src/HOL/List.thy
author nipkow
Fri, 20 Feb 2004 01:32:59 +0100
changeset 14402 4201e1916482
parent 14395 cc96cc06abf9
child 14495 e2a1c31cf6d3
permissions -rw-r--r--
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
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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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    License:    GPL (GNU GENERAL PUBLIC LICENSE)
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*)
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header {* The datatype of finite lists *}
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theory List = PreList:
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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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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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  fold_rel :: "('a * 'c * 'a) set => ('a * 'c list * 'a) set"
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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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  o2l :: "'a option => 'a list"
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  list_all:: "('a => bool) => ('a list => bool)"
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  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool"
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  map :: "('a=>'b) => ('a list => 'b list)"
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  mem :: "'a => 'a list => bool"    (infixl 55)
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  nth :: "'a list => nat => 'a"    (infixl "!" 100)
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  list_update :: "'a list => nat => 'a => 'a list"
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  take:: "nat => 'a list => 'a list"
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  drop:: "nat => 'a list => 'a list"
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  takeWhile :: "('a => bool) => 'a list => 'a list"
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  dropWhile :: "('a => bool) => 'a list => 'a list"
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  rev :: "'a list => 'a list"
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  zip :: "'a list => 'b list => ('a * 'b) list"
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  upt :: "nat => nat => nat list" ("(1[_../_'(])")
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  remdups :: "'a list => 'a list"
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  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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  postfix :: "'a list => 'a list => bool"
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syntax (xsymbols)
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  postfix :: "'a list => 'a list => bool"             ("(_/ \<sqsupseteq> _)" [51, 51] 50)
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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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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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 "o2l  None    = []"
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 "o2l (Some x) = [x]"
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primrec
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list_all_Nil:"list_all P [] = True"
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list_all_Cons: "list_all P (x#xs) = (P(x) \<and> list_all P xs)"
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primrec
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"map f [] = []"
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"map f (x#xs) = f(x)#map f xs"
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primrec
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append_Nil:"[]@ys = ys"
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append_Cons: "(x#xs)@ys = x#(xs@ys)"
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primrec
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"rev([]) = []"
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"rev(x#xs) = rev(xs) @ [x]"
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primrec
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"filter P [] = []"
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"filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
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primrec
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foldl_Nil:"foldl f a [] = a"
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foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
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primrec
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"foldr f [] a = a"
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"foldr f (x#xs) a = f x (foldr f xs a)"
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primrec
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"concat([]) = []"
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"concat(x#xs) = x @ concat(xs)"
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primrec
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drop_Nil:"drop n [] = []"
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drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
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-- {* Warning: simpset does not contain this definition *}
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-- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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take_Nil:"take n [] = []"
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take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
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-- {* Warning: simpset does not contain this definition *}
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-- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
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-- {* Warning: simpset does not contain this definition *}
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-- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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"[][i:=v] = []"
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"(x#xs)[i:=v] =
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(case i of 0 => v # xs
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| Suc j => x # xs[j:=v])"
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primrec
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"takeWhile P [] = []"
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"takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
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primrec
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"dropWhile P [] = []"
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"dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
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primrec
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"zip xs [] = []"
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zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
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-- {* Warning: simpset does not contain this definition *}
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-- {* but separate theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
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primrec
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upt_0: "[i..0(] = []"
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upt_Suc: "[i..(Suc j)(] = (if i <= j then [i..j(] @ [j] else [])"
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primrec
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"distinct [] = True"
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"distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
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primrec
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"remdups [] = []"
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"remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
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primrec
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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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 postfix_def: "postfix xs ys == \<exists>zs. xs = zs @ ys"
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defs
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 list_all2_def:
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 "list_all2 P xs ys == length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)"
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subsection {* Lexicographic orderings on lists *}
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consts
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lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
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primrec
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"lexn r 0 = {}"
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"lexn r (Suc n) =
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(prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
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{(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
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constdefs
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lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
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"lex r == \<Union>n. lexn r n"
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lexico :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
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"lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
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sublist :: "'a list => nat set => 'a list"
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"sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..size xs(]))"
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lemma not_Cons_self [simp]: "xs \<noteq> x # xs"
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by (induct xs) auto
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lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
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lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
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by (induct xs) auto
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lemma length_induct:
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"(!!xs. \<forall>ys. length ys < length xs --> P ys ==> P xs) ==> P xs"
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by (rule measure_induct [of length]) rules
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subsection {* @{text lists}: the list-forming operator over sets *}
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consts lists :: "'a set => 'a list set"
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inductive "lists A"
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intros
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Nil [intro!]: "[]: lists A"
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Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A"
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inductive_cases listsE [elim!]: "x#l : lists A"
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lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B"
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by (unfold lists.defs) (blast intro!: lfp_mono)
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lemma lists_IntI:
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  assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l
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  by induct blast+
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lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B"
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apply (rule mono_Int [THEN equalityI])
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apply (simp add: mono_def lists_mono)
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apply (blast intro!: lists_IntI)
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done
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lemma append_in_lists_conv [iff]:
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"(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"
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by (induct xs) auto
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subsection {* @{text length} *}
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text {*
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Needs to come before @{text "@"} because of theorem @{text
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append_eq_append_conv}.
