src/HOL/List.thy
author nipkow
Mon Nov 22 11:53:56 2004 +0100 (2004-11-22)
changeset 15305 0bd9eedaa301
parent 15304 3514ca74ac54
child 15307 10dd989282fd
permissions -rw-r--r--
added lemmas
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(*  Title:      HOL/List.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow
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*)
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header {* The datatype of finite lists *}
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theory List
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imports PreList
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begin
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datatype 'a list =
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    Nil    ("[]")
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  | Cons 'a  "'a list"    (infixr "#" 65)
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section{*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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  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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"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 *}
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-- {* but separate 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 *}
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-- {* but separate 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 *}
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-- {* but separate 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] =
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(case i of 0 => v # xs
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| 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 *}
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-- {* but separate 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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subsection {* @{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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subsection {* @{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 = [])"
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using append_same_eq [of _ _ "[]"] by auto
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lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
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using append_same_eq [of "[]"] by auto
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lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
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by (induct xs) auto
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lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
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by (induct xs) auto
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lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
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by (simp add: hd_append split: list.split)
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lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
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by (simp split: list.split)
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lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
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by (simp add: tl_append split: list.split)
wenzelm@13114
   334
wenzelm@13114
   335
nipkow@14300
   336
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   337
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   338
by(cases ys) auto
nipkow@14300
   339
nipkow@15281
   340
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   341
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   342
by(cases ys) auto
nipkow@15281
   343
nipkow@14300
   344
wenzelm@13142
   345
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   346
wenzelm@13114
   347
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   348
by simp
wenzelm@13114
   349
wenzelm@13142
   350
lemma Cons_eq_appendI:
nipkow@13145
   351
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   352
by (drule sym) simp
wenzelm@13114
   353
wenzelm@13142
   354
lemma append_eq_appendI:
nipkow@13145
   355
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   356
by (drule sym) simp
wenzelm@13114
   357
wenzelm@13114
   358
wenzelm@13142
   359
text {*
nipkow@13145
   360
Simplification procedure for all list equalities.
nipkow@13145
   361
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   362
- both lists end in a singleton list,
nipkow@13145
   363
- or both lists end in the same list.
wenzelm@13142
   364
*}
wenzelm@13142
   365
wenzelm@13142
   366
ML_setup {*
nipkow@3507
   367
local
nipkow@3507
   368
wenzelm@13122
   369
val append_assoc = thm "append_assoc";
wenzelm@13122
   370
val append_Nil = thm "append_Nil";
wenzelm@13122
   371
val append_Cons = thm "append_Cons";
wenzelm@13122
   372
val append1_eq_conv = thm "append1_eq_conv";
wenzelm@13122
   373
val append_same_eq = thm "append_same_eq";
wenzelm@13122
   374
wenzelm@13114
   375
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462
   376
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
wenzelm@13462
   377
  | last (Const("List.op @",_) $ _ $ ys) = last ys
wenzelm@13462
   378
  | last t = t;
wenzelm@13114
   379
wenzelm@13114
   380
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462
   381
  | list1 _ = false;
wenzelm@13114
   382
wenzelm@13114
   383
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462
   384
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
wenzelm@13462
   385
  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462
   386
  | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114
   387
wenzelm@13114
   388
val rearr_tac =
wenzelm@13462
   389
  simp_tac (HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]);
wenzelm@13114
   390
wenzelm@13114
   391
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   392
  let
wenzelm@13462
   393
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   394
    fun rearr conv =
wenzelm@13462
   395
      let
wenzelm@13462
   396
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   397
        val Type(_,listT::_) = eqT
wenzelm@13462
   398
        val appT = [listT,listT] ---> listT
wenzelm@13462
   399
        val app = Const("List.op @",appT)
wenzelm@13462
   400
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   401
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@13480
   402
        val thm = Tactic.prove sg [] [] eq (K (rearr_tac 1));
wenzelm@13462
   403
      in Some ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   404
wenzelm@13462
   405
  in
wenzelm@13462
   406
    if list1 lastl andalso list1 lastr then rearr append1_eq_conv
wenzelm@13462
   407
    else if lastl aconv lastr then rearr append_same_eq
wenzelm@13462
   408
    else None
wenzelm@13462
   409
  end;
wenzelm@13462
   410
wenzelm@13114
   411
in
wenzelm@13462
   412
wenzelm@13462
   413
val list_eq_simproc =
wenzelm@13462
   414
  Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq;
wenzelm@13462
   415
wenzelm@13114
   416
end;
wenzelm@13114
   417
wenzelm@13114
   418
Addsimprocs [list_eq_simproc];
wenzelm@13114
   419
*}
wenzelm@13114
   420
wenzelm@13114
   421
wenzelm@13142
   422
subsection {* @{text map} *}
wenzelm@13114
   423
wenzelm@13142
   424
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   425
by (induct xs) simp_all
wenzelm@13114
   426
wenzelm@13142
   427
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   428
by (rule ext, induct_tac xs) auto
wenzelm@13114
   429
wenzelm@13142
   430
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   431
by (induct xs) auto
wenzelm@13114
   432
wenzelm@13142
   433
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   434
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   435
wenzelm@13142
   436
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   437
by (induct xs) auto
wenzelm@13114
   438
nipkow@13737
   439
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   440
by (induct xs) auto
nipkow@13737
   441
wenzelm@13366
   442
lemma map_cong [recdef_cong]:
nipkow@13145
   443
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   444
-- {* a congruence rule for @{text map} *}
nipkow@13737
   445
by simp
wenzelm@13114
   446
wenzelm@13142
   447
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   448
by (cases xs) auto
wenzelm@13114
   449
wenzelm@13142
   450
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   451
by (cases xs) auto
wenzelm@13114
   452
nipkow@14025
   453
lemma map_eq_Cons_conv[iff]:
nipkow@14025
   454
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   455
by (cases xs) auto
wenzelm@13114
   456
nipkow@14025
   457
lemma Cons_eq_map_conv[iff]:
nipkow@14025
   458
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   459
by (cases ys) auto
nipkow@14025
   460
nipkow@14111
   461
lemma ex_map_conv:
nipkow@14111
   462
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
nipkow@14111
   463
by(induct ys, auto)
nipkow@14111
   464
nipkow@15110
   465
lemma map_eq_imp_length_eq:
nipkow@15110
   466
  "!!xs. map f xs = map f ys ==> length xs = length ys"
nipkow@15110
   467
apply (induct ys)
nipkow@15110
   468
 apply simp
nipkow@15110
   469
apply(simp (no_asm_use))
nipkow@15110
   470
apply clarify
nipkow@15110
   471
apply(simp (no_asm_use))
nipkow@15110
   472
apply fast
nipkow@15110
   473
done
nipkow@15110
   474
nipkow@15110
   475
lemma map_inj_on:
nipkow@15110
   476
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   477
  ==> xs = ys"
nipkow@15110
   478
apply(frule map_eq_imp_length_eq)
nipkow@15110
   479
apply(rotate_tac -1)
nipkow@15110
   480
apply(induct rule:list_induct2)
nipkow@15110
   481
 apply simp
nipkow@15110
   482
apply(simp)
nipkow@15110
   483
apply (blast intro:sym)
nipkow@15110
   484
done
nipkow@15110
   485
nipkow@15110
   486
lemma inj_on_map_eq_map:
nipkow@15110
   487
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   488
by(blast dest:map_inj_on)
nipkow@15110
   489
wenzelm@13114
   490
lemma map_injective:
nipkow@14338
   491
 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@14338
   492
by (induct ys) (auto dest!:injD)
wenzelm@13114
   493
nipkow@14339
   494
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   495
by(blast dest:map_injective)
nipkow@14339
   496
wenzelm@13114
   497
lemma inj_mapI: "inj f ==> inj (map f)"
paulson@13585
   498
by (rules dest: map_injective injD intro: inj_onI)
wenzelm@13114
   499
wenzelm@13114
   500
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   501
apply (unfold inj_on_def, clarify)
nipkow@13145
   502
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   503
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   504
apply blast
nipkow@13145
   505
done
wenzelm@13114
   506
nipkow@14339
   507
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   508
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   509
nipkow@15303
   510
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
   511
apply(rule inj_onI)
nipkow@15303
   512
apply(erule map_inj_on)
nipkow@15303
   513
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
