src/HOL/List.thy
author nipkow
Thu May 24 22:55:53 2007 +0200 (2007-05-24)
changeset 23096 423ad2fe9f76
parent 23060 0c0b03d0ec7e
child 23192 ec73b9707d48
permissions -rw-r--r--
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(*  Title:      HOL/List.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow
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*)
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header {* The datatype of finite lists *}
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theory List
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imports PreList
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uses "Tools/string_syntax.ML"
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begin
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datatype 'a list =
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    Nil    ("[]")
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  | Cons 'a  "'a list"    (infixr "#" 65)
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subsection{*Basic list processing functions*}
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consts
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  append :: "'a list => 'a list => 'a list" (infixr "@" 65)
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  filter:: "('a => bool) => 'a list => 'a list"
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  concat:: "'a list list => 'a list"
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  foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"
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  foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"
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  hd:: "'a list => 'a"
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  tl:: "'a list => 'a list"
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  last:: "'a list => 'a"
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  butlast :: "'a list => 'a list"
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  set :: "'a list => 'a set"
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  map :: "('a=>'b) => ('a list => 'b list)"
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  listsum ::  "'a list => 'a::monoid_add"
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  nth :: "'a list => nat => 'a"    (infixl "!" 100)
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  list_update :: "'a list => nat => 'a => 'a list"
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  take:: "nat => 'a list => 'a list"
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  drop:: "nat => 'a list => 'a list"
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  takeWhile :: "('a => bool) => 'a list => 'a list"
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  dropWhile :: "('a => bool) => 'a list => 'a list"
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  rev :: "'a list => 'a list"
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  zip :: "'a list => 'b list => ('a * 'b) list"
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  upt :: "nat => nat => nat list" ("(1[_..</_'])")
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  remdups :: "'a list => 'a list"
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  remove1 :: "'a => 'a list => 'a list"
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  "distinct":: "'a list => bool"
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  replicate :: "nat => 'a => 'a list"
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  splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
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  allpairs :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'c list"
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abbreviation
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  upto:: "nat => nat => nat list"  ("(1[_../_])") where
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  "[i..j] == [i..<(Suc j)]"
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nonterminals lupdbinds lupdbind
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syntax
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  -- {* list Enumeration *}
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  "@list" :: "args => 'a list"    ("[(_)]")
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  -- {* Special syntax for filter *}
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_:_./ _])")
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  -- {* list update *}
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  "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
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  "" :: "lupdbind => lupdbinds"    ("_")
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  "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
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  "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
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translations
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  "[x, xs]" == "x#[xs]"
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  "[x]" == "x#[]"
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  "[x:xs . P]"== "filter (%x. P) xs"
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  "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"
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  "xs[i:=x]" == "list_update xs i x"
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syntax (xsymbols)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
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syntax (HTML output)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
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text {*
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  Function @{text size} is overloaded for all datatypes. Users may
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  refer to the list version as @{text length}. *}
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abbreviation
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  length :: "'a list => nat" where
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  "length == size"
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primrec
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  "hd(x#xs) = x"
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primrec
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  "tl([]) = []"
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  "tl(x#xs) = xs"
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primrec
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  "last(x#xs) = (if xs=[] then x else last xs)"
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primrec
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  "butlast []= []"
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  "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
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primrec
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  "set [] = {}"
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  "set (x#xs) = insert x (set xs)"
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primrec
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  "map f [] = []"
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  "map f (x#xs) = f(x)#map f xs"
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primrec
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  append_Nil: "[]@ys = ys"
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  append_Cons: "(x#xs)@ys = x#(xs@ys)"
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primrec
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  "rev([]) = []"
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  "rev(x#xs) = rev(xs) @ [x]"
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primrec
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  "filter P [] = []"
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  "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
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primrec
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  foldl_Nil:"foldl f a [] = a"
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  foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
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primrec
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  "foldr f [] a = a"
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  "foldr f (x#xs) a = f x (foldr f xs a)"
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primrec
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  "concat([]) = []"
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  "concat(x#xs) = x @ concat(xs)"
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primrec
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"listsum [] = 0"
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"listsum (x # xs) = x + listsum xs"
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primrec
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  drop_Nil:"drop n [] = []"
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  drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  take_Nil:"take n [] = []"
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  take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  "[][i:=v] = []"
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  "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])"
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primrec
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  "takeWhile P [] = []"
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  "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
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primrec
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  "dropWhile P [] = []"
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  "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
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primrec
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  "zip xs [] = []"
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  zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
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primrec
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  upt_0: "[i..<0] = []"
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  upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
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primrec
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  "distinct [] = True"
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  "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
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primrec
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  "remdups [] = []"
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  "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
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primrec
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  "remove1 x [] = []"
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  "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)"
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primrec
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  replicate_0: "replicate 0 x = []"
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  replicate_Suc: "replicate (Suc n) x = x # replicate n x"
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definition
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  rotate1 :: "'a list \<Rightarrow> 'a list" where
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  "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"
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definition
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  rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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  "rotate n = rotate1 ^ n"
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definition
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  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
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  "list_all2 P xs ys =
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    (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"
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definition
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  sublist :: "'a list => nat set => 'a list" where
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  "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"
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primrec
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  "splice [] ys = ys"
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  "splice (x#xs) ys = (if ys=[] then x#xs else x # hd ys # splice xs (tl ys))"
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    -- {*Warning: simpset does not contain the second eqn but a derived one. *}
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primrec
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"allpairs f [] ys = []"
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"allpairs f (x # xs) ys = map (f x) ys @ allpairs f xs ys"
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subsubsection {* @{const Nil} and @{const Cons} *}
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lemma not_Cons_self [simp]:
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  "xs \<noteq> x # xs"
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by (induct xs) auto
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lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
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lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
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by (induct xs) auto
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lemma length_induct:
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  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
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by (rule measure_induct [of length]) iprover
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subsubsection {* @{const length} *}
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text {*
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  Needs to come before @{text "@"} because of theorem @{text
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  append_eq_append_conv}.
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*}
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lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
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by (induct xs) auto
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lemma length_map [simp]: "length (map f xs) = length xs"
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by (induct xs) auto
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lemma length_rev [simp]: "length (rev xs) = length xs"
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by (induct xs) auto
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lemma length_tl [simp]: "length (tl xs) = length xs - 1"
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by (cases xs) auto
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lemma length_allpairs[simp]:
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 "length(allpairs f xs ys) = length xs * length ys"
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by(induct xs) auto
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lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
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by (induct xs) auto
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lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
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by (induct xs) auto
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lemma length_Suc_conv:
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"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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by (induct xs) auto
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lemma Suc_length_conv:
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"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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apply (induct xs, simp, simp)
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apply blast
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done
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lemma impossible_Cons [rule_format]: 
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  "length xs <= length ys --> xs = x # ys = False"
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apply (induct xs)
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apply auto
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done
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lemma list_induct2[consumes 1]: "\<And>ys.
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 \<lbrakk> length xs = length ys;
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   P [] [];
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   \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
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 \<Longrightarrow> P xs ys"
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apply(induct xs)
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 apply simp
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apply(case_tac ys)
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 apply simp
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apply(simp)
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done
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lemma list_induct2': 
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  "\<lbrakk> P [] [];
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  \<And>x xs. P (x#xs) [];
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  \<And>y ys. P [] (y#ys);
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   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
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 \<Longrightarrow> P xs ys"
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by (induct xs arbitrary: ys) (case_tac x, auto)+
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lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
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apply(rule Eq_FalseI)
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by auto
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(*
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Reduces xs=ys to False if xs and ys cannot be of the same length.
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This is the case if the atomic sublists of one are a submultiset
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of those of the other list and there are fewer Cons's in one than the other.
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*)
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ML_setup {*
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local
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fun len (Const("List.list.Nil",_)) acc = acc
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  | len (Const("List.list.Cons",_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
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  | len (Const("List.append",_) $ xs $ ys) acc = len xs (len ys acc)
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  | len (Const("List.rev",_) $ xs) acc = len xs acc
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  | len (Const("List.map",_) $ _ $ xs) acc = len xs acc
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  | len t (ts,n) = (t::ts,n);
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fun list_eq ss (Const(_,eqT) $ lhs $ rhs) =
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  let
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    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
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    fun prove_neq() =
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      let
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        val Type(_,listT::_) = eqT;
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        val size = HOLogic.size_const listT;
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        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
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        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
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        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
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          (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
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      in SOME (thm RS @{thm neq_if_length_neq}) end
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  in
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    if m < n andalso gen_submultiset (op aconv) (ls,rs) orelse
nipkow@22143
   336
       n < m andalso gen_submultiset (op aconv) (rs,ls)
nipkow@22143
   337
    then prove_neq() else NONE
nipkow@22143
   338
  end;
nipkow@22143
   339
nipkow@22143
   340
in
nipkow@22143
   341
nipkow@22143
   342
val list_neq_simproc =
haftmann@22633
   343
  Simplifier.simproc @{theory} "list_neq" ["(xs::'a list) = ys"] (K list_eq);
nipkow@22143
   344
nipkow@22143
   345
end;
nipkow@22143
   346
nipkow@22143
   347
Addsimprocs [list_neq_simproc];
nipkow@22143
   348
*}
nipkow@22143
   349
nipkow@22143
   350
nipkow@15392
   351
subsubsection {* @{text "@"} -- append *}
wenzelm@13114
   352
wenzelm@13142
   353
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
nipkow@13145
   354
by (induct xs) auto
wenzelm@13114
   355
wenzelm@13142
   356
lemma append_Nil2 [simp]: "xs @ [] = xs"
nipkow@13145
   357
by (induct xs) auto
nipkow@3507
   358
wenzelm@13142
   359
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
nipkow@13145
   360
by (induct xs) auto
wenzelm@13114
   361
wenzelm@13142
   362
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
nipkow@13145
   363
by (induct xs) auto
wenzelm@13114
   364
wenzelm@13142
   365
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
nipkow@13145
   366
by (induct xs) auto
wenzelm@13114
   367
wenzelm@13142
   368
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
nipkow@13145
   369
by (induct xs) auto
wenzelm@13114
   370
berghofe@13883
   371
lemma append_eq_append_conv [simp]:
berghofe@13883
   372
 "!!ys. length xs = length ys \<or> length us = length vs
berghofe@13883
   373
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
berghofe@13883
   374
apply (induct xs)
paulson@14208
   375
 apply (case_tac ys, simp, force)
paulson@14208
   376
apply (case_tac ys, force, simp)
nipkow@13145
   377
done
wenzelm@13142
   378
nipkow@14495
   379
lemma append_eq_append_conv2: "!!ys zs ts.
nipkow@14495
   380
 (xs @ ys = zs @ ts) =
nipkow@14495
   381
 (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
nipkow@14495
   382
apply (induct xs)
nipkow@14495
   383
 apply fastsimp
nipkow@14495
   384
apply(case_tac zs)
nipkow@14495
   385
 apply simp
nipkow@14495
   386
apply fastsimp
nipkow@14495
   387
done
nipkow@14495
   388
wenzelm@13142
   389
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145
   390
by simp
wenzelm@13142
   391
wenzelm@13142
   392
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145
   393
by simp
wenzelm@13114
   394
wenzelm@13142
   395
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   396
by simp
wenzelm@13114
   397
wenzelm@13142
   398
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   399
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   400
wenzelm@13142
   401
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   402
using append_same_eq [of "[]"] by auto
wenzelm@13114
   403
wenzelm@13142
   404
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   405
by (induct xs) auto
wenzelm@13114
   406
wenzelm@13142
   407
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   408
by (induct xs) auto
wenzelm@13114
   409
wenzelm@13142
   410
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   411
by (simp add: hd_append split: list.split)
wenzelm@13114
   412
wenzelm@13142
   413
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   414
by (simp split: list.split)
wenzelm@13114
   415
wenzelm@13142
   416
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   417
by (simp add: tl_append split: list.split)
wenzelm@13114
   418
wenzelm@13114
   419
nipkow@14300
   420
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   421
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   422
by(cases ys) auto
nipkow@14300
   423
nipkow@15281
   424
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   425
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   426
by(cases ys) auto
nipkow@15281
   427
nipkow@14300
   428
wenzelm@13142
   429
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   430
wenzelm@13114
   431
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   432
by simp
wenzelm@13114
   433
wenzelm@13142
   434
lemma Cons_eq_appendI:
nipkow@13145
   435
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   436
by (drule sym) simp
wenzelm@13114
   437
wenzelm@13142
   438
lemma append_eq_appendI:
nipkow@13145
   439
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   440
by (drule sym) simp
wenzelm@13114
   441
wenzelm@13114
   442
wenzelm@13142
   443
text {*
nipkow@13145
   444
Simplification procedure for all list equalities.
nipkow@13145
   445
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   446
- both lists end in a singleton list,
nipkow@13145
   447
- or both lists end in the same list.