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*}
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lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
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by (induct xs) auto
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lemma length_map [simp]: "length (map f xs) = length xs"
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by (induct xs) auto
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lemma length_rev [simp]: "length (rev xs) = length xs"
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by (induct xs) auto
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lemma length_tl [simp]: "length (tl xs) = length xs - 1"
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by (cases xs) auto
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lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
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by (induct xs) auto
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lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
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by (induct xs) auto
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lemma length_Suc_conv:
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"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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by (induct xs) auto
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lemma Suc_length_conv:
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"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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apply (induct xs, simp, simp)
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apply blast
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done
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lemma impossible_Cons [rule_format]: 
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  "length xs <= length ys --> xs = x # ys = False"
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apply (induct xs, auto)
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done
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lemma list_induct2[consumes 1]: "\<And>ys.
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 \<lbrakk> length xs = length ys;
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   P [] [];
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   \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
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 \<Longrightarrow> P xs ys"
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apply(induct xs)
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 apply simp
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apply(case_tac ys)
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 apply simp
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apply(simp)
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done
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subsection {* @{text "@"} -- append *}
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lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
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by (induct xs) auto
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lemma append_Nil2 [simp]: "xs @ [] = xs"
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by (induct xs) auto
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lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
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by (induct xs) auto
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lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
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by (induct xs) auto
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lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
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by (induct xs) auto
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lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
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by (induct xs) auto
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lemma append_eq_append_conv [simp]:
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 "!!ys. length xs = length ys \<or> length us = length vs
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 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
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apply (induct xs)
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 apply (case_tac ys, simp, force)
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apply (case_tac ys, force, simp)
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done
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lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
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by simp
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lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
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by simp
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lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
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by simp
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lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
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using append_same_eq [of _ _ "[]"] by auto
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lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
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using append_same_eq [of "[]"] by auto
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lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
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by (induct xs) auto
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lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
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by (induct xs) auto
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lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
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by (simp add: hd_append split: list.split)
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lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
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by (simp split: list.split)
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lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
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by (simp add: tl_append split: list.split)
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   364
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lemma Cons_eq_append_conv: "x#xs = ys@zs =
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 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
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by(cases ys) auto
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text {* Trivial rules for solving @{text "@"}-equations automatically. *}
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lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
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by simp
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lemma Cons_eq_appendI:
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"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
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by (drule sym) simp
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lemma append_eq_appendI:
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"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
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by (drule sym) simp
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   384
text {*
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Simplification procedure for all list equalities.
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Currently only tries to rearrange @{text "@"} to see if
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- both lists end in a singleton list,
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- or both lists end in the same list.
