src/HOL/List.thy
author nipkow
Wed Jun 06 19:12:07 2007 +0200 (2007-06-06)
changeset 23279 e39dd93161d9
parent 23246 309a57cae012
child 23388 77645da0db85
permissions -rw-r--r--
tuned list comprehension, changed filter syntax from : to <-
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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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  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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  partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> (('a list) \<times> ('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[_\<leftarrow>_ ./ _])")
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syntax (HTML output)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
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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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function (*authentic syntax for append -- revert to primrec
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  as soon as "authentic" primrec is available*)
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  append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65)
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where
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  append_Nil: "[] @ ys = ys"
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  | append_Cons: "(x # xs) @ ys = x # (xs @ ys)"
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by (auto, case_tac a, auto)
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termination by (relation "measure (size o fst)") auto
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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{* Input syntax for Haskell-like list comprehension
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notation. Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"}, the list of all pairs of distinct elements from @{text xs} and @{text ys}.
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There are two differences to Haskell.  The general synatx is
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@{text"[e. p \<leftarrow> xs, \<dots>]"} rather than \verb![x| x <- xs, ...]!. Patterns in
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generators can only be tuples (at the moment).
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To avoid misunderstandings, the translation is not reversed upon
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output. You can add the inverse translations in your own theory if you
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desire.
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Hint: formulae containing complex list comprehensions may become quite
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unreadable after the simplifier has finished with them. It can be
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helpful to introduce definitions for such list comprehensions and
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treat them separately in suitable lemmas.
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*}
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(*
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Proper theorem proving support would be nice. For example, 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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*)
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nonterminals lc_qual lc_quals
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syntax
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"_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
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"_lc_gen" :: "pttrn \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ <- _")
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"_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
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"_lc_end" :: "lc_quals" ("]")
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"_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals" (", __")
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translations
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"[e. p<-xs]" => "map (%p. e) xs"
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"_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
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 => "concat (map (%p. _listcompr e Q Qs) xs)"
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"[e. P]" => "if P then [e] else []"
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"_listcompr e (_lc_test P) (_lc_quals Q Qs)"
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 => "if P then (_listcompr e Q Qs) else []"
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syntax (xsymbols)
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"_lc_gen" :: "pttrn \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
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syntax (HTML output)
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"_lc_gen" :: "pttrn \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
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(*
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term "[(x,y,z). b]"
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term "[(x,y,z). x \<leftarrow> xs]"
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term "[(x,y,z). x<a, x>b]"
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term "[(x,y,z). x<a, x\<leftarrow>xs]"
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term "[(x,y,z). x\<leftarrow>xs, x>b]"
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term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys]"
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term "[(x,y,z). x<a, x>b, x=d]"
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term "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
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term "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
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term "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
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term "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
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term "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
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term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
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term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
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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
nipkow@22828
   331
wenzelm@13142
   332
lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
nipkow@13145
   333
by (induct xs) auto
wenzelm@13114
   334
wenzelm@13142
   335
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
nipkow@13145
   336
by (induct xs) auto
wenzelm@13114
   337
wenzelm@13114
   338
lemma length_Suc_conv:
nipkow@13145
   339
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
nipkow@13145
   340
by (induct xs) auto
wenzelm@13142
   341
nipkow@14025
   342
lemma Suc_length_conv:
nipkow@14025
   343
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
paulson@14208
   344
apply (induct xs, simp, simp)
nipkow@14025
   345
apply blast
nipkow@14025
   346
done
nipkow@14025
   347
oheimb@14099
   348
lemma impossible_Cons [rule_format]: 
oheimb@14099
   349
  "length xs <= length ys --> xs = x # ys = False"
wenzelm@20503
   350
apply (induct xs)
wenzelm@20503
   351
apply auto
oheimb@14099
   352
done
oheimb@14099
   353
nipkow@14247
   354
lemma list_induct2[consumes 1]: "\<And>ys.
nipkow@14247
   355
 \<lbrakk> length xs = length ys;
nipkow@14247
   356
   P [] [];
nipkow@14247
   357
   \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
nipkow@14247
   358
 \<Longrightarrow> P xs ys"
nipkow@14247
   359
apply(induct xs)
nipkow@14247
   360
 apply simp
nipkow@14247
   361
apply(case_tac ys)
nipkow@14247
   362
 apply simp
nipkow@14247
   363
apply(simp)
nipkow@14247
   364
done
wenzelm@13114
   365
krauss@22493
   366
lemma list_induct2': 
krauss@22493
   367
  "\<lbrakk> P [] [];
krauss@22493
   368
  \<And>x xs. P (x#xs) [];
krauss@22493
   369
  \<And>y ys. P [] (y#ys);
krauss@22493
   370
   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
krauss@22493
   371
 \<Longrightarrow> P xs ys"
krauss@22493
   372
by (induct xs arbitrary: ys) (case_tac x, auto)+
krauss@22493
   373
nipkow@22143
   374
lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
nipkow@22143
   375
apply(rule Eq_FalseI)
nipkow@22143
   376
by auto
nipkow@22143
   377
nipkow@22143
   378
(*
nipkow@22143
   379
Reduces xs=ys to False if xs and ys cannot be of the same length.
nipkow@22143
   380
This is the case if the atomic sublists of one are a submultiset
nipkow@22143
   381
of those of the other list and there are fewer Cons's in one than the other.
nipkow@22143
   382
*)
nipkow@22143
   383
ML_setup {*
nipkow@22143
   384
local
nipkow@22143
   385
nipkow@22143
   386
fun len (Const("List.list.Nil",_)) acc = acc
nipkow@22143
   387
  | len (Const("List.list.Cons",_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
haftmann@23029
   388
  | len (Const("List.append",_) $ xs $ ys) acc = len xs (len ys acc)
nipkow@22143
   389
  | len (Const("List.rev",_) $ xs) acc = len xs acc
nipkow@22143
   390
  | len (Const("List.map",_) $ _ $ xs) acc = len xs acc
nipkow@22143
   391
  | len t (ts,n) = (t::ts,n);
nipkow@22143
   392
nipkow@22143
   393
fun list_eq ss (Const(_,eqT) $ lhs $ rhs) =
nipkow@22143
   394
  let
nipkow@22143
   395
    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
nipkow@22143
   396
    fun prove_neq() =
nipkow@22143
   397
      let
nipkow@22143
   398
        val Type(_,listT::_) = eqT;
haftmann@22994
   399
        val size = HOLogic.size_const listT;
nipkow@22143
   400
        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
nipkow@22143
   401
        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
nipkow@22143
   402
        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
haftmann@22633
   403
          (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
haftmann@22633
   404
      in SOME (thm RS @{thm neq_if_length_neq}) end
nipkow@22143
   405
  in
wenzelm@23214
   406
    if m < n andalso submultiset (op aconv) (ls,rs) orelse
wenzelm@23214
   407
       n < m andalso submultiset (op aconv) (rs,ls)
nipkow@22143
   408
    then prove_neq() else NONE
nipkow@22143
   409
  end;
nipkow@22143
   410
nipkow@22143
   411
in
nipkow@22143
   412
nipkow@22143
   413
val list_neq_simproc =
haftmann@22633
   414
  Simplifier.simproc @{theory} "list_neq" ["(xs::'a list) = ys"] (K list_eq);
nipkow@22143
   415
nipkow@22143
   416
end;
nipkow@22143
   417
nipkow@22143
   418
Addsimprocs [list_neq_simproc];
nipkow@22143
   419
*}
nipkow@22143
   420
nipkow@22143
   421
nipkow@15392
   422
subsubsection {* @{text "@"} -- append *}
wenzelm@13114
   423
wenzelm@13142
   424
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
nipkow@13145
   425
by (induct xs) auto
wenzelm@13114
   426
wenzelm@13142
   427
lemma append_Nil2 [simp]: "xs @ [] = xs"
nipkow@13145
   428
by (induct xs) auto
nipkow@3507
   429
wenzelm@13142
   430
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
nipkow@13145
   431
by (induct xs) auto
wenzelm@13114
   432
wenzelm@13142
   433
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
nipkow@13145
   434
by (induct xs) auto
wenzelm@13114
   435
wenzelm@13142
   436
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
nipkow@13145
   437
by (induct xs) auto
wenzelm@13114
   438
wenzelm@13142
   439
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
nipkow@13145
   440
by (induct xs) auto
wenzelm@13114
   441
berghofe@13883
   442
lemma append_eq_append_conv [simp]:
berghofe@13883
   443
 "!!ys. length xs = length ys \<or> length us = length vs
berghofe@13883
   444
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
berghofe@13883
   445
apply (induct xs)
paulson@14208
   446
 apply (case_tac ys, simp, force)
paulson@14208
   447
apply (case_tac ys, force, simp)
nipkow@13145
   448
done
wenzelm@13142
   449
nipkow@14495
   450
lemma append_eq_append_conv2: "!!ys zs ts.
nipkow@14495
   451
 (xs @ ys = zs @ ts) =
nipkow@14495
   452
 (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
nipkow@14495
   453
apply (induct xs)
nipkow@14495
   454
 apply fastsimp
nipkow@14495
   455
apply(case_tac zs)
nipkow@14495
   456
 apply simp
nipkow@14495
   457
apply fastsimp
nipkow@14495
   458
done
nipkow@14495
   459
wenzelm@13142
   460
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145
   461
by simp
wenzelm@13142
   462
wenzelm@13142
   463
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145
   464
by simp
wenzelm@13114
   465
wenzelm@13142
   466
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   467
by simp
wenzelm@13114
   468
wenzelm@13142
   469
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   470
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   471
wenzelm@13142
   472
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   473
using append_same_eq [of "[]"] by auto
wenzelm@13114
   474
wenzelm@13142
   475
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   476
by (induct xs) auto
wenzelm@13114
   477
wenzelm@13142
   478
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   479
by (induct xs) auto
wenzelm@13114
   480
wenzelm@13142
   481
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   482
by (simp add: hd_append split: list.split)
wenzelm@13114
   483
wenzelm@13142
   484
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   485
by (simp split: list.split)
wenzelm@13114
   486
wenzelm@13142
   487
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   488
by (simp add: tl_append split: list.split)
wenzelm@13114
   489
wenzelm@13114
   490
nipkow@14300
   491
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   492
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   493
by(cases ys) auto
nipkow@14300
   494
nipkow@15281
   495
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   496
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   497
by(cases ys) auto
nipkow@15281
   498
nipkow@14300
   499
wenzelm@13142
   500
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   501
wenzelm@13114
   502
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   503
by simp
wenzelm@13114
   504
wenzelm@13142
   505
lemma Cons_eq_appendI:
nipkow@13145
   506
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   507
by (drule sym) simp
wenzelm@13114
   508
wenzelm@13142
   509
lemma append_eq_appendI:
nipkow@13145
   510
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   511
by (drule sym) simp
wenzelm@13114
   512
wenzelm@13114
   513
wenzelm@13142
   514
text {*
nipkow@13145
   515
Simplification procedure for all list equalities.
nipkow@13145
   516
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   517
- both lists end in a singleton list,
nipkow@13145
   518
- or both lists end in the same list.
