
  1719
nipkow@15302
  1720
lemma drop_Cons':
nipkow@15302
  1721
     "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@15302
  1722
by (cases n) simp_all
nipkow@15302
  1723
nipkow@15302
  1724
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@15302
  1725
by (cases n) simp_all
nipkow@15302
  1726
nipkow@15302
  1727
lemmas [simp] = take_Cons'[of "number_of v",standard]
nipkow@15302
  1728
                drop_Cons'[of "number_of v",standard]
nipkow@15302
  1729
                nth_Cons'[of _ _ "number_of v",standard]
nipkow@15302
  1730
nipkow@15302
  1731
nipkow@15392
  1732
subsubsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  1733
wenzelm@13142
  1734
lemma distinct_append [simp]:
nipkow@13145
  1735
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  1736
by (induct xs) auto
wenzelm@13142
  1737
nipkow@15305
  1738
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
nipkow@15305
  1739
by(induct xs) auto
nipkow@15305
  1740
wenzelm@13142
  1741
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  1742
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  1743
wenzelm@13142
  1744
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  1745
by (induct xs) auto
wenzelm@13142
  1746
paulson@15072
  1747
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
paulson@15251
  1748
  by (induct x, auto) 
paulson@15072
  1749
paulson@15072
  1750
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
paulson@15251
  1751
  by (induct x, auto)
paulson@15072
  1752
nipkow@15245
  1753
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245
  1754
by (induct xs) auto
nipkow@15245
  1755
nipkow@15245
  1756
lemma length_remdups_eq[iff]:
nipkow@15245
  1757
  "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245
  1758
apply(induct xs)
nipkow@15245
  1759
 apply auto
nipkow@15245
  1760
apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245
  1761
 apply arith
nipkow@15245
  1762
apply(rule length_remdups_leq)
nipkow@15245
  1763
done
nipkow@15245
  1764
wenzelm@13142
  1765
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  1766
by (induct xs) auto
wenzelm@13114
  1767
nipkow@15304
  1768
lemma distinct_map_filterI:
nipkow@15304
  1769
 "distinct(map f xs) \<Longrightarrow> distinct(map f (filter P xs))"
nipkow@15304
  1770
apply(induct xs)
nipkow@15304
  1771
 apply simp
nipkow@15304
  1772
apply force
nipkow@15304
  1773
done
nipkow@15304
  1774
nipkow@17501
  1775
lemma distinct_upt[simp]: "distinct[i..<j]"
nipkow@17501
  1776
by (induct j) auto
nipkow@17501
  1777
nipkow@17501
  1778
lemma distinct_take[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (take i xs)"
nipkow@17501
  1779
apply(induct xs)
nipkow@17501
  1780
 apply simp
nipkow@17501
  1781
apply (case_tac i)
nipkow@17501
  1782
 apply simp_all
nipkow@17501
  1783
apply(blast dest:in_set_takeD)
nipkow@17501
  1784
done
nipkow@17501
  1785
nipkow@17501
  1786
lemma distinct_drop[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (drop i xs)"
nipkow@17501
  1787
apply(induct xs)
nipkow@17501
  1788
 apply simp
nipkow@17501
  1789
apply (case_tac i)
nipkow@17501
  1790
 apply simp_all
nipkow@17501
  1791
done
nipkow@17501
  1792
nipkow@17501
  1793
lemma distinct_list_update:
nipkow@17501
  1794
assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
nipkow@17501
  1795
shows "distinct (xs[i:=a])"
nipkow@17501
  1796
proof (cases "i < length xs")
nipkow@17501
  1797
  case True
nipkow@17501
  1798
  with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
nipkow@17501
  1799
    apply (drule_tac id_take_nth_drop) by simp
nipkow@17501
  1800
  with d True show ?thesis
nipkow@17501
  1801
    apply (simp add: upd_conv_take_nth_drop)
nipkow@17501
  1802
    apply (drule subst [OF id_take_nth_drop]) apply assumption
nipkow@17501
  1803
    apply simp apply (cases "a = xs!i") apply simp by blast
nipkow@17501
  1804
next
nipkow@17501
  1805
  case False with d show ?thesis by auto
nipkow@17501
  1806
qed
nipkow@17501
  1807
nipkow@17501
  1808
nipkow@17501
  1809
text {* It is best to avoid this indexed version of distinct, but
nipkow@17501
  1810
sometimes it is useful. *}
nipkow@17501
  1811
wenzelm@13142
  1812
lemma distinct_conv_nth:
nipkow@17501
  1813
"distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@15251
  1814
apply (induct xs, simp, simp)
paulson@14208
  1815
apply (rule iffI, clarsimp)
nipkow@13145
  1816
 apply (case_tac i)
paulson@14208
  1817
apply (case_tac j, simp)
nipkow@13145
  1818
apply (simp add: set_conv_nth)
nipkow@13145
  1819
 apply (case_tac j)
paulson@14208
  1820
apply (clarsimp simp add: set_conv_nth, simp)
nipkow@13145
  1821
apply (rule conjI)
nipkow@13145
  1822
 apply (clarsimp simp add: set_conv_nth)
nipkow@17501
  1823
 apply (erule_tac x = 0 in allE, simp)
paulson@14208
  1824
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
nipkow@17501
  1825
apply (erule_tac x = "Suc i" in allE, simp)
paulson@14208
  1826
apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145
  1827
done
wenzelm@13114
  1828
nipkow@15110
  1829
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
kleing@14388
  1830
  by (induct xs) auto
kleing@14388
  1831
nipkow@15110
  1832
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388
  1833
proof (induct xs)
kleing@14388
  1834
  case Nil thus ?case by simp
kleing@14388
  1835
next
kleing@14388
  1836
  case (Cons x xs)
kleing@14388
  1837
  show ?case
kleing@14388
  1838
  proof (cases "x \<in> set xs")
kleing@14388
  1839
    case False with Cons show ?thesis by simp
kleing@14388
  1840
  next
kleing@14388
  1841
    case True with Cons.prems
kleing@14388
  1842
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  1843
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  1844
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  1845
    ultimately have False by simp
kleing@14388
  1846
    thus ?thesis ..