   514
done
nipkow@15303
   515
kleing@14343
   516
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   517
by (induct xs, auto)
wenzelm@13114
   518
nipkow@14402
   519
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   520
by (induct xs) auto
nipkow@14402
   521
nipkow@15110
   522
lemma map_fst_zip[simp]:
nipkow@15110
   523
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   524
by (induct rule:list_induct2, simp_all)
nipkow@15110
   525
nipkow@15110
   526
lemma map_snd_zip[simp]:
nipkow@15110
   527
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   528
by (induct rule:list_induct2, simp_all)
nipkow@15110
   529
nipkow@15110
   530
wenzelm@13142
   531
subsection {* @{text rev} *}
wenzelm@13114
   532
wenzelm@13142
   533
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   534
by (induct xs) auto
wenzelm@13114
   535
wenzelm@13142
   536
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   537
by (induct xs) auto
wenzelm@13114
   538
wenzelm@13142
   539
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   540
by (induct xs) auto
wenzelm@13114
   541
wenzelm@13142
   542
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   543
by (induct xs) auto
wenzelm@13114
   544
wenzelm@13142
   545
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"
paulson@14208
   546
apply (induct xs, force)
paulson@14208
   547
apply (case_tac ys, simp, force)
nipkow@13145
   548
done
wenzelm@13114
   549
wenzelm@13366
   550
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   551
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
nipkow@13145
   552
apply(subst rev_rev_ident[symmetric])
nipkow@13145
   553
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   554
done
wenzelm@13114
   555
nipkow@13145
   556
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114
   557
wenzelm@13366
   558
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   559
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   560
by (induct xs rule: rev_induct) auto
wenzelm@13114
   561
wenzelm@13366
   562
lemmas rev_cases = rev_exhaust
wenzelm@13366
   563
wenzelm@13114
   564
wenzelm@13142
   565
subsection {* @{text set} *}
wenzelm@13114
   566
wenzelm@13142
   567
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   568
by (induct xs) auto
wenzelm@13114
   569
wenzelm@13142
   570
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   571
by (induct xs) auto
wenzelm@13114
   572
oheimb@14099
   573
lemma hd_in_set: "l = x#xs \<Longrightarrow> x\<in>set l"
paulson@14208
   574
by (case_tac l, auto)
oheimb@14099
   575
wenzelm@13142
   576
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   577
by auto
wenzelm@13114
   578
oheimb@14099
   579
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   580
by auto
oheimb@14099
   581
wenzelm@13142
   582
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   583
by (induct xs) auto
wenzelm@13114
   584
nipkow@15245
   585
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   586
by(induct xs) auto
nipkow@15245
   587
wenzelm@13142
   588
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   589
by (induct xs) auto
wenzelm@13114
   590
wenzelm@13142
   591
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   592
by (induct xs) auto
wenzelm@13114
   593
wenzelm@13142
   594
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   595
by (induct xs) auto
wenzelm@13114
   596
wenzelm@13142
   597
lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> k < j}"
paulson@14208
   598
apply (induct j, simp_all)
paulson@14208
   599
apply (erule ssubst, auto)
nipkow@13145
   600
done
wenzelm@13114
   601
wenzelm@13142
   602
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
paulson@15113
   603
proof (induct xs)
paulson@15113
   604
  case Nil show ?case by simp
paulson@15113
   605
  case (Cons a xs)
paulson@15113
   606
  show ?case
paulson@15113
   607
  proof 
paulson@15113
   608
    assume "x \<in> set (a # xs)"
paulson@15113
   609
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   610
      by (simp, blast intro: Cons_eq_appendI)
paulson@15113
   611
  next
paulson@15113
   612
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   613
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
paulson@15113
   614
    show "x \<in> set (a # xs)" 
paulson@15113
   615
      by (cases ys, auto simp add: eq)
paulson@15113
   616
  qed
paulson@15113
   617
qed
wenzelm@13142
   618
paulson@13508
   619
lemma finite_list: "finite A ==> EX l. set l = A"
paulson@13508
   620
apply (erule finite_induct, auto)
paulson@13508
   621
apply (rule_tac x="x#l" in exI, auto)
paulson@13508
   622
done
paulson@13508
   623
kleing@14388
   624
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
   625
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
   626
paulson@15168
   627
wenzelm@13142
   628
subsection {* @{text mem} *}
wenzelm@13114
   629
nipkow@15302
   630
text{* Only use @{text mem} for generating executable code.  Otherwise
nipkow@15302
   631
use @{prop"x : set xs"} instead --- it is much easier to reason
nipkow@15302
   632
about. *}
nipkow@15302
   633
wenzelm@13114
   634
lemma set_mem_eq: "(x mem xs) = (x : set xs)"
nipkow@13145
   635
by (induct xs) auto
wenzelm@13114
   636
wenzelm@13114
   637
wenzelm@13142
   638
subsection {* @{text list_all} *}
wenzelm@13114
   639
wenzelm@13142
   640
lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)"
nipkow@13145
   641
by (induct xs) auto
wenzelm@13114
   642
wenzelm@13142
   643
lemma list_all_append [simp]:
nipkow@13145
   644
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
nipkow@13145
   645
by (induct xs) auto
wenzelm@13114
   646
wenzelm@13114
   647
wenzelm@13142
   648
subsection {* @{text filter} *}
wenzelm@13114
   649
wenzelm@13142
   650
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   651
by (induct xs) auto
wenzelm@13114
   652
nipkow@15305
   653
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
   654
by (induct xs) simp_all
nipkow@15305
   655
wenzelm@13142
   656
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   657
by (induct xs) auto
wenzelm@13114
   658
wenzelm@13142
   659
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   660
by (induct xs) auto
wenzelm@13114
   661
wenzelm@13142
   662
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   663
by (induct xs) auto
wenzelm@13114
   664
nipkow@15246
   665
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@13145
   666
by (induct xs) (auto simp add: le_SucI)
wenzelm@13114
   667
wenzelm@13142
   668
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   669
by auto
wenzelm@13114
   670
nipkow@15246
   671
lemma length_filter_less:
nipkow@15246
   672
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
   673
proof (induct xs)
nipkow@15246
   674
  case Nil thus ?case by simp
nipkow@15246
   675
next
nipkow@15246
   676
  case (Cons x xs) thus ?case
nipkow@15246
   677
    apply (auto split:split_if_asm)
nipkow@15246
   678
    using length_filter_le[of P xs] apply arith
nipkow@15246
   679
  done
nipkow@15246
   680
qed
wenzelm@13114
   681
nipkow@15281
   682
lemma length_filter_conv_card:
nipkow@15281
   683
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
   684
proof (induct xs)
nipkow@15281
   685
  case Nil thus ?case by simp
nipkow@15281
   686
next
nipkow@15281
   687
  case (Cons x xs)
nipkow@15281
   688
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
   689
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
   690
  show ?case (is "?l = card ?S'")
nipkow@15281
   691
  proof (cases)
nipkow@15281
   692
    assume "p x"
nipkow@15281
   693
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@15281
   694
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   695
    have "length (filter p (x # xs)) = Suc(card ?S)"
nipkow@15281
   696
      using Cons by simp
nipkow@15281
   697
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
nipkow@15281
   698
      by (simp add: card_image inj_Suc)
nipkow@15281
   699
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   700
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
   701
    finally show ?thesis .
nipkow@15281
   702
  next
nipkow@15281
   703
    assume "\<not> p x"
nipkow@15281
   704
    hence eq: "?S' = Suc ` ?S"
nipkow@15281
   705
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   706
    have "length (filter p (x # xs)) = card ?S"
nipkow@15281
   707
      using Cons by simp
nipkow@15281
   708
    also have "\<dots> = card(Suc ` ?S)" using fin
nipkow@15281
   709
      by (simp add: card_image inj_Suc)
nipkow@15281
   710
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   711
      by (simp add:card_insert_if)
nipkow@15281
   712
    finally show ?thesis .
nipkow@15281
   713
  qed
nipkow@15281
   714
qed
nipkow@15281
   715
nipkow@15281
   716
wenzelm@13142
   717
subsection {* @{text concat} *}
wenzelm@13114
   718
wenzelm@13142
   719
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
   720
by (induct xs) auto
wenzelm@13114
   721
wenzelm@13142
   722
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   723
by (induct xss) auto
wenzelm@13114
   724
wenzelm@13142
   725
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   726
by (induct xss) auto
wenzelm@13114
   727
wenzelm@13142
   728
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
nipkow@13145
   729
by (induct xs) auto
wenzelm@13114
   730
wenzelm@13142
   731
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
   732
by (induct xs) auto
wenzelm@13114
   733
wenzelm@13142
   734
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
   735
by (induct xs) auto
wenzelm@13114
   736
wenzelm@13142
   737
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
   738
by (induct xs) auto
wenzelm@13114
   739
wenzelm@13114
   740
wenzelm@13142
   741
subsection {* @{text nth} *}
wenzelm@13114
   742
wenzelm@13142
   743
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
   744
by auto
wenzelm@13114
   745
wenzelm@13142
   746
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
   747
by auto
wenzelm@13114
   748
wenzelm@13142
   749
declare nth.simps [simp del]
wenzelm@13114
   750
wenzelm@13114
   751
lemma nth_append:
nipkow@13145
   752
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
paulson@14208
   753
apply (induct "xs", simp)
paulson@14208
   754
apply (case_tac n, auto)
nipkow@13145
   755
done
wenzelm@13114
   756
nipkow@14402
   757
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
nipkow@14402
   758
by (induct "xs") auto
nipkow@14402
   759
nipkow@14402
   760
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
nipkow@14402
   761
by (induct "xs") auto
nipkow@14402
   762
wenzelm@13142
   763
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
paulson@14208
   764
apply (induct xs, simp)
paulson@14208
   765
apply (case_tac n, auto)
nipkow@13145
   766
done
wenzelm@13114
   767
wenzelm@13142
   768
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
   769
apply (induct xs, simp, simp)
nipkow@13145
   770
apply safe
paulson@14208
   771
apply (rule_tac x = 0 in exI, simp)
paulson@14208
   772
 apply (rule_tac x = "Suc i" in exI, simp)
paulson@14208
   773
apply (case_tac i, simp)
nipkow@13145
   774
apply (rename_tac j)
paulson@14208
   775