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*}
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   391
ML_setup {*
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   392
local
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   393
13122
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val append_assoc = thm "append_assoc";
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   395
val append_Nil = thm "append_Nil";
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   396
val append_Cons = thm "append_Cons";
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   397
val append1_eq_conv = thm "append1_eq_conv";
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   398
val append_same_eq = thm "append_same_eq";
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diff changeset
   399
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fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
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  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
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   402
  | last (Const("List.op @",_) $ _ $ ys) = last ys
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   403
  | last t = t;
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   404
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   405
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
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  | list1 _ = false;
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   407
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   408
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
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  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
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   410
  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
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  | butlast xs = Const("List.list.Nil",fastype_of xs);
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   413
val rearr_tac =
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   414
  simp_tac (HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]);
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   415
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   416
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
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   417
  let
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   418
    val lastl = last lhs and lastr = last rhs;
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   419
    fun rearr conv =
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   420
      let
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   421
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
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   422
        val Type(_,listT::_) = eqT
56610e2ba220 sane interface for simprocs;
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   423
        val appT = [listT,listT] ---> listT
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   424
        val app = Const("List.op @",appT)
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   425
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
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diff changeset
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        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
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diff changeset
   427
        val thm = Tactic.prove sg [] [] eq (K (rearr_tac 1));
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diff changeset
   428
      in Some ((conv RS (thm RS trans)) RS eq_reflection) end;
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diff changeset
   429
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diff changeset
   430
  in
56610e2ba220 sane interface for simprocs;
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   431
    if list1 lastl andalso list1 lastr then rearr append1_eq_conv
56610e2ba220 sane interface for simprocs;
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   432
    else if lastl aconv lastr then rearr append_same_eq
56610e2ba220 sane interface for simprocs;
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   433
    else None
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diff changeset
   434
  end;
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   435
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   436
in
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diff changeset
   437
56610e2ba220 sane interface for simprocs;
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   438
val list_eq_simproc =
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   439
  Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq;
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   440
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   441
end;
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   442
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   443
Addsimprocs [list_eq_simproc];
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   444
*}
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   445
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   446
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   447
subsection {* @{text map} *}
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diff changeset
   448
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   449
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
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   450
by (induct xs) simp_all
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diff changeset
   451
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   452
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
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   453
by (rule ext, induct_tac xs) auto
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diff changeset
   454
13142
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   455
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
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   456
by (induct xs) auto
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   457
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   458
lemma map_compose: "map (f o g) xs = map f (map g xs)"
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   459
by (induct xs) (auto simp add: o_def)
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   460
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diff changeset
   461
lemma rev_map: "rev (map f xs) = map f (rev xs)"
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   462
by (induct xs) auto
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   463
13737
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nipkow
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   464
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
e564c3d2d174 added a few lemmas
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   465
by (induct xs) auto
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nipkow
parents: 13601
diff changeset
   466
13366
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diff changeset
   467
lemma map_cong [recdef_cong]:
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   468
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
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diff changeset
   469
-- {* a congruence rule for @{text map} *}
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diff changeset
   470
by simp
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diff changeset
   471
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parents: 13124
diff changeset
   472
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
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diff changeset
   473
by (cases xs) auto
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diff changeset
   474
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   475
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
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   476
by (cases xs) auto
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diff changeset
   477
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   478
lemma map_eq_Cons_conv[iff]:
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   479
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
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diff changeset
   480
by (cases xs) auto
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parents: 12887
diff changeset
   481
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   482
lemma Cons_eq_map_conv[iff]:
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nipkow
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diff changeset
   483
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
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parents: 13913
diff changeset
   484
by (cases ys) auto
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nipkow
parents: 13913
diff changeset
   485
14111
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nipkow
parents: 14099
diff changeset
   486
lemma ex_map_conv:
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   487
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   488
by(induct ys, auto)
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   489
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   490
lemma map_injective:
14338
a1add2de7601 *** empty log message ***
nipkow
parents: 14328
diff changeset
   491
 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
a1add2de7601 *** empty log message ***
nipkow
parents: 14328
diff changeset
   492
by (induct ys) (auto dest!:injD)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   493
14339
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   494
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   495
by(blast dest:map_injective)
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   496
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   497
lemma inj_mapI: "inj f ==> inj (map f)"
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 13508
diff changeset
   498
by (rules dest: map_injective injD intro: inj_onI)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   499
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   500
lemma inj_mapD: "inj (map f) ==> inj f"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   501
apply (unfold inj_on_def, clarify)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   502
apply (erule_tac x = "[x]" in ballE)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   503
 apply (erule_tac x = "[y]" in ballE, simp, blast)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   504
apply blast
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   505
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   506
14339
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   507
lemma inj_map[iff]: "inj (map f) = inj f"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   508
by (blast dest: inj_mapD intro: inj_mapI)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   509
14343
6bc647f472b9 map_idI
kleing
parents: 14339
diff changeset
   510
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
6bc647f472b9 map_idI
kleing
parents: 14339
diff changeset
   511
by (induct xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   512
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   513
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   514
by (induct xs) auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   515
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   516
subsection {* @{text rev} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   517
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   518
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   519
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   520
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   521
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   522
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   523
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   524
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   525
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   526
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   527
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   528