kleing@14388
  1847
  qed
kleing@14388
  1848
qed
kleing@14388
  1849
nipkow@15110
  1850
lemma inj_on_setI: "distinct(map f xs) ==> inj_on f (set xs)"
nipkow@15110
  1851
apply(induct xs)
nipkow@15110
  1852
 apply simp
nipkow@15110
  1853
apply fastsimp
nipkow@15110
  1854
done
nipkow@15110
  1855
nipkow@15110
  1856
lemma inj_on_set_conv:
nipkow@15110
  1857
 "distinct xs \<Longrightarrow> inj_on f (set xs) = distinct(map f xs)"
nipkow@15110
  1858
apply(induct xs)
nipkow@15110
  1859
 apply simp
nipkow@15110
  1860
apply fastsimp
nipkow@15110
  1861
done
nipkow@15110
  1862
nipkow@15110
  1863
nipkow@17906
  1864
lemma nth_eq_iff_index_eq:
nipkow@17906
  1865
 "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
nipkow@17906
  1866
by(auto simp: distinct_conv_nth)
nipkow@17906
  1867
nipkow@17906
  1868
nipkow@15392
  1869
subsubsection {* @{text remove1} *}
nipkow@15110
  1870
nipkow@18049
  1871
lemma remove1_append:
nipkow@18049
  1872
  "remove1 x (xs @ ys) =
nipkow@18049
  1873
  (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
nipkow@18049
  1874
by (induct xs) auto
nipkow@18049
  1875
nipkow@15110
  1876
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110
  1877
apply(induct xs)
nipkow@15110
  1878
 apply simp
nipkow@15110
  1879
apply simp
nipkow@15110
  1880
apply blast
nipkow@15110
  1881
done
nipkow@15110
  1882
paulson@17724
  1883
lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110
  1884
apply(induct xs)
nipkow@15110
  1885
 apply simp
nipkow@15110
  1886
apply simp
nipkow@15110
  1887
apply blast
nipkow@15110
  1888
done
nipkow@15110
  1889
nipkow@18049
  1890
lemma remove1_filter_not[simp]:
nipkow@18049
  1891
  "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
nipkow@18049
  1892
by(induct xs) auto
nipkow@18049
  1893
nipkow@15110
  1894
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110
  1895
apply(insert set_remove1_subset)
nipkow@15110
  1896
apply fast
nipkow@15110
  1897
done
nipkow@15110
  1898
nipkow@15110
  1899
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110
  1900
by (induct xs) simp_all
nipkow@15110
  1901
wenzelm@13114
  1902
nipkow@15392
  1903
subsubsection {* @{text replicate} *}
wenzelm@13114
  1904
wenzelm@13142
  1905
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  1906
by (induct n) auto
nipkow@13124
  1907
wenzelm@13142
  1908
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  1909
by (induct n) auto
wenzelm@13114
  1910
wenzelm@13114
  1911
lemma replicate_app_Cons_same:
nipkow@13145
  1912
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  1913
by (induct n) auto
wenzelm@13114
  1914
wenzelm@13142
  1915
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208
  1916
apply (induct n, simp)
nipkow@13145
  1917
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  1918
done
wenzelm@13114
  1919
wenzelm@13142
  1920
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  1921
by (induct n) auto
wenzelm@13114
  1922
nipkow@16397
  1923
text{* Courtesy of Matthias Daum: *}
nipkow@16397
  1924
lemma append_replicate_commute:
nipkow@16397
  1925
  "replicate n x @ replicate k x = replicate k x @ replicate n x"
nipkow@16397
  1926
apply (simp add: replicate_add [THEN sym])
nipkow@16397
  1927
apply (simp add: add_commute)
nipkow@16397
  1928
done
nipkow@16397
  1929
wenzelm@13142
  1930
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  1931
by (induct n) auto
wenzelm@13114
  1932
wenzelm@13142
  1933
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  1934
by (induct n) auto
wenzelm@13114
  1935
wenzelm@13142
  1936
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  1937
by (atomize (full), induct n) auto
wenzelm@13114
  1938
wenzelm@13142
  1939
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
paulson@14208
  1940
apply (induct n, simp)
nipkow@13145
  1941
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  1942
done
wenzelm@13114
  1943
nipkow@16397
  1944
text{* Courtesy of Matthias Daum (2 lemmas): *}
nipkow@16397
  1945
lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
nipkow@16397
  1946
apply (case_tac "k \<le> i")
nipkow@16397
  1947
 apply  (simp add: min_def)
nipkow@16397
  1948
apply (drule not_leE)
nipkow@16397
  1949
apply (simp add: min_def)
nipkow@16397
  1950
apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
nipkow@16397
  1951
 apply  simp
nipkow@16397
  1952
apply (simp add: replicate_add [symmetric])
nipkow@16397
  1953
done
nipkow@16397
  1954
nipkow@16397
  1955
lemma drop_replicate[simp]: "!!i. drop i (replicate k x) = replicate (k-i) x"
nipkow@16397
  1956
apply (induct k)
nipkow@16397
  1957
 apply simp
nipkow@16397
  1958
apply clarsimp
nipkow@16397
  1959
apply (case_tac i)
nipkow@16397
  1960
 apply simp
nipkow@16397
  1961
apply clarsimp
nipkow@16397
  1962
done
nipkow@16397
  1963
nipkow@16397
  1964
wenzelm@13142
  1965
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  1966
by (induct n) auto
wenzelm@13114
  1967
wenzelm@13142
  1968
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  1969
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  1970
wenzelm@13142
  1971
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  1972
by auto
wenzelm@13114
  1973
wenzelm@13142
  1974
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  1975
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  1976
wenzelm@13114
  1977
nipkow@15392
  1978
subsubsection{*@{text rotate1} and @{text rotate}*}
nipkow@15302
  1979
nipkow@15302
  1980
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
nipkow@15302
  1981
by(simp add:rotate1_def)
nipkow@15302
  1982
nipkow@15302
  1983
lemma rotate0[simp]: "rotate 0 = id"
nipkow@15302
  1984
by(simp add:rotate_def)
nipkow@15302
  1985
nipkow@15302
  1986
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
nipkow@15302
  1987
by(simp add:rotate_def)
nipkow@15302
  1988
nipkow@15302
  1989
lemma rotate_add:
nipkow@15302
  1990