apply (rule_tac x = j in exI, simp)
nipkow@13145
   776
done
wenzelm@13114
   777
nipkow@13145
   778
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
   779
by (auto simp add: set_conv_nth)
wenzelm@13114
   780
wenzelm@13142
   781
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
   782
by (auto simp add: set_conv_nth)
wenzelm@13114
   783
wenzelm@13114
   784
lemma all_nth_imp_all_set:
nipkow@13145
   785
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
   786
by (auto simp add: set_conv_nth)
wenzelm@13114
   787
wenzelm@13114
   788
lemma all_set_conv_all_nth:
nipkow@13145
   789
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
   790
by (auto simp add: set_conv_nth)
wenzelm@13114
   791
wenzelm@13114
   792
wenzelm@13142
   793
subsection {* @{text list_update} *}
wenzelm@13114
   794
wenzelm@13142
   795
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
nipkow@13145
   796
by (induct xs) (auto split: nat.split)
wenzelm@13114
   797
wenzelm@13114
   798
lemma nth_list_update:
nipkow@13145
   799
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@13145
   800
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   801
wenzelm@13142
   802
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
   803
by (simp add: nth_list_update)
wenzelm@13114
   804
wenzelm@13142
   805
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145
   806
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   807
wenzelm@13142
   808
lemma list_update_overwrite [simp]:
nipkow@13145
   809
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145
   810
by (induct xs) (auto split: nat.split)
wenzelm@13114
   811
nipkow@14402
   812
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
paulson@14208
   813
apply (induct xs, simp)
nipkow@14187
   814
apply(simp split:nat.splits)
nipkow@14187
   815
done
nipkow@14187
   816
wenzelm@13114
   817
lemma list_update_same_conv:
nipkow@13145
   818
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145
   819
by (induct xs) (auto split: nat.split)
wenzelm@13114
   820
nipkow@14187
   821
lemma list_update_append1:
nipkow@14187
   822
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
paulson@14208
   823
apply (induct xs, simp)
nipkow@14187
   824
apply(simp split:nat.split)
nipkow@14187
   825
done
nipkow@14187
   826
nipkow@14402
   827
lemma list_update_length [simp]:
nipkow@14402
   828
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
   829
by (induct xs, auto)
nipkow@14402
   830
wenzelm@13114
   831
lemma update_zip:
nipkow@13145
   832
"!!i xy xs. length xs = length ys ==>
nipkow@13145
   833
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145
   834
by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114
   835
wenzelm@13114
   836
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145
   837
by (induct xs) (auto split: nat.split)
wenzelm@13114
   838
wenzelm@13114
   839
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
   840
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
   841
wenzelm@13114
   842
wenzelm@13142
   843
subsection {* @{text last} and @{text butlast} *}
wenzelm@13114
   844
wenzelm@13142
   845
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
   846
by (induct xs) auto
wenzelm@13114
   847
wenzelm@13142
   848
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
   849
by (induct xs) auto
wenzelm@13114
   850
nipkow@14302
   851
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
   852
by(simp add:last.simps)
nipkow@14302
   853
nipkow@14302
   854
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
   855
by(simp add:last.simps)
nipkow@14302
   856
nipkow@14302
   857
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
   858
by (induct xs) (auto)
nipkow@14302
   859
nipkow@14302
   860
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
   861
by(simp add:last_append)
nipkow@14302
   862
nipkow@14302
   863
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
   864
by(simp add:last_append)
nipkow@14302
   865
nipkow@14302
   866
wenzelm@13142
   867
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
   868
by (induct xs rule: rev_induct) auto
wenzelm@13114
   869
wenzelm@13114
   870
lemma butlast_append:
nipkow@13145
   871
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145
   872
by (induct xs) auto
wenzelm@13114
   873
wenzelm@13142
   874
lemma append_butlast_last_id [simp]:
nipkow@13145
   875
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
   876
by (induct xs) auto
wenzelm@13114
   877
wenzelm@13142
   878
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
   879
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
   880
wenzelm@13114
   881
lemma in_set_butlast_appendI:
nipkow@13145
   882
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
   883
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
   884
wenzelm@13142
   885
wenzelm@13142
   886
subsection {* @{text take} and @{text drop} *}
wenzelm@13114
   887
wenzelm@13142
   888
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
   889
by (induct xs) auto
wenzelm@13114
   890
wenzelm@13142
   891
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
   892
by (induct xs) auto
wenzelm@13114
   893
wenzelm@13142
   894
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
   895
by simp
wenzelm@13114
   896
wenzelm@13142
   897
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
   898
by simp
wenzelm@13114
   899
wenzelm@13142
   900
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
   901
nipkow@15110
   902
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
   903
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
   904
nipkow@14187
   905
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
   906
by(cases xs, simp_all)
nipkow@14187
   907
nipkow@14187
   908
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
nipkow@14187
   909
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@14187
   910
nipkow@14187
   911
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
paulson@14208
   912
apply (induct xs, simp)
nipkow@14187
   913
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
   914
done
nipkow@14187
   915
nipkow@13913
   916
lemma take_Suc_conv_app_nth:
nipkow@13913
   917
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
paulson@14208
   918
apply (induct xs, simp)
paulson@14208
   919
apply (case_tac i, auto)
nipkow@13913
   920
done
nipkow@13913
   921
mehta@14591
   922
lemma drop_Suc_conv_tl:
mehta@14591
   923
  "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
mehta@14591
   924
apply (induct xs, simp)
mehta@14591
   925
apply (case_tac i, auto)
mehta@14591
   926
done
mehta@14591
   927
wenzelm@13142
   928
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
nipkow@13145
   929
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   930
wenzelm@13142
   931
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
nipkow@13145
   932
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   933
wenzelm@13142
   934
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
nipkow@13145
   935
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   936
wenzelm@13142
   937
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
nipkow@13145
   938
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   939
wenzelm@13142
   940
lemma take_append [simp]:
nipkow@13145
   941
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@13145
   942
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   943
wenzelm@13142
   944
lemma drop_append [simp]:
nipkow@13145
   945
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@13145
   946
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   947
wenzelm@13142
   948
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
paulson@14208
   949
apply (induct m, auto)
paulson@14208
   950
apply (case_tac xs, auto)
nipkow@15236
   951
apply (case_tac n, auto)
nipkow@13145
   952
done
wenzelm@13114
   953
wenzelm@13142
   954
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
paulson@14208
   955
apply (induct m, auto)
paulson@14208
   956
apply (case_tac xs, auto)
nipkow@13145
   957
done
wenzelm@13114
   958
wenzelm@13114
   959
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
paulson@14208
   960
apply (induct m, auto)
paulson@14208
   961
apply (case_tac xs, auto)
nipkow@13145
   962
done
wenzelm@13114
   963
nipkow@14802
   964
lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@14802
   965
apply(induct xs)
nipkow@14802
   966
 apply simp
nipkow@14802
   967
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
   968
done
nipkow@14802
   969
wenzelm@13142
   970
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
paulson@14208
   971
apply (induct n, auto)
paulson@14208
   972
apply (case_tac xs, auto)
nipkow@13145
   973
done
wenzelm@13114
   974
nipkow@15110
   975
lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@15110
   976
apply(induct xs)
nipkow@15110
   977
 apply simp
nipkow@15110
   978
apply(simp add:take_Cons split:nat.split)
nipkow@15110
   979
done
nipkow@15110
   980
nipkow@15110
   981
lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"
nipkow@15110
   982
apply(induct xs)
nipkow@15110
   983
apply simp
nipkow@15110
   984
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
   985
done
nipkow@15110
   986
wenzelm@13114
   987
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
paulson@14208
   988
apply (induct n, auto)
paulson@14208
   989
apply (case_tac xs, auto)
nipkow@13145
   990
done
wenzelm@13114
   991
wenzelm@13142
   992
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
paulson@14208
   993
apply (induct n, auto)
paulson@14208
   994
apply (case_tac xs, auto)
nipkow@13145
   995
done
wenzelm@13114
   996
wenzelm@13114
   997
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
paulson@14208
   998
apply (induct xs, auto)
paulson@14208
   999
apply (case_tac i, auto)
nipkow@13145
  1000
done
wenzelm@13114
  1001
wenzelm@13114
  1002
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
paulson@14208
  1003
apply (induct xs, auto)
paulson@14208
  1004
apply (case_tac i, auto)
nipkow@13145
  1005
done
wenzelm@13114
  1006
wenzelm@13142
  1007
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
paulson@14208
  1008
apply (induct xs, auto)
paulson@14208
  1009
apply (case_tac n, blast)
paulson@14208
  1010
apply (case_tac i, auto)
nipkow@13145
  1011
done
wenzelm@13114
  1012
wenzelm@13142
  1013
lemma nth_drop [simp]:
nipkow@13145
  1014
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
paulson@14208
  1015
apply (induct n, auto)
paulson@14208
  1016
apply (case_tac xs, auto)
nipkow@13145
  1017
done
nipkow@3507
  1018
nipkow@14025
  1019
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
nipkow@14025
  1020
by(induct xs)(auto simp:take_Cons split:nat.split)
nipkow@14025
  1021
nipkow@14025
  1022
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
nipkow@14025
  1023
by(induct xs)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1024
nipkow@14187
  1025
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1026
using set_take_subset by fast
nipkow@14187
  1027
nipkow@14187
  1028
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1029
using set_drop_subset by fast
nipkow@14187
  1030
wenzelm@13114
  1031
lemma append_eq_conv_conj:
nipkow@13145
  1032
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
paulson@14208
  1033
apply (induct xs, simp, clarsimp)
paulson@14208
  1034
apply (case_tac zs, auto)
nipkow@13145
  1035
done
wenzelm@13142
  1036
paulson@14050
  1037
lemma take_add [rule_format]: 
paulson@14050
  1038
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
paulson@14050
  1039
apply (induct xs, auto) 
paulson@14050
  1040
apply (case_tac i, simp_all) 
paulson@14050
  1041
done
paulson@14050
  1042
nipkow@14300
  1043
lemma append_eq_append_conv_if:
nipkow@14300
  1044
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1045
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1046
   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
  1047
   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
  1048
apply(induct xs\<^isub>1)
nipkow@14300
  1049
 apply simp
nipkow@14300
  1050
apply(case_tac ys\<^isub>1)
nipkow@14300
  1051
apply simp_all
nipkow@14300
  1052
done
nipkow@14300
  1053
nipkow@15110
  1054
lemma take_hd_drop:
nipkow@15110
  1055
  "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
nipkow@15110
  1056
apply(induct xs)
nipkow@15110
  1057
apply simp
nipkow@15110
  1058
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1059
done
nipkow@15110
  1060
wenzelm@13114
  1061
wenzelm@13142
  1062
subsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1063
wenzelm@13142
  1064
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1065
by (induct xs) auto
wenzelm@13114
  1066
wenzelm@13142
  1067
lemma takeWhile_append1 [simp]:
nipkow@13145
  1068
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1069
by (induct xs) auto
wenzelm@13114
  1070
wenzelm@13142
  1071
lemma takeWhile_append2 [simp]:
nipkow@13145
  1072
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1073
by (induct xs) auto
wenzelm@13114
  1074
wenzelm@13142
  1075
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1076
by (induct xs) auto
wenzelm@13114
  1077
wenzelm@13142
  1078
lemma dropWhile_append1 [simp]:
nipkow@13145
  1079
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1080
by (induct xs) auto
wenzelm@13114
  1081
wenzelm@13142
  1082
lemma dropWhile_append2 [simp]:
nipkow@13145
  1083
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1084
by (induct xs) auto
wenzelm@13114
  1085
wenzelm@13142
  1086
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1087
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1088
nipkow@13913
  1089
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1090
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1091
by(induct xs, auto)
nipkow@13913
  1092
nipkow@13913
  1093
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1094
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1095
by(induct xs, auto)
nipkow@13913
  1096
nipkow@13913
  1097
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1098
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1099
by(induct xs, auto)
nipkow@13913
  1100
wenzelm@13114
  1101
wenzelm@13142
  1102
subsection {* @{text zip} *}
wenzelm@13114
  1103
wenzelm@13142
  1104
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1105
by (induct ys) auto
wenzelm@13114
  1106
wenzelm@13142
  1107
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1108
by simp
wenzelm@13114
  1109
wenzelm@13142
  1110
declare zip_Cons [simp del]
wenzelm@13114
  1111
nipkow@15281
  1112
lemma zip_Cons1:
nipkow@15281
  1113
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  1114
by(auto split:list.split)
nipkow@15281
  1115
wenzelm@13142
  1116
lemma length_zip [simp]:
nipkow@13145
  1117
"!!xs. length (zip xs ys) = min (length xs) (length ys)"
paulson@14208
  1118
apply (induct ys, simp)
paulson@14208
  1119
apply (case_tac xs, auto)
nipkow@13145
  1120
done
wenzelm@13114
  1121
wenzelm@13114
  1122
lemma zip_append1:
nipkow@13145
  1123
"!!xs. zip (xs @ ys) zs =
nipkow@13145
  1124
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
paulson@14208
  1125
apply (induct zs, simp)
paulson@14208
  1126
apply (case_tac xs, simp_all)
nipkow@13145
  1127
done
wenzelm@13114
  1128
wenzelm@13114
  1129
lemma zip_append2:
nipkow@13145
  1130
"!!ys. zip xs (ys @ zs) =
nipkow@13145
  1131
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
paulson@14208
  1132
apply (induct xs, simp)
paulson@14208
  1133
apply (case_tac ys, simp_all)
nipkow@13145
  1134
done
wenzelm@13114
  1135
wenzelm@13142
  1136
lemma zip_append [simp]:
wenzelm@13142
  1137
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1138
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1139
by (simp add: zip_append1)
wenzelm@13114
  1140
wenzelm@13114
  1141
lemma zip_rev:
nipkow@14247
  1142
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1143
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1144
wenzelm@13142
  1145
lemma nth_zip [simp]:
nipkow@13145
  1146
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
paulson@14208
  1147
apply (induct ys, simp)
nipkow@13145
  1148
apply (case_tac xs)
nipkow@13145
  1149
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1150
done
wenzelm@13114
  1151
wenzelm@13114
  1152
lemma set_zip:
nipkow@13145
  1153
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
  1154
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1155
wenzelm@13114
  1156
lemma zip_update:
nipkow@13145
  1157
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
  1158
by (rule sym, simp add: update_zip)
wenzelm@13114
  1159
wenzelm@13142
  1160
lemma zip_replicate [simp]:
nipkow@13145
  1161
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
paulson@14208
  1162
apply (induct i, auto)
paulson@14208
  1163
apply (case_tac j, auto)
nipkow@13145
  1164
done
wenzelm@13114
  1165
wenzelm@13142
  1166
wenzelm@13142
  1167
subsection {* @{text list_all2} *}
wenzelm@13114
  1168
kleing@14316
  1169
lemma list_all2_lengthD [intro?]: 
kleing@14316
  1170
  "list_all2 P xs ys ==> length xs = length ys"
nipkow@13145
  1171
by (simp add: list_all2_def)
wenzelm@13114
  1172
wenzelm@13142
  1173
lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])"
nipkow@13145
  1174
by (simp add: list_all2_def)
wenzelm@13114
  1175
wenzelm@13142
  1176
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"
nipkow@13145
  1177
by (simp add: list_all2_def)
wenzelm@13114
  1178
wenzelm@13142
  1179
lemma list_all2_Cons [iff]:
nipkow@13145
  1180
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@13145
  1181
by (auto simp add: list_all2_def)
wenzelm@13114
  1182
wenzelm@13114
  1183
lemma list_all2_Cons1:
nipkow@13145
  1184
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1185
by (cases ys) auto
wenzelm@13114
  1186
wenzelm@13114
  1187
lemma list_all2_Cons2:
nipkow@13145
  1188
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1189
by (cases xs) auto
wenzelm@13114
  1190
wenzelm@13142
  1191
lemma list_all2_rev [iff]:
nipkow@13145
  1192
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1193
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1194
kleing@13863
  1195
lemma list_all2_rev1:
kleing@13863
  1196
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1197
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1198
wenzelm@13114
  1199
lemma list_all2_append1:
nipkow@13145
  1200
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1201
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1202
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1203
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1204
apply (rule iffI)
nipkow@13145
  1205
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1206
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  1207
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1208
apply (simp add: ball_Un)
nipkow@13145
  1209
done
wenzelm@13114
  1210
wenzelm@13114
  1211
lemma list_all2_append2:
nipkow@13145
  1212
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1213
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1214
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1215
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1216
apply (rule iffI)
nipkow@13145
  1217
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1218
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  1219
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1220
apply (simp add: ball_Un)
nipkow@13145
  1221
done
wenzelm@13114
  1222
kleing@13863
  1223
lemma list_all2_append:
nipkow@14247
  1224
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  1225
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  1226
by (induct rule:list_induct2, simp_all)
kleing@13863
  1227
kleing@13863
  1228
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1229
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
kleing@13863
  1230
  by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1231
wenzelm@13114
  1232
lemma list_all2_conv_all_nth:
nipkow@13145
  1233
"list_all2 P xs ys =
nipkow@13145
  1234
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1235
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1236
berghofe@13883
  1237
lemma list_all2_trans:
berghofe@13883
  1238
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1239
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1240
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1241
proof (induct as)
berghofe@13883
  1242
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1243
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1244
  proof (induct bs)
berghofe@13883
  1245
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1246
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1247
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1248
  qed simp
berghofe@13883
  1249
qed simp
berghofe@13883
  1250
kleing@13863
  1251
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1252
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
kleing@13863
  1253
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1254
paulson@14395
  1255
lemma list_all2I:
paulson@14395
  1256
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
paulson@14395
  1257
  by (simp add: list_all2_def)
paulson@14395
  1258
kleing@14328
  1259
lemma list_all2_nthD:
kleing@13863
  1260
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
kleing@13863
  1261
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1262
nipkow@14302
  1263
lemma list_all2_nthD2:
nipkow@14302
  1264
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@14302
  1265
  by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  1266
kleing@13863
  1267
lemma list_all2_map1: 
kleing@13863
  1268
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
kleing@13863
  1269
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1270
kleing@13863
  1271
lemma list_all2_map2: 
kleing@13863
  1272
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
kleing@13863
  1273
  by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1274
kleing@14316
  1275
lemma list_all2_refl [intro?]:
kleing@13863
  1276
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
kleing@13863
  1277
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1278
kleing@13863
  1279
lemma list_all2_update_cong:
kleing@13863
  1280
  "\<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
  1281
  by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1282