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   529
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   530
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   531
apply (induct xs, force)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   532
apply (case_tac ys, simp, force)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   533
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   534
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   535
lemma rev_induct [case_names Nil snoc]:
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   536
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   537
apply(subst rev_rev_ident[symmetric])
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   538
apply(rule_tac list = "rev xs" in list.induct, simp_all)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   539
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   540
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   541
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   542
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   543
lemma rev_exhaust [case_names Nil snoc]:
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   544
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   545
by (induct xs rule: rev_induct) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   546
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   547
lemmas rev_cases = rev_exhaust
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   548
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   549
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   550
subsection {* @{text set} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   551
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   552
lemma finite_set [iff]: "finite (set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   553
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   554
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   555
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   556
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   557
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   558
lemma hd_in_set: "l = x#xs \<Longrightarrow> x\<in>set l"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   559
by (case_tac l, auto)
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   560
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   561
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   562
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   563
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   564
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   565
by auto
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   566
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   567
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   568
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   569
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   570
lemma set_rev [simp]: "set (rev xs) = set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   571
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   572
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   573
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   574
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   575
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   576
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   577
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   578
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   579
lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> k < j}"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   580
apply (induct j, simp_all)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   581
apply (erule ssubst, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   582
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   583
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   584
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   585
apply (induct xs, simp, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   586
apply (rule iffI)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   587
 apply (blast intro: eq_Nil_appendI Cons_eq_appendI)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   588
apply (erule exE)+
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   589
apply (case_tac ys, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   590
done
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   591
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   592
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   593
-- {* eliminate @{text lists} in favour of @{text set} *}
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   594
by (induct xs) auto
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   595
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   596
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   597
by (rule in_lists_conv_set [THEN iffD1])
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   598
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   599
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   600
by (rule in_lists_conv_set [THEN iffD2])
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   601
13508
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   602
lemma finite_list: "finite A ==> EX l. set l = A"
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   603
apply (erule finite_induct, auto)
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   604
apply (rule_tac x="x#l" in exI, auto)
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   605
done
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   606
14388
04f04408b99b lemmas about card (set xs)
kleing
parents: 14343
diff changeset
   607
lemma card_length: "card (set xs) \<le> length xs"
04f04408b99b lemmas about card (set xs)
kleing
parents: 14343
diff changeset
   608
by (induct xs) (auto simp add: card_insert_if)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   609
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   610
subsection {* @{text mem} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   611
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   612
lemma set_mem_eq: "(x mem xs) = (x : set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   613
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   614
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   615
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   616
subsection {* @{text list_all} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   617
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   618
lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   619
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   620
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   621
lemma list_all_append [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   622
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   623
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   624
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   625
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   626
subsection {* @{text filter} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   627
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   628
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   629
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   630
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   631
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   632
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   633
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   634
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   635
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   636
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   637
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   638
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   639
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   640
lemma length_filter [simp]: "length (filter P xs) \<le> length xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   641
by (induct xs) (auto simp add: le_SucI)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   642
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   643
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   644
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   645
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   646
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   647
subsection {* @{text concat} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   648
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   649
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   650
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   651
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   652
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   653
by (induct xss) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   654
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   655
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   656
by (induct xss) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   657
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   658
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   659
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   660
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   661
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   662
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   663
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   664
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   665
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   666
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   667
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   668
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   669
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   670
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   671
subsection {* @{text nth} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   672
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   673
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   674
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   675
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   676
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   677
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   678
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   679
declare nth.simps [simp del]
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   680
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   681
lemma nth_append:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   682
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   683
apply (induct "xs", simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   684
apply (case_tac n, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   685
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   686
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   687
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   688
by (induct "xs") auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   689
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   690
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   691
by (induct "xs") auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   692
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   693
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   694
apply (induct xs, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   695
apply (case_tac n, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   696
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   697
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   698
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   699
apply (induct_tac xs, simp, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   700
apply safe
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   701
apply (rule_tac x = 0 in exI, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   702
 apply (rule_tac x = "Suc i" in exI, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   703
apply (case_tac i, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   704
apply (rename_tac j)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   705
apply (rule_tac x = j in exI, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   706
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   707