  "rotate (m+n) = rotate m o rotate n"
nipkow@15302
  1991
by(simp add:rotate_def funpow_add)
nipkow@15302
  1992
nipkow@15302
  1993
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
nipkow@15302
  1994
by(simp add:rotate_add)
nipkow@15302
  1995
nipkow@18049
  1996
lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
nipkow@18049
  1997
by(simp add:rotate_def funpow_swap1)
nipkow@18049
  1998
nipkow@15302
  1999
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
nipkow@15302
  2000
by(cases xs) simp_all
nipkow@15302
  2001
nipkow@15302
  2002
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2003
apply(induct n)
nipkow@15302
  2004
 apply simp
nipkow@15302
  2005
apply (simp add:rotate_def)
nipkow@13145
  2006
done
wenzelm@13114
  2007
nipkow@15302
  2008
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
nipkow@15302
  2009
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2010
nipkow@15302
  2011
lemma rotate_drop_take:
nipkow@15302
  2012
  "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
nipkow@15302
  2013
apply(induct n)
nipkow@15302
  2014
 apply simp
nipkow@15302
  2015
apply(simp add:rotate_def)
nipkow@15302
  2016
apply(cases "xs = []")
nipkow@15302
  2017
 apply (simp)
nipkow@15302
  2018
apply(case_tac "n mod length xs = 0")
nipkow@15302
  2019
 apply(simp add:mod_Suc)
nipkow@15302
  2020
 apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
nipkow@15302
  2021
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
nipkow@15302
  2022
                take_hd_drop linorder_not_le)
nipkow@13145
  2023
done
wenzelm@13114
  2024
nipkow@15302
  2025
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
nipkow@15302
  2026
by(simp add:rotate_drop_take)
nipkow@15302
  2027
nipkow@15302
  2028
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2029
by(simp add:rotate_drop_take)
nipkow@15302
  2030
nipkow@15302
  2031
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
nipkow@15302
  2032
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2033
nipkow@15302
  2034
lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs"
nipkow@15302
  2035
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2036
nipkow@15302
  2037
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
nipkow@15302
  2038
by(simp add:rotate1_def split:list.split) blast
nipkow@15302
  2039
nipkow@15302
  2040
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
nipkow@15302
  2041
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2042
nipkow@15302
  2043
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
nipkow@15302
  2044
by(simp add:rotate_drop_take take_map drop_map)
nipkow@15302
  2045
nipkow@15302
  2046
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
nipkow@15302
  2047
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2048
nipkow@15302
  2049
lemma set_rotate[simp]: "set(rotate n xs) = set xs"
nipkow@15302
  2050
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2051
nipkow@15302
  2052
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
nipkow@15302
  2053
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2054
nipkow@15302
  2055
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
nipkow@15302
  2056
by (induct n) (simp_all add:rotate_def)
wenzelm@13114
  2057
nipkow@15439
  2058
lemma rotate_rev:
nipkow@15439
  2059
  "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
nipkow@15439
  2060
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2061
apply(cases "length xs = 0")
nipkow@15439
  2062
 apply simp
nipkow@15439
  2063
apply(cases "n mod length xs = 0")
nipkow@15439
  2064
 apply simp
nipkow@15439
  2065
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2066
done
nipkow@15439
  2067
nipkow@18423
  2068
lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
nipkow@18423
  2069
apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
nipkow@18423
  2070
apply(subgoal_tac "length xs \<noteq> 0")
nipkow@18423
  2071
 prefer 2 apply simp
nipkow@18423
  2072
using mod_less_divisor[of "length xs" n] by arith
nipkow@18423
  2073
wenzelm@13114
  2074
nipkow@15392
  2075
subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  2076
wenzelm@13142
  2077
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  2078
by (auto simp add: sublist_def)
wenzelm@13114
  2079
wenzelm@13142
  2080
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  2081
by (auto simp add: sublist_def)
wenzelm@13114
  2082
nipkow@15281
  2083
lemma length_sublist:
nipkow@15281
  2084
  "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
nipkow@15281
  2085
by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
nipkow@15281
  2086
nipkow@15281
  2087
lemma sublist_shift_lemma_Suc:
nipkow@15281
  2088
  "!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
nipkow@15281
  2089
         map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
nipkow@15281
  2090
apply(induct xs)
nipkow@15281
  2091
 apply simp
nipkow@15281
  2092
apply (case_tac "is")
nipkow@15281
  2093
 apply simp
nipkow@15281
  2094
apply simp
nipkow@15281
  2095
done
nipkow@15281
  2096
wenzelm@13114
  2097
lemma sublist_shift_lemma:
nipkow@15425
  2098
     "map fst [p:zip xs [i..<i + length xs] . snd p : A] =
nipkow@15425
  2099
      map fst [p:zip xs [0..<length xs] . snd p + i : A]"
nipkow@13145
  2100
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  2101
wenzelm@13114
  2102
lemma sublist_append:
paulson@15168
  2103
     "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  2104
apply (unfold sublist_def)
paulson@14208
  2105
apply (induct l' rule: rev_induct, simp)
nipkow@13145
  2106
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  2107
apply (simp add: add_commute)
nipkow@13145
  2108
done
wenzelm@13114
  2109
wenzelm@13114
  2110
lemma sublist_Cons:
nipkow@13145
  2111
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  2112
apply (induct l rule: rev_induct)
nipkow@13145
  2113
 apply (simp add: sublist_def)
