src/HOL/List.thy
author nipkow
Tue Aug 02 13:13:18 2005 +0200 (2005-08-02)
changeset 16998 e0050191e2d1
parent 16973 b2a894562b8f
child 17086 0eb0c9259dd7
permissions -rw-r--r--
Added filter lemma
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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_all:: "('a => bool) => ('a list => bool)"
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  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool"
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  list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
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  map :: "('a=>'b) => ('a list => 'b list)"
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  mem :: "'a => 'a list => bool"    (infixl 55)
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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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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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  "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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  "set [] = {}"
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  "set (x#xs) = insert x (set xs)"
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primrec
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  list_all_Nil:"list_all P [] = True"
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  list_all_Cons: "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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  "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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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]) rules
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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)"
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apply (induct xs)
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 apply (case_tac ys, simp, force)
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apply (case_tac ys, force, simp)
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done
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lemma append_eq_append_conv2: "!!ys zs ts.
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 (xs @ ys = zs @ ts) =
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 (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
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apply (induct xs)
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 apply fastsimp
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apply(case_tac zs)
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 apply simp
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apply fastsimp
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done
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lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
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by simp
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lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
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by simp
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lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
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by simp
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lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   345
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   346
wenzelm@13142
   347
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   348
using append_same_eq [of "[]"] by auto
wenzelm@13114
   349
wenzelm@13142
   350
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   351
by (induct xs) auto
wenzelm@13114
   352
wenzelm@13142
   353
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   354
by (induct xs) auto
wenzelm@13114
   355
wenzelm@13142
   356
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   357
by (simp add: hd_append split: list.split)
wenzelm@13114
   358
wenzelm@13142
   359
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   360
by (simp split: list.split)
wenzelm@13114
   361
wenzelm@13142
   362
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   363
by (simp add: tl_append split: list.split)
wenzelm@13114
   364
wenzelm@13114
   365
nipkow@14300
   366
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   367
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   368
by(cases ys) auto
nipkow@14300
   369
nipkow@15281
   370
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   371
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   372
by(cases ys) auto
nipkow@15281
   373
nipkow@14300
   374
wenzelm@13142
   375
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   376
wenzelm@13114
   377
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   378
by simp
wenzelm@13114
   379
wenzelm@13142
   380
lemma Cons_eq_appendI:
nipkow@13145
   381
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   382
by (drule sym) simp
wenzelm@13114
   383
wenzelm@13142
   384
lemma append_eq_appendI:
nipkow@13145
   385
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   386
by (drule sym) simp
wenzelm@13114
   387
wenzelm@13114
   388
wenzelm@13142
   389
text {*
nipkow@13145
   390
Simplification procedure for all list equalities.
nipkow@13145
   391
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   392
- both lists end in a singleton list,
nipkow@13145
   393
- or both lists end in the same list.
wenzelm@13142
   394
*}
wenzelm@13142
   395
wenzelm@13142
   396
ML_setup {*
nipkow@3507
   397
local
nipkow@3507
   398
wenzelm@13122
   399
val append_assoc = thm "append_assoc";
wenzelm@13122
   400
val append_Nil = thm "append_Nil";
wenzelm@13122
   401
val append_Cons = thm "append_Cons";
wenzelm@13122
   402
val append1_eq_conv = thm "append1_eq_conv";
wenzelm@13122
   403
val append_same_eq = thm "append_same_eq";
wenzelm@13122
   404
wenzelm@13114
   405
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462
   406
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
wenzelm@13462
   407
  | last (Const("List.op @",_) $ _ $ ys) = last ys
wenzelm@13462
   408
  | last t = t;
wenzelm@13114
   409
wenzelm@13114
   410
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462
   411
  | list1 _ = false;
wenzelm@13114
   412
wenzelm@13114
   413
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462
   414
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
wenzelm@13462
   415
  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462
   416
  | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114
   417
wenzelm@16973
   418
val rearr_ss = HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons];
wenzelm@16973
   419
wenzelm@16973
   420
fun list_eq sg ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   421
  let
wenzelm@13462
   422
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   423
    fun rearr conv =
wenzelm@13462
   424
      let
wenzelm@13462
   425
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   426
        val Type(_,listT::_) = eqT
wenzelm@13462
   427
        val appT = [listT,listT] ---> listT
wenzelm@13462
   428
        val app = Const("List.op @",appT)
wenzelm@13462
   429
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   430
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@16973
   431
        val thm = Tactic.prove sg [] [] eq
wenzelm@16973
   432
          (K (simp_tac (Simplifier.inherit_bounds ss rearr_ss) 1));
skalberg@15531
   433
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   434
wenzelm@13462
   435
  in
wenzelm@13462
   436
    if list1 lastl andalso list1 lastr then rearr append1_eq_conv
wenzelm@13462
   437
    else if lastl aconv lastr then rearr append_same_eq
skalberg@15531
   438
    else NONE
wenzelm@13462
   439
  end;
wenzelm@13462
   440
wenzelm@13114
   441
in
wenzelm@13462
   442
wenzelm@13462
   443
val list_eq_simproc =
wenzelm@13462
   444
  Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq;
wenzelm@13462
   445
wenzelm@13114
   446
end;
wenzelm@13114
   447
wenzelm@13114
   448
Addsimprocs [list_eq_simproc];
wenzelm@13114
   449
*}
wenzelm@13114
   450
wenzelm@13114
   451
nipkow@15392
   452
subsubsection {* @{text map} *}
wenzelm@13114
   453
wenzelm@13142
   454
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   455
by (induct xs) simp_all
wenzelm@13114
   456
wenzelm@13142
   457
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   458
by (rule ext, induct_tac xs) auto
wenzelm@13114
   459
wenzelm@13142
   460
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   461
by (induct xs) auto
wenzelm@13114
   462
wenzelm@13142
   463
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   464
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   465
wenzelm@13142
   466
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   467
by (induct xs) auto
wenzelm@13114
   468
nipkow@13737
   469
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   470
by (induct xs) auto
nipkow@13737
   471
wenzelm@13366
   472
lemma map_cong [recdef_cong]:
nipkow@13145
   473
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   474
-- {* a congruence rule for @{text map} *}
nipkow@13737
   475
by simp
wenzelm@13114
   476
wenzelm@13142
   477
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   478
by (cases xs) auto
wenzelm@13114
   479
wenzelm@13142
   480
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   481
by (cases xs) auto
wenzelm@13114
   482
nipkow@14025
   483
lemma map_eq_Cons_conv[iff]:
nipkow@14025
   484
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   485
by (cases xs) auto
wenzelm@13114
   486
nipkow@14025
   487
lemma Cons_eq_map_conv[iff]:
nipkow@14025
   488
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   489
by (cases ys) auto
nipkow@14025
   490
nipkow@14111
   491
lemma ex_map_conv:
nipkow@14111
   492
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
nipkow@14111
   493
by(induct ys, auto)
nipkow@14111
   494
nipkow@15110
   495
lemma map_eq_imp_length_eq:
nipkow@15110
   496
  "!!xs. map f xs = map f ys ==> length xs = length ys"
nipkow@15110
   497
apply (induct ys)
nipkow@15110
   498
 apply simp
nipkow@15110
   499
apply(simp (no_asm_use))
nipkow@15110
   500
apply clarify
nipkow@15110
   501
apply(simp (no_asm_use))
nipkow@15110
   502
apply fast
nipkow@15110
   503
done
nipkow@15110
   504
nipkow@15110
   505
lemma map_inj_on:
nipkow@15110
   506
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   507
  ==> xs = ys"
nipkow@15110
   508
apply(frule map_eq_imp_length_eq)
nipkow@15110
   509
apply(rotate_tac -1)
nipkow@15110
   510
apply(induct rule:list_induct2)
nipkow@15110
   511
 apply simp
nipkow@15110
   512
apply(simp)
nipkow@15110
   513
apply (blast intro:sym)
nipkow@15110
   514
done
nipkow@15110
   515
nipkow@15110
   516
lemma inj_on_map_eq_map:
nipkow@15110
   517
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   518
by(blast dest:map_inj_on)
nipkow@15110
   519
wenzelm@13114
   520
lemma map_injective:
nipkow@14338
   521
 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@14338
   522
by (induct ys) (auto dest!:injD)
wenzelm@13114
   523
nipkow@14339
   524
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   525
by(blast dest:map_injective)
nipkow@14339
   526
wenzelm@13114
   527
lemma inj_mapI: "inj f ==> inj (map f)"
paulson@13585
   528
by (rules dest: map_injective injD intro: inj_onI)
wenzelm@13114
   529
wenzelm@13114
   530
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   531
apply (unfold inj_on_def, clarify)
nipkow@13145
   532
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   533
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   534
apply blast
nipkow@13145
   535
done
wenzelm@13114
   536
nipkow@14339
   537
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   538
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   539
nipkow@15303
   540
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
   541
apply(rule inj_onI)
nipkow@15303
   542
apply(erule map_inj_on)
nipkow@15303
   543
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
   544
done
nipkow@15303
   545
kleing@14343
   546
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   547
by (induct xs, auto)
wenzelm@13114
   548
nipkow@14402
   549
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   550
by (induct xs) auto
nipkow@14402
   551
nipkow@15110
   552
lemma map_fst_zip[simp]:
nipkow@15110
   553
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   554
by (induct rule:list_induct2, simp_all)
nipkow@15110
   555
nipkow@15110
   556
lemma map_snd_zip[simp]:
nipkow@15110
   557
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   558
by (induct rule:list_induct2, simp_all)
nipkow@15110
   559
nipkow@15110
   560
nipkow@15392
   561
subsubsection {* @{text rev} *}
wenzelm@13114
   562
wenzelm@13142
   563
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   564
by (induct xs) auto
wenzelm@13114
   565
wenzelm@13142
   566
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   567
by (induct xs) auto
wenzelm@13114
   568
kleing@15870
   569
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870
   570
by auto
kleing@15870
   571
wenzelm@13142
   572
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   573
by (induct xs) auto
wenzelm@13114
   574
wenzelm@13142
   575
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   576
by (induct xs) auto
wenzelm@13114
   577
kleing@15870
   578
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870
   579
by (cases xs) auto
kleing@15870
   580
kleing@15870
   581
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870
   582
by (cases xs) auto
kleing@15870
   583
wenzelm@13142
   584
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"
paulson@14208
   585
apply (induct xs, force)
paulson@14208
   586
apply (case_tac ys, simp, force)
nipkow@13145
   587
done
wenzelm@13114
   588
nipkow@15439
   589
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
   590
by(simp add:inj_on_def)
nipkow@15439
   591
wenzelm@13366
   592
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   593
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489
   594
apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145
   595
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   596
done
wenzelm@13114
   597
nipkow@13145
   598
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114
   599
wenzelm@13366
   600
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   601
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   602
by (induct xs rule: rev_induct) auto
wenzelm@13114
   603
wenzelm@13366
   604
lemmas rev_cases = rev_exhaust
wenzelm@13366
   605
wenzelm@13114
   606
nipkow@15392
   607
subsubsection {* @{text set} *}
wenzelm@13114
   608
wenzelm@13142
   609
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   610
by (induct xs) auto
wenzelm@13114
   611
wenzelm@13142
   612
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   613
by (induct xs) auto
wenzelm@13114
   614
oheimb@14099
   615
lemma hd_in_set: "l = x#xs \<Longrightarrow> x\<in>set l"
paulson@14208
   616
by (case_tac l, auto)
oheimb@14099
   617
wenzelm@13142
   618
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   619
by auto
wenzelm@13114
   620
oheimb@14099
   621
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   622
by auto
oheimb@14099
   623
wenzelm@13142
   624
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   625
by (induct xs) auto
wenzelm@13114
   626
nipkow@15245
   627
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   628
by(induct xs) auto
nipkow@15245
   629
wenzelm@13142
   630
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   631
by (induct xs) auto
wenzelm@13114
   632
wenzelm@13142
   633
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   634
by (induct xs) auto
wenzelm@13114
   635
wenzelm@13142
   636
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   637
by (induct xs) auto
wenzelm@13114
   638
nipkow@15425
   639
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
paulson@14208
   640
apply (induct j, simp_all)
paulson@14208
   641
apply (erule ssubst, auto)
nipkow@13145
   642
done
wenzelm@13114
   643
wenzelm@13142
   644
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
paulson@15113
   645
proof (induct xs)
paulson@15113
   646
  case Nil show ?case by simp
paulson@15113
   647
  case (Cons a xs)
paulson@15113
   648
  show ?case
paulson@15113
   649
  proof 
paulson@15113
   650
    assume "x \<in> set (a # xs)"
paulson@15113
   651
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   652
      by (simp, blast intro: Cons_eq_appendI)
paulson@15113
   653
  next
paulson@15113
   654
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   655
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
paulson@15113
   656
    show "x \<in> set (a # xs)" 
paulson@15113
   657
      by (cases ys, auto simp add: eq)
paulson@15113
   658
  qed
paulson@15113
   659
qed
wenzelm@13142
   660
paulson@13508
   661
lemma finite_list: "finite A ==> EX l. set l = A"
paulson@13508
   662
apply (erule finite_induct, auto)
paulson@13508
   663
apply (rule_tac x="x#l" in exI, auto)
paulson@13508
   664
done
paulson@13508
   665
kleing@14388
   666
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
   667
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
   668
paulson@15168
   669
nipkow@15439
   670
subsubsection {* @{text mem}, @{text list_all} and @{text list_ex} *}
wenzelm@13114
   671
nipkow@15302
   672
text{* Only use @{text mem} for generating executable code.  Otherwise
nipkow@15439
   673
use @{prop"x : set xs"} instead --- it is much easier to reason about.
