src/HOL/List.thy
author nipkow
Fri Jan 14 12:00:27 2005 +0100 (2005-01-14)
changeset 15439 71c0f98e31f1
parent 15426 f41e3e654706
child 15489 d136af442665
permissions -rw-r--r--
made diff_less a simp rule
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(*  Title:      HOL/List.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow
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*)
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header {* The datatype of finite lists *}
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theory List
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imports PreList
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begin
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datatype 'a list =
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    Nil    ("[]")
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  | Cons 'a  "'a list"    (infixr "#" 65)
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subsection{*Basic list processing functions*}
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consts
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  "@" :: "'a list => 'a list => 'a list"    (infixr 65)
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  filter:: "('a => bool) => 'a list => 'a list"
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  concat:: "'a list list => 'a list"
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  foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"
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  foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"
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  hd:: "'a list => 'a"
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  tl:: "'a list => 'a list"
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  last:: "'a list => 'a"
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  butlast :: "'a list => 'a list"
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  set :: "'a list => 'a set"
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  list_all:: "('a => bool) => ('a list => bool)"
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  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool"
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  list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
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  map :: "('a=>'b) => ('a list => 'b list)"
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  mem :: "'a => 'a list => bool"    (infixl 55)
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  nth :: "'a list => nat => 'a"    (infixl "!" 100)
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  list_update :: "'a list => nat => 'a => 'a list"
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  take:: "nat => 'a list => 'a list"
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  drop:: "nat => 'a list => 'a list"
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  takeWhile :: "('a => bool) => 'a list => 'a list"
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  dropWhile :: "('a => bool) => 'a list => 'a list"
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  rev :: "'a list => 'a list"
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  zip :: "'a list => 'b list => ('a * 'b) list"
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  upt :: "nat => nat => nat list" ("(1[_..</_'])")
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  remdups :: "'a list => 'a list"
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  remove1 :: "'a => 'a list => 'a list"
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  null:: "'a list => bool"
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  "distinct":: "'a list => bool"
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  replicate :: "nat => 'a => 'a list"
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  rotate1 :: "'a list \<Rightarrow> 'a list"
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  rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
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  sublist :: "'a list => nat set => 'a list"
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nonterminals lupdbinds lupdbind
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syntax
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  -- {* list Enumeration *}
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  "@list" :: "args => 'a list"    ("[(_)]")
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  -- {* Special syntax for filter *}
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_:_./ _])")
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  -- {* list update *}
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  "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
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  "" :: "lupdbind => lupdbinds"    ("_")
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  "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
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  "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
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  upto:: "nat => nat => nat list"    ("(1[_../_])")
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translations
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  "[x, xs]" == "x#[xs]"
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  "[x]" == "x#[]"
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  "[x:xs . P]"== "filter (%x. P) xs"
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  "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"
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  "xs[i:=x]" == "list_update xs i x"
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  "[i..j]" == "[i..<(Suc j)]"
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syntax (xsymbols)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
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syntax (HTML output)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
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text {*
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  Function @{text size} is overloaded for all datatypes. Users may
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  refer to the list version as @{text length}. *}
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syntax length :: "'a list => nat"
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translations "length" => "size :: _ list => nat"
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typed_print_translation {*
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  let
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    fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] =
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          Syntax.const "length" $ t
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      | size_tr' _ _ _ = raise Match;
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  in [("size", size_tr')] end
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*}
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primrec
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  "hd(x#xs) = x"
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primrec
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  "tl([]) = []"
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  "tl(x#xs) = xs"
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primrec
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  "null([]) = True"
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  "null(x#xs) = False"
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primrec
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  "last(x#xs) = (if xs=[] then x else last xs)"
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primrec
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  "butlast []= []"
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  "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
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primrec
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  "x mem [] = False"
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  "x mem (y#ys) = (if y=x then True else x mem ys)"
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primrec
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  "set [] = {}"
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  "set (x#xs) = insert x (set xs)"
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primrec
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  list_all_Nil:"list_all P [] = True"
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  list_all_Cons: "list_all P (x#xs) = (P(x) \<and> list_all P xs)"
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primrec
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"list_ex P [] = False"
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"list_ex P (x#xs) = (P x \<or> list_ex P xs)"
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primrec
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  "map f [] = []"
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  "map f (x#xs) = f(x)#map f xs"
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primrec
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  append_Nil:"[]@ys = ys"
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  append_Cons: "(x#xs)@ys = x#(xs@ys)"
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primrec
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  "rev([]) = []"
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  "rev(x#xs) = rev(xs) @ [x]"
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primrec
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  "filter P [] = []"
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  "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
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primrec
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  foldl_Nil:"foldl f a [] = a"
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  foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
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primrec
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  "foldr f [] a = a"
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  "foldr f (x#xs) a = f x (foldr f xs a)"
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primrec
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  "concat([]) = []"
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  "concat(x#xs) = x @ concat(xs)"
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primrec
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  drop_Nil:"drop n [] = []"
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  drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  take_Nil:"take n [] = []"
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  take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  "[][i:=v] = []"
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  "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])"
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primrec
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  "takeWhile P [] = []"
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  "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
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primrec
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  "dropWhile P [] = []"
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  "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
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primrec
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  "zip xs [] = []"
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  zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
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primrec
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  upt_0: "[i..<0] = []"
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  upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
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primrec
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  "distinct [] = True"
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  "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
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primrec
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  "remdups [] = []"
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  "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
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primrec
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  "remove1 x [] = []"
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  "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)"
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primrec
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  replicate_0: "replicate 0 x = []"
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  replicate_Suc: "replicate (Suc n) x = x # replicate n x"
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defs
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rotate1_def: "rotate1 xs == (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"
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rotate_def:  "rotate n == rotate1 ^ n"
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list_all2_def:
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 "list_all2 P xs ys ==
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  length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)"
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sublist_def:
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 "sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..<size xs]))"
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lemma not_Cons_self [simp]: "xs \<noteq> x # xs"
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by (induct xs) auto
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lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
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lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
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by (induct xs) auto
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lemma length_induct:
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"(!!xs. \<forall>ys. length ys < length xs --> P ys ==> P xs) ==> P xs"
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by (rule measure_induct [of length]) rules
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subsubsection {* @{text length} *}
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text {*
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Needs to come before @{text "@"} because of theorem @{text
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append_eq_append_conv}.
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*}
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lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
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by (induct xs) auto
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lemma length_map [simp]: "length (map f xs) = length xs"
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by (induct xs) auto
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lemma length_rev [simp]: "length (rev xs) = length xs"
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by (induct xs) auto
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lemma length_tl [simp]: "length (tl xs) = length xs - 1"
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by (cases xs) auto
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lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
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by (induct xs) auto
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lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
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by (induct xs) auto
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lemma length_Suc_conv:
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"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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by (induct xs) auto
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lemma Suc_length_conv:
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"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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apply (induct xs, simp, simp)
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apply blast
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done
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lemma impossible_Cons [rule_format]: 
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  "length xs <= length ys --> xs = x # ys = False"
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apply (induct xs, auto)
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done
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lemma list_induct2[consumes 1]: "\<And>ys.
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 \<lbrakk> length xs = length ys;
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   P [] [];
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   \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
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 \<Longrightarrow> P xs ys"
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apply(induct xs)
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 apply simp
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apply(case_tac ys)
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 apply simp
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apply(simp)
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done
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subsubsection {* @{text "@"} -- append *}
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lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
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by (induct xs) auto
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lemma append_Nil2 [simp]: "xs @ [] = xs"
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by (induct xs) auto
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lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
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by (induct xs) auto
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lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
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by (induct xs) auto
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lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
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by (induct xs) auto
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lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
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by (induct xs) auto
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lemma append_eq_append_conv [simp]:
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 "!!ys. length xs = length ys \<or> length us = length vs
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 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
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apply (induct xs)
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 apply (case_tac ys, simp, force)
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apply (case_tac ys, force, simp)
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done
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lemma append_eq_append_conv2: "!!ys zs ts.
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 (xs @ ys = zs @ ts) =
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 (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
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apply (induct xs)
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 apply fastsimp
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apply(case_tac zs)
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 apply simp
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apply fastsimp
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done
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lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
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by simp
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lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
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by simp
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lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
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by simp
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lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   345
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   346
wenzelm@13142
   347
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   348
using append_same_eq [of "[]"] by auto
wenzelm@13114
   349
wenzelm@13142
   350
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   351
by (induct xs) auto
wenzelm@13114
   352
wenzelm@13142
   353
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   354
by (induct xs) auto
wenzelm@13114
   355
wenzelm@13142
   356
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   357
by (simp add: hd_append split: list.split)
wenzelm@13114
   358
wenzelm@13142
   359
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   360
by (simp split: list.split)
wenzelm@13114
   361
wenzelm@13142
   362
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   363
by (simp add: tl_append split: list.split)
wenzelm@13114
   364
wenzelm@13114
   365
nipkow@14300
   366
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   367
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   368
by(cases ys) auto
nipkow@14300
   369
nipkow@15281
   370
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   371
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   372
by(cases ys) auto
nipkow@15281
   373
nipkow@14300
   374
wenzelm@13142
   375
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   376
wenzelm@13114
   377
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   378
by simp
wenzelm@13114
   379
wenzelm@13142
   380
lemma Cons_eq_appendI:
nipkow@13145
   381
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   382
by (drule sym) simp
wenzelm@13114
   383
wenzelm@13142
   384
lemma append_eq_appendI:
nipkow@13145
   385
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   386
by (drule sym) simp
wenzelm@13114
   387
wenzelm@13114
   388
wenzelm@13142
   389
text {*
nipkow@13145
   390
Simplification procedure for all list equalities.
nipkow@13145
   391
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   392
- both lists end in a singleton list,
nipkow@13145
   393
- or both lists end in the same list.