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   708
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   709
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   710
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   711
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   712
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   713
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   714
lemma all_nth_imp_all_set:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   715
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   716
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   717
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   718
lemma all_set_conv_all_nth:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   719
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   720
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   721
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   722
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   723
subsection {* @{text list_update} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   724
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   725
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   726
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   727
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   728
lemma nth_list_update:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   729
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   730
by (induct xs) (auto simp add: nth_Cons split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   731
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   732
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   733
by (simp add: nth_list_update)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   734
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   735
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   736
by (induct xs) (auto simp add: nth_Cons split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   737
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   738
lemma list_update_overwrite [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   739
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   740
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   741
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   742
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   743
apply (induct xs, simp)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   744
apply(simp split:nat.splits)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   745
done
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   746
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   747
lemma list_update_same_conv:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   748
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   749
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   750
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   751
lemma list_update_append1:
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   752
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   753
apply (induct xs, simp)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   754
apply(simp split:nat.split)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   755
done
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   756
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   757
lemma list_update_length [simp]:
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   758
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   759
by (induct xs, auto)
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   760
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   761
lemma update_zip:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   762
"!!i xy xs. length xs = length ys ==>
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   763
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   764
by (induct ys) (auto, case_tac xs, auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   765
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   766
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   767
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   768
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   769
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   770
by (blast dest!: set_update_subset_insert [THEN subsetD])
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   771
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   772
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   773
subsection {* @{text last} and @{text butlast} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   774
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   775
lemma last_snoc [simp]: "last (xs @ [x]) = x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   776
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   777
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   778
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   779
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   780
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
   781
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
   782
by(simp add:last.simps)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
   783
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
   784
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
   785
by(simp add:last.simps)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
   786
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
   787
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
   788
by (induct xs) (auto)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
   789
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
   790
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
   791
by(simp add:last_append)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
   792
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
   793
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
   794
by(simp add:last_append)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
   795
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
   796
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
   797
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   798
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   799
by (induct xs rule: rev_induct) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   800
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   801
lemma butlast_append:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   802
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   803
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   804
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   805
lemma append_butlast_last_id [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   806
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   807
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   808
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   809
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   810
by (induct xs) (auto split: split_if_asm)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   811
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   812
lemma in_set_butlast_appendI:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   813
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   814
by (auto dest: in_set_butlastD simp add: butlast_append)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   815
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   816
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   817
subsection {* @{text take} and @{text drop} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   818
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   819
lemma take_0 [simp]: "take 0 xs = []"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   820
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   821
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   822
lemma drop_0 [simp]: "drop 0 xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   823
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   824
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   825
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   826
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   827
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   828
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   829
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   830
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   831
declare take_Cons [simp del] and drop_Cons [simp del]
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   832
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   833
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   834
by(cases xs, simp_all)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   835
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   836
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   837
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   838
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   839
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   840
apply (induct xs, simp)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   841
apply(simp add:drop_Cons nth_Cons split:nat.splits)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   842
done
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   843
13913
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
   844
lemma take_Suc_conv_app_nth:
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
   845
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   846
apply (induct xs, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   847
apply (case_tac i, auto)
13913
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
   848
done
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
   849
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   850
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   851
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   852
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   853
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   854
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   855
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   856
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   857
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   858
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   859
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   860
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   861
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   862
lemma take_append [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   863
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   864
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   865
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   866
lemma drop_append [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   867
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   868
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   869
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   870
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   871
apply (induct m, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   872
apply (case_tac xs, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   873
apply (case_tac na, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   874
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   875
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   876
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   877
apply (induct m, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   878
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   879
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   880
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   881
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   882
apply (induct m, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   883
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   884
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   885
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   886
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   887
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   888
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   889
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   890