wenzelm@13142
   519
*}
wenzelm@13142
   520
wenzelm@13142
   521
ML_setup {*
nipkow@3507
   522
local
nipkow@3507
   523
wenzelm@13114
   524
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462
   525
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
haftmann@23029
   526
  | last (Const("List.append",_) $ _ $ ys) = last ys
wenzelm@13462
   527
  | last t = t;
wenzelm@13114
   528
wenzelm@13114
   529
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462
   530
  | list1 _ = false;
wenzelm@13114
   531
wenzelm@13114
   532
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462
   533
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
haftmann@23029
   534
  | butlast ((app as Const("List.append",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462
   535
  | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114
   536
haftmann@22633
   537
val rearr_ss = HOL_basic_ss addsimps [@{thm append_assoc},
haftmann@22633
   538
  @{thm append_Nil}, @{thm append_Cons}];
wenzelm@16973
   539
wenzelm@20044
   540
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   541
  let
wenzelm@13462
   542
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   543
    fun rearr conv =
wenzelm@13462
   544
      let
wenzelm@13462
   545
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   546
        val Type(_,listT::_) = eqT
wenzelm@13462
   547
        val appT = [listT,listT] ---> listT
haftmann@23029
   548
        val app = Const("List.append",appT)
wenzelm@13462
   549
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   550
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@20044
   551
        val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
wenzelm@17877
   552
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
skalberg@15531
   553
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   554
wenzelm@13462
   555
  in
haftmann@22633
   556
    if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
haftmann@22633
   557
    else if lastl aconv lastr then rearr @{thm append_same_eq}
skalberg@15531
   558
    else NONE
wenzelm@13462
   559
  end;
wenzelm@13462
   560
wenzelm@13114
   561
in
wenzelm@13462
   562
wenzelm@13462
   563
val list_eq_simproc =
haftmann@22633
   564
  Simplifier.simproc @{theory} "list_eq" ["(xs::'a list) = ys"] (K list_eq);
wenzelm@13462
   565
wenzelm@13114
   566
end;
wenzelm@13114
   567
wenzelm@13114
   568
Addsimprocs [list_eq_simproc];
wenzelm@13114
   569
*}
wenzelm@13114
   570
wenzelm@13114
   571
nipkow@15392
   572
subsubsection {* @{text map} *}
wenzelm@13114
   573
wenzelm@13142
   574
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   575
by (induct xs) simp_all
wenzelm@13114
   576
wenzelm@13142
   577
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   578
by (rule ext, induct_tac xs) auto
wenzelm@13114
   579
wenzelm@13142
   580
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   581
by (induct xs) auto
wenzelm@13114
   582
wenzelm@13142
   583
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   584
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   585
wenzelm@13142
   586
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   587
by (induct xs) auto
wenzelm@13114
   588
nipkow@13737
   589
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   590
by (induct xs) auto
nipkow@13737
   591
krauss@19770
   592
lemma map_cong [fundef_cong, recdef_cong]:
nipkow@13145
   593
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   594
-- {* a congruence rule for @{text map} *}
nipkow@13737
   595
by simp
wenzelm@13114
   596
wenzelm@13142
   597
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   598
by (cases xs) auto
wenzelm@13114
   599
wenzelm@13142
   600
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   601
by (cases xs) auto
wenzelm@13114
   602
paulson@18447
   603
lemma map_eq_Cons_conv:
nipkow@14025
   604
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   605
by (cases xs) auto
wenzelm@13114
   606
paulson@18447
   607
lemma Cons_eq_map_conv:
nipkow@14025
   608
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   609
by (cases ys) auto
nipkow@14025
   610
paulson@18447
   611
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
paulson@18447
   612
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
paulson@18447
   613
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
paulson@18447
   614
nipkow@14111
   615
lemma ex_map_conv:
nipkow@14111
   616
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
paulson@18447
   617
by(induct ys, auto simp add: Cons_eq_map_conv)
nipkow@14111
   618
nipkow@15110
   619
lemma map_eq_imp_length_eq:
nipkow@15110
   620
  "!!xs. map f xs = map f ys ==> length xs = length ys"
nipkow@15110
   621
apply (induct ys)
nipkow@15110
   622
 apply simp
nipkow@15110
   623
apply(simp (no_asm_use))
nipkow@15110
   624
apply clarify
nipkow@15110
   625
apply(simp (no_asm_use))
nipkow@15110
   626
apply fast
nipkow@15110
   627
done
nipkow@15110
   628
nipkow@15110
   629
lemma map_inj_on:
nipkow@15110
   630
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   631
  ==> xs = ys"
nipkow@15110
   632
apply(frule map_eq_imp_length_eq)
nipkow@15110
   633
apply(rotate_tac -1)
nipkow@15110
   634
apply(induct rule:list_induct2)
nipkow@15110
   635
 apply simp
nipkow@15110
   636
apply(simp)
nipkow@15110
   637
apply (blast intro:sym)
nipkow@15110
   638
done
nipkow@15110
   639
nipkow@15110
   640
lemma inj_on_map_eq_map:
nipkow@15110
   641
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   642
by(blast dest:map_inj_on)
nipkow@15110
   643
wenzelm@13114
   644
lemma map_injective:
nipkow@14338
   645
 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@14338
   646
by (induct ys) (auto dest!:injD)
wenzelm@13114
   647
nipkow@14339
   648
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   649
by(blast dest:map_injective)
nipkow@14339
   650
wenzelm@13114
   651
lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@17589
   652
by (iprover dest: map_injective injD intro: inj_onI)
wenzelm@13114
   653
wenzelm@13114
   654
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   655
apply (unfold inj_on_def, clarify)
nipkow@13145
   656
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   657
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   658
apply blast
nipkow@13145
   659
done
wenzelm@13114
   660
nipkow@14339
   661
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   662
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   663
nipkow@15303
   664
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
   665
apply(rule inj_onI)
nipkow@15303
   666
apply(erule map_inj_on)
nipkow@15303
   667
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
   668
done
nipkow@15303
   669
kleing@14343
   670
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   671
by (induct xs, auto)
wenzelm@13114
   672
nipkow@14402
   673
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   674
by (induct xs) auto
nipkow@14402
   675
nipkow@15110
   676
lemma map_fst_zip[simp]:
nipkow@15110
   677
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   678
by (induct rule:list_induct2, simp_all)
nipkow@15110
   679
nipkow@15110
   680
lemma map_snd_zip[simp]:
nipkow@15110
   681
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   682
by (induct rule:list_induct2, simp_all)
nipkow@15110
   683
nipkow@15110
   684
nipkow@15392
   685
subsubsection {* @{text rev} *}
wenzelm@13114
   686
wenzelm@13142
   687
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   688
by (induct xs) auto
wenzelm@13114
   689
wenzelm@13142
   690
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   691
by (induct xs) auto
wenzelm@13114
   692
kleing@15870
   693
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870
   694
by auto
kleing@15870
   695
wenzelm@13142
   696
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   697
by (induct xs) auto
wenzelm@13114
   698
wenzelm@13142
   699
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   700
by (induct xs) auto
wenzelm@13114
   701
kleing@15870
   702
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870
   703
by (cases xs) auto
kleing@15870
   704
kleing@15870
   705
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870
   706
by (cases xs) auto
kleing@15870
   707
haftmann@21061
   708
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
haftmann@21061
   709
apply (induct xs arbitrary: ys, force)
paulson@14208
   710
apply (case_tac ys, simp, force)
nipkow@13145
   711
done
wenzelm@13114
   712
nipkow@15439
   713
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
   714
by(simp add:inj_on_def)
nipkow@15439
   715
wenzelm@13366
   716
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   717
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489
   718
apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145
   719
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   720
done
wenzelm@13114
   721
nipkow@13145
   722
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114
   723
wenzelm@13366
   724
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   725
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   726
by (induct xs rule: rev_induct) auto
wenzelm@13114
   727
wenzelm@13366
   728
lemmas rev_cases = rev_exhaust
wenzelm@13366
   729
nipkow@18423
   730
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
nipkow@18423
   731
by(rule rev_cases[of xs]) auto
nipkow@18423
   732
wenzelm@13114
   733
nipkow@15392
   734
subsubsection {* @{text set} *}
wenzelm@13114
   735
wenzelm@13142
   736
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   737
by (induct xs) auto
wenzelm@13114
   738
wenzelm@13142
   739
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   740
by (induct xs) auto
wenzelm@13114
   741
nipkow@17830
   742
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
nipkow@17830
   743
by(cases xs) auto
oheimb@14099
   744
wenzelm@13142
   745
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   746
by auto
wenzelm@13114
   747
oheimb@14099
   748
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   749
by auto
oheimb@14099
   750
wenzelm@13142
   751
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   752
by (induct xs) auto
wenzelm@13114
   753
nipkow@15245
   754
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   755
by(induct xs) auto
nipkow@15245
   756
wenzelm@13142
   757
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   758
by (induct xs) auto
wenzelm@13114
   759
wenzelm@13142
   760
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   761
by (induct xs) auto
wenzelm@13114
   762
nipkow@22828
   763
lemma set_allpairs[simp]:
nipkow@22830
   764
 "set(allpairs f xs ys) = {z. EX x : set xs. EX y : set ys. z = f x y}"
nipkow@22828
   765
by(induct xs) auto
nipkow@22828
   766
wenzelm@13142
   767
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   768
by (induct xs) auto
wenzelm@13114
   769
nipkow@15425
   770
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
paulson@14208
   771
apply (induct j, simp_all)
paulson@14208
   772
apply (erule ssubst, auto)
nipkow@13145
   773
done
wenzelm@13114
   774
wenzelm@13142
   775
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
paulson@15113
   776
proof (induct xs)
paulson@15113
   777
  case Nil show ?case by simp
paulson@15113
   778
  case (Cons a xs)
paulson@15113
   779
  show ?case
paulson@15113
   780
  proof 
paulson@15113
   781
    assume "x \<in> set (a # xs)"
paulson@15113
   782
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   783
      by (simp, blast intro: Cons_eq_appendI)
paulson@15113
   784
  next
paulson@15113
   785
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   786
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
paulson@15113
   787
    show "x \<in> set (a # xs)" 
paulson@15113
   788
      by (cases ys, auto simp add: eq)
paulson@15113
   789
  qed
paulson@15113
   790
qed
wenzelm@13142
   791
nipkow@18049
   792
lemma in_set_conv_decomp_first:
nipkow@18049
   793
 "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
nipkow@18049
   794
proof (induct xs)
nipkow@18049
   795
  case Nil show ?case by simp
nipkow@18049
   796
next
nipkow@18049
   797
  case (Cons a xs)
nipkow@18049
   798
  show ?case
nipkow@18049
   799
  proof cases
nipkow@18049
   800
    assume "x = a" thus ?case using Cons by force
nipkow@18049
   801
  next
nipkow@18049
   802
    assume "x \<noteq> a"
nipkow@18049
   803
    show ?case
nipkow@18049
   804
    proof
nipkow@18049
   805
      assume "x \<in> set (a # xs)"
nipkow@18049
   806
      from prems show "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@18049
   807
	by(fastsimp intro!: Cons_eq_appendI)
nipkow@18049
   808
    next
nipkow@18049
   809
      assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@18049
   810
      then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
nipkow@18049
   811
      show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq)
nipkow@18049
   812
    qed
nipkow@18049
   813
  qed
nipkow@18049
   814
qed
nipkow@18049
   815
nipkow@18049
   816
lemmas split_list       = in_set_conv_decomp[THEN iffD1, standard]
nipkow@18049
   817
lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard]
nipkow@18049
   818
nipkow@18049
   819
paulson@13508
   820
lemma finite_list: "finite A ==> EX l. set l = A"
paulson@13508
   821
apply (erule finite_induct, auto)
paulson@13508
   822
apply (rule_tac x="x#l" in exI, auto)
paulson@13508
   823
done
paulson@13508
   824
kleing@14388
   825
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
   826
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
   827
paulson@15168
   828
nipkow@15392
   829
subsubsection {* @{text filter} *}
wenzelm@13114
   830
wenzelm@13142
   831
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   832
by (induct xs) auto
wenzelm@13114
   833
nipkow@15305
   834
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
   835
by (induct xs) simp_all
nipkow@15305
   836
wenzelm@13142
   837
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   838
by (induct xs) auto
wenzelm@13114
   839
nipkow@16998
   840
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998
   841
by (induct xs) (auto simp add: le_SucI)
nipkow@16998
   842
nipkow@18423
   843
lemma sum_length_filter_compl:
nipkow@18423
   844
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
nipkow@18423
   845
by(induct xs) simp_all
nipkow@18423
   846
wenzelm@13142
   847
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   848
by (induct xs) auto
wenzelm@13114
   849
wenzelm@13142
   850
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   851
by (induct xs) auto
wenzelm@13114
   852
nipkow@16998
   853
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@16998
   854
  by (induct xs) simp_all
nipkow@16998
   855
nipkow@16998
   856
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998
   857
apply (induct xs)
nipkow@16998
   858
 apply auto
nipkow@16998
   859
apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998
   860
apply simp
nipkow@16998
   861
done
wenzelm@13114
   862
nipkow@16965
   863
lemma filter_map:
nipkow@16965
   864
  "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965
   865
by (induct xs) simp_all
nipkow@16965
   866
nipkow@16965
   867
lemma length_filter_map[simp]:
nipkow@16965
   868
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965
   869
by (simp add:filter_map)
nipkow@16965
   870
wenzelm@13142
   871
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   872
by auto
wenzelm@13114
   873
nipkow@15246
   874
lemma length_filter_less:
nipkow@15246
   875
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
   876
proof (induct xs)
nipkow@15246
   877
  case Nil thus ?case by simp
nipkow@15246
   878
next
nipkow@15246
   879
  case (Cons x xs) thus ?case
nipkow@15246
   880
    apply (auto split:split_if_asm)
nipkow@15246
   881
    using length_filter_le[of P xs] apply arith
nipkow@15246
   882
  done
nipkow@15246
   883
qed
wenzelm@13114
   884
nipkow@15281
   885
lemma length_filter_conv_card:
nipkow@15281
   886
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
   887
proof (induct xs)
nipkow@15281
   888
  case Nil thus ?case by simp
nipkow@15281
   889
next
nipkow@15281
   890
  case (Cons x xs)
nipkow@15281
   891
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
   892
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
   893
  show ?case (is "?l = card ?S'")
nipkow@15281
   894
  proof (cases)
nipkow@15281
   895
    assume "p x"
nipkow@15281
   896
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@15281
   897
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   898
    have "length (filter p (x # xs)) = Suc(card ?S)"
nipkow@15281
   899
      using Cons by simp
nipkow@15281
   900
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
nipkow@15281
   901
      by (simp add: card_image inj_Suc)
nipkow@15281
   902
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   903
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
   904
    finally show ?thesis .
nipkow@15281
   905
  next
nipkow@15281
   906
    assume "\<not> p x"
nipkow@15281
   907
    hence eq: "?S' = Suc ` ?S"
nipkow@15281
   908
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   909
    have "length (filter p (x # xs)) = card ?S"
nipkow@15281
   910
      using Cons by simp
nipkow@15281
   911
    also have "\<dots> = card(Suc ` ?S)" using fin
nipkow@15281
   912
      by (simp add: card_image inj_Suc)
nipkow@15281
   913
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   914
      by (simp add:card_insert_if)
nipkow@15281
   915
    finally show ?thesis .