wenzelm@13142
   394
*}
wenzelm@13142
   395
wenzelm@13142
   396
ML_setup {*
nipkow@3507
   397
local
nipkow@3507
   398
wenzelm@13122
   399
val append_assoc = thm "append_assoc";
wenzelm@13122
   400
val append_Nil = thm "append_Nil";
wenzelm@13122
   401
val append_Cons = thm "append_Cons";
wenzelm@13122
   402
val append1_eq_conv = thm "append1_eq_conv";
wenzelm@13122
   403
val append_same_eq = thm "append_same_eq";
wenzelm@13122
   404
wenzelm@13114
   405
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462
   406
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
wenzelm@13462
   407
  | last (Const("List.op @",_) $ _ $ ys) = last ys
wenzelm@13462
   408
  | last t = t;
wenzelm@13114
   409
wenzelm@13114
   410
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462
   411
  | list1 _ = false;
wenzelm@13114
   412
wenzelm@13114
   413
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462
   414
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
wenzelm@13462
   415
  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462
   416
  | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114
   417
wenzelm@13114
   418
val rearr_tac =
wenzelm@13462
   419
  simp_tac (HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]);
wenzelm@13114
   420
wenzelm@13114
   421
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   422
  let
wenzelm@13462
   423
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   424
    fun rearr conv =
wenzelm@13462
   425
      let
wenzelm@13462
   426
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   427
        val Type(_,listT::_) = eqT
wenzelm@13462
   428
        val appT = [listT,listT] ---> listT
wenzelm@13462
   429
        val app = Const("List.op @",appT)
wenzelm@13462
   430
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   431
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@13480
   432
        val thm = Tactic.prove sg [] [] eq (K (rearr_tac 1));
wenzelm@13462
   433
      in Some ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   434
wenzelm@13462
   435
  in
wenzelm@13462
   436
    if list1 lastl andalso list1 lastr then rearr append1_eq_conv
wenzelm@13462
   437
    else if lastl aconv lastr then rearr append_same_eq
wenzelm@13462
   438
    else None
wenzelm@13462
   439
  end;
wenzelm@13462
   440
wenzelm@13114
   441
in
wenzelm@13462
   442
wenzelm@13462
   443
val list_eq_simproc =
wenzelm@13462
   444
  Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq;
wenzelm@13462
   445
wenzelm@13114
   446
end;
wenzelm@13114
   447
wenzelm@13114
   448
Addsimprocs [list_eq_simproc];
wenzelm@13114
   449
*}
wenzelm@13114
   450
wenzelm@13114
   451
nipkow@15392
   452
subsubsection {* @{text map} *}
wenzelm@13114
   453
wenzelm@13142
   454
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   455
by (induct xs) simp_all
wenzelm@13114
   456
wenzelm@13142
   457
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   458
by (rule ext, induct_tac xs) auto
wenzelm@13114
   459
wenzelm@13142
   460
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   461
by (induct xs) auto
wenzelm@13114
   462
wenzelm@13142
   463
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   464
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   465
wenzelm@13142
   466
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   467
by (induct xs) auto
wenzelm@13114
   468
nipkow@13737
   469
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   470
by (induct xs) auto
nipkow@13737
   471
wenzelm@13366
   472
lemma map_cong [recdef_cong]:
nipkow@13145
   473
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   474
-- {* a congruence rule for @{text map} *}
nipkow@13737
   475
by simp
wenzelm@13114
   476
wenzelm@13142
   477
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   478
by (cases xs) auto
wenzelm@13114
   479
wenzelm@13142
   480
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   481
by (cases xs) auto
wenzelm@13114
   482
nipkow@14025
   483
lemma map_eq_Cons_conv[iff]:
nipkow@14025
   484
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   485
by (cases xs) auto
wenzelm@13114
   486
nipkow@14025
   487
lemma Cons_eq_map_conv[iff]:
nipkow@14025
   488
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   489
by (cases ys) auto
nipkow@14025
   490
nipkow@14111
   491
lemma ex_map_conv:
nipkow@14111
   492
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
nipkow@14111
   493
by(induct ys, auto)
nipkow@14111
   494
nipkow@15110
   495
lemma map_eq_imp_length_eq:
nipkow@15110
   496
  "!!xs. map f xs = map f ys ==> length xs = length ys"
nipkow@15110
   497
apply (induct ys)
nipkow@15110
   498
 apply simp
nipkow@15110
   499
apply(simp (no_asm_use))
nipkow@15110
   500
apply clarify
nipkow@15110
   501
apply(simp (no_asm_use))
nipkow@15110
   502
apply fast
nipkow@15110
   503
done
nipkow@15110
   504
nipkow@15110
   505
lemma map_inj_on:
nipkow@15110
   506
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   507
  ==> xs = ys"
nipkow@15110
   508
apply(frule map_eq_imp_length_eq)
nipkow@15110
   509
apply(rotate_tac -1)
nipkow@15110
   510
apply(induct rule:list_induct2)
nipkow@15110
   511
 apply simp
nipkow@15110
   512
apply(simp)
nipkow@15110
   513
apply (blast intro:sym)
nipkow@15110
   514
done
nipkow@15110
   515
nipkow@15110
   516
lemma inj_on_map_eq_map:
nipkow@15110
   517
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   518
by(blast dest:map_inj_on)
nipkow@15110
   519
wenzelm@13114
   520
lemma map_injective:
nipkow@14338
   521
 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@14338
   522
by (induct ys) (auto dest!:injD)
wenzelm@13114
   523
nipkow@14339
   524
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   525
by(blast dest:map_injective)
nipkow@14339
   526
wenzelm@13114
   527
lemma inj_mapI: "inj f ==> inj (map f)"
paulson@13585
   528
by (rules dest: map_injective injD intro: inj_onI)
wenzelm@13114
   529
wenzelm@13114
   530
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   531
apply (unfold inj_on_def, clarify)
nipkow@13145
   532
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   533
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   534
apply blast
nipkow@13145
   535
done
wenzelm@13114
   536
nipkow@14339
   537
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   538
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   539
nipkow@15303
   540
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
   541
apply(rule inj_onI)
nipkow@15303
   542
apply(erule map_inj_on)
nipkow@15303
   543
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
   544
done
nipkow@15303
   545
kleing@14343
   546
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   547
by (induct xs, auto)
wenzelm@13114
   548
nipkow@14402
   549
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   550
by (induct xs) auto
nipkow@14402
   551
nipkow@15110
   552
lemma map_fst_zip[simp]:
nipkow@15110
   553
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   554
by (induct rule:list_induct2, simp_all)
nipkow@15110
   555
nipkow@15110
   556
lemma map_snd_zip[simp]:
nipkow@15110
   557
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   558
by (induct rule:list_induct2, simp_all)
nipkow@15110
   559
nipkow@15110
   560
nipkow@15392
   561
subsubsection {* @{text rev} *}
wenzelm@13114
   562
wenzelm@13142
   563
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   564
by (induct xs) auto
wenzelm@13114
   565
wenzelm@13142
   566
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   567
by (induct xs) auto
wenzelm@13114
   568
wenzelm@13142
   569
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   570
by (induct xs) auto
wenzelm@13114
   571
wenzelm@13142
   572
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   573
by (induct xs) auto
wenzelm@13114
   574
wenzelm@13142
   575
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"
paulson@14208
   576
apply (induct xs, force)
paulson@14208
   577
apply (case_tac ys, simp, force)
nipkow@13145
   578
done
wenzelm@13114
   579
nipkow@15439
   580
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
   581
by(simp add:inj_on_def)
nipkow@15439
   582
wenzelm@13366
   583
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   584
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
nipkow@13145
   585
apply(subst rev_rev_ident[symmetric])
nipkow@13145
   586
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   587
done
wenzelm@13114
   588
nipkow@13145
   589
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114
   590
wenzelm@13366
   591
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   592
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   593
by (induct xs rule: rev_induct) auto
wenzelm@13114
   594
wenzelm@13366
   595
lemmas rev_cases = rev_exhaust
wenzelm@13366
   596
wenzelm@13114
   597
nipkow@15392
   598
subsubsection {* @{text set} *}
wenzelm@13114
   599
wenzelm@13142
   600
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   601
by (induct xs) auto
wenzelm@13114
   602
wenzelm@13142
   603
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   604
by (induct xs) auto
wenzelm@13114
   605
oheimb@14099
   606
lemma hd_in_set: "l = x#xs \<Longrightarrow> x\<in>set l"
paulson@14208
   607
by (case_tac l, auto)
oheimb@14099
   608
wenzelm@13142
   609
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   610
by auto
wenzelm@13114
   611
oheimb@14099
   612
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   613
by auto
oheimb@14099
   614
wenzelm@13142
   615
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   616
by (induct xs) auto
wenzelm@13114
   617
nipkow@15245
   618
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   619
by(induct xs) auto
nipkow@15245
   620
wenzelm@13142
   621
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   622
by (induct xs) auto
wenzelm@13114
   623
wenzelm@13142
   624
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   625
by (induct xs) auto
wenzelm@13114
   626
wenzelm@13142
   627
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   628
by (induct xs) auto
wenzelm@13114
   629
nipkow@15425
   630
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
paulson@14208
   631
apply (induct j, simp_all)
paulson@14208
   632
apply (erule ssubst, auto)
nipkow@13145
   633
done
wenzelm@13114
   634
wenzelm@13142
   635
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
paulson@15113
   636
proof (induct xs)
paulson@15113
   637
  case Nil show ?case by simp
paulson@15113
   638
  case (Cons a xs)
paulson@15113
   639
  show ?case
paulson@15113
   640
  proof 
paulson@15113
   641
    assume "x \<in> set (a # xs)"
paulson@15113
   642
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   643
      by (simp, blast intro: Cons_eq_appendI)
paulson@15113
   644
  next
paulson@15113
   645
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   646
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
paulson@15113
   647
    show "x \<in> set (a # xs)" 
paulson@15113
   648
      by (cases ys, auto simp add: eq)
paulson@15113
   649
  qed
paulson@15113
   650
qed
wenzelm@13142
   651
paulson@13508
   652
lemma finite_list: "finite A ==> EX l. set l = A"
paulson@13508
   653
apply (erule finite_induct, auto)
paulson@13508
   654
apply (rule_tac x="x#l" in exI, auto)
paulson@13508
   655
done
paulson@13508
   656
kleing@14388
   657
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
   658
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
   659
paulson@15168
   660
nipkow@15439
   661
subsubsection {* @{text mem}, @{text list_all} and @{text list_ex} *}
wenzelm@13114
   662
nipkow@15302
   663
text{* Only use @{text mem} for generating executable code.  Otherwise
nipkow@15439
   664
use @{prop"x : set xs"} instead --- it is much easier to reason about.
