src/HOL/List.thy
author nipkow
Sun Nov 21 12:52:03 2004 +0100 (2004-11-21)
changeset 15302 a643fcbc3468
parent 15281 bd4611956c7b
child 15303 eedbb8d22ca2
permissions -rw-r--r--
Restructured List and added "rotate"
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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
kleing@14343
   510
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   511
by (induct xs, auto)
wenzelm@13114
   512
nipkow@14402
   513
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   514
by (induct xs) auto
nipkow@14402
   515
nipkow@15110
   516
lemma map_fst_zip[simp]:
nipkow@15110
   517
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   518
by (induct rule:list_induct2, simp_all)
nipkow@15110
   519
nipkow@15110
   520
lemma map_snd_zip[simp]:
nipkow@15110
   521
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   522
by (induct rule:list_induct2, simp_all)
nipkow@15110
   523
nipkow@15110
   524
wenzelm@13142
   525
subsection {* @{text rev} *}
wenzelm@13114
   526
wenzelm@13142
   527
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   528
by (induct xs) auto
wenzelm@13114
   529
wenzelm@13142
   530
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   531
by (induct xs) auto
wenzelm@13114
   532
wenzelm@13142
   533
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   534
by (induct xs) auto
wenzelm@13114
   535
wenzelm@13142
   536
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   537
by (induct xs) auto
wenzelm@13114
   538
wenzelm@13142
   539
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"
paulson@14208
   540
apply (induct xs, force)
paulson@14208
   541
apply (case_tac ys, simp, force)
nipkow@13145
   542
done
wenzelm@13114
   543
wenzelm@13366
   544
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   545
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
nipkow@13145
   546
apply(subst rev_rev_ident[symmetric])
nipkow@13145
   547
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   548
done
wenzelm@13114
   549
nipkow@13145
   550
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114
   551
wenzelm@13366
   552
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   553
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   554
by (induct xs rule: rev_induct) auto
wenzelm@13114
   555
wenzelm@13366
   556
lemmas rev_cases = rev_exhaust
wenzelm@13366
   557
wenzelm@13114
   558
wenzelm@13142
   559
subsection {* @{text set} *}
wenzelm@13114
   560
wenzelm@13142
   561
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   562
by (induct xs) auto
wenzelm@13114
   563
wenzelm@13142
   564
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   565
by (induct xs) auto
wenzelm@13114
   566
oheimb@14099
   567
lemma hd_in_set: "l = x#xs \<Longrightarrow> x\<in>set l"
paulson@14208
   568
by (case_tac l, auto)
oheimb@14099
   569
wenzelm@13142
   570
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   571
by auto
wenzelm@13114
   572
oheimb@14099
   573
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   574
by auto
oheimb@14099
   575
wenzelm@13142
   576
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   577
by (induct xs) auto
wenzelm@13114
   578
nipkow@15245
   579
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   580
by(induct xs) auto
nipkow@15245
   581
wenzelm@13142
   582
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   583
by (induct xs) auto
wenzelm@13114
   584
wenzelm@13142
   585
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   586
by (induct xs) auto
wenzelm@13114
   587
wenzelm@13142
   588
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   589
by (induct xs) auto
wenzelm@13114
   590
wenzelm@13142
   591
lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> k < j}"
paulson@14208
   592
apply (induct j, simp_all)
paulson@14208
   593
apply (erule ssubst, auto)
nipkow@13145
   594
done
wenzelm@13114
   595
wenzelm@13142
   596
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
paulson@15113
   597
proof (induct xs)
paulson@15113
   598
  case Nil show ?case by simp
paulson@15113
   599
  case (Cons a xs)
paulson@15113
   600
  show ?case
paulson@15113
   601
  proof 
paulson@15113
   602
    assume "x \<in> set (a # xs)"
paulson@15113
   603
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   604
      by (simp, blast intro: Cons_eq_appendI)
paulson@15113
   605
  next
paulson@15113
   606
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   607
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
paulson@15113
   608
    show "x \<in> set (a # xs)" 
paulson@15113
   609
      by (cases ys, auto simp add: eq)
paulson@15113
   610
  qed
paulson@15113
   611
qed
wenzelm@13142
   612
paulson@13508
   613
lemma finite_list: "finite A ==> EX l. set l = A"
paulson@13508
   614
apply (erule finite_induct, auto)
paulson@13508
   615
apply (rule_tac x="x#l" in exI, auto)
paulson@13508
   616
done
paulson@13508
   617
kleing@14388
   618
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
   619
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
   620
paulson@15168
   621
wenzelm@13142
   622
subsection {* @{text mem} *}
wenzelm@13114
   623
nipkow@15302
   624
text{* Only use @{text mem} for generating executable code.  Otherwise
nipkow@15302
   625
use @{prop"x : set xs"} instead --- it is much easier to reason
nipkow@15302
   626
about. *}
nipkow@15302
   627
wenzelm@13114
   628
lemma set_mem_eq: "(x mem xs) = (x : set xs)"
nipkow@13145
   629
by (induct xs) auto
wenzelm@13114
   630
wenzelm@13114
   631
wenzelm@13142
   632
subsection {* @{text list_all} *}
wenzelm@13114
   633
wenzelm@13142
   634
lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)"
nipkow@13145
   635
by (induct xs) auto
wenzelm@13114
   636
wenzelm@13142
   637
lemma list_all_append [simp]:
nipkow@13145
   638
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
nipkow@13145
   639
by (induct xs) auto
wenzelm@13114
   640
wenzelm@13114
   641
wenzelm@13142
   642
subsection {* @{text filter} *}
wenzelm@13114
   643
wenzelm@13142
   644
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   645
by (induct xs) auto
wenzelm@13114
   646
wenzelm@13142
   647
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   648
by (induct xs) auto
wenzelm@13114
   649
wenzelm@13142
   650
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   651
by (induct xs) auto
wenzelm@13114
   652
wenzelm@13142
   653
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   654
by (induct xs) auto
wenzelm@13114
   655
nipkow@15246
   656
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@13145
   657
by (induct xs) (auto simp add: le_SucI)
wenzelm@13114
   658
wenzelm@13142
   659
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   660
by auto
wenzelm@13114
   661
nipkow@15246
   662
lemma length_filter_less:
nipkow@15246
   663
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
   664
proof (induct xs)
nipkow@15246
   665
  case Nil thus ?case by simp
nipkow@15246
   666
next
nipkow@15246
   667
  case (Cons x xs) thus ?case
nipkow@15246
   668
    apply (auto split:split_if_asm)
nipkow@15246
   669
    using length_filter_le[of P xs] apply arith
nipkow@15246
   670
  done
nipkow@15246
   671
qed
wenzelm@13114
   672
nipkow@15281
   673
lemma length_filter_conv_card:
nipkow@15281
   674
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
   675
proof (induct xs)
nipkow@15281
   676
  case Nil thus ?case by simp
nipkow@15281
   677
next
nipkow@15281
   678
  case (Cons x xs)
nipkow@15281
   679
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
   680
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
   681
  show ?case (is "?l = card ?S'")
nipkow@15281
   682
  proof (cases)
nipkow@15281
   683
    assume "p x"
nipkow@15281
   684
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@15281
   685
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   686
    have "length (filter p (x # xs)) = Suc(card ?S)"
nipkow@15281
   687
      using Cons by simp
nipkow@15281
   688
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
nipkow@15281
   689
      by (simp add: card_image inj_Suc)
nipkow@15281
   690
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   691
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
   692
    finally show ?thesis .
nipkow@15281
   693
  next
nipkow@15281
   694
    assume "\<not> p x"
nipkow@15281
   695
    hence eq: "?S' = Suc ` ?S"
nipkow@15281
   696
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   697
    have "length (filter p (x # xs)) = card ?S"
nipkow@15281
   698
      using Cons by simp
nipkow@15281
   699
    also have "\<dots> = card(Suc ` ?S)" using fin
nipkow@15281
   700
      by (simp add: card_image inj_Suc)
nipkow@15281
   701
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   702
      by (simp add:card_insert_if)
nipkow@15281
   703
    finally show ?thesis .