wenzelm@13142
   448
*}
wenzelm@13142
   449
wenzelm@13142
   450
ML_setup {*
nipkow@3507
   451
local
nipkow@3507
   452
wenzelm@13114
   453
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462
   454
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
haftmann@23029
   455
  | last (Const("List.append",_) $ _ $ ys) = last ys
wenzelm@13462
   456
  | last t = t;
wenzelm@13114
   457
wenzelm@13114
   458
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462
   459
  | list1 _ = false;
wenzelm@13114
   460
wenzelm@13114
   461
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462
   462
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
haftmann@23029
   463
  | butlast ((app as Const("List.append",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462
   464
  | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114
   465
haftmann@22633
   466
val rearr_ss = HOL_basic_ss addsimps [@{thm append_assoc},
haftmann@22633
   467
  @{thm append_Nil}, @{thm append_Cons}];
wenzelm@16973
   468
wenzelm@20044
   469
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   470
  let
wenzelm@13462
   471
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   472
    fun rearr conv =
wenzelm@13462
   473
      let
wenzelm@13462
   474
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   475
        val Type(_,listT::_) = eqT
wenzelm@13462
   476
        val appT = [listT,listT] ---> listT
haftmann@23029
   477
        val app = Const("List.append",appT)
wenzelm@13462
   478
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   479
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@20044
   480
        val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
wenzelm@17877
   481
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
skalberg@15531
   482
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   483
wenzelm@13462
   484
  in
haftmann@22633
   485
    if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
haftmann@22633
   486
    else if lastl aconv lastr then rearr @{thm append_same_eq}
skalberg@15531
   487
    else NONE
wenzelm@13462
   488
  end;
wenzelm@13462
   489
wenzelm@13114
   490
in
wenzelm@13462
   491
wenzelm@13462
   492
val list_eq_simproc =
haftmann@22633
   493
  Simplifier.simproc @{theory} "list_eq" ["(xs::'a list) = ys"] (K list_eq);
wenzelm@13462
   494
wenzelm@13114
   495
end;
wenzelm@13114
   496
wenzelm@13114
   497
Addsimprocs [list_eq_simproc];
wenzelm@13114
   498
*}
wenzelm@13114
   499
wenzelm@13114
   500
nipkow@15392
   501
subsubsection {* @{text map} *}
wenzelm@13114
   502
wenzelm@13142
   503
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   504
by (induct xs) simp_all
wenzelm@13114
   505
wenzelm@13142
   506
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   507
by (rule ext, induct_tac xs) auto
wenzelm@13114
   508
wenzelm@13142
   509
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   510
by (induct xs) auto
wenzelm@13114
   511
wenzelm@13142
   512
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   513
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   514
wenzelm@13142
   515
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   516
by (induct xs) auto
wenzelm@13114
   517
nipkow@13737
   518
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   519
by (induct xs) auto
nipkow@13737
   520
krauss@19770
   521
lemma map_cong [fundef_cong, recdef_cong]:
nipkow@13145
   522
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   523
-- {* a congruence rule for @{text map} *}
nipkow@13737
   524
by simp
wenzelm@13114
   525
wenzelm@13142
   526
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   527
by (cases xs) auto
wenzelm@13114
   528
wenzelm@13142
   529
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   530
by (cases xs) auto
wenzelm@13114
   531
paulson@18447
   532
lemma map_eq_Cons_conv:
nipkow@14025
   533
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   534
by (cases xs) auto
wenzelm@13114
   535
paulson@18447
   536
lemma Cons_eq_map_conv:
nipkow@14025
   537
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   538
by (cases ys) auto
nipkow@14025
   539
paulson@18447
   540
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
paulson@18447
   541
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
paulson@18447
   542
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
paulson@18447
   543
nipkow@14111
   544
lemma ex_map_conv:
nipkow@14111
   545
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
paulson@18447
   546
by(induct ys, auto simp add: Cons_eq_map_conv)
nipkow@14111
   547
nipkow@15110
   548
lemma map_eq_imp_length_eq:
nipkow@15110
   549
  "!!xs. map f xs = map f ys ==> length xs = length ys"
nipkow@15110
   550
apply (induct ys)
nipkow@15110
   551
 apply simp
nipkow@15110
   552
apply(simp (no_asm_use))
nipkow@15110
   553
apply clarify
nipkow@15110
   554
apply(simp (no_asm_use))
nipkow@15110
   555
apply fast
nipkow@15110
   556
done
nipkow@15110
   557
nipkow@15110
   558
lemma map_inj_on:
nipkow@15110
   559
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   560
  ==> xs = ys"
nipkow@15110
   561
apply(frule map_eq_imp_length_eq)
nipkow@15110
   562
apply(rotate_tac -1)
nipkow@15110
   563
apply(induct rule:list_induct2)
nipkow@15110
   564
 apply simp
nipkow@15110
   565
apply(simp)
nipkow@15110
   566
apply (blast intro:sym)
nipkow@15110
   567
done
nipkow@15110
   568
nipkow@15110
   569
lemma inj_on_map_eq_map:
nipkow@15110
   570
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   571
by(blast dest:map_inj_on)
nipkow@15110
   572
wenzelm@13114
   573
lemma map_injective:
nipkow@14338
   574
 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@14338
   575
by (induct ys) (auto dest!:injD)
wenzelm@13114
   576
nipkow@14339
   577
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   578
by(blast dest:map_injective)
nipkow@14339
   579
wenzelm@13114
   580
lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@17589
   581
by (iprover dest: map_injective injD intro: inj_onI)
wenzelm@13114
   582
wenzelm@13114
   583
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   584
apply (unfold inj_on_def, clarify)
nipkow@13145
   585
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   586
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   587
apply blast
nipkow@13145
   588
done
wenzelm@13114
   589
nipkow@14339
   590
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   591
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   592
nipkow@15303
   593
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
   594
apply(rule inj_onI)
nipkow@15303
   595
apply(erule map_inj_on)
nipkow@15303
   596
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
   597
done
nipkow@15303
   598
kleing@14343
   599
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   600
by (induct xs, auto)
wenzelm@13114
   601
nipkow@14402
   602
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   603
by (induct xs) auto
nipkow@14402
   604
nipkow@15110
   605
lemma map_fst_zip[simp]:
nipkow@15110
   606
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   607
by (induct rule:list_induct2, simp_all)
nipkow@15110
   608
nipkow@15110
   609
lemma map_snd_zip[simp]:
nipkow@15110
   610
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   611
by (induct rule:list_induct2, simp_all)
nipkow@15110
   612
nipkow@15110
   613
nipkow@15392
   614
subsubsection {* @{text rev} *}
wenzelm@13114
   615
wenzelm@13142
   616
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   617
by (induct xs) auto
wenzelm@13114
   618
wenzelm@13142
   619
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   620
by (induct xs) auto
wenzelm@13114
   621
kleing@15870
   622
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870
   623
by auto
kleing@15870
   624
wenzelm@13142
   625
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   626
by (induct xs) auto
wenzelm@13114
   627
wenzelm@13142
   628
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   629
by (induct xs) auto
wenzelm@13114
   630
kleing@15870
   631
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870
   632
by (cases xs) auto
kleing@15870
   633
kleing@15870
   634
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870
   635
by (cases xs) auto
kleing@15870
   636
haftmann@21061
   637
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
haftmann@21061
   638
apply (induct xs arbitrary: ys, force)
paulson@14208
   639
apply (case_tac ys, simp, force)
nipkow@13145
   640
done
wenzelm@13114
   641
nipkow@15439
   642
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
   643
by(simp add:inj_on_def)
nipkow@15439
   644
wenzelm@13366
   645
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   646
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489
   647
apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145
   648
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   649
done
wenzelm@13114
   650
nipkow@13145
   651
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114
   652
wenzelm@13366
   653
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   654
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   655
by (induct xs rule: rev_induct) auto
wenzelm@13114
   656
wenzelm@13366
   657
lemmas rev_cases = rev_exhaust
wenzelm@13366
   658
nipkow@18423
   659
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
nipkow@18423
   660
by(rule rev_cases[of xs]) auto
nipkow@18423
   661
wenzelm@13114
   662
nipkow@15392
   663
subsubsection {* @{text set} *}
wenzelm@13114
   664
wenzelm@13142
   665
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   666
by (induct xs) auto
wenzelm@13114
   667
wenzelm@13142
   668
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   669
by (induct xs) auto
wenzelm@13114
   670
nipkow@17830
   671
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
nipkow@17830
   672
by(cases xs) auto
oheimb@14099
   673
wenzelm@13142
   674
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   675
by auto
wenzelm@13114
   676
oheimb@14099
   677
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   678
by auto
oheimb@14099
   679
wenzelm@13142
   680
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   681
by (induct xs) auto
wenzelm@13114
   682
nipkow@15245
   683
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   684
by(induct xs) auto
nipkow@15245
   685
wenzelm@13142
   686
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   687
by (induct xs) auto
wenzelm@13114
   688
wenzelm@13142
   689
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   690
by (induct xs) auto
wenzelm@13114
   691
nipkow@22828
   692
lemma set_allpairs[simp]:
nipkow@22830
   693
 "set(allpairs f xs ys) = {z. EX x : set xs. EX y : set ys. z = f x y}"
nipkow@22828
   694
by(induct xs) auto
nipkow@22828
   695
wenzelm@13142
   696
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   697
by (induct xs) auto
wenzelm@13114
   698
nipkow@15425
   699
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
paulson@14208
   700
apply (induct j, simp_all)
paulson@14208
   701
apply (erule ssubst, auto)
nipkow@13145
   702
done
wenzelm@13114
   703
wenzelm@13142
   704
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
paulson@15113
   705
proof (induct xs)
paulson@15113
   706
  case Nil show ?case by simp
paulson@15113
   707
  case (Cons a xs)
paulson@15113
   708
  show ?case
paulson@15113
   709
  proof 
paulson@15113
   710
    assume "x \<in> set (a # xs)"
paulson@15113
   711
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   712
      by (simp, blast intro: Cons_eq_appendI)
paulson@15113
   713
  next
paulson@15113
   714
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   715
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
paulson@15113
   716
    show "x \<in> set (a # xs)" 
paulson@15113
   717
      by (cases ys, auto simp add: eq)
paulson@15113
   718
  qed
paulson@15113
   719
qed
wenzelm@13142
   720
nipkow@18049
   721
lemma in_set_conv_decomp_first:
nipkow@18049
   722
 "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
nipkow@18049
   723
proof (induct xs)
nipkow@18049
   724
  case Nil show ?case by simp
nipkow@18049
   725
next
nipkow@18049
   726
  case (Cons a xs)
nipkow@18049
   727
  show ?case
nipkow@18049
   728
  proof cases
nipkow@18049
   729
    assume "x = a" thus ?case using Cons by force
nipkow@18049
   730
  next
nipkow@18049
   731
    assume "x \<noteq> a"
nipkow@18049
   732
    show ?case
nipkow@18049
   733
    proof
nipkow@18049
   734
      assume "x \<in> set (a # xs)"
nipkow@18049
   735
      from prems show "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@18049
   736
	by(fastsimp intro!: Cons_eq_appendI)
nipkow@18049
   737
    next
nipkow@18049
   738
      assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@18049
   739
      then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
nipkow@18049
   740
      show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq)
nipkow@18049
   741
    qed
nipkow@18049
   742
  qed
nipkow@18049
   743
qed
nipkow@18049
   744
nipkow@18049
   745
lemmas split_list       = in_set_conv_decomp[THEN iffD1, standard]
nipkow@18049
   746
lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard]
nipkow@18049
   747
nipkow@18049
   748
paulson@13508
   749
lemma finite_list: "finite A ==> EX l. set l = A"
paulson@13508
   750
apply (erule finite_induct, auto)
paulson@13508
   751
apply (rule_tac x="x#l" in exI, auto)
paulson@13508
   752
done
paulson@13508
   753
kleing@14388
   754
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
   755
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
   756
paulson@15168
   757
nipkow@15392
   758
subsubsection {* @{text filter} *}
wenzelm@13114
   759
wenzelm@13142
   760
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   761
by (induct xs) auto
wenzelm@13114
   762
nipkow@15305
   763
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
   764
by (induct xs) simp_all
nipkow@15305
   765
wenzelm@13142
   766
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   767
by (induct xs) auto
wenzelm@13114
   768
nipkow@16998
   769
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998
   770
by (induct xs) (auto simp add: le_SucI)
nipkow@16998
   771
nipkow@18423
   772
lemma sum_length_filter_compl:
nipkow@18423
   773
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
nipkow@18423
   774
by(induct xs) simp_all
nipkow@18423
   775
wenzelm@13142
   776
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   777
by (induct xs) auto
wenzelm@13114
   778
wenzelm@13142
   779
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   780
by (induct xs) auto
wenzelm@13114
   781
nipkow@16998
   782
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@16998
   783
  by (induct xs) simp_all
nipkow@16998
   784
nipkow@16998
   785
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998
   786
apply (induct xs)
nipkow@16998
   787
 apply auto
nipkow@16998
   788
apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998
   789
apply simp
nipkow@16998
   790
done
wenzelm@13114
   791
nipkow@16965
   792
lemma filter_map:
nipkow@16965
   793
  "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965
   794
by (induct xs) simp_all
nipkow@16965
   795
nipkow@16965
   796
lemma length_filter_map[simp]:
nipkow@16965
   797
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965
   798
by (simp add:filter_map)
nipkow@16965
   799
wenzelm@13142
   800
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   801
by auto
wenzelm@13114
   802
nipkow@15246
   803
lemma length_filter_less:
nipkow@15246
   804
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
   805
proof (induct xs)
nipkow@15246
   806
  case Nil thus ?case by simp
nipkow@15246
   807
next
nipkow@15246
   808
  case (Cons x xs) thus ?case
nipkow@15246
   809
    apply (auto split:split_if_asm)
nipkow@15246
   810
    using length_filter_le[of P xs] apply arith
nipkow@15246
   811
  done
nipkow@15246
   812
qed
wenzelm@13114
   813
nipkow@15281
   814
lemma length_filter_conv_card:
nipkow@15281
   815
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
   816
proof (induct xs)
nipkow@15281
   817
  case Nil thus ?case by simp
nipkow@15281
   818
next
nipkow@15281
   819
  case (Cons x xs)
nipkow@15281
   820
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
   821
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
   822
  show ?case (is "?l = card ?S'")
nipkow@15281
   823
  proof (cases)
nipkow@15281
   824
    assume "p x"
nipkow@15281
   825
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@15281
   826
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   827
    have "length (filter p (x # xs)) = Suc(card ?S)"
nipkow@15281
   828
      using Cons by simp
nipkow@15281
   829
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
nipkow@15281
   830
      by (simp add: card_image inj_Suc)
nipkow@15281
   831
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   832
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
   833
    finally show ?thesis .
nipkow@15281
   834
  next
nipkow@15281
   835
    assume "\<not> p x"
nipkow@15281
   836
    hence eq: "?S' = Suc ` ?S"
nipkow@15281
   837
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   838
    have "length (filter p (x # xs)) = card ?S"
nipkow@15281
   839
      using Cons by simp
nipkow@15281
   840
    also have "\<dots> = card(Suc ` ?S)" using fin
nipkow@15281
   841
      by (simp add: card_image inj_Suc)
nipkow@15281
   842
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   843
      by (simp add:card_insert_if)
nipkow@15281
   844
    finally show ?thesis .