nipkow@15439
   665
The same is true for @{text list_all} and @{text list_ex}: write
nipkow@15439
   666
@{text"\<forall>x\<in>set xs"} and @{text"\<exists>x\<in>set xs"} instead because the HOL
nipkow@15439
   667
quantifiers are aleady known to the automatic provers. For the purpose
nipkow@15439
   668
of generating executable code use the theorems @{text set_mem_eq},
nipkow@15439
   669
@{text list_all_conv} and @{text list_ex_iff} to get rid off or
nipkow@15439
   670
introduce the combinators. *}
nipkow@15302
   671
wenzelm@13114
   672
lemma set_mem_eq: "(x mem xs) = (x : set xs)"
nipkow@13145
   673
by (induct xs) auto
wenzelm@13114
   674
wenzelm@13142
   675
lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)"
nipkow@13145
   676
by (induct xs) auto
wenzelm@13114
   677
wenzelm@13142
   678
lemma list_all_append [simp]:
nipkow@13145
   679
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
nipkow@13145
   680
by (induct xs) auto
wenzelm@13114
   681
kleing@15426
   682
lemma list_all_rev [simp]: "list_all P (rev xs) = list_all P xs"
kleing@15426
   683
by (simp add: list_all_conv)
kleing@15426
   684
nipkow@15439
   685
lemma list_ex_iff: "list_ex P xs = (\<exists>x \<in> set xs. P x)"
nipkow@15439
   686
by (induct xs) simp_all
kleing@15426
   687
wenzelm@13114
   688
nipkow@15392
   689
subsubsection {* @{text filter} *}
wenzelm@13114
   690
wenzelm@13142
   691
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   692
by (induct xs) auto
wenzelm@13114
   693
nipkow@15305
   694
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
   695
by (induct xs) simp_all
nipkow@15305
   696
wenzelm@13142
   697
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   698
by (induct xs) auto
wenzelm@13114
   699
wenzelm@13142
   700
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   701
by (induct xs) auto
wenzelm@13114
   702
wenzelm@13142
   703
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   704
by (induct xs) auto
wenzelm@13114
   705
nipkow@15246
   706
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@13145
   707
by (induct xs) (auto simp add: le_SucI)
wenzelm@13114
   708
wenzelm@13142
   709
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   710
by auto
wenzelm@13114
   711
nipkow@15246
   712
lemma length_filter_less:
nipkow@15246
   713
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
   714
proof (induct xs)
nipkow@15246
   715
  case Nil thus ?case by simp
nipkow@15246
   716
next
nipkow@15246
   717
  case (Cons x xs) thus ?case
nipkow@15246
   718
    apply (auto split:split_if_asm)
nipkow@15246
   719
    using length_filter_le[of P xs] apply arith
nipkow@15246
   720
  done
nipkow@15246
   721
qed
wenzelm@13114
   722
nipkow@15281
   723
lemma length_filter_conv_card:
nipkow@15281
   724
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
   725
proof (induct xs)
nipkow@15281
   726
  case Nil thus ?case by simp
nipkow@15281
   727
next
nipkow@15281
   728
  case (Cons x xs)
nipkow@15281
   729
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
   730
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
   731
  show ?case (is "?l = card ?S'")
nipkow@15281
   732
  proof (cases)
nipkow@15281
   733
    assume "p x"
nipkow@15281
   734
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@15281
   735
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   736
    have "length (filter p (x # xs)) = Suc(card ?S)"
nipkow@15281
   737
      using Cons by simp
nipkow@15281
   738
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
nipkow@15281
   739
      by (simp add: card_image inj_Suc)
nipkow@15281
   740
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   741
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
   742
    finally show ?thesis .
nipkow@15281
   743
  next
nipkow@15281
   744
    assume "\<not> p x"
nipkow@15281
   745
    hence eq: "?S' = Suc ` ?S"
nipkow@15281
   746
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   747
    have "length (filter p (x # xs)) = card ?S"
nipkow@15281
   748
      using Cons by simp
nipkow@15281
   749
    also have "\<dots> = card(Suc ` ?S)" using fin
nipkow@15281
   750
      by (simp add: card_image inj_Suc)
nipkow@15281
   751
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   752
      by (simp add:card_insert_if)
nipkow@15281
   753
    finally show ?thesis .
nipkow@15281
   754
  qed
nipkow@15281
   755
qed
nipkow@15281
   756
nipkow@15281
   757
nipkow@15392
   758
subsubsection {* @{text concat} *}
wenzelm@13114
   759
wenzelm@13142
   760
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
   761
by (induct xs) auto
wenzelm@13114
   762
wenzelm@13142
   763
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   764
by (induct xss) auto
wenzelm@13114
   765
wenzelm@13142
   766
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   767
by (induct xss) auto
wenzelm@13114
   768
wenzelm@13142
   769
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
nipkow@13145
   770
by (induct xs) auto
wenzelm@13114
   771
wenzelm@13142
   772
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
   773
by (induct xs) auto
wenzelm@13114
   774
wenzelm@13142
   775
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
   776
by (induct xs) auto
wenzelm@13114
   777
wenzelm@13142
   778
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
   779
by (induct xs) auto
wenzelm@13114
   780
wenzelm@13114
   781
nipkow@15392
   782
subsubsection {* @{text nth} *}
wenzelm@13114
   783
wenzelm@13142
   784
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
   785
by auto
wenzelm@13114
   786
wenzelm@13142
   787
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
   788
by auto
wenzelm@13114
   789
wenzelm@13142
   790
declare nth.simps [simp del]
wenzelm@13114
   791
wenzelm@13114
   792
lemma nth_append:
nipkow@13145
   793
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
paulson@14208
   794
apply (induct "xs", simp)
paulson@14208
   795
apply (case_tac n, auto)
nipkow@13145
   796
done
wenzelm@13114
   797
nipkow@14402
   798
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
nipkow@14402
   799
by (induct "xs") auto
nipkow@14402
   800
nipkow@14402
   801
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
nipkow@14402
   802
by (induct "xs") auto
nipkow@14402
   803
wenzelm@13142
   804
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
paulson@14208
   805
apply (induct xs, simp)
paulson@14208
   806
apply (case_tac n, auto)
nipkow@13145
   807
done
wenzelm@13114
   808
wenzelm@13142
   809
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
   810
apply (induct xs, simp, simp)
nipkow@13145
   811
apply safe
paulson@14208
   812
apply (rule_tac x = 0 in exI, simp)
paulson@14208
   813
 apply (rule_tac x = "Suc i" in exI, simp)
paulson@14208
   814
apply (case_tac i, simp)
nipkow@13145
   815
apply (rename_tac j)
paulson@14208
   816
apply (rule_tac x = j in exI, simp)
nipkow@13145
   817
done
wenzelm@13114
   818
nipkow@13145
   819
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
   820
by (auto simp add: set_conv_nth)
wenzelm@13114
   821
wenzelm@13142
   822
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
   823
by (auto simp add: set_conv_nth)
wenzelm@13114
   824
wenzelm@13114
   825
lemma all_nth_imp_all_set:
nipkow@13145
   826
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
   827
by (auto simp add: set_conv_nth)
wenzelm@13114
   828
wenzelm@13114
   829
lemma all_set_conv_all_nth:
nipkow@13145
   830
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
   831
by (auto simp add: set_conv_nth)
wenzelm@13114
   832
wenzelm@13114
   833
nipkow@15392
   834
subsubsection {* @{text list_update} *}
wenzelm@13114
   835
wenzelm@13142
   836
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
nipkow@13145
   837
by (induct xs) (auto split: nat.split)
wenzelm@13114
   838
wenzelm@13114
   839
lemma nth_list_update:
nipkow@13145
   840
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@13145
   841
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   842
wenzelm@13142
   843
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
   844
by (simp add: nth_list_update)
wenzelm@13114
   845
wenzelm@13142
   846
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145
   847
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   848
wenzelm@13142
   849
lemma list_update_overwrite [simp]:
nipkow@13145
   850
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145
   851
by (induct xs) (auto split: nat.split)
wenzelm@13114
   852
nipkow@14402
   853
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
paulson@14208
   854
apply (induct xs, simp)
nipkow@14187
   855
apply(simp split:nat.splits)
nipkow@14187
   856
done
nipkow@14187
   857
wenzelm@13114
   858
lemma list_update_same_conv:
nipkow@13145
   859
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145
   860
by (induct xs) (auto split: nat.split)
wenzelm@13114
   861
nipkow@14187
   862
lemma list_update_append1:
nipkow@14187
   863
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
paulson@14208
   864
apply (induct xs, simp)
nipkow@14187
   865
apply(simp split:nat.split)
nipkow@14187
   866
done
nipkow@14187
   867
nipkow@14402
   868
lemma list_update_length [simp]:
nipkow@14402
   869
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
   870
by (induct xs, auto)
nipkow@14402
   871
wenzelm@13114
   872
lemma update_zip:
nipkow@13145
   873
"!!i xy xs. length xs = length ys ==>
nipkow@13145
   874
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145
   875
by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114
   876
wenzelm@13114
   877
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145
   878
by (induct xs) (auto split: nat.split)
wenzelm@13114
   879
wenzelm@13114
   880
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
   881
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
   882
wenzelm@13114
   883
nipkow@15392
   884
subsubsection {* @{text last} and @{text butlast} *}
wenzelm@13114
   885
wenzelm@13142
   886
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
   887
by (induct xs) auto
wenzelm@13114
   888
wenzelm@13142
   889
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
   890
by (induct xs) auto
wenzelm@13114
   891
nipkow@14302
   892
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
   893
by(simp add:last.simps)
nipkow@14302
   894
nipkow@14302
   895
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
   896
by(simp add:last.simps)
nipkow@14302
   897
nipkow@14302
   898
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
   899
by (induct xs) (auto)
nipkow@14302
   900
nipkow@14302
   901
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
   902
by(simp add:last_append)
nipkow@14302
   903
nipkow@14302
   904
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
   905
by(simp add:last_append)
nipkow@14302
   906
nipkow@14302
   907
wenzelm@13142
   908
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
   909
by (induct xs rule: rev_induct) auto
wenzelm@13114
   910
wenzelm@13114
   911
lemma butlast_append:
nipkow@13145
   912
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145
   913
by (induct xs) auto
wenzelm@13114
   914
wenzelm@13142
   915
lemma append_butlast_last_id [simp]:
nipkow@13145
   916
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
   917
by (induct xs) auto
wenzelm@13114
   918
wenzelm@13142
   919
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
   920
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
   921
wenzelm@13114
   922
lemma in_set_butlast_appendI:
nipkow@13145
   923
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
   924
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
   925
wenzelm@13142
   926
nipkow@15392
   927
subsubsection {* @{text take} and @{text drop} *}
wenzelm@13114
   928
wenzelm@13142
   929
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
   930
by (induct xs) auto
wenzelm@13114
   931
wenzelm@13142
   932
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
   933
by (induct xs) auto
wenzelm@13114
   934
wenzelm@13142
   935
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
   936
by simp
wenzelm@13114
   937
wenzelm@13142
   938
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
   939
by simp
wenzelm@13114
   940
wenzelm@13142
   941
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
   942
nipkow@15110
   943
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
   944
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
   945
nipkow@14187
   946
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
   947
by(cases xs, simp_all)
nipkow@14187
   948
nipkow@14187
   949
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
nipkow@14187
   950
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@14187
   951
nipkow@14187
   952
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
paulson@14208
   953
apply (induct xs, simp)
nipkow@14187
   954
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
   955
done
nipkow@14187
   956
nipkow@13913
   957
lemma take_Suc_conv_app_nth:
nipkow@13913
   958
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
paulson@14208
   959
apply (induct xs, simp)
paulson@14208
   960
apply (case_tac i, auto)
nipkow@13913
   961
done
nipkow@13913
   962
mehta@14591
   963
lemma drop_Suc_conv_tl:
mehta@14591
   964
  "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
mehta@14591
   965
apply (induct xs, simp)
mehta@14591
   966
apply (case_tac i, auto)
mehta@14591
   967
done
mehta@14591
   968
wenzelm@13142
