src/HOL/List.thy
author nipkow
Fri Jun 01 22:09:16 2007 +0200 (2007-06-01)
changeset 23192 ec73b9707d48
parent 23096 423ad2fe9f76
child 23209 098a23702aba
permissions -rw-r--r--
Moved list comprehension into List
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(*  Title:      HOL/List.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow
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*)
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header {* The datatype of finite lists *}
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theory List
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imports PreList
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uses "Tools/string_syntax.ML"
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begin
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datatype 'a list =
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    Nil    ("[]")
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  | Cons 'a  "'a list"    (infixr "#" 65)
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subsection{*Basic list processing functions*}
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consts
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  append :: "'a list => 'a list => 'a list" (infixr "@" 65)
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  filter:: "('a => bool) => 'a list => 'a list"
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  concat:: "'a list list => 'a list"
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  foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"
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  foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"
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  hd:: "'a list => 'a"
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  tl:: "'a list => 'a list"
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  last:: "'a list => 'a"
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  butlast :: "'a list => 'a list"
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  set :: "'a list => 'a set"
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  map :: "('a=>'b) => ('a list => 'b list)"
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  listsum ::  "'a list => 'a::monoid_add"
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  nth :: "'a list => nat => 'a"    (infixl "!" 100)
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  list_update :: "'a list => nat => 'a => 'a list"
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  take:: "nat => 'a list => 'a list"
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  drop:: "nat => 'a list => 'a list"
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  takeWhile :: "('a => bool) => 'a list => 'a list"
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  dropWhile :: "('a => bool) => 'a list => 'a list"
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  rev :: "'a list => 'a list"
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  zip :: "'a list => 'b list => ('a * 'b) list"
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  upt :: "nat => nat => nat list" ("(1[_..</_'])")
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  remdups :: "'a list => 'a list"
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  remove1 :: "'a => 'a list => 'a list"
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  "distinct":: "'a list => bool"
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  replicate :: "nat => 'a => 'a list"
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  splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
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  allpairs :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'c list"
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abbreviation
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  upto:: "nat => nat => nat list"  ("(1[_../_])") where
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  "[i..j] == [i..<(Suc j)]"
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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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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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syntax (xsymbols)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
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syntax (HTML output)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
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text {*
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  Function @{text size} is overloaded for all datatypes. Users may
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  refer to the list version as @{text length}. *}
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abbreviation
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  length :: "'a list => nat" where
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  "length == size"
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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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  "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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  "set [] = {}"
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  "set (x#xs) = insert x (set 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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"listsum [] = 0"
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"listsum (x # xs) = x + listsum xs"
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primrec
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  drop_Nil:"drop n [] = []"
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  drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  take_Nil:"take n [] = []"
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  take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  "[][i:=v] = []"
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  "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])"
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primrec
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  "takeWhile P [] = []"
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  "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
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primrec
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  "dropWhile P [] = []"
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  "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
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primrec
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  "zip xs [] = []"
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  zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
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primrec
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  upt_0: "[i..<0] = []"
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  upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
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primrec
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  "distinct [] = True"
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  "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
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primrec
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  "remdups [] = []"
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  "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
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primrec
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  "remove1 x [] = []"
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  "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)"
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primrec
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  replicate_0: "replicate 0 x = []"
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  replicate_Suc: "replicate (Suc n) x = x # replicate n x"
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definition
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  rotate1 :: "'a list \<Rightarrow> 'a list" where
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  "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"
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definition
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  rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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  "rotate n = rotate1 ^ n"
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definition
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  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
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  "list_all2 P xs ys =
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    (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"
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definition
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  sublist :: "'a list => nat set => 'a list" where
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  "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"
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primrec
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  "splice [] ys = ys"
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  "splice (x#xs) ys = (if ys=[] then x#xs else x # hd ys # splice xs (tl ys))"
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    -- {*Warning: simpset does not contain the second eqn but a derived one. *}
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primrec
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"allpairs f [] ys = []"
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"allpairs f (x # xs) ys = map (f x) ys @ allpairs f xs ys"
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subsubsection {* List comprehehsion *}
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text{* At the moment this theory provides only the input syntax for
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list comprehension: @{text"[x. x \<leftarrow> xs, \<dots>]"} rather than
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\verb![x| x <- xs, ...]! as in Haskell.
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The print translation from internal form to list comprehensions would
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be nice. Unfortunately one cannot just turn the translations around
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because in the final equality @{text p} occurs twice on the
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right-hand side. Due to this restriction, the translation must be hand-coded.
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A more substantial extension would be proper theorem proving
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support. For example, it would be nice if
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@{text"set[f x y. x \<leftarrow> xs, y \<leftarrow> ys, P x y]"}
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produced something like
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@{term"{z. EX x: set xs. EX y:set ys. P x y \<and> z = f x y}"}.  *}
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nonterminals lc_gentest
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syntax
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"_listcompr" :: "'a \<Rightarrow> idt \<Rightarrow> 'b list \<Rightarrow> lc_gentest \<Rightarrow> 'a list" ("[_ . _ \<leftarrow> __")
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"_listcompr" :: "'a \<Rightarrow> idt \<Rightarrow> 'b list \<Rightarrow> lc_gentest \<Rightarrow> 'a list" ("[_ . _ <- __")
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"_lc_end" :: "lc_gentest" ("]")
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"_lc_gen" :: "idt \<Rightarrow> 'a list \<Rightarrow> lc_gentest \<Rightarrow> lc_gentest" (",_ \<leftarrow> __")
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"_lc_gen" :: "idt \<Rightarrow> 'a list \<Rightarrow> lc_gentest \<Rightarrow> lc_gentest" (",_ <- __")
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"_lc_test" :: "bool \<Rightarrow> lc_gentest \<Rightarrow> lc_gentest" (",__")
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translations
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"[e. p\<leftarrow>xs]" => "map (%p. e) xs"
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"_listcompr e p xs (_lc_gen q ys GT)" =>
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 "concat (map (%p. _listcompr e q ys GT) xs)"
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"_listcompr e p xs (_lc_test P GT)" => "_listcompr e p (filter (%p. P) xs) GT"
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(* Some examples:
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term "[(x,y). x \<leftarrow> xs, x<y]"
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term "[(x,y). x \<leftarrow> xs, x<y, z\<leftarrow>zs]"
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term "[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x<y]"
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term "[(x,y,z). x \<leftarrow> xs, y \<leftarrow> ys, z\<leftarrow> zs]"
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term "[x. x \<leftarrow> xs, x < a, x > b]"
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*)
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subsubsection {* @{const Nil} and @{const Cons} *}
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lemma not_Cons_self [simp]:
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  "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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  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
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by (rule measure_induct [of length]) iprover
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subsubsection {* @{const 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_allpairs[simp]:
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 "length(allpairs f xs ys) = length xs * length ys"
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by(induct 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)
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apply 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)
nipkow@14247
   335
done
wenzelm@13114
   336
krauss@22493
   337
lemma list_induct2': 
krauss@22493
   338
  "\<lbrakk> P [] [];
krauss@22493
   339
  \<And>x xs. P (x#xs) [];
krauss@22493
   340
  \<And>y ys. P [] (y#ys);
krauss@22493
   341
   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
krauss@22493
   342
 \<Longrightarrow> P xs ys"
krauss@22493
   343
by (induct xs arbitrary: ys) (case_tac x, auto)+
krauss@22493
   344
nipkow@22143
   345
lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
nipkow@22143
   346
apply(rule Eq_FalseI)
nipkow@22143
   347
by auto
nipkow@22143
   348
nipkow@22143
   349
(*
nipkow@22143
   350
Reduces xs=ys to False if xs and ys cannot be of the same length.
nipkow@22143
   351
This is the case if the atomic sublists of one are a submultiset
nipkow@22143
   352
of those of the other list and there are fewer Cons's in one than the other.
nipkow@22143
   353
*)
nipkow@22143
   354
ML_setup {*
nipkow@22143
   355
local
nipkow@22143
   356
nipkow@22143
   357
fun len (Const("List.list.Nil",_)) acc = acc
nipkow@22143
   358
  | len (Const("List.list.Cons",_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
haftmann@23029
   359
  | len (Const("List.append",_) $ xs $ ys) acc = len xs (len ys acc)
nipkow@22143
   360
  | len (Const("List.rev",_) $ xs) acc = len xs acc
nipkow@22143
   361
  | len (Const("List.map",_) $ _ $ xs) acc = len xs acc
nipkow@22143
   362
  | len t (ts,n) = (t::ts,n);
nipkow@22143
   363
nipkow@22143
   364
fun list_eq ss (Const(_,eqT) $ lhs $ rhs) =
nipkow@22143
   365
  let
nipkow@22143
   366
    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
nipkow@22143
   367
    fun prove_neq() =
nipkow@22143
   368
      let
nipkow@22143
   369
        val Type(_,listT::_) = eqT;
haftmann@22994
   370
        val size = HOLogic.size_const listT;
nipkow@22143
   371
        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
nipkow@22143
   372
        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
nipkow@22143
   373
        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
haftmann@22633
   374
          (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
haftmann@22633
   375
      in SOME (thm RS @{thm neq_if_length_neq}) end
nipkow@22143
   376
  in
nipkow@22143
   377
    if m < n andalso gen_submultiset (op aconv) (ls,rs) orelse
nipkow@22143
   378
       n < m andalso gen_submultiset (op aconv) (rs,ls)
nipkow@22143
   379
    then prove_neq() else NONE
nipkow@22143
   380
  end;
nipkow@22143
   381
nipkow@22143
   382
in
nipkow@22143
   383
nipkow@22143
   384
val list_neq_simproc =
haftmann@22633
   385
  Simplifier.simproc @{theory} "list_neq" ["(xs::'a list) = ys"] (K list_eq);
nipkow@22143
   386
nipkow@22143
   387
end;
nipkow@22143
   388
nipkow@22143
   389
Addsimprocs [list_neq_simproc];
nipkow@22143
   390
*}
nipkow@22143
   391
nipkow@22143
   392
nipkow@15392
   393
subsubsection {* @{text "@"} -- append *}
wenzelm@13114
   394
wenzelm@13142
   395
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
nipkow@13145
   396
by (induct xs) auto
wenzelm@13114
   397
wenzelm@13142
   398
lemma append_Nil2 [simp]: "xs @ [] = xs"
nipkow@13145
   399
by (induct xs) auto
nipkow@3507
   400
wenzelm@13142
   401
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
nipkow@13145
   402
by (induct xs) auto
wenzelm@13114
   403
wenzelm@13142
   404
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
nipkow@13145
   405
by (induct xs) auto
wenzelm@13114
   406
wenzelm@13142
   407
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
nipkow@13145
   408
by (induct xs) auto
wenzelm@13114
   409
wenzelm@13142
   410
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
nipkow@13145
   411
by (induct xs) auto
wenzelm@13114
   412
berghofe@13883
   413
lemma append_eq_append_conv [simp]:
berghofe@13883
   414
 "!!ys. length xs = length ys \<or> length us = length vs
berghofe@13883
   415
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
berghofe@13883
   416
apply (induct xs)
paulson@14208
   417
 apply (case_tac ys, simp, force)
paulson@14208
   418
apply (case_tac ys, force, simp)
nipkow@13145
   419
done
wenzelm@13142
   420
nipkow@14495
   421
lemma append_eq_append_conv2: "!!ys zs ts.
nipkow@14495
   422
 (xs @ ys = zs @ ts) =
nipkow@14495
   423
 (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
nipkow@14495
   424
apply (induct xs)
nipkow@14495
   425
 apply fastsimp
nipkow@14495
   426
apply(case_tac zs)
nipkow@14495
   427
 apply simp
nipkow@14495
   428
apply fastsimp
nipkow@14495
   429
done
nipkow@14495
   430
wenzelm@13142
   431
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145
   432
by simp
wenzelm@13142
   433
wenzelm@13142
   434
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145
   435
by simp
wenzelm@13114
   436
wenzelm@13142
   437
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   438
by simp
wenzelm@13114
   439
wenzelm@13142
   440
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   441
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   442
wenzelm@13142
   443
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   444
using append_same_eq [of "[]"] by auto
wenzelm@13114
   445
wenzelm@13142
   446
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   447
by (induct xs) auto
wenzelm@13114
   448
wenzelm@13142
   449
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   450
by (induct xs) auto
wenzelm@13114
   451
wenzelm@13142
   452
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   453
by (simp add: hd_append split: list.split)
wenzelm@13114
   454
wenzelm@13142
   455
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   456
by (simp split: list.split)
wenzelm@13114
   457
wenzelm@13142
   458
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   459
by (simp add: tl_append split: list.split)
wenzelm@13114
   460
wenzelm@13114
   461
nipkow@14300
   462
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   463
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   464
by(cases ys) auto
nipkow@14300
   465
nipkow@15281
   466
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   467
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   468
by(cases ys) auto
nipkow@15281
   469
nipkow@14300
   470
wenzelm@13142
   471
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   472
wenzelm@13114
   473
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   474
by simp
wenzelm@13114
   475
wenzelm@13142
   476
lemma Cons_eq_appendI:
nipkow@13145
   477
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   478
by (drule sym) simp
wenzelm@13114
   479
wenzelm@13142
   480
lemma append_eq_appendI:
nipkow@13145
   481
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   482
by (drule sym) simp
wenzelm@13114
   483
wenzelm@13114
   484
wenzelm@13142
   485
text {*
nipkow@13145
   486
Simplification procedure for all list equalities.
nipkow@13145
   487
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   488
- both lists end in a singleton list,
nipkow@13145
   489
- or both lists end in the same list.