nipkow@15281
   845
  qed
nipkow@15281
   846
qed
nipkow@15281
   847
nipkow@17629
   848
lemma Cons_eq_filterD:
nipkow@17629
   849
 "x#xs = filter P ys \<Longrightarrow>
nipkow@17629
   850
  \<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
   851
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
nipkow@17629
   852
proof(induct ys)
nipkow@17629
   853
  case Nil thus ?case by simp
nipkow@17629
   854
next
nipkow@17629
   855
  case (Cons y ys)
nipkow@17629
   856
  show ?case (is "\<exists>x. ?Q x")
nipkow@17629
   857
  proof cases
nipkow@17629
   858
    assume Py: "P y"
nipkow@17629
   859
    show ?thesis
nipkow@17629
   860
    proof cases
nipkow@17629
   861
      assume xy: "x = y"
nipkow@17629
   862
      show ?thesis
nipkow@17629
   863
      proof from Py xy Cons(2) show "?Q []" by simp qed
nipkow@17629
   864
    next
nipkow@17629
   865
      assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp
nipkow@17629
   866
    qed
nipkow@17629
   867
  next
nipkow@17629
   868
    assume Py: "\<not> P y"
nipkow@17629
   869
    with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp
nipkow@17629
   870
    show ?thesis (is "? us. ?Q us")
nipkow@17629
   871
    proof show "?Q (y#us)" using 1 by simp qed
nipkow@17629
   872
  qed
nipkow@17629
   873
qed
nipkow@17629
   874
nipkow@17629
   875
lemma filter_eq_ConsD:
nipkow@17629
   876
 "filter P ys = x#xs \<Longrightarrow>
nipkow@17629
   877
  \<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
   878
by(rule Cons_eq_filterD) simp
nipkow@17629
   879
nipkow@17629
   880
lemma filter_eq_Cons_iff:
nipkow@17629
   881
 "(filter P ys = x#xs) =
nipkow@17629
   882
  (\<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
   883
by(auto dest:filter_eq_ConsD)
nipkow@17629
   884
nipkow@17629
   885
lemma Cons_eq_filter_iff:
nipkow@17629
   886
 "(x#xs = filter P ys) =
nipkow@17629
   887
  (\<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
   888
by(auto dest:Cons_eq_filterD)
nipkow@17629
   889
krauss@19770
   890
lemma filter_cong[fundef_cong, recdef_cong]:
nipkow@17501
   891
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
nipkow@17501
   892
apply simp
nipkow@17501
   893
apply(erule thin_rl)
nipkow@17501
   894
by (induct ys) simp_all
nipkow@17501
   895
nipkow@15281
   896
nipkow@15392
   897
subsubsection {* @{text concat} *}
wenzelm@13114
   898
wenzelm@13142
   899
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
   900
by (induct xs) auto
wenzelm@13114
   901
paulson@18447
   902
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   903
by (induct xss) auto
wenzelm@13114
   904
paulson@18447
   905
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   906
by (induct xss) auto
wenzelm@13114
   907
wenzelm@13142
   908
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
nipkow@13145
   909
by (induct xs) auto
wenzelm@13114
   910
wenzelm@13142
   911
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
   912
by (induct xs) auto
wenzelm@13114
   913
wenzelm@13142
   914
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
   915
by (induct xs) auto
wenzelm@13114
   916
wenzelm@13142
   917
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
   918
by (induct xs) auto
wenzelm@13114
   919
wenzelm@13114
   920
nipkow@15392
   921
subsubsection {* @{text nth} *}
wenzelm@13114
   922
wenzelm@13142
   923
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
   924
by auto
wenzelm@13114
   925
wenzelm@13142
   926
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
   927
by auto
wenzelm@13114
   928
wenzelm@13142
   929
declare nth.simps [simp del]
wenzelm@13114
   930
wenzelm@13114
   931
lemma nth_append:
nipkow@13145
   932
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
paulson@14208
   933
apply (induct "xs", simp)
paulson@14208
   934
apply (case_tac n, auto)
nipkow@13145
   935
done
wenzelm@13114
   936
nipkow@14402
   937
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
nipkow@14402
   938
by (induct "xs") auto
nipkow@14402
   939
nipkow@14402
   940
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
nipkow@14402
   941
by (induct "xs") auto
nipkow@14402
   942
wenzelm@13142
   943
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
paulson@14208
   944
apply (induct xs, simp)
paulson@14208
   945
apply (case_tac n, auto)
nipkow@13145
   946
done
wenzelm@13114
   947
nipkow@18423
   948
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
nipkow@18423
   949
by(cases xs) simp_all
nipkow@18423
   950
nipkow@18049
   951
nipkow@18049
   952
lemma list_eq_iff_nth_eq:
nipkow@18049
   953
 "!!ys. (xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
nipkow@18049
   954
apply(induct xs)
nipkow@18049
   955
 apply simp apply blast
nipkow@18049
   956
apply(case_tac ys)
nipkow@18049
   957
 apply simp
nipkow@18049
   958
apply(simp add:nth_Cons split:nat.split)apply blast
nipkow@18049
   959
done
nipkow@18049
   960
wenzelm@13142
   961
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
   962
apply (induct xs, simp, simp)
nipkow@13145
   963
apply safe
paulson@14208
   964
apply (rule_tac x = 0 in exI, simp)
paulson@14208
   965
 apply (rule_tac x = "Suc i" in exI, simp)
paulson@14208
   966
apply (case_tac i, simp)
nipkow@13145
   967
apply (rename_tac j)
paulson@14208
   968
apply (rule_tac x = j in exI, simp)
nipkow@13145
   969
done
wenzelm@13114
   970
nipkow@17501
   971
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
nipkow@17501
   972
by(auto simp:set_conv_nth)
nipkow@17501
   973
nipkow@13145
   974
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
   975
by (auto simp add: set_conv_nth)
wenzelm@13114
   976
wenzelm@13142
   977
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
   978
by (auto simp add: set_conv_nth)
wenzelm@13114
   979
wenzelm@13114
   980
lemma all_nth_imp_all_set:
nipkow@13145
   981
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
   982
by (auto simp add: set_conv_nth)
wenzelm@13114
   983
wenzelm@13114
   984
lemma all_set_conv_all_nth:
nipkow@13145
   985
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
   986
by (auto simp add: set_conv_nth)
wenzelm@13114
   987
wenzelm@13114
   988
nipkow@15392
   989
subsubsection {* @{text list_update} *}
wenzelm@13114
   990
wenzelm@13142
   991
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
nipkow@13145
   992
by (induct xs) (auto split: nat.split)
wenzelm@13114
   993
wenzelm@13114
   994
lemma nth_list_update:
nipkow@13145
   995
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@13145
   996
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   997
wenzelm@13142
   998
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
   999
by (simp add: nth_list_update)
wenzelm@13114
  1000
wenzelm@13142
  1001
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145
  1002
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1003
wenzelm@13142
  1004
lemma list_update_overwrite [simp]:
nipkow@13145
  1005
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145
  1006
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1007
nipkow@14402
  1008
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
paulson@14208
  1009
apply (induct xs, simp)
nipkow@14187
  1010
apply(simp split:nat.splits)
nipkow@14187
  1011
done
nipkow@14187
  1012
nipkow@17501
  1013
lemma list_update_beyond[simp]: "\<And>i. length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
nipkow@17501
  1014
apply (induct xs)
nipkow@17501
  1015
 apply simp
nipkow@17501
  1016
apply (case_tac i)
nipkow@17501
  1017
apply simp_all
nipkow@17501
  1018
done
nipkow@17501
  1019
wenzelm@13114
  1020
lemma list_update_same_conv:
nipkow@13145
  1021
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145
  1022
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1023
nipkow@14187
  1024
lemma list_update_append1:
nipkow@14187
  1025
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
paulson@14208
  1026
apply (induct xs, simp)
nipkow@14187
  1027
apply(simp split:nat.split)
nipkow@14187
  1028
done
nipkow@14187
  1029
kleing@15868
  1030
lemma list_update_append:
kleing@15868
  1031
  "!!n. (xs @ ys) [n:= x] = 
kleing@15868
  1032
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
kleing@15868
  1033
by (induct xs) (auto split:nat.splits)
kleing@15868
  1034
nipkow@14402
  1035
lemma list_update_length [simp]:
nipkow@14402
  1036
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
  1037
by (induct xs, auto)
nipkow@14402
  1038
wenzelm@13114
  1039
lemma update_zip:
nipkow@13145
  1040
"!!i xy xs. length xs = length ys ==>
nipkow@13145
  1041
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145
  1042
by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114
  1043
wenzelm@13114
  1044
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145
  1045
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1046
wenzelm@13114
  1047
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
  1048
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
  1049
kleing@15868
  1050
lemma set_update_memI: "!!n. n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
kleing@15868
  1051
by (induct xs) (auto split:nat.splits)
kleing@15868
  1052
wenzelm@13114
  1053
nipkow@15392
  1054
subsubsection {* @{text last} and @{text butlast} *}
wenzelm@13114
  1055
wenzelm@13142
  1056
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
  1057
by (induct xs) auto
wenzelm@13114
  1058
wenzelm@13142
  1059
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
  1060
by (induct xs) auto
wenzelm@13114
  1061
nipkow@14302
  1062
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
  1063
by(simp add:last.simps)
nipkow@14302
  1064
nipkow@14302
  1065
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
  1066
by(simp add:last.simps)
nipkow@14302
  1067
nipkow@14302
  1068
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
  1069
by (induct xs) (auto)
nipkow@14302
  1070
nipkow@14302
  1071
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
  1072
by(simp add:last_append)
nipkow@14302
  1073
nipkow@14302
  1074
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
  1075
by(simp add:last_append)
nipkow@14302
  1076
nipkow@17762
  1077
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
nipkow@17762
  1078
by(rule rev_exhaust[of xs]) simp_all
nipkow@17762
  1079
nipkow@17762
  1080
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
nipkow@17762
  1081
by(cases xs) simp_all
nipkow@17762
  1082
nipkow@17765
  1083
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
nipkow@17765
  1084
by (induct as) auto
nipkow@17762
  1085
wenzelm@13142
  1086
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
  1087
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1088
wenzelm@13114
  1089
lemma butlast_append:
nipkow@13145
  1090
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145
  1091
by (induct xs) auto
wenzelm@13114
  1092
wenzelm@13142
  1093
lemma append_butlast_last_id [simp]:
nipkow@13145
  1094
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
  1095
by (induct xs) auto
wenzelm@13114
  1096
wenzelm@13142
  1097
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
  1098
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1099
wenzelm@13114
  1100
lemma in_set_butlast_appendI:
nipkow@13145
  1101
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
  1102
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
  1103
nipkow@17501
  1104
lemma last_drop[simp]: "!!n. n < length xs \<Longrightarrow> last (drop n xs) = last xs"
nipkow@17501
  1105
apply (induct xs)
nipkow@17501
  1106
 apply simp
nipkow@17501
  1107
apply (auto split:nat.split)
nipkow@17501
  1108
done
nipkow@17501
  1109
nipkow@17589
  1110
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
nipkow@17589
  1111
by(induct xs)(auto simp:neq_Nil_conv)
nipkow@17589
  1112
nipkow@15392
  1113
subsubsection {* @{text take} and @{text drop} *}
wenzelm@13114
  1114
wenzelm@13142
  1115
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
  1116
by (induct xs) auto
wenzelm@13114
  1117
wenzelm@13142
  1118
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
  1119
by (induct xs) auto
wenzelm@13114
  1120
wenzelm@13142
  1121
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
  1122
by simp
wenzelm@13114
  1123
wenzelm@13142
  1124
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
  1125
by simp
wenzelm@13114
  1126
wenzelm@13142
  1127
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
  1128
nipkow@15110
  1129
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
  1130
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
  1131
nipkow@14187
  1132
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
  1133
by(cases xs, simp_all)
nipkow@14187
  1134
nipkow@14187
  1135
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
nipkow@14187
  1136
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@14187
  1137
nipkow@14187
  1138
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
paulson@14208
  1139
apply (induct xs, simp)
nipkow@14187
  1140
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
  1141
done
nipkow@14187
  1142
nipkow@13913
  1143
lemma take_Suc_conv_app_nth:
nipkow@13913
  1144
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
paulson@14208
  1145
apply (induct xs, simp)
paulson@14208
  1146
apply (case_tac i, auto)
nipkow@13913
  1147
done
nipkow@13913
  1148
mehta@14591
  1149
lemma drop_Suc_conv_tl:
mehta@14591
  1150
  "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
mehta@14591
  1151
apply (induct xs, simp)
mehta@14591
  1152
apply (case_tac i, auto)
mehta@14591
  1153
done
mehta@14591
  1154
wenzelm@13142
  1155
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
nipkow@13145
  1156
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1157
wenzelm@13142
  1158
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
nipkow@13145
  1159
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1160
wenzelm@13142
  1161
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
nipkow@13145
  1162
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1163
wenzelm@13142
  1164
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
nipkow@13145
  1165
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1166
wenzelm@13142
  1167
lemma take_append [simp]:
nipkow@13145
  1168
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@13145
  1169
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1170
wenzelm@13142
  1171
lemma drop_append [simp]:
nipkow@13145
  1172
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@13145
  1173
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1174
wenzelm@13142
  1175
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
paulson@14208
  1176
apply (induct m, auto)
paulson@14208
  1177
apply (case_tac xs, auto)
nipkow@15236
  1178
apply (case_tac n, auto)
nipkow@13145
  1179
done
wenzelm@13114
  1180
wenzelm@13142
  1181
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
paulson@14208
  1182
apply (induct m, auto)
paulson@14208
  1183
apply (case_tac xs, auto)
nipkow@13145
  1184
done
wenzelm@13114
  1185
wenzelm@13114
  1186
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
paulson@14208
  1187
apply (induct m, auto)
paulson@14208
  1188
apply (case_tac xs, auto)
nipkow@13145
  1189
done
wenzelm@13114
  1190
nipkow@14802
  1191
lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@14802
  1192
apply(induct xs)
nipkow@14802
  1193
 apply simp
nipkow@14802
  1194
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  1195
done
nipkow@14802
  1196
wenzelm@13142
  1197
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
paulson@14208
  1198
apply (induct n, auto)
paulson@14208
  1199
apply (case_tac xs, auto)
nipkow@13145
  1200
done
wenzelm@13114
  1201
nipkow@15110
  1202
lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@15110
  1203
apply(induct xs)
nipkow@15110
  1204
 apply simp
nipkow@15110
  1205
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  1206
done
nipkow@15110
  1207
nipkow@15110
  1208
lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"
nipkow@15110
  1209
apply(induct xs)
nipkow@15110
  1210
apply simp
nipkow@15110
  1211
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1212
done
nipkow@15110
  1213
wenzelm@13114
  1214
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
paulson@14208
  1215
apply (induct n, auto)
paulson@14208
  1216
apply (case_tac xs, auto)
nipkow@13145
  1217
done
wenzelm@13114
  1218
wenzelm@13142
  1219
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
paulson@14208
  1220
apply (induct n, auto)
paulson@14208
  1221
apply (case_tac xs, auto)
nipkow@13145
  1222
done
wenzelm@13114
  1223
wenzelm@13114
  1224
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
paulson@14208
  1225
apply (induct xs, auto)
paulson@14208
  1226
apply (case_tac i, auto)
nipkow@13145
  1227
done
wenzelm@13114
  1228
wenzelm@13114
  1229
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
paulson@14208
  1230
apply (induct xs, auto)
paulson@14208
  1231
apply (case_tac i, auto)
nipkow@13145
  1232
done
wenzelm@13114
  1233
wenzelm@13142
  1234
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
paulson@14208
  1235
apply (induct xs, auto)
paulson@14208
  1236
apply (case_tac n, blast)
paulson@14208
  1237
apply (case_tac i, auto)
nipkow@13145
  1238
done
wenzelm@13114
  1239
wenzelm@13142
  1240
lemma nth_drop [simp]:
nipkow@13145
  1241
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
paulson@14208
  1242
apply (induct n, auto)
paulson@14208
  1243
apply (case_tac xs, auto)
nipkow@13145
  1244
done
nipkow@3507
  1245
nipkow@18423
  1246
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
nipkow@18423
  1247
by(simp add: hd_conv_nth)
nipkow@18423
  1248
nipkow@14025
  1249
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
nipkow@14025
  1250
by(induct xs)(auto simp:take_Cons split:nat.split)
nipkow@14025
  1251
nipkow@14025
  1252
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
nipkow@14025
  1253
by(induct xs)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1254
nipkow@14187
  1255
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1256
using set_take_subset by fast
nipkow@14187
  1257
nipkow@14187
  1258
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1259
using set_drop_subset by fast
nipkow@14187
  1260
wenzelm@13114
  1261
lemma append_eq_conv_conj:
nipkow@13145
  1262
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
paulson@14208
  1263
apply (induct xs, simp, clarsimp)
paulson@14208
  1264
apply (case_tac zs, auto)
nipkow@13145
  1265
done
wenzelm@13142
  1266
paulson@14050
  1267
lemma take_add [rule_format]: 
paulson@14050
  1268
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
paulson@14050
  1269
apply (induct xs, auto) 
paulson@14050
  1270
apply (case_tac i, simp_all) 
paulson@14050
  1271
done
paulson@14050
  1272
nipkow@14300
  1273
lemma append_eq_append_conv_if:
nipkow@14300
  1274
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1275
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1276
   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
  1277
   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
  1278
apply(induct xs\<^isub>1)
nipkow@14300
  1279
 apply simp
nipkow@14300
  1280
apply(case_tac ys\<^isub>1)
nipkow@14300
  1281
apply simp_all
nipkow@14300
  1282
done
nipkow@14300
  1283
nipkow@15110
  1284
lemma take_hd_drop:
nipkow@15110
  1285
  "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
nipkow@15110
  1286
apply(induct xs)
nipkow@15110
  1287
apply simp
nipkow@15110
  1288
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1289
done
nipkow@15110
  1290
nipkow@17501
  1291
lemma id_take_nth_drop:
nipkow@17501
  1292
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
nipkow@17501
  1293
proof -
nipkow@17501
  1294
  assume si: "i < length xs"
nipkow@17501
  1295
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
nipkow@17501
  1296
  moreover
nipkow@17501
  1297
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
nipkow@17501
  1298
    apply (rule_tac take_Suc_conv_app_nth) by arith
nipkow@17501
  1299
  ultimately show ?thesis by auto
nipkow@17501
  1300
qed
nipkow@17501
  1301
  
nipkow@17501
  1302
lemma upd_conv_take_nth_drop:
nipkow@17501
  1303
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1304
proof -
nipkow@17501
  1305
  assume i: "i < length xs"
nipkow@17501
  1306
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
nipkow@17501
  1307
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
nipkow@17501
  1308
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1309
    using i by (simp add: list_update_append)
nipkow@17501
  1310
  finally show ?thesis .