   969
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
nipkow@13145
   970
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   971
wenzelm@13142
   972
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
nipkow@13145
   973
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   974
wenzelm@13142
   975
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
nipkow@13145
   976
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   977
wenzelm@13142
   978
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
nipkow@13145
   979
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   980
wenzelm@13142
   981
lemma take_append [simp]:
nipkow@13145
   982
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@13145
   983
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   984
wenzelm@13142
   985
lemma drop_append [simp]:
nipkow@13145
   986
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@13145
   987
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   988
wenzelm@13142
   989
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
paulson@14208
   990
apply (induct m, auto)
paulson@14208
   991
apply (case_tac xs, auto)
nipkow@15236
   992
apply (case_tac n, auto)
nipkow@13145
   993
done
wenzelm@13114
   994
wenzelm@13142
   995
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
paulson@14208
   996
apply (induct m, auto)
paulson@14208
   997
apply (case_tac xs, auto)
nipkow@13145
   998
done
wenzelm@13114
   999
wenzelm@13114
  1000
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
paulson@14208
  1001
apply (induct m, auto)
paulson@14208
  1002
apply (case_tac xs, auto)
nipkow@13145
  1003
done
wenzelm@13114
  1004
nipkow@14802
  1005
lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@14802
  1006
apply(induct xs)
nipkow@14802
  1007
 apply simp
nipkow@14802
  1008
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  1009
done
nipkow@14802
  1010
wenzelm@13142
  1011
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
paulson@14208
  1012
apply (induct n, auto)
paulson@14208
  1013
apply (case_tac xs, auto)
nipkow@13145
  1014
done
wenzelm@13114
  1015
nipkow@15110
  1016
lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@15110
  1017
apply(induct xs)
nipkow@15110
  1018
 apply simp
nipkow@15110
  1019
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  1020
done
nipkow@15110
  1021
nipkow@15110
  1022
lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"
nipkow@15110
  1023
apply(induct xs)
nipkow@15110
  1024
apply simp
nipkow@15110
  1025
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1026
done
nipkow@15110
  1027
wenzelm@13114
  1028
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
paulson@14208
  1029
apply (induct n, auto)
paulson@14208
  1030
apply (case_tac xs, auto)
nipkow@13145
  1031
done
wenzelm@13114
  1032
wenzelm@13142
  1033
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
paulson@14208
  1034
apply (induct n, auto)
paulson@14208
  1035
apply (case_tac xs, auto)
nipkow@13145
  1036
done
wenzelm@13114
  1037
wenzelm@13114
  1038
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
paulson@14208
  1039
apply (induct xs, auto)
paulson@14208
  1040
apply (case_tac i, auto)
nipkow@13145
  1041
done
wenzelm@13114
  1042
wenzelm@13114
  1043
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
paulson@14208
  1044
apply (induct xs, auto)
paulson@14208
  1045
apply (case_tac i, auto)
nipkow@13145
  1046
done
wenzelm@13114
  1047
wenzelm@13142
  1048
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
paulson@14208
  1049
apply (induct xs, auto)
paulson@14208
  1050
apply (case_tac n, blast)
paulson@14208
  1051
apply (case_tac i, auto)
nipkow@13145
  1052
done
wenzelm@13114
  1053
wenzelm@13142
  1054
lemma nth_drop [simp]:
nipkow@13145
  1055
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
paulson@14208
  1056
apply (induct n, auto)
paulson@14208
  1057
apply (case_tac xs, auto)
nipkow@13145
  1058
done
nipkow@3507
  1059
nipkow@14025
  1060
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
nipkow@14025
  1061
by(induct xs)(auto simp:take_Cons split:nat.split)
nipkow@14025
  1062
nipkow@14025
  1063
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
nipkow@14025
  1064
by(induct xs)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1065
nipkow@14187
  1066
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1067
using set_take_subset by fast
nipkow@14187
  1068
nipkow@14187
  1069
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1070
using set_drop_subset by fast
nipkow@14187
  1071
wenzelm@13114
  1072
lemma append_eq_conv_conj:
nipkow@13145
  1073
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
paulson@14208
  1074
apply (induct xs, simp, clarsimp)
paulson@14208
  1075
apply (case_tac zs, auto)
nipkow@13145
  1076
done
wenzelm@13142
  1077
paulson@14050
  1078
lemma take_add [rule_format]: 
paulson@14050
  1079
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
paulson@14050
  1080
apply (induct xs, auto) 
paulson@14050
  1081
apply (case_tac i, simp_all) 
paulson@14050
  1082
done
paulson@14050
  1083
nipkow@14300
  1084
lemma append_eq_append_conv_if:
nipkow@14300
  1085
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1086
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1087
   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
  1088
   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
  1089
apply(induct xs\<^isub>1)
nipkow@14300
  1090
 apply simp
nipkow@14300
  1091
apply(case_tac ys\<^isub>1)
nipkow@14300
  1092
apply simp_all
nipkow@14300
  1093
done
nipkow@14300
  1094
nipkow@15110
  1095
lemma take_hd_drop:
nipkow@15110
  1096
  "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
nipkow@15110
  1097
apply(induct xs)
nipkow@15110
  1098
apply simp
nipkow@15110
  1099
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1100
done
nipkow@15110
  1101
wenzelm@13114
  1102
nipkow@15392
  1103
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1104
wenzelm@13142
  1105
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1106
by (induct xs) auto
wenzelm@13114
  1107
wenzelm@13142
  1108
lemma takeWhile_append1 [simp]:
nipkow@13145
  1109
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1110
by (induct xs) auto
wenzelm@13114
  1111
wenzelm@13142
  1112
lemma takeWhile_append2 [simp]:
nipkow@13145
  1113
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1114
by (induct xs) auto
wenzelm@13114
  1115
wenzelm@13142
  1116
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1117
by (induct xs) auto
wenzelm@13114
  1118
wenzelm@13142
  1119
lemma dropWhile_append1 [simp]:
nipkow@13145
  1120
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1121
by (induct xs) auto
wenzelm@13114
  1122
wenzelm@13142
  1123
lemma dropWhile_append2 [simp]:
nipkow@13145
  1124
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1125
by (induct xs) auto
wenzelm@13114
  1126
wenzelm@13142
  1127
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1128
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1129
nipkow@13913
  1130
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1131
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1132
by(induct xs, auto)
nipkow@13913
  1133
nipkow@13913
  1134
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1135
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1136
by(induct xs, auto)
nipkow@13913
  1137
nipkow@13913
  1138
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1139
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1140
by(induct xs, auto)
nipkow@13913
  1141
wenzelm@13114
  1142
nipkow@15392
  1143
subsubsection {* @{text zip} *}
wenzelm@13114
  1144
wenzelm@13142
  1145
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1146
by (induct ys) auto
wenzelm@13114
  1147
wenzelm@13142
  1148
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1149
by simp
wenzelm@13114
  1150
wenzelm@13142
  1151
declare zip_Cons [simp del]
wenzelm@13114
  1152
nipkow@15281
  1153
lemma zip_Cons1:
nipkow@15281
  1154
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  1155
by(auto split:list.split)
nipkow@15281
  1156
wenzelm@13142
  1157
lemma length_zip [simp]:
nipkow@13145
  1158
"!!xs. length (zip xs ys) = min (length xs) (length ys)"
paulson@14208
  1159
apply (induct ys, simp)
paulson@14208
  1160
apply (case_tac xs, auto)
nipkow@13145
  1161
done
wenzelm@13114
  1162
wenzelm@13114
  1163
lemma zip_append1:
nipkow@13145
  1164
"!!xs. zip (xs @ ys) zs =
nipkow@13145
  1165
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
paulson@14208
  1166
apply (induct zs, simp)
paulson@14208
  1167
apply (case_tac xs, simp_all)
nipkow@13145
  1168
done
wenzelm@13114
  1169
wenzelm@13114
  1170
lemma zip_append2:
nipkow@13145
  1171
"!!ys. zip xs (ys @ zs) =
nipkow@13145
  1172
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
paulson@14208
  1173
apply (induct xs, simp)
paulson@14208
  1174
apply (case_tac ys, simp_all)
nipkow@13145
  1175
done
wenzelm@13114
  1176
wenzelm@13142
  1177
lemma zip_append [simp]:
wenzelm@13142
  1178
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1179
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1180
by (simp add: zip_append1)
wenzelm@13114
  1181
wenzelm@13114
  1182
lemma zip_rev:
nipkow@14247
  1183
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1184
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1185
wenzelm@13142
  1186
lemma nth_zip [simp]:
nipkow@13145
  1187
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
paulson@14208
  1188
apply (induct ys, simp)
nipkow@13145
  1189
apply (case_tac xs)
nipkow@13145
  1190
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1191
done
wenzelm@13114
  1192
wenzelm@13114
  1193
lemma set_zip:
nipkow@13145
  1194
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
  1195
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1196
wenzelm@13114
  1197
lemma zip_update:
nipkow@13145
  1198
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
  1199
by (rule sym, simp add: update_zip)
wenzelm@13114
  1200
wenzelm@13142
  1201
lemma zip_replicate [simp]:
nipkow@13145
  1202
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
paulson@14208
  1203
apply (induct i, auto)
paulson@14208
  1204
apply (case_tac j, auto)
nipkow@13145
  1205
done
wenzelm@13114
  1206
wenzelm@13142
  1207
nipkow@15392
  1208
subsubsection {* @{text list_all2} *}
wenzelm@13114
  1209
kleing@14316
  1210
lemma list_all2_lengthD [intro?]: 
kleing@14316
  1211
  "list_all2 P xs ys ==> length xs = length ys"
nipkow@13145
  1212
by (simp add: list_all2_def)
wenzelm@13114
  1213
wenzelm@13142
  1214
lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])"
nipkow@13145
  1215
by (simp add: list_all2_def)
wenzelm@13114
  1216
wenzelm@13142
  1217
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"
nipkow@13145
  1218
by (simp add: list_all2_def)
wenzelm@13114
  1219
wenzelm@13142
  1220
lemma list_all2_Cons [iff]:
nipkow@13145
  1221
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@13145
  1222
by (auto simp add: list_all2_def)
wenzelm@13114
  1223
wenzelm@13114
  1224
lemma list_all2_Cons1:
nipkow@13145
  1225
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1226
by (cases ys) auto
wenzelm@13114
  1227
wenzelm@13114
  1228
lemma list_all2_Cons2:
nipkow@13145
  1229
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1230
by (cases xs) auto
wenzelm@13114
  1231
wenzelm@13142
  1232
lemma list_all2_rev [iff]:
nipkow@13145
  1233
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1234
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1235
kleing@13863
  1236
lemma list_all2_rev1:
kleing@13863
  1237
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1238
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1239
wenzelm@13114
  1240
lemma list_all2_append1:
nipkow@13145
  1241
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1242
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1243
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1244
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1245
apply (rule iffI)
nipkow@13145
  1246
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1247
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  1248
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1249
apply (simp add: ball_Un)
nipkow@13145
  1250
done
wenzelm@13114
  1251
wenzelm@13114
  1252
lemma list_all2_append2:
nipkow@13145
  1253
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1254
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1255
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1256
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1257
apply (rule iffI)
nipkow@13145
  1258
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1259
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  1260
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1261
apply (simp add: ball_Un)
nipkow@13145
  1262
done
wenzelm@13114
  1263
kleing@13863
  1264
lemma list_all2_append:
nipkow@14247
  1265
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  1266
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  1267
by (induct rule:list_induct2, simp_all)
kleing@13863
  1268
kleing@13863
  1269
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1270
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
kleing@13863
  1271
  by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1272
wenzelm@13114
  1273
lemma list_all2_conv_all_nth:
nipkow@13145
  1274
"list_all2 P xs ys =
nipkow@13145
  1275
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1276