nipkow@15281
   916
  qed
nipkow@15281
   917
qed
nipkow@15281
   918
nipkow@17629
   919
lemma Cons_eq_filterD:
nipkow@17629
   920
 "x#xs = filter P ys \<Longrightarrow>
nipkow@17629
   921
  \<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
   922
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
nipkow@17629
   923
proof(induct ys)
nipkow@17629
   924
  case Nil thus ?case by simp
nipkow@17629
   925
next
nipkow@17629
   926
  case (Cons y ys)
nipkow@17629
   927
  show ?case (is "\<exists>x. ?Q x")
nipkow@17629
   928
  proof cases
nipkow@17629
   929
    assume Py: "P y"
nipkow@17629
   930
    show ?thesis
nipkow@17629
   931
    proof cases
nipkow@17629
   932
      assume xy: "x = y"
nipkow@17629
   933
      show ?thesis
nipkow@17629
   934
      proof from Py xy Cons(2) show "?Q []" by simp qed
nipkow@17629
   935
    next
nipkow@17629
   936
      assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp
nipkow@17629
   937
    qed
nipkow@17629
   938
  next
nipkow@17629
   939
    assume Py: "\<not> P y"
nipkow@17629
   940
    with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp
nipkow@17629
   941
    show ?thesis (is "? us. ?Q us")
nipkow@17629
   942
    proof show "?Q (y#us)" using 1 by simp qed
nipkow@17629
   943
  qed
nipkow@17629
   944
qed
nipkow@17629
   945
nipkow@17629
   946
lemma filter_eq_ConsD:
nipkow@17629
   947
 "filter P ys = x#xs \<Longrightarrow>
nipkow@17629
   948
  \<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
   949
by(rule Cons_eq_filterD) simp
nipkow@17629
   950
nipkow@17629
   951
lemma filter_eq_Cons_iff:
nipkow@17629
   952
 "(filter P ys = x#xs) =
nipkow@17629
   953
  (\<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
   954
by(auto dest:filter_eq_ConsD)
nipkow@17629
   955
nipkow@17629
   956
lemma Cons_eq_filter_iff:
nipkow@17629
   957
 "(x#xs = filter P ys) =
nipkow@17629
   958
  (\<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
   959
by(auto dest:Cons_eq_filterD)
nipkow@17629
   960
krauss@19770
   961
lemma filter_cong[fundef_cong, recdef_cong]:
nipkow@17501
   962
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
nipkow@17501
   963
apply simp
nipkow@17501
   964
apply(erule thin_rl)
nipkow@17501
   965
by (induct ys) simp_all
nipkow@17501
   966
nipkow@15281
   967
nipkow@15392
   968
subsubsection {* @{text concat} *}
wenzelm@13114
   969
wenzelm@13142
   970
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
   971
by (induct xs) auto
wenzelm@13114
   972
paulson@18447
   973
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   974
by (induct xss) auto
wenzelm@13114
   975
paulson@18447
   976
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   977
by (induct xss) auto
wenzelm@13114
   978
wenzelm@13142
   979
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
nipkow@13145
   980
by (induct xs) auto
wenzelm@13114
   981
wenzelm@13142
   982
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
   983
by (induct xs) auto
wenzelm@13114
   984
wenzelm@13142
   985
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
   986
by (induct xs) auto
wenzelm@13114
   987
wenzelm@13142
   988
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
   989
by (induct xs) auto
wenzelm@13114
   990
wenzelm@13114
   991
nipkow@15392
   992
subsubsection {* @{text nth} *}
wenzelm@13114
   993
wenzelm@13142
   994
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
   995
by auto
wenzelm@13114
   996
wenzelm@13142
   997
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
   998
by auto
wenzelm@13114
   999
wenzelm@13142
  1000
declare nth.simps [simp del]
wenzelm@13114
  1001
wenzelm@13114
  1002
lemma nth_append:
nipkow@13145
  1003
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
paulson@14208
  1004
apply (induct "xs", simp)
paulson@14208
  1005
apply (case_tac n, auto)
nipkow@13145
  1006
done
wenzelm@13114
  1007
nipkow@14402
  1008
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
nipkow@14402
  1009
by (induct "xs") auto
nipkow@14402
  1010
nipkow@14402
  1011
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
nipkow@14402
  1012
by (induct "xs") auto
nipkow@14402
  1013
wenzelm@13142
  1014
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
paulson@14208
  1015
apply (induct xs, simp)
paulson@14208
  1016
apply (case_tac n, auto)
nipkow@13145
  1017
done
wenzelm@13114
  1018
nipkow@18423
  1019
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
nipkow@18423
  1020
by(cases xs) simp_all
nipkow@18423
  1021
nipkow@18049
  1022
nipkow@18049
  1023
lemma list_eq_iff_nth_eq:
nipkow@18049
  1024
 "!!ys. (xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
nipkow@18049
  1025
apply(induct xs)
nipkow@18049
  1026
 apply simp apply blast
nipkow@18049
  1027
apply(case_tac ys)
nipkow@18049
  1028
 apply simp
nipkow@18049
  1029
apply(simp add:nth_Cons split:nat.split)apply blast
nipkow@18049
  1030
done
nipkow@18049
  1031
wenzelm@13142
  1032
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
  1033
apply (induct xs, simp, simp)
nipkow@13145
  1034
apply safe
paulson@14208
  1035
apply (rule_tac x = 0 in exI, simp)
paulson@14208
  1036
 apply (rule_tac x = "Suc i" in exI, simp)
paulson@14208
  1037
apply (case_tac i, simp)
nipkow@13145
  1038
apply (rename_tac j)
paulson@14208
  1039
apply (rule_tac x = j in exI, simp)
nipkow@13145
  1040
done
wenzelm@13114
  1041
nipkow@17501
  1042
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
nipkow@17501
  1043
by(auto simp:set_conv_nth)
nipkow@17501
  1044
nipkow@13145
  1045
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
  1046
by (auto simp add: set_conv_nth)
wenzelm@13114
  1047
wenzelm@13142
  1048
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
  1049
by (auto simp add: set_conv_nth)
wenzelm@13114
  1050
wenzelm@13114
  1051
lemma all_nth_imp_all_set:
nipkow@13145
  1052
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
  1053
by (auto simp add: set_conv_nth)
wenzelm@13114
  1054
wenzelm@13114
  1055
lemma all_set_conv_all_nth:
nipkow@13145
  1056
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
  1057
by (auto simp add: set_conv_nth)
wenzelm@13114
  1058
wenzelm@13114
  1059
nipkow@15392
  1060
subsubsection {* @{text list_update} *}
wenzelm@13114
  1061
wenzelm@13142
  1062
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
nipkow@13145
  1063
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1064
wenzelm@13114
  1065
lemma nth_list_update:
nipkow@13145
  1066
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@13145
  1067
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1068
wenzelm@13142
  1069
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
  1070
by (simp add: nth_list_update)
wenzelm@13114
  1071
wenzelm@13142
  1072
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145
  1073
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1074
wenzelm@13142
  1075
lemma list_update_overwrite [simp]:
nipkow@13145
  1076
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145
  1077
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1078
nipkow@14402
  1079
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
paulson@14208
  1080
apply (induct xs, simp)
nipkow@14187
  1081
apply(simp split:nat.splits)
nipkow@14187
  1082
done
nipkow@14187
  1083
nipkow@17501
  1084
lemma list_update_beyond[simp]: "\<And>i. length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
nipkow@17501
  1085
apply (induct xs)
nipkow@17501
  1086
 apply simp
nipkow@17501
  1087
apply (case_tac i)
nipkow@17501
  1088
apply simp_all
nipkow@17501
  1089
done
nipkow@17501
  1090
wenzelm@13114
  1091
lemma list_update_same_conv:
nipkow@13145
  1092
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145
  1093
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1094
nipkow@14187
  1095
lemma list_update_append1:
nipkow@14187
  1096
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
paulson@14208
  1097
apply (induct xs, simp)
nipkow@14187
  1098
apply(simp split:nat.split)
nipkow@14187
  1099
done
nipkow@14187
  1100
kleing@15868
  1101
lemma list_update_append:
kleing@15868
  1102
  "!!n. (xs @ ys) [n:= x] = 
kleing@15868
  1103
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
kleing@15868
  1104
by (induct xs) (auto split:nat.splits)
kleing@15868
  1105
nipkow@14402
  1106
lemma list_update_length [simp]:
nipkow@14402
  1107
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
  1108
by (induct xs, auto)
nipkow@14402
  1109
wenzelm@13114
  1110
lemma update_zip:
nipkow@13145
  1111
"!!i xy xs. length xs = length ys ==>
nipkow@13145
  1112
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145
  1113
by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114
  1114
wenzelm@13114
  1115
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145
  1116
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1117
wenzelm@13114
  1118
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
  1119
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
  1120
kleing@15868
  1121
lemma set_update_memI: "!!n. n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
kleing@15868
  1122
by (induct xs) (auto split:nat.splits)
kleing@15868
  1123
wenzelm@13114
  1124
nipkow@15392
  1125
subsubsection {* @{text last} and @{text butlast} *}
wenzelm@13114
  1126
wenzelm@13142
  1127
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
  1128
by (induct xs) auto
wenzelm@13114
  1129
wenzelm@13142
  1130
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
  1131
by (induct xs) auto
wenzelm@13114
  1132
nipkow@14302
  1133
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
  1134
by(simp add:last.simps)
nipkow@14302
  1135
nipkow@14302
  1136
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
  1137
by(simp add:last.simps)
nipkow@14302
  1138
nipkow@14302
  1139
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
  1140
by (induct xs) (auto)
nipkow@14302
  1141
nipkow@14302
  1142
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
  1143
by(simp add:last_append)
nipkow@14302
  1144
nipkow@14302
  1145
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
  1146
by(simp add:last_append)
nipkow@14302
  1147
nipkow@17762
  1148
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
nipkow@17762
  1149
by(rule rev_exhaust[of xs]) simp_all
nipkow@17762
  1150
nipkow@17762
  1151
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
nipkow@17762
  1152
by(cases xs) simp_all
nipkow@17762
  1153
nipkow@17765
  1154
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
nipkow@17765
  1155
by (induct as) auto
nipkow@17762
  1156
wenzelm@13142
  1157
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
  1158
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1159
wenzelm@13114
  1160
lemma butlast_append:
nipkow@13145
  1161
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145
  1162
by (induct xs) auto
wenzelm@13114
  1163
wenzelm@13142
  1164
lemma append_butlast_last_id [simp]:
nipkow@13145
  1165
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
  1166
by (induct xs) auto
wenzelm@13114
  1167
wenzelm@13142
  1168
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
  1169
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1170
wenzelm@13114
  1171
lemma in_set_butlast_appendI:
nipkow@13145
  1172
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
  1173
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
  1174
nipkow@17501
  1175
lemma last_drop[simp]: "!!n. n < length xs \<Longrightarrow> last (drop n xs) = last xs"
nipkow@17501
  1176
apply (induct xs)
nipkow@17501
  1177
 apply simp
nipkow@17501
  1178
apply (auto split:nat.split)
nipkow@17501
  1179
done
nipkow@17501
  1180
nipkow@17589
  1181
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
nipkow@17589
  1182
by(induct xs)(auto simp:neq_Nil_conv)
nipkow@17589
  1183
nipkow@15392
  1184
subsubsection {* @{text take} and @{text drop} *}
wenzelm@13114
  1185
wenzelm@13142
  1186
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
  1187
by (induct xs) auto
wenzelm@13114
  1188
wenzelm@13142
  1189
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
  1190
by (induct xs) auto
wenzelm@13114
  1191
wenzelm@13142
  1192
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
  1193
by simp
wenzelm@13114
  1194
wenzelm@13142
  1195
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
  1196
by simp
wenzelm@13114
  1197
wenzelm@13142
  1198
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
  1199
nipkow@15110
  1200
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
  1201
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
  1202
nipkow@14187
  1203
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
  1204
by(cases xs, simp_all)
nipkow@14187
  1205
nipkow@14187
  1206
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
nipkow@14187
  1207
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@14187
  1208
nipkow@14187
  1209
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
paulson@14208
  1210
apply (induct xs, simp)
nipkow@14187
  1211
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
  1212
done
nipkow@14187
  1213
nipkow@13913
  1214
lemma take_Suc_conv_app_nth:
nipkow@13913
  1215
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
paulson@14208
  1216
apply (induct xs, simp)
paulson@14208
  1217
apply (case_tac i, auto)
nipkow@13913
  1218
done
nipkow@13913
  1219
mehta@14591
  1220
lemma drop_Suc_conv_tl:
mehta@14591
  1221
  "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
mehta@14591
  1222
apply (induct xs, simp)
mehta@14591
  1223
apply (case_tac i, auto)
mehta@14591
  1224
done
mehta@14591
  1225
wenzelm@13142
  1226
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
nipkow@13145
  1227
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1228
wenzelm@13142
  1229
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
nipkow@13145
  1230
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1231
wenzelm@13142
  1232
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
nipkow@13145
  1233
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1234
wenzelm@13142
  1235
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
nipkow@13145
  1236
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1237
wenzelm@13142
  1238
lemma take_append [simp]:
nipkow@13145
  1239
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@13145
  1240
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1241
wenzelm@13142
  1242
lemma drop_append [simp]:
nipkow@13145
  1243
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@13145
  1244
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1245
wenzelm@13142
  1246
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
paulson@14208
  1247
apply (induct m, auto)
paulson@14208
  1248
apply (case_tac xs, auto)
nipkow@15236
  1249
apply (case_tac n, auto)
nipkow@13145
  1250
done
wenzelm@13114
  1251
wenzelm@13142
  1252
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
paulson@14208
  1253
apply (induct m, auto)
paulson@14208
  1254
apply (case_tac xs, auto)
nipkow@13145
  1255
done
wenzelm@13114
  1256
wenzelm@13114
  1257