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   891
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   892
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   893
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   894
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   895
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   896
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   897
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   898
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   899
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   900
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   901
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   902
apply (induct xs, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   903
apply (case_tac i, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   904
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   905
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   906
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   907
apply (induct xs, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   908
apply (case_tac i, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   909
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   910
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   911
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   912
apply (induct xs, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   913
apply (case_tac n, blast)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   914
apply (case_tac i, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   915
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   916
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   917
lemma nth_drop [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   918
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   919
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   920
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   921
done
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   922
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   923
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   924
by(induct xs)(auto simp:take_Cons split:nat.split)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   925
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   926
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   927
by(induct xs)(auto simp:drop_Cons split:nat.split)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   928
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   929
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   930
using set_take_subset by fast
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   931
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   932
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   933
using set_drop_subset by fast
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   934
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   935
lemma append_eq_conv_conj:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   936
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   937
apply (induct xs, simp, clarsimp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   938
apply (case_tac zs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   939
done
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   940
14050
826037db30cd new theorem
paulson
parents: 14025
diff changeset
   941
lemma take_add [rule_format]: 
826037db30cd new theorem
paulson
parents: 14025
diff changeset
   942
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
826037db30cd new theorem
paulson
parents: 14025
diff changeset
   943
apply (induct xs, auto) 
826037db30cd new theorem
paulson
parents: 14025
diff changeset
   944
apply (case_tac i, simp_all) 
826037db30cd new theorem
paulson
parents: 14025
diff changeset
   945
done
826037db30cd new theorem
paulson
parents: 14025
diff changeset
   946
14300
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   947
lemma append_eq_append_conv_if:
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   948
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   949
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   950
   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
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   951
   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)"
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   952
apply(induct xs\<^isub>1)
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   953
 apply simp
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   954
apply(case_tac ys\<^isub>1)
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   955
apply simp_all
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   956
done
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   957
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   958
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   959
subsection {* @{text takeWhile} and @{text dropWhile} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   960
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   961
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   962
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   963
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   964
lemma takeWhile_append1 [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   965
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   966
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   967
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   968
lemma takeWhile_append2 [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   969
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   970
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   971
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   972
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   973
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   974
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   975
lemma dropWhile_append1 [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   976
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   977
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   978
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   979
lemma dropWhile_append2 [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   980
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   981
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   982
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   983
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   984
by (induct xs) (auto split: split_if_asm)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   985
13913
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
   986
lemma takeWhile_eq_all_conv[simp]:
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
   987
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
   988
by(induct xs, auto)
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
   989
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
   990
lemma dropWhile_eq_Nil_conv[simp]:
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
   991
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
   992
by(induct xs, auto)
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
   993
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
   994
lemma dropWhile_eq_Cons_conv:
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
   995
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
   996
by(induct xs, auto)
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
   997
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   998
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   999
subsection {* @{text zip} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1000
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1001
lemma zip_Nil [simp]: "zip [] ys = []"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1002
by (induct ys) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1003
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1004
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1005
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1006
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1007
declare zip_Cons [simp del]
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1008
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1009
lemma length_zip [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1010
"!!xs. length (zip xs ys) = min (length xs) (length ys)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1011
apply (induct ys, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1012
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1013
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1014
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1015
lemma zip_append1:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1016
"!!xs. zip (xs @ ys) zs =
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1017
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1018
apply (induct zs, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1019
apply (case_tac xs, simp_all)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1020
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1021
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1022
lemma zip_append2:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1023
"!!ys. zip xs (ys @ zs) =
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1024
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1025
apply (induct xs, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1026
apply (case_tac ys, simp_all)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1027
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1028
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1029
lemma zip_append [simp]:
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1030
 "[| length xs = length us; length ys = length vs |] ==>
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1031
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1032
by (simp add: zip_append1)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1033
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1034
lemma zip_rev:
14247
cb32eb89bddd *** empty log message ***
nipkow
parents: 14208
diff changeset
  1035
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
cb32eb89bddd *** empty log message ***
nipkow
parents: 14208
diff changeset
  1036
by (induct rule:list_induct2, simp_all)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1037
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1038
lemma nth_zip [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1039
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1040
apply (induct ys, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1041
apply (case_tac xs)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1042
 apply (simp_all add: nth.simps split: nat.split)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1043
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1044
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1045
lemma set_zip:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1046
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1047
by (simp add: set_conv_nth cong: rev_conj_cong)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1048
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1049
lemma zip_update:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1050
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1051
by (rule sym, simp add: update_zip)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1052
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1053
lemma zip_replicate [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1054
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1055
apply (induct i, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1056
apply (case_tac j, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1057
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1058
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1059
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1060
subsection {* @{text list_all2} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1061
14316
91b897b9a2dc added some [intro?] and [trans] for list_all2 lemmas
kleing
parents: 14302
diff changeset
  1062
lemma list_all2_lengthD [intro?]: 
91b897b9a2dc added some [intro?] and [trans] for list_all2 lemmas
kleing
parents: 14302