nipkow@15281
   704
  qed
nipkow@15281
   705
qed
nipkow@15281
   706
nipkow@15281
   707
wenzelm@13142
   708
subsection {* @{text concat} *}
wenzelm@13114
   709
wenzelm@13142
   710
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
   711
by (induct xs) auto
wenzelm@13114
   712
wenzelm@13142
   713
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   714
by (induct xss) auto
wenzelm@13114
   715
wenzelm@13142
   716
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   717
by (induct xss) auto
wenzelm@13114
   718
wenzelm@13142
   719
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
nipkow@13145
   720
by (induct xs) auto
wenzelm@13114
   721
wenzelm@13142
   722
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
   723
by (induct xs) auto
wenzelm@13114
   724
wenzelm@13142
   725
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
   726
by (induct xs) auto
wenzelm@13114
   727
wenzelm@13142
   728
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
   729
by (induct xs) auto
wenzelm@13114
   730
wenzelm@13114
   731
wenzelm@13142
   732
subsection {* @{text nth} *}
wenzelm@13114
   733
wenzelm@13142
   734
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
   735
by auto
wenzelm@13114
   736
wenzelm@13142
   737
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
   738
by auto
wenzelm@13114
   739
wenzelm@13142
   740
declare nth.simps [simp del]
wenzelm@13114
   741
wenzelm@13114
   742
lemma nth_append:
nipkow@13145
   743
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
paulson@14208
   744
apply (induct "xs", simp)
paulson@14208
   745
apply (case_tac n, auto)
nipkow@13145
   746
done
wenzelm@13114
   747
nipkow@14402
   748
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
nipkow@14402
   749
by (induct "xs") auto
nipkow@14402
   750
nipkow@14402
   751
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
nipkow@14402
   752
by (induct "xs") auto
nipkow@14402
   753
wenzelm@13142
   754
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
paulson@14208
   755
apply (induct xs, simp)
paulson@14208
   756
apply (case_tac n, auto)
nipkow@13145
   757
done
wenzelm@13114
   758
wenzelm@13142
   759
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
   760
apply (induct xs, simp, simp)
nipkow@13145
   761
apply safe
paulson@14208
   762
apply (rule_tac x = 0 in exI, simp)
paulson@14208
   763
 apply (rule_tac x = "Suc i" in exI, simp)
paulson@14208
   764
apply (case_tac i, simp)
nipkow@13145
   765
apply (rename_tac j)
paulson@14208
   766
apply (rule_tac x = j in exI, simp)
nipkow@13145
   767
done
wenzelm@13114
   768
nipkow@13145
   769
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
   770
by (auto simp add: set_conv_nth)
wenzelm@13114
   771
wenzelm@13142
   772
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
   773
by (auto simp add: set_conv_nth)
wenzelm@13114
   774
wenzelm@13114
   775
lemma all_nth_imp_all_set:
nipkow@13145
   776
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
   777
by (auto simp add: set_conv_nth)
wenzelm@13114
   778
wenzelm@13114
   779
lemma all_set_conv_all_nth:
nipkow@13145
   780
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
   781
by (auto simp add: set_conv_nth)
wenzelm@13114
   782
wenzelm@13114
   783
wenzelm@13142
   784
subsection {* @{text list_update} *}
wenzelm@13114
   785
wenzelm@13142
   786
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
nipkow@13145
   787
by (induct xs) (auto split: nat.split)
wenzelm@13114
   788
wenzelm@13114
   789
lemma nth_list_update:
nipkow@13145
   790
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@13145
   791
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   792
wenzelm@13142
   793
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
   794
by (simp add: nth_list_update)
wenzelm@13114
   795
wenzelm@13142
   796
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145
   797
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   798
wenzelm@13142
   799
lemma list_update_overwrite [simp]:
nipkow@13145
   800
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145
   801
by (induct xs) (auto split: nat.split)
wenzelm@13114
   802
nipkow@14402
   803
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
paulson@14208
   804
apply (induct xs, simp)
nipkow@14187
   805
apply(simp split:nat.splits)
nipkow@14187
   806
done
nipkow@14187
   807
wenzelm@13114
   808
lemma list_update_same_conv:
nipkow@13145
   809
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145
   810
by (induct xs) (auto split: nat.split)
wenzelm@13114
   811
nipkow@14187
   812
lemma list_update_append1:
nipkow@14187
   813
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
paulson@14208
   814
apply (induct xs, simp)
nipkow@14187
   815
apply(simp split:nat.split)
nipkow@14187
   816
done
nipkow@14187
   817
nipkow@14402
   818
lemma list_update_length [simp]:
nipkow@14402
   819
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
   820
by (induct xs, auto)
nipkow@14402
   821
wenzelm@13114
   822
lemma update_zip:
nipkow@13145
   823
"!!i xy xs. length xs = length ys ==>
nipkow@13145
   824
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145
   825
by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114
   826
wenzelm@13114
   827
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145
   828
by (induct xs) (auto split: nat.split)
wenzelm@13114
   829
wenzelm@13114
   830
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
   831
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
   832
wenzelm@13114
   833
wenzelm@13142
   834
subsection {* @{text last} and @{text butlast} *}
wenzelm@13114
   835
wenzelm@13142
   836
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
   837
by (induct xs) auto
wenzelm@13114
   838
wenzelm@13142
   839
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
   840
by (induct xs) auto
wenzelm@13114
   841
nipkow@14302
   842
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
   843
by(simp add:last.simps)
nipkow@14302
   844
nipkow@14302
   845
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
   846
by(simp add:last.simps)
nipkow@14302
   847
nipkow@14302
   848
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
   849
by (induct xs) (auto)
nipkow@14302
   850
nipkow@14302
   851
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
   852
by(simp add:last_append)
nipkow@14302
   853
nipkow@14302
   854
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
   855
by(simp add:last_append)
nipkow@14302
   856
nipkow@14302
   857
wenzelm@13142
   858
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
   859
by (induct xs rule: rev_induct) auto
wenzelm@13114
   860
wenzelm@13114
   861
lemma butlast_append:
nipkow@13145
   862
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145
   863
by (induct xs) auto
wenzelm@13114
   864
wenzelm@13142
   865
lemma append_butlast_last_id [simp]:
nipkow@13145
   866
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
   867
by (induct xs) auto
wenzelm@13114
   868
wenzelm@13142
   869
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
   870
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
   871
wenzelm@13114
   872
lemma in_set_butlast_appendI:
nipkow@13145
   873
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
   874
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
   875
wenzelm@13142
   876
wenzelm@13142
   877
subsection {* @{text take} and @{text drop} *}
wenzelm@13114
   878
wenzelm@13142
   879
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
   880
by (induct xs) auto
wenzelm@13114
   881
wenzelm@13142
   882
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
   883
by (induct xs) auto
wenzelm@13114
   884
wenzelm@13142
   885
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
   886
by simp
wenzelm@13114
   887
wenzelm@13142
   888
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
   889
by simp
wenzelm@13114
   890
wenzelm@13142
   891
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
   892
nipkow@15110
   893
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
   894
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
   895
nipkow@14187
   896
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
   897
by(cases xs, simp_all)
nipkow@14187
   898
nipkow@14187
   899
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
nipkow@14187
   900
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@14187
   901
nipkow@14187
   902
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
paulson@14208
   903
apply (induct xs, simp)
nipkow@14187
   904
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
   905
done
nipkow@14187
   906
nipkow@13913
   907
lemma take_Suc_conv_app_nth:
nipkow@13913
   908
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
paulson@14208
   909
apply (induct xs, simp)
paulson@14208
   910
apply (case_tac i, auto)
nipkow@13913
   911
done
nipkow@13913
   912
mehta@14591
   913
lemma drop_Suc_conv_tl:
mehta@14591
   914