nipkow@17501
  1311
qed
nipkow@17501
  1312
wenzelm@13114
  1313
nipkow@15392
  1314
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1315
wenzelm@13142
  1316
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1317
by (induct xs) auto
wenzelm@13114
  1318
wenzelm@13142
  1319
lemma takeWhile_append1 [simp]:
nipkow@13145
  1320
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1321
by (induct xs) auto
wenzelm@13114
  1322
wenzelm@13142
  1323
lemma takeWhile_append2 [simp]:
nipkow@13145
  1324
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1325
by (induct xs) auto
wenzelm@13114
  1326
wenzelm@13142
  1327
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1328
by (induct xs) auto
wenzelm@13114
  1329
wenzelm@13142
  1330
lemma dropWhile_append1 [simp]:
nipkow@13145
  1331
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1332
by (induct xs) auto
wenzelm@13114
  1333
wenzelm@13142
  1334
lemma dropWhile_append2 [simp]:
nipkow@13145
  1335
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1336
by (induct xs) auto
wenzelm@13114
  1337
wenzelm@13142
  1338
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1339
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1340
nipkow@13913
  1341
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1342
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1343
by(induct xs, auto)
nipkow@13913
  1344
nipkow@13913
  1345
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1346
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1347
by(induct xs, auto)
nipkow@13913
  1348
nipkow@13913
  1349
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1350
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1351
by(induct xs, auto)
nipkow@13913
  1352
nipkow@17501
  1353
text{* The following two lemmmas could be generalized to an arbitrary
nipkow@17501
  1354
property. *}
nipkow@17501
  1355
nipkow@17501
  1356
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1357
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
nipkow@17501
  1358
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
nipkow@17501
  1359
nipkow@17501
  1360
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1361
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
nipkow@17501
  1362
apply(induct xs)
nipkow@17501
  1363
 apply simp
nipkow@17501
  1364
apply auto
nipkow@17501
  1365
apply(subst dropWhile_append2)
nipkow@17501
  1366
apply auto
nipkow@17501
  1367
done
nipkow@17501
  1368
nipkow@18423
  1369
lemma takeWhile_not_last:
nipkow@18423
  1370
 "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
nipkow@18423
  1371
apply(induct xs)
nipkow@18423
  1372
 apply simp
nipkow@18423
  1373
apply(case_tac xs)
nipkow@18423
  1374
apply(auto)
nipkow@18423
  1375
done
nipkow@18423
  1376
krauss@19770
  1377
lemma takeWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1378
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1379
  ==> takeWhile P l = takeWhile Q k"
wenzelm@20503
  1380
  by (induct k arbitrary: l) (simp_all)
krauss@18336
  1381
krauss@19770
  1382
lemma dropWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1383
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1384
  ==> dropWhile P l = dropWhile Q k"
wenzelm@20503
  1385
  by (induct k arbitrary: l, simp_all)
krauss@18336
  1386
wenzelm@13114
  1387
nipkow@15392
  1388
subsubsection {* @{text zip} *}
wenzelm@13114
  1389
wenzelm@13142
  1390
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1391
by (induct ys) auto
wenzelm@13114
  1392
wenzelm@13142
  1393
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1394
by simp
wenzelm@13114
  1395
wenzelm@13142
  1396
declare zip_Cons [simp del]
wenzelm@13114
  1397
nipkow@15281
  1398
lemma zip_Cons1:
nipkow@15281
  1399
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  1400
by(auto split:list.split)
nipkow@15281
  1401
wenzelm@13142
  1402
lemma length_zip [simp]:
krauss@22493
  1403
"length (zip xs ys) = min (length xs) (length ys)"
krauss@22493
  1404
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1405
wenzelm@13114
  1406
lemma zip_append1:
krauss@22493
  1407
"zip (xs @ ys) zs =
nipkow@13145
  1408
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
krauss@22493
  1409
by (induct xs zs rule:list_induct2') auto
wenzelm@13114
  1410
wenzelm@13114
  1411
lemma zip_append2:
krauss@22493
  1412
"zip xs (ys @ zs) =
nipkow@13145
  1413
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
krauss@22493
  1414
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1415
wenzelm@13142
  1416
lemma zip_append [simp]:
wenzelm@13142
  1417
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1418
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1419
by (simp add: zip_append1)
wenzelm@13114
  1420
wenzelm@13114
  1421
lemma zip_rev:
nipkow@14247
  1422
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1423
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1424
nipkow@23096
  1425
lemma map_zip_map:
nipkow@23096
  1426
 "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
nipkow@23096
  1427
apply(induct xs arbitrary:ys) apply simp
nipkow@23096
  1428
apply(case_tac ys)
nipkow@23096
  1429
apply simp_all
nipkow@23096
  1430
done
nipkow@23096
  1431
nipkow@23096
  1432
lemma map_zip_map2:
nipkow@23096
  1433
 "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
nipkow@23096
  1434
apply(induct xs arbitrary:ys) apply simp
nipkow@23096
  1435
apply(case_tac ys)
nipkow@23096
  1436
apply simp_all
nipkow@23096
  1437
done
nipkow@23096
  1438
wenzelm@13142
  1439
lemma nth_zip [simp]:
nipkow@13145
  1440
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
paulson@14208
  1441
apply (induct ys, simp)
nipkow@13145
  1442
apply (case_tac xs)
nipkow@13145
  1443
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1444
done
wenzelm@13114
  1445
wenzelm@13114
  1446
lemma set_zip:
nipkow@13145
  1447
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
  1448
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1449
wenzelm@13114
  1450
lemma zip_update:
nipkow@13145
  1451
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
  1452
by (rule sym, simp add: update_zip)
wenzelm@13114
  1453
wenzelm@13142
  1454
lemma zip_replicate [simp]:
nipkow@13145
  1455
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
paulson@14208
  1456
apply (induct i, auto)
paulson@14208
  1457
apply (case_tac j, auto)
nipkow@13145
  1458
done
wenzelm@13114
  1459
nipkow@19487
  1460
lemma take_zip:
nipkow@19487
  1461
 "!!xs ys. take n (zip xs ys) = zip (take n xs) (take n ys)"
nipkow@19487
  1462
apply (induct n)
nipkow@19487
  1463
 apply simp
nipkow@19487
  1464
apply (case_tac xs, simp)
nipkow@19487
  1465
apply (case_tac ys, simp_all)
nipkow@19487
  1466
done
nipkow@19487
  1467
nipkow@19487
  1468
lemma drop_zip:
nipkow@19487
  1469
 "!!xs ys. drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
nipkow@19487
  1470
apply (induct n)
nipkow@19487
  1471
 apply simp
nipkow@19487
  1472
apply (case_tac xs, simp)
nipkow@19487
  1473
apply (case_tac ys, simp_all)
nipkow@19487
  1474
done
nipkow@19487
  1475
krauss@22493
  1476
lemma set_zip_leftD:
krauss@22493
  1477
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
krauss@22493
  1478
by (induct xs ys rule:list_induct2') auto
krauss@22493
  1479
krauss@22493
  1480
lemma set_zip_rightD:
krauss@22493
  1481
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
krauss@22493
  1482
by (induct xs ys rule:list_induct2') auto
wenzelm@13142
  1483
nipkow@15392
  1484
subsubsection {* @{text list_all2} *}
wenzelm@13114
  1485
kleing@14316
  1486
lemma list_all2_lengthD [intro?]: 
kleing@14316
  1487
  "list_all2 P xs ys ==> length xs = length ys"
haftmann@19607
  1488
  by (simp add: list_all2_def)
haftmann@19607
  1489
haftmann@19787
  1490
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
haftmann@19607
  1491
  by (simp add: list_all2_def)
haftmann@19607
  1492
haftmann@19787
  1493
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
haftmann@19787
  1494
  by (simp add: list_all2_def)
haftmann@19607
  1495
haftmann@19607
  1496
lemma list_all2_Cons [iff, code]:
haftmann@19607
  1497
  "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
haftmann@19607
  1498
  by (auto simp add: list_all2_def)
wenzelm@13114
  1499
wenzelm@13114
  1500
lemma list_all2_Cons1:
nipkow@13145
  1501
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1502
by (cases ys) auto
wenzelm@13114
  1503
wenzelm@13114
  1504
lemma list_all2_Cons2:
nipkow@13145
  1505
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1506
by (cases xs) auto
wenzelm@13114
  1507
wenzelm@13142
  1508
lemma list_all2_rev [iff]:
nipkow@13145
  1509
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1510
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1511
kleing@13863
  1512
lemma list_all2_rev1:
kleing@13863
  1513
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1514
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1515
wenzelm@13114
  1516
lemma list_all2_append1:
nipkow@13145
  1517
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1518
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1519
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1520
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1521
apply (rule iffI)
nipkow@13145
  1522
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1523
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  1524
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1525
apply (simp add: ball_Un)
nipkow@13145
  1526
done
wenzelm@13114
  1527
wenzelm@13114
  1528
lemma list_all2_append2:
nipkow@13145
  1529
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1530
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1531
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1532
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1533
apply (rule iffI)
nipkow@13145
  1534
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1535
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  1536
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1537
apply (simp add: ball_Un)
nipkow@13145
  1538
done
wenzelm@13114
  1539
kleing@13863
  1540
lemma list_all2_append:
nipkow@14247
  1541
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  1542
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  1543
by (induct rule:list_induct2, simp_all)
kleing@13863
  1544
kleing@13863
  1545
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1546
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
kleing@13863
  1547
  by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1548
wenzelm@13114
  1549
lemma list_all2_conv_all_nth:
nipkow@13145
  1550
"list_all2 P xs ys =
nipkow@13145
  1551
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1552
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1553
berghofe@13883
  1554
lemma list_all2_trans:
berghofe@13883
  1555
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1556
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1557
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1558
proof (induct as)
berghofe@13883
  1559
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1560
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1561
  proof (induct bs)
berghofe@13883
  1562
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1563
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1564
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1565
  qed simp
berghofe@13883
  1566
qed simp
berghofe@13883
  1567
kleing@13863
  1568
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1569
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
kleing@13863
  1570
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1571
paulson@14395
  1572
lemma list_all2I:
paulson@14395
  1573
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
paulson@14395
  1574
  by (simp add: list_all2_def)
paulson@14395
  1575
kleing@14328
  1576
lemma list_all2_nthD:
kleing@13863
  1577
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
kleing@13863
  1578
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1579
nipkow@14302
  1580
lemma list_all2_nthD2:
nipkow@14302
  1581
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@14302
  1582
  by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  1583
kleing@13863
  1584
lemma list_all2_map1: 
kleing@13863
  1585
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
kleing@13863
  1586
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1587
kleing@13863
  1588
lemma list_all2_map2: 
kleing@13863
  1589
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
kleing@13863
  1590
  by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1591
kleing@14316
  1592
lemma list_all2_refl [intro?]:
kleing@13863
  1593
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
kleing@13863
  1594
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1595
kleing@13863
  1596
lemma list_all2_update_cong:
kleing@13863
  1597
  "\<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
  1598
  by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1599
kleing@13863
  1600
lemma list_all2_update_cong2:
kleing@13863
  1601
  "\<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
  1602
  by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1603
nipkow@14302
  1604
lemma list_all2_takeI [simp,intro?]:
nipkow@14302
  1605
  "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@14302
  1606
  apply (induct xs)
nipkow@14302
  1607
   apply simp
nipkow@14302
  1608
  apply (clarsimp simp add: list_all2_Cons1)
nipkow@14302
  1609
  apply (case_tac n)
nipkow@14302
  1610
  apply auto
nipkow@14302
  1611
  done
nipkow@14302
  1612
nipkow@14302
  1613
lemma list_all2_dropI [simp,intro?]:
kleing@13863
  1614
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
paulson@14208
  1615
  apply (induct as, simp)
kleing@13863
  1616
  apply (clarsimp simp add: list_all2_Cons1)
paulson@14208
  1617
  apply (case_tac n, simp, simp)
kleing@13863
  1618
  done
kleing@13863
  1619
kleing@14327
  1620
lemma list_all2_mono [intro?]:
kleing@13863
  1621
  "\<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
  1622
  apply (induct x, simp)
paulson@14208
  1623
  apply (case_tac y, auto)
kleing@13863
  1624
  done
kleing@13863
  1625
haftmann@22551
  1626
lemma list_all2_eq:
haftmann@22551
  1627
  "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
haftmann@22551
  1628
  by (induct xs ys rule: list_induct2') auto
haftmann@22551
  1629
wenzelm@13142
  1630
nipkow@15392
  1631
subsubsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  1632
wenzelm@13142
  1633
lemma foldl_append [simp]:
nipkow@13145
  1634
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145
  1635
by (induct xs) auto
wenzelm@13142
  1636
nipkow@14402
  1637
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  1638
by (induct xs) auto
nipkow@14402
  1639
nipkow@23096
  1640
lemma foldr_map: "foldr g (map f xs) a = foldr (g o f) xs a"
nipkow@23096
  1641
by(induct xs) simp_all
nipkow@23096
  1642
nipkow@23096
  1643
lemma foldl_map: "foldl g a (map f xs) = foldl (%a x. g a (f x)) a xs"