diff changeset
  1063
  "list_all2 P xs ys ==> length xs = length ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1064
by (simp add: list_all2_def)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1065
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1066
lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1067
by (simp add: list_all2_def)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1068
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1069
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1070
by (simp add: list_all2_def)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1071
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1072
lemma list_all2_Cons [iff]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1073
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1074
by (auto simp add: list_all2_def)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1075
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1076
lemma list_all2_Cons1:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1077
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1078
by (cases ys) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1079
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1080
lemma list_all2_Cons2:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1081
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1082
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1083
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1084
lemma list_all2_rev [iff]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1085
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1086
by (simp add: list_all2_def zip_rev cong: conj_cong)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1087
13863
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1088
lemma list_all2_rev1:
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1089
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1090
by (subst list_all2_rev [symmetric]) simp
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1091
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1092
lemma list_all2_append1:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1093
"list_all2 P (xs @ ys) zs =
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1094
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1095
list_all2 P xs us \<and> list_all2 P ys vs)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1096
apply (simp add: list_all2_def zip_append1)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1097
apply (rule iffI)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1098
 apply (rule_tac x = "take (length xs) zs" in exI)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1099
 apply (rule_tac x = "drop (length xs) zs" in exI)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1100
 apply (force split: nat_diff_split simp add: min_def, clarify)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1101
apply (simp add: ball_Un)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1102
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1103
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1104
lemma list_all2_append2:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1105
"list_all2 P xs (ys @ zs) =
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1106
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1107
list_all2 P us ys \<and> list_all2 P vs zs)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1108
apply (simp add: list_all2_def zip_append2)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1109
apply (rule iffI)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1110
 apply (rule_tac x = "take (length ys) xs" in exI)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1111
 apply (rule_tac x = "drop (length ys) xs" in exI)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1112
 apply (force split: nat_diff_split simp add: min_def, clarify)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1113
apply (simp add: ball_Un)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1114
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1115
13863
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1116
lemma list_all2_append:
14247
cb32eb89bddd *** empty log message ***
nipkow
parents: 14208
diff changeset
  1117
  "length xs = length ys \<Longrightarrow>
cb32eb89bddd *** empty log message ***
nipkow
parents: 14208
diff changeset
  1118
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
cb32eb89bddd *** empty log message ***
nipkow
parents: 14208
diff changeset
  1119
by (induct rule:list_induct2, simp_all)
13863
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1120
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1121
lemma list_all2_appendI [intro?, trans]:
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1122
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1123
  by (simp add: list_all2_append list_all2_lengthD)
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1124
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1125
lemma list_all2_conv_all_nth:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1126
"list_all2 P xs ys =
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1127
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1128
by (force simp add: list_all2_def set_zip)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1129
13883
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
  1130
lemma list_all2_trans:
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
  1131
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
  1132
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
  1133
        (is "!!bs cs. PROP ?Q as bs cs")
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
  1134
proof (induct as)
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
  1135
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
  1136
  show "!!cs. PROP ?Q (x # xs) bs cs"
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
  1137
  proof (induct bs)
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
  1138
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
  1139
    show "PROP ?Q (x # xs) (y # ys) cs"
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
  1140
      by (induct cs) (auto intro: tr I1 I2)
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
  1141
  qed simp
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
  1142
qed simp
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
  1143
13863
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1144
lemma list_all2_all_nthI [intro?]:
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1145
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1146
  by (simp add: list_all2_conv_all_nth)
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1147
14395
cc96cc06abf9 new theorem
paulson
parents: 14388
diff changeset
  1148
lemma list_all2I:
cc96cc06abf9 new theorem
paulson
parents: 14388
diff changeset
  1149
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
cc96cc06abf9 new theorem
paulson
parents: 14388
diff changeset
  1150
  by (simp add: list_all2_def)
cc96cc06abf9 new theorem
paulson
parents: 14388
diff changeset
  1151
14328
fd063037fdf5 list_all2_nthD no good as [intro?]
kleing
parents: 14327
diff changeset
  1152
lemma list_all2_nthD:
13863
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1153
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1154
  by (simp add: list_all2_conv_all_nth)
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1155
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1156
lemma list_all2_nthD2:
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1157
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1158
  by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1159
13863
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1160
lemma list_all2_map1: 
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1161
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1162
  by (simp add: list_all2_conv_all_nth)
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1163
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1164
lemma list_all2_map2: 
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1165
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1166
  by (auto simp add: list_all2_conv_all_nth)
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1167
14316
91b897b9a2dc added some [intro?] and [trans] for list_all2 lemmas
kleing
parents: 14302
diff changeset
  1168
lemma list_all2_refl [intro?]:
13863
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1169
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1170
  by (simp add: list_all2_conv_all_nth)
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1171
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1172
lemma list_all2_update_cong:
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1173
  "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1174
  by (simp add: list_all2_conv_all_nth nth_list_update)
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1175
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1176
lemma list_all2_update_cong2:
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1177
  "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1178
  by (simp add: list_all2_lengthD list_all2_update_cong)
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1179
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1180
lemma list_all2_takeI [simp,intro?]:
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1181
  "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1182
  apply (induct xs)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1183
   apply simp
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1184
  apply (clarsimp simp add: list_all2_Cons1)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1185
  apply (case_tac n)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1186
  apply auto
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1187
  done
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1188
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1189
lemma list_all2_dropI [simp,intro?]:
13863
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1190
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1191
  apply (induct as, simp)
13863
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1192
  apply (clarsimp simp add: list_all2_Cons1)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1193
  apply (case_tac n, simp, simp)
13863
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1194
  done
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1195
14327
9cd4dea593e3 list_all2_mono should not be [trans]
kleing
parents: 14316
diff changeset
  1196
lemma list_all2_mono [intro?]:
13863
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1197
  "\<And>y. list_all2 P x y \<Longrightarrow> (\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> list_all2 Q x y"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1198
  apply (induct x, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1199
  apply (case_tac y, auto)
13863
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1200
  done
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1201
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1202
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1203
subsection {* @{text foldl} and @{text foldr} *}
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1204
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1205
lemma foldl_append [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1206
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1207
by (induct xs) auto
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1208
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1209
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1210
by (induct xs) auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1211
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1212
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1213
by (induct xs) auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1214
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1215
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1216
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1217
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1218
text {*
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1219
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1220
difficult to use because it requires an additional transitivity step.