  "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
mehta@14591
   915
apply (induct xs, simp)
mehta@14591
   916
apply (case_tac i, auto)
mehta@14591
   917
done
mehta@14591
   918
wenzelm@13142
   919
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
nipkow@13145
   920
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   921
wenzelm@13142
   922
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
nipkow@13145
   923
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   924
wenzelm@13142
   925
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
nipkow@13145
   926
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   927
wenzelm@13142
   928
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
nipkow@13145
   929
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   930
wenzelm@13142
   931
lemma take_append [simp]:
nipkow@13145
   932
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@13145
   933
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   934
wenzelm@13142
   935
lemma drop_append [simp]:
nipkow@13145
   936
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@13145
   937
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   938
wenzelm@13142
   939
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
paulson@14208
   940
apply (induct m, auto)
paulson@14208
   941
apply (case_tac xs, auto)
nipkow@15236
   942
apply (case_tac n, auto)
nipkow@13145
   943
done
wenzelm@13114
   944
wenzelm@13142
   945
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
paulson@14208
   946
apply (induct m, auto)
paulson@14208
   947
apply (case_tac xs, auto)
nipkow@13145
   948
done
wenzelm@13114
   949
wenzelm@13114
   950
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
paulson@14208
   951
apply (induct m, auto)
paulson@14208
   952
apply (case_tac xs, auto)
nipkow@13145
   953
done
wenzelm@13114
   954
nipkow@14802
   955
lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@14802
   956
apply(induct xs)
nipkow@14802
   957
 apply simp
nipkow@14802
   958
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
   959
done
nipkow@14802
   960
wenzelm@13142
   961
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
paulson@14208
   962
apply (induct n, auto)
paulson@14208
   963
apply (case_tac xs, auto)
nipkow@13145
   964
done
wenzelm@13114
   965
nipkow@15110
   966
lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@15110
   967
apply(induct xs)
nipkow@15110
   968
 apply simp
nipkow@15110
   969
apply(simp add:take_Cons split:nat.split)
nipkow@15110
   970
done
nipkow@15110
   971
nipkow@15110
   972
lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"
nipkow@15110
   973
apply(induct xs)
nipkow@15110
   974
apply simp
nipkow@15110
   975
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
   976
done
nipkow@15110
   977
wenzelm@13114
   978
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
paulson@14208
   979
apply (induct n, auto)
paulson@14208
   980
apply (case_tac xs, auto)
nipkow@13145
   981
done
wenzelm@13114
   982
wenzelm@13142
   983
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
paulson@14208
   984
apply (induct n, auto)
paulson@14208
   985
apply (case_tac xs, auto)
nipkow@13145
   986
done
wenzelm@13114
   987
wenzelm@13114
   988
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
paulson@14208
   989
apply (induct xs, auto)
paulson@14208
   990
apply (case_tac i, auto)
nipkow@13145
   991
done
wenzelm@13114
   992
wenzelm@13114
   993
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
paulson@14208
   994
apply (induct xs, auto)
paulson@14208
   995
apply (case_tac i, auto)
nipkow@13145
   996
done
wenzelm@13114
   997
wenzelm@13142
   998
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
paulson@14208
   999
apply (induct xs, auto)
paulson@14208
  1000
apply (case_tac n, blast)
paulson@14208
  1001
apply (case_tac i, auto)
nipkow@13145
  1002
done
wenzelm@13114
  1003
wenzelm@13142
  1004
lemma nth_drop [simp]:
nipkow@13145
  1005
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
paulson@14208
  1006
apply (induct n, auto)
paulson@14208
  1007
apply (case_tac xs, auto)
nipkow@13145
  1008
done
nipkow@3507
  1009
nipkow@14025
  1010
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
nipkow@14025
  1011
by(induct xs)(auto simp:take_Cons split:nat.split)
nipkow@14025
  1012
nipkow@14025
  1013
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
nipkow@14025
  1014
by(induct xs)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1015
nipkow@14187
  1016
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1017
using set_take_subset by fast
nipkow@14187
  1018
nipkow@14187
  1019
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1020
using set_drop_subset by fast
nipkow@14187
  1021
wenzelm@13114
  1022
lemma append_eq_conv_conj:
nipkow@13145
  1023
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
paulson@14208
  1024
apply (induct xs, simp, clarsimp)
paulson@14208
  1025
apply (case_tac zs, auto)
nipkow@13145
  1026
done
wenzelm@13142
  1027
paulson@14050
  1028
lemma take_add [rule_format]: 
paulson@14050
  1029
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
paulson@14050
  1030
apply (induct xs, auto) 
paulson@14050
  1031
apply (case_tac i, simp_all) 
paulson@14050
  1032
done
paulson@14050
  1033
nipkow@14300
  1034
lemma append_eq_append_conv_if:
nipkow@14300
  1035
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1036
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1037
   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
  1038
   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
  1039
apply(induct xs\<^isub>1)
nipkow@14300
  1040
 apply simp
nipkow@14300
  1041
apply(case_tac ys\<^isub>1)
nipkow@14300
  1042
apply simp_all
nipkow@14300
  1043
done
nipkow@14300
  1044
nipkow@15110
  1045
lemma take_hd_drop:
nipkow@15110
  1046
  "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
nipkow@15110
  1047
apply(induct xs)
nipkow@15110
  1048
apply simp
nipkow@15110
  1049
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1050
done
nipkow@15110
  1051
wenzelm@13114
  1052
wenzelm@13142
  1053
subsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1054
wenzelm@13142
  1055
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1056
by (induct xs) auto
wenzelm@13114
  1057
wenzelm@13142
  1058
lemma takeWhile_append1 [simp]:
nipkow@13145
  1059
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1060
by (induct xs) auto
wenzelm@13114
  1061
wenzelm@13142
  1062
lemma takeWhile_append2 [simp]:
nipkow@13145
  1063
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1064
by (induct xs) auto
wenzelm@13114
  1065
wenzelm@13142
  1066
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1067
by (induct xs) auto
wenzelm@13114
  1068
wenzelm@13142
  1069
lemma dropWhile_append1 [simp]:
nipkow@13145
  1070
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1071
by (induct xs) auto
wenzelm@13114
  1072
wenzelm@13142
  1073
lemma dropWhile_append2 [simp]:
nipkow@13145
  1074
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1075
by (induct xs) auto
wenzelm@13114
  1076
wenzelm@13142
  1077
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1078
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1079
nipkow@13913
  1080
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1081
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1082
by(induct xs, auto)
nipkow@13913
  1083
nipkow@13913
  1084
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1085
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1086
by(induct xs, auto)
nipkow@13913
  1087
nipkow@13913
  1088
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1089
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1090
by(induct xs, auto)
nipkow@13913
  1091
wenzelm@13114
  1092
wenzelm@13142
  1093
subsection {* @{text zip} *}
wenzelm@13114
  1094
wenzelm@13142
  1095
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1096
by (induct ys) auto
wenzelm@13114
  1097
wenzelm@13142
  1098
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1099
by simp
wenzelm@13114
  1100
wenzelm@13142
  1101
declare zip_Cons [simp del]
wenzelm@13114
  1102
nipkow@15281
  1103
lemma zip_Cons1:
nipkow@15281
  1104
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  1105
by(auto split:list.split)
nipkow@15281
  1106
wenzelm@13142
  1107
lemma length_zip [simp]:
nipkow@13145
  1108
"!!xs. length (zip xs ys) = min (length xs) (length ys)"
paulson@14208
  1109
apply (induct ys, simp)
paulson@14208
  1110
apply (case_tac xs, auto)
nipkow@13145
  1111
done
wenzelm@13114
  1112
wenzelm@13114
  1113
lemma zip_append1:
nipkow@13145
  1114
"!!xs. zip (xs @ ys) zs =
nipkow@13145
  1115
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
paulson@14208
  1116
apply (induct zs, simp)
paulson@14208
  1117
apply (case_tac xs, simp_all)
nipkow@13145
  1118
done
wenzelm@13114
  1119
wenzelm@13114
  1120
lemma zip_append2:
nipkow@13145
  1121
"!!ys. zip xs (ys @ zs) =
nipkow@13145
  1122
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