nipkow@13145
  2114
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  2115
done
wenzelm@13114
  2116
nipkow@15281
  2117
lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
nipkow@15281
  2118
apply(induct xs)
nipkow@15281
  2119
 apply simp
nipkow@15281
  2120
apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE)
nipkow@15281
  2121
 apply(erule lessE)
nipkow@15281
  2122
  apply auto
nipkow@15281
  2123
apply(erule lessE)
nipkow@15281
  2124
apply auto
nipkow@15281
  2125
done
nipkow@15281
  2126
nipkow@15281
  2127
lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
nipkow@15281
  2128
by(auto simp add:set_sublist)
nipkow@15281
  2129
nipkow@15281
  2130
lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
nipkow@15281
  2131
by(auto simp add:set_sublist)
nipkow@15281
  2132
nipkow@15281
  2133
lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
nipkow@15281
  2134
by(auto simp add:set_sublist)
nipkow@15281
  2135
wenzelm@13142
  2136
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  2137
by (simp add: sublist_Cons)
wenzelm@13114
  2138
nipkow@15281
  2139
nipkow@15281
  2140
lemma distinct_sublistI[simp]: "!!I. distinct xs \<Longrightarrow> distinct(sublist xs I)"
nipkow@15281
  2141
apply(induct xs)
nipkow@15281
  2142
 apply simp
nipkow@15281
  2143
apply(auto simp add:sublist_Cons)
nipkow@15281
  2144
done
nipkow@15281
  2145
nipkow@15281
  2146
nipkow@15045
  2147
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
paulson@14208
  2148
apply (induct l rule: rev_induct, simp)
nipkow@13145
  2149
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  2150
done
wenzelm@13114
  2151
nipkow@17501
  2152
lemma filter_in_sublist: "\<And>s. distinct xs \<Longrightarrow>
nipkow@17501
  2153
  filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
nipkow@17501
  2154
proof (induct xs)
nipkow@17501
  2155
  case Nil thus ?case by simp
nipkow@17501
  2156
next
nipkow@17501
  2157
  case (Cons a xs)
nipkow@17501
  2158
  moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
nipkow@17501
  2159
  ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
nipkow@17501
  2160
qed
nipkow@17501
  2161
wenzelm@13114
  2162
nipkow@15392
  2163
subsubsection{*Sets of Lists*}
nipkow@15392
  2164
nipkow@15392
  2165
subsubsection {* @{text lists}: the list-forming operator over sets *}
nipkow@15302
  2166
nipkow@15302
  2167
consts lists :: "'a set => 'a list set"
nipkow@15302
  2168
inductive "lists A"
nipkow@15302
  2169
 intros
nipkow@15302
  2170
  Nil [intro!]: "[]: lists A"
nipkow@15302
  2171
  Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A"
nipkow@15302
  2172
nipkow@15302
  2173
inductive_cases listsE [elim!]: "x#l : lists A"
nipkow@15302
  2174
nipkow@15302
  2175
lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B"
nipkow@15302
  2176
by (unfold lists.defs) (blast intro!: lfp_mono)
nipkow@15302
  2177
nipkow@15302
  2178
lemma lists_IntI:
nipkow@15302
  2179
  assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l
nipkow@15302
  2180
  by induct blast+
nipkow@15302
  2181
nipkow@15302
  2182
lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B"
nipkow@15302
  2183
proof (rule mono_Int [THEN equalityI])
nipkow@15302
  2184
  show "mono lists" by (simp add: mono_def lists_mono)
nipkow@15302
  2185
  show "lists A \<inter> lists B \<subseteq> lists (A \<inter> B)" by (blast intro: lists_IntI)
kleing@14388
  2186
qed
kleing@14388
  2187
nipkow@15302
  2188
lemma append_in_lists_conv [iff]:
nipkow@15302
  2189
     "(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"
nipkow@15302
  2190
by (induct xs) auto
nipkow@15302
  2191
nipkow@15302
  2192
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"
nipkow@15302
  2193
-- {* eliminate @{text lists} in favour of @{text set} *}
nipkow@15302
  2194
by (induct xs) auto
nipkow@15302
  2195
nipkow@15302
  2196
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"
nipkow@15302
  2197
by (rule in_lists_conv_set [THEN iffD1])
nipkow@15302
  2198
nipkow@15302
  2199
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"
nipkow@15302
  2200
by (rule in_lists_conv_set [THEN iffD2])
nipkow@15302
  2201
nipkow@15302
  2202
lemma lists_UNIV [simp]: "lists UNIV = UNIV"
nipkow@15302
  2203
by auto
nipkow@15302
  2204
nipkow@17086
  2205
subsubsection {* For efficiency *}
nipkow@17086
  2206
nipkow@17086
  2207
text{* Only use @{text mem} for generating executable code.  Otherwise
nipkow@17086
  2208
use @{prop"x : set xs"} instead --- it is much easier to reason about.
nipkow@17086
  2209
The same is true for @{const list_all} and @{const list_ex}: write
nipkow@17086
  2210
@{text"\<forall>x\<in>set xs"} and @{text"\<exists>x\<in>set xs"} instead because the HOL
nipkow@17090
  2211
quantifiers are aleady known to the automatic provers. In fact, the declarations in the Code subsection make sure that @{text"\<in>"}, @{text"\<forall>x\<in>set xs"}
nipkow@17090
  2212
and @{text"\<exists>x\<in>set xs"} are implemented efficiently.
nipkow@17086
  2213
nipkow@17086
  2214
The functions @{const itrev}, @{const filtermap} and @{const
nipkow@17086
  2215
map_filter} are just there to generate efficient code. Do not use them
nipkow@17086
  2216
for modelling and proving. *}
nipkow@17086
  2217
nipkow@17086
  2218
lemma mem_iff: "(x mem xs) = (x : set xs)"
nipkow@17086
  2219
by (induct xs) auto
nipkow@17086
  2220
nipkow@17086
  2221
lemma list_inter_conv: "set(list_inter xs ys) = set xs \<inter> set ys"
nipkow@17086
  2222
by (induct xs) auto
nipkow@17086
  2223
nipkow@17086
  2224
lemma list_all_iff: "list_all P xs = (\<forall>x \<in> set xs. P x)"
nipkow@17086
  2225
by (induct xs) auto
nipkow@17086
  2226
nipkow@17086
  2227
lemma list_all_append [simp]:
nipkow@17086
  2228
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
nipkow@17086
  2229
by (induct xs) auto
nipkow@17086
  2230
nipkow@17086
  2231