wenzelm@13142
   490
*}
wenzelm@13142
   491
wenzelm@13142
   492
ML_setup {*
nipkow@3507
   493
local
nipkow@3507
   494
wenzelm@13114
   495
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462
   496
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
haftmann@23029
   497
  | last (Const("List.append",_) $ _ $ ys) = last ys
wenzelm@13462
   498
  | last t = t;
wenzelm@13114
   499
wenzelm@13114
   500
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462
   501
  | list1 _ = false;
wenzelm@13114
   502
wenzelm@13114
   503
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462
   504
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
haftmann@23029
   505
  | butlast ((app as Const("List.append",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462
   506
  | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114
   507
haftmann@22633
   508
val rearr_ss = HOL_basic_ss addsimps [@{thm append_assoc},
haftmann@22633
   509
  @{thm append_Nil}, @{thm append_Cons}];
wenzelm@16973
   510
wenzelm@20044
   511
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   512
  let
wenzelm@13462
   513
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   514
    fun rearr conv =
wenzelm@13462
   515
      let
wenzelm@13462
   516
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   517
        val Type(_,listT::_) = eqT
wenzelm@13462
   518
        val appT = [listT,listT] ---> listT
haftmann@23029
   519
        val app = Const("List.append",appT)
wenzelm@13462
   520
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   521
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@20044
   522
        val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
wenzelm@17877
   523
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
skalberg@15531
   524
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   525
wenzelm@13462
   526
  in
haftmann@22633
   527
    if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
haftmann@22633
   528
    else if lastl aconv lastr then rearr @{thm append_same_eq}
skalberg@15531
   529
    else NONE
wenzelm@13462
   530
  end;
wenzelm@13462
   531
wenzelm@13114
   532
in
wenzelm@13462
   533
wenzelm@13462
   534
val list_eq_simproc =
haftmann@22633
   535
  Simplifier.simproc @{theory} "list_eq" ["(xs::'a list) = ys"] (K list_eq);
wenzelm@13462
   536
wenzelm@13114
   537
end;
wenzelm@13114
   538
wenzelm@13114
   539
Addsimprocs [list_eq_simproc];
wenzelm@13114
   540
*}
wenzelm@13114
   541
wenzelm@13114
   542
nipkow@15392
   543
subsubsection {* @{text map} *}
wenzelm@13114
   544
wenzelm@13142
   545
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   546
by (induct xs) simp_all
wenzelm@13114
   547
wenzelm@13142
   548
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   549
by (rule ext, induct_tac xs) auto
wenzelm@13114
   550
wenzelm@13142
   551
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   552
by (induct xs) auto
wenzelm@13114
   553
wenzelm@13142
   554
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   555
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   556
wenzelm@13142
   557
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   558
by (induct xs) auto
wenzelm@13114
   559
nipkow@13737
   560
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   561
by (induct xs) auto
nipkow@13737
   562
krauss@19770
   563
lemma map_cong [fundef_cong, recdef_cong]:
nipkow@13145
   564
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   565
-- {* a congruence rule for @{text map} *}
nipkow@13737
   566
by simp
wenzelm@13114
   567
wenzelm@13142
   568
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   569
by (cases xs) auto
wenzelm@13114
   570
wenzelm@13142
   571
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   572
by (cases xs) auto
wenzelm@13114
   573
paulson@18447
   574
lemma map_eq_Cons_conv:
nipkow@14025
   575
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   576
by (cases xs) auto
wenzelm@13114
   577
paulson@18447
   578
lemma Cons_eq_map_conv:
nipkow@14025
   579
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   580
by (cases ys) auto
nipkow@14025
   581
paulson@18447
   582
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
paulson@18447
   583
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
paulson@18447
   584
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
paulson@18447
   585
nipkow@14111
   586
lemma ex_map_conv:
nipkow@14111
   587
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
paulson@18447
   588
by(induct ys, auto simp add: Cons_eq_map_conv)
nipkow@14111
   589
nipkow@15110
   590
lemma map_eq_imp_length_eq:
nipkow@15110
   591
  "!!xs. map f xs = map f ys ==> length xs = length ys"
nipkow@15110
   592
apply (induct ys)
nipkow@15110
   593
 apply simp
nipkow@15110
   594
apply(simp (no_asm_use))
nipkow@15110
   595
apply clarify
nipkow@15110
   596
apply(simp (no_asm_use))
nipkow@15110
   597
apply fast
nipkow@15110
   598
done
nipkow@15110
   599
nipkow@15110
   600
lemma map_inj_on:
nipkow@15110
   601
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   602
  ==> xs = ys"
nipkow@15110
   603
apply(frule map_eq_imp_length_eq)
nipkow@15110
   604
apply(rotate_tac -1)
nipkow@15110
   605
apply(induct rule:list_induct2)
nipkow@15110
   606
 apply simp
nipkow@15110
   607
apply(simp)
nipkow@15110
   608
apply (blast intro:sym)
nipkow@15110
   609
done
nipkow@15110
   610
nipkow@15110
   611
lemma inj_on_map_eq_map:
nipkow@15110
   612
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   613
by(blast dest:map_inj_on)
nipkow@15110
   614
wenzelm@13114
   615
lemma map_injective:
nipkow@14338
   616
 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@14338
   617
by (induct ys) (auto dest!:injD)
wenzelm@13114
   618
nipkow@14339
   619
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   620
by(blast dest:map_injective)
nipkow@14339
   621
wenzelm@13114
   622
lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@17589
   623
by (iprover dest: map_injective injD intro: inj_onI)
wenzelm@13114
   624
wenzelm@13114
   625
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   626
apply (unfold inj_on_def, clarify)
nipkow@13145
   627
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   628
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   629
apply blast
nipkow@13145
   630
done
wenzelm@13114
   631
nipkow@14339
   632
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   633
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   634
nipkow@15303
   635
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
   636
apply(rule inj_onI)
nipkow@15303
   637
apply(erule map_inj_on)
nipkow@15303
   638
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
   639
done
nipkow@15303
   640
kleing@14343
   641
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   642
by (induct xs, auto)
wenzelm@13114
   643
nipkow@14402
   644
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   645
by (induct xs) auto
nipkow@14402
   646
nipkow@15110
   647
lemma map_fst_zip[simp]:
nipkow@15110
   648
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   649
by (induct rule:list_induct2, simp_all)
nipkow@15110
   650
nipkow@15110
   651
lemma map_snd_zip[simp]:
nipkow@15110
   652
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   653
by (induct rule:list_induct2, simp_all)
nipkow@15110
   654
nipkow@15110
   655
nipkow@15392
   656
subsubsection {* @{text rev} *}
wenzelm@13114
   657
wenzelm@13142
   658
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   659
by (induct xs) auto
wenzelm@13114
   660
wenzelm@13142
   661
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   662
by (induct xs) auto
wenzelm@13114
   663
kleing@15870
   664
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870
   665
by auto
kleing@15870
   666
wenzelm@13142
   667
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   668
by (induct xs) auto
wenzelm@13114
   669
wenzelm@13142
   670
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   671
by (induct xs) auto
wenzelm@13114
   672
kleing@15870
   673
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870
   674
by (cases xs) auto
kleing@15870
   675
kleing@15870
   676
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870
   677
by (cases xs) auto
kleing@15870
   678
haftmann@21061
   679
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
haftmann@21061
   680
apply (induct xs arbitrary: ys, force)
paulson@14208
   681
apply (case_tac ys, simp, force)
nipkow@13145
   682
done
wenzelm@13114
   683
nipkow@15439
   684
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
   685
by(simp add:inj_on_def)
nipkow@15439
   686
wenzelm@13366
   687
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   688
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489
   689
apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145
   690
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   691
done
wenzelm@13114
   692
nipkow@13145
   693
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114
   694
wenzelm@13366
   695
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   696
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   697
by (induct xs rule: rev_induct) auto
wenzelm@13114
   698
wenzelm@13366
   699
lemmas rev_cases = rev_exhaust
wenzelm@13366
   700
nipkow@18423
   701
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
nipkow@18423
   702
by(rule rev_cases[of xs]) auto
nipkow@18423
   703
wenzelm@13114
   704
nipkow@15392
   705
subsubsection {* @{text set} *}
wenzelm@13114
   706
wenzelm@13142
   707
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   708
by (induct xs) auto
wenzelm@13114
   709
wenzelm@13142
   710
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   711
by (induct xs) auto
wenzelm@13114
   712
nipkow@17830
   713
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
nipkow@17830
   714
by(cases xs) auto
oheimb@14099
   715
wenzelm@13142
   716
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   717
by auto
wenzelm@13114
   718
oheimb@14099
   719
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   720
by auto
oheimb@14099
   721
wenzelm@13142
   722
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   723
by (induct xs) auto
wenzelm@13114
   724
nipkow@15245
   725
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   726
by(induct xs) auto
nipkow@15245
   727
wenzelm@13142
   728
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   729
by (induct xs) auto
wenzelm@13114
   730
wenzelm@13142
   731
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   732
by (induct xs) auto
wenzelm@13114
   733
nipkow@22828
   734
lemma set_allpairs[simp]:
nipkow@22830
   735
 "set(allpairs f xs ys) = {z. EX x : set xs. EX y : set ys. z = f x y}"
nipkow@22828
   736
by(induct xs) auto
nipkow@22828
   737
wenzelm@13142
   738
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   739
by (induct xs) auto
wenzelm@13114
   740
nipkow@15425
   741
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
paulson@14208
   742
apply (induct j, simp_all)
paulson@14208
   743
apply (erule ssubst, auto)
nipkow@13145
   744
done
wenzelm@13114
   745
wenzelm@13142
   746
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
paulson@15113
   747
proof (induct xs)
paulson@15113
   748
  case Nil show ?case by simp
paulson@15113
   749
  case (Cons a xs)
paulson@15113
   750
  show ?case
paulson@15113
   751
  proof 
paulson@15113
   752
    assume "x \<in> set (a # xs)"
paulson@15113
   753
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   754
      by (simp, blast intro: Cons_eq_appendI)
paulson@15113
   755
  next
paulson@15113
   756
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   757
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
paulson@15113
   758
    show "x \<in> set (a # xs)" 
paulson@15113
   759
      by (cases ys, auto simp add: eq)
paulson@15113
   760
  qed
paulson@15113
   761
qed
wenzelm@13142
   762
nipkow@18049
   763
lemma in_set_conv_decomp_first:
nipkow@18049
   764
 "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
nipkow@18049
   765
proof (induct xs)
nipkow@18049
   766
  case Nil show ?case by simp
nipkow@18049
   767
next
nipkow@18049
   768
  case (Cons a xs)
nipkow@18049
   769
  show ?case
nipkow@18049
   770
  proof cases
nipkow@18049
   771
    assume "x = a" thus ?case using Cons by force
nipkow@18049
   772
  next
nipkow@18049
   773
    assume "x \<noteq> a"
nipkow@18049
   774
    show ?case
nipkow@18049
   775
    proof
nipkow@18049
   776
      assume "x \<in> set (a # xs)"
nipkow@18049
   777
      from prems show "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@18049
   778
	by(fastsimp intro!: Cons_eq_appendI)
nipkow@18049
   779
    next
nipkow@18049
   780
      assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@18049
   781
      then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
nipkow@18049
   782
      show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq)
nipkow@18049
   783
    qed
nipkow@18049
   784
  qed
nipkow@18049
   785
qed
nipkow@18049
   786
nipkow@18049
   787
lemmas split_list       = in_set_conv_decomp[THEN iffD1, standard]
nipkow@18049
   788
lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard]
nipkow@18049
   789
nipkow@18049
   790
paulson@13508
   791
lemma finite_list: "finite A ==> EX l. set l = A"
paulson@13508
   792
apply (erule finite_induct, auto)
paulson@13508
   793
apply (rule_tac x="x#l" in exI, auto)
paulson@13508
   794
done
paulson@13508
   795
kleing@14388
   796
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
   797
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
   798
paulson@15168
   799
nipkow@15392
   800
subsubsection {* @{text filter} *}
wenzelm@13114
   801
wenzelm@13142
   802
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   803
by (induct xs) auto
wenzelm@13114
   804
nipkow@15305
   805
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
   806
by (induct xs) simp_all
nipkow@15305
   807
wenzelm@13142
   808
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   809
by (induct xs) auto
wenzelm@13114
   810
nipkow@16998
   811
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998
   812
by (induct xs) (auto simp add: le_SucI)
nipkow@16998
   813
nipkow@18423
   814
lemma sum_length_filter_compl:
nipkow@18423
   815
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
nipkow@18423
   816
by(induct xs) simp_all
nipkow@18423
   817
wenzelm@13142
   818
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   819
by (induct xs) auto
wenzelm@13114
   820
wenzelm@13142
   821
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   822
by (induct xs) auto
wenzelm@13114
   823
nipkow@16998
   824
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@16998
   825
  by (induct xs) simp_all
nipkow@16998
   826
nipkow@16998
   827
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998
   828
apply (induct xs)
nipkow@16998
   829
 apply auto
nipkow@16998
   830
apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998
   831
apply simp
nipkow@16998
   832
done
wenzelm@13114
   833
nipkow@16965
   834
lemma filter_map:
nipkow@16965
   835
  "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965
   836
by (induct xs) simp_all
nipkow@16965
   837
nipkow@16965
   838
lemma length_filter_map[simp]:
nipkow@16965
   839
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965
   840
by (simp add:filter_map)
nipkow@16965
   841
wenzelm@13142
   842
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   843
by auto
wenzelm@13114
   844
nipkow@15246
   845
lemma length_filter_less:
nipkow@15246
   846
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
   847
proof (induct xs)
nipkow@15246
   848
  case Nil thus ?case by simp
nipkow@15246
   849
next
nipkow@15246
   850
  case (Cons x xs) thus ?case
nipkow@15246
   851
    apply (auto split:split_if_asm)
nipkow@15246
   852
    using length_filter_le[of P xs] apply arith
nipkow@15246
   853
  done
nipkow@15246
   854
qed
wenzelm@13114
   855
nipkow@15281
   856
lemma length_filter_conv_card:
nipkow@15281
   857
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
   858
proof (induct xs)
nipkow@15281
   859
  case Nil thus ?case by simp
nipkow@15281
   860
next
nipkow@15281
   861
  case (Cons x xs)
nipkow@15281
   862
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
   863
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
   864
  show ?case (is "?l = card ?S'")
nipkow@15281
   865
  proof (cases)
nipkow@15281
   866
    assume "p x"
nipkow@15281
   867
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@15281
   868
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   869
    have "length (filter p (x # xs)) = Suc(card ?S)"
nipkow@15281
   870
      using Cons by simp
nipkow@15281
   871
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
nipkow@15281
   872
      by (simp add: card_image inj_Suc)
nipkow@15281
   873
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   874
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
   875
    finally show ?thesis .