paulson@14208
  1123
apply (induct xs, simp)
paulson@14208
  1124
apply (case_tac ys, simp_all)
nipkow@13145
  1125
done
wenzelm@13114
  1126
wenzelm@13142
  1127
lemma zip_append [simp]:
wenzelm@13142
  1128
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1129
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1130
by (simp add: zip_append1)
wenzelm@13114
  1131
wenzelm@13114
  1132
lemma zip_rev:
nipkow@14247
  1133
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1134
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1135
wenzelm@13142
  1136
lemma nth_zip [simp]:
nipkow@13145
  1137
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
paulson@14208
  1138
apply (induct ys, simp)
nipkow@13145
  1139
apply (case_tac xs)
nipkow@13145
  1140
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1141
done
wenzelm@13114
  1142
wenzelm@13114
  1143
lemma set_zip:
nipkow@13145
  1144
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
  1145
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1146
wenzelm@13114
  1147
lemma zip_update:
nipkow@13145
  1148
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
  1149
by (rule sym, simp add: update_zip)
wenzelm@13114
  1150
wenzelm@13142
  1151
lemma zip_replicate [simp]:
nipkow@13145
  1152
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
paulson@14208
  1153
apply (induct i, auto)
paulson@14208
  1154
apply (case_tac j, auto)
nipkow@13145
  1155
done
wenzelm@13114
  1156
wenzelm@13142
  1157
wenzelm@13142
  1158
subsection {* @{text list_all2} *}
wenzelm@13114
  1159
kleing@14316
  1160
lemma list_all2_lengthD [intro?]: 
kleing@14316
  1161
  "list_all2 P xs ys ==> length xs = length ys"
nipkow@13145
  1162
by (simp add: list_all2_def)
wenzelm@13114
  1163
wenzelm@13142
  1164
lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])"
nipkow@13145
  1165
by (simp add: list_all2_def)
wenzelm@13114
  1166
wenzelm@13142
  1167
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"
nipkow@13145
  1168
by (simp add: list_all2_def)
wenzelm@13114
  1169
wenzelm@13142
  1170
lemma list_all2_Cons [iff]:
nipkow@13145
  1171
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@13145
  1172
by (auto simp add: list_all2_def)
wenzelm@13114
  1173
wenzelm@13114
  1174
lemma list_all2_Cons1:
nipkow@13145
  1175
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1176
by (cases ys) auto
wenzelm@13114
  1177
wenzelm@13114
  1178
lemma list_all2_Cons2:
nipkow@13145
  1179
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1180
by (cases xs) auto
wenzelm@13114
  1181
wenzelm@13142
  1182
lemma list_all2_rev [iff]:
nipkow@13145
  1183
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1184
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1185
kleing@13863
  1186
lemma list_all2_rev1:
kleing@13863
  1187
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1188
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1189
wenzelm@13114
  1190
lemma list_all2_append1:
nipkow@13145
  1191
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1192
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1193
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1194
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1195
apply (rule iffI)
nipkow@13145
  1196
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1197
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  1198
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1199
apply (simp add: ball_Un)
nipkow@13145
  1200
done
wenzelm@13114
  1201
wenzelm@13114
  1202
lemma list_all2_append2:
nipkow@13145
  1203
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1204
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1205
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1206
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1207
apply (rule iffI)
nipkow@13145
  1208
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1209
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  1210
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1211
apply (simp add: ball_Un)
nipkow@13145
  1212
done
wenzelm@13114
  1213
kleing@13863
  1214
lemma list_all2_append:
nipkow@14247
  1215
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  1216
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  1217
by (induct rule:list_induct2, simp_all)
kleing@13863
  1218
kleing@13863
  1219
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1220
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
kleing@13863
  1221
  by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1222
wenzelm@13114
  1223
lemma list_all2_conv_all_nth:
nipkow@13145
  1224
"list_all2 P xs ys =
nipkow@13145
  1225
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1226
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1227
berghofe@13883
  1228
lemma list_all2_trans:
berghofe@13883
  1229
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1230
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1231
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1232
proof (induct as)
berghofe@13883
  1233
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1234
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1235
  proof (induct bs)
berghofe@13883
  1236
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1237
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1238
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1239
  qed simp
berghofe@13883
  1240
qed simp
berghofe@13883
  1241
kleing@13863
  1242
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1243
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
kleing@13863
  1244
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1245
paulson@14395
  1246
lemma list_all2I:
paulson@14395
  1247
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
paulson@14395
  1248
  by (simp add: list_all2_def)
paulson@14395
  1249
kleing@14328
  1250
lemma list_all2_nthD:
kleing@13863
  1251
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
kleing@13863
  1252
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1253
nipkow@14302
  1254
lemma list_all2_nthD2:
nipkow@14302
  1255
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@14302
  1256
  by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  1257
kleing@13863
  1258
lemma list_all2_map1: 
kleing@13863
  1259
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
kleing@13863
  1260
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1261
kleing@13863
  1262
lemma list_all2_map2: 
kleing@13863
  1263
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
kleing@13863
  1264
  by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1265
kleing@14316
  1266
lemma list_all2_refl [intro?]:
kleing@13863
  1267
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
kleing@13863
  1268
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1269
kleing@13863
  1270
lemma list_all2_update_cong:
kleing@13863
  1271
  "\<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
  1272
  by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1273
kleing@13863
  1274
lemma list_all2_update_cong2:
kleing@13863
  1275
  "\<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
  1276
  by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1277
nipkow@14302
  1278
lemma list_all2_takeI [simp,intro?]:
nipkow@14302
  1279
  "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@14302
  1280
  apply (induct xs)
nipkow@14302
  1281
   apply simp
nipkow@14302
  1282
  apply (clarsimp simp add: list_all2_Cons1)
nipkow@14302
  1283
  apply (case_tac n)
nipkow@14302
  1284
  apply auto
nipkow@14302
  1285
  done
nipkow@14302
  1286
nipkow@14302
  1287
lemma list_all2_dropI [simp,intro?]:
kleing@13863
  1288
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
paulson@14208
  1289
  apply (induct as, simp)
kleing@13863
  1290
  apply (clarsimp simp add: list_all2_Cons1)
paulson@14208
  1291
  apply (case_tac n, simp, simp)
kleing@13863
  1292
  done
kleing@13863
  1293
kleing@14327
  1294
lemma list_all2_mono [intro?]:
kleing@13863
  1295
  "\<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
  1296
  apply (induct x, simp)
paulson@14208
  1297
  apply (case_tac y, auto)
kleing@13863
  1298
  done
kleing@13863
  1299
wenzelm@13142
  1300
nipkow@14402
  1301
subsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  1302
wenzelm@13142
  1303
lemma foldl_append [simp]:
nipkow@13145
  1304
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145
  1305
by (induct xs) auto
wenzelm@13142
  1306
nipkow@14402
  1307
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  1308
by (induct xs) auto
nipkow@14402
  1309
nipkow@14402
  1310
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  1311
by (induct xs) auto
nipkow@14402
  1312
nipkow@14402
  1313
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  1314
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  1315
wenzelm@13142
  1316
text {*
nipkow@13145
  1317
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1318
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1319