lemma list_all_rev [simp]: "list_all P (rev xs) = list_all P xs"
nipkow@17086
  2232
by (simp add: list_all_iff)
nipkow@17086
  2233
nipkow@17086
  2234
lemma list_ex_iff: "list_ex P xs = (\<exists>x \<in> set xs. P x)"
nipkow@17086
  2235
by (induct xs) simp_all
nipkow@17086
  2236
nipkow@17086
  2237
lemma itrev[simp]: "ALL ys. itrev xs ys = rev xs @ ys"
nipkow@17086
  2238
by (induct xs) simp_all
nipkow@17086
  2239
nipkow@17086
  2240
lemma filtermap_conv:
nipkow@17086
  2241
 "filtermap f xs = map (%x. the(f x)) (filter (%x. f x \<noteq> None) xs)"
nipkow@17086
  2242
by (induct xs) auto
nipkow@17086
  2243
nipkow@17086
  2244
lemma map_filter_conv[simp]: "map_filter f P xs = map f (filter P xs)"
nipkow@17086
  2245
by (induct xs) auto
nipkow@17086
  2246
nipkow@17086
  2247
nipkow@17086
  2248
subsubsection {* Code generation *}
nipkow@17086
  2249
nipkow@17086
  2250
text{* Defaults for generating efficient code for some standard functions. *}
nipkow@17086
  2251
nipkow@17090
  2252
lemmas in_set_code[code unfold] = mem_iff[symmetric, THEN eq_reflection]
nipkow@17090
  2253
nipkow@17090
  2254
lemma rev_code[code unfold]: "rev xs == itrev xs []"
nipkow@17086
  2255
by simp
nipkow@17086
  2256
nipkow@17090
  2257
lemma distinct_Cons_mem[code]: "distinct (x#xs) = (~(x mem xs) \<and> distinct xs)"
nipkow@17086
  2258
by (simp add:mem_iff)
nipkow@17086
  2259
nipkow@17090
  2260
lemma remdups_Cons_mem[code]:
nipkow@17086
  2261
 "remdups (x#xs) = (if x mem xs then remdups xs else x # remdups xs)"
nipkow@17086
  2262
by (simp add:mem_iff)
nipkow@17086
  2263
nipkow@17090
  2264
lemma list_inter_Cons_mem[code]:  "list_inter (a#as) bs =
nipkow@17086
  2265
  (if a mem bs then a#(list_inter as bs) else list_inter as bs)"
nipkow@17086
  2266
by(simp add:mem_iff)
nipkow@17086
  2267
nipkow@17090
  2268
text{* For implementing bounded quantifiers over lists by
nipkow@17090
  2269
@{const list_ex}/@{const list_all}: *}
nipkow@17090
  2270
nipkow@17090
  2271
lemmas list_bex_code[code unfold] = list_ex_iff[symmetric, THEN eq_reflection]
nipkow@17090
  2272
lemmas list_ball_code[code unfold] = list_all_iff[symmetric, THEN eq_reflection]
nipkow@17086
  2273
nipkow@17086
  2274
nipkow@17086
  2275
subsubsection{* Inductive definition for membership *}
nipkow@17086
  2276
nipkow@17086
  2277
consts ListMem :: "('a \<times> 'a list)set"
nipkow@17086
  2278
inductive ListMem
nipkow@17086
  2279
intros
nipkow@17086
  2280
 elem:  "(x,x#xs) \<in> ListMem"
nipkow@17086
  2281
 insert:  "(x,xs) \<in> ListMem \<Longrightarrow> (x,y#xs) \<in> ListMem"
nipkow@17086
  2282
nipkow@17086
  2283
lemma ListMem_iff: "((x,xs) \<in> ListMem) = (x \<in> set xs)"
nipkow@17086
  2284
apply (rule iffI)
nipkow@17086
  2285
 apply (induct set: ListMem)
nipkow@17086
  2286
  apply auto
nipkow@17086
  2287
apply (induct xs)
nipkow@17086
  2288
 apply (auto intro: ListMem.intros)
nipkow@17086
  2289
done
nipkow@17086
  2290
nipkow@17086
  2291
nipkow@17086
  2292
nipkow@15392
  2293
subsubsection{*Lists as Cartesian products*}
nipkow@15302
  2294
nipkow@15302
  2295
text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
nipkow@15302
  2296
@{term A} and tail drawn from @{term Xs}.*}
nipkow@15302
  2297
nipkow@15302
  2298
constdefs
nipkow@15302
  2299
  set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
nipkow@15302
  2300
  "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"
nipkow@15302
  2301
paulson@17724
  2302
lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
nipkow@15302
  2303
by (auto simp add: set_Cons_def)
nipkow@15302
  2304
nipkow@15302
  2305
text{*Yields the set of lists, all of the same length as the argument and
nipkow@15302
  2306
with elements drawn from the corresponding element of the argument.*}
nipkow@15302
  2307
nipkow@15302
  2308
consts  listset :: "'a set list \<Rightarrow> 'a list set"
nipkow@15302
  2309
primrec
nipkow@15302
  2310
   "listset []    = {[]}"
nipkow@15302
  2311
   "listset(A#As) = set_Cons A (listset As)"
nipkow@15302
  2312
nipkow@15302
  2313
paulson@15656
  2314
subsection{*Relations on Lists*}
paulson@15656
  2315
paulson@15656
  2316
subsubsection {* Length Lexicographic Ordering *}
paulson@15656
  2317
paulson@15656
  2318
text{*These orderings preserve well-foundedness: shorter lists 
paulson@15656
  2319
  precede longer lists. These ordering are not used in dictionaries.*}
paulson@15656
  2320
paulson@15656
  2321
consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
paulson@15656
  2322
        --{*The lexicographic ordering for lists of the specified length*}
nipkow@15302
  2323
primrec
paulson@15656
  2324
  "lexn r 0 = {}"
paulson@15656
  2325
  "lexn r (Suc n) =
paulson@15656
  2326
    (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
paulson@15656
  2327
    {(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
nipkow@15302
  2328
nipkow@15302
  2329
constdefs
paulson@15656
  2330
  lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
paulson@15656
  2331
    "lex r == \<Union>n. lexn r n"
paulson@15656
  2332
        --{*Holds only between lists of the same length*}
paulson@15656
  2333
nipkow@15693
  2334
  lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@15693
  2335
    "lenlex r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
paulson@15656
  2336
        --{*Compares lists by their length and then lexicographically*}
nipkow@15302
  2337
nipkow@15302
  2338
nipkow@15302
  2339
lemma wf_lexn: "wf r ==> wf (lexn r n)"
nipkow@15302
  2340
apply (induct n, simp, simp)
nipkow@15302
  2341
apply(rule wf_subset)
nipkow@15302
  2342
 prefer 2 apply (rule Int_lower1)
nipkow@15302
  2343
apply(rule wf_prod_fun_image)
nipkow@15302
  2344
 prefer 2 apply (rule inj_onI, auto)
nipkow@15302
  2345
done
nipkow@15302
  2346
nipkow@15302
  2347
lemma lexn_length:
nipkow@15302
  2348
     "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@15302
  2349
by (induct n) auto