nipkow@15281
   876
  next
nipkow@15281
   877
    assume "\<not> p x"
nipkow@15281
   878
    hence eq: "?S' = Suc ` ?S"
nipkow@15281
   879
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   880
    have "length (filter p (x # xs)) = card ?S"
nipkow@15281
   881
      using Cons by simp
nipkow@15281
   882
    also have "\<dots> = card(Suc ` ?S)" using fin
nipkow@15281
   883
      by (simp add: card_image inj_Suc)
nipkow@15281
   884
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   885
      by (simp add:card_insert_if)
nipkow@15281
   886
    finally show ?thesis .
nipkow@15281
   887
  qed
nipkow@15281
   888
qed
nipkow@15281
   889
nipkow@17629
   890
lemma Cons_eq_filterD:
nipkow@17629
   891
 "x#xs = filter P ys \<Longrightarrow>
nipkow@17629
   892
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
wenzelm@19585
   893
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
nipkow@17629
   894
proof(induct ys)
nipkow@17629
   895
  case Nil thus ?case by simp
nipkow@17629
   896
next
nipkow@17629
   897
  case (Cons y ys)
nipkow@17629
   898
  show ?case (is "\<exists>x. ?Q x")
nipkow@17629
   899
  proof cases
nipkow@17629
   900
    assume Py: "P y"
nipkow@17629
   901
    show ?thesis
nipkow@17629
   902
    proof cases
nipkow@17629
   903
      assume xy: "x = y"
nipkow@17629
   904
      show ?thesis
nipkow@17629
   905
      proof from Py xy Cons(2) show "?Q []" by simp qed
nipkow@17629
   906
    next
nipkow@17629
   907
      assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp
nipkow@17629
   908
    qed
nipkow@17629
   909
  next
nipkow@17629
   910
    assume Py: "\<not> P y"
nipkow@17629
   911
    with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp
nipkow@17629
   912
    show ?thesis (is "? us. ?Q us")
nipkow@17629
   913
    proof show "?Q (y#us)" using 1 by simp qed
nipkow@17629
   914
  qed
nipkow@17629
   915
qed
nipkow@17629
   916
nipkow@17629
   917
lemma filter_eq_ConsD:
nipkow@17629
   918
 "filter P ys = x#xs \<Longrightarrow>
nipkow@17629
   919
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
nipkow@17629
   920
by(rule Cons_eq_filterD) simp
nipkow@17629
   921
nipkow@17629
   922
lemma filter_eq_Cons_iff:
nipkow@17629
   923
 "(filter P ys = x#xs) =
nipkow@17629
   924
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
nipkow@17629
   925
by(auto dest:filter_eq_ConsD)
nipkow@17629
   926
nipkow@17629
   927
lemma Cons_eq_filter_iff:
nipkow@17629
   928
 "(x#xs = filter P ys) =
nipkow@17629
   929
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
nipkow@17629
   930
by(auto dest:Cons_eq_filterD)
nipkow@17629
   931
krauss@19770
   932
lemma filter_cong[fundef_cong, recdef_cong]:
nipkow@17501
   933
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
nipkow@17501
   934
apply simp
nipkow@17501
   935
apply(erule thin_rl)
nipkow@17501
   936
by (induct ys) simp_all
nipkow@17501
   937
nipkow@15281
   938
nipkow@15392
   939
subsubsection {* @{text concat} *}
wenzelm@13114
   940
wenzelm@13142
   941
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
   942
by (induct xs) auto
wenzelm@13114
   943
paulson@18447
   944
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   945
by (induct xss) auto
wenzelm@13114
   946
paulson@18447
   947
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   948
by (induct xss) auto
wenzelm@13114
   949
wenzelm@13142
   950
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
nipkow@13145
   951
by (induct xs) auto
wenzelm@13114
   952
wenzelm@13142
   953
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
   954
by (induct xs) auto
wenzelm@13114
   955
wenzelm@13142
   956
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
   957
by (induct xs) auto
wenzelm@13114
   958
wenzelm@13142
   959
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
   960
by (induct xs) auto
wenzelm@13114
   961
wenzelm@13114
   962
nipkow@15392
   963
subsubsection {* @{text nth} *}
wenzelm@13114
   964
wenzelm@13142
   965
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
   966
by auto
wenzelm@13114
   967
wenzelm@13142
   968
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
   969
by auto
wenzelm@13114
   970
wenzelm@13142
   971
declare nth.simps [simp del]
wenzelm@13114
   972
wenzelm@13114
   973
lemma nth_append:
nipkow@13145
   974
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
paulson@14208
   975
apply (induct "xs", simp)
paulson@14208
   976
apply (case_tac n, auto)
nipkow@13145
   977
done
wenzelm@13114
   978
nipkow@14402
   979
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
nipkow@14402
   980
by (induct "xs") auto
nipkow@14402
   981
nipkow@14402
   982
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
nipkow@14402
   983
by (induct "xs") auto
nipkow@14402
   984
wenzelm@13142
   985
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
paulson@14208
   986
apply (induct xs, simp)
paulson@14208
   987
apply (case_tac n, auto)
nipkow@13145
   988
done
wenzelm@13114
   989
nipkow@18423
   990
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
nipkow@18423
   991
by(cases xs) simp_all
nipkow@18423
   992
nipkow@18049
   993
nipkow@18049
   994
lemma list_eq_iff_nth_eq:
nipkow@18049
   995
 "!!ys. (xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
nipkow@18049
   996
apply(induct xs)
nipkow@18049
   997
 apply simp apply blast
nipkow@18049
   998
apply(case_tac ys)
nipkow@18049
   999
 apply simp
nipkow@18049
  1000
apply(simp add:nth_Cons split:nat.split)apply blast
nipkow@18049
  1001
done
nipkow@18049
  1002
wenzelm@13142
  1003
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
  1004
apply (induct xs, simp, simp)
nipkow@13145
  1005
apply safe
paulson@14208
  1006
apply (rule_tac x = 0 in exI, simp)
paulson@14208
  1007
 apply (rule_tac x = "Suc i" in exI, simp)
paulson@14208
  1008
apply (case_tac i, simp)
nipkow@13145
  1009
apply (rename_tac j)
paulson@14208
  1010
apply (rule_tac x = j in exI, simp)
nipkow@13145
  1011
done
wenzelm@13114
  1012
nipkow@17501
  1013
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
nipkow@17501
  1014
by(auto simp:set_conv_nth)
nipkow@17501
  1015
nipkow@13145
  1016
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
  1017
by (auto simp add: set_conv_nth)
wenzelm@13114
  1018
wenzelm@13142
  1019
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
  1020
by (auto simp add: set_conv_nth)
wenzelm@13114
  1021
wenzelm@13114
  1022
lemma all_nth_imp_all_set:
nipkow@13145
  1023
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
  1024
by (auto simp add: set_conv_nth)
wenzelm@13114
  1025
wenzelm@13114
  1026
lemma all_set_conv_all_nth:
nipkow@13145
  1027
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
  1028
by (auto simp add: set_conv_nth)
wenzelm@13114
  1029
wenzelm@13114
  1030
nipkow@15392
  1031
subsubsection {* @{text list_update} *}
wenzelm@13114
  1032
wenzelm@13142
  1033
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
nipkow@13145
  1034
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1035
wenzelm@13114
  1036
lemma nth_list_update:
nipkow@13145
  1037
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@13145
  1038
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1039
wenzelm@13142
  1040
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
  1041
by (simp add: nth_list_update)
wenzelm@13114
  1042
wenzelm@13142
  1043
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145
  1044
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1045
wenzelm@13142
  1046
lemma list_update_overwrite [simp]:
nipkow@13145
  1047
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145
  1048
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1049
nipkow@14402
  1050
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
paulson@14208
  1051
apply (induct xs, simp)
nipkow@14187
  1052
apply(simp split:nat.splits)
nipkow@14187
  1053
done
nipkow@14187
  1054
nipkow@17501
  1055
lemma list_update_beyond[simp]: "\<And>i. length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
nipkow@17501
  1056
apply (induct xs)
nipkow@17501
  1057
 apply simp
nipkow@17501
  1058
apply (case_tac i)
nipkow@17501
  1059
apply simp_all
nipkow@17501
  1060
done
nipkow@17501
  1061
wenzelm@13114
  1062
lemma list_update_same_conv:
nipkow@13145
  1063
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145
  1064
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1065
nipkow@14187
  1066
lemma list_update_append1:
nipkow@14187
  1067
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
paulson@14208
  1068
apply (induct xs, simp)
nipkow@14187
  1069
apply(simp split:nat.split)
nipkow@14187
  1070
done
nipkow@14187
  1071
kleing@15868
  1072
lemma list_update_append:
kleing@15868
  1073
  "!!n. (xs @ ys) [n:= x] = 
kleing@15868
  1074
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
kleing@15868
  1075
by (induct xs) (auto split:nat.splits)
kleing@15868
  1076
nipkow@14402
  1077
lemma list_update_length [simp]:
nipkow@14402
  1078
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
  1079
by (induct xs, auto)
nipkow@14402
  1080
wenzelm@13114
  1081
lemma update_zip:
nipkow@13145
  1082
"!!i xy xs. length xs = length ys ==>
nipkow@13145
  1083
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145
  1084
by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114
  1085
wenzelm@13114
  1086
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145
  1087
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1088
wenzelm@13114
  1089
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
  1090
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
  1091
kleing@15868
  1092
lemma set_update_memI: "!!n. n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
kleing@15868
  1093
by (induct xs) (auto split:nat.splits)
kleing@15868
  1094
wenzelm@13114
  1095
nipkow@15392
  1096
subsubsection {* @{text last} and @{text butlast} *}
wenzelm@13114
  1097
wenzelm@13142
  1098
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
  1099
by (induct xs) auto
wenzelm@13114
  1100
wenzelm@13142
  1101
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
  1102
by (induct xs) auto
wenzelm@13114
  1103
nipkow@14302
  1104
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
  1105
by(simp add:last.simps)
nipkow@14302
  1106
nipkow@14302
  1107
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
  1108
by(simp add:last.simps)
nipkow@14302
  1109
nipkow@14302
  1110
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
  1111
by (induct xs) (auto)
nipkow@14302
  1112
nipkow@14302
  1113
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
  1114
by(simp add:last_append)
nipkow@14302
  1115
nipkow@14302
  1116
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
  1117
by(simp add:last_append)
nipkow@14302
  1118
nipkow@17762
  1119
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
nipkow@17762
  1120
by(rule rev_exhaust[of xs]) simp_all
nipkow@17762
  1121
nipkow@17762
  1122
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
nipkow@17762
  1123
by(cases xs) simp_all
nipkow@17762
  1124
nipkow@17765
  1125
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
nipkow@17765
  1126
by (induct as) auto
nipkow@17762
  1127
wenzelm@13142
  1128
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
  1129
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1130
wenzelm@13114
  1131
lemma butlast_append:
nipkow@13145
  1132
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145
  1133
by (induct xs) auto
wenzelm@13114
  1134
wenzelm@13142
  1135
lemma append_butlast_last_id [simp]:
nipkow@13145
  1136
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
  1137
by (induct xs) auto
wenzelm@13114
  1138
wenzelm@13142
  1139
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
  1140
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1141
wenzelm@13114
  1142
lemma in_set_butlast_appendI:
nipkow@13145
  1143
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
  1144
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
  1145
nipkow@17501
  1146
lemma last_drop[simp]: "!!n. n < length xs \<Longrightarrow> last (drop n xs) = last xs"
nipkow@17501
  1147
apply (induct xs)
nipkow@17501
  1148
 apply simp
nipkow@17501
  1149
apply (auto split:nat.split)
nipkow@17501
  1150
done
nipkow@17501
  1151
nipkow@17589
  1152
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
nipkow@17589
  1153
by(induct xs)(auto simp:neq_Nil_conv)
nipkow@17589
  1154
nipkow@15392
  1155
subsubsection {* @{text take} and @{text drop} *}
wenzelm@13114
  1156
wenzelm@13142
  1157
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
  1158
by (induct xs) auto
wenzelm@13114
  1159
wenzelm@13142
  1160
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
  1161
by (induct xs) auto
wenzelm@13114
  1162
wenzelm@13142
  1163
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
  1164
by simp
wenzelm@13114
  1165
wenzelm@13142
  1166
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
  1167
by simp
wenzelm@13114
  1168
wenzelm@13142
  1169
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
  1170
nipkow@15110
  1171
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
  1172
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
  1173
nipkow@14187
  1174
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
  1175
by(cases xs, simp_all)
nipkow@14187
  1176
nipkow@14187
  1177
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
nipkow@14187
  1178
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@14187
  1179
nipkow@14187
  1180
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
paulson@14208
  1181
apply (induct xs, simp)
nipkow@14187
  1182
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
  1183
done
nipkow@14187
  1184
nipkow@13913
  1185
lemma take_Suc_conv_app_nth:
nipkow@13913
  1186
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
paulson@14208
  1187
apply (induct xs, simp)
paulson@14208
  1188
apply (case_tac i, auto)
nipkow@13913
  1189
done
nipkow@13913
  1190
mehta@14591
  1191
lemma drop_Suc_conv_tl:
mehta@14591
  1192
  "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
mehta@14591
  1193
apply (induct xs, simp)
mehta@14591
  1194
apply (case_tac i, auto)
mehta@14591
  1195
done
mehta@14591
  1196
wenzelm@13142
  1197
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
nipkow@13145
  1198
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1199
wenzelm@13142
  1200
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
nipkow@13145
  1201
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1202
wenzelm@13142
  1203
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
nipkow@13145
  1204
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1205
wenzelm@13142
  1206
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
nipkow@13145
  1207
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1208
wenzelm@13142
  1209
lemma take_append [simp]:
nipkow@13145
  1210
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@13145
  1211
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1212
wenzelm@13142
  1213
lemma drop_append [simp]:
nipkow@13145
  1214
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@13145
  1215
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1216
wenzelm@13142
  1217
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
paulson@14208
  1218