*}
wenzelm@13142
  1320
wenzelm@13142
  1321
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
nipkow@13145
  1322
by (induct ns) auto
wenzelm@13142
  1323
wenzelm@13142
  1324
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1325
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1326
wenzelm@13142
  1327
lemma sum_eq_0_conv [iff]:
nipkow@13145
  1328
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145
  1329
by (induct ns) auto
wenzelm@13114
  1330
wenzelm@13114
  1331
wenzelm@13142
  1332
subsection {* @{text upto} *}
wenzelm@13114
  1333
wenzelm@13142
  1334
lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])"
nipkow@13145
  1335
-- {* Does not terminate! *}
nipkow@13145
  1336
by (induct j) auto
wenzelm@13142
  1337
wenzelm@13142
  1338
lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []"
nipkow@13145
  1339
by (subst upt_rec) simp
wenzelm@13114
  1340
nipkow@15281
  1341
lemma upt_eq_Nil_conv[simp]: "([i..j(] = []) = (j = 0 \<or> j <= i)"
nipkow@15281
  1342
by(induct j)simp_all
nipkow@15281
  1343
nipkow@15281
  1344
lemma upt_eq_Cons_conv:
nipkow@15281
  1345
 "!!x xs. ([i..j(] = x#xs) = (i < j & i = x & [i+1..j(] = xs)"
nipkow@15281
  1346
apply(induct j)
nipkow@15281
  1347
 apply simp
nipkow@15281
  1348
apply(clarsimp simp add: append_eq_Cons_conv)
nipkow@15281
  1349
apply arith
nipkow@15281
  1350
done
nipkow@15281
  1351
wenzelm@13142
  1352
lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]"
nipkow@13145
  1353
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1354
by simp
wenzelm@13114
  1355
wenzelm@13142
  1356
lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]"
nipkow@13145
  1357
apply(rule trans)
nipkow@13145
  1358
apply(subst upt_rec)
paulson@14208
  1359
 prefer 2 apply (rule refl, simp)
nipkow@13145
  1360
done
wenzelm@13114
  1361
wenzelm@13142
  1362
lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]"
nipkow@13145
  1363
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1364
by (induct k) auto
wenzelm@13114
  1365
wenzelm@13142
  1366
lemma length_upt [simp]: "length [i..j(] = j - i"
nipkow@13145
  1367
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1368
wenzelm@13142
  1369
lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k"
nipkow@13145
  1370
apply (induct j)
nipkow@13145
  1371
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1372
done
wenzelm@13114
  1373
wenzelm@13142
  1374
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]"
paulson@14208
  1375
apply (induct m, simp)
nipkow@13145
  1376
apply (subst upt_rec)
nipkow@13145
  1377
apply (rule sym)
nipkow@13145
  1378
apply (subst upt_rec)
nipkow@13145
  1379
apply (simp del: upt.simps)
nipkow@13145
  1380
done
nipkow@3507
  1381
wenzelm@13114
  1382
lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]"
nipkow@13145
  1383
by (induct n) auto
wenzelm@13114
  1384
wenzelm@13114
  1385
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..n(]) ! i = f(m+i)"
nipkow@13145
  1386
apply (induct n m rule: diff_induct)
nipkow@13145
  1387
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1388
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1389
done
wenzelm@13114
  1390
berghofe@13883
  1391
lemma nth_take_lemma:
berghofe@13883
  1392
  "!!xs ys. k <= length xs ==> k <= length ys ==>
berghofe@13883
  1393
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
berghofe@13883
  1394
apply (atomize, induct k)
paulson@14208
  1395
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  1396
txt {* Both lists must be non-empty *}
paulson@14208
  1397
apply (case_tac xs, simp)
paulson@14208
  1398
apply (case_tac ys, clarify)
nipkow@13145
  1399
 apply (simp (no_asm_use))
nipkow@13145
  1400
apply clarify
nipkow@13145
  1401
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  1402
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  1403
apply blast
nipkow@13145
  1404
done
wenzelm@13114
  1405
wenzelm@13114
  1406
lemma nth_equalityI:
wenzelm@13114
  1407
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  1408
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  1409
apply (simp_all add: take_all)
nipkow@13145
  1410
done
wenzelm@13142
  1411
kleing@13863
  1412
(* needs nth_equalityI *)
kleing@13863
  1413
lemma list_all2_antisym:
kleing@13863
  1414
  "\<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
  1415
  \<Longrightarrow> xs = ys"
kleing@13863
  1416
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  1417
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  1418
  done
kleing@13863
  1419
wenzelm@13142
  1420
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  1421
-- {* The famous take-lemma. *}
nipkow@13145
  1422
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  1423
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  1424
done
wenzelm@13142
  1425
wenzelm@13142
  1426
nipkow@15302
  1427
lemma take_Cons':
nipkow@15302
  1428
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@15302
  1429
by (cases n) simp_all
nipkow@15302
  1430
nipkow@15302
  1431
lemma drop_Cons':
nipkow@15302
  1432
     "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@15302
  1433
by (cases n) simp_all
nipkow@15302
  1434
nipkow@15302
  1435
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@15302
  1436
by (cases n) simp_all
nipkow@15302
  1437
nipkow@15302
  1438
lemmas [simp] = take_Cons'[of "number_of v",standard]
nipkow@15302
  1439
                drop_Cons'[of "number_of v",standard]
nipkow@15302
  1440
                nth_Cons'[of _ _ "number_of v",standard]
nipkow@15302
  1441
nipkow@15302
  1442
wenzelm@13142
  1443
subsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  1444
wenzelm@13142
  1445
lemma distinct_append [simp]:
nipkow@13145
  1446
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  1447
by (induct xs) auto
wenzelm@13142
  1448
wenzelm@13142
  1449
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  1450
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  1451
wenzelm@13142
  1452
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  1453
by (induct xs) auto
wenzelm@13142
  1454
paulson@15072
  1455
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
paulson@15251
  1456
  by (induct x, auto) 
paulson@15072
  1457
paulson@15072
  1458
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
paulson@15251
  1459
  by (induct x, auto)
paulson@15072
  1460
nipkow@15245
  1461
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245
  1462
by (induct xs) auto
nipkow@15245
  1463
nipkow@15245
  1464
lemma length_remdups_eq[iff]:
nipkow@15245
  1465
  "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245
  1466
apply(induct xs)
nipkow@15245
  1467
 apply auto
nipkow@15245
  1468
apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245
  1469
 apply arith
nipkow@15245
  1470
apply(rule length_remdups_leq)
nipkow@15245
  1471
done
nipkow@15245
  1472
wenzelm@13142
  1473
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  1474
by (induct xs) auto
wenzelm@13114
  1475
wenzelm@13142
  1476
text {*
nipkow@13145
  1477
It is best to avoid this indexed version of distinct, but sometimes
nipkow@13145
  1478
it is useful. *}
wenzelm@13142
  1479
lemma distinct_conv_nth:
nipkow@13145
  1480
"distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@15251
  1481
apply (induct xs, simp, simp)
paulson@14208
  1482
apply (rule iffI, clarsimp)
nipkow@13145
  1483
 apply (case_tac i)
paulson@14208
  1484
apply (case_tac j, simp)
nipkow@13145
  1485
apply (simp add: set_conv_nth)
nipkow@13145
  1486
 apply (case_tac j)
paulson@14208
  1487
apply (clarsimp simp add: set_conv_nth, simp)
nipkow@13145
  1488
apply (rule conjI)
nipkow@13145
  1489
 apply (clarsimp simp add: set_conv_nth)
nipkow@13145
  1490
 apply (erule_tac x = 0 in allE)
paulson@14208
  1491
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
nipkow@13145
  1492
apply (erule_tac x = "Suc i" in allE)
paulson@14208
  1493
apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145
  1494
done
wenzelm@13114
  1495
nipkow@15110
  1496
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
kleing@14388
  1497
  by (induct xs) auto
kleing@14388
  1498
nipkow@15110
  1499
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388
  1500
proof (induct xs)
kleing@14388
  1501
  case Nil thus ?case by simp
kleing@14388
  1502
next
kleing@14388
  1503
  case (Cons x xs)
kleing@14388
  1504
  show ?case
kleing@14388
  1505
  proof (cases "x \<in> set xs")
kleing@14388
  1506
    case False with Cons show ?thesis by simp
kleing@14388
  1507
  next
kleing@14388
  1508
    case True with Cons.prems
kleing@14388
  1509
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  1510
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  1511
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  1512
    ultimately have False by simp
kleing@14388
  1513
    thus ?thesis ..
kleing@14388
  1514
  qed
kleing@14388
  1515
qed
kleing@14388
  1516
nipkow@15110
  1517
lemma inj_on_setI: "distinct(map f xs) ==> inj_on f (set xs)"
nipkow@15110