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
paulson@14208
  1258
apply (induct m, auto)
paulson@14208
  1259
apply (case_tac xs, auto)
nipkow@13145
  1260
done
wenzelm@13114
  1261
nipkow@14802
  1262
lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@14802
  1263
apply(induct xs)
nipkow@14802
  1264
 apply simp
nipkow@14802
  1265
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  1266
done
nipkow@14802
  1267
wenzelm@13142
  1268
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
paulson@14208
  1269
apply (induct n, auto)
paulson@14208
  1270
apply (case_tac xs, auto)
nipkow@13145
  1271
done
wenzelm@13114
  1272
nipkow@15110
  1273
lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@15110
  1274
apply(induct xs)
nipkow@15110
  1275
 apply simp
nipkow@15110
  1276
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  1277
done
nipkow@15110
  1278
nipkow@15110
  1279
lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"
nipkow@15110
  1280
apply(induct xs)
nipkow@15110
  1281
apply simp
nipkow@15110
  1282
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1283
done
nipkow@15110
  1284
wenzelm@13114
  1285
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
paulson@14208
  1286
apply (induct n, auto)
paulson@14208
  1287
apply (case_tac xs, auto)
nipkow@13145
  1288
done
wenzelm@13114
  1289
wenzelm@13142
  1290
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
paulson@14208
  1291
apply (induct n, auto)
paulson@14208
  1292
apply (case_tac xs, auto)
nipkow@13145
  1293
done
wenzelm@13114
  1294
wenzelm@13114
  1295
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
paulson@14208
  1296
apply (induct xs, auto)
paulson@14208
  1297
apply (case_tac i, auto)
nipkow@13145
  1298
done
wenzelm@13114
  1299
wenzelm@13114
  1300
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
paulson@14208
  1301
apply (induct xs, auto)
paulson@14208
  1302
apply (case_tac i, auto)
nipkow@13145
  1303
done
wenzelm@13114
  1304
wenzelm@13142
  1305
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
paulson@14208
  1306
apply (induct xs, auto)
paulson@14208
  1307
apply (case_tac n, blast)
paulson@14208
  1308
apply (case_tac i, auto)
nipkow@13145
  1309
done
wenzelm@13114
  1310
wenzelm@13142
  1311
lemma nth_drop [simp]:
nipkow@13145
  1312
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
paulson@14208
  1313
apply (induct n, auto)
paulson@14208
  1314
apply (case_tac xs, auto)
nipkow@13145
  1315
done
nipkow@3507
  1316
nipkow@18423
  1317
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
nipkow@18423
  1318
by(simp add: hd_conv_nth)
nipkow@18423
  1319
nipkow@14025
  1320
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
nipkow@14025
  1321
by(induct xs)(auto simp:take_Cons split:nat.split)
nipkow@14025
  1322
nipkow@14025
  1323
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
nipkow@14025
  1324
by(induct xs)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1325
nipkow@14187
  1326
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1327
using set_take_subset by fast
nipkow@14187
  1328
nipkow@14187
  1329
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1330
using set_drop_subset by fast
nipkow@14187
  1331
wenzelm@13114
  1332
lemma append_eq_conv_conj:
nipkow@13145
  1333
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
paulson@14208
  1334
apply (induct xs, simp, clarsimp)
paulson@14208
  1335
apply (case_tac zs, auto)
nipkow@13145
  1336
done
wenzelm@13142
  1337
paulson@14050
  1338
lemma take_add [rule_format]: 
paulson@14050
  1339
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
paulson@14050
  1340
apply (induct xs, auto) 
paulson@14050
  1341
apply (case_tac i, simp_all) 
paulson@14050
  1342
done
paulson@14050
  1343
nipkow@14300
  1344
lemma append_eq_append_conv_if:
nipkow@14300
  1345
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1346
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1347
   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
  1348
   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
  1349
apply(induct xs\<^isub>1)
nipkow@14300
  1350
 apply simp
nipkow@14300
  1351
apply(case_tac ys\<^isub>1)
nipkow@14300
  1352
apply simp_all
nipkow@14300
  1353
done
nipkow@14300
  1354
nipkow@15110
  1355
lemma take_hd_drop:
nipkow@15110
  1356
  "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
nipkow@15110
  1357
apply(induct xs)
nipkow@15110
  1358
apply simp
nipkow@15110
  1359
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1360
done
nipkow@15110
  1361
nipkow@17501
  1362
lemma id_take_nth_drop:
nipkow@17501
  1363
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
nipkow@17501
  1364
proof -
nipkow@17501
  1365
  assume si: "i < length xs"
nipkow@17501
  1366
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
nipkow@17501
  1367
  moreover
nipkow@17501
  1368
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
nipkow@17501
  1369
    apply (rule_tac take_Suc_conv_app_nth) by arith
nipkow@17501
  1370
  ultimately show ?thesis by auto
nipkow@17501
  1371
qed
nipkow@17501
  1372
  
nipkow@17501
  1373
lemma upd_conv_take_nth_drop:
nipkow@17501
  1374
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1375
proof -
nipkow@17501
  1376
  assume i: "i < length xs"
nipkow@17501
  1377
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
nipkow@17501
  1378
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
nipkow@17501
  1379
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1380
    using i by (simp add: list_update_append)
nipkow@17501
  1381
  finally show ?thesis .
nipkow@17501
  1382
qed
nipkow@17501
  1383
wenzelm@13114
  1384
nipkow@15392
  1385
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1386
wenzelm@13142
  1387
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1388
by (induct xs) auto
wenzelm@13114
  1389
wenzelm@13142
  1390
lemma takeWhile_append1 [simp]:
nipkow@13145
  1391
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1392
by (induct xs) auto
wenzelm@13114
  1393
wenzelm@13142
  1394
lemma takeWhile_append2 [simp]:
nipkow@13145
  1395
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1396
by (induct xs) auto
wenzelm@13114
  1397
wenzelm@13142
  1398
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1399
by (induct xs) auto
wenzelm@13114
  1400
wenzelm@13142
  1401
lemma dropWhile_append1 [simp]:
nipkow@13145
  1402
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1403
by (induct xs) auto
wenzelm@13114
  1404
wenzelm@13142
  1405
lemma dropWhile_append2 [simp]:
nipkow@13145
  1406
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1407
by (induct xs) auto
wenzelm@13114
  1408
wenzelm@13142
  1409
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1410
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1411
nipkow@13913
  1412
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1413
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1414
by(induct xs, auto)
nipkow@13913
  1415
nipkow@13913
  1416
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1417
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1418
by(induct xs, auto)
nipkow@13913
  1419
nipkow@13913
  1420
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1421
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1422
by(induct xs, auto)
nipkow@13913
  1423
nipkow@17501
  1424
text{* The following two lemmmas could be generalized to an arbitrary
nipkow@17501
  1425
property. *}
nipkow@17501
  1426
nipkow@17501
  1427
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1428
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
nipkow@17501
  1429
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
nipkow@17501
  1430
nipkow@17501
  1431
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1432
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
nipkow@17501
  1433
apply(induct xs)
nipkow@17501
  1434
 apply simp
nipkow@17501
  1435
apply auto
nipkow@17501
  1436
apply(subst dropWhile_append2)
nipkow@17501
  1437
apply auto
nipkow@17501
  1438
done
nipkow@17501
  1439
nipkow@18423
  1440
lemma takeWhile_not_last:
nipkow@18423
  1441
 "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
nipkow@18423
  1442
apply(induct xs)
nipkow@18423
  1443
 apply simp
nipkow@18423
  1444
apply(case_tac xs)
nipkow@18423
  1445
apply(auto)
nipkow@18423
  1446
done
nipkow@18423
  1447
krauss@19770
  1448
lemma takeWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1449
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1450
  ==> takeWhile P l = takeWhile Q k"
wenzelm@20503
  1451
  by (induct k arbitrary: l) (simp_all)
krauss@18336
  1452
krauss@19770
  1453
lemma dropWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1454
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1455
  ==> dropWhile P l = dropWhile Q k"
wenzelm@20503
  1456
  by (induct k arbitrary: l, simp_all)
krauss@18336
  1457
wenzelm@13114
  1458
nipkow@15392
  1459
subsubsection {* @{text zip} *}
wenzelm@13114
  1460
wenzelm@13142
  1461
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1462
by (induct ys) auto
wenzelm@13114
  1463
wenzelm@13142
  1464
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1465
by simp
wenzelm@13114
  1466
wenzelm@13142
  1467
declare zip_Cons [simp del]
wenzelm@13114
  1468
nipkow@15281
  1469
lemma zip_Cons1:
nipkow@15281
  1470
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  1471
by(auto split:list.split)
nipkow@15281
  1472
wenzelm@13142
  1473
lemma length_zip [simp]:
krauss@22493
  1474
"length (zip xs ys) = min (length xs) (length ys)"
krauss@22493
  1475
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1476
wenzelm@13114
  1477
lemma zip_append1:
krauss@22493
  1478
"zip (xs @ ys) zs =
nipkow@13145
  1479
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
krauss@22493
  1480
by (induct xs zs rule:list_induct2') auto
wenzelm@13114
  1481
wenzelm@13114
  1482
lemma zip_append2:
krauss@22493
  1483
"zip xs (ys @ zs) =
nipkow@13145
  1484
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
krauss@22493
  1485
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1486
wenzelm@13142
  1487
lemma zip_append [simp]:
wenzelm@13142
  1488
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1489
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1490
by (simp add: zip_append1)
wenzelm@13114
  1491
wenzelm@13114
  1492
lemma zip_rev:
nipkow@14247
  1493
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1494
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1495
nipkow@23096
  1496
lemma map_zip_map:
nipkow@23096
  1497
 "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
nipkow@23096
  1498
apply(induct xs arbitrary:ys) apply simp
nipkow@23096
  1499
apply(case_tac ys)
nipkow@23096
  1500
apply simp_all
nipkow@23096
  1501
done
nipkow@23096
  1502
nipkow@23096
  1503
lemma map_zip_map2:
nipkow@23096
  1504
 "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
nipkow@23096
  1505
apply(induct xs arbitrary:ys) apply simp
nipkow@23096
  1506
apply(case_tac ys)
nipkow@23096
  1507
apply simp_all
nipkow@23096
  1508
done
nipkow@23096
  1509
wenzelm@13142
  1510
lemma nth_zip [simp]:
nipkow@13145
  1511
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
paulson@14208
  1512
apply (induct ys, simp)
nipkow@13145
  1513
apply (case_tac xs)
nipkow@13145
  1514
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1515
done
wenzelm@13114
  1516
wenzelm@13114
  1517
lemma set_zip:
nipkow@13145
  1518
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
  1519
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1520
wenzelm@13114
  1521
lemma zip_update:
nipkow@13145
  1522
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
  1523
by (rule sym, simp add: update_zip)
wenzelm@13114
  1524
wenzelm@13142
  1525
lemma zip_replicate [simp]:
nipkow@13145
  1526
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
paulson@14208
  1527
apply (induct i, auto)
paulson@14208
  1528
apply (case_tac j, auto)
nipkow@13145
  1529
done
wenzelm@13114
  1530
nipkow@19487
  1531
lemma take_zip:
nipkow@19487
  1532
 "!!xs ys. take n (zip xs ys) = zip (take n xs) (take n ys)"
nipkow@19487
  1533
apply (induct n)
nipkow@19487
  1534
 apply simp
nipkow@19487
  1535
apply (case_tac xs, simp)
nipkow@19487
  1536
apply (case_tac ys, simp_all)
nipkow@19487
  1537
done
nipkow@19487
  1538
nipkow@19487
  1539
lemma drop_zip:
nipkow@19487
  1540
 "!!xs ys. drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
nipkow@19487
  1541
apply (induct n)
nipkow@19487
  1542
 apply simp
nipkow@19487
  1543
apply (case_tac xs, simp)
nipkow@19487
  1544
apply (case_tac ys, simp_all)
nipkow@19487
  1545
done
nipkow@19487
  1546
krauss@22493
  1547
lemma set_zip_leftD:
krauss@22493
  1548
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
krauss@22493
  1549
by (induct xs ys rule:list_induct2') auto
krauss@22493
  1550
krauss@22493
  1551
lemma set_zip_rightD:
krauss@22493
  1552
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
krauss@22493
  1553
by (induct xs ys rule:list_induct2') auto
wenzelm@13142
  1554
nipkow@15392
  1555
subsubsection {* @{text list_all2} *}
wenzelm@13114
  1556
kleing@14316
  1557
lemma list_all2_lengthD [intro?]: 
kleing@14316
  1558
  "list_all2 P xs ys ==> length xs = length ys"
haftmann@19607
  1559
  by (simp add: list_all2_def)
haftmann@19607
  1560
haftmann@19787
  1561
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
haftmann@19607
  1562
  by (simp add: list_all2_def)
haftmann@19607
  1563
haftmann@19787
  1564
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
haftmann@19787
  1565
  by (simp add: list_all2_def)
haftmann@19607
  1566
haftmann@19607
  1567
lemma list_all2_Cons [iff, code]:
haftmann@19607
  1568
  "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
haftmann@19607
  1569
  by (auto simp add: list_all2_def)
wenzelm@13114
  1570
wenzelm@13114
  1571
lemma list_all2_Cons1:
nipkow@13145
  1572
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1573
by (cases ys) auto
wenzelm@13114
  1574
wenzelm@13114
  1575
lemma list_all2_Cons2:
nipkow@13145
  1576
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1577
by (cases xs) auto
wenzelm@13114
  1578
wenzelm@13142
  1579
lemma list_all2_rev [iff]:
nipkow@13145
  1580
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1581
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1582
kleing@13863
  1583
lemma list_all2_rev1:
kleing@13863
  1584
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1585
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1586
wenzelm@13114
  1587
lemma list_all2_append1:
nipkow@13145
  1588
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1589
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1590
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1591
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1592
apply (rule iffI)
nipkow@13145
  1593
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1594
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  1595
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1596
apply (simp add: ball_Un)
nipkow@13145
  1597
done
wenzelm@13114
  1598
wenzelm@13114
  1599