apply (induct m, auto)
paulson@14208
  1219
apply (case_tac xs, auto)
nipkow@15236
  1220
apply (case_tac n, auto)
nipkow@13145
  1221
done
wenzelm@13114
  1222
wenzelm@13142
  1223
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
paulson@14208
  1224
apply (induct m, auto)
paulson@14208
  1225
apply (case_tac xs, auto)
nipkow@13145
  1226
done
wenzelm@13114
  1227
wenzelm@13114
  1228
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
paulson@14208
  1229
apply (induct m, auto)
paulson@14208
  1230
apply (case_tac xs, auto)
nipkow@13145
  1231
done
wenzelm@13114
  1232
nipkow@14802
  1233
lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@14802
  1234
apply(induct xs)
nipkow@14802
  1235
 apply simp
nipkow@14802
  1236
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  1237
done
nipkow@14802
  1238
wenzelm@13142
  1239
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
paulson@14208
  1240
apply (induct n, auto)
paulson@14208
  1241
apply (case_tac xs, auto)
nipkow@13145
  1242
done
wenzelm@13114
  1243
nipkow@15110
  1244
lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@15110
  1245
apply(induct xs)
nipkow@15110
  1246
 apply simp
nipkow@15110
  1247
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  1248
done
nipkow@15110
  1249
nipkow@15110
  1250
lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"
nipkow@15110
  1251
apply(induct xs)
nipkow@15110
  1252
apply simp
nipkow@15110
  1253
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1254
done
nipkow@15110
  1255
wenzelm@13114
  1256
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
paulson@14208
  1257
apply (induct n, auto)
paulson@14208
  1258
apply (case_tac xs, auto)
nipkow@13145
  1259
done
wenzelm@13114
  1260
wenzelm@13142
  1261
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
paulson@14208
  1262
apply (induct n, auto)
paulson@14208
  1263
apply (case_tac xs, auto)
nipkow@13145
  1264
done
wenzelm@13114
  1265
wenzelm@13114
  1266
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
paulson@14208
  1267
apply (induct xs, auto)
paulson@14208
  1268
apply (case_tac i, auto)
nipkow@13145
  1269
done
wenzelm@13114
  1270
wenzelm@13114
  1271
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
paulson@14208
  1272
apply (induct xs, auto)
paulson@14208
  1273
apply (case_tac i, auto)
nipkow@13145
  1274
done
wenzelm@13114
  1275
wenzelm@13142
  1276
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
paulson@14208
  1277
apply (induct xs, auto)
paulson@14208
  1278
apply (case_tac n, blast)
paulson@14208
  1279
apply (case_tac i, auto)
nipkow@13145
  1280
done
wenzelm@13114
  1281
wenzelm@13142
  1282
lemma nth_drop [simp]:
nipkow@13145
  1283
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
paulson@14208
  1284
apply (induct n, auto)
paulson@14208
  1285
apply (case_tac xs, auto)
nipkow@13145
  1286
done
nipkow@3507
  1287
nipkow@18423
  1288
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
nipkow@18423
  1289
by(simp add: hd_conv_nth)
nipkow@18423
  1290
nipkow@14025
  1291
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
nipkow@14025
  1292
by(induct xs)(auto simp:take_Cons split:nat.split)
nipkow@14025
  1293
nipkow@14025
  1294
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
nipkow@14025
  1295
by(induct xs)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1296
nipkow@14187
  1297
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1298
using set_take_subset by fast
nipkow@14187
  1299
nipkow@14187
  1300
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1301
using set_drop_subset by fast
nipkow@14187
  1302
wenzelm@13114
  1303
lemma append_eq_conv_conj:
nipkow@13145
  1304
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
paulson@14208
  1305
apply (induct xs, simp, clarsimp)
paulson@14208
  1306
apply (case_tac zs, auto)
nipkow@13145
  1307
done
wenzelm@13142
  1308
paulson@14050
  1309
lemma take_add [rule_format]: 
paulson@14050
  1310
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
paulson@14050
  1311
apply (induct xs, auto) 
paulson@14050
  1312
apply (case_tac i, simp_all) 
paulson@14050
  1313
done
paulson@14050
  1314
nipkow@14300
  1315
lemma append_eq_append_conv_if:
nipkow@14300
  1316
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1317
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1318
   then xs\<^isub>1 = take (size xs\<^isub>1) ys\<^isub>1 \<and> xs\<^isub>2 = drop (size xs\<^isub>1) ys\<^isub>1 @ ys\<^isub>2
nipkow@14300
  1319
   else take (size ys\<^isub>1) xs\<^isub>1 = ys\<^isub>1 \<and> drop (size ys\<^isub>1) xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>2)"
nipkow@14300
  1320
apply(induct xs\<^isub>1)
nipkow@14300
  1321
 apply simp
nipkow@14300
  1322
apply(case_tac ys\<^isub>1)
nipkow@14300
  1323
apply simp_all
nipkow@14300
  1324
done
nipkow@14300
  1325
nipkow@15110
  1326
lemma take_hd_drop:
nipkow@15110
  1327
  "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
nipkow@15110
  1328
apply(induct xs)
nipkow@15110
  1329
apply simp
nipkow@15110
  1330
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1331
done
nipkow@15110
  1332
nipkow@17501
  1333
lemma id_take_nth_drop:
nipkow@17501
  1334
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
nipkow@17501
  1335
proof -
nipkow@17501
  1336
  assume si: "i < length xs"
nipkow@17501
  1337
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
nipkow@17501
  1338
  moreover
nipkow@17501
  1339
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
nipkow@17501
  1340
    apply (rule_tac take_Suc_conv_app_nth) by arith
nipkow@17501
  1341
  ultimately show ?thesis by auto
nipkow@17501
  1342
qed
nipkow@17501
  1343
  
nipkow@17501
  1344
lemma upd_conv_take_nth_drop:
nipkow@17501
  1345
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1346
proof -
nipkow@17501
  1347
  assume i: "i < length xs"
nipkow@17501
  1348
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
nipkow@17501
  1349
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
nipkow@17501
  1350
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1351
    using i by (simp add: list_update_append)
nipkow@17501
  1352
  finally show ?thesis .
nipkow@17501
  1353
qed
nipkow@17501
  1354
wenzelm@13114
  1355
nipkow@15392
  1356
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1357
wenzelm@13142
  1358
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1359
by (induct xs) auto
wenzelm@13114
  1360
wenzelm@13142
  1361
lemma takeWhile_append1 [simp]:
nipkow@13145
  1362
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1363
by (induct xs) auto
wenzelm@13114
  1364
wenzelm@13142
  1365
lemma takeWhile_append2 [simp]:
nipkow@13145
  1366
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1367
by (induct xs) auto
wenzelm@13114
  1368
wenzelm@13142
  1369
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1370
by (induct xs) auto
wenzelm@13114
  1371
wenzelm@13142
  1372
lemma dropWhile_append1 [simp]:
nipkow@13145
  1373
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1374
by (induct xs) auto
wenzelm@13114
  1375
wenzelm@13142
  1376
lemma dropWhile_append2 [simp]:
nipkow@13145
  1377
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1378
by (induct xs) auto
wenzelm@13114
  1379
wenzelm@13142
  1380
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1381
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1382
nipkow@13913
  1383
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1384
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1385
by(induct xs, auto)
nipkow@13913
  1386
nipkow@13913
  1387
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1388
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1389
by(induct xs, auto)
nipkow@13913
  1390
nipkow@13913
  1391
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1392
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1393
by(induct xs, auto)
nipkow@13913
  1394
nipkow@17501
  1395
text{* The following two lemmmas could be generalized to an arbitrary
nipkow@17501
  1396
property. *}
nipkow@17501
  1397
nipkow@17501
  1398
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1399
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
nipkow@17501
  1400
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
nipkow@17501
  1401
nipkow@17501
  1402
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1403
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
nipkow@17501
  1404
apply(induct xs)
nipkow@17501
  1405
 apply simp
nipkow@17501
  1406
apply auto
nipkow@17501
  1407
apply(subst dropWhile_append2)
nipkow@17501
  1408
apply auto
nipkow@17501
  1409
done
nipkow@17501
  1410
nipkow@18423
  1411
lemma takeWhile_not_last:
nipkow@18423
  1412
 "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
nipkow@18423
  1413
apply(induct xs)
nipkow@18423
  1414
 apply simp
nipkow@18423
  1415
apply(case_tac xs)
nipkow@18423
  1416
apply(auto)
nipkow@18423
  1417
done
nipkow@18423
  1418
krauss@19770
  1419
lemma takeWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1420
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1421
  ==> takeWhile P l = takeWhile Q k"
wenzelm@20503
  1422
  by (induct k arbitrary: l) (simp_all)
krauss@18336
  1423
krauss@19770
  1424
lemma dropWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1425
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1426
  ==> dropWhile P l = dropWhile Q k"
wenzelm@20503
  1427
  by (induct k arbitrary: l, simp_all)
krauss@18336
  1428
wenzelm@13114
  1429
nipkow@15392
  1430
subsubsection {* @{text zip} *}
wenzelm@13114
  1431
wenzelm@13142
  1432
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1433
by (induct ys) auto
wenzelm@13114
  1434
wenzelm@13142
  1435
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1436
by simp
wenzelm@13114
  1437
wenzelm@13142
  1438
declare zip_Cons [simp del]
wenzelm@13114
  1439
nipkow@15281
  1440
lemma zip_Cons1:
nipkow@15281
  1441
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  1442
by(auto split:list.split)
nipkow@15281
  1443
wenzelm@13142
  1444
lemma length_zip [simp]:
krauss@22493
  1445
"length (zip xs ys) = min (length xs) (length ys)"
krauss@22493
  1446
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1447
wenzelm@13114
  1448
lemma zip_append1:
krauss@22493
  1449
"zip (xs @ ys) zs =
nipkow@13145
  1450
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
krauss@22493
  1451
by (induct xs zs rule:list_induct2') auto
wenzelm@13114
  1452
wenzelm@13114
  1453
lemma zip_append2:
krauss@22493
  1454
"zip xs (ys @ zs) =
nipkow@13145
  1455
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
krauss@22493
  1456
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1457
wenzelm@13142
  1458
lemma zip_append [simp]:
wenzelm@13142
  1459
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1460
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1461
by (simp add: zip_append1)
wenzelm@13114
  1462
wenzelm@13114
  1463
lemma zip_rev:
nipkow@14247
  1464
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1465
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1466
nipkow@23096
  1467
lemma map_zip_map:
nipkow@23096
  1468
 "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
nipkow@23096
  1469
apply(induct xs arbitrary:ys) apply simp
nipkow@23096
  1470
apply(case_tac ys)
nipkow@23096
  1471
apply simp_all
nipkow@23096
  1472
done
nipkow@23096
  1473
nipkow@23096
  1474
lemma map_zip_map2:
nipkow@23096
  1475
 "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
nipkow@23096
  1476
apply(induct xs arbitrary:ys) apply simp
nipkow@23096
  1477
apply(case_tac ys)
nipkow@23096
  1478
apply simp_all
nipkow@23096
  1479
done
nipkow@23096
  1480
wenzelm@13142
  1481
lemma nth_zip [simp]:
nipkow@13145
  1482
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
paulson@14208
  1483
apply (induct ys, simp)
nipkow@13145
  1484
apply (case_tac xs)
nipkow@13145
  1485
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1486
done
wenzelm@13114
  1487
wenzelm@13114
  1488
lemma set_zip:
nipkow@13145
  1489
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
  1490
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1491
wenzelm@13114
  1492
lemma zip_update:
nipkow@13145
  1493
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
  1494
by (rule sym, simp add: update_zip)
wenzelm@13114
  1495
wenzelm@13142
  1496
lemma zip_replicate [simp]:
nipkow@13145
  1497
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
paulson@14208
  1498
apply (induct i, auto)
paulson@14208
  1499
apply (case_tac j, auto)
nipkow@13145
  1500
done
wenzelm@13114
  1501
nipkow@19487
  1502
lemma take_zip:
nipkow@19487
  1503
 "!!xs ys. take n (zip xs ys) = zip (take n xs) (take n ys)"
nipkow@19487
  1504
apply (induct n)
nipkow@19487
  1505
 apply simp
nipkow@19487
  1506
apply (case_tac xs, simp)
nipkow@19487
  1507
apply (case_tac ys, simp_all)
nipkow@19487
  1508
done
nipkow@19487
  1509
nipkow@19487
  1510
lemma drop_zip:
nipkow@19487
  1511
 "!!xs ys. drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
nipkow@19487
  1512
apply (induct n)
nipkow@19487
  1513
 apply simp
nipkow@19487
  1514
apply (case_tac xs, simp)
nipkow@19487
  1515
apply (case_tac ys, simp_all)
nipkow@19487
  1516
done
nipkow@19487
  1517
krauss@22493
  1518
lemma set_zip_leftD:
krauss@22493
  1519
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
krauss@22493
  1520
by (induct xs ys rule:list_induct2') auto
krauss@22493
  1521
krauss@22493
  1522
lemma set_zip_rightD:
krauss@22493
  1523
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
krauss@22493
  1524
by (induct xs ys rule:list_induct2') auto
wenzelm@13142
  1525
nipkow@15392
  1526
subsubsection {* @{text list_all2} *}
wenzelm@13114
  1527
kleing@14316
  1528
lemma list_all2_lengthD [intro?]: 
kleing@14316
  1529
  "list_all2 P xs ys ==> length xs = length ys"
haftmann@19607
  1530
  by (simp add: list_all2_def)
haftmann@19607
  1531
haftmann@19787
  1532
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
haftmann@19607
  1533
  by (simp add: list_all2_def)
haftmann@19607
  1534
haftmann@19787
  1535
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
haftmann@19787
  1536
  by (simp add: list_all2_def)
haftmann@19607
  1537
haftmann@19607
  1538
lemma list_all2_Cons [iff, code]:
haftmann@19607
  1539
  "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
haftmann@19607
  1540
  by (auto simp add: list_all2_def)
wenzelm@13114
  1541
wenzelm@13114
  1542
lemma list_all2_Cons1:
nipkow@13145
  1543
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1544
by (cases ys) auto
wenzelm@13114
  1545
wenzelm@13114
  1546
lemma list_all2_Cons2:
nipkow@13145
  1547
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1548
by (cases xs) auto
wenzelm@13114
  1549
wenzelm@13142
  1550
lemma list_all2_rev [iff]:
nipkow@13145
  1551
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1552
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1553
kleing@13863
  1554
lemma list_all2_rev1:
kleing@13863
  1555
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1556
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1557
wenzelm@13114
  1558
lemma list_all2_append1:
nipkow@13145
  1559
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1560
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1561
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1562