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1277
berghofe@13883
  1278
lemma list_all2_trans:
berghofe@13883
  1279
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1280
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1281
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1282
proof (induct as)
berghofe@13883
  1283
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1284
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1285
  proof (induct bs)
berghofe@13883
  1286
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1287
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1288
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1289
  qed simp
berghofe@13883
  1290
qed simp
berghofe@13883
  1291
kleing@13863
  1292
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1293
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
kleing@13863
  1294
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1295
paulson@14395
  1296
lemma list_all2I:
paulson@14395
  1297
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
paulson@14395
  1298
  by (simp add: list_all2_def)
paulson@14395
  1299
kleing@14328
  1300
lemma list_all2_nthD:
kleing@13863
  1301
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
kleing@13863
  1302
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1303
nipkow@14302
  1304
lemma list_all2_nthD2:
nipkow@14302
  1305
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@14302
  1306
  by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  1307
kleing@13863
  1308
lemma list_all2_map1: 
kleing@13863
  1309
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
kleing@13863
  1310
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1311
kleing@13863
  1312
lemma list_all2_map2: 
kleing@13863
  1313
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
kleing@13863
  1314
  by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1315
kleing@14316
  1316
lemma list_all2_refl [intro?]:
kleing@13863
  1317
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
kleing@13863
  1318
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1319
kleing@13863
  1320
lemma list_all2_update_cong:
kleing@13863
  1321
  "\<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
  1322
  by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1323
kleing@13863
  1324
lemma list_all2_update_cong2:
kleing@13863
  1325
  "\<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
  1326
  by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1327
nipkow@14302
  1328
lemma list_all2_takeI [simp,intro?]:
nipkow@14302
  1329
  "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@14302
  1330
  apply (induct xs)
nipkow@14302
  1331
   apply simp
nipkow@14302
  1332
  apply (clarsimp simp add: list_all2_Cons1)
nipkow@14302
  1333
  apply (case_tac n)
nipkow@14302
  1334
  apply auto
nipkow@14302
  1335
  done
nipkow@14302
  1336
nipkow@14302
  1337
lemma list_all2_dropI [simp,intro?]:
kleing@13863
  1338
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
paulson@14208
  1339
  apply (induct as, simp)
kleing@13863
  1340
  apply (clarsimp simp add: list_all2_Cons1)
paulson@14208
  1341
  apply (case_tac n, simp, simp)
kleing@13863
  1342
  done
kleing@13863
  1343
kleing@14327
  1344
lemma list_all2_mono [intro?]:
kleing@13863
  1345
  "\<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
  1346
  apply (induct x, simp)
paulson@14208
  1347
  apply (case_tac y, auto)
kleing@13863
  1348
  done
kleing@13863
  1349
wenzelm@13142
  1350
nipkow@15392
  1351
subsubsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  1352
wenzelm@13142
  1353
lemma foldl_append [simp]:
nipkow@13145
  1354
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145
  1355
by (induct xs) auto
wenzelm@13142
  1356
nipkow@14402
  1357
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  1358
by (induct xs) auto
nipkow@14402
  1359
nipkow@14402
  1360
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  1361
by (induct xs) auto
nipkow@14402
  1362
nipkow@14402
  1363
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  1364
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  1365
wenzelm@13142
  1366
text {*
nipkow@13145
  1367
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1368
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1369
*}
wenzelm@13142
  1370
wenzelm@13142
  1371
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
nipkow@13145
  1372
by (induct ns) auto
wenzelm@13142
  1373
wenzelm@13142
  1374
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1375
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1376
wenzelm@13142
  1377
lemma sum_eq_0_conv [iff]:
nipkow@13145
  1378
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145
  1379
by (induct ns) auto
wenzelm@13114
  1380
wenzelm@13114
  1381
nipkow@15392
  1382
subsubsection {* @{text upto} *}
wenzelm@13114
  1383
nipkow@15425
  1384
lemma upt_rec: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
nipkow@13145
  1385
-- {* Does not terminate! *}
nipkow@13145
  1386
by (induct j) auto
wenzelm@13142
  1387
nipkow@15425
  1388
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
nipkow@13145
  1389
by (subst upt_rec) simp
wenzelm@13114
  1390
nipkow@15425
  1391
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
nipkow@15281
  1392
by(induct j)simp_all
nipkow@15281
  1393
nipkow@15281
  1394
lemma upt_eq_Cons_conv:
nipkow@15425
  1395
 "!!x xs. ([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
nipkow@15281
  1396
apply(induct j)
nipkow@15281
  1397
 apply simp
nipkow@15281
  1398
apply(clarsimp simp add: append_eq_Cons_conv)
nipkow@15281
  1399
apply arith
nipkow@15281
  1400
done
nipkow@15281
  1401
nipkow@15425
  1402
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
nipkow@13145
  1403
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1404
by simp
wenzelm@13114
  1405
nipkow@15425
  1406
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
nipkow@13145
  1407
apply(rule trans)
nipkow@13145
  1408
apply(subst upt_rec)
paulson@14208
  1409
 prefer 2 apply (rule refl, simp)
nipkow@13145
  1410
done
wenzelm@13114
  1411
nipkow@15425
  1412
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
nipkow@13145
  1413
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1414
by (induct k) auto
wenzelm@13114
  1415
nipkow@15425
  1416
lemma length_upt [simp]: "length [i..<j] = j - i"
nipkow@13145
  1417
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1418
nipkow@15425
  1419
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
nipkow@13145
  1420
apply (induct j)
nipkow@13145
  1421
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1422
done
wenzelm@13114
  1423
nipkow@15425
  1424
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..<n] = [i..<i+m]"
paulson@14208
  1425
apply (induct m, simp)
nipkow@13145
  1426
apply (subst upt_rec)
nipkow@13145
  1427
apply (rule sym)
nipkow@13145
  1428
apply (subst upt_rec)
nipkow@13145
  1429
apply (simp del: upt.simps)
nipkow@13145
  1430
done
nipkow@3507
  1431
nipkow@15425
  1432
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..n]"
nipkow@13145
  1433
by (induct n) auto
wenzelm@13114
  1434
nipkow@15425
  1435
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
nipkow@13145
  1436
apply (induct n m rule: diff_induct)
nipkow@13145
  1437
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1438
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1439
done
wenzelm@13114
  1440
berghofe@13883
  1441
lemma nth_take_lemma:
berghofe@13883
  1442
  "!!xs ys. k <= length xs ==> k <= length ys ==>
berghofe@13883
  1443
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
berghofe@13883
  1444
apply (atomize, induct k)
paulson@14208
  1445
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  1446
txt {* Both lists must be non-empty *}
paulson@14208
  1447
apply (case_tac xs, simp)
paulson@14208
  1448
apply (case_tac ys, clarify)
nipkow@13145
  1449
 apply (simp (no_asm_use))
nipkow@13145
  1450
apply clarify
nipkow@13145
  1451
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  1452
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  1453
apply blast
nipkow@13145
  1454
done
wenzelm@13114
  1455
wenzelm@13114
  1456
lemma nth_equalityI:
wenzelm@13114
  1457
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  1458
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  1459
apply (simp_all add: take_all)
nipkow@13145
  1460
done
wenzelm@13142
  1461
kleing@13863
  1462
(* needs nth_equalityI *)
kleing@13863
  1463
lemma list_all2_antisym:
kleing@13863
  1464
  "\<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
  1465
  \<Longrightarrow> xs = ys"
kleing@13863
  1466
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  1467
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  1468
  done
kleing@13863
  1469
wenzelm@13142
  1470
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  1471
-- {* The famous take-lemma. *}
nipkow@13145
  1472
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  1473
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  1474
done
wenzelm@13142
  1475
wenzelm@13142
  1476
nipkow@15302
  1477
lemma take_Cons':
nipkow@15302
  1478
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@15302
  1479
by (cases n) simp_all
nipkow@15302
  1480
nipkow@15302
  1481
lemma drop_Cons':
nipkow@15302
  1482
     "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@15302
  1483
by (cases n) simp_all
nipkow@15302
  1484
nipkow@15302
  1485
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@15302
  1486
by (cases n) simp_all
nipkow@15302
  1487
nipkow@15302
  1488
lemmas [simp] = take_Cons'[of "number_of v",standard]
nipkow@15302
  1489
                drop_Cons'[of "number_of v",standard]
nipkow@15302
  1490
                nth_Cons'[of _ _ "number_of v",standard]
nipkow@15302
  1491
nipkow@15302
  1492
nipkow@15392
  1493
subsubsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  1494
wenzelm@13142
  1495
lemma distinct_append [simp]:
nipkow@13145
  1496
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  1497
by (induct xs) auto
wenzelm@13142
  1498
nipkow@15305
  1499
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
nipkow@15305
  1500
by(induct xs) auto
nipkow@15305
  1501
wenzelm@13142
  1502
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  1503
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  1504
wenzelm@13142
  1505
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  1506
by (induct xs) auto
wenzelm@13142
  1507
paulson@15072
  1508
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
paulson@15251
  1509
  by (induct x, auto) 
paulson@15072
  1510
paulson@15072
  1511
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
paulson@15251
  1512
  by (induct x, auto)
paulson@15072
  1513
nipkow@15245
  1514
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245
  1515
by (induct xs) auto
nipkow@15245
  1516
nipkow@15245
  1517
lemma length_remdups_eq[iff]:
nipkow@15245
  1518
  "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245
  1519
apply(induct xs)
nipkow@15245
  1520
 apply auto
nipkow@15245
  1521
apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245
  1522
 apply arith
nipkow@15245
  1523
apply(rule length_remdups_leq)
nipkow@15245
  1524
done
nipkow@15245
  1525
wenzelm@13142
  1526
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  1527
by (induct xs) auto
wenzelm@13114
  1528
nipkow@15304
  1529
lemma distinct_map_filterI:
nipkow@15304
  1530
 "distinct(map f xs) \<Longrightarrow> distinct(map f (filter P xs))"
nipkow@15304
  1531
apply(induct xs)
nipkow@15304
  1532
 apply simp
nipkow@15304
  1533
apply force
nipkow@15304
  1534
done
nipkow@15304
  1535
wenzelm@13142
  1536
text {*
nipkow@13145
  1537
It is best to avoid this indexed version of distinct, but sometimes
nipkow@13145
  1538
it is useful. *}
wenzelm@13142
  1539
lemma distinct_conv_nth:
nipkow@13145
  1540
"distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@15251
  1541
apply (induct xs, simp, simp)
paulson@14208
  1542
apply (rule iffI, clarsimp)
nipkow@13145
  1543
 apply (case_tac i)
paulson@14208
  1544
apply (case_tac j, simp)
nipkow@13145
  1545
apply (simp add: set_conv_nth)
nipkow@13145
  1546
 apply (case_tac j)
paulson@14208
  1547
apply (clarsimp simp add: set_conv_nth, simp)
nipkow@13145
  1548
apply (rule conjI)
nipkow@13145
  1549
 apply (clarsimp simp add: set_conv_nth)
nipkow@13145
  1550
 apply (erule_tac x = 0 in allE)
paulson@14208
  1551
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
nipkow@13145
  1552
apply (erule_tac x = "Suc i" in allE)
paulson@14208
  1553
apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145
  1554
done
wenzelm@13114
  1555
nipkow@15110
  1556
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
kleing@14388
  1557
  by (induct xs) auto
kleing@14388
  1558
nipkow@15110
  1559
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388
  1560
proof (induct xs)
kleing@14388
  1561
  case Nil thus ?case by simp
kleing@14388
  1562
next
kleing@14388
  1563
  case (Cons x xs)
kleing@14388
  1564
  show ?case
kleing@14388
  1565
  proof (cases "x \<in> set xs")
kleing@14388
  1566
    case False with Cons show ?thesis by simp
kleing@14388
  1567
  next
kleing@14388
  1568
    case True with Cons.prems
kleing@14388
  1569
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  1570
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  1571
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  1572
    ultimately have False by simp
kleing@14388
  1573
    thus ?thesis ..