lemma list_all2_append2:
nipkow@13145
  1600
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1601
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1602
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1603
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1604
apply (rule iffI)
nipkow@13145
  1605
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1606
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  1607
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1608
apply (simp add: ball_Un)
nipkow@13145
  1609
done
wenzelm@13114
  1610
kleing@13863
  1611
lemma list_all2_append:
nipkow@14247
  1612
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  1613
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  1614
by (induct rule:list_induct2, simp_all)
kleing@13863
  1615
kleing@13863
  1616
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1617
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
kleing@13863
  1618
  by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1619
wenzelm@13114
  1620
lemma list_all2_conv_all_nth:
nipkow@13145
  1621
"list_all2 P xs ys =
nipkow@13145
  1622
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1623
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1624
berghofe@13883
  1625
lemma list_all2_trans:
berghofe@13883
  1626
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1627
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1628
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1629
proof (induct as)
berghofe@13883
  1630
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1631
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1632
  proof (induct bs)
berghofe@13883
  1633
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1634
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1635
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1636
  qed simp
berghofe@13883
  1637
qed simp
berghofe@13883
  1638
kleing@13863
  1639
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1640
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
kleing@13863
  1641
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1642
paulson@14395
  1643
lemma list_all2I:
paulson@14395
  1644
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
paulson@14395
  1645
  by (simp add: list_all2_def)
paulson@14395
  1646
kleing@14328
  1647
lemma list_all2_nthD:
kleing@13863
  1648
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
kleing@13863
  1649
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1650
nipkow@14302
  1651
lemma list_all2_nthD2:
nipkow@14302
  1652
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@14302
  1653
  by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  1654
kleing@13863
  1655
lemma list_all2_map1: 
kleing@13863
  1656
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
kleing@13863
  1657
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1658
kleing@13863
  1659
lemma list_all2_map2: 
kleing@13863
  1660
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
kleing@13863
  1661
  by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1662
kleing@14316
  1663
lemma list_all2_refl [intro?]:
kleing@13863
  1664
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
kleing@13863
  1665
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1666
kleing@13863
  1667
lemma list_all2_update_cong:
kleing@13863
  1668
  "\<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
  1669
  by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1670
kleing@13863
  1671
lemma list_all2_update_cong2:
kleing@13863
  1672
  "\<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
  1673
  by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1674
nipkow@14302
  1675
lemma list_all2_takeI [simp,intro?]:
nipkow@14302
  1676
  "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@14302
  1677
  apply (induct xs)
nipkow@14302
  1678
   apply simp
nipkow@14302
  1679
  apply (clarsimp simp add: list_all2_Cons1)
nipkow@14302
  1680
  apply (case_tac n)
nipkow@14302
  1681
  apply auto
nipkow@14302
  1682
  done
nipkow@14302
  1683
nipkow@14302
  1684
lemma list_all2_dropI [simp,intro?]:
kleing@13863
  1685
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
paulson@14208
  1686
  apply (induct as, simp)
kleing@13863
  1687
  apply (clarsimp simp add: list_all2_Cons1)
paulson@14208
  1688
  apply (case_tac n, simp, simp)
kleing@13863
  1689
  done
kleing@13863
  1690
kleing@14327
  1691
lemma list_all2_mono [intro?]:
kleing@13863
  1692
  "\<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
  1693
  apply (induct x, simp)
paulson@14208
  1694
  apply (case_tac y, auto)
kleing@13863
  1695
  done
kleing@13863
  1696
haftmann@22551
  1697
lemma list_all2_eq:
haftmann@22551
  1698
  "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
haftmann@22551
  1699
  by (induct xs ys rule: list_induct2') auto
haftmann@22551
  1700
wenzelm@13142
  1701
nipkow@15392
  1702
subsubsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  1703
wenzelm@13142
  1704
lemma foldl_append [simp]:
nipkow@13145
  1705
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145
  1706
by (induct xs) auto
wenzelm@13142
  1707
nipkow@14402
  1708
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  1709
by (induct xs) auto
nipkow@14402
  1710
nipkow@23096
  1711
lemma foldr_map: "foldr g (map f xs) a = foldr (g o f) xs a"
nipkow@23096
  1712
by(induct xs) simp_all
nipkow@23096
  1713
nipkow@23096
  1714
lemma foldl_map: "foldl g a (map f xs) = foldl (%a x. g a (f x)) a xs"
nipkow@23096
  1715
by(induct xs arbitrary:a) simp_all
nipkow@23096
  1716
krauss@19770
  1717
lemma foldl_cong [fundef_cong, recdef_cong]:
krauss@18336
  1718
  "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] 
krauss@18336
  1719
  ==> foldl f a l = foldl g b k"
wenzelm@20503
  1720
  by (induct k arbitrary: a b l) simp_all
krauss@18336
  1721
krauss@19770
  1722
lemma foldr_cong [fundef_cong, recdef_cong]:
krauss@18336
  1723
  "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] 
krauss@18336
  1724
  ==> foldr f l a = foldr g k b"
wenzelm@20503
  1725
  by (induct k arbitrary: a b l) simp_all
krauss@18336
  1726
nipkow@23096
  1727
text{* The ``First Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096
  1728
nipkow@23096
  1729
lemma foldl_foldr1_lemma:
nipkow@23096
  1730
 "foldl op + a xs = a + foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096
  1731
by (induct xs arbitrary: a) (auto simp:add_assoc)
nipkow@23096
  1732
nipkow@23096
  1733
corollary foldl_foldr1:
nipkow@23096
  1734
 "foldl op + 0 xs = foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096
  1735
by (simp add:foldl_foldr1_lemma)
nipkow@23096
  1736
nipkow@23096
  1737
nipkow@23096
  1738
text{* The ``Third Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096
  1739
nipkow@14402
  1740
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  1741
by (induct xs) auto
nipkow@14402
  1742
nipkow@14402
  1743
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  1744
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  1745
wenzelm@13142
  1746
text {*
nipkow@13145
  1747
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1748
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1749
*}
wenzelm@13142
  1750
wenzelm@13142
  1751
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
nipkow@13145
  1752
by (induct ns) auto
wenzelm@13142
  1753
wenzelm@13142
  1754
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1755
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1756
wenzelm@13142
  1757
lemma sum_eq_0_conv [iff]:
nipkow@13145
  1758
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145
  1759
by (induct ns) auto
wenzelm@13114
  1760
nipkow@23096
  1761
subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
nipkow@23096
  1762
nipkow@23096
  1763
lemma listsum_foldr:
nipkow@23096
  1764
 "listsum xs = foldr (op +) xs 0"
nipkow@23096
  1765
by(induct xs) auto
nipkow@23096
  1766
nipkow@23096
  1767
(* for efficient code generation *)
nipkow@23096
  1768
lemma listsum[code]: "listsum xs = foldl (op +) 0 xs"
nipkow@23096
  1769
by(simp add:listsum_foldr foldl_foldr1)
nipkow@23096
  1770
nipkow@23096
  1771
text{* Some syntactic sugar for summing a function over a list: *}
nipkow@23096
  1772
nipkow@23096
  1773
syntax
nipkow@23096
  1774
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
nipkow@23096
  1775
syntax (xsymbols)
nipkow@23096
  1776
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096
  1777
syntax (HTML output)
nipkow@23096
  1778
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096
  1779
nipkow@23096
  1780
translations -- {* Beware of argument permutation! *}
nipkow@23096
  1781
  "SUM x<-xs. b" == "CONST listsum (map (%x. b) xs)"
nipkow@23096
  1782
  "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (map (%x. b) xs)"
nipkow@23096
  1783
nipkow@23096
  1784
lemma listsum_0 [simp]: "(\<Sum>x\<leftarrow>xs. 0) = 0"
nipkow@23096
  1785
by (induct xs) simp_all
nipkow@23096
  1786
nipkow@23096
  1787
text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
nipkow@23096
  1788
lemma uminus_listsum_map:
nipkow@23096
  1789
 "- listsum (map f xs) = (listsum (map (uminus o f) xs) :: 'a::ab_group_add)"
nipkow@23096
  1790
by(induct xs) simp_all
nipkow@23096
  1791
wenzelm@13114
  1792
nipkow@15392
  1793
subsubsection {* @{text upto} *}
wenzelm@13114
  1794
nipkow@17090
  1795
lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
nipkow@17090
  1796
-- {* simp does not terminate! *}
nipkow@13145
  1797
by (induct j) auto
wenzelm@13142
  1798
nipkow@15425
  1799
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
nipkow@13145
  1800
by (subst upt_rec) simp
wenzelm@13114
  1801
nipkow@15425
  1802
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
nipkow@15281
  1803
by(induct j)simp_all
nipkow@15281
  1804
nipkow@15281
  1805
lemma upt_eq_Cons_conv:
nipkow@15425
  1806
 "!!x xs. ([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
nipkow@15281
  1807
apply(induct j)
nipkow@15281
  1808
 apply simp
nipkow@15281
  1809
apply(clarsimp simp add: append_eq_Cons_conv)
nipkow@15281
  1810
apply arith
nipkow@15281
  1811
done
nipkow@15281
  1812
nipkow@15425
  1813
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
nipkow@13145
  1814
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1815
by simp
wenzelm@13114
  1816
nipkow@15425
  1817
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
nipkow@13145
  1818
apply(rule trans)
nipkow@13145
  1819
apply(subst upt_rec)
paulson@14208
  1820
 prefer 2 apply (rule refl, simp)
nipkow@13145
  1821
done
wenzelm@13114
  1822
nipkow@15425
  1823
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
nipkow@13145
  1824
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1825
by (induct k) auto
wenzelm@13114
  1826
nipkow@15425
  1827
lemma length_upt [simp]: "length [i..<j] = j - i"
nipkow@13145
  1828
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1829
nipkow@15425
  1830
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
nipkow@13145
  1831
apply (induct j)
nipkow@13145
  1832
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1833
done
wenzelm@13114
  1834
nipkow@17906
  1835
nipkow@17906
  1836
lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
nipkow@17906
  1837
by(simp add:upt_conv_Cons)
nipkow@17906
  1838
nipkow@17906
  1839
lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
nipkow@17906
  1840
apply(cases j)
nipkow@17906
  1841
 apply simp
nipkow@17906
  1842
by(simp add:upt_Suc_append)
nipkow@17906
  1843
nipkow@15425
  1844
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..<n] = [i..<i+m]"
paulson@14208
  1845
apply (induct m, simp)
nipkow@13145
  1846
apply (subst upt_rec)
nipkow@13145
  1847
apply (rule sym)
nipkow@13145
  1848
apply (subst upt_rec)
nipkow@13145
  1849
apply (simp del: upt.simps)
nipkow@13145
  1850
done
nipkow@3507
  1851
nipkow@17501
  1852
lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
nipkow@17501
  1853
apply(induct j)
nipkow@17501
  1854
apply auto
nipkow@17501
  1855
done
nipkow@17501
  1856
nipkow@15425
  1857
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..n]"
nipkow@13145
  1858
by (induct n) auto
wenzelm@13114
  1859
nipkow@15425
  1860
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
nipkow@13145
  1861
apply (induct n m rule: diff_induct)
nipkow@13145
  1862
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1863
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1864
done
wenzelm@13114
  1865
berghofe@13883
  1866
lemma nth_take_lemma:
berghofe@13883
  1867
  "!!xs ys. k <= length xs ==> k <= length ys ==>
berghofe@13883
  1868
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
berghofe@13883
  1869
apply (atomize, induct k)
paulson@14208
  1870
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  1871
txt {* Both lists must be non-empty *}
paulson@14208
  1872
apply (case_tac xs, simp)
paulson@14208
  1873
apply (case_tac ys, clarify)
nipkow@13145
  1874
 apply (simp (no_asm_use))
nipkow@13145
  1875
apply clarify
nipkow@13145
  1876
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  1877
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  1878
apply blast
nipkow@13145
  1879
done
wenzelm@13114
  1880
wenzelm@13114
  1881
lemma nth_equalityI:
wenzelm@13114
  1882
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  1883
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  1884
apply (simp_all add: take_all)
nipkow@13145
  1885
done
wenzelm@13142
  1886
kleing@13863
  1887
(* needs nth_equalityI *)
kleing@13863
  1888
lemma list_all2_antisym:
kleing@13863
  1889
  "\<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
  1890
  \<Longrightarrow> xs = ys"
kleing@13863
  1891
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  1892
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  1893
  done
kleing@13863
  1894
wenzelm@13142
  1895
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  1896
-- {* The famous take-lemma. *}
nipkow@13145
  1897
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  1898
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  1899
done
wenzelm@13142
  1900
wenzelm@13142
  1901
nipkow@15302
  1902
lemma take_Cons':
nipkow@15302
  1903
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@15302
  1904
by (cases n) simp_all
nipkow@15302
  1905
nipkow@15302
  1906
lemma drop_Cons':
nipkow@15302
  1907
     "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@15302
  1908
by (cases n) simp_all
nipkow@15302
  1909
nipkow@15302
  1910
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@15302
  1911
by (cases n) simp_all
nipkow@15302
  1912
paulson@18622
  1913
lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard]
paulson@18622
  1914
lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard]
paulson@18622
  1915
lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard]
paulson@18622
  1916
paulson@18622
  1917
declare take_Cons_number_of [simp] 
paulson@18622
  1918
        drop_Cons_number_of [simp] 
paulson@18622
  1919
        nth_Cons_number_of [simp] 
nipkow@15302
  1920
nipkow@15302
  1921
nipkow@15392
  1922
subsubsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  1923
wenzelm@13142
  1924
lemma distinct_append [simp]:
nipkow@13145
  1925
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  1926
by (induct xs) auto
wenzelm@13142
  1927
nipkow@15305
  1928
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
nipkow@15305
  1929
by(induct xs) auto
nipkow@15305
  1930
wenzelm@13142
  1931
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  1932
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  1933
wenzelm@13142
  1934
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  1935
by (induct xs) auto
wenzelm@13142
  1936
paulson@15072
  1937
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
paulson@15251
  1938
  by (induct x, auto) 
paulson@15072
  1939
paulson@15072
  1940
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
paulson@15251
  1941
  by (induct x, auto)
paulson@15072
  1942
nipkow@15245
  1943
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245
  1944
by (induct xs) auto
nipkow@15245
  1945
nipkow@15245
  1946
lemma length_remdups_eq[iff]:
nipkow@15245
  1947
  "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245
  1948
apply(induct xs)
nipkow@15245
  1949
 apply auto
nipkow@15245
  1950
apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245
  1951
 apply arith
nipkow@15245
  1952
apply(rule length_remdups_leq)
nipkow@15245
  1953
done
nipkow@15245
  1954
nipkow@18490
  1955
nipkow@18490
  1956
lemma distinct_map:
nipkow@18490
  1957
  "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
nipkow@18490
  1958
by (induct xs) auto
nipkow@18490
  1959
nipkow@18490
  1960
wenzelm@13142
  1961
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  1962
by (induct xs) auto
wenzelm@13114
  1963
nipkow@17501
  1964
lemma distinct_upt[simp]: "distinct[i..<j]"
nipkow@17501
  1965
by (induct j) auto
nipkow@17501
  1966
nipkow@17501
  1967
lemma distinct_take[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (take i xs)"
nipkow@17501
  1968
apply(induct xs)
nipkow@17501
  1969
 apply simp
nipkow@17501
  1970
apply (case_tac i)
nipkow@17501
  1971
 apply simp_all
nipkow@17501
  1972
apply(blast dest:in_set_takeD)
nipkow@17501
  1973
done
nipkow@17501
  1974
nipkow@17501
  1975
lemma distinct_drop[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (drop i xs)"
nipkow@17501
  1976
apply(induct xs)
nipkow@17501
  1977
 apply simp
nipkow@17501
  1978
apply (case_tac i)
nipkow@17501
  1979
 apply simp_all
nipkow@17501
  1980
done
nipkow@17501
  1981
nipkow@17501
  1982
lemma distinct_list_update:
nipkow@17501
  1983
assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
nipkow@17501
  1984
shows "distinct (xs[i:=a])"
nipkow@17501
  1985
proof (cases "i < length xs")
nipkow@17501
  1986
  case True
nipkow@17501
  1987
  with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
nipkow@17501
  1988
    apply (drule_tac id_take_nth_drop) by simp
nipkow@17501
  1989
  with d True show ?thesis
nipkow@17501
  1990
    apply (simp add: upd_conv_take_nth_drop)
nipkow@17501
  1991
    apply (drule subst [OF id_take_nth_drop]) apply assumption
nipkow@17501
  1992
    apply simp apply (cases "a = xs!i") apply simp by blast
nipkow@17501
  1993
next
nipkow@17501
  1994
  case False with d show ?thesis by auto
nipkow@17501
  1995
qed
nipkow@17501
  1996
nipkow@17501
  1997
nipkow@17501
  1998
text {* It is best to avoid this indexed version of distinct, but
nipkow@17501
  1999
sometimes it is useful. *}
nipkow@17501
  2000
wenzelm@13142
  2001
lemma distinct_conv_nth:
nipkow@17501
  2002
"distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@15251
  2003
apply (induct xs, simp, simp)
paulson@14208
  2004
apply (rule iffI, clarsimp)
nipkow@13145
  2005
 apply (case_tac i)
paulson@14208
  2006
apply (case_tac j, simp)
nipkow@13145
  2007
apply (simp add: set_conv_nth)
nipkow@13145
  2008
 apply (case_tac j)
paulson@14208
  2009
apply (clarsimp simp add: set_conv_nth, simp)
nipkow@13145
  2010
apply (rule conjI)
nipkow@13145
  2011
 apply (clarsimp simp add: set_conv_nth)
nipkow@17501
  2012
 apply (erule_tac x = 0 in allE, simp)
paulson@14208
  2013
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
nipkow@17501
  2014
apply (erule_tac x = "Suc i" in allE, simp)
paulson@14208
  2015
apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145
  2016
done
wenzelm@13114
  2017
nipkow@18490
  2018
lemma nth_eq_iff_index_eq:
nipkow@18490
  2019
 "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
nipkow@18490
  2020
by(auto simp: distinct_conv_nth)
nipkow@18490
  2021
nipkow@15110
  2022
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
kleing@14388
  2023
  by (induct xs) auto
kleing@14388
  2024
nipkow@15110
  2025
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388
  2026
proof (induct xs)
kleing@14388
  2027
  case Nil thus ?case by simp
kleing@14388
  2028
next
kleing@14388
  2029
  case (Cons x xs)
kleing@14388
  2030
  show ?case
kleing@14388
  2031
  proof (cases "x \<in> set xs")
kleing@14388
  2032
    case False with Cons show ?thesis by simp
kleing@14388
  2033
  next
kleing@14388
  2034
    case True with Cons.prems
kleing@14388
  2035
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  2036
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  2037
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  2038
    ultimately have False by simp
kleing@14388
  2039
    thus ?thesis ..
kleing@14388
  2040
  qed
kleing@14388
  2041
qed
kleing@14388
  2042
nipkow@18490
  2043
nipkow@18490
  2044
lemma length_remdups_concat:
nipkow@18490
  2045
 "length(remdups(concat xss)) = card(\<Union>xs \<in> set xss. set xs)"
nipkow@18490
  2046
by(simp add: distinct_card[symmetric])
nipkow@17906
  2047
nipkow@17906
  2048
nipkow@15392
  2049
subsubsection {* @{text remove1} *}
nipkow@15110
  2050
nipkow@18049
  2051
lemma remove1_append:
nipkow@18049
  2052
  "remove1 x (xs @ ys) =
nipkow@18049
  2053
  (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
nipkow@18049
  2054
by (induct xs) auto
nipkow@18049
  2055
nipkow@15110
  2056
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110
  2057
apply(induct xs)
nipkow@15110
  2058
 apply simp
nipkow@15110
  2059
apply simp
nipkow@15110
  2060
apply blast
nipkow@15110
  2061
done
nipkow@15110
  2062
paulson@17724
  2063
lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110
  2064
apply(induct xs)
nipkow@15110
  2065
 apply simp
nipkow@15110
  2066
apply simp
nipkow@15110
  2067
apply blast
nipkow@15110
  2068
done
nipkow@15110
  2069
nipkow@18049
  2070
lemma remove1_filter_not[simp]:
nipkow@18049
  2071
  "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
nipkow@18049
  2072
by(induct xs) auto
nipkow@18049
  2073
nipkow@15110
  2074
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110
  2075
apply(insert set_remove1_subset)
nipkow@15110
  2076
apply fast
nipkow@15110
  2077
done
nipkow@15110
  2078
nipkow@15110
  2079
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110
  2080
by (induct xs) simp_all
nipkow@15110
  2081
wenzelm@13114
  2082
nipkow@15392
  2083
subsubsection {* @{text replicate} *}
wenzelm@13114
  2084
wenzelm@13142
  2085
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  2086
by (induct n) auto
nipkow@13124
  2087
wenzelm@13142
  2088
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  2089
by (induct n) auto
wenzelm@13114
  2090
wenzelm@13114
  2091
lemma replicate_app_Cons_same:
nipkow@13145
  2092
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  2093
by (induct n) auto
wenzelm@13114
  2094
wenzelm@13142
  2095
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208
  2096
apply (induct n, simp)
nipkow@13145
  2097
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  2098
done
wenzelm@13114
  2099
wenzelm@13142
  2100
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  2101
by (induct n) auto
wenzelm@13114
  2102
nipkow@16397
  2103
text{* Courtesy of Matthias Daum: *}
nipkow@16397
  2104
lemma append_replicate_commute:
nipkow@16397
  2105
  "replicate n x @ replicate k x = replicate k x @ replicate n x"
nipkow@16397
  2106
apply (simp add: replicate_add [THEN sym])
nipkow@16397
  2107
apply (simp add: add_commute)
nipkow@16397
  2108
done
nipkow@16397
  2109
wenzelm@13142
  2110
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  2111
by (induct n) auto
wenzelm@13114
  2112
wenzelm@13142
  2113
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  2114
by (induct n) auto
wenzelm@13114
  2115
wenzelm@13142
  2116
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  2117
by (atomize (full), induct n) auto
wenzelm@13114
  2118
wenzelm@13142
  2119
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
paulson@14208
  2120
apply (induct n, simp)
nipkow@13145
  2121
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  2122
done
wenzelm@13114
  2123
nipkow@16397
  2124
text{* Courtesy of Matthias Daum (2 lemmas): *}
nipkow@16397
  2125
lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
nipkow@16397
  2126
apply (case_tac "k \<le> i")
nipkow@16397
  2127
 apply  (simp add: min_def)
nipkow@16397
  2128
apply (drule not_leE)
nipkow@16397
  2129
apply (simp add: min_def)
nipkow@16397
  2130
apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
nipkow@16397
  2131
 apply  simp
nipkow@16397
  2132
apply (simp add: replicate_add [symmetric])
nipkow@16397
  2133
done
nipkow@16397
  2134
nipkow@16397
  2135
lemma drop_replicate[simp]: "!!i. drop i (replicate k x) = replicate (k-i) x"
nipkow@16397
  2136
apply (induct k)
nipkow@16397
  2137
 apply simp
nipkow@16397
  2138
apply clarsimp
nipkow@16397
  2139
apply (case_tac i)
nipkow@16397
  2140
 apply simp
nipkow@16397
  2141
apply clarsimp
nipkow@16397
  2142
done
nipkow@16397
  2143
nipkow@16397
  2144
wenzelm@13142
  2145
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  2146
by (induct n) auto
wenzelm@13114
  2147
wenzelm@13142
  2148
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  2149
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  2150
wenzelm@13142
  2151
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  2152
by auto
wenzelm@13114
  2153
wenzelm@13142
  2154
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  2155
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  2156
wenzelm@13114
  2157
nipkow@15392
  2158
subsubsection{*@{text rotate1} and @{text rotate}*}
nipkow@15302
  2159
nipkow@15302
  2160
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
nipkow@15302
  2161
by(simp add:rotate1_def)
nipkow@15302
  2162
nipkow@15302
  2163
lemma rotate0[simp]: "rotate 0 = id"
nipkow@15302
  2164
by(simp add:rotate_def)
nipkow@15302
  2165
nipkow@15302
  2166
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
nipkow@15302
  2167
by(simp add:rotate_def)
nipkow@15302
  2168
nipkow@15302
  2169
lemma rotate_add:
nipkow@15302
  2170
  "rotate (m+n) = rotate m o rotate n"
nipkow@15302
  2171
by(simp add:rotate_def funpow_add)
nipkow@15302
  2172
nipkow@15302
  2173
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
nipkow@15302
  2174
by(simp add:rotate_add)
nipkow@15302
  2175
nipkow@18049
  2176
lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
nipkow@18049
  2177
by(simp add:rotate_def funpow_swap1)
nipkow@18049
  2178
nipkow@15302
  2179
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
nipkow@15302
  2180
by(cases xs) simp_all
nipkow@15302
  2181
nipkow@15302
  2182
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2183
apply(induct n)
nipkow@15302
  2184
 apply simp
nipkow@15302
  2185
apply (simp add:rotate_def)
nipkow@13145
  2186
done
wenzelm@13114
  2187
nipkow@15302
  2188
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
nipkow@15302
  2189
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2190
nipkow@15302
  2191
lemma rotate_drop_take:
nipkow@15302
  2192
  "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
nipkow@15302
  2193
apply(induct n)
nipkow@15302
  2194
 apply simp
nipkow@15302
  2195
apply(simp add:rotate_def)
nipkow@15302
  2196
apply(cases "xs = []")
nipkow@15302
  2197
 apply (simp)
nipkow@15302
  2198
apply(case_tac "n mod length xs = 0")
nipkow@15302
  2199
 apply(simp add:mod_Suc)
nipkow@15302
  2200
 apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
nipkow@15302
  2201
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
nipkow@15302
  2202
                take_hd_drop linorder_not_le)
nipkow@13145
  2203
done
wenzelm@13114
  2204
nipkow@15302
  2205
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
nipkow@15302
  2206
by(simp add:rotate_drop_take)
nipkow@15302
  2207
nipkow@15302
  2208
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2209
by(simp add:rotate_drop_take)
nipkow@15302
  2210
nipkow@15302
  2211
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
nipkow@15302
  2212
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2213
nipkow@15302
  2214
lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs"
nipkow@15302
  2215
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2216
nipkow@15302
  2217
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
nipkow@15302
  2218
by(simp add:rotate1_def split:list.split) blast
nipkow@15302
  2219
nipkow@15302
  2220
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
nipkow@15302
  2221
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2222
nipkow@15302
  2223
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
nipkow@15302
  2224
by(simp add:rotate_drop_take take_map drop_map)
nipkow@15302
  2225
nipkow@15302
  2226
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
nipkow@15302
  2227
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2228
nipkow@15302
  2229
lemma set_rotate[simp]: "set(rotate n xs) = set xs"
nipkow@15302
  2230
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2231
nipkow@15302
  2232
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
nipkow@15302
  2233
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2234
nipkow@15302
  2235
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
nipkow@15302
  2236
by (induct n) (simp_all add:rotate_def)
wenzelm@13114
  2237
nipkow@15439
  2238
lemma rotate_rev:
nipkow@15439
  2239
  "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
nipkow@15439
  2240
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2241
apply(cases "length xs = 0")
nipkow@15439
  2242
 apply simp
nipkow@15439
  2243
apply(cases "n mod length xs = 0")
nipkow@15439
  2244
 apply simp
nipkow@15439
  2245
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2246
done
nipkow@15439
  2247
nipkow@18423
  2248
lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
nipkow@18423
  2249
apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
nipkow@18423
  2250
apply(subgoal_tac "length xs \<noteq> 0")
nipkow@18423
  2251
 prefer 2 apply simp
nipkow@18423
  2252
using mod_less_divisor[of "length xs" n] by arith
nipkow@18423
  2253
wenzelm@13114
  2254
nipkow@15392
  2255
subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  2256
wenzelm@13142
  2257
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  2258
by (auto simp add: sublist_def)
wenzelm@13114
  2259
wenzelm@13142
  2260