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1563
apply (rule iffI)
nipkow@13145
  1564
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1565
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  1566
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1567
apply (simp add: ball_Un)
nipkow@13145
  1568
done
wenzelm@13114
  1569
wenzelm@13114
  1570
lemma list_all2_append2:
nipkow@13145
  1571
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1572
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1573
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1574
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1575
apply (rule iffI)
nipkow@13145
  1576
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1577
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  1578
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1579
apply (simp add: ball_Un)
nipkow@13145
  1580
done
wenzelm@13114
  1581
kleing@13863
  1582
lemma list_all2_append:
nipkow@14247
  1583
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  1584
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  1585
by (induct rule:list_induct2, simp_all)
kleing@13863
  1586
kleing@13863
  1587
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1588
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
kleing@13863
  1589
  by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1590
wenzelm@13114
  1591
lemma list_all2_conv_all_nth:
nipkow@13145
  1592
"list_all2 P xs ys =
nipkow@13145
  1593
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1594
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1595
berghofe@13883
  1596
lemma list_all2_trans:
berghofe@13883
  1597
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1598
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1599
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1600
proof (induct as)
berghofe@13883
  1601
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1602
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1603
  proof (induct bs)
berghofe@13883
  1604
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1605
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1606
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1607
  qed simp
berghofe@13883
  1608
qed simp
berghofe@13883
  1609
kleing@13863
  1610
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1611
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
kleing@13863
  1612
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1613
paulson@14395
  1614
lemma list_all2I:
paulson@14395
  1615
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
paulson@14395
  1616
  by (simp add: list_all2_def)
paulson@14395
  1617
kleing@14328
  1618
lemma list_all2_nthD:
kleing@13863
  1619
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
kleing@13863
  1620
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1621
nipkow@14302
  1622
lemma list_all2_nthD2:
nipkow@14302
  1623
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@14302
  1624
  by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  1625
kleing@13863
  1626
lemma list_all2_map1: 
kleing@13863
  1627
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
kleing@13863
  1628
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1629
kleing@13863
  1630
lemma list_all2_map2: 
kleing@13863
  1631
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
kleing@13863
  1632
  by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1633
kleing@14316
  1634
lemma list_all2_refl [intro?]:
kleing@13863
  1635
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
kleing@13863
  1636
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1637
kleing@13863
  1638
lemma list_all2_update_cong:
kleing@13863
  1639
  "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
kleing@13863
  1640
  by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1641
kleing@13863
  1642
lemma list_all2_update_cong2:
kleing@13863
  1643
  "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
kleing@13863
  1644
  by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1645
nipkow@14302
  1646
lemma list_all2_takeI [simp,intro?]:
nipkow@14302
  1647
  "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@14302
  1648
  apply (induct xs)
nipkow@14302
  1649
   apply simp
nipkow@14302
  1650
  apply (clarsimp simp add: list_all2_Cons1)
nipkow@14302
  1651
  apply (case_tac n)
nipkow@14302
  1652
  apply auto
nipkow@14302
  1653
  done
nipkow@14302
  1654
nipkow@14302
  1655
lemma list_all2_dropI [simp,intro?]:
kleing@13863
  1656
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
paulson@14208
  1657
  apply (induct as, simp)
kleing@13863
  1658
  apply (clarsimp simp add: list_all2_Cons1)
paulson@14208
  1659
  apply (case_tac n, simp, simp)
kleing@13863
  1660
  done
kleing@13863
  1661
kleing@14327
  1662
lemma list_all2_mono [intro?]:
kleing@13863
  1663
  "\<And>y. list_all2 P x y \<Longrightarrow> (\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> list_all2 Q x y"
paulson@14208
  1664
  apply (induct x, simp)
paulson@14208
  1665
  apply (case_tac y, auto)
kleing@13863
  1666
  done
kleing@13863
  1667
haftmann@22551
  1668
lemma list_all2_eq:
haftmann@22551
  1669
  "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
haftmann@22551
  1670
  by (induct xs ys rule: list_induct2') auto
haftmann@22551
  1671
wenzelm@13142
  1672
nipkow@15392
  1673
subsubsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  1674
wenzelm@13142
  1675
lemma foldl_append [simp]:
nipkow@13145
  1676
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145
  1677
by (induct xs) auto
wenzelm@13142
  1678
nipkow@14402
  1679
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  1680
by (induct xs) auto
nipkow@14402
  1681
nipkow@23096
  1682
lemma foldr_map: "foldr g (map f xs) a = foldr (g o f) xs a"
nipkow@23096
  1683
by(induct xs) simp_all
nipkow@23096
  1684
nipkow@23096
  1685
lemma foldl_map: "foldl g a (map f xs) = foldl (%a x. g a (f x)) a xs"
nipkow@23096
  1686
by(induct xs arbitrary:a) simp_all
nipkow@23096
  1687
krauss@19770
  1688
lemma foldl_cong [fundef_cong, recdef_cong]:
krauss@18336
  1689
  "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] 
krauss@18336
  1690
  ==> foldl f a l = foldl g b k"
wenzelm@20503
  1691
  by (induct k arbitrary: a b l) simp_all
krauss@18336
  1692
krauss@19770
  1693
lemma foldr_cong [fundef_cong, recdef_cong]:
krauss@18336
  1694
  "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] 
krauss@18336
  1695
  ==> foldr f l a = foldr g k b"
wenzelm@20503
  1696
  by (induct k arbitrary: a b l) simp_all
krauss@18336
  1697
nipkow@23096
  1698
text{* The ``First Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096
  1699
nipkow@23096
  1700
lemma foldl_foldr1_lemma:
nipkow@23096
  1701
 "foldl op + a xs = a + foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096
  1702
by (induct xs arbitrary: a) (auto simp:add_assoc)
nipkow@23096
  1703
nipkow@23096
  1704
corollary foldl_foldr1:
nipkow@23096
  1705
 "foldl op + 0 xs = foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096
  1706
by (simp add:foldl_foldr1_lemma)
nipkow@23096
  1707
nipkow@23096
  1708
nipkow@23096
  1709
text{* The ``Third Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096
  1710
nipkow@14402
  1711
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  1712
by (induct xs) auto
nipkow@14402
  1713
nipkow@14402
  1714
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  1715
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  1716
wenzelm@13142
  1717
text {*
nipkow@13145
  1718
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1719
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1720
*}
wenzelm@13142
  1721
wenzelm@13142
  1722
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
nipkow@13145
  1723
by (induct ns) auto
wenzelm@13142
  1724
wenzelm@13142
  1725
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1726
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1727
wenzelm@13142
  1728
lemma sum_eq_0_conv [iff]:
nipkow@13145
  1729
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145
  1730
by (induct ns) auto
wenzelm@13114
  1731
nipkow@23096
  1732
subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
nipkow@23096
  1733
nipkow@23096
  1734
lemma listsum_foldr:
nipkow@23096
  1735
 "listsum xs = foldr (op +) xs 0"
nipkow@23096
  1736
by(induct xs) auto
nipkow@23096
  1737
nipkow@23096
  1738
(* for efficient code generation *)
nipkow@23096
  1739
lemma listsum[code]: "listsum xs = foldl (op +) 0 xs"
nipkow@23096
  1740
by(simp add:listsum_foldr foldl_foldr1)
nipkow@23096
  1741
nipkow@23096
  1742
text{* Some syntactic sugar for summing a function over a list: *}
nipkow@23096
  1743
nipkow@23096
  1744
syntax
nipkow@23096
  1745
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
nipkow@23096
  1746
syntax (xsymbols)
nipkow@23096
  1747
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096
  1748
syntax (HTML output)
nipkow@23096
  1749
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096
  1750
nipkow@23096
  1751
translations -- {* Beware of argument permutation! *}
nipkow@23096
  1752
  "SUM x<-xs. b" == "CONST listsum (map (%x. b) xs)"
nipkow@23096
  1753
  "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (map (%x. b) xs)"
nipkow@23096
  1754
nipkow@23096
  1755
lemma listsum_0 [simp]: "(\<Sum>x\<leftarrow>xs. 0) = 0"
nipkow@23096
  1756
by (induct xs) simp_all
nipkow@23096
  1757
nipkow@23096
  1758
text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
nipkow@23096
  1759
lemma uminus_listsum_map:
nipkow@23096
  1760
 "- listsum (map f xs) = (listsum (map (uminus o f) xs) :: 'a::ab_group_add)"
nipkow@23096
  1761
by(induct xs) simp_all
nipkow@23096
  1762
wenzelm@13114
  1763
nipkow@15392
  1764
subsubsection {* @{text upto} *}
wenzelm@13114
  1765
nipkow@17090
  1766
lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
nipkow@17090
  1767
-- {* simp does not terminate! *}
nipkow@13145
  1768
by (induct j) auto
wenzelm@13142
  1769
nipkow@15425
  1770
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
nipkow@13145
  1771
by (subst upt_rec) simp
wenzelm@13114
  1772
nipkow@15425
  1773
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
nipkow@15281
  1774
by(induct j)simp_all
nipkow@15281
  1775
nipkow@15281
  1776
lemma upt_eq_Cons_conv:
nipkow@15425
  1777
 "!!x xs. ([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
nipkow@15281
  1778
apply(induct j)
nipkow@15281
  1779
 apply simp
nipkow@15281
  1780
apply(clarsimp simp add: append_eq_Cons_conv)
nipkow@15281
  1781
apply arith
nipkow@15281
  1782
done
nipkow@15281
  1783
nipkow@15425
  1784
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
nipkow@13145
  1785
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1786
by simp
wenzelm@13114
  1787
nipkow@15425
  1788
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
nipkow@13145
  1789
apply(rule trans)
nipkow@13145
  1790
apply(subst upt_rec)
paulson@14208
  1791
 prefer 2 apply (rule refl, simp)
nipkow@13145
  1792
done
wenzelm@13114
  1793
nipkow@15425
  1794
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
nipkow@13145
  1795
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1796
by (induct k) auto
wenzelm@13114
  1797
nipkow@15425
  1798
lemma length_upt [simp]: "length [i..<j] = j - i"
nipkow@13145
  1799
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1800
nipkow@15425
  1801
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
nipkow@13145
  1802
apply (induct j)
nipkow@13145
  1803
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1804
done
wenzelm@13114
  1805
nipkow@17906
  1806
nipkow@17906
  1807
lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
nipkow@17906
  1808
by(simp add:upt_conv_Cons)
nipkow@17906
  1809
nipkow@17906
  1810
lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
nipkow@17906
  1811
apply(cases j)
nipkow@17906
  1812
 apply simp
nipkow@17906
  1813
by(simp add:upt_Suc_append)
nipkow@17906
  1814
nipkow@15425
  1815
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..<n] = [i..<i+m]"
paulson@14208
  1816
apply (induct m, simp)
nipkow@13145
  1817
apply (subst upt_rec)
nipkow@13145
  1818
apply (rule sym)
nipkow@13145
  1819
apply (subst upt_rec)
nipkow@13145
  1820
apply (simp del: upt.simps)
nipkow@13145
  1821
done
nipkow@3507
  1822
nipkow@17501
  1823
lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
nipkow@17501
  1824
apply(induct j)
nipkow@17501
  1825
apply auto
nipkow@17501
  1826
done
nipkow@17501
  1827
nipkow@15425
  1828
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..n]"
nipkow@13145
  1829
by (induct n) auto
wenzelm@13114
  1830
nipkow@15425
  1831
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
nipkow@13145
  1832
apply (induct n m rule: diff_induct)
nipkow@13145
  1833
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1834
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1835
done
wenzelm@13114
  1836
berghofe@13883
  1837
lemma nth_take_lemma:
berghofe@13883
  1838
  "!!xs ys. k <= length xs ==> k <= length ys ==>
berghofe@13883
  1839
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
berghofe@13883
  1840
apply (atomize, induct k)
paulson@14208
  1841
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  1842
txt {* Both lists must be non-empty *}
paulson@14208
  1843
apply (case_tac xs, simp)
paulson@14208
  1844
apply (case_tac ys, clarify)
nipkow@13145
  1845
 apply (simp (no_asm_use))
nipkow@13145
  1846
apply clarify
nipkow@13145
  1847
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  1848
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  1849
apply blast
nipkow@13145
  1850
done
wenzelm@13114
  1851
wenzelm@13114
  1852
lemma nth_equalityI:
wenzelm@13114
  1853
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  1854
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  1855
apply (simp_all add: take_all)
nipkow@13145
  1856
done
wenzelm@13142
  1857
kleing@13863
  1858
(* needs nth_equalityI *)
kleing@13863
  1859
lemma list_all2_antisym:
kleing@13863
  1860
  "\<lbrakk> (\<And>x y. \<lbrakk>P x y; Q y x\<rbrakk> \<Longrightarrow> x = y); list_all2 P xs ys; list_all2 Q ys xs \<rbrakk> 
kleing@13863
  1861
  \<Longrightarrow> xs = ys"
kleing@13863
  1862
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  1863
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  1864
  done
kleing@13863
  1865
wenzelm@13142
  1866
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  1867
-- {* The famous take-lemma. *}
nipkow@13145
  1868
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  1869
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  1870
done
wenzelm@13142
  1871
wenzelm@13142
  1872
nipkow@15302
  1873
lemma take_Cons':
nipkow@15302
  1874
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@15302
  1875
by (cases n) simp_all
nipkow@15302
  1876
nipkow@15302
  1877
lemma drop_Cons':
nipkow@15302
  1878
     "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@15302
  1879
by (cases n) simp_all
nipkow@15302
  1880
nipkow@15302
  1881
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@15302
  1882
by (cases n) simp_all
nipkow@15302
  1883
paulson@18622
  1884
lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard]
paulson@18622
  1885
lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard]