nipkow@23096
  1644
by(induct xs arbitrary:a) simp_all
nipkow@23096
  1645
krauss@19770
  1646
lemma foldl_cong [fundef_cong, recdef_cong]:
krauss@18336
  1647
  "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] 
krauss@18336
  1648
  ==> foldl f a l = foldl g b k"
wenzelm@20503
  1649
  by (induct k arbitrary: a b l) simp_all
krauss@18336
  1650
krauss@19770
  1651
lemma foldr_cong [fundef_cong, recdef_cong]:
krauss@18336
  1652
  "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] 
krauss@18336
  1653
  ==> foldr f l a = foldr g k b"
wenzelm@20503
  1654
  by (induct k arbitrary: a b l) simp_all
krauss@18336
  1655
nipkow@23096
  1656
text{* The ``First Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096
  1657
nipkow@23096
  1658
lemma foldl_foldr1_lemma:
nipkow@23096
  1659
 "foldl op + a xs = a + foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096
  1660
by (induct xs arbitrary: a) (auto simp:add_assoc)
nipkow@23096
  1661
nipkow@23096
  1662
corollary foldl_foldr1:
nipkow@23096
  1663
 "foldl op + 0 xs = foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096
  1664
by (simp add:foldl_foldr1_lemma)
nipkow@23096
  1665
nipkow@23096
  1666
nipkow@23096
  1667
text{* The ``Third Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096
  1668
nipkow@14402
  1669
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  1670
by (induct xs) auto
nipkow@14402
  1671
nipkow@14402
  1672
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  1673
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  1674
wenzelm@13142
  1675
text {*
nipkow@13145
  1676
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1677
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1678
*}
wenzelm@13142
  1679
wenzelm@13142
  1680
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
nipkow@13145
  1681
by (induct ns) auto
wenzelm@13142
  1682
wenzelm@13142
  1683
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1684
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1685
wenzelm@13142
  1686
lemma sum_eq_0_conv [iff]:
nipkow@13145
  1687
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145
  1688
by (induct ns) auto
wenzelm@13114
  1689
nipkow@23096
  1690
subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
nipkow@23096
  1691
nipkow@23096
  1692
lemma listsum_foldr:
nipkow@23096
  1693
 "listsum xs = foldr (op +) xs 0"
nipkow@23096
  1694
by(induct xs) auto
nipkow@23096
  1695
nipkow@23096
  1696
(* for efficient code generation *)
nipkow@23096
  1697
lemma listsum[code]: "listsum xs = foldl (op +) 0 xs"
nipkow@23096
  1698
by(simp add:listsum_foldr foldl_foldr1)
nipkow@23096
  1699
nipkow@23096
  1700
text{* Some syntactic sugar for summing a function over a list: *}
nipkow@23096
  1701
nipkow@23096
  1702
syntax
nipkow@23096
  1703
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
nipkow@23096
  1704
syntax (xsymbols)
nipkow@23096
  1705
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096
  1706
syntax (HTML output)
nipkow@23096
  1707
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096
  1708
nipkow@23096
  1709
translations -- {* Beware of argument permutation! *}
nipkow@23096
  1710
  "SUM x<-xs. b" == "CONST listsum (map (%x. b) xs)"
nipkow@23096
  1711
  "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (map (%x. b) xs)"
nipkow@23096
  1712
nipkow@23096
  1713
lemma listsum_0 [simp]: "(\<Sum>x\<leftarrow>xs. 0) = 0"
nipkow@23096
  1714
by (induct xs) simp_all
nipkow@23096
  1715
nipkow@23096
  1716
text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
nipkow@23096
  1717
lemma uminus_listsum_map:
nipkow@23096
  1718
 "- listsum (map f xs) = (listsum (map (uminus o f) xs) :: 'a::ab_group_add)"
nipkow@23096
  1719
by(induct xs) simp_all
nipkow@23096
  1720
wenzelm@13114
  1721
nipkow@15392
  1722
subsubsection {* @{text upto} *}
wenzelm@13114
  1723
nipkow@17090
  1724
lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
nipkow@17090
  1725
-- {* simp does not terminate! *}
nipkow@13145
  1726
by (induct j) auto
wenzelm@13142
  1727
nipkow@15425
  1728
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
nipkow@13145
  1729
by (subst upt_rec) simp
wenzelm@13114
  1730
nipkow@15425
  1731
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
nipkow@15281
  1732
by(induct j)simp_all
nipkow@15281
  1733
nipkow@15281
  1734
lemma upt_eq_Cons_conv:
nipkow@15425
  1735
 "!!x xs. ([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
nipkow@15281
  1736
apply(induct j)
nipkow@15281
  1737
 apply simp
nipkow@15281
  1738
apply(clarsimp simp add: append_eq_Cons_conv)
nipkow@15281
  1739
apply arith
nipkow@15281
  1740
done
nipkow@15281
  1741
nipkow@15425
  1742
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
nipkow@13145
  1743
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1744
by simp
wenzelm@13114
  1745
nipkow@15425
  1746
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
nipkow@13145
  1747
apply(rule trans)
nipkow@13145
  1748
apply(subst upt_rec)
paulson@14208
  1749
 prefer 2 apply (rule refl, simp)
nipkow@13145
  1750
done
wenzelm@13114
  1751
nipkow@15425
  1752
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
nipkow@13145
  1753
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1754
by (induct k) auto
wenzelm@13114
  1755
nipkow@15425
  1756
lemma length_upt [simp]: "length [i..<j] = j - i"
nipkow@13145
  1757
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1758
nipkow@15425
  1759
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
nipkow@13145
  1760
apply (induct j)
nipkow@13145
  1761
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1762
done
wenzelm@13114
  1763
nipkow@17906
  1764
nipkow@17906
  1765
lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
nipkow@17906
  1766
by(simp add:upt_conv_Cons)
nipkow@17906
  1767
nipkow@17906
  1768
lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
nipkow@17906
  1769
apply(cases j)
nipkow@17906
  1770
 apply simp
nipkow@17906
  1771
by(simp add:upt_Suc_append)
nipkow@17906
  1772
nipkow@15425
  1773
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..<n] = [i..<i+m]"
paulson@14208
  1774
apply (induct m, simp)
nipkow@13145
  1775
apply (subst upt_rec)
nipkow@13145
  1776
apply (rule sym)
nipkow@13145
  1777
apply (subst upt_rec)
nipkow@13145
  1778
apply (simp del: upt.simps)
nipkow@13145
  1779
done
nipkow@3507
  1780
nipkow@17501
  1781
lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
nipkow@17501
  1782
apply(induct j)
nipkow@17501
  1783
apply auto
nipkow@17501
  1784
done
nipkow@17501
  1785
nipkow@15425
  1786
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..n]"
nipkow@13145
  1787
by (induct n) auto
wenzelm@13114
  1788
nipkow@15425
  1789
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
nipkow@13145
  1790
apply (induct n m rule: diff_induct)
nipkow@13145
  1791
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1792
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1793
done
wenzelm@13114
  1794
berghofe@13883
  1795
lemma nth_take_lemma:
berghofe@13883
  1796
  "!!xs ys. k <= length xs ==> k <= length ys ==>
berghofe@13883
  1797
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
berghofe@13883
  1798
apply (atomize, induct k)
paulson@14208
  1799
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  1800
txt {* Both lists must be non-empty *}
paulson@14208
  1801
apply (case_tac xs, simp)
paulson@14208
  1802
apply (case_tac ys, clarify)
nipkow@13145
  1803
 apply (simp (no_asm_use))
nipkow@13145
  1804
apply clarify
nipkow@13145
  1805
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  1806
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  1807
apply blast
nipkow@13145
  1808
done
wenzelm@13114
  1809
wenzelm@13114
  1810
lemma nth_equalityI:
wenzelm@13114
  1811
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  1812
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  1813
apply (simp_all add: take_all)
nipkow@13145
  1814
done
wenzelm@13142
  1815
kleing@13863
  1816
(* needs nth_equalityI *)
kleing@13863
  1817
lemma list_all2_antisym:
kleing@13863
  1818
  "\<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
  1819
  \<Longrightarrow> xs = ys"
kleing@13863
  1820
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  1821
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  1822
  done
kleing@13863
  1823
wenzelm@13142
  1824
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  1825
-- {* The famous take-lemma. *}
nipkow@13145
  1826
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  1827
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  1828
done
wenzelm@13142
  1829
wenzelm@13142
  1830
nipkow@15302
  1831
lemma take_Cons':
nipkow@15302
  1832
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@15302
  1833
by (cases n) simp_all
nipkow@15302
  1834
nipkow@15302
  1835
lemma drop_Cons':
nipkow@15302
  1836
     "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@15302
  1837
by (cases n) simp_all
nipkow@15302
  1838
nipkow@15302
  1839
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@15302
  1840
by (cases n) simp_all
nipkow@15302
  1841
paulson@18622
  1842
lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard]
paulson@18622
  1843
lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard]
paulson@18622
  1844
lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard]
paulson@18622
  1845
paulson@18622
  1846
declare take_Cons_number_of [simp] 
paulson@18622
  1847
        drop_Cons_number_of [simp] 
paulson@18622
  1848
        nth_Cons_number_of [simp] 
nipkow@15302
  1849
nipkow@15302
  1850
nipkow@15392
  1851
subsubsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  1852
wenzelm@13142
  1853
lemma distinct_append [simp]:
nipkow@13145
  1854
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  1855
by (induct xs) auto
wenzelm@13142
  1856
nipkow@15305
  1857
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
nipkow@15305
  1858
by(induct xs) auto
nipkow@15305
  1859
wenzelm@13142
  1860
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  1861
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  1862
wenzelm@13142
  1863
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  1864
by (induct xs) auto
wenzelm@13142
  1865
paulson@15072
  1866
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
paulson@15251
  1867
  by (induct x, auto) 
paulson@15072
  1868
paulson@15072
  1869
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
paulson@15251
  1870
  by (induct x, auto)
paulson@15072
  1871
nipkow@15245
  1872
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245
  1873
by (induct xs) auto
nipkow@15245
  1874
nipkow@15245
  1875
lemma length_remdups_eq[iff]:
nipkow@15245
  1876
  "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245
  1877
apply(induct xs)
nipkow@15245
  1878
 apply auto
nipkow@15245
  1879
apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245
  1880
 apply arith
nipkow@15245
  1881
apply(rule length_remdups_leq)
nipkow@15245
  1882
done
nipkow@15245
  1883
nipkow@18490
  1884
nipkow@18490
  1885
lemma distinct_map:
nipkow@18490
  1886
  "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
nipkow@18490
  1887
by (induct xs) auto
nipkow@18490
  1888
nipkow@18490
  1889
wenzelm@13142
  1890
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  1891
by (induct xs) auto
wenzelm@13114
  1892
nipkow@17501
  1893
lemma distinct_upt[simp]: "distinct[i..<j]"
nipkow@17501
  1894
by (induct j) auto
nipkow@17501
  1895
nipkow@17501
  1896
lemma distinct_take[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (take i xs)"
nipkow@17501
  1897
apply(induct xs)
nipkow@17501
  1898
 apply simp
nipkow@17501
  1899
apply (case_tac i)
nipkow@17501
  1900
 apply simp_all
nipkow@17501
  1901
apply(blast dest:in_set_takeD)
nipkow@17501
  1902
done
nipkow@17501
  1903
nipkow@17501
  1904
lemma distinct_drop[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (drop i xs)"
nipkow@17501
  1905
apply(induct xs)
nipkow@17501
  1906
 apply simp
nipkow@17501
  1907
apply (case_tac i)
nipkow@17501
  1908
 apply simp_all
nipkow@17501
  1909
done
nipkow@17501
  1910
nipkow@17501
  1911
lemma distinct_list_update:
nipkow@17501
  1912
assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
nipkow@17501
  1913
shows "distinct (xs[i:=a])"
nipkow@17501
  1914
proof (cases "i < length xs")
nipkow@17501
  1915
  case True
nipkow@17501
  1916
  with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
nipkow@17501
  1917
    apply (drule_tac id_take_nth_drop) by simp
nipkow@17501
  1918
  with d True show ?thesis
nipkow@17501
  1919
    apply (simp add: upd_conv_take_nth_drop)
nipkow@17501
  1920
    apply (drule subst [OF id_take_nth_drop]) apply assumption
nipkow@17501
  1921
    apply simp apply (cases "a = xs!i") apply simp by blast
nipkow@17501
  1922
next
nipkow@17501
  1923
  case False with d show ?thesis by auto
nipkow@17501
  1924
qed
nipkow@17501
  1925
nipkow@17501
  1926
nipkow@17501
  1927
text {* It is best to avoid this indexed version of distinct, but
nipkow@17501
  1928
sometimes it is useful. *}
nipkow@17501
  1929
wenzelm@13142
  1930
lemma distinct_conv_nth:
nipkow@17501
  1931
"distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@15251
  1932
apply (induct xs, simp, simp)
paulson@14208
  1933
apply (rule iffI, clarsimp)
nipkow@13145
  1934
 apply (case_tac i)
paulson@14208
  1935
apply (case_tac j, simp)
nipkow@13145
  1936
apply (simp add: set_conv_nth)
nipkow@13145
  1937
 apply (case_tac j)
paulson@14208
  1938
apply (clarsimp simp add: set_conv_nth, simp)
nipkow@13145
  1939
apply (rule conjI)
nipkow@13145
  1940
 apply (clarsimp simp add: set_conv_nth)
nipkow@17501
  1941
 apply (erule_tac x = 0 in allE, simp)
paulson@14208
  1942
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
nipkow@17501
  1943
apply (erule_tac x = "Suc i" in allE, simp)
paulson@14208
  1944
apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145
  1945
done
wenzelm@13114
  1946
nipkow@18490
  1947
lemma nth_eq_iff_index_eq:
nipkow@18490
  1948
 "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
nipkow@18490
  1949
by(auto simp: distinct_conv_nth)
nipkow@18490
  1950
nipkow@15110
  1951
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
kleing@14388
  1952
  by (induct xs) auto
kleing@14388
  1953
nipkow@15110
  1954
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388
  1955
proof (induct xs)
kleing@14388
  1956
  case Nil thus ?case by simp
kleing@14388
  1957
next
kleing@14388
  1958
  case (Cons x xs)
kleing@14388
  1959
  show ?case
kleing@14388
  1960
  proof (cases "x \<in> set xs")
kleing@14388
  1961
    case False with Cons show ?thesis by simp
kleing@14388
  1962
  next
kleing@14388
  1963
    case True with Cons.prems
kleing@14388
  1964
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  1965
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  1966
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  1967
    ultimately have False by simp
kleing@14388
  1968
    thus ?thesis ..