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  2261
by (auto simp add: sublist_def)
wenzelm@13114
  2262
nipkow@15281
  2263
lemma length_sublist:
nipkow@15281
  2264
  "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
nipkow@15281
  2265
by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
nipkow@15281
  2266
nipkow@15281
  2267
lemma sublist_shift_lemma_Suc:
nipkow@15281
  2268
  "!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
nipkow@15281
  2269
         map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
nipkow@15281
  2270
apply(induct xs)
nipkow@15281
  2271
 apply simp
nipkow@15281
  2272
apply (case_tac "is")
nipkow@15281
  2273
 apply simp
nipkow@15281
  2274
apply simp
nipkow@15281
  2275
done
nipkow@15281
  2276
wenzelm@13114
  2277
lemma sublist_shift_lemma:
nipkow@23279
  2278
     "map fst [p<-zip xs [i..<i + length xs] . snd p : A] =
nipkow@23279
  2279
      map fst [p<-zip xs [0..<length xs] . snd p + i : A]"
nipkow@13145
  2280
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  2281
wenzelm@13114
  2282
lemma sublist_append:
paulson@15168
  2283
     "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  2284
apply (unfold sublist_def)
paulson@14208
  2285
apply (induct l' rule: rev_induct, simp)
nipkow@13145
  2286
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  2287
apply (simp add: add_commute)
nipkow@13145
  2288
done
wenzelm@13114
  2289
wenzelm@13114
  2290
lemma sublist_Cons:
nipkow@13145
  2291
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  2292
apply (induct l rule: rev_induct)
nipkow@13145
  2293
 apply (simp add: sublist_def)
nipkow@13145
  2294
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  2295
done
wenzelm@13114
  2296
nipkow@15281
  2297
lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
nipkow@15281
  2298
apply(induct xs)
nipkow@15281
  2299
 apply simp
nipkow@15281
  2300
apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE)
nipkow@15281
  2301
 apply(erule lessE)
nipkow@15281
  2302
  apply auto
nipkow@15281
  2303
apply(erule lessE)
nipkow@15281
  2304
apply auto
nipkow@15281
  2305
done
nipkow@15281
  2306
nipkow@15281
  2307
lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
nipkow@15281
  2308
by(auto simp add:set_sublist)
nipkow@15281
  2309
nipkow@15281
  2310
lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
nipkow@15281
  2311
by(auto simp add:set_sublist)
nipkow@15281
  2312
nipkow@15281
  2313
lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
nipkow@15281
  2314
by(auto simp add:set_sublist)
nipkow@15281
  2315
wenzelm@13142
  2316
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  2317
by (simp add: sublist_Cons)
wenzelm@13114
  2318
nipkow@15281
  2319
nipkow@15281
  2320
lemma distinct_sublistI[simp]: "!!I. distinct xs \<Longrightarrow> distinct(sublist xs I)"
nipkow@15281
  2321
apply(induct xs)
nipkow@15281
  2322
 apply simp
nipkow@15281
  2323
apply(auto simp add:sublist_Cons)
nipkow@15281
  2324
done
nipkow@15281
  2325
nipkow@15281
  2326
nipkow@15045
  2327
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
paulson@14208
  2328
apply (induct l rule: rev_induct, simp)
nipkow@13145
  2329
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  2330
done
wenzelm@13114
  2331
nipkow@17501
  2332
lemma filter_in_sublist: "\<And>s. distinct xs \<Longrightarrow>
nipkow@17501
  2333
  filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
nipkow@17501
  2334
proof (induct xs)
nipkow@17501
  2335
  case Nil thus ?case by simp
nipkow@17501
  2336
next
nipkow@17501
  2337
  case (Cons a xs)
nipkow@17501
  2338
  moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
nipkow@17501
  2339
  ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
nipkow@17501
  2340
qed
nipkow@17501
  2341
wenzelm@13114
  2342
nipkow@19390
  2343
subsubsection {* @{const splice} *}
nipkow@19390
  2344
haftmann@19607
  2345
lemma splice_Nil2 [simp, code]:
nipkow@19390
  2346
 "splice xs [] = xs"
nipkow@19390
  2347
by (cases xs) simp_all
nipkow@19390
  2348
haftmann@19607
  2349
lemma splice_Cons_Cons [simp, code]:
nipkow@19390
  2350
 "splice (x#xs) (y#ys) = x # y # splice xs ys"
nipkow@19390
  2351
by simp
nipkow@19390
  2352
haftmann@19607
  2353
declare splice.simps(2) [simp del, code del]
nipkow@19390
  2354
nipkow@22793
  2355
lemma length_splice[simp]: "!!ys. length(splice xs ys) = length xs + length ys"
nipkow@22793
  2356
apply(induct xs) apply simp
nipkow@22793
  2357
apply(case_tac ys)
nipkow@22793
  2358
 apply auto
nipkow@22793
  2359
done
nipkow@22793
  2360
nipkow@22828
  2361
subsubsection {* @{const allpairs} *}
nipkow@22828
  2362
nipkow@22940
  2363
lemma allpairs_conv_concat:
nipkow@22940
  2364
 "allpairs f xs ys = concat(map (%x. map (f x) ys) xs)"
nipkow@22940
  2365
by(induct xs) auto
nipkow@22940
  2366
nipkow@22828
  2367
lemma allpairs_append:
nipkow@22830
  2368
 "allpairs f (xs @ ys) zs = allpairs f xs zs @ allpairs f ys zs"
nipkow@22828
  2369
by(induct xs) auto
nipkow@22828
  2370
nipkow@15392
  2371
subsubsection {* @{text lists}: the list-forming operator over sets *}
nipkow@15302
  2372
berghofe@22262
  2373
inductive2
berghofe@22262
  2374
  listsp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
berghofe@22262
  2375
  for A :: "'a \<Rightarrow> bool"
berghofe@22262
  2376
where
berghofe@22262
  2377
    Nil [intro!]: "listsp A []"
berghofe@22262
  2378
  | Cons [intro!]: "[| A a; listsp A l |] ==> listsp A (a # l)"
berghofe@22262
  2379
berghofe@22262
  2380
constdefs
berghofe@22262
  2381
  lists :: "'a set => 'a list set"
berghofe@22262
  2382
  "lists A == Collect (listsp (member A))"
berghofe@22262
  2383
berghofe@22262
  2384
lemma listsp_lists_eq [pred_set_conv]: "listsp (member A) = member (lists A)"
berghofe@22262
  2385
  by (simp add: lists_def)
berghofe@22262
  2386
berghofe@22262
  2387
lemmas lists_intros [intro!] = listsp.intros [to_set]
berghofe@22262
  2388
berghofe@22262
  2389
lemmas lists_induct [consumes 1, case_names Nil Cons, induct set: lists] =
berghofe@22262
  2390
  listsp.induct [to_set]
berghofe@22262
  2391
berghofe@22262
  2392
inductive_cases2 listspE [elim!]: "listsp A (x # l)"
berghofe@22262
  2393
berghofe@22262
  2394
lemmas listsE [elim!] = listspE [to_set]
berghofe@22262
  2395
berghofe@22262
  2396
lemma listsp_mono [mono2]: "A \<le> B ==> listsp A \<le> listsp B"
berghofe@22262
  2397
  by (clarify, erule listsp.induct, blast+)
berghofe@22262
  2398
berghofe@22262
  2399
lemmas lists_mono [mono] = listsp_mono [to_set]
berghofe@22262
  2400
haftmann@22422
  2401
lemma listsp_infI:
haftmann@22422
  2402
  assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
nipkow@15302
  2403
  by induct blast+
nipkow@15302
  2404
haftmann@22422
  2405
lemmas lists_IntI = listsp_infI [to_set]
haftmann@22422
  2406
haftmann@22422
  2407
lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
haftmann@22422
  2408
proof (rule mono_inf [where f=listsp, THEN order_antisym])
berghofe@22262
  2409
  show "mono listsp" by (simp add: mono_def listsp_mono)
haftmann@22422
  2410
  show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro: listsp_infI)
kleing@14388
  2411
qed
kleing@14388
  2412
haftmann@22422
  2413
lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_eq inf_bool_eq]
haftmann@22422
  2414
haftmann@22422
  2415
lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set]
berghofe@22262
  2416
berghofe@22262
  2417
lemma append_in_listsp_conv [iff]:
berghofe@22262
  2418
     "(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
nipkow@15302
  2419
by (induct xs) auto
nipkow@15302
  2420
berghofe@22262
  2421
lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]
berghofe@22262
  2422
berghofe@22262
  2423
lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
berghofe@22262
  2424
-- {* eliminate @{text listsp} in favour of @{text set} *}
nipkow@15302
  2425
by (induct xs) auto
nipkow@15302
  2426
berghofe@22262
  2427
lemmas in_lists_conv_set = in_listsp_conv_set [to_set]
berghofe@22262
  2428
berghofe@22262
  2429
lemma in_listspD [dest!]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
berghofe@22262
  2430
by (rule in_listsp_conv_set [THEN iffD1])
berghofe@22262
  2431
berghofe@22262
  2432
lemmas in_listsD [dest!] = in_listspD [to_set]
berghofe@22262
  2433
berghofe@22262
  2434
lemma in_listspI [intro!]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
berghofe@22262
  2435
by (rule in_listsp_conv_set [THEN iffD2])
berghofe@22262
  2436
berghofe@22262
  2437
lemmas in_listsI [intro!] = in_listspI [to_set]
nipkow@15302
  2438
nipkow@15302
  2439
lemma lists_UNIV [simp]: "lists UNIV = UNIV"
nipkow@15302
  2440
by auto
nipkow@15302
  2441
nipkow@17086
  2442
nipkow@17086
  2443
nipkow@17086
  2444
subsubsection{* Inductive definition for membership *}
nipkow@17086
  2445
berghofe@22262
  2446
inductive2 ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
berghofe@22262
  2447
where
berghofe@22262
  2448
    elem:  "ListMem x (x # xs)"
berghofe@22262
  2449
  | insert:  "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"
berghofe@22262
  2450
berghofe@22262
  2451
lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
nipkow@17086
  2452
apply (rule iffI)
nipkow@17086
  2453
 apply (induct set: ListMem)
nipkow@17086
  2454
  apply auto
nipkow@17086
  2455
apply (induct xs)
nipkow@17086
  2456
 apply (auto intro: ListMem.intros)
nipkow@17086
  2457
done
nipkow@17086
  2458
nipkow@17086
  2459
nipkow@17086
  2460
nipkow@15392
  2461
subsubsection{*Lists as Cartesian products*}
nipkow@15302
  2462
nipkow@15302
  2463
text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
nipkow@15302
  2464
@{term A} and tail drawn from @{term Xs}.*}
nipkow@15302
  2465
nipkow@15302
  2466
constdefs
nipkow@15302
  2467
  set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
nipkow@15302
  2468
  "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"
nipkow@15302
  2469
paulson@17724
  2470
lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
nipkow@15302
  2471
by (auto simp add: set_Cons_def)
nipkow@15302
  2472
nipkow@15302
  2473
text{*Yields the set of lists, all of the same length as the argument and
nipkow@15302
  2474
with elements drawn from the corresponding element of the argument.*}
nipkow@15302
  2475
nipkow@15302
  2476
consts  listset :: "'a set list \<Rightarrow> 'a list set"
nipkow@15302
  2477
primrec
nipkow@15302
  2478
   "listset []    = {[]}"
nipkow@15302
  2479
   "listset(A#As) = set_Cons A (listset As)"
nipkow@15302
  2480
nipkow@15302
  2481
paulson@15656
  2482
subsection{*Relations on Lists*}
paulson@15656
  2483
paulson@15656
  2484
subsubsection {* Length Lexicographic Ordering *}
paulson@15656
  2485
paulson@15656
  2486
text{*These orderings preserve well-foundedness: shorter lists 
paulson@15656
  2487
  precede longer lists. These ordering are not used in dictionaries.*}
paulson@15656
  2488
paulson@15656
  2489
consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
paulson@15656
  2490
        --{*The lexicographic ordering for lists of the specified length*}
nipkow@15302
  2491
primrec
paulson@15656
  2492
  "lexn r 0 = {}"
paulson@15656
  2493
  "lexn r (Suc n) =
paulson@15656
  2494
    (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
paulson@15656
  2495
    {(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
nipkow@15302
  2496
nipkow@15302
  2497
constdefs
paulson@15656
  2498
  lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
paulson@15656
  2499
    "lex r == \<Union>n. lexn r n"
paulson@15656
  2500
        --{*Holds only between lists of the same length*}
paulson@15656
  2501
nipkow@15693
  2502
  lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@15693
  2503
    "lenlex r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
paulson@15656
  2504
        --{*Compares lists by their length and then lexicographically*}
nipkow@15302
  2505
nipkow@15302
  2506
nipkow@15302
  2507
lemma wf_lexn: "wf r ==> wf (lexn r n)"
nipkow@15302
  2508
apply (induct n, simp, simp)
nipkow@15302
  2509
apply(rule wf_subset)
nipkow@15302
  2510
 prefer 2 apply (rule Int_lower1)
nipkow@15302
  2511
apply(rule wf_prod_fun_image)
nipkow@15302
  2512
 prefer 2 apply (rule inj_onI, auto)
nipkow@15302
  2513
done
nipkow@15302
  2514
nipkow@15302
  2515
lemma lexn_length:
nipkow@15302
  2516
     "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@15302
  2517
by (induct n) auto
nipkow@15302
  2518
nipkow@15302
  2519
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@15302
  2520
apply (unfold lex_def)
nipkow@15302
  2521
apply (rule wf_UN)
nipkow@15302
  2522
apply (blast intro: wf_lexn, clarify)
nipkow@15302
  2523
apply (rename_tac m n)
nipkow@15302
  2524
apply (subgoal_tac "m \<noteq> n")
nipkow@15302
  2525
 prefer 2 apply blast
nipkow@15302
  2526
apply (blast dest: lexn_length not_sym)
nipkow@15302
  2527
done
nipkow@15302
  2528
nipkow@15302
  2529
lemma lexn_conv:
paulson@15656
  2530
  "lexn r n =
paulson@15656
  2531
    {(xs,ys). length xs = n \<and> length ys = n \<and>
paulson@15656
  2532
    (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
nipkow@18423
  2533
apply (induct n, simp)
nipkow@15302
  2534
apply (simp add: image_Collect lex_prod_def, safe, blast)
nipkow@15302
  2535
 apply (rule_tac x = "ab # xys" in exI, simp)
nipkow@15302
  2536
apply (case_tac xys, simp_all, blast)
nipkow@15302
  2537
done
nipkow@15302
  2538
nipkow@15302
  2539
lemma lex_conv:
paulson@15656
  2540
  "lex r =
paulson@15656
  2541
    {(xs,ys). length xs = length ys \<and>
paulson@15656
  2542
    (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@15302
  2543
by (force simp add: lex_def lexn_conv)
nipkow@15302
  2544
nipkow@15693
  2545
lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
nipkow@15693
  2546
by (unfold lenlex_def) blast
nipkow@15693
  2547
nipkow@15693
  2548
lemma lenlex_conv:
nipkow@15693
  2549
    "lenlex r = {(xs,ys). length xs < length ys |
paulson@15656
  2550
                 length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@19623
  2551
by (simp add: lenlex_def diag_def lex_prod_def inv_image_def)
nipkow@15302
  2552
nipkow@15302
  2553
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@15302
  2554
by (simp add: lex_conv)
nipkow@15302
  2555
nipkow@15302
  2556
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@15302
  2557
by (simp add:lex_conv)
nipkow@15302
  2558
paulson@18447
  2559
lemma Cons_in_lex [simp]:
paulson@15656
  2560
    "((x # xs, y # ys) : lex r) =
paulson@15656
  2561
      ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@15302
  2562
apply (simp add: lex_conv)
nipkow@15302
  2563
apply (rule iffI)
nipkow@15302
  2564
 prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
nipkow@15302
  2565
apply (case_tac xys, simp, simp)
nipkow@15302
  2566
apply blast
nipkow@15302
  2567
done
nipkow@15302
  2568
nipkow@15302
  2569
paulson@15656
  2570
subsubsection {* Lexicographic Ordering *}
paulson@15656
  2571
paulson@15656
  2572
text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
paulson@15656
  2573
    This ordering does \emph{not} preserve well-foundedness.