paulson@18622
  1886
lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard]
paulson@18622
  1887
paulson@18622
  1888
declare take_Cons_number_of [simp] 
paulson@18622
  1889
        drop_Cons_number_of [simp] 
paulson@18622
  1890
        nth_Cons_number_of [simp] 
nipkow@15302
  1891
nipkow@15302
  1892
nipkow@15392
  1893
subsubsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  1894
wenzelm@13142
  1895
lemma distinct_append [simp]:
nipkow@13145
  1896
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  1897
by (induct xs) auto
wenzelm@13142
  1898
nipkow@15305
  1899
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
nipkow@15305
  1900
by(induct xs) auto
nipkow@15305
  1901
wenzelm@13142
  1902
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  1903
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  1904
wenzelm@13142
  1905
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  1906
by (induct xs) auto
wenzelm@13142
  1907
paulson@15072
  1908
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
paulson@15251
  1909
  by (induct x, auto) 
paulson@15072
  1910
paulson@15072
  1911
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
paulson@15251
  1912
  by (induct x, auto)
paulson@15072
  1913
nipkow@15245
  1914
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245
  1915
by (induct xs) auto
nipkow@15245
  1916
nipkow@15245
  1917
lemma length_remdups_eq[iff]:
nipkow@15245
  1918
  "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245
  1919
apply(induct xs)
nipkow@15245
  1920
 apply auto
nipkow@15245
  1921
apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245
  1922
 apply arith
nipkow@15245
  1923
apply(rule length_remdups_leq)
nipkow@15245
  1924
done
nipkow@15245
  1925
nipkow@18490
  1926
nipkow@18490
  1927
lemma distinct_map:
nipkow@18490
  1928
  "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
nipkow@18490
  1929
by (induct xs) auto
nipkow@18490
  1930
nipkow@18490
  1931
wenzelm@13142
  1932
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  1933
by (induct xs) auto
wenzelm@13114
  1934
nipkow@17501
  1935
lemma distinct_upt[simp]: "distinct[i..<j]"
nipkow@17501
  1936
by (induct j) auto
nipkow@17501
  1937
nipkow@17501
  1938
lemma distinct_take[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (take i xs)"
nipkow@17501
  1939
apply(induct xs)
nipkow@17501
  1940
 apply simp
nipkow@17501
  1941
apply (case_tac i)
nipkow@17501
  1942
 apply simp_all
nipkow@17501
  1943
apply(blast dest:in_set_takeD)
nipkow@17501
  1944
done
nipkow@17501
  1945
nipkow@17501
  1946
lemma distinct_drop[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (drop i xs)"
nipkow@17501
  1947
apply(induct xs)
nipkow@17501
  1948
 apply simp
nipkow@17501
  1949
apply (case_tac i)
nipkow@17501
  1950
 apply simp_all
nipkow@17501
  1951
done
nipkow@17501
  1952
nipkow@17501
  1953
lemma distinct_list_update:
nipkow@17501
  1954
assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
nipkow@17501
  1955
shows "distinct (xs[i:=a])"
nipkow@17501
  1956
proof (cases "i < length xs")
nipkow@17501
  1957
  case True
nipkow@17501
  1958
  with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
nipkow@17501
  1959
    apply (drule_tac id_take_nth_drop) by simp
nipkow@17501
  1960
  with d True show ?thesis
nipkow@17501
  1961
    apply (simp add: upd_conv_take_nth_drop)
nipkow@17501
  1962
    apply (drule subst [OF id_take_nth_drop]) apply assumption
nipkow@17501
  1963
    apply simp apply (cases "a = xs!i") apply simp by blast
nipkow@17501
  1964
next
nipkow@17501
  1965
  case False with d show ?thesis by auto
nipkow@17501
  1966
qed
nipkow@17501
  1967
nipkow@17501
  1968
nipkow@17501
  1969
text {* It is best to avoid this indexed version of distinct, but
nipkow@17501
  1970
sometimes it is useful. *}
nipkow@17501
  1971
wenzelm@13142
  1972
lemma distinct_conv_nth:
nipkow@17501
  1973
"distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@15251
  1974
apply (induct xs, simp, simp)
paulson@14208
  1975
apply (rule iffI, clarsimp)
nipkow@13145
  1976
 apply (case_tac i)
paulson@14208
  1977
apply (case_tac j, simp)
nipkow@13145
  1978
apply (simp add: set_conv_nth)
nipkow@13145
  1979
 apply (case_tac j)
paulson@14208
  1980
apply (clarsimp simp add: set_conv_nth, simp)
nipkow@13145
  1981
apply (rule conjI)
nipkow@13145
  1982
 apply (clarsimp simp add: set_conv_nth)
nipkow@17501
  1983
 apply (erule_tac x = 0 in allE, simp)
paulson@14208
  1984
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
nipkow@17501
  1985
apply (erule_tac x = "Suc i" in allE, simp)
paulson@14208
  1986
apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145
  1987
done
wenzelm@13114
  1988
nipkow@18490
  1989
lemma nth_eq_iff_index_eq:
nipkow@18490
  1990
 "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
nipkow@18490
  1991
by(auto simp: distinct_conv_nth)
nipkow@18490
  1992
nipkow@15110
  1993
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
kleing@14388
  1994
  by (induct xs) auto
kleing@14388
  1995
nipkow@15110
  1996
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388
  1997
proof (induct xs)
kleing@14388
  1998
  case Nil thus ?case by simp
kleing@14388
  1999
next
kleing@14388
  2000
  case (Cons x xs)
kleing@14388
  2001
  show ?case
kleing@14388
  2002
  proof (cases "x \<in> set xs")
kleing@14388
  2003
    case False with Cons show ?thesis by simp
kleing@14388
  2004
  next
kleing@14388
  2005
    case True with Cons.prems
kleing@14388
  2006
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  2007
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  2008
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  2009
    ultimately have False by simp
kleing@14388
  2010
    thus ?thesis ..
kleing@14388
  2011
  qed
kleing@14388
  2012
qed
kleing@14388
  2013
nipkow@18490
  2014
nipkow@18490
  2015
lemma length_remdups_concat:
nipkow@18490
  2016
 "length(remdups(concat xss)) = card(\<Union>xs \<in> set xss. set xs)"
nipkow@18490
  2017
by(simp add: distinct_card[symmetric])
nipkow@17906
  2018
nipkow@17906
  2019
nipkow@15392
  2020
subsubsection {* @{text remove1} *}
nipkow@15110
  2021
nipkow@18049
  2022
lemma remove1_append:
nipkow@18049
  2023
  "remove1 x (xs @ ys) =
nipkow@18049
  2024
  (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
nipkow@18049
  2025
by (induct xs) auto
nipkow@18049
  2026
nipkow@15110
  2027
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110
  2028
apply(induct xs)
nipkow@15110
  2029
 apply simp
nipkow@15110
  2030
apply simp
nipkow@15110
  2031
apply blast
nipkow@15110
  2032
done
nipkow@15110
  2033
paulson@17724
  2034
lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110
  2035
apply(induct xs)
nipkow@15110
  2036
 apply simp
nipkow@15110
  2037
apply simp
nipkow@15110
  2038
apply blast
nipkow@15110
  2039
done
nipkow@15110
  2040
nipkow@18049
  2041
lemma remove1_filter_not[simp]:
nipkow@18049
  2042
  "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
nipkow@18049
  2043
by(induct xs) auto
nipkow@18049
  2044
nipkow@15110
  2045
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110
  2046
apply(insert set_remove1_subset)
nipkow@15110
  2047
apply fast
nipkow@15110
  2048
done
nipkow@15110
  2049
nipkow@15110
  2050
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110
  2051
by (induct xs) simp_all
nipkow@15110
  2052
wenzelm@13114
  2053
nipkow@15392
  2054
subsubsection {* @{text replicate} *}
wenzelm@13114
  2055
wenzelm@13142
  2056
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  2057
by (induct n) auto
nipkow@13124
  2058
wenzelm@13142
  2059
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  2060
by (induct n) auto
wenzelm@13114
  2061
wenzelm@13114
  2062
lemma replicate_app_Cons_same:
nipkow@13145
  2063
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  2064
by (induct n) auto
wenzelm@13114
  2065
wenzelm@13142
  2066
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208
  2067
apply (induct n, simp)
nipkow@13145
  2068
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  2069
done
wenzelm@13114
  2070
wenzelm@13142
  2071
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  2072
by (induct n) auto
wenzelm@13114
  2073
nipkow@16397
  2074
text{* Courtesy of Matthias Daum: *}
nipkow@16397
  2075
lemma append_replicate_commute:
nipkow@16397
  2076
  "replicate n x @ replicate k x = replicate k x @ replicate n x"
nipkow@16397
  2077
apply (simp add: replicate_add [THEN sym])
nipkow@16397
  2078
apply (simp add: add_commute)
nipkow@16397
  2079
done
nipkow@16397
  2080
wenzelm@13142
  2081
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  2082
by (induct n) auto
wenzelm@13114
  2083
wenzelm@13142
  2084
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  2085
by (induct n) auto
wenzelm@13114
  2086
wenzelm@13142
  2087
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  2088
by (atomize (full), induct n) auto
wenzelm@13114
  2089
wenzelm@13142
  2090
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
paulson@14208
  2091
apply (induct n, simp)
nipkow@13145
  2092
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  2093
done
wenzelm@13114
  2094
nipkow@16397
  2095
text{* Courtesy of Matthias Daum (2 lemmas): *}
nipkow@16397
  2096
lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
nipkow@16397
  2097
apply (case_tac "k \<le> i")
nipkow@16397
  2098
 apply  (simp add: min_def)
nipkow@16397
  2099
apply (drule not_leE)
nipkow@16397
  2100
apply (simp add: min_def)
nipkow@16397
  2101
apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
nipkow@16397
  2102
 apply  simp
nipkow@16397
  2103
apply (simp add: replicate_add [symmetric])
nipkow@16397
  2104
done
nipkow@16397
  2105
nipkow@16397
  2106
lemma drop_replicate[simp]: "!!i. drop i (replicate k x) = replicate (k-i) x"
nipkow@16397
  2107
apply (induct k)
nipkow@16397
  2108
 apply simp
nipkow@16397
  2109
apply clarsimp
nipkow@16397
  2110
apply (case_tac i)
nipkow@16397
  2111
 apply simp
nipkow@16397
  2112
apply clarsimp
nipkow@16397
  2113
done
nipkow@16397
  2114
nipkow@16397
  2115
wenzelm@13142
  2116
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  2117
by (induct n) auto
wenzelm@13114
  2118
wenzelm@13142
  2119
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  2120
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  2121
wenzelm@13142
  2122
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  2123
by auto
wenzelm@13114
  2124
wenzelm@13142
  2125
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  2126
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  2127
wenzelm@13114
  2128
nipkow@15392
  2129
subsubsection{*@{text rotate1} and @{text rotate}*}
nipkow@15302
  2130
nipkow@15302
  2131
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
nipkow@15302
  2132
by(simp add:rotate1_def)
nipkow@15302
  2133
nipkow@15302
  2134
lemma rotate0[simp]: "rotate 0 = id"
nipkow@15302
  2135
by(simp add:rotate_def)
nipkow@15302
  2136
nipkow@15302
  2137
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
nipkow@15302
  2138
by(simp add:rotate_def)
nipkow@15302
  2139
nipkow@15302
  2140
lemma rotate_add:
nipkow@15302
  2141
  "rotate (m+n) = rotate m o rotate n"
nipkow@15302
  2142
by(simp add:rotate_def funpow_add)
nipkow@15302
  2143
nipkow@15302
  2144
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
nipkow@15302
  2145
by(simp add:rotate_add)
nipkow@15302
  2146
nipkow@18049
  2147
lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
nipkow@18049
  2148
by(simp add:rotate_def funpow_swap1)
nipkow@18049
  2149
nipkow@15302
  2150
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
nipkow@15302
  2151
by(cases xs) simp_all
nipkow@15302
  2152
nipkow@15302
  2153
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2154
apply(induct n)
nipkow@15302
  2155
 apply simp
nipkow@15302
  2156
apply (simp add:rotate_def)
nipkow@13145
  2157
done
wenzelm@13114
  2158
nipkow@15302
  2159
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
nipkow@15302
  2160
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2161
nipkow@15302
  2162
lemma rotate_drop_take:
nipkow@15302
  2163
  "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
nipkow@15302
  2164
apply(induct n)
nipkow@15302
  2165
 apply simp
nipkow@15302
  2166
apply(simp add:rotate_def)
nipkow@15302
  2167
apply(cases "xs = []")
nipkow@15302
  2168
 apply (simp)
nipkow@15302
  2169
apply(case_tac "n mod length xs = 0")
nipkow@15302
  2170
 apply(simp add:mod_Suc)
nipkow@15302
  2171
 apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
nipkow@15302
  2172
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
nipkow@15302
  2173
                take_hd_drop linorder_not_le)
nipkow@13145
  2174
done
wenzelm@13114
  2175
nipkow@15302
  2176
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
nipkow@15302
  2177
by(simp add:rotate_drop_take)
nipkow@15302
  2178
nipkow@15302
  2179
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2180
by(simp add:rotate_drop_take)
nipkow@15302
  2181
nipkow@15302
  2182
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
nipkow@15302
  2183
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2184
nipkow@15302
  2185
lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs"
nipkow@15302
  2186
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2187
nipkow@15302
  2188
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
nipkow@15302
  2189
by(simp add:rotate1_def split:list.split) blast
nipkow@15302
  2190
nipkow@15302
  2191
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
nipkow@15302
  2192
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2193
nipkow@15302
  2194
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
nipkow@15302
  2195
by(simp add:rotate_drop_take take_map drop_map)
nipkow@15302
  2196
nipkow@15302
  2197
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
nipkow@15302
  2198
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2199
nipkow@15302
  2200
lemma set_rotate[simp]: "set(rotate n xs) = set xs"
nipkow@15302
  2201
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2202
nipkow@15302
  2203
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
nipkow@15302
  2204
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2205
nipkow@15302
  2206
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
nipkow@15302
  2207
by (induct n) (simp_all add:rotate_def)
wenzelm@13114
  2208
nipkow@15439
  2209
lemma rotate_rev:
nipkow@15439
  2210
  "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
nipkow@15439
  2211
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2212
apply(cases "length xs = 0")
nipkow@15439
  2213
 apply simp
nipkow@15439
  2214
apply(cases "n mod length xs = 0")
nipkow@15439
  2215
 apply simp
nipkow@15439
  2216
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2217
done
nipkow@15439
  2218
nipkow@18423
  2219
lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
nipkow@18423
  2220
apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
nipkow@18423
  2221
apply(subgoal_tac "length xs \<noteq> 0")
nipkow@18423
  2222
 prefer 2 apply simp
nipkow@18423
  2223
using mod_less_divisor[of "length xs" n] by arith
nipkow@18423
  2224
wenzelm@13114
  2225
nipkow@15392
  2226
subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  2227
wenzelm@13142
  2228