kleing@14388
  1574
  qed
kleing@14388
  1575
qed
kleing@14388
  1576
nipkow@15110
  1577
lemma inj_on_setI: "distinct(map f xs) ==> inj_on f (set xs)"
nipkow@15110
  1578
apply(induct xs)
nipkow@15110
  1579
 apply simp
nipkow@15110
  1580
apply fastsimp
nipkow@15110
  1581
done
nipkow@15110
  1582
nipkow@15110
  1583
lemma inj_on_set_conv:
nipkow@15110
  1584
 "distinct xs \<Longrightarrow> inj_on f (set xs) = distinct(map f xs)"
nipkow@15110
  1585
apply(induct xs)
nipkow@15110
  1586
 apply simp
nipkow@15110
  1587
apply fastsimp
nipkow@15110
  1588
done
nipkow@15110
  1589
nipkow@15110
  1590
nipkow@15392
  1591
subsubsection {* @{text remove1} *}
nipkow@15110
  1592
nipkow@15110
  1593
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110
  1594
apply(induct xs)
nipkow@15110
  1595
 apply simp
nipkow@15110
  1596
apply simp
nipkow@15110
  1597
apply blast
nipkow@15110
  1598
done
nipkow@15110
  1599
nipkow@15110
  1600
lemma [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110
  1601
apply(induct xs)
nipkow@15110
  1602
 apply simp
nipkow@15110
  1603
apply simp
nipkow@15110
  1604
apply blast
nipkow@15110
  1605
done
nipkow@15110
  1606
nipkow@15110
  1607
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110
  1608
apply(insert set_remove1_subset)
nipkow@15110
  1609
apply fast
nipkow@15110
  1610
done
nipkow@15110
  1611
nipkow@15110
  1612
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110
  1613
by (induct xs) simp_all
nipkow@15110
  1614
wenzelm@13114
  1615
nipkow@15392
  1616
subsubsection {* @{text replicate} *}
wenzelm@13114
  1617
wenzelm@13142
  1618
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  1619
by (induct n) auto
nipkow@13124
  1620
wenzelm@13142
  1621
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  1622
by (induct n) auto
wenzelm@13114
  1623
wenzelm@13114
  1624
lemma replicate_app_Cons_same:
nipkow@13145
  1625
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  1626
by (induct n) auto
wenzelm@13114
  1627
wenzelm@13142
  1628
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208
  1629
apply (induct n, simp)
nipkow@13145
  1630
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  1631
done
wenzelm@13114
  1632
wenzelm@13142
  1633
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  1634
by (induct n) auto
wenzelm@13114
  1635
wenzelm@13142
  1636
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  1637
by (induct n) auto
wenzelm@13114
  1638
wenzelm@13142
  1639
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  1640
by (induct n) auto
wenzelm@13114
  1641
wenzelm@13142
  1642
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  1643
by (atomize (full), induct n) auto
wenzelm@13114
  1644
wenzelm@13142
  1645
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
paulson@14208
  1646
apply (induct n, simp)
nipkow@13145
  1647
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  1648
done
wenzelm@13114
  1649
wenzelm@13142
  1650
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  1651
by (induct n) auto
wenzelm@13114
  1652
wenzelm@13142
  1653
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  1654
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  1655
wenzelm@13142
  1656
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  1657
by auto
wenzelm@13114
  1658
wenzelm@13142
  1659
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  1660
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  1661
wenzelm@13114
  1662
nipkow@15392
  1663
subsubsection{*@{text rotate1} and @{text rotate}*}
nipkow@15302
  1664
nipkow@15302
  1665
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
nipkow@15302
  1666
by(simp add:rotate1_def)
nipkow@15302
  1667
nipkow@15302
  1668
lemma rotate0[simp]: "rotate 0 = id"
nipkow@15302
  1669
by(simp add:rotate_def)
nipkow@15302
  1670
nipkow@15302
  1671
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
nipkow@15302
  1672
by(simp add:rotate_def)
nipkow@15302
  1673
nipkow@15302
  1674
lemma rotate_add:
nipkow@15302
  1675
  "rotate (m+n) = rotate m o rotate n"
nipkow@15302
  1676
by(simp add:rotate_def funpow_add)
nipkow@15302
  1677
nipkow@15302
  1678
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
nipkow@15302
  1679
by(simp add:rotate_add)
nipkow@15302
  1680
nipkow@15302
  1681
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
nipkow@15302
  1682
by(cases xs) simp_all
nipkow@15302
  1683
nipkow@15302
  1684
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  1685
apply(induct n)
nipkow@15302
  1686
 apply simp
nipkow@15302
  1687
apply (simp add:rotate_def)
nipkow@13145
  1688
done
wenzelm@13114
  1689
nipkow@15302
  1690
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
nipkow@15302
  1691
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1692
nipkow@15302
  1693
lemma rotate_drop_take:
nipkow@15302
  1694
  "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
nipkow@15302
  1695
apply(induct n)
nipkow@15302
  1696
 apply simp
nipkow@15302
  1697
apply(simp add:rotate_def)
nipkow@15302
  1698
apply(cases "xs = []")
nipkow@15302
  1699
 apply (simp)
nipkow@15302
  1700
apply(case_tac "n mod length xs = 0")
nipkow@15302
  1701
 apply(simp add:mod_Suc)
nipkow@15302
  1702
 apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
nipkow@15302
  1703
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
nipkow@15302
  1704
                take_hd_drop linorder_not_le)
nipkow@13145
  1705
done
wenzelm@13114
  1706
nipkow@15302
  1707
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
nipkow@15302
  1708
by(simp add:rotate_drop_take)
nipkow@15302
  1709
nipkow@15302
  1710
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  1711
by(simp add:rotate_drop_take)
nipkow@15302
  1712
nipkow@15302
  1713
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
nipkow@15302
  1714
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1715
nipkow@15302
  1716
lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs"
nipkow@15302
  1717
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  1718
nipkow@15302
  1719
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
nipkow@15302
  1720
by(simp add:rotate1_def split:list.split) blast
nipkow@15302
  1721
nipkow@15302
  1722
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
nipkow@15302
  1723
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  1724
nipkow@15302
  1725
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
nipkow@15302
  1726
by(simp add:rotate_drop_take take_map drop_map)
nipkow@15302
  1727
nipkow@15302
  1728
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
nipkow@15302
  1729
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1730
nipkow@15302
  1731
lemma set_rotate[simp]: "set(rotate n xs) = set xs"
nipkow@15302
  1732
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  1733
nipkow@15302
  1734
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
nipkow@15302
  1735
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1736
nipkow@15302
  1737
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
nipkow@15302
  1738
by (induct n) (simp_all add:rotate_def)
wenzelm@13114
  1739
nipkow@15439
  1740
lemma rotate_rev:
nipkow@15439
  1741
  "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
nipkow@15439
  1742
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  1743
apply(cases "length xs = 0")
nipkow@15439
  1744
 apply simp
nipkow@15439
  1745
apply(cases "n mod length xs = 0")
nipkow@15439
  1746
 apply simp
nipkow@15439
  1747
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  1748
done
nipkow@15439
  1749
wenzelm@13114
  1750
nipkow@15392
  1751
subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  1752
wenzelm@13142
  1753
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  1754
by (auto simp add: sublist_def)
wenzelm@13114
  1755
wenzelm@13142
  1756
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  1757
by (auto simp add: sublist_def)
wenzelm@13114
  1758
nipkow@15281
  1759
lemma length_sublist:
nipkow@15281
  1760
  "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
nipkow@15281
  1761
by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
nipkow@15281
  1762
nipkow@15281
  1763
lemma sublist_shift_lemma_Suc:
nipkow@15281
  1764
  "!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
nipkow@15281
  1765
         map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
nipkow@15281
  1766
apply(induct xs)
nipkow@15281
  1767
 apply simp
nipkow@15281
  1768
apply (case_tac "is")
nipkow@15281
  1769
 apply simp
nipkow@15281
  1770
apply simp
nipkow@15281
  1771
done
nipkow@15281
  1772
wenzelm@13114
  1773
lemma sublist_shift_lemma:
nipkow@15425
  1774
     "map fst [p:zip xs [i..<i + length xs] . snd p : A] =
nipkow@15425
  1775
      map fst [p:zip xs [0..<length xs] . snd p + i : A]"
nipkow@13145
  1776
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  1777
wenzelm@13114
  1778
lemma sublist_append:
paulson@15168
  1779
     "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  1780
apply (unfold sublist_def)
paulson@14208
  1781
apply (induct l' rule: rev_induct, simp)
nipkow@13145
  1782
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  1783
apply (simp add: add_commute)
nipkow@13145
  1784
done
wenzelm@13114