nipkow@15439
   674
The same is true for @{text list_all} and @{text list_ex}: write
nipkow@15439
   675
@{text"\<forall>x\<in>set xs"} and @{text"\<exists>x\<in>set xs"} instead because the HOL
nipkow@15439
   676
quantifiers are aleady known to the automatic provers. For the purpose
nipkow@15439
   677
of generating executable code use the theorems @{text set_mem_eq},
nipkow@15439
   678
@{text list_all_conv} and @{text list_ex_iff} to get rid off or
nipkow@15439
   679
introduce the combinators. *}
nipkow@15302
   680
wenzelm@13114
   681
lemma set_mem_eq: "(x mem xs) = (x : set xs)"
nipkow@13145
   682
by (induct xs) auto
wenzelm@13114
   683
wenzelm@13142
   684
lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)"
nipkow@13145
   685
by (induct xs) auto
wenzelm@13114
   686
wenzelm@13142
   687
lemma list_all_append [simp]:
nipkow@13145
   688
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
nipkow@13145
   689
by (induct xs) auto
wenzelm@13114
   690
kleing@15426
   691
lemma list_all_rev [simp]: "list_all P (rev xs) = list_all P xs"
kleing@15426
   692
by (simp add: list_all_conv)
kleing@15426
   693
nipkow@15439
   694
lemma list_ex_iff: "list_ex P xs = (\<exists>x \<in> set xs. P x)"
nipkow@15439
   695
by (induct xs) simp_all
kleing@15426
   696
wenzelm@13114
   697
nipkow@15392
   698
subsubsection {* @{text filter} *}
wenzelm@13114
   699
wenzelm@13142
   700
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   701
by (induct xs) auto
wenzelm@13114
   702
nipkow@15305
   703
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
   704
by (induct xs) simp_all
nipkow@15305
   705
wenzelm@13142
   706
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   707
by (induct xs) auto
wenzelm@13114
   708
nipkow@16998
   709
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998
   710
by (induct xs) (auto simp add: le_SucI)
nipkow@16998
   711
wenzelm@13142
   712
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   713
by (induct xs) auto
wenzelm@13114
   714
wenzelm@13142
   715
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   716
by (induct xs) auto
wenzelm@13114
   717
nipkow@16998
   718
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@16998
   719
  by (induct xs) simp_all
nipkow@16998
   720
nipkow@16998
   721
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998
   722
apply (induct xs)
nipkow@16998
   723
 apply auto
nipkow@16998
   724
apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998
   725
apply simp
nipkow@16998
   726
done
wenzelm@13114
   727
nipkow@16965
   728
lemma filter_map:
nipkow@16965
   729
  "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965
   730
by (induct xs) simp_all
nipkow@16965
   731
nipkow@16965
   732
lemma length_filter_map[simp]:
nipkow@16965
   733
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965
   734
by (simp add:filter_map)
nipkow@16965
   735
wenzelm@13142
   736
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   737
by auto
wenzelm@13114
   738
nipkow@15246
   739
lemma length_filter_less:
nipkow@15246
   740
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
   741
proof (induct xs)
nipkow@15246
   742
  case Nil thus ?case by simp
nipkow@15246
   743
next
nipkow@15246
   744
  case (Cons x xs) thus ?case
nipkow@15246
   745
    apply (auto split:split_if_asm)
nipkow@15246
   746
    using length_filter_le[of P xs] apply arith
nipkow@15246
   747
  done
nipkow@15246
   748
qed
wenzelm@13114
   749
nipkow@15281
   750
lemma length_filter_conv_card:
nipkow@15281
   751
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
   752
proof (induct xs)
nipkow@15281
   753
  case Nil thus ?case by simp
nipkow@15281
   754
next
nipkow@15281
   755
  case (Cons x xs)
nipkow@15281
   756
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
   757
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
   758
  show ?case (is "?l = card ?S'")
nipkow@15281
   759
  proof (cases)
nipkow@15281
   760
    assume "p x"
nipkow@15281
   761
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@15281
   762
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   763
    have "length (filter p (x # xs)) = Suc(card ?S)"
nipkow@15281
   764
      using Cons by simp
nipkow@15281
   765
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
nipkow@15281
   766
      by (simp add: card_image inj_Suc)
nipkow@15281
   767
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   768
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
   769
    finally show ?thesis .
nipkow@15281
   770
  next
nipkow@15281
   771
    assume "\<not> p x"
nipkow@15281
   772
    hence eq: "?S' = Suc ` ?S"
nipkow@15281
   773
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   774
    have "length (filter p (x # xs)) = card ?S"
nipkow@15281
   775
      using Cons by simp
nipkow@15281
   776
    also have "\<dots> = card(Suc ` ?S)" using fin
nipkow@15281
   777
      by (simp add: card_image inj_Suc)
nipkow@15281
   778
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   779
      by (simp add:card_insert_if)
nipkow@15281
   780
    finally show ?thesis .
nipkow@15281
   781
  qed
nipkow@15281
   782
qed
nipkow@15281
   783
nipkow@15281
   784
nipkow@15392
   785
subsubsection {* @{text concat} *}
wenzelm@13114
   786
wenzelm@13142
   787
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
   788
by (induct xs) auto
wenzelm@13114
   789
wenzelm@13142
   790
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   791
by (induct xss) auto
wenzelm@13114
   792
wenzelm@13142
   793
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   794
by (induct xss) auto
wenzelm@13114
   795
wenzelm@13142
   796
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
nipkow@13145
   797
by (induct xs) auto
wenzelm@13114
   798
wenzelm@13142
   799
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
   800
by (induct xs) auto
wenzelm@13114
   801
wenzelm@13142
   802
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
   803
by (induct xs) auto
wenzelm@13114
   804
wenzelm@13142
   805
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
   806
by (induct xs) auto
wenzelm@13114
   807
wenzelm@13114
   808
nipkow@15392
   809
subsubsection {* @{text nth} *}
wenzelm@13114
   810
wenzelm@13142
   811
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
   812
by auto
wenzelm@13114
   813
wenzelm@13142
   814
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
   815
by auto
wenzelm@13114
   816
wenzelm@13142
   817
declare nth.simps [simp del]
wenzelm@13114
   818
wenzelm@13114
   819
lemma nth_append:
nipkow@13145
   820
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
paulson@14208
   821
apply (induct "xs", simp)
paulson@14208
   822
apply (case_tac n, auto)
nipkow@13145
   823
done
wenzelm@13114
   824
nipkow@14402
   825
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
nipkow@14402
   826
by (induct "xs") auto
nipkow@14402
   827
nipkow@14402
   828
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
nipkow@14402
   829
by (induct "xs") auto
nipkow@14402
   830
wenzelm@13142
   831
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
paulson@14208
   832
apply (induct xs, simp)
paulson@14208
   833
apply (case_tac n, auto)
nipkow@13145
   834
done
wenzelm@13114
   835
wenzelm@13142
   836
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
   837
apply (induct xs, simp, simp)
nipkow@13145
   838
apply safe
paulson@14208
   839
apply (rule_tac x = 0 in exI, simp)
paulson@14208
   840
 apply (rule_tac x = "Suc i" in exI, simp)
paulson@14208
   841
apply (case_tac i, simp)
nipkow@13145
   842
apply (rename_tac j)
paulson@14208
   843
apply (rule_tac x = j in exI, simp)
nipkow@13145
   844
done
wenzelm@13114
   845
nipkow@13145
   846
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
   847
by (auto simp add: set_conv_nth)
wenzelm@13114
   848
wenzelm@13142
   849
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
   850
by (auto simp add: set_conv_nth)
wenzelm@13114
   851
wenzelm@13114
   852
lemma all_nth_imp_all_set:
nipkow@13145
   853
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
   854
by (auto simp add: set_conv_nth)
wenzelm@13114
   855
wenzelm@13114
   856
lemma all_set_conv_all_nth:
nipkow@13145
   857
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
   858
by (auto simp add: set_conv_nth)
wenzelm@13114
   859
wenzelm@13114
   860
nipkow@15392
   861
subsubsection {* @{text list_update} *}
wenzelm@13114
   862
wenzelm@13142
   863
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
nipkow@13145
   864
by (induct xs) (auto split: nat.split)
wenzelm@13114
   865
wenzelm@13114
   866
lemma nth_list_update:
nipkow@13145
   867
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@13145
   868
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   869
wenzelm@13142
   870
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
   871
by (simp add: nth_list_update)
wenzelm@13114
   872
wenzelm@13142
   873
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145
   874
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   875
wenzelm@13142
   876
lemma list_update_overwrite [simp]:
nipkow@13145
   877
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145
   878
by (induct xs) (auto split: nat.split)
wenzelm@13114
   879
nipkow@14402
   880
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
paulson@14208
   881
apply (induct xs, simp)
nipkow@14187
   882
apply(simp split:nat.splits)
nipkow@14187
   883
done
nipkow@14187
   884
wenzelm@13114
   885
lemma list_update_same_conv:
nipkow@13145
   886
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145
   887
by (induct xs) (auto split: nat.split)
wenzelm@13114
   888
nipkow@14187
   889
lemma list_update_append1:
nipkow@14187
   890
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
paulson@14208
   891
apply (induct xs, simp)
nipkow@14187
   892
apply(simp split:nat.split)
nipkow@14187
   893
done
nipkow@14187
   894
kleing@15868
   895
lemma list_update_append:
kleing@15868
   896
  "!!n. (xs @ ys) [n:= x] = 
kleing@15868
   897
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
kleing@15868
   898
by (induct xs) (auto split:nat.splits)
kleing@15868
   899
nipkow@14402
   900
lemma list_update_length [simp]:
nipkow@14402
   901
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
   902
by (induct xs, auto)
nipkow@14402
   903
wenzelm@13114
   904
lemma update_zip:
nipkow@13145
   905
"!!i xy xs. length xs = length ys ==>
nipkow@13145
   906
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145
   907
by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114
   908
wenzelm@13114
   909
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145
   910
by (induct xs) (auto split: nat.split)
wenzelm@13114
   911
wenzelm@13114
   912
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
   913
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
   914
kleing@15868
   915
lemma set_update_memI: "!!n. n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