nipkow@15302
  2350
nipkow@15302
  2351
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@15302
  2352
apply (unfold lex_def)
nipkow@15302
  2353
apply (rule wf_UN)
nipkow@15302
  2354
apply (blast intro: wf_lexn, clarify)
nipkow@15302
  2355
apply (rename_tac m n)
nipkow@15302
  2356
apply (subgoal_tac "m \<noteq> n")
nipkow@15302
  2357
 prefer 2 apply blast
nipkow@15302
  2358
apply (blast dest: lexn_length not_sym)
nipkow@15302
  2359
done
nipkow@15302
  2360
nipkow@15302
  2361
lemma lexn_conv:
paulson@15656
  2362
  "lexn r n =
paulson@15656
  2363
    {(xs,ys). length xs = n \<and> length ys = n \<and>
paulson@15656
  2364
    (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
nipkow@18423
  2365
apply (induct n, simp)
nipkow@15302
  2366
apply (simp add: image_Collect lex_prod_def, safe, blast)
nipkow@15302
  2367
 apply (rule_tac x = "ab # xys" in exI, simp)
nipkow@15302
  2368
apply (case_tac xys, simp_all, blast)
nipkow@15302
  2369
done
nipkow@15302
  2370
nipkow@15302
  2371
lemma lex_conv:
paulson@15656
  2372
  "lex r =
paulson@15656
  2373
    {(xs,ys). length xs = length ys \<and>
paulson@15656
  2374
    (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@15302
  2375
by (force simp add: lex_def lexn_conv)
nipkow@15302
  2376
nipkow@15693
  2377
lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
nipkow@15693
  2378
by (unfold lenlex_def) blast
nipkow@15693
  2379
nipkow@15693
  2380
lemma lenlex_conv:
nipkow@15693
  2381
    "lenlex r = {(xs,ys). length xs < length ys |
paulson@15656
  2382
                 length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@15693
  2383
by (simp add: lenlex_def diag_def lex_prod_def measure_def inv_image_def)
nipkow@15302
  2384
nipkow@15302
  2385
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@15302
  2386
by (simp add: lex_conv)
nipkow@15302
  2387
nipkow@15302
  2388
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@15302
  2389
by (simp add:lex_conv)
nipkow@15302
  2390
nipkow@15302
  2391
lemma Cons_in_lex [iff]:
paulson@15656
  2392
    "((x # xs, y # ys) : lex r) =
paulson@15656
  2393
      ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@15302
  2394
apply (simp add: lex_conv)
nipkow@15302
  2395
apply (rule iffI)
nipkow@15302
  2396
 prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
nipkow@15302
  2397
apply (case_tac xys, simp, simp)
nipkow@15302
  2398
apply blast
nipkow@15302
  2399
done
nipkow@15302
  2400
nipkow@15302
  2401
paulson@15656
  2402
subsubsection {* Lexicographic Ordering *}
paulson@15656
  2403
paulson@15656
  2404
text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
paulson@15656
  2405
    This ordering does \emph{not} preserve well-foundedness.
nipkow@17090
  2406
     Author: N. Voelker, March 2005. *} 
paulson@15656
  2407
paulson@15656
  2408
constdefs 
paulson@15656
  2409
  lexord :: "('a * 'a)set \<Rightarrow> ('a list * 'a list) set" 
paulson@15656
  2410
  "lexord  r == {(x,y). \<exists> a v. y = x @ a # v \<or> 
paulson@15656
  2411
            (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
paulson@15656
  2412
paulson@15656
  2413
lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
paulson@15656
  2414
  by (unfold lexord_def, induct_tac y, auto) 
paulson@15656
  2415
paulson@15656
  2416
lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
paulson@15656
  2417
  by (unfold lexord_def, induct_tac x, auto)
paulson@15656
  2418
paulson@15656
  2419
lemma lexord_cons_cons[simp]:
paulson@15656
  2420
     "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
paulson@15656
  2421
  apply (unfold lexord_def, safe, simp_all)
paulson@15656
  2422
  apply (case_tac u, simp, simp)
paulson@15656
  2423
  apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
paulson@15656
  2424
  apply (erule_tac x="b # u" in allE)
paulson@15656
  2425
  by force
paulson@15656
  2426
paulson@15656
  2427
lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
paulson@15656
  2428
paulson@15656
  2429
lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
paulson@15656
  2430
  by (induct_tac x, auto)  
paulson@15656
  2431
paulson@15656
  2432
lemma lexord_append_left_rightI:
paulson@15656
  2433
     "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
paulson@15656
  2434
  by (induct_tac u, auto)
paulson@15656
  2435
paulson@15656
  2436
lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
paulson@15656
  2437
  by (induct x, auto)
paulson@15656
  2438
paulson@15656
  2439
lemma lexord_append_leftD:
paulson@15656
  2440
     "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
paulson@15656
  2441
  by (erule rev_mp, induct_tac x, auto)
paulson@15656
  2442
paulson@15656
  2443
lemma lexord_take_index_conv: 
paulson@15656
  2444
   "((x,y) : lexord r) = 
paulson@15656
  2445
    ((length x < length y \<and> take (length x) y = x) \<or> 
paulson@15656
  2446
     (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
paulson@15656
  2447
  apply (unfold lexord_def Let_def, clarsimp) 
paulson@15656
  2448
  apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
paulson@15656
  2449
  apply auto 
paulson@15656
  2450
  apply (rule_tac x="hd (drop (length x) y)" in exI)
paulson@15656
  2451
  apply (rule_tac x="tl (drop (length x) y)" in exI)
paulson@15656
  2452
  apply (erule subst, simp add: min_def) 
paulson@15656
  2453
  apply (rule_tac x ="length u" in exI, simp) 
paulson@15656
  2454
  apply (rule_tac x ="take i x" in exI) 
paulson@15656
  2455
  apply (rule_tac x ="x ! i" in exI) 
paulson@15656
  2456
  apply (rule_tac x ="y ! i" in exI, safe) 
paulson@15656
  2457
  apply (rule_tac x="drop (Suc i) x" in exI)
paulson@15656
  2458
  apply (drule sym, simp add: drop_Suc_conv_tl) 
paulson@15656
  2459
  apply (rule_tac x="drop (Suc i) y" in exI)
paulson@15656
  2460
  by (simp add: drop_Suc_conv_tl) 
paulson@15656
  2461
paulson@15656
  2462