  1785
wenzelm@13114
  1786
lemma sublist_Cons:
nipkow@13145
  1787
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  1788
apply (induct l rule: rev_induct)
nipkow@13145
  1789
 apply (simp add: sublist_def)
nipkow@13145
  1790
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  1791
done
wenzelm@13114
  1792
nipkow@15281
  1793
lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
nipkow@15281
  1794
apply(induct xs)
nipkow@15281
  1795
 apply simp
nipkow@15281
  1796
apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE)
nipkow@15281
  1797
 apply(erule lessE)
nipkow@15281
  1798
  apply auto
nipkow@15281
  1799
apply(erule lessE)
nipkow@15281
  1800
apply auto
nipkow@15281
  1801
done
nipkow@15281
  1802
nipkow@15281
  1803
lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
nipkow@15281
  1804
by(auto simp add:set_sublist)
nipkow@15281
  1805
nipkow@15281
  1806
lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
nipkow@15281
  1807
by(auto simp add:set_sublist)
nipkow@15281
  1808
nipkow@15281
  1809
lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
nipkow@15281
  1810
by(auto simp add:set_sublist)
nipkow@15281
  1811
wenzelm@13142
  1812
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  1813
by (simp add: sublist_Cons)
wenzelm@13114
  1814
nipkow@15281
  1815
nipkow@15281
  1816
lemma distinct_sublistI[simp]: "!!I. distinct xs \<Longrightarrow> distinct(sublist xs I)"
nipkow@15281
  1817
apply(induct xs)
nipkow@15281
  1818
 apply simp
nipkow@15281
  1819
apply(auto simp add:sublist_Cons)
nipkow@15281
  1820
done
nipkow@15281
  1821
nipkow@15281
  1822
nipkow@15045
  1823
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
paulson@14208
  1824
apply (induct l rule: rev_induct, simp)
nipkow@13145
  1825
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  1826
done
wenzelm@13114
  1827
wenzelm@13114
  1828
nipkow@15392
  1829
subsubsection{*Sets of Lists*}
nipkow@15392
  1830
nipkow@15392
  1831
subsubsection {* @{text lists}: the list-forming operator over sets *}
nipkow@15302
  1832
nipkow@15302
  1833
consts lists :: "'a set => 'a list set"
nipkow@15302
  1834
inductive "lists A"
nipkow@15302
  1835
 intros
nipkow@15302
  1836
  Nil [intro!]: "[]: lists A"
nipkow@15302
  1837
  Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A"
nipkow@15302
  1838
nipkow@15302
  1839
inductive_cases listsE [elim!]: "x#l : lists A"
nipkow@15302
  1840
nipkow@15302
  1841
lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B"
nipkow@15302
  1842
by (unfold lists.defs) (blast intro!: lfp_mono)
nipkow@15302
  1843
nipkow@15302
  1844
lemma lists_IntI:
nipkow@15302
  1845
  assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l
nipkow@15302
  1846
  by induct blast+
nipkow@15302
  1847
nipkow@15302
  1848
lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B"
nipkow@15302
  1849
proof (rule mono_Int [THEN equalityI])
nipkow@15302
  1850
  show "mono lists" by (simp add: mono_def lists_mono)
nipkow@15302
  1851
  show "lists A \<inter> lists B \<subseteq> lists (A \<inter> B)" by (blast intro: lists_IntI)
kleing@14388
  1852
qed
kleing@14388
  1853
nipkow@15302
  1854
lemma append_in_lists_conv [iff]:
nipkow@15302
  1855
     "(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"
nipkow@15302
  1856
by (induct xs) auto
nipkow@15302
  1857
nipkow@15302
  1858
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"
nipkow@15302
  1859
-- {* eliminate @{text lists} in favour of @{text set} *}
nipkow@15302
  1860
by (induct xs) auto
nipkow@15302
  1861
nipkow@15302
  1862
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"
nipkow@15302
  1863
by (rule in_lists_conv_set [THEN iffD1])
nipkow@15302
  1864
nipkow@15302
  1865
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"
nipkow@15302
  1866
by (rule in_lists_conv_set [THEN iffD2])
nipkow@15302
  1867
nipkow@15302
  1868
lemma lists_UNIV [simp]: "lists UNIV = UNIV"
nipkow@15302
  1869
by auto
nipkow@15302
  1870
nipkow@15392
  1871
subsubsection{*Lists as Cartesian products*}
nipkow@15302
  1872
nipkow@15302
  1873
text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
nipkow@15302
  1874
@{term A} and tail drawn from @{term Xs}.*}
nipkow@15302
  1875
nipkow@15302
  1876
constdefs
nipkow@15302
  1877
  set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
nipkow@15302
  1878
  "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"
nipkow@15302
  1879
nipkow@15302
  1880
lemma [simp]: "set_Cons A {[]} = (%x. [x])`A"
nipkow@15302
  1881
by (auto simp add: set_Cons_def)
nipkow@15302
  1882
nipkow@15302
  1883
text{*Yields the set of lists, all of the same length as the argument and
nipkow@15302
  1884
with elements drawn from the corresponding element of the argument.*}
nipkow@15302
  1885
nipkow@15302
  1886
consts  listset :: "'a set list \<Rightarrow> 'a list set"
nipkow@15302
  1887
primrec
nipkow@15302
  1888
   "listset []    = {[]}"
nipkow@15302
  1889
   "listset(A#As) = set_Cons A (listset As)"
nipkow@15302
  1890
nipkow@15302
  1891
nipkow@15392
  1892
subsection{*Relations on lists*}
nipkow@15392
  1893
nipkow@15392
  1894
subsubsection {* Lexicographic orderings on lists *}
nipkow@15302
  1895
nipkow@15302
  1896
consts
nipkow@15302
  1897
lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
nipkow@15302
  1898
primrec
nipkow@15302
  1899
"lexn r 0 = {}"
nipkow@15302
  1900
"lexn r (Suc n) =
nipkow@15302
  1901
(prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
nipkow@15302
  1902
{(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
nipkow@15302
  1903
nipkow@15302
  1904
constdefs
nipkow@15302
  1905
lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@15302
  1906
"lex r == \<Union>n. lexn r n"
nipkow@15302
  1907
nipkow@15302
  1908
lexico :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@15302
  1909
"lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
nipkow@15302
  1910
nipkow@15302
  1911
nipkow@15302
  1912
lemma wf_lexn: "wf r ==> wf (lexn r n)"
nipkow@15302
  1913
apply (induct n, simp, simp)
nipkow@15302
  1914
apply(rule wf_subset)
nipkow@15302
  1915
 prefer 2 apply (rule Int_lower1)
nipkow@15302
  1916
apply(rule wf_prod_fun_image)
nipkow@15302
  1917
 prefer 2 apply (rule inj_onI, auto)
nipkow@15302
  1918
done
nipkow@15302
  1919
nipkow@15302
  1920
lemma lexn_length:
nipkow@15302
  1921
     "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@15302
  1922
by (induct n) auto
nipkow@15302
  1923
nipkow@15302
  1924
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@15302
  1925
apply (unfold lex_def)
nipkow@15302
  1926
apply (rule wf_UN)
nipkow@15302
  1927
apply (blast intro: wf_lexn, clarify)
nipkow@15302
  1928
apply (rename_tac m n)
nipkow@15302
  1929
apply (subgoal_tac "m \<noteq> n")
nipkow@15302
  1930
 prefer 2 apply blast
nipkow@15302
  1931
apply (blast dest: lexn_length not_sym)
nipkow@15302
  1932
done
nipkow@15302
  1933
nipkow@15302
  1934
lemma lexn_conv:
nipkow@15302
  1935
"lexn r n =
nipkow@15302
  1936
{(xs,ys). length xs = n \<and> length ys = n \<and>
nipkow@15302
  1937
(\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
nipkow@15302
  1938
apply (induct n, simp, blast)
nipkow@15302
  1939
apply (simp add: image_Collect lex_prod_def, safe, blast)
nipkow@15302
  1940
 apply (rule_tac x = "ab # xys" in exI, simp)
nipkow@15302
  1941
apply (case_tac xys, simp_all, blast)
nipkow@15302
  1942
done
nipkow@15302
  1943
nipkow@15302
  1944
lemma lex_conv:
nipkow@15302
  1945
"lex r =
nipkow@15302
  1946
{(xs,ys). length xs = length ys \<and>
nipkow@15302
  1947
(\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@15302
  1948
by (force simp add: lex_def lexn_conv)
nipkow@15302
  1949
nipkow@15302
  1950
lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)"
nipkow@15302
  1951
by (unfold lexico_def) blast
nipkow@15302
  1952
nipkow@15302
  1953
lemma lexico_conv:
nipkow@15302
  1954
"lexico r = {(xs,ys). length xs < length ys |
nipkow@15302
  1955
length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@15302
  1956
by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def)
nipkow@15302
  1957
nipkow@15302
  1958
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@15302
  1959
by (simp add: lex_conv)
nipkow@15302
  1960
nipkow@15302
  1961
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@15302
  1962
by (simp add:lex_conv)
nipkow@15302
  1963
nipkow@15302
  1964
lemma Cons_in_lex [iff]:
nipkow@15302
  1965
"((x # xs, y # ys) : lex r) =
nipkow@15302
  1966
((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@15302
  1967
apply (simp add: lex_conv)
nipkow@15302
  1968
apply (rule iffI)
nipkow@15302
  1969
 prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
nipkow@15302
  1970
apply (case_tac xys, simp, simp)
nipkow@15302
  1971
apply blast
nipkow@15302
  1972
done
nipkow@15302
  1973
nipkow@15302
  1974
nipkow@15392
  1975
subsubsection{*Lifting a Relation on List Elements to the Lists*}
nipkow@15302
  1976
nipkow@15302
  1977
consts  listrel :: "('a * 'a)set => ('a list * 'a list)set"
nipkow@15302
  1978
nipkow@15302
  1979
inductive "listrel(r)"
nipkow@15302
  1980
 intros
nipkow@15302
  1981
   Nil:  "([],[]) \<in> listrel r"
nipkow@15302
  1982
   Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