kleing@15868
   916
by (induct xs) (auto split:nat.splits)
kleing@15868
   917
wenzelm@13114
   918
nipkow@15392
   919
subsubsection {* @{text last} and @{text butlast} *}
wenzelm@13114
   920
wenzelm@13142
   921
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
   922
by (induct xs) auto
wenzelm@13114
   923
wenzelm@13142
   924
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
   925
by (induct xs) auto
wenzelm@13114
   926
nipkow@14302
   927
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
   928
by(simp add:last.simps)
nipkow@14302
   929
nipkow@14302
   930
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
   931
by(simp add:last.simps)
nipkow@14302
   932
nipkow@14302
   933
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
   934
by (induct xs) (auto)
nipkow@14302
   935
nipkow@14302
   936
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
   937
by(simp add:last_append)
nipkow@14302
   938
nipkow@14302
   939
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
   940
by(simp add:last_append)
nipkow@14302
   941
nipkow@14302
   942
wenzelm@13142
   943
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
   944
by (induct xs rule: rev_induct) auto
wenzelm@13114
   945
wenzelm@13114
   946
lemma butlast_append:
nipkow@13145
   947
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145
   948
by (induct xs) auto
wenzelm@13114
   949
wenzelm@13142
   950
lemma append_butlast_last_id [simp]:
nipkow@13145
   951
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
   952
by (induct xs) auto
wenzelm@13114
   953
wenzelm@13142
   954
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
   955
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
   956
wenzelm@13114
   957
lemma in_set_butlast_appendI:
nipkow@13145
   958
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
   959
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
   960
wenzelm@13142
   961
nipkow@15392
   962
subsubsection {* @{text take} and @{text drop} *}
wenzelm@13114
   963
wenzelm@13142
   964
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
   965
by (induct xs) auto
wenzelm@13114
   966
wenzelm@13142
   967
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
   968
by (induct xs) auto
wenzelm@13114
   969
wenzelm@13142
   970
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
   971
by simp
wenzelm@13114
   972
wenzelm@13142
   973
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
   974
by simp
wenzelm@13114
   975
wenzelm@13142
   976
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
   977
nipkow@15110
   978
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
   979
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
   980
nipkow@14187
   981
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
   982
by(cases xs, simp_all)
nipkow@14187
   983
nipkow@14187
   984
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
nipkow@14187
   985
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@14187
   986
nipkow@14187
   987
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
paulson@14208
   988
apply (induct xs, simp)
nipkow@14187
   989
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
   990
done
nipkow@14187
   991
nipkow@13913
   992
lemma take_Suc_conv_app_nth:
nipkow@13913
   993
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
paulson@14208
   994
apply (induct xs, simp)
paulson@14208
   995
apply (case_tac i, auto)
nipkow@13913
   996
done
nipkow@13913
   997
mehta@14591
   998
lemma drop_Suc_conv_tl:
mehta@14591
   999
  "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
mehta@14591
  1000
apply (induct xs, simp)
mehta@14591
  1001
apply (case_tac i, auto)
mehta@14591
  1002
done
mehta@14591
  1003
wenzelm@13142
  1004
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
nipkow@13145
  1005
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1006
wenzelm@13142
  1007
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
nipkow@13145
  1008
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1009
wenzelm@13142
  1010
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
nipkow@13145
  1011
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1012
wenzelm@13142
  1013
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
nipkow@13145
  1014
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1015
wenzelm@13142
  1016
lemma take_append [simp]:
nipkow@13145
  1017
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@13145
  1018
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1019
wenzelm@13142
  1020
lemma drop_append [simp]:
nipkow@13145
  1021
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@13145
  1022
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1023
wenzelm@13142
  1024
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
paulson@14208
  1025
apply (induct m, auto)
paulson@14208
  1026
apply (case_tac xs, auto)
nipkow@15236
  1027
apply (case_tac n, auto)
nipkow@13145
  1028
done
wenzelm@13114
  1029
wenzelm@13142
  1030
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
paulson@14208
  1031
apply (induct m, auto)
paulson@14208
  1032
apply (case_tac xs, auto)
nipkow@13145
  1033
done
wenzelm@13114
  1034
wenzelm@13114
  1035
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
paulson@14208
  1036
apply (induct m, auto)
paulson@14208
  1037
apply (case_tac xs, auto)
nipkow@13145
  1038
done
wenzelm@13114
  1039
nipkow@14802
  1040
lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@14802
  1041
apply(induct xs)
nipkow@14802
  1042
 apply simp
nipkow@14802
  1043
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  1044
done
nipkow@14802
  1045
wenzelm@13142
  1046
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
paulson@14208
  1047
apply (induct n, auto)
paulson@14208
  1048
apply (case_tac xs, auto)
nipkow@13145
  1049
done
wenzelm@13114
  1050
nipkow@15110
  1051
lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@15110
  1052
apply(induct xs)
nipkow@15110
  1053
 apply simp
nipkow@15110
  1054
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  1055
done
nipkow@15110
  1056
nipkow@15110
  1057
lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"
nipkow@15110
  1058
apply(induct xs)
nipkow@15110
  1059
apply simp
nipkow@15110
  1060
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1061
done
nipkow@15110
  1062
wenzelm@13114
  1063
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
paulson@14208
  1064
apply (induct n, auto)
paulson@14208
  1065
apply (case_tac xs, auto)
nipkow@13145
  1066
done
wenzelm@13114
  1067
wenzelm@13142
  1068
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
paulson@14208
  1069
apply (induct n, auto)
paulson@14208
  1070
apply (case_tac xs, auto)
nipkow@13145
  1071
done
wenzelm@13114
  1072
wenzelm@13114
  1073
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
paulson@14208
  1074
apply (induct xs, auto)
paulson@14208
  1075
apply (case_tac i, auto)
nipkow@13145
  1076
done
wenzelm@13114
  1077
wenzelm@13114
  1078
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
paulson@14208
  1079
apply (induct xs, auto)
paulson@14208
  1080
apply (case_tac i, auto)
nipkow@13145
  1081
done
wenzelm@13114
  1082
wenzelm@13142
  1083
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
paulson@14208
  1084
apply (induct xs, auto)
paulson@14208
  1085
apply (case_tac n, blast)
paulson@14208
  1086
apply (case_tac i, auto)
nipkow@13145
  1087
done
wenzelm@13114
  1088
wenzelm@13142
  1089
lemma nth_drop [simp]:
nipkow@13145
  1090
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
paulson@14208
  1091
apply (induct n, auto)
paulson@14208
  1092
apply (case_tac xs, auto)
nipkow@13145
  1093
done
nipkow@3507
  1094
nipkow@14025
  1095
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
nipkow@14025
  1096
by(induct xs)(auto simp:take_Cons split:nat.split)
nipkow@14025
  1097
nipkow@14025
  1098
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
nipkow@14025
  1099
by(induct xs)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1100
nipkow@14187
  1101
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1102
using set_take_subset by fast
nipkow@14187
  1103
nipkow@14187
  1104
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1105
using set_drop_subset by fast
nipkow@14187
  1106
wenzelm@13114
  1107
lemma append_eq_conv_conj:
nipkow@13145
  1108
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
paulson@14208
  1109
apply (induct xs, simp, clarsimp)
paulson@14208
  1110
apply (case_tac zs, auto)
nipkow@13145
  1111
done
wenzelm@13142
  1112
paulson@14050
  1113
lemma take_add [rule_format]: 
paulson@14050
  1114
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
paulson@14050
  1115
apply (induct xs, auto) 
paulson@14050
  1116
apply (case_tac i, simp_all) 
paulson@14050
  1117
done
paulson@14050
  1118
nipkow@14300
  1119
lemma append_eq_append_conv_if:
nipkow@14300
  1120
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1121
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1122
   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
  1123
   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
  1124
apply(induct xs\<^isub>1)
nipkow@14300
  1125
 apply simp
nipkow@14300
  1126
apply(case_tac ys\<^isub>1)
nipkow@14300
  1127
apply simp_all
nipkow@14300
  1128
done
nipkow@14300
  1129
nipkow@15110
  1130
lemma take_hd_drop:
nipkow@15110
  1131
  "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
nipkow@15110
  1132
apply(induct xs)
nipkow@15110
  1133
apply simp
nipkow@15110
  1134
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1135
done
nipkow@15110
  1136
wenzelm@13114
  1137
nipkow@15392
  1138
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1139
wenzelm@13142
  1140
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1141
by (induct xs) auto
wenzelm@13114
  1142
wenzelm@13142
  1143
lemma takeWhile_append1 [simp]:
nipkow@13145
  1144
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1145
by (induct xs) auto
wenzelm@13114
  1146
wenzelm@13142
  1147
lemma takeWhile_append2 [simp]:
nipkow@13145
  1148
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1149
by (induct xs) auto
wenzelm@13114
  1150
wenzelm@13142
  1151
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1152
by (induct xs) auto
wenzelm@13114
  1153
wenzelm@13142
  1154
lemma dropWhile_append1 [simp]:
nipkow@13145
  1155
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1156
by (induct xs) auto
wenzelm@13114
  1157
wenzelm@13142
  1158
lemma dropWhile_append2 [simp]:
nipkow@13145
  1159
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1160
by (induct xs) auto
wenzelm@13114
  1161
wenzelm@13142
  1162
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1163
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1164
nipkow@13913
  1165
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1166
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1167
by(induct xs, auto)
nipkow@13913
  1168
nipkow@13913
  1169