kleing@13863
  1283
lemma list_all2_update_cong2:
kleing@13863
  1284
  "\<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
  1285
  by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1286
nipkow@14302
  1287
lemma list_all2_takeI [simp,intro?]:
nipkow@14302
  1288
  "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@14302
  1289
  apply (induct xs)
nipkow@14302
  1290
   apply simp
nipkow@14302
  1291
  apply (clarsimp simp add: list_all2_Cons1)
nipkow@14302
  1292
  apply (case_tac n)
nipkow@14302
  1293
  apply auto
nipkow@14302
  1294
  done
nipkow@14302
  1295
nipkow@14302
  1296
lemma list_all2_dropI [simp,intro?]:
kleing@13863
  1297
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
paulson@14208
  1298
  apply (induct as, simp)
kleing@13863
  1299
  apply (clarsimp simp add: list_all2_Cons1)
paulson@14208
  1300
  apply (case_tac n, simp, simp)
kleing@13863
  1301
  done
kleing@13863
  1302
kleing@14327
  1303
lemma list_all2_mono [intro?]:
kleing@13863
  1304
  "\<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
  1305
  apply (induct x, simp)
paulson@14208
  1306
  apply (case_tac y, auto)
kleing@13863
  1307
  done
kleing@13863
  1308
wenzelm@13142
  1309
nipkow@14402
  1310
subsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  1311
wenzelm@13142
  1312
lemma foldl_append [simp]:
nipkow@13145
  1313
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145
  1314
by (induct xs) auto
wenzelm@13142
  1315
nipkow@14402
  1316
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  1317
by (induct xs) auto
nipkow@14402
  1318
nipkow@14402
  1319
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  1320
by (induct xs) auto
nipkow@14402
  1321
nipkow@14402
  1322
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  1323
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  1324
wenzelm@13142
  1325
text {*
nipkow@13145
  1326
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1327
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1328
*}
wenzelm@13142
  1329
wenzelm@13142
  1330
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
nipkow@13145
  1331
by (induct ns) auto
wenzelm@13142
  1332
wenzelm@13142
  1333
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1334
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1335
wenzelm@13142
  1336
lemma sum_eq_0_conv [iff]:
nipkow@13145
  1337
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145
  1338
by (induct ns) auto
wenzelm@13114
  1339
wenzelm@13114
  1340
wenzelm@13142
  1341
subsection {* @{text upto} *}
wenzelm@13114
  1342
wenzelm@13142
  1343
lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])"
nipkow@13145
  1344
-- {* Does not terminate! *}
nipkow@13145
  1345
by (induct j) auto
wenzelm@13142
  1346
wenzelm@13142
  1347
lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []"
nipkow@13145
  1348
by (subst upt_rec) simp
wenzelm@13114
  1349
nipkow@15281
  1350
lemma upt_eq_Nil_conv[simp]: "([i..j(] = []) = (j = 0 \<or> j <= i)"
nipkow@15281
  1351
by(induct j)simp_all
nipkow@15281
  1352
nipkow@15281
  1353
lemma upt_eq_Cons_conv:
nipkow@15281
  1354
 "!!x xs. ([i..j(] = x#xs) = (i < j & i = x & [i+1..j(] = xs)"
nipkow@15281
  1355
apply(induct j)
nipkow@15281
  1356
 apply simp
nipkow@15281
  1357
apply(clarsimp simp add: append_eq_Cons_conv)
nipkow@15281
  1358
apply arith
nipkow@15281
  1359
done
nipkow@15281
  1360
wenzelm@13142
  1361
lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]"
nipkow@13145
  1362
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1363
by simp
wenzelm@13114
  1364
wenzelm@13142
  1365
lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]"
nipkow@13145
  1366
apply(rule trans)
nipkow@13145
  1367
apply(subst upt_rec)
paulson@14208
  1368
 prefer 2 apply (rule refl, simp)
nipkow@13145
  1369
done
wenzelm@13114
  1370
wenzelm@13142
  1371
lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]"
nipkow@13145
  1372
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1373
by (induct k) auto
wenzelm@13114
  1374
wenzelm@13142
  1375
lemma length_upt [simp]: "length [i..j(] = j - i"
nipkow@13145
  1376
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1377
wenzelm@13142
  1378
lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k"
nipkow@13145
  1379
apply (induct j)
nipkow@13145
  1380
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1381
done
wenzelm@13114
  1382
wenzelm@13142
  1383
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]"
paulson@14208
  1384
apply (induct m, simp)
nipkow@13145
  1385
apply (subst upt_rec)
nipkow@13145
  1386
apply (rule sym)
nipkow@13145
  1387
apply (subst upt_rec)
nipkow@13145
  1388
apply (simp del: upt.simps)
nipkow@13145
  1389
done
nipkow@3507
  1390
wenzelm@13114
  1391
lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]"
nipkow@13145
  1392
by (induct n) auto
wenzelm@13114
  1393
wenzelm@13114
  1394
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..n(]) ! i = f(m+i)"
nipkow@13145
  1395
apply (induct n m rule: diff_induct)
nipkow@13145
  1396
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1397
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1398
done
wenzelm@13114
  1399
berghofe@13883
  1400
lemma nth_take_lemma:
berghofe@13883
  1401
  "!!xs ys. k <= length xs ==> k <= length ys ==>
berghofe@13883
  1402
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
berghofe@13883
  1403
apply (atomize, induct k)
paulson@14208
  1404
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  1405
txt {* Both lists must be non-empty *}
paulson@14208
  1406
apply (case_tac xs, simp)
paulson@14208
  1407
apply (case_tac ys, clarify)
nipkow@13145
  1408
 apply (simp (no_asm_use))
nipkow@13145
  1409
apply clarify
nipkow@13145
  1410
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  1411
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  1412
apply blast
nipkow@13145
  1413
done
wenzelm@13114
  1414
wenzelm@13114
  1415
lemma nth_equalityI:
wenzelm@13114
  1416
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  1417
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  1418
apply (simp_all add: take_all)
nipkow@13145
  1419
done
wenzelm@13142
  1420
kleing@13863
  1421
(* needs nth_equalityI *)
kleing@13863
  1422
lemma list_all2_antisym:
kleing@13863
  1423
  "\<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
  1424
  \<Longrightarrow> xs = ys"
kleing@13863
  1425
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  1426
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  1427
  done
kleing@13863
  1428
wenzelm@13142
  1429
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  1430
-- {* The famous take-lemma. *}
nipkow@13145
  1431
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  1432
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  1433
done
wenzelm@13142
  1434
wenzelm@13142
  1435
nipkow@15302
  1436
lemma take_Cons':
nipkow@15302
  1437
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@15302
  1438
by (cases n) simp_all
nipkow@15302
  1439
nipkow@15302
  1440
lemma drop_Cons':
nipkow@15302
  1441
     "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@15302
  1442
by (cases n) simp_all
nipkow@15302
  1443
nipkow@15302
  1444
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@15302
  1445
by (cases n) simp_all
nipkow@15302
  1446
nipkow@15302
  1447
lemmas [simp] = take_Cons'[of "number_of v",standard]
nipkow@15302
  1448
                drop_Cons'[of "number_of v",standard]
nipkow@15302
  1449
                nth_Cons'[of _ _ "number_of v",standard]
nipkow@15302
  1450
nipkow@15302
  1451
wenzelm@13142
  1452
subsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  1453
wenzelm@13142
  1454
lemma distinct_append [simp]:
nipkow@13145
  1455
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  1456
by (induct xs) auto
wenzelm@13142
  1457
nipkow@15305
  1458
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
nipkow@15305
  1459
by(induct xs) auto
nipkow@15305
  1460
wenzelm@13142
  1461
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  1462
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  1463
wenzelm@13142
  1464
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  1465
by (induct xs) auto
wenzelm@13142
  1466
paulson@15072
  1467
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
paulson@15251
  1468
  by (induct x, auto) 
paulson@15072
  1469
paulson@15072
  1470
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
paulson@15251
  1471
  by (induct x, auto)
paulson@15072
  1472
nipkow@15245
  1473
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245
  1474
by (induct xs) auto
nipkow@15245
  1475
nipkow@15245
  1476
lemma length_remdups_eq[iff]:
nipkow@15245
  1477
  "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245
  1478
apply(induct xs)
nipkow@15245
  1479
 apply auto
nipkow@15245
  1480
apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245
  1481
 apply arith
nipkow@15245
  1482
apply(rule length_remdups_leq)
nipkow@15245
  1483
done
nipkow@15245
  1484
wenzelm@13142
  1485
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  1486
by (induct xs) auto
wenzelm@13114
  1487
nipkow@15304
  1488
lemma distinct_map_filterI:
nipkow@15304
  1489
 "distinct(map f xs) \<Longrightarrow> distinct(map f (filter P xs))"
nipkow@15304
  1490
apply(induct xs)
nipkow@15304
  1491
 apply simp
nipkow@15304
  1492
apply force
nipkow@15304
  1493
done
nipkow@15304
  1494
wenzelm@13142
  1495
text {*
nipkow@13145
  1496
It is best to avoid this indexed version of distinct, but sometimes
nipkow@13145
  1497
it is useful. *}
wenzelm@13142
  1498
lemma distinct_conv_nth:
nipkow@13145
  1499
"distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@15251
  1500
apply (induct xs, simp, simp)
paulson@14208
  1501
apply (rule iffI, clarsimp)
nipkow@13145
  1502
 apply (case_tac i)
paulson@14208
  1503
apply (case_tac j, simp)
nipkow@13145
  1504
apply (simp add: set_conv_nth)
nipkow@13145
  1505
 apply (case_tac j)
paulson@14208
  1506
apply (clarsimp simp add: set_conv_nth, simp)
nipkow@13145
  1507
apply (rule conjI)
nipkow@13145
  1508
 apply (clarsimp simp add: set_conv_nth)
nipkow@13145
  1509
 apply (erule_tac x = 0 in allE)
paulson@14208
  1510
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
nipkow@13145
  1511
apply (erule_tac x = "Suc i" in allE)
paulson@14208
  1512
apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145
  1513
done
wenzelm@13114
  1514
nipkow@15110
  1515
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
kleing@14388
  1516
  by (induct xs) auto
kleing@14388
  1517
nipkow@15110
  1518
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388
  1519
proof (induct xs)
kleing@14388
  1520
  case Nil thus ?case by simp
kleing@14388
  1521
next
kleing@14388
  1522
  case (Cons x xs)
kleing@14388
  1523
  show ?case
kleing@14388
  1524
  proof (cases "x \<in> set xs")
kleing@14388
  1525
    case False with Cons show ?thesis by simp
kleing@14388
  1526
  next
kleing@14388
  1527
    case True with Cons.prems
kleing@14388
  1528
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  1529
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  1530
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  1531
    ultimately have False by simp
kleing@14388
  1532
    thus ?thesis ..