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  2229
by (auto simp add: sublist_def)
wenzelm@13114
  2230
wenzelm@13142
  2231
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  2232
by (auto simp add: sublist_def)
wenzelm@13114
  2233
nipkow@15281
  2234
lemma length_sublist:
nipkow@15281
  2235
  "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
nipkow@15281
  2236
by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
nipkow@15281
  2237
nipkow@15281
  2238
lemma sublist_shift_lemma_Suc:
nipkow@15281
  2239
  "!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
nipkow@15281
  2240
         map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
nipkow@15281
  2241
apply(induct xs)
nipkow@15281
  2242
 apply simp
nipkow@15281
  2243
apply (case_tac "is")
nipkow@15281
  2244
 apply simp
nipkow@15281
  2245
apply simp
nipkow@15281
  2246
done
nipkow@15281
  2247
wenzelm@13114
  2248
lemma sublist_shift_lemma:
nipkow@15425
  2249
     "map fst [p:zip xs [i..<i + length xs] . snd p : A] =
nipkow@15425
  2250
      map fst [p:zip xs [0..<length xs] . snd p + i : A]"
nipkow@13145
  2251
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  2252
wenzelm@13114
  2253
lemma sublist_append:
paulson@15168
  2254
     "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  2255
apply (unfold sublist_def)
paulson@14208
  2256
apply (induct l' rule: rev_induct, simp)
nipkow@13145
  2257
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  2258
apply (simp add: add_commute)
nipkow@13145
  2259
done
wenzelm@13114
  2260
wenzelm@13114
  2261
lemma sublist_Cons:
nipkow@13145
  2262
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  2263
apply (induct l rule: rev_induct)
nipkow@13145
  2264
 apply (simp add: sublist_def)
nipkow@13145
  2265
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  2266
done
wenzelm@13114
  2267
nipkow@15281
  2268
lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
nipkow@15281
  2269
apply(induct xs)
nipkow@15281
  2270
 apply simp
nipkow@15281
  2271
apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE)
nipkow@15281
  2272
 apply(erule lessE)
nipkow@15281
  2273
  apply auto
nipkow@15281
  2274
apply(erule lessE)
nipkow@15281
  2275
apply auto
nipkow@15281
  2276
done
nipkow@15281
  2277
nipkow@15281
  2278
lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
nipkow@15281
  2279
by(auto simp add:set_sublist)
nipkow@15281
  2280
nipkow@15281
  2281
lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
nipkow@15281
  2282
by(auto simp add:set_sublist)
nipkow@15281
  2283
nipkow@15281
  2284
lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
nipkow@15281
  2285
by(auto simp add:set_sublist)
nipkow@15281
  2286
wenzelm@13142
  2287
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  2288
by (simp add: sublist_Cons)
wenzelm@13114
  2289
nipkow@15281
  2290
nipkow@15281
  2291
lemma distinct_sublistI[simp]: "!!I. distinct xs \<Longrightarrow> distinct(sublist xs I)"
nipkow@15281
  2292
apply(induct xs)
nipkow@15281
  2293
 apply simp
nipkow@15281
  2294
apply(auto simp add:sublist_Cons)
nipkow@15281
  2295
done
nipkow@15281
  2296
nipkow@15281
  2297
nipkow@15045
  2298
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
paulson@14208
  2299
apply (induct l rule: rev_induct, simp)
nipkow@13145
  2300
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  2301
done
wenzelm@13114
  2302
nipkow@17501
  2303
lemma filter_in_sublist: "\<And>s. distinct xs \<Longrightarrow>
nipkow@17501
  2304
  filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
nipkow@17501
  2305
proof (induct xs)
nipkow@17501
  2306
  case Nil thus ?case by simp
nipkow@17501
  2307
next
nipkow@17501
  2308
  case (Cons a xs)
nipkow@17501
  2309
  moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
nipkow@17501
  2310
  ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
nipkow@17501
  2311
qed
nipkow@17501
  2312
wenzelm@13114
  2313
nipkow@19390
  2314
subsubsection {* @{const splice} *}
nipkow@19390
  2315
haftmann@19607
  2316
lemma splice_Nil2 [simp, code]:
nipkow@19390
  2317
 "splice xs [] = xs"
nipkow@19390
  2318
by (cases xs) simp_all
nipkow@19390
  2319
haftmann@19607
  2320
lemma splice_Cons_Cons [simp, code]:
nipkow@19390
  2321
 "splice (x#xs) (y#ys) = x # y # splice xs ys"
nipkow@19390
  2322
by simp
nipkow@19390
  2323
haftmann@19607
  2324
declare splice.simps(2) [simp del, code del]
nipkow@19390
  2325
nipkow@22793
  2326
lemma length_splice[simp]: "!!ys. length(splice xs ys) = length xs + length ys"
nipkow@22793
  2327
apply(induct xs) apply simp
nipkow@22793
  2328
apply(case_tac ys)
nipkow@22793
  2329
 apply auto
nipkow@22793
  2330
done
nipkow@22793
  2331
nipkow@22828
  2332
nipkow@22828
  2333
subsubsection {* @{const allpairs} *}
nipkow@22828
  2334
nipkow@22940
  2335
lemma allpairs_conv_concat:
nipkow@22940
  2336
 "allpairs f xs ys = concat(map (%x. map (f x) ys) xs)"
nipkow@22940
  2337
by(induct xs) auto
nipkow@22940
  2338
nipkow@22828
  2339
lemma allpairs_append:
nipkow@22830
  2340
 "allpairs f (xs @ ys) zs = allpairs f xs zs @ allpairs f ys zs"
nipkow@22828
  2341
by(induct xs) auto
nipkow@22828
  2342
nipkow@22828
  2343
nipkow@15392
  2344
subsubsection{*Sets of Lists*}
nipkow@15392
  2345
nipkow@15392
  2346
subsubsection {* @{text lists}: the list-forming operator over sets *}
nipkow@15302
  2347
berghofe@22262
  2348
inductive2
berghofe@22262
  2349
  listsp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
berghofe@22262
  2350
  for A :: "'a \<Rightarrow> bool"
berghofe@22262
  2351
where
berghofe@22262
  2352
    Nil [intro!]: "listsp A []"
berghofe@22262
  2353
  | Cons [intro!]: "[| A a; listsp A l |] ==> listsp A (a # l)"
berghofe@22262
  2354
berghofe@22262
  2355
constdefs
berghofe@22262
  2356
  lists :: "'a set => 'a list set"
berghofe@22262
  2357
  "lists A == Collect (listsp (member A))"
berghofe@22262
  2358
berghofe@22262
  2359
lemma listsp_lists_eq [pred_set_conv]: "listsp (member A) = member (lists A)"
berghofe@22262
  2360
  by (simp add: lists_def)
berghofe@22262
  2361
berghofe@22262
  2362
lemmas lists_intros [intro!] = listsp.intros [to_set]
berghofe@22262
  2363
berghofe@22262
  2364
lemmas lists_induct [consumes 1, case_names Nil Cons, induct set: lists] =
berghofe@22262
  2365
  listsp.induct [to_set]
berghofe@22262
  2366
berghofe@22262
  2367
inductive_cases2 listspE [elim!]: "listsp A (x # l)"
berghofe@22262
  2368
berghofe@22262
  2369
lemmas listsE [elim!] = listspE [to_set]
berghofe@22262
  2370
berghofe@22262
  2371
lemma listsp_mono [mono2]: "A \<le> B ==> listsp A \<le> listsp B"
berghofe@22262
  2372
  by (clarify, erule listsp.induct, blast+)
berghofe@22262
  2373
berghofe@22262
  2374
lemmas lists_mono [mono] = listsp_mono [to_set]
berghofe@22262
  2375
haftmann@22422
  2376
lemma listsp_infI:
haftmann@22422
  2377
  assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
nipkow@15302
  2378
  by induct blast+
nipkow@15302
  2379
haftmann@22422
  2380
lemmas lists_IntI = listsp_infI [to_set]
haftmann@22422
  2381
haftmann@22422
  2382
lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
haftmann@22422
  2383
proof (rule mono_inf [where f=listsp, THEN order_antisym])
berghofe@22262
  2384
  show "mono listsp" by (simp add: mono_def listsp_mono)
haftmann@22422
  2385
  show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro: listsp_infI)
kleing@14388
  2386
qed
kleing@14388
  2387
haftmann@22422
  2388
lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_eq inf_bool_eq]
haftmann@22422
  2389
haftmann@22422
  2390
lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set]
berghofe@22262
  2391
berghofe@22262
  2392
lemma append_in_listsp_conv [iff]:
berghofe@22262
  2393
     "(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
nipkow@15302
  2394
by (induct xs) auto
nipkow@15302
  2395
berghofe@22262
  2396
lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]
berghofe@22262
  2397
berghofe@22262
  2398
lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
berghofe@22262
  2399
-- {* eliminate @{text listsp} in favour of @{text set} *}
nipkow@15302
  2400
by (induct xs) auto
nipkow@15302
  2401
berghofe@22262
  2402
lemmas in_lists_conv_set = in_listsp_conv_set [to_set]
berghofe@22262
  2403
berghofe@22262
  2404
lemma in_listspD [dest!]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
berghofe@22262
  2405
by (rule in_listsp_conv_set [THEN iffD1])
berghofe@22262
  2406
berghofe@22262
  2407
lemmas in_listsD [dest!] = in_listspD [to_set]
berghofe@22262
  2408
berghofe@22262
  2409
lemma in_listspI [intro!]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
berghofe@22262
  2410
by (rule in_listsp_conv_set [THEN iffD2])
berghofe@22262
  2411
berghofe@22262
  2412
lemmas in_listsI [intro!] = in_listspI [to_set]
nipkow@15302
  2413
nipkow@15302
  2414
lemma lists_UNIV [simp]: "lists UNIV = UNIV"
nipkow@15302
  2415
by auto
nipkow@15302
  2416
nipkow@17086
  2417
nipkow@17086
  2418
nipkow@17086
  2419
subsubsection{* Inductive definition for membership *}
nipkow@17086
  2420
berghofe@22262
  2421
inductive2 ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
berghofe@22262
  2422
where
berghofe@22262
  2423
    elem:  "ListMem x (x # xs)"
berghofe@22262
  2424
  | insert:  "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"
berghofe@22262
  2425
berghofe@22262
  2426
lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
nipkow@17086
  2427
apply (rule iffI)
nipkow@17086
  2428
 apply (induct set: ListMem)
nipkow@17086
  2429
  apply auto
nipkow@17086
  2430
apply (induct xs)
nipkow@17086
  2431
 apply (auto intro: ListMem.intros)
nipkow@17086
  2432
done
nipkow@17086
  2433
nipkow@17086
  2434
nipkow@17086
  2435
nipkow@15392
  2436
subsubsection{*Lists as Cartesian products*}
nipkow@15302
  2437
nipkow@15302
  2438
text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
nipkow@15302
  2439
@{term A} and tail drawn from @{term Xs}.*}
nipkow@15302
  2440
nipkow@15302
  2441
constdefs
nipkow@15302
  2442
  set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
nipkow@15302
  2443
  "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"
nipkow@15302
  2444
paulson@17724
  2445
lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
nipkow@15302
  2446
by (auto simp add: set_Cons_def)
nipkow@15302
  2447
nipkow@15302
  2448
text{*Yields the set of lists, all of the same length as the argument and
nipkow@15302
  2449
with elements drawn from the corresponding element of the argument.*}
nipkow@15302
  2450
nipkow@15302
  2451
consts  listset :: "'a set list \<Rightarrow> 'a list set"
nipkow@15302
  2452
primrec
nipkow@15302
  2453
   "listset []    = {[]}"
nipkow@15302
  2454
   "listset(A#As) = set_Cons A (listset As)"
nipkow@15302
  2455
nipkow@15302
  2456
paulson@15656
  2457
subsection{*Relations on Lists*}
paulson@15656
  2458
paulson@15656
  2459
subsubsection {* Length Lexicographic Ordering *}
paulson@15656
  2460
paulson@15656
  2461
text{*These orderings preserve well-foundedness: shorter lists 
paulson@15656
  2462
  precede longer lists. These ordering are not used in dictionaries.*}
paulson@15656
  2463
paulson@15656
  2464
consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
paulson@15656
  2465
        --{*The lexicographic ordering for lists of the specified length*}
nipkow@15302
  2466
primrec
paulson@15656
  2467
  "lexn r 0 = {}"
paulson@15656
  2468
  "lexn r (Suc n) =
paulson@15656
  2469
    (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
paulson@15656
  2470
    {(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
nipkow@15302
  2471
nipkow@15302
  2472
constdefs
paulson@15656
  2473
  lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
paulson@15656
  2474
    "lex r == \<Union>n. lexn r n"
paulson@15656
  2475
        --{*Holds only between lists of the same length*}
paulson@15656
  2476
nipkow@15693
  2477
  lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@15693
  2478
    "lenlex r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
paulson@15656
  2479
        --{*Compares lists by their length and then lexicographically*}
nipkow@15302
  2480
nipkow@15302
  2481
nipkow@15302
  2482
lemma wf_lexn: "wf r ==> wf (lexn r n)"
nipkow@15302
  2483
apply (induct n, simp, simp)
nipkow@15302
  2484
apply(rule wf_subset)
nipkow@15302
  2485
 prefer 2 apply (rule Int_lower1)
nipkow@15302
  2486
apply(rule wf_prod_fun_image)
nipkow@15302
  2487
 prefer 2 apply (rule inj_onI, auto)
nipkow@15302
  2488
done
nipkow@15302
  2489
nipkow@15302
  2490
lemma lexn_length:
nipkow@15302
  2491
     "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@15302
  2492
by (induct n) auto
nipkow@15302
  2493
nipkow@15302
  2494
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@15302
  2495
apply (unfold lex_def)
nipkow@15302
  2496
apply (rule wf_UN)
nipkow@15302
  2497
apply (blast intro: wf_lexn, clarify)
nipkow@15302
  2498
apply (rename_tac m n)
nipkow@15302
  2499
apply (subgoal_tac "m \<noteq> n")
nipkow@15302
  2500
 prefer 2 apply blast
nipkow@15302
  2501
apply (blast dest: lexn_length not_sym)
nipkow@15302
  2502
done
nipkow@15302
  2503
nipkow@15302
  2504
lemma lexn_conv:
paulson@15656
  2505
  "lexn r n =
paulson@15656
  2506
    {(xs,ys). length xs = n \<and> length ys = n \<and>
paulson@15656
  2507
    (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
nipkow@18423
  2508
apply (induct n, simp)
nipkow@15302
  2509
apply (simp add: image_Collect lex_prod_def, safe, blast)
nipkow@15302
  2510
 apply (rule_tac x = "ab # xys" in exI, simp)
nipkow@15302
  2511
apply (case_tac xys, simp_all, blast)
nipkow@15302
  2512
done
nipkow@15302
  2513
nipkow@15302
  2514
lemma lex_conv:
paulson@15656
  2515
  "lex r =
paulson@15656
  2516
    {(xs,ys). length xs = length ys \<and>
paulson@15656
  2517
    (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@15302
  2518
by (force simp add: lex_def lexn_conv)
nipkow@15302
  2519
nipkow@15693
  2520
lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
nipkow@15693
  2521
by (unfold lenlex_def) blast
nipkow@15693
  2522
nipkow@15693
  2523
lemma lenlex_conv:
nipkow@15693
  2524
    "lenlex r = {(xs,ys). length xs < length ys |
paulson@15656
  2525
                 length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@19623
  2526
by (simp add: lenlex_def diag_def lex_prod_def inv_image_def)
nipkow@15302
  2527
nipkow@15302
  2528
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@15302
  2529
by (simp add: lex_conv)
nipkow@15302
  2530
nipkow@15302
  2531
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@15302
  2532
by (simp add:lex_conv)
nipkow@15302
  2533
paulson@18447
  2534
lemma Cons_in_lex [simp]:
paulson@15656
  2535
    "((x # xs, y # ys) : lex r) =
paulson@15656
  2536
      ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@15302
  2537
apply (simp add: lex_conv)
nipkow@15302
  2538
apply (rule iffI)
nipkow@15302
  2539
 prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
nipkow@15302
  2540
apply (case_tac xys, simp, simp)
nipkow@15302
  2541
apply blast
nipkow@15302
  2542
done
nipkow@15302
  2543
nipkow@15302
  2544
paulson@15656
  2545
subsubsection {* Lexicographic Ordering *}
paulson@15656
  2546
paulson@15656
  2547
text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
paulson@15656
  2548
    This ordering does \emph{not} preserve well-foundedness.