kleing@14388
  1969
  qed
kleing@14388
  1970
qed
kleing@14388
  1971
nipkow@18490
  1972
nipkow@18490
  1973
lemma length_remdups_concat:
nipkow@18490
  1974
 "length(remdups(concat xss)) = card(\<Union>xs \<in> set xss. set xs)"
nipkow@18490
  1975
by(simp add: distinct_card[symmetric])
nipkow@17906
  1976
nipkow@17906
  1977
nipkow@15392
  1978
subsubsection {* @{text remove1} *}
nipkow@15110
  1979
nipkow@18049
  1980
lemma remove1_append:
nipkow@18049
  1981
  "remove1 x (xs @ ys) =
nipkow@18049
  1982
  (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
nipkow@18049
  1983
by (induct xs) auto
nipkow@18049
  1984
nipkow@15110
  1985
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110
  1986
apply(induct xs)
nipkow@15110
  1987
 apply simp
nipkow@15110
  1988
apply simp
nipkow@15110
  1989
apply blast
nipkow@15110
  1990
done
nipkow@15110
  1991
paulson@17724
  1992
lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110
  1993
apply(induct xs)
nipkow@15110
  1994
 apply simp
nipkow@15110
  1995
apply simp
nipkow@15110
  1996
apply blast
nipkow@15110
  1997
done
nipkow@15110
  1998
nipkow@18049
  1999
lemma remove1_filter_not[simp]:
nipkow@18049
  2000
  "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
nipkow@18049
  2001
by(induct xs) auto
nipkow@18049
  2002
nipkow@15110
  2003
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110
  2004
apply(insert set_remove1_subset)
nipkow@15110
  2005
apply fast
nipkow@15110
  2006
done
nipkow@15110
  2007
nipkow@15110
  2008
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110
  2009
by (induct xs) simp_all
nipkow@15110
  2010
wenzelm@13114
  2011
nipkow@15392
  2012
subsubsection {* @{text replicate} *}
wenzelm@13114
  2013
wenzelm@13142
  2014
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  2015
by (induct n) auto
nipkow@13124
  2016
wenzelm@13142
  2017
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  2018
by (induct n) auto
wenzelm@13114
  2019
wenzelm@13114
  2020
lemma replicate_app_Cons_same:
nipkow@13145
  2021
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  2022
by (induct n) auto
wenzelm@13114
  2023
wenzelm@13142
  2024
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208
  2025
apply (induct n, simp)
nipkow@13145
  2026
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  2027
done
wenzelm@13114
  2028
wenzelm@13142
  2029
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  2030
by (induct n) auto
wenzelm@13114
  2031
nipkow@16397
  2032
text{* Courtesy of Matthias Daum: *}
nipkow@16397
  2033
lemma append_replicate_commute:
nipkow@16397
  2034
  "replicate n x @ replicate k x = replicate k x @ replicate n x"
nipkow@16397
  2035
apply (simp add: replicate_add [THEN sym])
nipkow@16397
  2036
apply (simp add: add_commute)
nipkow@16397
  2037
done
nipkow@16397
  2038
wenzelm@13142
  2039
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  2040
by (induct n) auto
wenzelm@13114
  2041
wenzelm@13142
  2042
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  2043
by (induct n) auto
wenzelm@13114
  2044
wenzelm@13142
  2045
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  2046
by (atomize (full), induct n) auto
wenzelm@13114
  2047
wenzelm@13142
  2048
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
paulson@14208
  2049
apply (induct n, simp)
nipkow@13145
  2050
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  2051
done
wenzelm@13114
  2052
nipkow@16397
  2053
text{* Courtesy of Matthias Daum (2 lemmas): *}
nipkow@16397
  2054
lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
nipkow@16397
  2055
apply (case_tac "k \<le> i")
nipkow@16397
  2056
 apply  (simp add: min_def)
nipkow@16397
  2057
apply (drule not_leE)
nipkow@16397
  2058
apply (simp add: min_def)
nipkow@16397
  2059
apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
nipkow@16397
  2060
 apply  simp
nipkow@16397
  2061
apply (simp add: replicate_add [symmetric])
nipkow@16397
  2062
done
nipkow@16397
  2063
nipkow@16397
  2064
lemma drop_replicate[simp]: "!!i. drop i (replicate k x) = replicate (k-i) x"
nipkow@16397
  2065
apply (induct k)
nipkow@16397
  2066
 apply simp
nipkow@16397
  2067
apply clarsimp
nipkow@16397
  2068
apply (case_tac i)
nipkow@16397
  2069
 apply simp
nipkow@16397
  2070
apply clarsimp
nipkow@16397
  2071
done
nipkow@16397
  2072
nipkow@16397
  2073
wenzelm@13142
  2074
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  2075
by (induct n) auto
wenzelm@13114
  2076
wenzelm@13142
  2077
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  2078
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  2079
wenzelm@13142
  2080
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  2081
by auto
wenzelm@13114
  2082
wenzelm@13142
  2083
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  2084
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  2085
wenzelm@13114
  2086
nipkow@15392
  2087
subsubsection{*@{text rotate1} and @{text rotate}*}
nipkow@15302
  2088
nipkow@15302
  2089
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
nipkow@15302
  2090
by(simp add:rotate1_def)
nipkow@15302
  2091
nipkow@15302
  2092
lemma rotate0[simp]: "rotate 0 = id"
nipkow@15302
  2093
by(simp add:rotate_def)
nipkow@15302
  2094
nipkow@15302
  2095
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
nipkow@15302
  2096
by(simp add:rotate_def)
nipkow@15302
  2097
nipkow@15302
  2098
lemma rotate_add:
nipkow@15302
  2099
  "rotate (m+n) = rotate m o rotate n"
nipkow@15302
  2100
by(simp add:rotate_def funpow_add)
nipkow@15302
  2101
nipkow@15302
  2102
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
nipkow@15302
  2103
by(simp add:rotate_add)
nipkow@15302
  2104
nipkow@18049
  2105
lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
nipkow@18049
  2106
by(simp add:rotate_def funpow_swap1)
nipkow@18049
  2107
nipkow@15302
  2108
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
nipkow@15302
  2109
by(cases xs) simp_all
nipkow@15302
  2110
nipkow@15302
  2111
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2112
apply(induct n)
nipkow@15302
  2113
 apply simp
nipkow@15302
  2114
apply (simp add:rotate_def)
nipkow@13145
  2115
done
wenzelm@13114
  2116
nipkow@15302
  2117
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
nipkow@15302
  2118
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2119
nipkow@15302
  2120
lemma rotate_drop_take:
nipkow@15302
  2121
  "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
nipkow@15302
  2122
apply(induct n)
nipkow@15302
  2123
 apply simp
nipkow@15302
  2124
apply(simp add:rotate_def)
nipkow@15302
  2125
apply(cases "xs = []")
nipkow@15302
  2126
 apply (simp)
nipkow@15302
  2127
apply(case_tac "n mod length xs = 0")
nipkow@15302
  2128
 apply(simp add:mod_Suc)
nipkow@15302
  2129
 apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
nipkow@15302
  2130
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
nipkow@15302
  2131
                take_hd_drop linorder_not_le)
nipkow@13145
  2132
done
wenzelm@13114
  2133
nipkow@15302
  2134
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
nipkow@15302
  2135
by(simp add:rotate_drop_take)
nipkow@15302
  2136
nipkow@15302
  2137
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2138
by(simp add:rotate_drop_take)
nipkow@15302
  2139
nipkow@15302
  2140
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
nipkow@15302
  2141
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2142
nipkow@15302
  2143
lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs"
nipkow@15302
  2144
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2145
nipkow@15302
  2146
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
nipkow@15302
  2147
by(simp add:rotate1_def split:list.split) blast
nipkow@15302
  2148
nipkow@15302
  2149
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
nipkow@15302
  2150
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2151
nipkow@15302
  2152
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
nipkow@15302
  2153
by(simp add:rotate_drop_take take_map drop_map)
nipkow@15302
  2154
nipkow@15302
  2155
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
nipkow@15302
  2156
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2157
nipkow@15302
  2158
lemma set_rotate[simp]: "set(rotate n xs) = set xs"
nipkow@15302
  2159
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2160
nipkow@15302
  2161
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
nipkow@15302
  2162
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2163
nipkow@15302
  2164
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
nipkow@15302
  2165
by (induct n) (simp_all add:rotate_def)
wenzelm@13114
  2166
nipkow@15439
  2167
lemma rotate_rev:
nipkow@15439
  2168
  "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
nipkow@15439
  2169
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2170
apply(cases "length xs = 0")
nipkow@15439
  2171
 apply simp
nipkow@15439
  2172
apply(cases "n mod length xs = 0")
nipkow@15439
  2173
 apply simp
nipkow@15439
  2174
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2175
done
nipkow@15439
  2176
nipkow@18423
  2177
lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
nipkow@18423
  2178
apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
nipkow@18423
  2179
apply(subgoal_tac "length xs \<noteq> 0")
nipkow@18423
  2180
 prefer 2 apply simp
nipkow@18423
  2181
using mod_less_divisor[of "length xs" n] by arith
nipkow@18423
  2182
wenzelm@13114
  2183
nipkow@15392
  2184
subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  2185
wenzelm@13142
  2186
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  2187
by (auto simp add: sublist_def)
wenzelm@13114
  2188
wenzelm@13142
  2189
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  2190
by (auto simp add: sublist_def)
wenzelm@13114
  2191
nipkow@15281
  2192
lemma length_sublist:
nipkow@15281
  2193
  "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
nipkow@15281
  2194
by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
nipkow@15281
  2195
nipkow@15281
  2196
lemma sublist_shift_lemma_Suc:
nipkow@15281
  2197
  "!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
nipkow@15281
  2198
         map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
nipkow@15281
  2199
apply(induct xs)
nipkow@15281
  2200
 apply simp
nipkow@15281
  2201
apply (case_tac "is")
nipkow@15281
  2202
 apply simp
nipkow@15281
  2203
apply simp
nipkow@15281
  2204
done
nipkow@15281
  2205
wenzelm@13114
  2206
lemma sublist_shift_lemma:
nipkow@15425
  2207
     "map fst [p:zip xs [i..<i + length xs] . snd p : A] =
nipkow@15425
  2208
      map fst [p:zip xs [0..<length xs] . snd p + i : A]"
nipkow@13145
  2209
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  2210
wenzelm@13114
  2211
lemma sublist_append:
paulson@15168
  2212
     "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  2213
apply (unfold sublist_def)
paulson@14208
  2214
apply (induct l' rule: rev_induct, simp)
nipkow@13145
  2215
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  2216
apply (simp add: add_commute)
nipkow@13145
  2217
done
wenzelm@13114
  2218
wenzelm@13114
  2219
lemma sublist_Cons:
nipkow@13145
  2220
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  2221
apply (induct l rule: rev_induct)
nipkow@13145
  2222
 apply (simp add: sublist_def)
nipkow@13145
  2223
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  2224
done
wenzelm@13114
  2225
nipkow@15281
  2226
lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
nipkow@15281
  2227
apply(induct xs)
nipkow@15281
  2228
 apply simp
nipkow@15281
  2229
apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE)
nipkow@15281
  2230
 apply(erule lessE)
nipkow@15281
  2231
  apply auto
nipkow@15281
  2232
apply(erule lessE)
nipkow@15281
  2233
apply auto
nipkow@15281
  2234
done
nipkow@15281
  2235
nipkow@15281
  2236
lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
nipkow@15281
  2237
by(auto simp add:set_sublist)
nipkow@15281
  2238
nipkow@15281
  2239
lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
nipkow@15281
  2240
by(auto simp add:set_sublist)
nipkow@15281
  2241
nipkow@15281
  2242
lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
nipkow@15281
  2243
by(auto simp add:set_sublist)
nipkow@15281
  2244
wenzelm@13142
  2245
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  2246
by (simp add: sublist_Cons)
wenzelm@13114
  2247
nipkow@15281
  2248
nipkow@15281
  2249