  1518
apply(induct xs)
nipkow@15110
  1519
 apply simp
nipkow@15110
  1520
apply fastsimp
nipkow@15110
  1521
done
nipkow@15110
  1522
nipkow@15110
  1523
lemma inj_on_set_conv:
nipkow@15110
  1524
 "distinct xs \<Longrightarrow> inj_on f (set xs) = distinct(map f xs)"
nipkow@15110
  1525
apply(induct xs)
nipkow@15110
  1526
 apply simp
nipkow@15110
  1527
apply fastsimp
nipkow@15110
  1528
done
nipkow@15110
  1529
nipkow@15110
  1530
nipkow@15110
  1531
subsection {* @{text remove1} *}
nipkow@15110
  1532
nipkow@15110
  1533
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110
  1534
apply(induct xs)
nipkow@15110
  1535
 apply simp
nipkow@15110
  1536
apply simp
nipkow@15110
  1537
apply blast
nipkow@15110
  1538
done
nipkow@15110
  1539
nipkow@15110
  1540
lemma [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110
  1541
apply(induct xs)
nipkow@15110
  1542
 apply simp
nipkow@15110
  1543
apply simp
nipkow@15110
  1544
apply blast
nipkow@15110
  1545
done
nipkow@15110
  1546
nipkow@15110
  1547
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110
  1548
apply(insert set_remove1_subset)
nipkow@15110
  1549
apply fast
nipkow@15110
  1550
done
nipkow@15110
  1551
nipkow@15110
  1552
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110
  1553
by (induct xs) simp_all
nipkow@15110
  1554
wenzelm@13114
  1555
wenzelm@13142
  1556
subsection {* @{text replicate} *}
wenzelm@13114
  1557
wenzelm@13142
  1558
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  1559
by (induct n) auto
nipkow@13124
  1560
wenzelm@13142
  1561
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  1562
by (induct n) auto
wenzelm@13114
  1563
wenzelm@13114
  1564
lemma replicate_app_Cons_same:
nipkow@13145
  1565
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  1566
by (induct n) auto
wenzelm@13114
  1567
wenzelm@13142
  1568
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208
  1569
apply (induct n, simp)
nipkow@13145
  1570
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  1571
done
wenzelm@13114
  1572
wenzelm@13142
  1573
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  1574
by (induct n) auto
wenzelm@13114
  1575
wenzelm@13142
  1576
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  1577
by (induct n) auto
wenzelm@13114
  1578
wenzelm@13142
  1579
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  1580
by (induct n) auto
wenzelm@13114
  1581
wenzelm@13142
  1582
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  1583
by (atomize (full), induct n) auto
wenzelm@13114
  1584
wenzelm@13142
  1585
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
paulson@14208
  1586
apply (induct n, simp)
nipkow@13145
  1587
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  1588
done
wenzelm@13114
  1589
wenzelm@13142
  1590
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  1591
by (induct n) auto
wenzelm@13114
  1592
wenzelm@13142
  1593
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  1594
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  1595
wenzelm@13142
  1596
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  1597
by auto
wenzelm@13114
  1598
wenzelm@13142
  1599
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  1600
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  1601
wenzelm@13114
  1602
nipkow@15302
  1603
subsection{*@{text rotate1} and @{text rotate}*}
nipkow@15302
  1604
nipkow@15302
  1605
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
nipkow@15302
  1606
by(simp add:rotate1_def)
nipkow@15302
  1607
nipkow@15302
  1608
lemma rotate0[simp]: "rotate 0 = id"
nipkow@15302
  1609
by(simp add:rotate_def)
nipkow@15302
  1610
nipkow@15302
  1611
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
nipkow@15302
  1612
by(simp add:rotate_def)
nipkow@15302
  1613
nipkow@15302
  1614
lemma rotate_add:
nipkow@15302
  1615
  "rotate (m+n) = rotate m o rotate n"
nipkow@15302
  1616
by(simp add:rotate_def funpow_add)
nipkow@15302
  1617
nipkow@15302
  1618
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
nipkow@15302
  1619
by(simp add:rotate_add)
nipkow@15302
  1620
nipkow@15302
  1621
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
nipkow@15302
  1622
by(cases xs) simp_all
nipkow@15302
  1623
nipkow@15302
  1624
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  1625
apply(induct n)
nipkow@15302
  1626
 apply simp
nipkow@15302
  1627
apply (simp add:rotate_def)
nipkow@13145
  1628
done
wenzelm@13114
  1629
nipkow@15302
  1630
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
nipkow@15302
  1631
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1632
nipkow@15302
  1633
lemma rotate_drop_take:
nipkow@15302
  1634
  "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
nipkow@15302
  1635
apply(induct n)
nipkow@15302
  1636
 apply simp
nipkow@15302
  1637
apply(simp add:rotate_def)
nipkow@15302
  1638
apply(cases "xs = []")
nipkow@15302
  1639
 apply (simp)
nipkow@15302
  1640
apply(case_tac "n mod length xs = 0")
nipkow@15302
  1641
 apply(simp add:mod_Suc)
nipkow@15302
  1642
 apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
nipkow@15302
  1643
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
nipkow@15302
  1644
                take_hd_drop linorder_not_le)
nipkow@13145
  1645
done
wenzelm@13114
  1646
nipkow@15302
  1647
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
nipkow@15302
  1648
by(simp add:rotate_drop_take)
nipkow@15302
  1649
nipkow@15302
  1650
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  1651
by(simp add:rotate_drop_take)
nipkow@15302
  1652
nipkow@15302
  1653
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
nipkow@15302
  1654
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1655
nipkow@15302
  1656
lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs"
nipkow@15302
  1657
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  1658
nipkow@15302
  1659
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
nipkow@15302
  1660
by(simp add:rotate1_def split:list.split) blast
nipkow@15302
  1661
nipkow@15302
  1662
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
nipkow@15302
  1663
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  1664
nipkow@15302
  1665
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
nipkow@15302
  1666
by(simp add:rotate_drop_take take_map drop_map)
nipkow@15302
  1667
nipkow@15302
  1668
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
nipkow@15302
  1669
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1670
nipkow@15302
  1671
lemma set_rotate[simp]: "set(rotate n xs) = set xs"
nipkow@15302
  1672
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  1673
nipkow@15302
  1674
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
nipkow@15302
  1675
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1676
nipkow@15302
  1677
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
nipkow@15302
  1678
by (induct n) (simp_all add:rotate_def)
wenzelm@13114
  1679
wenzelm@13114
  1680
wenzelm@13142
  1681
subsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  1682
wenzelm@13142
  1683
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  1684
by (auto simp add: sublist_def)
wenzelm@13114
  1685
wenzelm@13142
  1686
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  1687
by (auto simp add: sublist_def)
wenzelm@13114
  1688
nipkow@15281
  1689
lemma length_sublist:
nipkow@15281
  1690
  "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
nipkow@15281
  1691
by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
nipkow@15281
  1692
nipkow@15281
  1693
lemma sublist_shift_lemma_Suc:
nipkow@15281
  1694
  "!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
nipkow@15281
  1695
         map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
nipkow@15281
  1696
apply(induct xs)
nipkow@15281
  1697
 apply simp
nipkow@15281
  1698
apply (case_tac "is")
nipkow@15281
  1699
 apply simp
nipkow@15281
  1700
apply simp
nipkow@15281
  1701
done
nipkow@15281
  1702
wenzelm@13114
  1703
lemma sublist_shift_lemma:
paulson@15168
  1704
     "map fst [p:zip xs [i..i + length xs(] . snd p : A] =
paulson@15168
  1705
      map fst [p:zip xs [0..length xs(] . snd p + i : A]"
nipkow@13145
  1706
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  1707
wenzelm@13114
  1708
lemma sublist_append:
paulson@15168
  1709
     "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  1710
apply (unfold sublist_def)
paulson@14208
  1711
apply (induct l' rule: rev_induct, simp)
nipkow@13145
  1712
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  1713
apply (simp add: add_commute)
nipkow@13145
  1714
done
wenzelm@13114
  1715
wenzelm@13114
  1716
lemma sublist_Cons:
nipkow@13145
  1717
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  1718
apply (induct l rule: rev_induct)
nipkow@13145
  1719
 apply (simp add: sublist_def)
nipkow@13145
  1720
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  1721
done
wenzelm@13114
  1722
nipkow@15281
  1723
lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