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1170
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1171
by(induct xs, auto)
nipkow@13913
  1172
nipkow@13913
  1173
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1174
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1175
by(induct xs, auto)
nipkow@13913
  1176
wenzelm@13114
  1177
nipkow@15392
  1178
subsubsection {* @{text zip} *}
wenzelm@13114
  1179
wenzelm@13142
  1180
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1181
by (induct ys) auto
wenzelm@13114
  1182
wenzelm@13142
  1183
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1184
by simp
wenzelm@13114
  1185
wenzelm@13142
  1186
declare zip_Cons [simp del]
wenzelm@13114
  1187
nipkow@15281
  1188
lemma zip_Cons1:
nipkow@15281
  1189
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  1190
by(auto split:list.split)
nipkow@15281
  1191
wenzelm@13142
  1192
lemma length_zip [simp]:
nipkow@13145
  1193
"!!xs. length (zip xs ys) = min (length xs) (length ys)"
paulson@14208
  1194
apply (induct ys, simp)
paulson@14208
  1195
apply (case_tac xs, auto)
nipkow@13145
  1196
done
wenzelm@13114
  1197
wenzelm@13114
  1198
lemma zip_append1:
nipkow@13145
  1199
"!!xs. zip (xs @ ys) zs =
nipkow@13145
  1200
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
paulson@14208
  1201
apply (induct zs, simp)
paulson@14208
  1202
apply (case_tac xs, simp_all)
nipkow@13145
  1203
done
wenzelm@13114
  1204
wenzelm@13114
  1205
lemma zip_append2:
nipkow@13145
  1206
"!!ys. zip xs (ys @ zs) =
nipkow@13145
  1207
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
paulson@14208
  1208
apply (induct xs, simp)
paulson@14208
  1209
apply (case_tac ys, simp_all)
nipkow@13145
  1210
done
wenzelm@13114
  1211
wenzelm@13142
  1212
lemma zip_append [simp]:
wenzelm@13142
  1213
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1214
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1215
by (simp add: zip_append1)
wenzelm@13114
  1216
wenzelm@13114
  1217
lemma zip_rev:
nipkow@14247
  1218
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1219
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1220
wenzelm@13142
  1221
lemma nth_zip [simp]:
nipkow@13145
  1222
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
paulson@14208
  1223
apply (induct ys, simp)
nipkow@13145
  1224
apply (case_tac xs)
nipkow@13145
  1225
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1226
done
wenzelm@13114
  1227
wenzelm@13114
  1228
lemma set_zip:
nipkow@13145
  1229
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
  1230
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1231
wenzelm@13114
  1232
lemma zip_update:
nipkow@13145
  1233
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
  1234
by (rule sym, simp add: update_zip)
wenzelm@13114
  1235
wenzelm@13142
  1236
lemma zip_replicate [simp]:
nipkow@13145
  1237
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
paulson@14208
  1238
apply (induct i, auto)
paulson@14208
  1239
apply (case_tac j, auto)
nipkow@13145
  1240
done
wenzelm@13114
  1241
wenzelm@13142
  1242
nipkow@15392
  1243
subsubsection {* @{text list_all2} *}
wenzelm@13114
  1244
kleing@14316
  1245
lemma list_all2_lengthD [intro?]: 
kleing@14316
  1246
  "list_all2 P xs ys ==> length xs = length ys"
nipkow@13145
  1247
by (simp add: list_all2_def)
wenzelm@13114
  1248
wenzelm@13142
  1249
lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])"
nipkow@13145
  1250
by (simp add: list_all2_def)
wenzelm@13114
  1251
wenzelm@13142
  1252
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"
nipkow@13145
  1253
by (simp add: list_all2_def)
wenzelm@13114
  1254
wenzelm@13142
  1255
lemma list_all2_Cons [iff]:
nipkow@13145
  1256
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@13145
  1257
by (auto simp add: list_all2_def)
wenzelm@13114
  1258
wenzelm@13114
  1259
lemma list_all2_Cons1:
nipkow@13145
  1260
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1261
by (cases ys) auto
wenzelm@13114
  1262
wenzelm@13114
  1263
lemma list_all2_Cons2:
nipkow@13145
  1264
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1265
by (cases xs) auto
wenzelm@13114
  1266
wenzelm@13142
  1267
lemma list_all2_rev [iff]:
nipkow@13145
  1268
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1269
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1270
kleing@13863
  1271
lemma list_all2_rev1:
kleing@13863
  1272
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1273
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1274
wenzelm@13114
  1275
lemma list_all2_append1:
nipkow@13145
  1276
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1277
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1278
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1279
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1280
apply (rule iffI)
nipkow@13145
  1281
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1282
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  1283
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1284
apply (simp add: ball_Un)
nipkow@13145
  1285
done
wenzelm@13114
  1286
wenzelm@13114
  1287
lemma list_all2_append2:
nipkow@13145
  1288
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1289
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1290
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1291
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1292
apply (rule iffI)
nipkow@13145
  1293
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1294
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  1295
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1296
apply (simp add: ball_Un)
nipkow@13145
  1297
done
wenzelm@13114
  1298
kleing@13863
  1299
lemma list_all2_append:
nipkow@14247
  1300
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  1301
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  1302
by (induct rule:list_induct2, simp_all)
kleing@13863
  1303
kleing@13863
  1304
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1305
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
kleing@13863
  1306
  by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1307
wenzelm@13114
  1308
lemma list_all2_conv_all_nth:
nipkow@13145
  1309
"list_all2 P xs ys =
nipkow@13145
  1310
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1311
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1312
berghofe@13883
  1313
lemma list_all2_trans:
berghofe@13883
  1314
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1315
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1316
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1317
proof (induct as)
berghofe@13883
  1318
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1319
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1320
  proof (induct bs)
berghofe@13883
  1321
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1322
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1323
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1324
  qed simp
berghofe@13883
  1325
qed simp
berghofe@13883
  1326
kleing@13863
  1327
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1328
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
kleing@13863
  1329
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1330
paulson@14395
  1331
lemma list_all2I:
paulson@14395
  1332
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
paulson@14395
  1333
  by (simp add: list_all2_def)
paulson@14395
  1334
kleing@14328
  1335
lemma list_all2_nthD:
kleing@13863
  1336
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
kleing@13863
  1337
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1338
nipkow@14302
  1339
lemma list_all2_nthD2:
nipkow@14302
  1340
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@14302
  1341
  by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  1342
kleing@13863
  1343
lemma list_all2_map1: 
kleing@13863
  1344
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
kleing@13863
  1345
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1346
kleing@13863
  1347
lemma list_all2_map2: 
kleing@13863
  1348
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
kleing@13863
  1349
  by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1350
kleing@14316
  1351
lemma list_all2_refl [intro?]:
kleing@13863
  1352
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
kleing@13863
  1353
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1354
kleing@13863
  1355
lemma list_all2_update_cong:
kleing@13863
  1356
  "\<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
  1357
  by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1358
kleing@13863
  1359
lemma list_all2_update_cong2:
kleing@13863
  1360
  "\<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
  1361
  by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1362
nipkow@14302
  1363
lemma list_all2_takeI [simp,intro?]:
nipkow@14302
  1364
  "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@14302
  1365
  apply (induct xs)
nipkow@14302
  1366
   apply simp
nipkow@14302
  1367
  apply (clarsimp simp add: list_all2_Cons1)
nipkow@14302
  1368
  apply (case_tac n)
nipkow@14302
  1369
  apply auto
nipkow@14302
  1370
  done
nipkow@14302
  1371
nipkow@14302
  1372
lemma list_all2_dropI [simp,intro?]:
kleing@13863
  1373
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
paulson@14208
  1374
  apply (induct as, simp)
kleing@13863
  1375
  apply (clarsimp simp add: list_all2_Cons1)
paulson@14208
  1376
  apply (case_tac n, simp, simp)
kleing@13863
  1377
  done
kleing@13863
  1378
kleing@14327
  1379
lemma list_all2_mono [intro?]:
kleing@13863
  1380
  "\<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
  1381
  apply (induct x, simp)
paulson@14208
  1382
  apply (case_tac y, auto)
kleing@13863
  1383
  done
kleing@13863
  1384
wenzelm@13142
  1385
nipkow@15392
  1386
subsubsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  1387
wenzelm@13142
  1388
lemma foldl_append [simp]:
nipkow@13145
  1389
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145
  1390
by (induct xs) auto
wenzelm@13142
  1391
nipkow@14402
  1392
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  1393
by (induct xs) auto
nipkow@14402
  1394
nipkow@14402
  1395
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  1396
by (induct xs) auto
nipkow@14402
  1397
nipkow@14402
  1398
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  1399
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  1400
wenzelm@13142
  1401
text {*
nipkow@13145
  1402
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1403
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1404
*}
wenzelm@13142
  1405
wenzelm@13142
  1406
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
nipkow@13145
  1407
by (induct ns) auto
wenzelm@13142
  1408
wenzelm@13142
  1409
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1410
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1411
wenzelm@13142
  1412
lemma sum_eq_0_conv [iff]:
nipkow@13145
  1413
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145
  1414
by (induct ns) auto
wenzelm@13114
  1415
wenzelm@13114
  1416
nipkow@15392
  1417
subsubsection {* @{text upto} *}
wenzelm@13114
  1418
nipkow@15425
  1419
lemma upt_rec: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
nipkow@13145
  1420
-- {* Does not terminate! *}
nipkow@13145
  1421
by (induct j) auto
wenzelm@13142
  1422
nipkow@15425
  1423
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
nipkow@13145
  1424
by (subst upt_rec) simp
wenzelm@13114
  1425
nipkow@15425
  1426
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
nipkow@15281
  1427
by(induct j)simp_all
nipkow@15281
  1428
nipkow@15281
  1429
lemma upt_eq_Cons_conv:
nipkow@15425
  1430
 "!!x xs. ([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
nipkow@15281
  1431
apply(induct j)
nipkow@15281
  1432
 apply simp
nipkow@15281
  1433
apply(clarsimp simp add: append_eq_Cons_conv)
nipkow@15281
  1434
apply arith
nipkow@15281
  1435
done
nipkow@15281
  1436
nipkow@15425
  1437
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
nipkow@13145
  1438
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1439
by simp
wenzelm@13114
  1440
nipkow@15425
  1441
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
nipkow@13145
  1442
apply(rule trans)
nipkow@13145
  1443
apply(subst upt_rec)
paulson@14208
  1444
 prefer 2 apply (rule refl, simp)
nipkow@13145
  1445
done
wenzelm@13114
  1446
nipkow@15425
  1447
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
nipkow@13145
  1448
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1449
by (induct k) auto
wenzelm@13114
  1450
nipkow@15425
  1451
lemma length_upt [simp]: "length [i..<j] = j - i"
nipkow@13145
  1452
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1453
nipkow@15425
  1454
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
nipkow@13145
  1455
apply (induct j)
nipkow@13145
  1456
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1457
done
wenzelm@13114
  1458
nipkow@15425
  1459
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..<n] = [i..<i+m]"
paulson@14208
  1460
apply (induct m, simp)
nipkow@13145
  1461
apply (subst upt_rec)
nipkow@13145
  1462
apply (rule sym)
nipkow@13145
  1463
apply (subst upt_rec)
nipkow@13145
  1464
apply (simp del: upt.simps)
nipkow@13145
  1465
done
nipkow@3507
  1466
nipkow@15425
  1467
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..n]"
nipkow@13145
  1468
by (induct n) auto
wenzelm@13114
  1469
nipkow@15425
  1470
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
nipkow@13145
  1471
apply (induct n m rule: diff_induct)
nipkow@13145
  1472
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1473
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1474
done
wenzelm@13114
  1475
berghofe@13883
  1476
lemma nth_take_lemma:
berghofe@13883
  1477
  "!!xs ys. k <= length xs ==> k <= length ys ==>
berghofe@13883
  1478
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
berghofe@13883
  1479
apply (atomize, induct k)
paulson@14208
  1480
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  1481
txt {* Both lists must be non-empty *}
paulson@14208
  1482
apply (case_tac xs, simp)
paulson@14208
  1483
apply (case_tac ys, clarify)
nipkow@13145
  1484
 apply (simp (no_asm_use))
nipkow@13145
  1485
apply clarify
nipkow@13145
  1486
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  1487
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  1488
apply blast
nipkow@13145
  1489
done
wenzelm@13114
  1490
wenzelm@13114
  1491
lemma nth_equalityI:
wenzelm@13114
  1492
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  1493
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  1494
apply (simp_all add: take_all)
nipkow@13145
  1495
done
wenzelm@13142
  1496
kleing@13863
  1497
(* needs nth_equalityI *)
kleing@13863
  1498
lemma list_all2_antisym:
kleing@13863
  1499
  "\<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
  1500
  \<Longrightarrow> xs = ys"
kleing@13863
  1501
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  1502
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  1503
  done
kleing@13863
  1504
wenzelm@13142
  1505
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  1506
-- {* The famous take-lemma. *}
nipkow@13145
  1507
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  1508
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  1509
done
wenzelm@13142
  1510
wenzelm@13142
  1511
nipkow@15302
  1512
lemma take_Cons':
nipkow@15302
  1513
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@15302
  1514
by (cases n) simp_all
nipkow@15302
  1515
nipkow@15302
  1516
lemma drop_Cons':
nipkow@15302
  1517
     "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@15302
  1518
by (cases n) simp_all
nipkow@15302
  1519
nipkow@15302
  1520
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@15302
  1521
by (cases n) simp_all
nipkow@15302
  1522
nipkow@15302
  1523
lemmas [simp] = take_Cons'[of "number_of v",standard]
nipkow@15302
  1524
                drop_Cons'[of "number_of v",standard]
nipkow@15302
  1525
                nth_Cons'[of _ _ "number_of v",standard]
nipkow@15302
  1526
nipkow@15302
  1527
nipkow@15392
  1528
subsubsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  1529
wenzelm@13142
  1530
lemma distinct_append [simp]:
nipkow@13145
  1531
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  1532
by (induct xs) auto
wenzelm@13142
  1533
nipkow@15305
  1534
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
nipkow@15305
  1535
by(induct xs) auto
nipkow@15305
  1536
wenzelm@13142
  1537
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  1538
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  1539
wenzelm@13142
  1540
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  1541
by (induct xs) auto
wenzelm@13142
  1542
paulson@15072
  1543
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
paulson@15251
  1544
  by (induct x, auto) 
paulson@15072
  1545
paulson@15072
  1546
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
paulson@15251
  1547
  by (induct x, auto)
paulson@15072
  1548
nipkow@15245
  1549
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245
  1550
by (induct xs) auto
nipkow@15245
  1551
nipkow@15245
  1552
lemma length_remdups_eq[iff]:
nipkow@15245
  1553
  "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245
  1554
apply(induct xs)
nipkow@15245
  1555
 apply auto
nipkow@15245
  1556
apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245
  1557
 apply arith
nipkow@15245
  1558
apply(rule length_remdups_leq)
nipkow@15245
  1559
done
nipkow@15245
  1560
wenzelm@13142
  1561
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  1562
by (induct xs) auto
wenzelm@13114
  1563
nipkow@15304
  1564
lemma distinct_map_filterI:
nipkow@15304
  1565
 "distinct(map f xs) \<Longrightarrow> distinct(map f (filter P xs))"
nipkow@15304
  1566
apply(induct xs)
nipkow@15304
  1567
 apply simp
nipkow@15304
  1568
apply force
nipkow@15304
  1569
done
nipkow@15304
  1570
wenzelm@13142
  1571
text {*
nipkow@13145
  1572
It is best to avoid this indexed version of distinct, but sometimes
nipkow@13145
  1573
it is useful. *}
wenzelm@13142
  1574
lemma distinct_conv_nth:
nipkow@13145
  1575
"distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@15251
  1576
apply (induct xs, simp, simp)
paulson@14208
  1577
apply (rule iffI, clarsimp)
nipkow@13145
  1578
 apply (case_tac i)
paulson@14208
  1579
apply (case_tac j, simp)
nipkow@13145
  1580
apply (simp add: set_conv_nth)
nipkow@13145
  1581
 apply (case_tac j)
paulson@14208
  1582
apply (clarsimp simp add: set_conv_nth, simp)
nipkow@13145
  1583
apply (rule conjI)
nipkow@13145
  1584
 apply (clarsimp simp add: set_conv_nth)
nipkow@13145
  1585
 apply (erule_tac x = 0 in allE)
paulson@14208
  1586
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
nipkow@13145
  1587
apply (erule_tac x = "Suc i" in allE)
paulson@14208
  1588
apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145
  1589
done
wenzelm@13114
  1590
nipkow@15110
  1591
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
kleing@14388
  1592
  by (induct xs) auto
kleing@14388
  1593
nipkow@15110
  1594
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388
  1595
proof (induct xs)
kleing@14388
  1596
  case Nil thus ?case by simp
kleing@14388
  1597
next
kleing@14388
  1598
  case (Cons x xs)
kleing@14388
  1599
  show ?case
kleing@14388
  1600
  proof (cases "x \<in> set xs")
kleing@14388
  1601
    case False with Cons show ?thesis by simp
kleing@14388
  1602
  next
kleing@14388
  1603
    case True with Cons.prems
kleing@14388
  1604
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  1605
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  1606
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  1607
    ultimately have False by simp
kleing@14388
  1608
    thus ?thesis ..
kleing@14388
  1609
  qed
kleing@14388
  1610
qed
kleing@14388
  1611
nipkow@15110
  1612
lemma inj_on_setI: "distinct(map f xs) ==> inj_on f (set xs)"
nipkow@15110
  1613
apply(induct xs)
nipkow@15110
  1614
 apply simp
nipkow@15110
  1615
apply fastsimp
nipkow@15110
  1616
done
nipkow@15110
  1617
nipkow@15110
  1618
lemma inj_on_set_conv:
nipkow@15110
  1619
 "distinct xs \<Longrightarrow> inj_on f (set xs) = distinct(map f xs)"
nipkow@15110
  1620
apply(induct xs)
nipkow@15110
  1621
 apply simp
nipkow@15110
  1622
apply fastsimp
nipkow@15110
  1623
done
nipkow@15110
  1624
nipkow@15110
  1625
nipkow@15392
  1626
subsubsection {* @{text remove1} *}
nipkow@15110
  1627
nipkow@15110
  1628
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110
  1629
apply(induct xs)
nipkow@15110
  1630
 apply simp
nipkow@15110
  1631
apply simp
nipkow@15110
  1632
apply blast
nipkow@15110
  1633
done
nipkow@15110
  1634
nipkow@15110
  1635
lemma [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110
  1636
apply(induct xs)
nipkow@15110
  1637
 apply simp
nipkow@15110
  1638
apply simp
nipkow@15110
  1639
apply blast
nipkow@15110
  1640
done
nipkow@15110
  1641
nipkow@15110
  1642
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110
  1643
apply(insert set_remove1_subset)
nipkow@15110
  1644
apply fast
nipkow@15110
  1645
done
nipkow@15110
  1646
nipkow@15110
  1647
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110
  1648
by (induct xs) simp_all
nipkow@15110
  1649
wenzelm@13114
  1650
nipkow@15392
  1651
subsubsection {* @{text replicate} *}
wenzelm@13114
  1652
wenzelm@13142
  1653
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  1654
by (induct n) auto
nipkow@13124
  1655
wenzelm@13142
  1656
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  1657
by (induct n) auto
wenzelm@13114
  1658
wenzelm@13114
  1659
lemma replicate_app_Cons_same:
nipkow@13145
  1660
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  1661
by (induct n) auto
wenzelm@13114
  1662
wenzelm@13142
  1663
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208
  1664