nipkow@17090
  2574
     Author: N. Voelker, March 2005. *} 
paulson@15656
  2575
paulson@15656
  2576
constdefs 
paulson@15656
  2577
  lexord :: "('a * 'a)set \<Rightarrow> ('a list * 'a list) set" 
paulson@15656
  2578
  "lexord  r == {(x,y). \<exists> a v. y = x @ a # v \<or> 
paulson@15656
  2579
            (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
paulson@15656
  2580
paulson@15656
  2581
lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
paulson@15656
  2582
  by (unfold lexord_def, induct_tac y, auto) 
paulson@15656
  2583
paulson@15656
  2584
lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
paulson@15656
  2585
  by (unfold lexord_def, induct_tac x, auto)
paulson@15656
  2586
paulson@15656
  2587
lemma lexord_cons_cons[simp]:
paulson@15656
  2588
     "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
paulson@15656
  2589
  apply (unfold lexord_def, safe, simp_all)
paulson@15656
  2590
  apply (case_tac u, simp, simp)
paulson@15656
  2591
  apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
paulson@15656
  2592
  apply (erule_tac x="b # u" in allE)
paulson@15656
  2593
  by force
paulson@15656
  2594
paulson@15656
  2595
lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
paulson@15656
  2596
paulson@15656
  2597
lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
paulson@15656
  2598
  by (induct_tac x, auto)  
paulson@15656
  2599
paulson@15656
  2600
lemma lexord_append_left_rightI:
paulson@15656
  2601
     "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
paulson@15656
  2602
  by (induct_tac u, auto)
paulson@15656
  2603
paulson@15656
  2604
lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
paulson@15656
  2605
  by (induct x, auto)
paulson@15656
  2606
paulson@15656
  2607
lemma lexord_append_leftD:
paulson@15656
  2608
     "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
paulson@15656
  2609
  by (erule rev_mp, induct_tac x, auto)
paulson@15656
  2610
paulson@15656
  2611
lemma lexord_take_index_conv: 
paulson@15656
  2612
   "((x,y) : lexord r) = 
paulson@15656
  2613
    ((length x < length y \<and> take (length x) y = x) \<or> 
paulson@15656
  2614
     (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
paulson@15656
  2615
  apply (unfold lexord_def Let_def, clarsimp) 
paulson@15656
  2616
  apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
paulson@15656
  2617
  apply auto 
paulson@15656
  2618
  apply (rule_tac x="hd (drop (length x) y)" in exI)
paulson@15656
  2619
  apply (rule_tac x="tl (drop (length x) y)" in exI)
paulson@15656
  2620
  apply (erule subst, simp add: min_def) 
paulson@15656
  2621
  apply (rule_tac x ="length u" in exI, simp) 
paulson@15656
  2622
  apply (rule_tac x ="take i x" in exI) 
paulson@15656
  2623
  apply (rule_tac x ="x ! i" in exI) 
paulson@15656
  2624
  apply (rule_tac x ="y ! i" in exI, safe) 
paulson@15656
  2625
  apply (rule_tac x="drop (Suc i) x" in exI)
paulson@15656
  2626
  apply (drule sym, simp add: drop_Suc_conv_tl) 
paulson@15656
  2627
  apply (rule_tac x="drop (Suc i) y" in exI)
paulson@15656
  2628
  by (simp add: drop_Suc_conv_tl) 
paulson@15656
  2629
paulson@15656
  2630
-- {* lexord is extension of partial ordering List.lex *} 
paulson@15656
  2631
lemma lexord_lex: " (x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
paulson@15656
  2632
  apply (rule_tac x = y in spec) 
paulson@15656
  2633
  apply (induct_tac x, clarsimp) 
paulson@15656
  2634
  by (clarify, case_tac x, simp, force)
paulson@15656
  2635
paulson@15656
  2636
lemma lexord_irreflexive: "(! x. (x,x) \<notin> r) \<Longrightarrow> (y,y) \<notin> lexord r"
paulson@15656
  2637
  by (induct y, auto)
paulson@15656
  2638
paulson@15656
  2639
lemma lexord_trans: 
paulson@15656
  2640
    "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
paulson@15656
  2641
   apply (erule rev_mp)+
paulson@15656
  2642
   apply (rule_tac x = x in spec) 
paulson@15656
  2643
  apply (rule_tac x = z in spec) 
paulson@15656
  2644
  apply ( induct_tac y, simp, clarify)
paulson@15656
  2645
  apply (case_tac xa, erule ssubst) 
paulson@15656
  2646
  apply (erule allE, erule allE) -- {* avoid simp recursion *} 
paulson@15656
  2647
  apply (case_tac x, simp, simp) 
paulson@15656
  2648
  apply (case_tac x, erule allE, erule allE, simp) 
paulson@15656
  2649
  apply (erule_tac x = listb in allE) 
paulson@15656
  2650
  apply (erule_tac x = lista in allE, simp)
paulson@15656
  2651
  apply (unfold trans_def)
paulson@15656
  2652
  by blast
paulson@15656
  2653
paulson@15656
  2654
lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
paulson@15656
  2655
  by (rule transI, drule lexord_trans, blast) 
paulson@15656
  2656
paulson@15656
  2657
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
  2658
  apply (rule_tac x = y in spec) 
paulson@15656
  2659
  apply (induct_tac x, rule allI) 
paulson@15656
  2660
  apply (case_tac x, simp, simp) 
paulson@15656
  2661
  apply (rule allI, case_tac x, simp, simp) 
paulson@15656
  2662
  by blast
paulson@15656
  2663
paulson@15656
  2664
krauss@21103
  2665
subsection {* Lexicographic combination of measure functions *}
krauss@21103
  2666
krauss@21103
  2667
text {* These are useful for termination proofs *}
krauss@21103
  2668
krauss@21103
  2669
definition
krauss@21103
  2670
  "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"
krauss@21103
  2671
krauss@21106
  2672
lemma wf_measures[recdef_wf, simp]: "wf (measures fs)"
krauss@21103
  2673
  unfolding measures_def
krauss@21103
  2674
  by blast
krauss@21103
  2675
krauss@21103
  2676
lemma in_measures[simp]: 
krauss@21103
  2677
  "(x, y) \<in> measures [] = False"
krauss@21103
  2678
  "(x, y) \<in> measures (f # fs)
krauss@21103
  2679
         = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"  
krauss@21103
  2680
  unfolding measures_def
krauss@21103
  2681
  by auto
krauss@21103
  2682
krauss@21103
  2683
lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
krauss@21103
  2684
  by simp
krauss@21103
  2685
krauss@21103
  2686
lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
krauss@21103
  2687
  by auto
krauss@21103
  2688
krauss@21211
  2689
(* install the lexicographic_order method and the "fun" command *)
bulwahn@21131
  2690
use "Tools/function_package/lexicographic_order.ML"
krauss@21211
  2691
use "Tools/function_package/fundef_datatype.ML"
krauss@21211
  2692
setup LexicographicOrder.setup
krauss@21211
  2693
setup FundefDatatype.setup
krauss@21211
  2694
krauss@21103
  2695
nipkow@15392
  2696
subsubsection{*Lifting a Relation on List Elements to the Lists*}
nipkow@15302
  2697
berghofe@22262
  2698
inductive2
berghofe@22262
  2699
  list_all2' :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool"
berghofe@22262
  2700
  for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
berghofe@22262
  2701
where
berghofe@22262
  2702
    Nil:  "list_all2' r [] []"
berghofe@22262
  2703
  | Cons: "[| r x y; list_all2' r xs ys |] ==> list_all2' r (x#xs) (y#ys)"
berghofe@22262
  2704
berghofe@22262
  2705
constdefs
berghofe@22262
  2706
  listrel :: "('a * 'b) set => ('a list * 'b list) set"
berghofe@22262
  2707
  "listrel r == Collect2 (list_all2' (member2 r))"
berghofe@22262
  2708
berghofe@22262
  2709
lemma list_all2_listrel_eq [pred_set_conv]:
berghofe@22262
  2710
  "list_all2' (member2 r) = member2 (listrel r)"
berghofe@22262
  2711
  by (simp add: listrel_def)
berghofe@22262
  2712
berghofe@22262
  2713
lemmas listrel_induct [consumes 1, case_names Nil Cons, induct set: listrel] =
berghofe@22262
  2714
  list_all2'.induct [to_set]
berghofe@22262
  2715
berghofe@22262
  2716
lemmas listrel_intros = list_all2'.intros [to_set]
berghofe@22262
  2717
berghofe@22262
  2718
inductive_cases2 listrel_Nil1 [to_set, elim!]: "list_all2' r [] xs"
berghofe@22262
  2719
inductive_cases2 listrel_Nil2 [to_set, elim!]: "list_all2' r xs []"
berghofe@22262
  2720
inductive_cases2 listrel_Cons1 [to_set, elim!]: "list_all2' r  (y#ys) xs"
berghofe@22262
  2721
inductive_cases2 listrel_Cons2 [to_set, elim!]: "list_all2' r xs (y#ys)"
nipkow@15302
  2722
nipkow@15302
  2723
nipkow@15302
  2724
lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
nipkow@15302
  2725
apply clarify  
berghofe@22262
  2726
apply (erule listrel_induct)
berghofe@22262
  2727
apply (blast intro: listrel_intros)+
nipkow@15302
  2728
done
nipkow@15302
  2729
nipkow@15302
  2730
lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
nipkow@15302
  2731
apply clarify 
berghofe@22262
  2732
apply (erule listrel_induct, auto) 
nipkow@15302
  2733
done
nipkow@15302
  2734
nipkow@15302
  2735
lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" 
nipkow@15302
  2736
apply (simp add: refl_def listrel_subset Ball_def)
nipkow@15302
  2737
apply (rule allI) 
nipkow@15302
  2738
apply (induct_tac x) 
berghofe@22262
  2739
apply (auto intro: listrel_intros)
nipkow@15302
  2740
done
nipkow@15302
  2741
nipkow@15302
  2742
lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
nipkow@15302
  2743
apply (auto simp add: sym_def)
berghofe@22262
  2744
apply (erule listrel_induct) 
berghofe@22262
  2745
apply (blast intro: listrel_intros)+
nipkow@15302
  2746
done
nipkow@15302
  2747
nipkow@15302
  2748
lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
nipkow@15302
  2749
apply (simp add: trans_def)
nipkow@15302
  2750
apply (intro allI) 
nipkow@15302
  2751
apply (rule impI) 
berghofe@22262
  2752
apply (erule listrel_induct) 
berghofe@22262
  2753
apply (blast intro: listrel_intros)+
nipkow@15302
  2754
done
nipkow@15302
  2755
nipkow@15302
  2756
theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
nipkow@15302
  2757
by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) 
nipkow@15302
  2758
nipkow@15302
  2759
lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}