nipkow@17090
  2549
     Author: N. Voelker, March 2005. *} 
paulson@15656
  2550
paulson@15656
  2551
constdefs 
paulson@15656
  2552
  lexord :: "('a * 'a)set \<Rightarrow> ('a list * 'a list) set" 
paulson@15656
  2553
  "lexord  r == {(x,y). \<exists> a v. y = x @ a # v \<or> 
paulson@15656
  2554
            (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
paulson@15656
  2555
paulson@15656
  2556
lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
paulson@15656
  2557
  by (unfold lexord_def, induct_tac y, auto) 
paulson@15656
  2558
paulson@15656
  2559
lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
paulson@15656
  2560
  by (unfold lexord_def, induct_tac x, auto)
paulson@15656
  2561
paulson@15656
  2562
lemma lexord_cons_cons[simp]:
paulson@15656
  2563
     "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
paulson@15656
  2564
  apply (unfold lexord_def, safe, simp_all)
paulson@15656
  2565
  apply (case_tac u, simp, simp)
paulson@15656
  2566
  apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
paulson@15656
  2567
  apply (erule_tac x="b # u" in allE)
paulson@15656
  2568
  by force
paulson@15656
  2569
paulson@15656
  2570
lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
paulson@15656
  2571
paulson@15656
  2572
lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
paulson@15656
  2573
  by (induct_tac x, auto)  
paulson@15656
  2574
paulson@15656
  2575
lemma lexord_append_left_rightI:
paulson@15656
  2576
     "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
paulson@15656
  2577
  by (induct_tac u, auto)
paulson@15656
  2578
paulson@15656
  2579
lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
paulson@15656
  2580
  by (induct x, auto)
paulson@15656
  2581
paulson@15656
  2582
lemma lexord_append_leftD:
paulson@15656
  2583
     "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
paulson@15656
  2584
  by (erule rev_mp, induct_tac x, auto)
paulson@15656
  2585
paulson@15656
  2586
lemma lexord_take_index_conv: 
paulson@15656
  2587
   "((x,y) : lexord r) = 
paulson@15656
  2588
    ((length x < length y \<and> take (length x) y = x) \<or> 
paulson@15656
  2589
     (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
paulson@15656
  2590
  apply (unfold lexord_def Let_def, clarsimp) 
paulson@15656
  2591
  apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
paulson@15656
  2592
  apply auto 
paulson@15656
  2593
  apply (rule_tac x="hd (drop (length x) y)" in exI)
paulson@15656
  2594
  apply (rule_tac x="tl (drop (length x) y)" in exI)
paulson@15656
  2595
  apply (erule subst, simp add: min_def) 
paulson@15656
  2596
  apply (rule_tac x ="length u" in exI, simp) 
paulson@15656
  2597
  apply (rule_tac x ="take i x" in exI) 
paulson@15656
  2598
  apply (rule_tac x ="x ! i" in exI) 
paulson@15656
  2599
  apply (rule_tac x ="y ! i" in exI, safe) 
paulson@15656
  2600
  apply (rule_tac x="drop (Suc i) x" in exI)
paulson@15656
  2601
  apply (drule sym, simp add: drop_Suc_conv_tl) 
paulson@15656
  2602
  apply (rule_tac x="drop (Suc i) y" in exI)
paulson@15656
  2603
  by (simp add: drop_Suc_conv_tl) 
paulson@15656
  2604
paulson@15656
  2605
-- {* lexord is extension of partial ordering List.lex *} 
paulson@15656
  2606
lemma lexord_lex: " (x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
paulson@15656
  2607
  apply (rule_tac x = y in spec) 
paulson@15656
  2608
  apply (induct_tac x, clarsimp) 
paulson@15656
  2609
  by (clarify, case_tac x, simp, force)
paulson@15656
  2610
paulson@15656
  2611
lemma lexord_irreflexive: "(! x. (x,x) \<notin> r) \<Longrightarrow> (y,y) \<notin> lexord r"
paulson@15656
  2612
  by (induct y, auto)
paulson@15656
  2613
paulson@15656
  2614
lemma lexord_trans: 
paulson@15656
  2615
    "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
paulson@15656
  2616
   apply (erule rev_mp)+
paulson@15656
  2617
   apply (rule_tac x = x in spec) 
paulson@15656
  2618
  apply (rule_tac x = z in spec) 
paulson@15656
  2619
  apply ( induct_tac y, simp, clarify)
paulson@15656
  2620
  apply (case_tac xa, erule ssubst) 
paulson@15656
  2621
  apply (erule allE, erule allE) -- {* avoid simp recursion *} 
paulson@15656
  2622
  apply (case_tac x, simp, simp) 
paulson@15656
  2623
  apply (case_tac x, erule allE, erule allE, simp) 
paulson@15656
  2624
  apply (erule_tac x = listb in allE) 
paulson@15656
  2625
  apply (erule_tac x = lista in allE, simp)
paulson@15656
  2626
  apply (unfold trans_def)
paulson@15656
  2627
  by blast
paulson@15656
  2628
paulson@15656
  2629
lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
paulson@15656
  2630
  by (rule transI, drule lexord_trans, blast) 
paulson@15656
  2631
paulson@15656
  2632
lemma lexord_linear: "(! a b. (a,b)\<in> r | a = b | (b,a) \<in> r) \<Longrightarrow> (x,y) : lexord r | x = y | (y,x) : lexord r"
paulson@15656
  2633
  apply (rule_tac x = y in spec) 
paulson@15656
  2634
  apply (induct_tac x, rule allI) 
paulson@15656
  2635
  apply (case_tac x, simp, simp) 
paulson@15656
  2636
  apply (rule allI, case_tac x, simp, simp) 
paulson@15656
  2637
  by blast
paulson@15656
  2638
paulson@15656
  2639
krauss@21103
  2640
subsection {* Lexicographic combination of measure functions *}
krauss@21103
  2641
krauss@21103
  2642
text {* These are useful for termination proofs *}
krauss@21103
  2643
krauss@21103
  2644
definition
krauss@21103
  2645
  "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"
krauss@21103
  2646
krauss@21106
  2647
lemma wf_measures[recdef_wf, simp]: "wf (measures fs)"
krauss@21103
  2648
  unfolding measures_def
krauss@21103
  2649
  by blast
krauss@21103
  2650
krauss@21103
  2651
lemma in_measures[simp]: 
krauss@21103
  2652
  "(x, y) \<in> measures [] = False"
krauss@21103
  2653
  "(x, y) \<in> measures (f # fs)
krauss@21103
  2654
         = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"  
krauss@21103
  2655
  unfolding measures_def
krauss@21103
  2656
  by auto
krauss@21103
  2657
krauss@21103
  2658
lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
krauss@21103
  2659
  by simp
krauss@21103
  2660
krauss@21103
  2661
lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
krauss@21103
  2662
  by auto
krauss@21103
  2663
krauss@21211
  2664
(* install the lexicographic_order method and the "fun" command *)
bulwahn@21131
  2665
use "Tools/function_package/lexicographic_order.ML"
krauss@21211
  2666
use "Tools/function_package/fundef_datatype.ML"
krauss@21211
  2667
setup LexicographicOrder.setup
krauss@21211
  2668
setup FundefDatatype.setup
krauss@21211
  2669
krauss@21103
  2670
nipkow@15392
  2671
subsubsection{*Lifting a Relation on List Elements to the Lists*}
nipkow@15302
  2672
berghofe@22262
  2673
inductive2
berghofe@22262
  2674
  list_all2' :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool"
berghofe@22262
  2675
  for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
berghofe@22262
  2676
where
berghofe@22262
  2677
    Nil:  "list_all2' r [] []"
berghofe@22262
  2678
  | Cons: "[| r x y; list_all2' r xs ys |] ==> list_all2' r (x#xs) (y#ys)"
berghofe@22262
  2679
berghofe@22262
  2680
constdefs
berghofe@22262
  2681
  listrel :: "('a * 'b) set => ('a list * 'b list) set"
berghofe@22262
  2682
  "listrel r == Collect2 (list_all2' (member2 r))"
berghofe@22262
  2683
berghofe@22262
  2684
lemma list_all2_listrel_eq [pred_set_conv]:
berghofe@22262
  2685
  "list_all2' (member2 r) = member2 (listrel r)"
berghofe@22262
  2686
  by (simp add: listrel_def)
berghofe@22262
  2687
berghofe@22262
  2688
lemmas listrel_induct [consumes 1, case_names Nil Cons, induct set: listrel] =
berghofe@22262
  2689
  list_all2'.induct [to_set]
berghofe@22262
  2690
berghofe@22262
  2691
lemmas listrel_intros = list_all2'.intros [to_set]
berghofe@22262
  2692
berghofe@22262
  2693
inductive_cases2 listrel_Nil1 [to_set, elim!]: "list_all2' r [] xs"
berghofe@22262
  2694
inductive_cases2 listrel_Nil2 [to_set, elim!]: "list_all2' r xs []"
berghofe@22262
  2695
inductive_cases2 listrel_Cons1 [to_set, elim!]: "list_all2' r  (y#ys) xs"
berghofe@22262
  2696
inductive_cases2 listrel_Cons2 [to_set, elim!]: "list_all2' r xs (y#ys)"
nipkow@15302
  2697
nipkow@15302
  2698
nipkow@15302
  2699
lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
nipkow@15302
  2700
apply clarify  
berghofe@22262
  2701
apply (erule listrel_induct)
berghofe@22262
  2702
apply (blast intro: listrel_intros)+
nipkow@15302
  2703
done
nipkow@15302
  2704
nipkow@15302
  2705
lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
nipkow@15302
  2706
apply clarify 
berghofe@22262
  2707
apply (erule listrel_induct, auto) 
nipkow@15302
  2708
done
nipkow@15302
  2709
nipkow@15302
  2710
lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" 
nipkow@15302
  2711
apply (simp add: refl_def listrel_subset Ball_def)
nipkow@15302
  2712
apply (rule allI) 
nipkow@15302
  2713
apply (induct_tac x) 
berghofe@22262
  2714
apply (auto intro: listrel_intros)
nipkow@15302
  2715
done
nipkow@15302
  2716
nipkow@15302
  2717
lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
nipkow@15302
  2718
apply (auto simp add: sym_def)
berghofe@22262
  2719
apply (erule listrel_induct) 
berghofe@22262
  2720
apply (blast intro: listrel_intros)+
nipkow@15302
  2721
done
nipkow@15302
  2722
nipkow@15302
  2723
lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
nipkow@15302
  2724
apply (simp add: trans_def)
nipkow@15302
  2725
apply (intro allI) 
nipkow@15302
  2726
apply (rule impI) 
berghofe@22262
  2727
apply (erule listrel_induct) 
berghofe@22262
  2728
apply (blast intro: listrel_intros)+
nipkow@15302
  2729
done
nipkow@15302
  2730
nipkow@15302
  2731
theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
nipkow@15302
  2732
by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) 
nipkow@15302
  2733
nipkow@15302
  2734
lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
berghofe@22262
  2735
by (blast intro: listrel_intros)
nipkow@15302
  2736
nipkow@15302
  2737
lemma listrel_Cons:
nipkow@15302
  2738
     "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})";
berghofe@22262
  2739
by (auto simp add: set_Cons_def intro: listrel_intros) 
nipkow@15302
  2740
nipkow@15302
  2741
nipkow@15392
  2742
subsection{*Miscellany*}
nipkow@15392
  2743
nipkow@15392
  2744
subsubsection {* Characters and strings *}
wenzelm@13366
  2745
wenzelm@13366
  2746
datatype nibble =
wenzelm@13366
  2747
    Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
wenzelm@13366
  2748
  | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
wenzelm@13366
  2749
wenzelm@13366
  2750
datatype char = Char nibble nibble
wenzelm@13366
  2751
  -- "Note: canonical order of character encoding coincides with standard term ordering"
wenzelm@13366
  2752
wenzelm@13366
  2753
types string = "char list"
wenzelm@13366
  2754
wenzelm@13366
  2755
syntax
wenzelm@13366
  2756
  "_Char" :: "xstr => char"    ("CHR _")
wenzelm@13366
  2757
  "_String" :: "xstr => string"    ("_")
wenzelm@13366
  2758
wenzelm@21754
  2759
setup StringSyntax.setup
wenzelm@13366