apply (induct n, simp)
nipkow@13145
  1665
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  1666
done
wenzelm@13114
  1667
wenzelm@13142
  1668
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  1669
by (induct n) auto
wenzelm@13114
  1670
nipkow@16397
  1671
text{* Courtesy of Matthias Daum: *}
nipkow@16397
  1672
lemma append_replicate_commute:
nipkow@16397
  1673
  "replicate n x @ replicate k x = replicate k x @ replicate n x"
nipkow@16397
  1674
apply (simp add: replicate_add [THEN sym])
nipkow@16397
  1675
apply (simp add: add_commute)
nipkow@16397
  1676
done
nipkow@16397
  1677
wenzelm@13142
  1678
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  1679
by (induct n) auto
wenzelm@13114
  1680
wenzelm@13142
  1681
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  1682
by (induct n) auto
wenzelm@13114
  1683
wenzelm@13142
  1684
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  1685
by (atomize (full), induct n) auto
wenzelm@13114
  1686
wenzelm@13142
  1687
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
paulson@14208
  1688
apply (induct n, simp)
nipkow@13145
  1689
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  1690
done
wenzelm@13114
  1691
nipkow@16397
  1692
text{* Courtesy of Matthias Daum (2 lemmas): *}
nipkow@16397
  1693
lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
nipkow@16397
  1694
apply (case_tac "k \<le> i")
nipkow@16397
  1695
 apply  (simp add: min_def)
nipkow@16397
  1696
apply (drule not_leE)
nipkow@16397
  1697
apply (simp add: min_def)
nipkow@16397
  1698
apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
nipkow@16397
  1699
 apply  simp
nipkow@16397
  1700
apply (simp add: replicate_add [symmetric])
nipkow@16397
  1701
done
nipkow@16397
  1702
nipkow@16397
  1703
lemma drop_replicate[simp]: "!!i. drop i (replicate k x) = replicate (k-i) x"
nipkow@16397
  1704
apply (induct k)
nipkow@16397
  1705
 apply simp
nipkow@16397
  1706
apply clarsimp
nipkow@16397
  1707
apply (case_tac i)
nipkow@16397
  1708
 apply simp
nipkow@16397
  1709
apply clarsimp
nipkow@16397
  1710
done
nipkow@16397
  1711
nipkow@16397
  1712
wenzelm@13142
  1713
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  1714
by (induct n) auto
wenzelm@13114
  1715
wenzelm@13142
  1716
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  1717
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  1718
wenzelm@13142
  1719
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  1720
by auto
wenzelm@13114
  1721
wenzelm@13142
  1722
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  1723
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  1724
wenzelm@13114
  1725
nipkow@15392
  1726
subsubsection{*@{text rotate1} and @{text rotate}*}
nipkow@15302
  1727
nipkow@15302
  1728
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
nipkow@15302
  1729
by(simp add:rotate1_def)
nipkow@15302
  1730
nipkow@15302
  1731
lemma rotate0[simp]: "rotate 0 = id"
nipkow@15302
  1732
by(simp add:rotate_def)
nipkow@15302
  1733
nipkow@15302
  1734
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
nipkow@15302
  1735
by(simp add:rotate_def)
nipkow@15302
  1736
nipkow@15302
  1737
lemma rotate_add:
nipkow@15302
  1738
  "rotate (m+n) = rotate m o rotate n"
nipkow@15302
  1739
by(simp add:rotate_def funpow_add)
nipkow@15302
  1740
nipkow@15302
  1741
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
nipkow@15302
  1742
by(simp add:rotate_add)
nipkow@15302
  1743
nipkow@15302
  1744
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
nipkow@15302
  1745
by(cases xs) simp_all
nipkow@15302
  1746
nipkow@15302
  1747
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  1748
apply(induct n)
nipkow@15302
  1749
 apply simp
nipkow@15302
  1750
apply (simp add:rotate_def)
nipkow@13145
  1751
done
wenzelm@13114
  1752
nipkow@15302
  1753
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
nipkow@15302
  1754
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1755
nipkow@15302
  1756
lemma rotate_drop_take:
nipkow@15302
  1757
  "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
nipkow@15302
  1758
apply(induct n)
nipkow@15302
  1759
 apply simp
nipkow@15302
  1760
apply(simp add:rotate_def)
nipkow@15302
  1761
apply(cases "xs = []")
nipkow@15302
  1762
 apply (simp)
nipkow@15302
  1763
apply(case_tac "n mod length xs = 0")
nipkow@15302
  1764
 apply(simp add:mod_Suc)
nipkow@15302
  1765
 apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
nipkow@15302
  1766
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
nipkow@15302
  1767
                take_hd_drop linorder_not_le)
nipkow@13145
  1768
done
wenzelm@13114
  1769
nipkow@15302
  1770
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
nipkow@15302
  1771
by(simp add:rotate_drop_take)
nipkow@15302
  1772
nipkow@15302
  1773
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  1774
by(simp add:rotate_drop_take)
nipkow@15302
  1775
nipkow@15302
  1776
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
nipkow@15302
  1777
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1778
nipkow@15302
  1779
lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs"
nipkow@15302
  1780
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  1781
nipkow@15302
  1782
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
nipkow@15302
  1783
by(simp add:rotate1_def split:list.split) blast
nipkow@15302
  1784
nipkow@15302
  1785
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
nipkow@15302
  1786
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  1787
nipkow@15302
  1788
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
nipkow@15302
  1789
by(simp add:rotate_drop_take take_map drop_map)
nipkow@15302
  1790
nipkow@15302
  1791
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
nipkow@15302
  1792
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1793
nipkow@15302
  1794
lemma set_rotate[simp]: "set(rotate n xs) = set xs"
nipkow@15302
  1795
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  1796
nipkow@15302
  1797
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
nipkow@15302
  1798
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1799
nipkow@15302
  1800
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
nipkow@15302
  1801
by (induct n) (simp_all add:rotate_def)
wenzelm@13114
  1802
nipkow@15439
  1803
lemma rotate_rev:
nipkow@15439
  1804
  "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
nipkow@15439
  1805
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  1806
apply(cases "length xs = 0")
nipkow@15439
  1807
 apply simp
nipkow@15439
  1808
apply(cases "n mod length xs = 0")
nipkow@15439
  1809
 apply simp
nipkow@15439
  1810
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  1811
done
nipkow@15439
  1812
wenzelm@13114
  1813
nipkow@15392
  1814
subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  1815
wenzelm@13142
  1816
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  1817
by (auto simp add: sublist_def)
wenzelm@13114
  1818
wenzelm@13142
  1819
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  1820
by (auto simp add: sublist_def)
wenzelm@13114
  1821
nipkow@15281
  1822
lemma length_sublist:
nipkow@15281
  1823
  "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
nipkow@15281
  1824
by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
nipkow@15281
  1825
nipkow@15281
  1826
lemma sublist_shift_lemma_Suc:
nipkow@15281
  1827
  "!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
nipkow@15281
  1828
         map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
nipkow@15281
  1829
apply(induct xs)
nipkow@15281
  1830
 apply simp
nipkow@15281
  1831
apply (case_tac "is")
nipkow@15281
  1832
 apply simp
nipkow@15281
  1833
apply simp
nipkow@15281
  1834
done
nipkow@15281
  1835
wenzelm@13114
  1836
lemma sublist_shift_lemma:
nipkow@15425
  1837
     "map fst [p:zip xs [i..<i + length xs] . snd p : A] =
nipkow@15425
  1838
      map fst [p:zip xs [0..<length xs] . snd p + i : A]"
nipkow@13145
  1839
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  1840
wenzelm@13114
  1841
lemma sublist_append:
paulson@15168
  1842
     "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  1843
apply (unfold sublist_def)
paulson@14208
  1844
apply (induct l' rule: rev_induct, simp)
nipkow@13145
  1845
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  1846
apply (simp add: add_commute)
nipkow@13145
  1847
done
wenzelm@13114
  1848
wenzelm@13114
  1849
lemma sublist_Cons:
nipkow@13145
  1850
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  1851
apply (induct l rule: rev_induct)
nipkow@13145
  1852
 apply (simp add: sublist_def)
nipkow@13145
  1853
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  1854
done
wenzelm@13114
  1855
nipkow@15281
  1856
lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
nipkow@15281
  1857
apply(induct xs)
nipkow@15281
  1858
 apply simp
nipkow@15281
  1859
apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE)
nipkow@15281
  1860
 apply(erule lessE)
nipkow@15281
  1861
  apply auto
nipkow@15281
  1862
apply(erule lessE)
nipkow@15281
  1863
apply auto
nipkow@15281
  1864
done
nipkow@15281
  1865
nipkow@15281
  1866
lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
nipkow@15281
  1867
by(auto simp add:set_sublist)
nipkow@15281
  1868
nipkow@15281
  1869
lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
nipkow@15281
  1870
by(auto simp add:set_sublist)
nipkow@15281
  1871
nipkow@15281
  1872
lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
nipkow@15281
  1873
by(auto simp add:set_sublist)
nipkow@15281
  1874
wenzelm@13142
  1875
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  1876
by (simp add: sublist_Cons)
wenzelm@13114
  1877
nipkow@15281
  1878
nipkow@15281
  1879
lemma distinct_sublistI[simp]: "!!I. distinct xs \<Longrightarrow> distinct(sublist xs I)"
nipkow@15281
  1880
apply(induct xs)
nipkow@15281
  1881
 apply simp
nipkow@15281
  1882
apply(auto simp add:sublist_Cons)
nipkow@15281
  1883
done
nipkow@15281
  1884
nipkow@15281
  1885
nipkow@15045
  1886
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
paulson@14208
  1887
apply (induct l rule: rev_induct, simp)
nipkow@13145
  1888
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  1889
done
wenzelm@13114
  1890
wenzelm@13114
  1891
nipkow@15392
  1892
subsubsection{*Sets of Lists*}
nipkow@15392
  1893
nipkow@15392
  1894
subsubsection {* @{text lists}: the list-forming operator over sets *}
nipkow@15302
  1895
nipkow@15302
  1896
consts lists :: "'a set => 'a list set"
nipkow@15302
  1897
inductive "lists A"
nipkow@15302
  1898
 intros
nipkow@15302
  1899
  Nil [intro!]: "[]: lists A"
nipkow@15302
  1900
  Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A"
nipkow@15302
  1901
nipkow@15302
  1902
inductive_cases listsE [elim!]: "x#l : lists A"
nipkow@15302
  1903
nipkow@15302
  1904
lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B"
nipkow@15302
  1905
by (unfold lists.defs) (blast intro!: lfp_mono)