-- {* lexord is extension of partial ordering List.lex *} 
paulson@15656
  2463
lemma lexord_lex: " (x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
paulson@15656
  2464
  apply (rule_tac x = y in spec) 
paulson@15656
  2465
  apply (induct_tac x, clarsimp) 
paulson@15656
  2466
  by (clarify, case_tac x, simp, force)
paulson@15656
  2467
paulson@15656
  2468
lemma lexord_irreflexive: "(! x. (x,x) \<notin> r) \<Longrightarrow> (y,y) \<notin> lexord r"
paulson@15656
  2469
  by (induct y, auto)
paulson@15656
  2470
paulson@15656
  2471
lemma lexord_trans: 
paulson@15656
  2472
    "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
paulson@15656
  2473
   apply (erule rev_mp)+
paulson@15656
  2474
   apply (rule_tac x = x in spec) 
paulson@15656
  2475
  apply (rule_tac x = z in spec) 
paulson@15656
  2476
  apply ( induct_tac y, simp, clarify)
paulson@15656
  2477
  apply (case_tac xa, erule ssubst) 
paulson@15656
  2478
  apply (erule allE, erule allE) -- {* avoid simp recursion *} 
paulson@15656
  2479
  apply (case_tac x, simp, simp) 
paulson@15656
  2480
  apply (case_tac x, erule allE, erule allE, simp) 
paulson@15656
  2481
  apply (erule_tac x = listb in allE) 
paulson@15656
  2482
  apply (erule_tac x = lista in allE, simp)
paulson@15656
  2483
  apply (unfold trans_def)
paulson@15656
  2484
  by blast
paulson@15656
  2485
paulson@15656
  2486
lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
paulson@15656
  2487
  by (rule transI, drule lexord_trans, blast) 
paulson@15656
  2488
paulson@15656
  2489
lemma lexord_linear: "(! a b. (a,b)\<in> r | a = b | (b,a) \<in> r) \<Longrightarrow> (x,y) : lexord r | x = y | (y,x) : lexord r"
paulson@15656
  2490
  apply (rule_tac x = y in spec) 
paulson@15656
  2491
  apply (induct_tac x, rule allI) 
paulson@15656
  2492
  apply (case_tac x, simp, simp) 
paulson@15656
  2493
  apply (rule allI, case_tac x, simp, simp) 
paulson@15656
  2494
  by blast
paulson@15656
  2495
paulson@15656
  2496
nipkow@15392
  2497
subsubsection{*Lifting a Relation on List Elements to the Lists*}
nipkow@15302
  2498
nipkow@15302
  2499
consts  listrel :: "('a * 'a)set => ('a list * 'a list)set"
nipkow@15302
  2500
nipkow@15302
  2501
inductive "listrel(r)"
nipkow@15302
  2502
 intros
nipkow@15302
  2503
   Nil:  "([],[]) \<in> listrel r"
nipkow@15302
  2504
   Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
nipkow@15302
  2505
nipkow@15302
  2506
inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
nipkow@15302
  2507
inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
nipkow@15302
  2508
inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
nipkow@15302
  2509
inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
nipkow@15302
  2510
nipkow@15302
  2511
nipkow@15302
  2512
lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
nipkow@15302
  2513
apply clarify  
nipkow@15302
  2514
apply (erule listrel.induct)
nipkow@15302
  2515
apply (blast intro: listrel.intros)+
nipkow@15302
  2516
done
nipkow@15302
  2517
nipkow@15302
  2518
lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
nipkow@15302
  2519
apply clarify 
nipkow@15302
  2520
apply (erule listrel.induct, auto) 
nipkow@15302
  2521
done
nipkow@15302
  2522
nipkow@15302
  2523
lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" 
nipkow@15302
  2524
apply (simp add: refl_def listrel_subset Ball_def)
nipkow@15302
  2525
apply (rule allI) 
nipkow@15302
  2526
apply (induct_tac x) 
nipkow@15302
  2527
apply (auto intro: listrel.intros)
nipkow@15302
  2528
done
nipkow@15302
  2529
nipkow@15302
  2530
lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
nipkow@15302
  2531
apply (auto simp add: sym_def)
nipkow@15302
  2532
apply (erule listrel.induct) 
nipkow@15302
  2533
apply (blast intro: listrel.intros)+
nipkow@15302
  2534
done
nipkow@15302
  2535
nipkow@15302
  2536
lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
nipkow@15302
  2537
apply (simp add: trans_def)
nipkow@15302
  2538
apply (intro allI) 
nipkow@15302
  2539
apply (rule impI) 
nipkow@15302
  2540
apply (erule listrel.induct) 
nipkow@15302
  2541
apply (blast intro: listrel.intros)+
nipkow@15302
  2542
done
nipkow@15302
  2543
nipkow@15302
  2544
theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
nipkow@15302
  2545
by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) 
nipkow@15302
  2546
nipkow@15302
  2547
lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
nipkow@15302
  2548
by (blast intro: listrel.intros)
nipkow@15302
  2549
nipkow@15302
  2550
lemma listrel_Cons:
nipkow@15302
  2551
     "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})";
nipkow@15302
  2552
by (auto simp add: set_Cons_def intro: listrel.intros) 
nipkow@15302
  2553
nipkow@15302
  2554
nipkow@15392
  2555
subsection{*Miscellany*}
nipkow@15392
  2556
nipkow@15392
  2557
subsubsection {* Characters and strings *}
wenzelm@13366
  2558
wenzelm@13366
  2559
datatype nibble =
wenzelm@13366
  2560
    Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
wenzelm@13366
  2561
  | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
wenzelm@13366
  2562
wenzelm@13366
  2563
datatype char = Char nibble nibble
wenzelm@13366
  2564
  -- "Note: canonical order of character encoding coincides with standard term ordering"
wenzelm@13366
  2565
wenzelm@13366
  2566
types string = "char list"
wenzelm@13366
  2567
wenzelm@13366
  2568
syntax
wenzelm@13366
  2569
  "_Char" :: "xstr => char"    ("CHR _")
wenzelm@13366
  2570
  "_String" :: "xstr => string"    ("_")
wenzelm@13366
  2571
wenzelm@13366
  2572
parse_ast_translation {*
wenzelm@13366
  2573
  let
wenzelm@13366
  2574
    val constants = Syntax.Appl o map Syntax.Constant;
wenzelm@13366
  2575
wenzelm@13366
  2576
    fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10));
wenzelm@13366
  2577
    fun mk_char c =
wenzelm@13366
  2578
      if Symbol.is_ascii c andalso Symbol.is_printable c then
wenzelm@13366