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1221
*}
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1222
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1223
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1224
by (induct ns) auto
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1225
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1226
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1227
by (force intro: start_le_sum simp add: in_set_conv_decomp)
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1228
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1229
lemma sum_eq_0_conv [iff]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1230
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1231
by (induct ns) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1232
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1233
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
  1234
subsection {* folding a relation over a list *}
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
  1235
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
  1236
(*"fold_rel R cs \<equiv> foldl (%r c. r O {(x,y). (c,x,y):R}) Id cs"*)
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
  1237
inductive "fold_rel R" intros
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
  1238
  Nil:  "(a, [],a) : fold_rel R"
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
  1239
  Cons: "[|(a,x,b) : R; (b,xs,c) : fold_rel R|] ==> (a,x#xs,c) : fold_rel R"
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
  1240
inductive_cases fold_rel_elim_case [elim!]:
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1241
   "(a, [] , b) : fold_rel R"
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
  1242
   "(a, x#xs, b) : fold_rel R"
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
  1243
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
  1244
lemma fold_rel_Nil [intro!]: "a = b ==> (a, [], b) : fold_rel R" 
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
  1245
by (simp add: fold_rel.Nil)
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
  1246
declare fold_rel.Cons [intro!]
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
  1247
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
  1248
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1249
subsection {* @{text upto} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1250
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1251
lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1252
-- {* Does not terminate! *}
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1253
by (induct j) auto
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1254
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1255
lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1256
by (subst upt_rec) simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1257
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1258
lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1259
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1260
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1261
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1262
lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1263
apply(rule trans)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1264
apply(subst upt_rec)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1265
 prefer 2 apply (rule refl, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1266
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1267
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1268
lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1269
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1270
by (induct k) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1271
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1272
lemma length_upt [simp]: "length [i..j(] = j - i"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1273
by (induct j) (auto simp add: Suc_diff_le)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1274
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1275
lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1276
apply (induct j)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1277
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1278
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1279
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1280
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1281
apply (induct m, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1282
apply (subst upt_rec)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1283
apply (rule sym)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1284
apply (subst upt_rec)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1285
apply (simp del: upt.simps)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1286
done
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
  1287
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1288
lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1289
by (induct n) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1290
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1291
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..n(]) ! i = f(m+i)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1292
apply (induct n m rule: diff_induct)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1293
prefer 3 apply (subst map_Suc_upt[symmetric])
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1294
apply (auto simp add: less_diff_conv nth_upt)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1295
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1296
13883
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
  1297
lemma nth_take_lemma:
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
  1298
  "!!xs ys. k <= length xs ==> k <= length ys ==>
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
  1299
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
  1300
apply (atomize, induct k)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1301
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1302
txt {* Both lists must be non-empty *}
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1303
apply (case_tac xs, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1304
apply (case_tac ys, clarify)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1305
 apply (simp (no_asm_use))
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1306
apply clarify
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1307
txt {* prenexing's needed, not miniscoping *}
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1308
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1309
apply blast
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1310
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1311
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1312
lemma nth_equalityI:
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1313
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1314
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1315
apply (simp_all add: take_all)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1316
done
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1317
13863
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1318
(* needs nth_equalityI *)
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1319
lemma list_all2_antisym:
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1320
  "\<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> 
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1321
  \<Longrightarrow> xs = ys"
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1322
  apply (simp add: list_all2_conv_all_nth) 
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1323
  apply (rule nth_equalityI, blast, simp)
13863
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1324
  done
ec901a432ea1 more about list_all2
kleing
parents: 13737
diff changeset
  1325
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1326
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1327
-- {* The famous take-lemma. *}
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1328
apply (drule_tac x = "max (length xs) (length ys)" in spec)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1329
apply (simp add: le_max_iff_disj take_all)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1330
done
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1331
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1332
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1333
subsection {* @{text "distinct"} and @{text remdups} *}
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1334
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1335
lemma distinct_append [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1336
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1337
by (induct xs) auto
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1338
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1339
lemma set_remdups [simp]: "set (remdups xs) = set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1340
by (induct xs) (auto simp add: insert_absorb)
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1341
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1342
lemma distinct_remdups [iff]: "distinct (remdups xs)"