lemma distinct_sublistI[simp]: "!!I. distinct xs \<Longrightarrow> distinct(sublist xs I)"
nipkow@15281
  2250
apply(induct xs)
nipkow@15281
  2251
 apply simp
nipkow@15281
  2252
apply(auto simp add:sublist_Cons)
nipkow@15281
  2253
done
nipkow@15281
  2254
nipkow@15281
  2255
nipkow@15045
  2256
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
paulson@14208
  2257
apply (induct l rule: rev_induct, simp)
nipkow@13145
  2258
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  2259
done
wenzelm@13114
  2260
nipkow@17501
  2261
lemma filter_in_sublist: "\<And>s. distinct xs \<Longrightarrow>
nipkow@17501
  2262
  filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
nipkow@17501
  2263
proof (induct xs)
nipkow@17501
  2264
  case Nil thus ?case by simp
nipkow@17501
  2265
next
nipkow@17501
  2266
  case (Cons a xs)
nipkow@17501
  2267
  moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
nipkow@17501
  2268
  ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
nipkow@17501
  2269
qed
nipkow@17501
  2270
wenzelm@13114
  2271
nipkow@19390
  2272
subsubsection {* @{const splice} *}
nipkow@19390
  2273
haftmann@19607
  2274
lemma splice_Nil2 [simp, code]:
nipkow@19390
  2275
 "splice xs [] = xs"
nipkow@19390
  2276
by (cases xs) simp_all
nipkow@19390
  2277
haftmann@19607
  2278
lemma splice_Cons_Cons [simp, code]:
nipkow@19390
  2279
 "splice (x#xs) (y#ys) = x # y # splice xs ys"
nipkow@19390
  2280
by simp
nipkow@19390
  2281
haftmann@19607
  2282
declare splice.simps(2) [simp del, code del]
nipkow@19390
  2283
nipkow@22793
  2284
lemma length_splice[simp]: "!!ys. length(splice xs ys) = length xs + length ys"
nipkow@22793
  2285
apply(induct xs) apply simp
nipkow@22793
  2286
apply(case_tac ys)
nipkow@22793
  2287
 apply auto
nipkow@22793
  2288
done
nipkow@22793
  2289
nipkow@22828
  2290
nipkow@22828
  2291
subsubsection {* @{const allpairs} *}
nipkow@22828
  2292
nipkow@22940
  2293
lemma allpairs_conv_concat:
nipkow@22940
  2294
 "allpairs f xs ys = concat(map (%x. map (f x) ys) xs)"
nipkow@22940
  2295
by(induct xs) auto
nipkow@22940
  2296
nipkow@22828
  2297
lemma allpairs_append:
nipkow@22830
  2298
 "allpairs f (xs @ ys) zs = allpairs f xs zs @ allpairs f ys zs"
nipkow@22828
  2299
by(induct xs) auto
nipkow@22828
  2300
nipkow@22828
  2301
nipkow@15392
  2302
subsubsection{*Sets of Lists*}
nipkow@15392
  2303
nipkow@15392
  2304
subsubsection {* @{text lists}: the list-forming operator over sets *}
nipkow@15302
  2305
berghofe@22262
  2306
inductive2
berghofe@22262
  2307
  listsp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
berghofe@22262
  2308
  for A :: "'a \<Rightarrow> bool"
berghofe@22262
  2309
where
berghofe@22262
  2310
    Nil [intro!]: "listsp A []"
berghofe@22262
  2311
  | Cons [intro!]: "[| A a; listsp A l |] ==> listsp A (a # l)"
berghofe@22262
  2312
berghofe@22262
  2313
constdefs
berghofe@22262
  2314
  lists :: "'a set => 'a list set"
berghofe@22262
  2315
  "lists A == Collect (listsp (member A))"
berghofe@22262
  2316
berghofe@22262
  2317
lemma listsp_lists_eq [pred_set_conv]: "listsp (member A) = member (lists A)"
berghofe@22262
  2318
  by (simp add: lists_def)
berghofe@22262
  2319
berghofe@22262
  2320
lemmas lists_intros [intro!] = listsp.intros [to_set]
berghofe@22262
  2321
berghofe@22262
  2322
lemmas lists_induct [consumes 1, case_names Nil Cons, induct set: lists] =
berghofe@22262
  2323
  listsp.induct [to_set]
berghofe@22262
  2324
berghofe@22262
  2325
inductive_cases2 listspE [elim!]: "listsp A (x # l)"
berghofe@22262
  2326
berghofe@22262
  2327
lemmas listsE [elim!] = listspE [to_set]
berghofe@22262
  2328
berghofe@22262
  2329
lemma listsp_mono [mono2]: "A \<le> B ==> listsp A \<le> listsp B"
berghofe@22262
  2330
  by (clarify, erule listsp.induct, blast+)
berghofe@22262
  2331
berghofe@22262
  2332
lemmas lists_mono [mono] = listsp_mono [to_set]
berghofe@22262
  2333
haftmann@22422
  2334
lemma listsp_infI:
haftmann@22422
  2335
  assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
nipkow@15302
  2336
  by induct blast+
nipkow@15302
  2337
haftmann@22422
  2338
lemmas lists_IntI = listsp_infI [to_set]
haftmann@22422
  2339
haftmann@22422
  2340
lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
haftmann@22422
  2341
proof (rule mono_inf [where f=listsp, THEN order_antisym])
berghofe@22262
  2342
  show "mono listsp" by (simp add: mono_def listsp_mono)
haftmann@22422
  2343
  show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro: listsp_infI)
kleing@14388
  2344
qed
kleing@14388
  2345
haftmann@22422
  2346
lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_eq inf_bool_eq]
haftmann@22422
  2347
haftmann@22422
  2348
lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set]
berghofe@22262
  2349
berghofe@22262
  2350
lemma append_in_listsp_conv [iff]:
berghofe@22262
  2351
     "(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
nipkow@15302
  2352
by (induct xs) auto
nipkow@15302
  2353
berghofe@22262
  2354
lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]
berghofe@22262
  2355
berghofe@22262
  2356
lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
berghofe@22262
  2357
-- {* eliminate @{text listsp} in favour of @{text set} *}
nipkow@15302
  2358
by (induct xs) auto
nipkow@15302
  2359
berghofe@22262
  2360
lemmas in_lists_conv_set = in_listsp_conv_set [to_set]
berghofe@22262
  2361
berghofe@22262
  2362
lemma in_listspD [dest!]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
berghofe@22262
  2363
by (rule in_listsp_conv_set [THEN iffD1])
berghofe@22262
  2364
berghofe@22262
  2365
lemmas in_listsD [dest!] = in_listspD [to_set]
berghofe@22262
  2366
berghofe@22262
  2367
lemma in_listspI [intro!]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
berghofe@22262
  2368
by (rule in_listsp_conv_set [THEN iffD2])
berghofe@22262
  2369
berghofe@22262
  2370
lemmas in_listsI [intro!] = in_listspI [to_set]
nipkow@15302
  2371
nipkow@15302
  2372
lemma lists_UNIV [simp]: "lists UNIV = UNIV"
nipkow@15302
  2373
by auto
nipkow@15302
  2374
nipkow@17086
  2375
nipkow@17086
  2376
nipkow@17086
  2377
subsubsection{* Inductive definition for membership *}
nipkow@17086
  2378
berghofe@22262
  2379
inductive2 ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
berghofe@22262
  2380
where
berghofe@22262
  2381
    elem:  "ListMem x (x # xs)"
berghofe@22262
  2382
  | insert:  "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"
berghofe@22262
  2383
berghofe@22262
  2384
lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
nipkow@17086
  2385
apply (rule iffI)
nipkow@17086
  2386
 apply (induct set: ListMem)
nipkow@17086
  2387
  apply auto
nipkow@17086
  2388
apply (induct xs)
nipkow@17086
  2389
 apply (auto intro: ListMem.intros)
nipkow@17086
  2390
done
nipkow@17086
  2391
nipkow@17086
  2392
nipkow@17086
  2393
nipkow@15392
  2394
subsubsection{*Lists as Cartesian products*}
nipkow@15302
  2395
nipkow@15302
  2396
text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
nipkow@15302
  2397
@{term A} and tail drawn from @{term Xs}.*}
nipkow@15302
  2398
nipkow@15302
  2399
constdefs
nipkow@15302
  2400
  set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
nipkow@15302
  2401
  "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"
nipkow@15302
  2402
paulson@17724
  2403
lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
nipkow@15302
  2404
by (auto simp add: set_Cons_def)
nipkow@15302
  2405
nipkow@15302
  2406
text{*Yields the set of lists, all of the same length as the argument and
nipkow@15302
  2407
with elements drawn from the corresponding element of the argument.*}
nipkow@15302
  2408
nipkow@15302
  2409
consts  listset :: "'a set list \<Rightarrow> 'a list set"
nipkow@15302
  2410
primrec
nipkow@15302
  2411
   "listset []    = {[]}"
nipkow@15302
  2412
   "listset(A#As) = set_Cons A (listset As)"
nipkow@15302
  2413
nipkow@15302
  2414
paulson@15656
  2415
subsection{*Relations on Lists*}
paulson@15656
  2416
paulson@15656
  2417
subsubsection {* Length Lexicographic Ordering *}
paulson@15656
  2418
paulson@15656
  2419
text{*These orderings preserve well-foundedness: shorter lists 
paulson@15656
  2420
  precede longer lists. These ordering are not used in dictionaries.*}
paulson@15656
  2421
paulson@15656
  2422
consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
paulson@15656
  2423
        --{*The lexicographic ordering for lists of the specified length*}
nipkow@15302
  2424
primrec
paulson@15656
  2425
  "lexn r 0 = {}"
paulson@15656
  2426
  "lexn r (Suc n) =
paulson@15656
  2427
    (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
paulson@15656
  2428
    {(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
nipkow@15302
  2429
nipkow@15302
  2430
constdefs
paulson@15656
  2431
  lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
paulson@15656
  2432
    "lex r == \<Union>n. lexn r n"
paulson@15656
  2433
        --{*Holds only between lists of the same length*}
paulson@15656
  2434
nipkow@15693
  2435
  lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@15693
  2436
    "lenlex r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
paulson@15656
  2437
        --{*Compares lists by their length and then lexicographically*}
nipkow@15302
  2438
nipkow@15302
  2439
nipkow@15302
  2440
lemma wf_lexn: "wf r ==> wf (lexn r n)"
nipkow@15302
  2441
apply (induct n, simp, simp)
nipkow@15302
  2442
apply(rule wf_subset)
nipkow@15302
  2443
 prefer 2 apply (rule Int_lower1)
nipkow@15302
  2444
apply(rule wf_prod_fun_image)
nipkow@15302
  2445
 prefer 2 apply (rule inj_onI, auto)
nipkow@15302
  2446
done
nipkow@15302
  2447
nipkow@15302
  2448
lemma lexn_length:
nipkow@15302
  2449
     "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@15302
  2450
by (induct n) auto
nipkow@15302
  2451
nipkow@15302
  2452
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@15302
  2453
apply (unfold lex_def)
nipkow@15302
  2454
apply (rule wf_UN)
nipkow@15302
  2455
apply (blast intro: wf_lexn, clarify)
nipkow@15302
  2456
apply (rename_tac m n)
nipkow@15302
  2457
apply (subgoal_tac "m \<noteq> n")
nipkow@15302
  2458
 prefer 2 apply blast
nipkow@15302
  2459
apply (blast dest: lexn_length not_sym)
nipkow@15302
  2460
done
nipkow@15302
  2461
nipkow@15302
  2462
lemma lexn_conv:
paulson@15656
  2463
  "lexn r n =
paulson@15656
  2464
    {(xs,ys). length xs = n \<and> length ys = n \<and>
paulson@15656
  2465
    (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
nipkow@18423
  2466
apply (induct n, simp)
nipkow@15302
  2467
apply (simp add: image_Collect lex_prod_def, safe, blast)
nipkow@15302
  2468
 apply (rule_tac x = "ab # xys" in exI, simp)
nipkow@15302
  2469
apply (case_tac xys, simp_all, blast)
nipkow@15302
  2470
done
nipkow@15302
  2471
nipkow@15302
  2472
lemma lex_conv:
paulson@15656
  2473
  "lex r =
paulson@15656
  2474
    {(xs,ys). length xs = length ys \<and>
paulson@15656
  2475
    (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@15302
  2476
by (force simp add: lex_def lexn_conv)
nipkow@15302
  2477
nipkow@15693
  2478
lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
nipkow@15693
  2479
by (unfold lenlex_def) blast
nipkow@15693
  2480
nipkow@15693
  2481
lemma lenlex_conv:
nipkow@15693
  2482
    "lenlex r = {(xs,ys). length xs < length ys |
paulson@15656
  2483
                 length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@19623
  2484
by (simp add: lenlex_def diag_def lex_prod_def inv_image_def)
nipkow@15302
  2485
nipkow@15302
  2486
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@15302
  2487
by (simp add: lex_conv)
nipkow@15302
  2488
nipkow@15302
  2489
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@15302
  2490
by (simp add:lex_conv)
nipkow@15302
  2491
paulson@18447
  2492
lemma Cons_in_lex [simp]:
paulson@15656
  2493
    "((x # xs, y # ys) : lex r) =
paulson@15656
  2494
      ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@15302
  2495
apply (simp add: lex_conv)
nipkow@15302
  2496
apply (rule iffI)
nipkow@15302
  2497
 prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
nipkow@15302
  2498
apply (case_tac xys, simp, simp)
nipkow@15302
  2499
apply blast
nipkow@15302
  2500
done
nipkow@15302
  2501
nipkow@15302
  2502
paulson@15656
  2503
subsubsection {* Lexicographic Ordering *}
paulson@15656
  2504
paulson@15656
  2505
text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
paulson@15656
  2506
    This ordering does \emph{not} preserve well-foundedness.