  2579
        constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)]
wenzelm@13366
  2580
      else error ("Printable ASCII character expected: " ^ quote c);
wenzelm@13366
  2581
wenzelm@13366
  2582
    fun mk_string [] = Syntax.Constant "Nil"
wenzelm@13366
  2583
      | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs];
wenzelm@13366
  2584
wenzelm@13366
  2585
    fun char_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  2586
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  2587
          [c] => mk_char c
wenzelm@13366
  2588
        | _ => error ("Single character expected: " ^ xstr))
wenzelm@13366
  2589
      | char_ast_tr asts = raise AST ("char_ast_tr", asts);
wenzelm@13366
  2590
wenzelm@13366
  2591
    fun string_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  2592
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  2593
          [] => constants [Syntax.constrainC, "Nil", "string"]
wenzelm@13366
  2594
        | cs => mk_string cs)
wenzelm@13366
  2595
      | string_ast_tr asts = raise AST ("string_tr", asts);
wenzelm@13366
  2596
  in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end;
wenzelm@13366
  2597
*}
wenzelm@13366
  2598
berghofe@15064
  2599
ML {*
berghofe@15064
  2600
fun int_of_nibble h =
berghofe@15064
  2601
  if "0" <= h andalso h <= "9" then ord h - ord "0"
berghofe@15064
  2602
  else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10
berghofe@15064
  2603
  else raise Match;
berghofe@15064
  2604
berghofe@15064
  2605
fun nibble_of_int i =
berghofe@15064
  2606
  if i <= 9 then chr (ord "0" + i) else chr (ord "A" + i - 10);
berghofe@15064
  2607
*}
berghofe@15064
  2608
wenzelm@13366
  2609
print_ast_translation {*
wenzelm@13366
  2610
  let
wenzelm@13366
  2611
    fun dest_nib (Syntax.Constant c) =
wenzelm@13366
  2612
        (case explode c of
berghofe@15064
  2613
          ["N", "i", "b", "b", "l", "e", h] => int_of_nibble h
wenzelm@13366
  2614
        | _ => raise Match)
wenzelm@13366
  2615
      | dest_nib _ = raise Match;
wenzelm@13366
  2616
wenzelm@13366
  2617
    fun dest_chr c1 c2 =
wenzelm@13366
  2618
      let val c = chr (dest_nib c1 * 16 + dest_nib c2)
wenzelm@13366
  2619
      in if Symbol.is_printable c then c else raise Match end;
wenzelm@13366
  2620
wenzelm@13366
  2621
    fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2
wenzelm@13366
  2622
      | dest_char _ = raise Match;
wenzelm@13366
  2623
wenzelm@13366
  2624
    fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)];
wenzelm@13366
  2625
wenzelm@13366
  2626
    fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]]
wenzelm@13366
  2627
      | char_ast_tr' _ = raise Match;
wenzelm@13366
  2628
wenzelm@13366
  2629
    fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String",
wenzelm@13366
  2630
            xstr (map dest_char (Syntax.unfold_ast "_args" args))]
wenzelm@13366
  2631
      | list_ast_tr' ts = raise Match;
wenzelm@13366
  2632
  in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end;
wenzelm@13366
  2633
*}
wenzelm@13366
  2634
nipkow@15392
  2635
subsubsection {* Code generator setup *}
berghofe@15064
  2636
berghofe@15064
  2637
ML {*
berghofe@15064
  2638
local
berghofe@15064
  2639
berghofe@16634
  2640
fun list_codegen thy defs gr dep thyname b t =
berghofe@16634
  2641
  let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy defs dep thyname false)
berghofe@15064
  2642
    (gr, HOLogic.dest_list t)
skalberg@15531
  2643
  in SOME (gr', Pretty.list "[" "]" ps) end handle TERM _ => NONE;
berghofe@15064
  2644
berghofe@15064
  2645
fun dest_nibble (Const (s, _)) = int_of_nibble (unprefix "List.nibble.Nibble" s)
berghofe@15064
  2646
  | dest_nibble _ = raise Match;
berghofe@15064
  2647
berghofe@16634
  2648
fun char_codegen thy defs gr dep thyname b (Const ("List.char.Char", _) $ c1 $ c2) =
berghofe@15064
  2649
    (let val c = chr (dest_nibble c1 * 16 + dest_nibble c2)
skalberg@15531
  2650
     in if Symbol.is_printable c then SOME (gr, Pretty.quote (Pretty.str c))
skalberg@15531
  2651
       else NONE
skalberg@15570
  2652
     end handle Fail _ => NONE | Match => NONE)
berghofe@16634
  2653
  | char_codegen thy defs gr dep thyname b _ = NONE;
berghofe@15064
  2654
berghofe@15064
  2655
in
berghofe@15064
  2656
berghofe@15064
  2657
val list_codegen_setup =
berghofe@15064
  2658
  [Codegen.add_codegen "list_codegen" list_codegen,
berghofe@15064
  2659
   Codegen.add_codegen "char_codegen" char_codegen];
berghofe@15064
  2660
berghofe@15064
  2661
end;
berghofe@16770
  2662
*}
berghofe@16770
  2663
berghofe@16770
  2664
types_code
berghofe@16770
  2665
  "list" ("_ list")
berghofe@16770
  2666
attach (term_of) {*
berghofe@15064
  2667
val term_of_list = HOLogic.mk_list;
berghofe@16770
  2668
*}
berghofe@16770
  2669
attach (test) {*
berghofe@15064
  2670
fun gen_list' aG i j = frequency
berghofe@15064
  2671
  [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] ()
berghofe@15064
  2672
and gen_list aG i = gen_list' aG i i;
berghofe@16770
  2673
*}
berghofe@16770
  2674
  "char" ("string")
berghofe@16770
  2675
attach (term_of) {*
berghofe@15064
  2676
val nibbleT = Type ("List.nibble", []);
berghofe@15064
  2677
berghofe@15064
  2678
fun term_of_char c =
berghofe@15064
  2679
  Const ("List.char.Char", nibbleT --> nibbleT --> Type ("List.char", [])) $
berghofe@15064
  2680
    Const ("List.nibble.Nibble" ^ nibble_of_int (ord c div 16), nibbleT) $
berghofe@15064
  2681
    Const ("List.nibble.Nibble" ^ nibble_of_int (ord c mod 16), nibbleT);
berghofe@16770
  2682
*}
berghofe@16770
  2683
attach (test) {*
berghofe@15064
  2684
fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z")));
berghofe@15064
  2685
*}
berghofe@15064
  2686
berghofe@15064
  2687
consts_code "Cons" ("(_ ::/ _)")
berghofe@15064
  2688
berghofe@15064
  2689
setup list_codegen_setup
berghofe@15064
  2690
wenzelm@13122
  2691
end