nipkow@15302
  1983
nipkow@15302
  1984
inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
nipkow@15302
  1985
inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
nipkow@15302
  1986
inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
nipkow@15302
  1987
inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
nipkow@15302
  1988
nipkow@15302
  1989
nipkow@15302
  1990
lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
nipkow@15302
  1991
apply clarify  
nipkow@15302
  1992
apply (erule listrel.induct)
nipkow@15302
  1993
apply (blast intro: listrel.intros)+
nipkow@15302
  1994
done
nipkow@15302
  1995
nipkow@15302
  1996
lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
nipkow@15302
  1997
apply clarify 
nipkow@15302
  1998
apply (erule listrel.induct, auto) 
nipkow@15302
  1999
done
nipkow@15302
  2000
nipkow@15302
  2001
lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" 
nipkow@15302
  2002
apply (simp add: refl_def listrel_subset Ball_def)
nipkow@15302
  2003
apply (rule allI) 
nipkow@15302
  2004
apply (induct_tac x) 
nipkow@15302
  2005
apply (auto intro: listrel.intros)
nipkow@15302
  2006
done
nipkow@15302
  2007
nipkow@15302
  2008
lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
nipkow@15302
  2009
apply (auto simp add: sym_def)
nipkow@15302
  2010
apply (erule listrel.induct) 
nipkow@15302
  2011
apply (blast intro: listrel.intros)+
nipkow@15302
  2012
done
nipkow@15302
  2013
nipkow@15302
  2014
lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
nipkow@15302
  2015
apply (simp add: trans_def)
nipkow@15302
  2016
apply (intro allI) 
nipkow@15302
  2017
apply (rule impI) 
nipkow@15302
  2018
apply (erule listrel.induct) 
nipkow@15302
  2019
apply (blast intro: listrel.intros)+
nipkow@15302
  2020
done
nipkow@15302
  2021
nipkow@15302
  2022
theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
nipkow@15302
  2023
by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) 
nipkow@15302
  2024
nipkow@15302
  2025
lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
nipkow@15302
  2026
by (blast intro: listrel.intros)
nipkow@15302
  2027
nipkow@15302
  2028
lemma listrel_Cons:
nipkow@15302
  2029
     "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})";
nipkow@15302
  2030
by (auto simp add: set_Cons_def intro: listrel.intros) 
nipkow@15302
  2031
nipkow@15302
  2032
nipkow@15392
  2033
subsection{*Miscellany*}
nipkow@15392
  2034
nipkow@15392
  2035
subsubsection {* Characters and strings *}
wenzelm@13366
  2036
wenzelm@13366
  2037
datatype nibble =
wenzelm@13366
  2038
    Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
wenzelm@13366
  2039
  | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
wenzelm@13366
  2040
wenzelm@13366
  2041
datatype char = Char nibble nibble
wenzelm@13366
  2042
  -- "Note: canonical order of character encoding coincides with standard term ordering"
wenzelm@13366
  2043
wenzelm@13366
  2044
types string = "char list"
wenzelm@13366
  2045
wenzelm@13366
  2046
syntax
wenzelm@13366
  2047
  "_Char" :: "xstr => char"    ("CHR _")
wenzelm@13366
  2048
  "_String" :: "xstr => string"    ("_")
wenzelm@13366
  2049
wenzelm@13366
  2050
parse_ast_translation {*
wenzelm@13366
  2051
  let
wenzelm@13366
  2052
    val constants = Syntax.Appl o map Syntax.Constant;
wenzelm@13366
  2053
wenzelm@13366
  2054
    fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10));
wenzelm@13366
  2055
    fun mk_char c =
wenzelm@13366
  2056
      if Symbol.is_ascii c andalso Symbol.is_printable c then
wenzelm@13366
  2057
        constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)]
wenzelm@13366
  2058
      else error ("Printable ASCII character expected: " ^ quote c);
wenzelm@13366
  2059
wenzelm@13366
  2060
    fun mk_string [] = Syntax.Constant "Nil"
wenzelm@13366
  2061
      | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs];
wenzelm@13366
  2062
wenzelm@13366
  2063
    fun char_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  2064
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  2065
          [c] => mk_char c
wenzelm@13366
  2066
        | _ => error ("Single character expected: " ^ xstr))
wenzelm@13366
  2067
      | char_ast_tr asts = raise AST ("char_ast_tr", asts);
wenzelm@13366
  2068
wenzelm@13366
  2069
    fun string_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  2070
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  2071
          [] => constants [Syntax.constrainC, "Nil", "string"]
wenzelm@13366
  2072
        | cs => mk_string cs)
wenzelm@13366
  2073
      | string_ast_tr asts = raise AST ("string_tr", asts);
wenzelm@13366
  2074
  in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end;
wenzelm@13366
  2075
*}
wenzelm@13366
  2076
berghofe@15064
  2077
ML {*
berghofe@15064
  2078
fun int_of_nibble h =
berghofe@15064
  2079
  if "0" <= h andalso h <= "9" then ord h - ord "0"
berghofe@15064
  2080
  else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10
berghofe@15064
  2081
  else raise Match;
berghofe@15064
  2082
berghofe@15064
  2083
fun nibble_of_int i =
berghofe@15064
  2084
  if i <= 9 then chr (ord "0" + i) else chr (ord "A" + i - 10);
berghofe@15064
  2085
*}
berghofe@15064
  2086
wenzelm@13366
  2087
print_ast_translation {*
wenzelm@13366
  2088
  let
wenzelm@13366
  2089
    fun dest_nib (Syntax.Constant c) =
wenzelm@13366
  2090
        (case explode c of
berghofe@15064
  2091
          ["N", "i", "b", "b", "l", "e", h] => int_of_nibble h
wenzelm@13366
  2092
        | _ => raise Match)
wenzelm@13366
  2093
      | dest_nib _ = raise Match;
wenzelm@13366
  2094
wenzelm@13366
  2095
    fun dest_chr c1 c2 =
wenzelm@13366
  2096
      let val c = chr (dest_nib c1 * 16 + dest_nib c2)
wenzelm@13366
  2097
      in if Symbol.is_printable c then c else raise Match end;
wenzelm@13366
  2098
wenzelm@13366
  2099
    fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2
wenzelm@13366
  2100
      | dest_char _ = raise Match;
wenzelm@13366
  2101
wenzelm@13366
  2102
    fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)];
wenzelm@13366
  2103
wenzelm@13366
  2104
    fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]]
wenzelm@13366
  2105
      | char_ast_tr' _ = raise Match;
wenzelm@13366
  2106
wenzelm@13366
  2107
    fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String",
wenzelm@13366
  2108
            xstr (map dest_char (Syntax.unfold_ast "_args" args))]
wenzelm@13366
  2109
      | list_ast_tr' ts = raise Match;
wenzelm@13366
  2110
  in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end;
wenzelm@13366
  2111
*}
wenzelm@13366
  2112
nipkow@15392
  2113
subsubsection {* Code generator setup *}
berghofe@15064
  2114
berghofe@15064
  2115
ML {*
berghofe@15064
  2116
local
berghofe@15064
  2117
berghofe@15064
  2118
fun list_codegen thy gr dep b t =
berghofe@15064
  2119
  let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy dep false)
berghofe@15064
  2120
    (gr, HOLogic.dest_list t)
berghofe@15064
  2121
  in Some (gr', Pretty.list "[" "]" ps) end handle TERM _ => None;
berghofe@15064
  2122
berghofe@15064
  2123
fun dest_nibble (Const (s, _)) = int_of_nibble (unprefix "List.nibble.Nibble" s)
berghofe@15064
  2124
  | dest_nibble _ = raise Match;
berghofe@15064
  2125
berghofe@15064
  2126
fun char_codegen thy gr dep b (Const ("List.char.Char", _) $ c1 $ c2) =
berghofe@15064
  2127
    (let val c = chr (dest_nibble c1 * 16 + dest_nibble c2)
berghofe@15064
  2128
     in if Symbol.is_printable c then Some (gr, Pretty.quote (Pretty.str c))
berghofe@15064
  2129
       else None
berghofe@15064
  2130
     end handle LIST _ => None | Match => None)
berghofe@15064
  2131
  | char_codegen thy gr dep b _ = None;
berghofe@15064
  2132
berghofe@15064
  2133
in
berghofe@15064
  2134
berghofe@15064
  2135
val list_codegen_setup =
berghofe@15064
  2136
  [Codegen.add_codegen "list_codegen" list_codegen,
berghofe@15064
  2137
   Codegen.add_codegen "char_codegen" char_codegen];
berghofe@15064
  2138
berghofe@15064
  2139
end;
berghofe@15064
  2140
berghofe@15064
  2141
val term_of_list = HOLogic.mk_list;
berghofe@15064
  2142
berghofe@15064
  2143
fun gen_list' aG i j = frequency
berghofe@15064
  2144
  [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] ()
berghofe@15064
  2145
and gen_list aG i = gen_list' aG i i;
berghofe@15064
  2146
berghofe@15064
  2147
val nibbleT = Type ("List.nibble", []);
berghofe@15064
  2148
berghofe@15064
  2149
fun term_of_char c =
berghofe@15064
  2150
  Const ("List.char.Char", nibbleT --> nibbleT --> Type ("List.char", [])) $
berghofe@15064
  2151
    Const ("List.nibble.Nibble" ^ nibble_of_int (ord c div 16), nibbleT) $
berghofe@15064
  2152
    Const ("List.nibble.Nibble" ^ nibble_of_int (ord c mod 16), nibbleT);
berghofe@15064
  2153
berghofe@15064
  2154
fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z")));
berghofe@15064
  2155
*}
berghofe@15064
  2156
berghofe@15064
  2157
types_code
berghofe@15064
  2158
  "list" ("_ list")
berghofe@15064
  2159
  "char" ("string")
berghofe@15064
  2160
berghofe@15064
  2161
consts_code "Cons" ("(_ ::/ _)")
berghofe@15064
  2162
berghofe@15064
  2163
setup list_codegen_setup
berghofe@15064
  2164
wenzelm@13122
  2165
end