kleing@14388
  1533
  qed
kleing@14388
  1534
qed
kleing@14388
  1535
nipkow@15110
  1536
lemma inj_on_setI: "distinct(map f xs) ==> inj_on f (set xs)"
nipkow@15110
  1537
apply(induct xs)
nipkow@15110
  1538
 apply simp
nipkow@15110
  1539
apply fastsimp
nipkow@15110
  1540
done
nipkow@15110
  1541
nipkow@15110
  1542
lemma inj_on_set_conv:
nipkow@15110
  1543
 "distinct xs \<Longrightarrow> inj_on f (set xs) = distinct(map f xs)"
nipkow@15110
  1544
apply(induct xs)
nipkow@15110
  1545
 apply simp
nipkow@15110
  1546
apply fastsimp
nipkow@15110
  1547
done
nipkow@15110
  1548
nipkow@15110
  1549
nipkow@15110
  1550
subsection {* @{text remove1} *}
nipkow@15110
  1551
nipkow@15110
  1552
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110
  1553
apply(induct xs)
nipkow@15110
  1554
 apply simp
nipkow@15110
  1555
apply simp
nipkow@15110
  1556
apply blast
nipkow@15110
  1557
done
nipkow@15110
  1558
nipkow@15110
  1559
lemma [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110
  1560
apply(induct xs)
nipkow@15110
  1561
 apply simp
nipkow@15110
  1562
apply simp
nipkow@15110
  1563
apply blast
nipkow@15110
  1564
done
nipkow@15110
  1565
nipkow@15110
  1566
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110
  1567
apply(insert set_remove1_subset)
nipkow@15110
  1568
apply fast
nipkow@15110
  1569
done
nipkow@15110
  1570
nipkow@15110
  1571
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110
  1572
by (induct xs) simp_all
nipkow@15110
  1573
wenzelm@13114
  1574
wenzelm@13142
  1575
subsection {* @{text replicate} *}
wenzelm@13114
  1576
wenzelm@13142
  1577
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  1578
by (induct n) auto
nipkow@13124
  1579
wenzelm@13142
  1580
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  1581
by (induct n) auto
wenzelm@13114
  1582
wenzelm@13114
  1583
lemma replicate_app_Cons_same:
nipkow@13145
  1584
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  1585
by (induct n) auto
wenzelm@13114
  1586
wenzelm@13142
  1587
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208
  1588
apply (induct n, simp)
nipkow@13145
  1589
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  1590
done
wenzelm@13114
  1591
wenzelm@13142
  1592
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  1593
by (induct n) auto
wenzelm@13114
  1594
wenzelm@13142
  1595
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  1596
by (induct n) auto
wenzelm@13114
  1597
wenzelm@13142
  1598
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  1599
by (induct n) auto
wenzelm@13114
  1600
wenzelm@13142
  1601
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  1602
by (atomize (full), induct n) auto
wenzelm@13114
  1603
wenzelm@13142
  1604
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
paulson@14208
  1605
apply (induct n, simp)
nipkow@13145
  1606
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  1607
done
wenzelm@13114
  1608
wenzelm@13142
  1609
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  1610
by (induct n) auto
wenzelm@13114
  1611
wenzelm@13142
  1612
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  1613
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  1614
wenzelm@13142
  1615
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  1616
by auto
wenzelm@13114
  1617
wenzelm@13142
  1618
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  1619
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  1620
wenzelm@13114
  1621
nipkow@15302
  1622
subsection{*@{text rotate1} and @{text rotate}*}
nipkow@15302
  1623
nipkow@15302
  1624
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
nipkow@15302
  1625
by(simp add:rotate1_def)
nipkow@15302
  1626
nipkow@15302
  1627
lemma rotate0[simp]: "rotate 0 = id"
nipkow@15302
  1628
by(simp add:rotate_def)
nipkow@15302
  1629
nipkow@15302
  1630
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
nipkow@15302
  1631
by(simp add:rotate_def)
nipkow@15302
  1632
nipkow@15302
  1633
lemma rotate_add:
nipkow@15302
  1634
  "rotate (m+n) = rotate m o rotate n"
nipkow@15302
  1635
by(simp add:rotate_def funpow_add)
nipkow@15302
  1636
nipkow@15302
  1637
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
nipkow@15302
  1638
by(simp add:rotate_add)
nipkow@15302
  1639
nipkow@15302
  1640
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
nipkow@15302
  1641
by(cases xs) simp_all
nipkow@15302
  1642
nipkow@15302
  1643
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  1644
apply(induct n)
nipkow@15302
  1645
 apply simp
nipkow@15302
  1646
apply (simp add:rotate_def)
nipkow@13145
  1647
done
wenzelm@13114
  1648
nipkow@15302
  1649
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
nipkow@15302
  1650
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1651
nipkow@15302
  1652
lemma rotate_drop_take:
nipkow@15302
  1653
  "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
nipkow@15302
  1654
apply(induct n)
nipkow@15302
  1655
 apply simp
nipkow@15302
  1656
apply(simp add:rotate_def)
nipkow@15302
  1657
apply(cases "xs = []")
nipkow@15302
  1658
 apply (simp)
nipkow@15302
  1659
apply(case_tac "n mod length xs = 0")
nipkow@15302
  1660
 apply(simp add:mod_Suc)
nipkow@15302
  1661
 apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
nipkow@15302
  1662
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
nipkow@15302
  1663
                take_hd_drop linorder_not_le)
nipkow@13145
  1664
done
wenzelm@13114
  1665
nipkow@15302
  1666
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
nipkow@15302
  1667
by(simp add:rotate_drop_take)
nipkow@15302
  1668
nipkow@15302
  1669
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  1670
by(simp add:rotate_drop_take)
nipkow@15302
  1671
nipkow@15302
  1672
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
nipkow@15302
  1673
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1674
nipkow@15302
  1675
lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs"
nipkow@15302
  1676
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  1677
nipkow@15302
  1678
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
nipkow@15302
  1679
by(simp add:rotate1_def split:list.split) blast
nipkow@15302
  1680
nipkow@15302
  1681
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
nipkow@15302
  1682
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  1683
nipkow@15302
  1684
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
nipkow@15302
  1685
by(simp add:rotate_drop_take take_map drop_map)
nipkow@15302
  1686
nipkow@15302
  1687
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
nipkow@15302
  1688
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1689
nipkow@15302
  1690
lemma set_rotate[simp]: "set(rotate n xs) = set xs"
nipkow@15302
  1691
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  1692
nipkow@15302
  1693
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
nipkow@15302
  1694
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1695
nipkow@15302
  1696
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
nipkow@15302
  1697
by (induct n) (simp_all add:rotate_def)
wenzelm@13114
  1698
wenzelm@13114
  1699
wenzelm@13142
  1700
subsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  1701
wenzelm@13142
  1702
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  1703
by (auto simp add: sublist_def)
wenzelm@13114
  1704
wenzelm@13142
  1705
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  1706
by (auto simp add: sublist_def)
wenzelm@13114
  1707
nipkow@15281
  1708
lemma length_sublist:
nipkow@15281
  1709
  "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
nipkow@15281
  1710
by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
nipkow@15281
  1711
nipkow@15281
  1712
lemma sublist_shift_lemma_Suc:
nipkow@15281
  1713
  "!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
nipkow@15281
  1714
         map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
nipkow@15281
  1715
apply(induct xs)
nipkow@15281
  1716
 apply simp
nipkow@15281
  1717
apply (case_tac "is")
nipkow@15281
  1718
 apply simp
nipkow@15281
  1719
apply simp
nipkow@15281
  1720
done
nipkow@15281
  1721
wenzelm@13114
  1722
lemma sublist_shift_lemma:
paulson@15168
  1723
     "map fst [p:zip xs [i..i + length xs(] . snd p : A] =
paulson@15168
  1724
      map fst [p:zip xs [0..length xs(] . snd p + i : A]"
nipkow@13145
  1725
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  1726
wenzelm@13114
  1727
lemma sublist_append:
paulson@15168
  1728
     "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  1729
apply (unfold sublist_def)
paulson@14208
  1730
apply (induct l' rule: rev_induct, simp)
nipkow@13145
  1731
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  1732
apply (simp add: add_commute)
nipkow@13145
  1733
done
wenzelm@13114
  1734
wenzelm@13114
  1735
lemma sublist_Cons:
nipkow@13145
  1736
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  1737
apply (induct l rule: rev_induct)
nipkow@13145
  1738
 apply (simp add: sublist_def)
nipkow@13145
  1739
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  1740