nipkow@15281
  1724
apply(induct xs)
nipkow@15281
  1725
 apply simp
nipkow@15281
  1726
apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE)
nipkow@15281
  1727
 apply(erule lessE)
nipkow@15281
  1728
  apply auto
nipkow@15281
  1729
apply(erule lessE)
nipkow@15281
  1730
apply auto
nipkow@15281
  1731
done
nipkow@15281
  1732
nipkow@15281
  1733
lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
nipkow@15281
  1734
by(auto simp add:set_sublist)
nipkow@15281
  1735
nipkow@15281
  1736
lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
nipkow@15281
  1737
by(auto simp add:set_sublist)
nipkow@15281
  1738
nipkow@15281
  1739
lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
nipkow@15281
  1740
by(auto simp add:set_sublist)
nipkow@15281
  1741
wenzelm@13142
  1742
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  1743
by (simp add: sublist_Cons)
wenzelm@13114
  1744
nipkow@15281
  1745
nipkow@15281
  1746
lemma distinct_sublistI[simp]: "!!I. distinct xs \<Longrightarrow> distinct(sublist xs I)"
nipkow@15281
  1747
apply(induct xs)
nipkow@15281
  1748
 apply simp
nipkow@15281
  1749
apply(auto simp add:sublist_Cons)
nipkow@15281
  1750
done
nipkow@15281
  1751
nipkow@15281
  1752
nipkow@15045
  1753
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
paulson@14208
  1754
apply (induct l rule: rev_induct, simp)
nipkow@13145
  1755
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  1756
done
wenzelm@13114
  1757
wenzelm@13114
  1758
nipkow@15302
  1759
subsection{*Sets of Lists*}
nipkow@15302
  1760
nipkow@15302
  1761
subsection {* @{text lists}: the list-forming operator over sets *}
nipkow@15302
  1762
nipkow@15302
  1763
consts lists :: "'a set => 'a list set"
nipkow@15302
  1764
inductive "lists A"
nipkow@15302
  1765
 intros
nipkow@15302
  1766
  Nil [intro!]: "[]: lists A"
nipkow@15302
  1767
  Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A"
nipkow@15302
  1768
nipkow@15302
  1769
inductive_cases listsE [elim!]: "x#l : lists A"
nipkow@15302
  1770
nipkow@15302
  1771
lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B"
nipkow@15302
  1772
by (unfold lists.defs) (blast intro!: lfp_mono)
nipkow@15302
  1773
nipkow@15302
  1774
lemma lists_IntI:
nipkow@15302
  1775
  assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l
nipkow@15302
  1776
  by induct blast+
nipkow@15302
  1777
nipkow@15302
  1778
lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B"
nipkow@15302
  1779
proof (rule mono_Int [THEN equalityI])
nipkow@15302
  1780
  show "mono lists" by (simp add: mono_def lists_mono)
nipkow@15302
  1781
  show "lists A \<inter> lists B \<subseteq> lists (A \<inter> B)" by (blast intro: lists_IntI)
kleing@14388
  1782
qed
kleing@14388
  1783
nipkow@15302
  1784
lemma append_in_lists_conv [iff]:
nipkow@15302
  1785
     "(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"
nipkow@15302
  1786
by (induct xs) auto
nipkow@15302
  1787
nipkow@15302
  1788
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"
nipkow@15302
  1789
-- {* eliminate @{text lists} in favour of @{text set} *}
nipkow@15302
  1790
by (induct xs) auto
nipkow@15302
  1791
nipkow@15302
  1792
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"
nipkow@15302
  1793
by (rule in_lists_conv_set [THEN iffD1])
nipkow@15302
  1794
nipkow@15302
  1795
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"
nipkow@15302
  1796
by (rule in_lists_conv_set [THEN iffD2])
nipkow@15302
  1797
nipkow@15302
  1798
lemma lists_UNIV [simp]: "lists UNIV = UNIV"
nipkow@15302
  1799
by auto
nipkow@15302
  1800
nipkow@15302
  1801
subsection{*Lists as Cartesian products*}
nipkow@15302
  1802
nipkow@15302
  1803
text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
nipkow@15302
  1804
@{term A} and tail drawn from @{term Xs}.*}
nipkow@15302
  1805
nipkow@15302
  1806
constdefs
nipkow@15302
  1807
  set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
nipkow@15302
  1808
  "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"
nipkow@15302
  1809
nipkow@15302
  1810
lemma [simp]: "set_Cons A {[]} = (%x. [x])`A"
nipkow@15302
  1811
by (auto simp add: set_Cons_def)
nipkow@15302
  1812
nipkow@15302
  1813
text{*Yields the set of lists, all of the same length as the argument and
nipkow@15302
  1814
with elements drawn from the corresponding element of the argument.*}
nipkow@15302
  1815
nipkow@15302
  1816
consts  listset :: "'a set list \<Rightarrow> 'a list set"
nipkow@15302
  1817
primrec
nipkow@15302
  1818
   "listset []    = {[]}"
nipkow@15302
  1819
   "listset(A#As) = set_Cons A (listset As)"
nipkow@15302
  1820
nipkow@15302
  1821
nipkow@15302
  1822
section{*Relations on lists*}
nipkow@15302
  1823
nipkow@15302
  1824
subsection {* Lexicographic orderings on lists *}
nipkow@15302
  1825
nipkow@15302
  1826
consts
nipkow@15302
  1827
lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
nipkow@15302
  1828
primrec
nipkow@15302
  1829
"lexn r 0 = {}"
nipkow@15302
  1830
"lexn r (Suc n) =
nipkow@15302
  1831
(prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
nipkow@15302
  1832
{(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
nipkow@15302
  1833
nipkow@15302
  1834
constdefs
nipkow@15302
  1835
lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@15302
  1836
"lex r == \<Union>n. lexn r n"
nipkow@15302
  1837
nipkow@15302
  1838
lexico :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@15302
  1839
"lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
nipkow@15302
  1840
nipkow@15302
  1841
nipkow@15302
  1842
lemma wf_lexn: "wf r ==> wf (lexn r n)"
nipkow@15302
  1843
apply (induct n, simp, simp)
nipkow@15302
  1844
apply(rule wf_subset)
nipkow@15302
  1845
 prefer 2 apply (rule Int_lower1)
nipkow@15302
  1846
apply(rule wf_prod_fun_image)
nipkow@15302
  1847
 prefer 2 apply (rule inj_onI, auto)
nipkow@15302
  1848
done
nipkow@15302
  1849
nipkow@15302
  1850
lemma lexn_length:
nipkow@15302
  1851
     "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@15302
  1852
by (induct n) auto
nipkow@15302
  1853
nipkow@15302
  1854
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@15302
  1855
apply (unfold lex_def)
nipkow@15302
  1856
apply (rule wf_UN)
nipkow@15302
  1857
apply (blast intro: wf_lexn, clarify)
nipkow@15302
  1858
apply (rename_tac m n)
nipkow@15302
  1859
apply (subgoal_tac "m \<noteq> n")
nipkow@15302
  1860
 prefer 2 apply blast
nipkow@15302
  1861
apply (blast dest: lexn_length not_sym)
nipkow@15302
  1862
done
nipkow@15302
  1863
nipkow@15302
  1864
lemma lexn_conv:
nipkow@15302
  1865
"lexn r n =
nipkow@15302
  1866
{(xs,ys). length xs = n \<and> length ys = n \<and>
nipkow@15302
  1867
(\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
nipkow@15302
  1868
apply (induct n, simp, blast)
nipkow@15302
  1869
apply (simp add: image_Collect lex_prod_def, safe, blast)
nipkow@15302
  1870
 apply (rule_tac x = "ab # xys" in exI, simp)
nipkow@15302
  1871
apply (case_tac xys, simp_all, blast)
nipkow@15302
  1872
done
nipkow@15302
  1873
nipkow@15302
  1874
lemma lex_conv:
nipkow@15302
  1875
"lex r =
nipkow@15302
  1876
{(xs,ys). length xs = length ys \<and>
nipkow@15302
  1877
(\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@15302
  1878
by (force simp add: lex_def lexn_conv)
nipkow@15302
  1879
nipkow@15302
  1880
lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)"
nipkow@15302
  1881
by (unfold lexico_def) blast
nipkow@15302
  1882
nipkow@15302
  1883
lemma lexico_conv:
nipkow@15302
  1884
"lexico r = {(xs,ys). length xs < length ys |
nipkow@15302
  1885
length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@15302
  1886
by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def)
nipkow@15302
  1887
nipkow@15302
  1888
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@15302
  1889
by (simp add: lex_conv)
nipkow@15302
  1890
nipkow@15302
  1891
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@15302
  1892
by (simp add:lex_conv)
nipkow@15302
  1893
nipkow@15302
  1894
lemma Cons_in_lex [iff]:
nipkow@15302
  1895
"((x # xs, y # ys) : lex r) =
nipkow@15302
  1896
((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@15302
  1897
apply (simp add: lex_conv)
nipkow@15302
  1898
apply (rule iffI)
nipkow@15302
  1899
 prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
nipkow@15302
  1900
apply (case_tac xys, simp, simp)
nipkow@15302
  1901
apply blast
nipkow@15302
  1902
done
nipkow@15302
  1903
nipkow@15302
  1904
nipkow@15302
  1905
subsection{*Lifting a Relation on List Elements to the Lists*}
nipkow@15302
  1906
nipkow@15302
  1907
consts  listrel :: "('a * 'a)set => ('a list * 'a list)set"
nipkow@15302
  1908
nipkow@15302
  1909
inductive "listrel(r)"
nipkow@15302
  1910
 intros
nipkow@15302
  1911
   Nil:  "([],[]) \<in> listrel r"
nipkow@15302
  1912
   Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
nipkow@15302
  1913
nipkow@15302
  1914
inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
nipkow@15302
  1915
inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
nipkow@15302
  1916
inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
nipkow@15302
  1917
inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
nipkow@15302
  1918
nipkow@15302
  1919
nipkow@15302
  1920
lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
nipkow@15302
  1921
apply clarify  
nipkow@15302
  1922
apply (erule listrel.induct)
nipkow@15302
  1923
apply (blast intro: listrel.intros)+
nipkow@15302
  1924
done
nipkow@15302
  1925
nipkow@15302
  1926
lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
nipkow@15302
  1927
apply clarify 
nipkow@15302
  1928
apply (erule listrel.induct, auto) 
nipkow@15302
  1929
done
nipkow@15302
  1930
nipkow@15302
  1931
lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" 
nipkow@15302
  1932
apply (simp add: refl_def listrel_subset Ball_def)
nipkow@15302
  1933
apply (rule allI) 
nipkow@15302
  1934
apply (induct_tac x) 
nipkow@15302
  1935
apply (auto intro: listrel.intros)
nipkow@15302
  1936
done
nipkow@15302
  1937
nipkow@15302
  1938
lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
nipkow@15302
  1939
apply (auto simp add: sym_def)
nipkow@15302
  1940
apply (erule listrel.induct) 
nipkow@15302
  1941
apply (blast intro: listrel.intros)+
nipkow@15302
  1942
done
nipkow@15302
  1943
nipkow@15302
  1944
lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
nipkow@15302
  1945
apply (simp add: trans_def)
nipkow@15302
  1946
apply (intro allI) 
nipkow@15302
  1947
apply (rule impI) 
nipkow@15302
  1948
apply (erule listrel.induct) 
nipkow@15302
  1949
apply (blast intro: listrel.intros)+
nipkow@15302
  1950
done
nipkow@15302
  1951
nipkow@15302
  1952
theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
nipkow@15302
  1953
by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) 
nipkow@15302
  1954
nipkow@15302
  1955
lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
nipkow@15302
  1956
by (blast intro: listrel.intros)
nipkow@15302
  1957
nipkow@15302
  1958
lemma listrel_Cons:
nipkow@15302
  1959
     "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})";
nipkow@15302
  1960
by (auto simp add: set_Cons_def intro: listrel.intros) 
nipkow@15302
  1961
nipkow@15302
  1962
nipkow@15302
  1963
section{*Miscellany*}
nipkow@15302
  1964
wenzelm@13366
  1965
subsection {* Characters and strings *}
wenzelm@13366
  1966
wenzelm@13366
  1967
datatype nibble =
wenzelm@13366
  1968
    Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
wenzelm@13366
  1969
  | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
wenzelm@13366
  1970
wenzelm@13366
  1971
datatype char = Char nibble nibble
wenzelm@13366
  1972
  -- "Note: canonical order of character encoding coincides with standard term ordering"
wenzelm@13366
  1973
wenzelm@13366
  1974
types string = "char list"
wenzelm@13366
  1975
wenzelm@13366
  1976
syntax
wenzelm@13366
  1977
  "_Char" :: "xstr => char"    ("CHR _")
wenzelm@13366
  1978
  "_String" :: "xstr => string"    ("_")
wenzelm@13366
  1979
wenzelm@13366
  1980
parse_ast_translation {*
wenzelm@13366
  1981
  let
wenzelm@13366
  1982
    val constants = Syntax.Appl o map Syntax.Constant;
wenzelm@13366
  1983
wenzelm@13366
  1984
    fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10));
wenzelm@13366
  1985
    fun mk_char c =
wenzelm@13366
  1986
      if Symbol.is_ascii c andalso Symbol.is_printable c then
wenzelm@13366
  1987
        constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)]
wenzelm@13366
  1988
      else error ("Printable ASCII character expected: " ^ quote c);
wenzelm@13366
  1989
wenzelm@13366
  1990
    fun mk_string [] = Syntax.Constant "Nil"
wenzelm@13366
  1991
      | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs];
wenzelm@13366
  1992
wenzelm@13366
  1993
    fun char_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  1994
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  1995
          [c] => mk_char c
wenzelm@13366
  1996
        | _ => error ("Single character expected: " ^ xstr))
wenzelm@13366
  1997
      | char_ast_tr asts = raise AST ("char_ast_tr", asts);
wenzelm@13366
  1998
wenzelm@13366
  1999
    fun string_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  2000
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  2001
          [] => constants [Syntax.constrainC, "Nil", "string"]
wenzelm@13366
  2002
        | cs => mk_string cs)
wenzelm@13366
  2003
      | string_ast_tr asts = raise AST ("string_tr", asts);
wenzelm@13366
  2004
  in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end;
wenzelm@13366
  2005
*}
wenzelm@13366
  2006
berghofe@15064
  2007
ML {*
berghofe@15064
  2008
fun int_of_nibble h =
berghofe@15064
  2009
  if "0" <= h andalso h <= "9" then ord h - ord "0"
berghofe@15064
  2010
  else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10
berghofe@15064
  2011
  else raise Match;
berghofe@15064
  2012
berghofe@15064
  2013
fun nibble_of_int i =
berghofe@15064
  2014
  if i <= 9 then chr (ord "0" + i) else chr (ord "A" + i - 10);
berghofe@15064
  2015
*}
berghofe@15064
  2016
wenzelm@13366
  2017
print_ast_translation {*
wenzelm@13366
  2018
  let
wenzelm@13366
  2019
    fun dest_nib (Syntax.Constant c) =
wenzelm@13366
  2020
        (case explode c of
berghofe@15064
  2021
          ["N", "i", "b", "b", "l", "e", h] => int_of_nibble h
wenzelm@13366
  2022
        | _ => raise Match)
wenzelm@13366
  2023
      | dest_nib _ = raise Match;
wenzelm@13366
  2024
wenzelm@13366
  2025
    fun dest_chr c1 c2 =
wenzelm@13366
  2026
      let val c = chr (dest_nib c1 * 16 + dest_nib c2)
wenzelm@13366
  2027
      in if Symbol.is_printable c then c else raise Match end;
wenzelm@13366
  2028
wenzelm@13366
  2029
    fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2
wenzelm@13366
  2030
      | dest_char _ = raise Match;
wenzelm@13366
  2031
wenzelm@13366
  2032
    fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)];
wenzelm@13366
  2033
wenzelm@13366
  2034
    fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]]
wenzelm@13366
  2035
      | char_ast_tr' _ = raise Match;
wenzelm@13366
  2036
wenzelm@13366
  2037
    fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String",
wenzelm@13366
  2038
            xstr (map dest_char (Syntax.unfold_ast "_args" args))]
wenzelm@13366
  2039
      | list_ast_tr' ts = raise Match;
wenzelm@13366
  2040
  in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end;
wenzelm@13366
  2041
*}
wenzelm@13366
  2042
berghofe@15064
  2043
subsection {* Code generator setup *}
berghofe@15064
  2044
berghofe@15064
  2045
ML {*
berghofe@15064
  2046
local
berghofe@15064
  2047
berghofe@15064
  2048
fun list_codegen thy gr dep b t =
berghofe@15064
  2049
  let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy dep false)
berghofe@15064
  2050
    (gr, HOLogic.dest_list t)
berghofe@15064
  2051
  in Some (gr', Pretty.list "[" "]" ps) end handle TERM _ => None;
berghofe@15064
  2052
berghofe@15064
  2053
fun dest_nibble (Const (s, _)) = int_of_nibble (unprefix "List.nibble.Nibble" s)
berghofe@15064
  2054
  | dest_nibble _ = raise Match;
berghofe@15064
  2055
berghofe@15064
  2056
fun char_codegen thy gr dep b (Const ("List.char.Char", _) $ c1 $ c2) =
berghofe@15064
  2057
    (let val c = chr (dest_nibble c1 * 16 + dest_nibble c2)
berghofe@15064
  2058
     in if Symbol.is_printable c then Some (gr, Pretty.quote (Pretty.str c))
berghofe@15064
  2059
       else None
berghofe@15064
  2060
     end handle LIST _ => None | Match => None)
berghofe@15064
  2061
  | char_codegen thy gr dep b _ = None;
berghofe@15064
  2062
berghofe@15064
  2063
in
berghofe@15064
  2064
berghofe@15064
  2065
val list_codegen_setup =
berghofe@15064
  2066
  [Codegen.add_codegen "list_codegen" list_codegen,
berghofe@15064
  2067
   Codegen.add_codegen "char_codegen" char_codegen];
berghofe@15064
  2068
berghofe@15064
  2069
end;
berghofe@15064
  2070
berghofe@15064
  2071
val term_of_list = HOLogic.mk_list;
berghofe@15064
  2072
berghofe@15064
  2073
fun gen_list' aG i j = frequency
berghofe@15064
  2074
  [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] ()
berghofe@15064
  2075
and gen_list aG i = gen_list' aG i i;
berghofe@15064
  2076
berghofe@15064
  2077
val nibbleT = Type ("List.nibble", []);
berghofe@15064
  2078
berghofe@15064
  2079
fun term_of_char c =
berghofe@15064
  2080
  Const ("List.char.Char", nibbleT --> nibbleT --> Type ("List.char", [])) $
berghofe@15064
  2081
    Const ("List.nibble.Nibble" ^ nibble_of_int (ord c div 16), nibbleT) $
berghofe@15064
  2082
    Const ("List.nibble.Nibble" ^ nibble_of_int (ord c mod 16), nibbleT);
berghofe@15064
  2083
berghofe@15064
  2084
fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z")));
berghofe@15064
  2085
*}
berghofe@15064
  2086
berghofe@15064
  2087
types_code
berghofe@15064
  2088
  "list" ("_ list")
berghofe@15064
  2089
  "char" ("string")
berghofe@15064
  2090
berghofe@15064
  2091
consts_code "Cons" ("(_ ::/ _)")
berghofe@15064
  2092
berghofe@15064
  2093
setup list_codegen_setup
berghofe@15064
  2094
wenzelm@13122
  2095
end