nipkow@17090
  2507
     Author: N. Voelker, March 2005. *} 
paulson@15656
  2508
paulson@15656
  2509
constdefs 
paulson@15656
  2510
  lexord :: "('a * 'a)set \<Rightarrow> ('a list * 'a list) set" 
paulson@15656
  2511
  "lexord  r == {(x,y). \<exists> a v. y = x @ a # v \<or> 
paulson@15656
  2512
            (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
paulson@15656
  2513
paulson@15656
  2514
lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
paulson@15656
  2515
  by (unfold lexord_def, induct_tac y, auto) 
paulson@15656
  2516
paulson@15656
  2517
lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
paulson@15656
  2518
  by (unfold lexord_def, induct_tac x, auto)
paulson@15656
  2519
paulson@15656
  2520
lemma lexord_cons_cons[simp]:
paulson@15656
  2521
     "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
paulson@15656
  2522
  apply (unfold lexord_def, safe, simp_all)
paulson@15656
  2523
  apply (case_tac u, simp, simp)
paulson@15656
  2524
  apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
paulson@15656
  2525
  apply (erule_tac x="b # u" in allE)
paulson@15656
  2526
  by force
paulson@15656
  2527
paulson@15656
  2528
lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
paulson@15656
  2529
paulson@15656
  2530
lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
paulson@15656
  2531
  by (induct_tac x, auto)  
paulson@15656
  2532
paulson@15656
  2533
lemma lexord_append_left_rightI:
paulson@15656
  2534
     "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
paulson@15656
  2535
  by (induct_tac u, auto)
paulson@15656
  2536
paulson@15656
  2537
lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
paulson@15656
  2538
  by (induct x, auto)
paulson@15656
  2539
paulson@15656
  2540
lemma lexord_append_leftD:
paulson@15656
  2541
     "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
paulson@15656
  2542
  by (erule rev_mp, induct_tac x, auto)
paulson@15656
  2543
paulson@15656
  2544
lemma lexord_take_index_conv: 
paulson@15656
  2545
   "((x,y) : lexord r) = 
paulson@15656
  2546
    ((length x < length y \<and> take (length x) y = x) \<or> 
paulson@15656
  2547
     (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
paulson@15656
  2548
  apply (unfold lexord_def Let_def, clarsimp) 
paulson@15656
  2549
  apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
paulson@15656
  2550
  apply auto 
paulson@15656
  2551
  apply (rule_tac x="hd (drop (length x) y)" in exI)
paulson@15656
  2552
  apply (rule_tac x="tl (drop (length x) y)" in exI)
paulson@15656
  2553
  apply (erule subst, simp add: min_def) 
paulson@15656
  2554
  apply (rule_tac x ="length u" in exI, simp) 
paulson@15656
  2555
  apply (rule_tac x ="take i x" in exI) 
paulson@15656
  2556
  apply (rule_tac x ="x ! i" in exI) 
paulson@15656
  2557
  apply (rule_tac x ="y ! i" in exI, safe) 
paulson@15656
  2558
  apply (rule_tac x="drop (Suc i) x" in exI)
paulson@15656
  2559
  apply (drule sym, simp add: drop_Suc_conv_tl) 
paulson@15656
  2560
  apply (rule_tac x="drop (Suc i) y" in exI)
paulson@15656
  2561
  by (simp add: drop_Suc_conv_tl) 
paulson@15656
  2562
paulson@15656
  2563
-- {* lexord is extension of partial ordering List.lex *} 
paulson@15656
  2564
lemma lexord_lex: " (x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
paulson@15656
  2565
  apply (rule_tac x = y in spec) 
paulson@15656
  2566
  apply (induct_tac x, clarsimp) 
paulson@15656
  2567
  by (clarify, case_tac x, simp, force)
paulson@15656
  2568
paulson@15656
  2569
lemma lexord_irreflexive: "(! x. (x,x) \<notin> r) \<Longrightarrow> (y,y) \<notin> lexord r"
paulson@15656
  2570
  by (induct y, auto)
paulson@15656
  2571
paulson@15656
  2572
lemma lexord_trans: 
paulson@15656
  2573
    "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
paulson@15656
  2574
   apply (erule rev_mp)+
paulson@15656
  2575
   apply (rule_tac x = x in spec) 
paulson@15656
  2576
  apply (rule_tac x = z in spec) 
paulson@15656
  2577
  apply ( induct_tac y, simp, clarify)
paulson@15656
  2578
  apply (case_tac xa, erule ssubst) 
paulson@15656
  2579
  apply (erule allE, erule allE) -- {* avoid simp recursion *} 
paulson@15656
  2580
  apply (case_tac x, simp, simp) 
paulson@15656
  2581
  apply (case_tac x, erule allE, erule allE, simp) 
paulson@15656
  2582
  apply (erule_tac x = listb in allE) 
paulson@15656
  2583
  apply (erule_tac x = lista in allE, simp)
paulson@15656
  2584
  apply (unfold trans_def)
paulson@15656
  2585
  by blast
paulson@15656
  2586
paulson@15656
  2587
lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
paulson@15656
  2588
  by (rule transI, drule lexord_trans, blast) 
paulson@15656
  2589
paulson@15656
  2590
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
  2591
  apply (rule_tac x = y in spec) 
paulson@15656
  2592
  apply (induct_tac x, rule allI) 
paulson@15656
  2593
  apply (case_tac x, simp, simp) 
paulson@15656
  2594
  apply (rule allI, case_tac x, simp, simp) 
paulson@15656
  2595
  by blast
paulson@15656
  2596
paulson@15656
  2597
krauss@21103
  2598
subsection {* Lexicographic combination of measure functions *}
krauss@21103
  2599
krauss@21103
  2600
text {* These are useful for termination proofs *}
krauss@21103
  2601
krauss@21103
  2602
definition
krauss@21103
  2603
  "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"
krauss@21103
  2604
krauss@21106
  2605
lemma wf_measures[recdef_wf, simp]: "wf (measures fs)"
krauss@21103
  2606
  unfolding measures_def
krauss@21103
  2607
  by blast
krauss@21103
  2608
krauss@21103
  2609
lemma in_measures[simp]: 
krauss@21103
  2610
  "(x, y) \<in> measures [] = False"
krauss@21103
  2611
  "(x, y) \<in> measures (f # fs)
krauss@21103
  2612
         = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"  
krauss@21103
  2613
  unfolding measures_def
krauss@21103
  2614
  by auto
krauss@21103
  2615
krauss@21103
  2616
lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
krauss@21103
  2617
  by simp
krauss@21103
  2618
krauss@21103
  2619
lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
krauss@21103
  2620
  by auto
krauss@21103
  2621
krauss@21211
  2622
(* install the lexicographic_order method and the "fun" command *)
bulwahn@21131
  2623
use "Tools/function_package/lexicographic_order.ML"
krauss@21211
  2624
use "Tools/function_package/fundef_datatype.ML"
krauss@21211
  2625
setup LexicographicOrder.setup
krauss@21211
  2626
setup FundefDatatype.setup
krauss@21211
  2627
krauss@21103
  2628
nipkow@15392
  2629
subsubsection{*Lifting a Relation on List Elements to the Lists*}
nipkow@15302
  2630
berghofe@22262
  2631
inductive2
berghofe@22262
  2632
  list_all2' :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool"
berghofe@22262
  2633
  for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
berghofe@22262
  2634
where
berghofe@22262
  2635
    Nil:  "list_all2' r [] []"
berghofe@22262
  2636
  | Cons: "[| r x y; list_all2' r xs ys |] ==> list_all2' r (x#xs) (y#ys)"
berghofe@22262
  2637
berghofe@22262
  2638
constdefs
berghofe@22262
  2639
  listrel :: "('a * 'b) set => ('a list * 'b list) set"
berghofe@22262
  2640
  "listrel r == Collect2 (list_all2' (member2 r))"
berghofe@22262
  2641
berghofe@22262
  2642
lemma list_all2_listrel_eq [pred_set_conv]:
berghofe@22262
  2643
  "list_all2' (member2 r) = member2 (listrel r)"
berghofe@22262
  2644
  by (simp add: listrel_def)
berghofe@22262
  2645
berghofe@22262
  2646
lemmas listrel_induct [consumes 1, case_names Nil Cons, induct set: listrel] =
berghofe@22262
  2647
  list_all2'.induct [to_set]
berghofe@22262
  2648
berghofe@22262
  2649
lemmas listrel_intros = list_all2'.intros [to_set]
berghofe@22262
  2650
berghofe@22262
  2651
inductive_cases2 listrel_Nil1 [to_set, elim!]: "list_all2' r [] xs"
berghofe@22262
  2652
inductive_cases2 listrel_Nil2 [to_set, elim!]: "list_all2' r xs []"
berghofe@22262
  2653
inductive_cases2 listrel_Cons1 [to_set, elim!]: "list_all2' r  (y#ys) xs"
berghofe@22262
  2654
inductive_cases2 listrel_Cons2 [to_set, elim!]: "list_all2' r xs (y#ys)"
nipkow@15302
  2655
nipkow@15302
  2656
nipkow@15302
  2657
lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
nipkow@15302
  2658
apply clarify  
berghofe@22262
  2659
apply (erule listrel_induct)
berghofe@22262
  2660
apply (blast intro: listrel_intros)+
nipkow@15302
  2661
done
nipkow@15302
  2662
nipkow@15302
  2663
lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
nipkow@15302
  2664
apply clarify 
berghofe@22262
  2665
apply (erule listrel_induct, auto) 
nipkow@15302
  2666
done
nipkow@15302
  2667
nipkow@15302
  2668
lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" 
nipkow@15302
  2669
apply (simp add: refl_def listrel_subset Ball_def)
nipkow@15302
  2670
apply (rule allI) 
nipkow@15302
  2671
apply (induct_tac x) 
berghofe@22262
  2672
apply (auto intro: listrel_intros)
nipkow@15302
  2673
done
nipkow@15302
  2674
nipkow@15302
  2675
lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
nipkow@15302
  2676
apply (auto simp add: sym_def)
berghofe@22262
  2677
apply (erule listrel_induct) 
berghofe@22262
  2678
apply (blast intro: listrel_intros)+
nipkow@15302
  2679
done
nipkow@15302
  2680
nipkow@15302
  2681
lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
nipkow@15302
  2682
apply (simp add: trans_def)
nipkow@15302
  2683
apply (intro allI) 
nipkow@15302
  2684
apply (rule impI) 
berghofe@22262
  2685
apply (erule listrel_induct) 
berghofe@22262
  2686
apply (blast intro: listrel_intros)+
nipkow@15302
  2687
done
nipkow@15302
  2688
nipkow@15302
  2689
theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
nipkow@15302
  2690
by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) 
nipkow@15302
  2691
nipkow@15302
  2692
lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
berghofe@22262
  2693
by (blast intro: listrel_intros)
nipkow@15302
  2694
nipkow@15302
  2695
lemma listrel_Cons:
nipkow@15302
  2696
     "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})";
berghofe@22262
  2697
by (auto simp add: set_Cons_def intro: listrel_intros) 
nipkow@15302
  2698
nipkow@15302
  2699
nipkow@15392
  2700
subsection{*Miscellany*}
nipkow@15392
  2701
nipkow@15392
  2702
subsubsection {* Characters and strings *}
wenzelm@13366
  2703
wenzelm@13366
  2704
datatype nibble =
wenzelm@13366
  2705
    Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
wenzelm@13366
  2706
  | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
wenzelm@13366
  2707
wenzelm@13366
  2708
datatype char = Char nibble nibble
wenzelm@13366
  2709
  -- "Note: canonical order of character encoding coincides with standard term ordering"
wenzelm@13366
  2710
wenzelm@13366
  2711
types string = "char list"
wenzelm@13366
  2712
wenzelm@13366
  2713
syntax
wenzelm@13366
  2714
  "_Char" :: "xstr => char"    ("CHR _")
wenzelm@13366
  2715
  "_String" :: "xstr => string"    ("_")
wenzelm@13366
  2716
wenzelm@21754
  2717
setup StringSyntax.setup
wenzelm@13366
  2718
haftmann@20453
  2719
haftmann@21061
  2720
subsection {* Code generator *}
haftmann@21061
  2721
haftmann@21061
  2722
subsubsection {* Setup *}
berghofe@15064
  2723
berghofe@16770
  2724
types_code
berghofe@16770
  2725
  "list" ("_ list")
berghofe@16770
  2726
attach (term_of) {*
wenzelm@21760
  2727
fun term_of_list f T = HOLogic.mk_list T o map f;
berghofe@16770
  2728
*}
berghofe@16770
  2729
attach (test) {*
berghofe@15064
  2730
fun gen_list' aG i j = frequency
berghofe@15064
  2731
  [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] ()
berghofe@15064
  2732
and gen_list aG i = gen_list' aG i i;
berghofe@16770
  2733
*}
berghofe@16770
  2734
  "char" ("string")
berghofe@16770
  2735
attach (term_of) {*
haftmann@21455
  2736
val term_of_char = HOLogic.mk_char;
berghofe@16770
  2737
*}
berghofe@16770
  2738
attach (test) {*
berghofe@15064
  2739
fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z")));
berghofe@15064
  2740
*}
berghofe@15064
  2741
berghofe@15064
  2742
consts_code "Cons" ("(_ ::/ _)")
berghofe@15064
  2743
haftmann@20453
  2744
code_type list
haftmann@20453
  2745
  (SML "_ list")
haftmann@21911
  2746
  (OCaml "_ list")
haftmann@21113
  2747
  (Haskell "![_]")
haftmann@20453
  2748
haftmann@22799
  2749
code_reserved SML
haftmann@22799
  2750
  list
haftmann@22799
  2751
haftmann@22799
  2752
code_reserved OCaml
haftmann@22799
  2753
  list
haftmann@22799
  2754
haftmann@20453
  2755
code_const Nil
haftmann@21113
  2756
  (SML "[]")
haftmann@21911
  2757
  (OCaml "[]")