nipkow@15302
  1906
nipkow@15302
  1907
lemma lists_IntI:
nipkow@15302
  1908
  assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l
nipkow@15302
  1909
  by induct blast+
nipkow@15302
  1910
nipkow@15302
  1911
lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B"
nipkow@15302
  1912
proof (rule mono_Int [THEN equalityI])
nipkow@15302
  1913
  show "mono lists" by (simp add: mono_def lists_mono)
nipkow@15302
  1914
  show "lists A \<inter> lists B \<subseteq> lists (A \<inter> B)" by (blast intro: lists_IntI)
kleing@14388
  1915
qed
kleing@14388
  1916
nipkow@15302
  1917
lemma append_in_lists_conv [iff]:
nipkow@15302
  1918
     "(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"
nipkow@15302
  1919
by (induct xs) auto
nipkow@15302
  1920
nipkow@15302
  1921
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"
nipkow@15302
  1922
-- {* eliminate @{text lists} in favour of @{text set} *}
nipkow@15302
  1923
by (induct xs) auto
nipkow@15302
  1924
nipkow@15302
  1925
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"
nipkow@15302
  1926
by (rule in_lists_conv_set [THEN iffD1])
nipkow@15302
  1927
nipkow@15302
  1928
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"
nipkow@15302
  1929
by (rule in_lists_conv_set [THEN iffD2])
nipkow@15302
  1930
nipkow@15302
  1931
lemma lists_UNIV [simp]: "lists UNIV = UNIV"
nipkow@15302
  1932
by auto
nipkow@15302
  1933
nipkow@15392
  1934
subsubsection{*Lists as Cartesian products*}
nipkow@15302
  1935
nipkow@15302
  1936
text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
nipkow@15302
  1937
@{term A} and tail drawn from @{term Xs}.*}
nipkow@15302
  1938
nipkow@15302
  1939
constdefs
nipkow@15302
  1940
  set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
nipkow@15302
  1941
  "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"
nipkow@15302
  1942
nipkow@15302
  1943
lemma [simp]: "set_Cons A {[]} = (%x. [x])`A"
nipkow@15302
  1944
by (auto simp add: set_Cons_def)
nipkow@15302
  1945
nipkow@15302
  1946
text{*Yields the set of lists, all of the same length as the argument and
nipkow@15302
  1947
with elements drawn from the corresponding element of the argument.*}
nipkow@15302
  1948
nipkow@15302
  1949
consts  listset :: "'a set list \<Rightarrow> 'a list set"
nipkow@15302
  1950
primrec
nipkow@15302
  1951
   "listset []    = {[]}"
nipkow@15302
  1952
   "listset(A#As) = set_Cons A (listset As)"
nipkow@15302
  1953
nipkow@15302
  1954
paulson@15656
  1955
subsection{*Relations on Lists*}
paulson@15656
  1956
paulson@15656
  1957
subsubsection {* Length Lexicographic Ordering *}
paulson@15656
  1958
paulson@15656
  1959
text{*These orderings preserve well-foundedness: shorter lists 
paulson@15656
  1960
  precede longer lists. These ordering are not used in dictionaries.*}
paulson@15656
  1961
paulson@15656
  1962
consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
paulson@15656
  1963
        --{*The lexicographic ordering for lists of the specified length*}
nipkow@15302
  1964
primrec
paulson@15656
  1965
  "lexn r 0 = {}"
paulson@15656
  1966
  "lexn r (Suc n) =
paulson@15656
  1967
    (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
paulson@15656
  1968
    {(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
nipkow@15302
  1969
nipkow@15302
  1970
constdefs
paulson@15656
  1971
  lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
paulson@15656
  1972
    "lex r == \<Union>n. lexn r n"
paulson@15656
  1973
        --{*Holds only between lists of the same length*}
paulson@15656
  1974
nipkow@15693
  1975
  lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@15693
  1976
    "lenlex r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
paulson@15656
  1977
        --{*Compares lists by their length and then lexicographically*}
nipkow@15302
  1978
nipkow@15302
  1979
nipkow@15302
  1980
lemma wf_lexn: "wf r ==> wf (lexn r n)"
nipkow@15302
  1981
apply (induct n, simp, simp)
nipkow@15302
  1982
apply(rule wf_subset)
nipkow@15302
  1983
 prefer 2 apply (rule Int_lower1)
nipkow@15302
  1984
apply(rule wf_prod_fun_image)
nipkow@15302
  1985
 prefer 2 apply (rule inj_onI, auto)
nipkow@15302
  1986
done
nipkow@15302
  1987
nipkow@15302
  1988
lemma lexn_length:
nipkow@15302
  1989
     "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@15302
  1990
by (induct n) auto
nipkow@15302
  1991
nipkow@15302
  1992
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@15302
  1993
apply (unfold lex_def)
nipkow@15302
  1994
apply (rule wf_UN)
nipkow@15302
  1995
apply (blast intro: wf_lexn, clarify)
nipkow@15302
  1996
apply (rename_tac m n)
nipkow@15302
  1997
apply (subgoal_tac "m \<noteq> n")
nipkow@15302
  1998
 prefer 2 apply blast
nipkow@15302
  1999
apply (blast dest: lexn_length not_sym)
nipkow@15302
  2000
done
nipkow@15302
  2001
nipkow@15302
  2002
lemma lexn_conv:
paulson@15656
  2003
  "lexn r n =
paulson@15656
  2004
    {(xs,ys). length xs = n \<and> length ys = n \<and>
paulson@15656
  2005
    (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
nipkow@15302
  2006
apply (induct n, simp, blast)
nipkow@15302
  2007
apply (simp add: image_Collect lex_prod_def, safe, blast)
nipkow@15302
  2008
 apply (rule_tac x = "ab # xys" in exI, simp)
nipkow@15302
  2009
apply (case_tac xys, simp_all, blast)
nipkow@15302
  2010
done
nipkow@15302
  2011
nipkow@15302
  2012
lemma lex_conv:
paulson@15656
  2013
  "lex r =
paulson@15656
  2014
    {(xs,ys). length xs = length ys \<and>
paulson@15656
  2015
    (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@15302
  2016
by (force simp add: lex_def lexn_conv)
nipkow@15302
  2017
nipkow@15693
  2018
lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
nipkow@15693
  2019
by (unfold lenlex_def) blast
nipkow@15693
  2020
nipkow@15693
  2021
lemma lenlex_conv:
nipkow@15693
  2022
    "lenlex r = {(xs,ys). length xs < length ys |
paulson@15656
  2023
                 length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@15693
  2024
by (simp add: lenlex_def diag_def lex_prod_def measure_def inv_image_def)
nipkow@15302
  2025
nipkow@15302
  2026
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@15302
  2027
by (simp add: lex_conv)
nipkow@15302
  2028
nipkow@15302
  2029
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@15302
  2030
by (simp add:lex_conv)
nipkow@15302
  2031
nipkow@15302
  2032
lemma Cons_in_lex [iff]:
paulson@15656
  2033
    "((x # xs, y # ys) : lex r) =
paulson@15656
  2034
      ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@15302
  2035
apply (simp add: lex_conv)
nipkow@15302
  2036
apply (rule iffI)
nipkow@15302
  2037
 prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
nipkow@15302
  2038
apply (case_tac xys, simp, simp)
nipkow@15302
  2039
apply blast
nipkow@15302
  2040
done
nipkow@15302
  2041
nipkow@15302
  2042
paulson@15656
  2043
subsubsection {* Lexicographic Ordering *}
paulson@15656
  2044
paulson@15656
  2045
text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
paulson@15656
  2046
    This ordering does \emph{not} preserve well-foundedness.
paulson@15656
  2047
     Author: N. Voelker, March 2005 *} 
paulson@15656
  2048
paulson@15656
  2049
constdefs 
paulson@15656
  2050
  lexord :: "('a * 'a)set \<Rightarrow> ('a list * 'a list) set" 
paulson@15656
  2051
  "lexord  r == {(x,y). \<exists> a v. y = x @ a # v \<or> 
paulson@15656
  2052
            (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
paulson@15656
  2053
paulson@15656
  2054
lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
paulson@15656
  2055
  by (unfold lexord_def, induct_tac y, auto) 
paulson@15656
  2056
paulson@15656
  2057
lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
paulson@15656
  2058
  by (unfold lexord_def, induct_tac x, auto)
paulson@15656
  2059
paulson@15656
  2060
lemma lexord_cons_cons[simp]:
paulson@15656
  2061
     "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
paulson@15656
  2062
  apply (unfold lexord_def, safe, simp_all)
paulson@15656
  2063
  apply (case_tac u, simp, simp)
paulson@15656
  2064
  apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
paulson@15656
  2065
  apply (erule_tac x="b # u" in allE)
paulson@15656
  2066
  by force
paulson@15656
  2067
paulson@15656
  2068
lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
paulson@15656
  2069
paulson@15656
  2070
lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
paulson@15656
  2071
  by (induct_tac x, auto)  
paulson@15656
  2072
paulson@15656
  2073
lemma lexord_append_left_rightI:
paulson@15656
  2074
     "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
paulson@15656
  2075
  by (induct_tac u, auto)
paulson@15656
  2076
paulson@15656
  2077
lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
paulson@15656
  2078
  by (induct x, auto)
paulson@15656
  2079
paulson@15656
  2080
lemma lexord_append_leftD:
paulson@15656
  2081
     "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
paulson@15656
  2082
  by (erule rev_mp, induct_tac x, auto)
paulson@15656
  2083
paulson@15656
  2084
lemma lexord_take_index_conv: 
paulson@15656
  2085
   "((x,y) : lexord r) = 
paulson@15656
  2086
    ((length x < length y \<and> take (length x) y = x) \<or> 
paulson@15656
  2087
     (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
paulson@15656
  2088
  apply (unfold lexord_def Let_def, clarsimp) 
paulson@15656
  2089
  apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
paulson@15656
  2090
  apply auto 
paulson@15656
  2091
  apply (rule_tac x="hd (drop (length x) y)" in exI)
paulson@15656
  2092
  apply (rule_tac x="tl (drop (length x) y)" in exI)
paulson@15656
  2093
  apply (erule subst, simp add: min_def) 
paulson@15656
  2094
  apply (rule_tac x ="length u" in exI, simp) 
paulson@15656
  2095
  apply (rule_tac x ="take i x" in exI) 
paulson@15656
  2096
  apply (rule_tac x ="x ! i" in exI) 
paulson@15656
  2097
  apply (rule_tac x ="y ! i" in exI, safe) 
paulson@15656
  2098
  apply (rule_tac x="drop (Suc i) x" in exI)
paulson@15656
  2099
  apply (drule sym, simp add: drop_Suc_conv_tl) 
paulson@15656
  2100
  apply (rule_tac x="drop (Suc i) y" in exI)
paulson@15656
  2101
  by (simp add: drop_Suc_conv_tl) 
paulson@15656
  2102
paulson@15656
  2103
-- {* lexord is extension of partial ordering List.lex *} 
paulson@15656
  2104
lemma lexord_lex: " (x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
paulson@15656
  2105
  apply (rule_tac x = y in spec) 
paulson@15656
  2106
  apply (induct_tac x, clarsimp) 
paulson@15656
  2107
  by (clarify, case_tac x, simp, force)
paulson@15656
  2108
paulson@15656
  2109
lemma lexord_irreflexive: "(! x. (x,x) \<notin> r) \<Longrightarrow> (y,y) \<notin> lexord r"
paulson@15656
  2110
  by (induct y, auto)
paulson@15656
  2111
paulson@15656
  2112
lemma lexord_trans: 
paulson@15656
  2113
    "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
paulson@15656
  2114
   apply (erule rev_mp)+
paulson@15656
  2115
   apply (rule_tac x = x in spec) 
paulson@15656