done
wenzelm@13114
  1741
nipkow@15281
  1742
lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
nipkow@15281
  1743
apply(induct xs)
nipkow@15281
  1744
 apply simp
nipkow@15281
  1745
apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE)
nipkow@15281
  1746
 apply(erule lessE)
nipkow@15281
  1747
  apply auto
nipkow@15281
  1748
apply(erule lessE)
nipkow@15281
  1749
apply auto
nipkow@15281
  1750
done
nipkow@15281
  1751
nipkow@15281
  1752
lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
nipkow@15281
  1753
by(auto simp add:set_sublist)
nipkow@15281
  1754
nipkow@15281
  1755
lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
nipkow@15281
  1756
by(auto simp add:set_sublist)
nipkow@15281
  1757
nipkow@15281
  1758
lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
nipkow@15281
  1759
by(auto simp add:set_sublist)
nipkow@15281
  1760
wenzelm@13142
  1761
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  1762
by (simp add: sublist_Cons)
wenzelm@13114
  1763
nipkow@15281
  1764
nipkow@15281
  1765
lemma distinct_sublistI[simp]: "!!I. distinct xs \<Longrightarrow> distinct(sublist xs I)"
nipkow@15281
  1766
apply(induct xs)
nipkow@15281
  1767
 apply simp
nipkow@15281
  1768
apply(auto simp add:sublist_Cons)
nipkow@15281
  1769
done
nipkow@15281
  1770
nipkow@15281
  1771
nipkow@15045
  1772
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
paulson@14208
  1773
apply (induct l rule: rev_induct, simp)
nipkow@13145
  1774
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  1775
done
wenzelm@13114
  1776
wenzelm@13114
  1777
nipkow@15302
  1778
subsection{*Sets of Lists*}
nipkow@15302
  1779
nipkow@15302
  1780
subsection {* @{text lists}: the list-forming operator over sets *}
nipkow@15302
  1781
nipkow@15302
  1782
consts lists :: "'a set => 'a list set"
nipkow@15302
  1783
inductive "lists A"
nipkow@15302
  1784
 intros
nipkow@15302
  1785
  Nil [intro!]: "[]: lists A"
nipkow@15302
  1786
  Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A"
nipkow@15302
  1787
nipkow@15302
  1788
inductive_cases listsE [elim!]: "x#l : lists A"
nipkow@15302
  1789
nipkow@15302
  1790
lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B"
nipkow@15302
  1791
by (unfold lists.defs) (blast intro!: lfp_mono)
nipkow@15302
  1792
nipkow@15302
  1793
lemma lists_IntI:
nipkow@15302
  1794
  assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l
nipkow@15302
  1795
  by induct blast+
nipkow@15302
  1796
nipkow@15302
  1797
lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B"
nipkow@15302
  1798
proof (rule mono_Int [THEN equalityI])
nipkow@15302
  1799
  show "mono lists" by (simp add: mono_def lists_mono)
nipkow@15302
  1800
  show "lists A \<inter> lists B \<subseteq> lists (A \<inter> B)" by (blast intro: lists_IntI)
kleing@14388
  1801
qed
kleing@14388
  1802
nipkow@15302
  1803
lemma append_in_lists_conv [iff]:
nipkow@15302
  1804
     "(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"
nipkow@15302
  1805
by (induct xs) auto
nipkow@15302
  1806
nipkow@15302
  1807
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"
nipkow@15302
  1808
-- {* eliminate @{text lists} in favour of @{text set} *}
nipkow@15302
  1809
by (induct xs) auto
nipkow@15302
  1810
nipkow@15302
  1811
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"
nipkow@15302
  1812
by (rule in_lists_conv_set [THEN iffD1])
nipkow@15302
  1813
nipkow@15302
  1814
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"
nipkow@15302
  1815
by (rule in_lists_conv_set [THEN iffD2])
nipkow@15302
  1816
nipkow@15302
  1817
lemma lists_UNIV [simp]: "lists UNIV = UNIV"
nipkow@15302
  1818
by auto
nipkow@15302
  1819
nipkow@15302
  1820
subsection{*Lists as Cartesian products*}
nipkow@15302
  1821
nipkow@15302
  1822
text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
nipkow@15302
  1823
@{term A} and tail drawn from @{term Xs}.*}
nipkow@15302
  1824
nipkow@15302
  1825
constdefs
nipkow@15302
  1826
  set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
nipkow@15302
  1827
  "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"
nipkow@15302
  1828
nipkow@15302
  1829
lemma [simp]: "set_Cons A {[]} = (%x. [x])`A"
nipkow@15302
  1830
by (auto simp add: set_Cons_def)
nipkow@15302
  1831
nipkow@15302
  1832
text{*Yields the set of lists, all of the same length as the argument and
nipkow@15302
  1833
with elements drawn from the corresponding element of the argument.*}
nipkow@15302
  1834
nipkow@15302
  1835
consts  listset :: "'a set list \<Rightarrow> 'a list set"
nipkow@15302
  1836
primrec
nipkow@15302
  1837
   "listset []    = {[]}"
nipkow@15302
  1838
   "listset(A#As) = set_Cons A (listset As)"
nipkow@15302
  1839
nipkow@15302
  1840
nipkow@15302
  1841
section{*Relations on lists*}
nipkow@15302
  1842
nipkow@15302
  1843
subsection {* Lexicographic orderings on lists *}
nipkow@15302
  1844
nipkow@15302
  1845
consts
nipkow@15302
  1846
lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
nipkow@15302
  1847
primrec
nipkow@15302
  1848
"lexn r 0 = {}"
nipkow@15302
  1849
"lexn r (Suc n) =
nipkow@15302
  1850
(prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
nipkow@15302
  1851
{(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
nipkow@15302
  1852
nipkow@15302
  1853
constdefs
nipkow@15302
  1854
lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@15302
  1855
"lex r == \<Union>n. lexn r n"
nipkow@15302
  1856
nipkow@15302
  1857
lexico :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@15302
  1858
"lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
nipkow@15302
  1859
nipkow@15302
  1860
nipkow@15302
  1861
lemma wf_lexn: "wf r ==> wf (lexn r n)"
nipkow@15302
  1862
apply (induct n, simp, simp)
nipkow@15302
  1863
apply(rule wf_subset)
nipkow@15302
  1864
 prefer 2 apply (rule Int_lower1)
nipkow@15302
  1865
apply(rule wf_prod_fun_image)
nipkow@15302
  1866
 prefer 2 apply (rule inj_onI, auto)
nipkow@15302
  1867
done
nipkow@15302
  1868
nipkow@15302
  1869
lemma lexn_length:
nipkow@15302
  1870
     "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@15302
  1871
by (induct n) auto
nipkow@15302
  1872
nipkow@15302
  1873
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@15302
  1874
apply (unfold lex_def)
nipkow@15302
  1875
apply (rule wf_UN)
nipkow@15302
  1876
apply (blast intro: wf_lexn, clarify)
nipkow@15302
  1877
apply (rename_tac m n)
nipkow@15302
  1878
apply (subgoal_tac "m \<noteq> n")
nipkow@15302
  1879
 prefer 2 apply blast
nipkow@15302
  1880
apply (blast dest: lexn_length not_sym)
nipkow@15302
  1881
done
nipkow@15302
  1882
nipkow@15302
  1883
lemma lexn_conv:
nipkow@15302
  1884
"lexn r n =
nipkow@15302
  1885
{(xs,ys). length xs = n \<and> length ys = n \<and>
nipkow@15302
  1886
(\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
nipkow@15302
  1887
apply (induct n, simp, blast)
nipkow@15302
  1888
apply (simp add: image_Collect lex_prod_def, safe, blast)
nipkow@15302
  1889
 apply (rule_tac x = "ab # xys" in exI, simp)
nipkow@15302
  1890
apply (case_tac xys, simp_all, blast)
nipkow@15302
  1891
done
nipkow@15302
  1892
nipkow@15302
  1893
lemma lex_conv:
nipkow@15302
  1894
"lex r =
nipkow@15302
  1895
{(xs,ys). length xs = length ys \<and>
nipkow@15302
  1896
(\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@15302
  1897
by (force simp add: lex_def lexn_conv)
nipkow@15302
  1898
nipkow@15302
  1899
lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)"
nipkow@15302
  1900
by (unfold lexico_def) blast
nipkow@15302
  1901
nipkow@15302
  1902
lemma lexico_conv:
nipkow@15302
  1903
"lexico r = {(xs,ys). length xs < length ys |
nipkow@15302
  1904
length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@15302
  1905
by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def)
nipkow@15302
  1906
nipkow@15302
  1907
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@15302
  1908
by (simp add: lex_conv)
nipkow@15302
  1909
nipkow@15302
  1910
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@15302
  1911
by (simp add:lex_conv)
nipkow@15302
  1912
nipkow@15302
  1913
lemma Cons_in_lex [iff]:
nipkow@15302
  1914
"((x # xs, y # ys) : lex r) =
nipkow@15302
  1915
((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@15302
  1916
apply (simp add: lex_conv)
nipkow@15302
  1917
apply (rule iffI)
nipkow@15302
  1918
 prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
nipkow@15302
  1919
apply (case_tac xys, simp, simp)
nipkow@15302
  1920
apply blast
nipkow@15302
  1921
done
nipkow@15302
  1922
nipkow@15302
  1923
nipkow@15302
  1924
subsection{*Lifting a Relation on List Elements to the Lists*}
nipkow@15302
  1925
nipkow@15302
  1926
consts  listrel :: "('a * 'a)set => ('a list * 'a list)set"
nipkow@15302
  1927
nipkow@15302
  1928
inductive "listrel(r)"
nipkow@15302
  1929
 intros
nipkow@15302
  1930
   Nil:  "([],[]) \<in> listrel r"
nipkow@15302
  1931
   Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
nipkow@15302
  1932
nipkow@15302
  1933
inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
nipkow@15302
  1934
inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
nipkow@15302
  1935
inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
nipkow@15302
  1936
inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
nipkow@15302
  1937
nipkow@15302
  1938
nipkow@15302
  1939
lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
nipkow@15302
  1940
apply clarify  
nipkow@15302
  1941
apply (erule listrel.induct)
nipkow@15302
  1942
apply (blast intro: listrel.intros)+
nipkow@15302
  1943
done
nipkow@15302
  1944
nipkow@15302
  1945
lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
nipkow@15302
  1946
apply clarify 
nipkow@15302
  1947
apply (erule listrel.induct, auto) 
nipkow@15302
  1948
done
nipkow@15302
  1949
nipkow@15302
  1950
lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" 
nipkow@15302
  1951
apply (simp add: refl_def listrel_subset Ball_def)
nipkow@15302
  1952
apply (rule allI) 
nipkow@15302
  1953
apply (induct_tac x) 
nipkow@15302
  1954
apply (auto intro: listrel.intros)
nipkow@15302
  1955
done
nipkow@15302
  1956
nipkow@15302
  1957
lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
nipkow@15302
  1958
apply (auto simp add: sym_def)
nipkow@15302
  1959
apply (erule listrel.induct) 
nipkow@15302
  1960
apply (blast intro: listrel.intros)+
nipkow@15302
  1961
done
nipkow@15302
  1962
nipkow@15302
  1963
lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
nipkow@15302
  1964
apply (simp add: trans_def)
nipkow@15302
  1965
apply (intro allI) 
nipkow@15302
  1966
apply (rule impI) 
nipkow@15302
  1967
apply (erule listrel.induct) 
nipkow@15302
  1968
apply (blast intro: listrel.intros)+
nipkow@15302
  1969
done
nipkow@15302
  1970
nipkow@15302
  1971
theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
nipkow@15302
  1972
by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) 
nipkow@15302
  1973
nipkow@15302
  1974
lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
nipkow@15302
  1975
by (blast intro: listrel.intros)
nipkow@15302
  1976
nipkow@15302
  1977
lemma listrel_Cons:
nipkow@15302
  1978
     "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})";
nipkow@15302
  1979
by (auto simp add: set_Cons_def intro: listrel.intros) 
nipkow@15302
  1980
nipkow@15302
  1981
nipkow@15302
  1982
section{*Miscellany*}
nipkow@15302
  1983
wenzelm@13366
  1984
subsection {* Characters and strings *}
wenzelm@13366
  1985
wenzelm@13366
  1986
datatype nibble =
wenzelm@13366
  1987
    Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
wenzelm@13366
  1988
  | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
wenzelm@13366
  1989
wenzelm@13366
  1990
datatype char = Char nibble nibble
wenzelm@13366
  1991
  -- "Note: canonical order of character encoding coincides with standard term ordering"
wenzelm@13366
  1992
wenzelm@13366
  1993
types string = "char list"
wenzelm@13366
  1994
wenzelm@13366
  1995
syntax
wenzelm@13366
  1996
  "_Char" :: "xstr => char"    ("CHR _")
wenzelm@13366
  1997
  "_String" :: "xstr => string"    ("_")
wenzelm@13366
  1998
wenzelm@13366
  1999
parse_ast_translation {*
wenzelm@13366
  2000
  let
wenzelm@13366
  2001
    val constants = Syntax.Appl o map Syntax.Constant;
wenzelm@13366
  2002
wenzelm@13366
  2003
    fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10));
wenzelm@13366
  2004
    fun mk_char c =
wenzelm@13366
  2005
      if Symbol.is_ascii c andalso Symbol.is_printable c then
wenzelm@13366
  2006
        constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)]
wenzelm@13366
  2007
      else error ("Printable ASCII character expected: " ^ quote c);
wenzelm@13366
  2008
wenzelm@13366
  2009
    fun mk_string [] = Syntax.Constant "Nil"
wenzelm@13366
  2010
      | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs];
wenzelm@13366
  2011
wenzelm@13366
  2012
    fun char_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  2013
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  2014
          [c] => mk_char c
wenzelm@13366
  2015
        | _ => error ("Single character expected: " ^ xstr))
wenzelm@13366
  2016
      | char_ast_tr asts = raise AST ("char_ast_tr", asts);
wenzelm@13366
  2017
wenzelm@13366
  2018
    fun string_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  2019
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  2020
          [] => constants [Syntax.constrainC, "Nil", "string"]
wenzelm@13366
  2021
        | cs => mk_string cs)
wenzelm@13366
  2022
      | string_ast_tr asts = raise AST ("string_tr", asts);
wenzelm@13366
  2023
  in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end;
wenzelm@13366
  2024
*}
wenzelm@13366
  2025
berghofe@15064
  2026
ML {*
berghofe@15064
  2027
fun int_of_nibble h =
berghofe@15064
  2028
  if "0" <= h andalso h <= "9" then ord h - ord "0"
berghofe@15064
  2029
  else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10
berghofe@15064
  2030
  else raise Match;
berghofe@15064
  2031
berghofe@15064
  2032
fun nibble_of_int i =
berghofe@15064
  2033
  if i <= 9 then chr (ord "0" + i) else chr (ord "A" + i - 10);
berghofe@15064
  2034
*}
berghofe@15064
  2035
wenzelm@13366
  2036
print_ast_translation {*
wenzelm@13366
  2037
  let
wenzelm@13366
  2038
    fun dest_nib (Syntax.Constant c) =
wenzelm@13366
  2039
        (case explode c of
berghofe@15064
  2040
          ["N", "i", "b", "b", "l", "e", h] => int_of_nibble h
wenzelm@13366
  2041
        | _ => raise Match)
wenzelm@13366
  2042
      | dest_nib _ = raise Match;
wenzelm@13366
  2043
wenzelm@13366
  2044
    fun dest_chr c1 c2 =
wenzelm@13366
  2045
      let val c = chr (dest_nib c1 * 16 + dest_nib c2)
wenzelm@13366
  2046
      in if Symbol.is_printable c then c else raise Match end;
wenzelm@13366
  2047
wenzelm@13366
  2048
    fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2
wenzelm@13366
  2049
      | dest_char _ = raise Match;
wenzelm@13366
  2050
wenzelm@13366
  2051
    fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)];
wenzelm@13366
  2052
wenzelm@13366
  2053
    fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]]
wenzelm@13366
  2054
      | char_ast_tr' _ = raise Match;
wenzelm@13366
  2055
wenzelm@13366
  2056
    fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String",
wenzelm@13366
  2057
            xstr (map dest_char (Syntax.unfold_ast "_args" args))]
wenzelm@13366
  2058
      | list_ast_tr' ts = raise Match;
wenzelm@13366
  2059
  in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end;
wenzelm@13366
  2060
*}
wenzelm@13366
  2061
berghofe@15064
  2062
subsection {* Code generator setup *}
berghofe@15064
  2063
berghofe@15064
  2064
ML {*
berghofe@15064
  2065
local
berghofe@15064
  2066
berghofe@15064
  2067
fun list_codegen thy gr dep b t =
berghofe@15064
  2068
  let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy dep false)
berghofe@15064
  2069
    (gr, HOLogic.dest_list t)
berghofe@15064
  2070
  in Some (gr', Pretty.list "[" "]" ps) end handle TERM _ => None;
berghofe@15064
  2071
berghofe@15064
  2072
fun dest_nibble (Const (s, _)) = int_of_nibble (unprefix "List.nibble.Nibble" s)
berghofe@15064
  2073
  | dest_nibble _ = raise Match;
berghofe@15064
  2074
berghofe@15064
  2075
fun char_codegen thy gr dep b (Const ("List.char.Char", _) $ c1 $ c2) =
berghofe@15064
  2076
    (let val c = chr (dest_nibble c1 * 16 + dest_nibble c2)
berghofe@15064
  2077
     in if Symbol.is_printable c then Some (gr, Pretty.quote (Pretty.str c))
berghofe@15064
  2078
       else None
berghofe@15064
  2079
     end handle LIST _ => None | Match => None)
berghofe@15064
  2080
  | char_codegen thy gr dep b _ = None;
berghofe@15064
  2081
berghofe@15064
  2082
in
berghofe@15064
  2083
berghofe@15064
  2084
val list_codegen_setup =
berghofe@15064
  2085
  [Codegen.add_codegen "list_codegen" list_codegen,
berghofe@15064
  2086
   Codegen.add_codegen "char_codegen" char_codegen];
berghofe@15064
  2087
berghofe@15064
  2088
end;
berghofe@15064
  2089
berghofe@15064
  2090
val term_of_list = HOLogic.mk_list;
berghofe@15064
  2091
berghofe@15064
  2092
fun gen_list' aG i j = frequency
berghofe@15064
  2093
  [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] ()
berghofe@15064
  2094
and gen_list aG i = gen_list' aG i i;
berghofe@15064
  2095
berghofe@15064
  2096
val nibbleT = Type ("List.nibble", []);
berghofe@15064
  2097
berghofe@15064
  2098
fun term_of_char c =
berghofe@15064
  2099
  Const ("List.char.Char", nibbleT --> nibbleT --> Type ("List.char", [])) $
berghofe@15064
  2100
    Const ("List.nibble.Nibble" ^ nibble_of_int (ord c div 16), nibbleT) $
berghofe@15064
  2101
    Const ("List.nibble.Nibble" ^ nibble_of_int (ord c mod 16), nibbleT);
berghofe@15064
  2102
berghofe@15064
  2103
fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z")));
berghofe@15064
  2104
*}
berghofe@15064
  2105
berghofe@15064
  2106
types_code
berghofe@15064
  2107
  "list" ("_ list")
berghofe@15064
  2108
  "char" ("string")
berghofe@15064
  2109
berghofe@15064
  2110
consts_code "Cons" ("(_ ::/ _)")
berghofe@15064
  2111
berghofe@15064
  2112
setup list_codegen_setup
berghofe@15064
  2113
wenzelm@13122
  2114
end