  2116
  apply (rule_tac x = z in spec) 
paulson@15656
  2117
  apply ( induct_tac y, simp, clarify)
paulson@15656
  2118
  apply (case_tac xa, erule ssubst) 
paulson@15656
  2119
  apply (erule allE, erule allE) -- {* avoid simp recursion *} 
paulson@15656
  2120
  apply (case_tac x, simp, simp) 
paulson@15656
  2121
  apply (case_tac x, erule allE, erule allE, simp) 
paulson@15656
  2122
  apply (erule_tac x = listb in allE) 
paulson@15656
  2123
  apply (erule_tac x = lista in allE, simp)
paulson@15656
  2124
  apply (unfold trans_def)
paulson@15656
  2125
  by blast
paulson@15656
  2126
paulson@15656
  2127
lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
paulson@15656
  2128
  by (rule transI, drule lexord_trans, blast) 
paulson@15656
  2129
paulson@15656
  2130
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
  2131
  apply (rule_tac x = y in spec) 
paulson@15656
  2132
  apply (induct_tac x, rule allI) 
paulson@15656
  2133
  apply (case_tac x, simp, simp) 
paulson@15656
  2134
  apply (rule allI, case_tac x, simp, simp) 
paulson@15656
  2135
  by blast
paulson@15656
  2136
paulson@15656
  2137
nipkow@15392
  2138
subsubsection{*Lifting a Relation on List Elements to the Lists*}
nipkow@15302
  2139
nipkow@15302
  2140
consts  listrel :: "('a * 'a)set => ('a list * 'a list)set"
nipkow@15302
  2141
nipkow@15302
  2142
inductive "listrel(r)"
nipkow@15302
  2143
 intros
nipkow@15302
  2144
   Nil:  "([],[]) \<in> listrel r"
nipkow@15302
  2145
   Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
nipkow@15302
  2146
nipkow@15302
  2147
inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
nipkow@15302
  2148
inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
nipkow@15302
  2149
inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
nipkow@15302
  2150
inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
nipkow@15302
  2151
nipkow@15302
  2152
nipkow@15302
  2153
lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
nipkow@15302
  2154
apply clarify  
nipkow@15302
  2155
apply (erule listrel.induct)
nipkow@15302
  2156
apply (blast intro: listrel.intros)+
nipkow@15302
  2157
done
nipkow@15302
  2158
nipkow@15302
  2159
lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
nipkow@15302
  2160
apply clarify 
nipkow@15302
  2161
apply (erule listrel.induct, auto) 
nipkow@15302
  2162
done
nipkow@15302
  2163
nipkow@15302
  2164
lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" 
nipkow@15302
  2165
apply (simp add: refl_def listrel_subset Ball_def)
nipkow@15302
  2166
apply (rule allI) 
nipkow@15302
  2167
apply (induct_tac x) 
nipkow@15302
  2168
apply (auto intro: listrel.intros)
nipkow@15302
  2169
done
nipkow@15302
  2170
nipkow@15302
  2171
lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
nipkow@15302
  2172
apply (auto simp add: sym_def)
nipkow@15302
  2173
apply (erule listrel.induct) 
nipkow@15302
  2174
apply (blast intro: listrel.intros)+
nipkow@15302
  2175
done
nipkow@15302
  2176
nipkow@15302
  2177
lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
nipkow@15302
  2178
apply (simp add: trans_def)
nipkow@15302
  2179
apply (intro allI) 
nipkow@15302
  2180
apply (rule impI) 
nipkow@15302
  2181
apply (erule listrel.induct) 
nipkow@15302
  2182
apply (blast intro: listrel.intros)+
nipkow@15302
  2183
done
nipkow@15302
  2184
nipkow@15302
  2185
theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
nipkow@15302
  2186
by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) 
nipkow@15302
  2187
nipkow@15302
  2188
lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
nipkow@15302
  2189
by (blast intro: listrel.intros)
nipkow@15302
  2190
nipkow@15302
  2191
lemma listrel_Cons:
nipkow@15302
  2192
     "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})";
nipkow@15302
  2193
by (auto simp add: set_Cons_def intro: listrel.intros) 
nipkow@15302
  2194
nipkow@15302
  2195
nipkow@15392
  2196
subsection{*Miscellany*}
nipkow@15392
  2197
nipkow@15392
  2198
subsubsection {* Characters and strings *}
wenzelm@13366
  2199
wenzelm@13366
  2200
datatype nibble =
wenzelm@13366
  2201
    Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
wenzelm@13366
  2202
  | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
wenzelm@13366
  2203
wenzelm@13366
  2204
datatype char = Char nibble nibble
wenzelm@13366
  2205
  -- "Note: canonical order of character encoding coincides with standard term ordering"
wenzelm@13366
  2206
wenzelm@13366
  2207
types string = "char list"
wenzelm@13366
  2208
wenzelm@13366
  2209
syntax
wenzelm@13366
  2210
  "_Char" :: "xstr => char"    ("CHR _")
wenzelm@13366
  2211
  "_String" :: "xstr => string"    ("_")
wenzelm@13366
  2212
wenzelm@13366
  2213
parse_ast_translation {*
wenzelm@13366
  2214
  let
wenzelm@13366
  2215
    val constants = Syntax.Appl o map Syntax.Constant;
wenzelm@13366
  2216
wenzelm@13366
  2217
    fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10));
wenzelm@13366
  2218
    fun mk_char c =
wenzelm@13366
  2219
      if Symbol.is_ascii c andalso Symbol.is_printable c then
wenzelm@13366
  2220
        constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)]
wenzelm@13366
  2221
      else error ("Printable ASCII character expected: " ^ quote c);
wenzelm@13366
  2222
wenzelm@13366
  2223
    fun mk_string [] = Syntax.Constant "Nil"
wenzelm@13366
  2224
      | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs];
wenzelm@13366
  2225
wenzelm@13366
  2226
    fun char_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  2227
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  2228
          [c] => mk_char c
wenzelm@13366
  2229
        | _ => error ("Single character expected: " ^ xstr))
wenzelm@13366
  2230
      | char_ast_tr asts = raise AST ("char_ast_tr", asts);
wenzelm@13366
  2231
wenzelm@13366
  2232
    fun string_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  2233
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  2234
          [] => constants [Syntax.constrainC, "Nil", "string"]
wenzelm@13366
  2235
        | cs => mk_string cs)
wenzelm@13366
  2236
      | string_ast_tr asts = raise AST ("string_tr", asts);
wenzelm@13366
  2237
  in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end;
wenzelm@13366
  2238
*}
wenzelm@13366
  2239
berghofe@15064
  2240
ML {*
berghofe@15064
  2241
fun int_of_nibble h =
berghofe@15064
  2242
  if "0" <= h andalso h <= "9" then ord h - ord "0"
berghofe@15064
  2243
  else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10
berghofe@15064
  2244
  else raise Match;
berghofe@15064
  2245
berghofe@15064
  2246
fun nibble_of_int i =
berghofe@15064
  2247
  if i <= 9 then chr (ord "0" + i) else chr (ord "A" + i - 10);
berghofe@15064
  2248
*}
berghofe@15064
  2249
wenzelm@13366
  2250
print_ast_translation {*
wenzelm@13366
  2251
  let
wenzelm@13366
  2252
    fun dest_nib (Syntax.Constant c) =
wenzelm@13366
  2253
        (case explode c of
berghofe@15064
  2254
          ["N", "i", "b", "b", "l", "e", h] => int_of_nibble h
wenzelm@13366
  2255
        | _ => raise Match)
wenzelm@13366
  2256
      | dest_nib _ = raise Match;
wenzelm@13366
  2257
wenzelm@13366
  2258
    fun dest_chr c1 c2 =
wenzelm@13366
  2259
      let val c = chr (dest_nib c1 * 16 + dest_nib c2)
wenzelm@13366
  2260
      in if Symbol.is_printable c then c else raise Match end;
wenzelm@13366
  2261
wenzelm@13366
  2262
    fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2
wenzelm@13366
  2263
      | dest_char _ = raise Match;
wenzelm@13366
  2264
wenzelm@13366
  2265
    fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)];
wenzelm@13366
  2266
wenzelm@13366
  2267
    fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]]
wenzelm@13366
  2268
      | char_ast_tr' _ = raise Match;
wenzelm@13366
  2269
wenzelm@13366
  2270
    fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String",
wenzelm@13366
  2271
            xstr (map dest_char (Syntax.unfold_ast "_args" args))]
wenzelm@13366
  2272
      | list_ast_tr' ts = raise Match;
wenzelm@13366
  2273
  in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end;
wenzelm@13366
  2274
*}
wenzelm@13366
  2275
nipkow@15392
  2276
subsubsection {* Code generator setup *}
berghofe@15064
  2277
berghofe@15064
  2278
ML {*
berghofe@15064
  2279
local
berghofe@15064
  2280
berghofe@16634
  2281
fun list_codegen thy defs gr dep thyname b t =
berghofe@16634
  2282
  let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy defs dep thyname false)
berghofe@15064
  2283
    (gr, HOLogic.dest_list t)
skalberg@15531
  2284
  in SOME (gr', Pretty.list "[" "]" ps) end handle TERM _ => NONE;
berghofe@15064
  2285
berghofe@15064
  2286
fun dest_nibble (Const (s, _)) = int_of_nibble (unprefix "List.nibble.Nibble" s)
berghofe@15064
  2287
  | dest_nibble _ = raise Match;
berghofe@15064
  2288
berghofe@16634
  2289
fun char_codegen thy defs gr dep thyname b (Const ("List.char.Char", _) $ c1 $ c2) =
berghofe@15064
  2290
    (let val c = chr (dest_nibble c1 * 16 + dest_nibble c2)
skalberg@15531
  2291
     in if Symbol.is_printable c then SOME (gr, Pretty.quote (Pretty.str c))
skalberg@15531
  2292
       else NONE
skalberg@15570
  2293
     end handle Fail _ => NONE | Match => NONE)
berghofe@16634
  2294
  | char_codegen thy defs gr dep thyname b _ = NONE;
berghofe@15064
  2295
berghofe@15064
  2296
in
berghofe@15064
  2297
berghofe@15064
  2298
val list_codegen_setup =
berghofe@15064
  2299
  [Codegen.add_codegen "list_codegen" list_codegen,
berghofe@15064
  2300
   Codegen.add_codegen "char_codegen" char_codegen];
berghofe@15064
  2301
berghofe@15064
  2302
end;
berghofe@16770
  2303
*}
berghofe@16770
  2304
berghofe@16770
  2305
types_code
berghofe@16770
  2306
  "list" ("_ list")
berghofe@16770
  2307
attach (term_of) {*
berghofe@15064
  2308
val term_of_list = HOLogic.mk_list;
berghofe@16770
  2309
*}
berghofe@16770
  2310
attach (test) {*
berghofe@15064
  2311
fun gen_list' aG i j = frequency
berghofe@15064
  2312
  [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] ()
berghofe@15064
  2313
and gen_list aG i = gen_list' aG i i;
berghofe@16770
  2314
*}
berghofe@16770
  2315
  "char" ("string")
berghofe@16770
  2316
attach (term_of) {*
berghofe@15064
  2317
val nibbleT = Type ("List.nibble", []);
berghofe@15064
  2318
berghofe@15064
  2319
fun term_of_char c =
berghofe@15064
  2320
  Const ("List.char.Char", nibbleT --> nibbleT --> Type ("List.char", [])) $
berghofe@15064
  2321
    Const ("List.nibble.Nibble" ^ nibble_of_int (ord c div 16), nibbleT) $
berghofe@15064
  2322
    Const ("List.nibble.Nibble" ^ nibble_of_int (ord c mod 16), nibbleT);
berghofe@16770
  2323
*}
berghofe@16770
  2324
attach (test) {*
berghofe@15064
  2325
fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z")));
berghofe@15064
  2326
*}
berghofe@15064
  2327
berghofe@15064
  2328
consts_code "Cons" ("(_ ::/ _)")
berghofe@15064
  2329
berghofe@15064
  2330
setup list_codegen_setup
berghofe@15064
  2331
wenzelm@13122
  2332
end