src/HOL/List.thy
author nipkow
Sun Jun 24 15:17:54 2007 +0200 (2007-06-24)
changeset 23479 10adbdcdc65b
parent 23388 77645da0db85
child 23554 151d60fbfffe
permissions -rw-r--r--
new lemmas
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(*  Title:      HOL/List.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow
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*)
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header {* The datatype of finite lists *}
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theory List
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imports PreList
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uses "Tools/string_syntax.ML"
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begin
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datatype 'a list =
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    Nil    ("[]")
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  | Cons 'a  "'a list"    (infixr "#" 65)
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subsection{*Basic list processing functions*}
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consts
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  filter:: "('a => bool) => 'a list => 'a list"
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  concat:: "'a list list => 'a list"
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  foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"
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  foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"
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  hd:: "'a list => 'a"
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  tl:: "'a list => 'a list"
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  last:: "'a list => 'a"
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  butlast :: "'a list => 'a list"
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  set :: "'a list => 'a set"
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  map :: "('a=>'b) => ('a list => 'b list)"
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  listsum ::  "'a list => 'a::monoid_add"
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  nth :: "'a list => nat => 'a"    (infixl "!" 100)
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  list_update :: "'a list => nat => 'a => 'a list"
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  take:: "nat => 'a list => 'a list"
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  drop:: "nat => 'a list => 'a list"
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  takeWhile :: "('a => bool) => 'a list => 'a list"
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  dropWhile :: "('a => bool) => 'a list => 'a list"
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  rev :: "'a list => 'a list"
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  zip :: "'a list => 'b list => ('a * 'b) list"
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  upt :: "nat => nat => nat list" ("(1[_..</_'])")
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  remdups :: "'a list => 'a list"
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  remove1 :: "'a => 'a list => 'a list"
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  "distinct":: "'a list => bool"
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  replicate :: "nat => 'a => 'a list"
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  splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
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  allpairs :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'c list"
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abbreviation
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  upto:: "nat => nat => nat list"  ("(1[_../_])") where
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  "[i..j] == [i..<(Suc j)]"
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nonterminals lupdbinds lupdbind
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syntax
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  -- {* list Enumeration *}
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  "@list" :: "args => 'a list"    ("[(_)]")
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  -- {* Special syntax for filter *}
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_<-_./ _])")
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  -- {* list update *}
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  "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
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  "" :: "lupdbind => lupdbinds"    ("_")
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  "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
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  "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
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translations
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  "[x, xs]" == "x#[xs]"
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  "[x]" == "x#[]"
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  "[x<-xs . P]"== "filter (%x. P) xs"
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  "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"
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  "xs[i:=x]" == "list_update xs i x"
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syntax (xsymbols)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
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syntax (HTML output)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
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text {*
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  Function @{text size} is overloaded for all datatypes. Users may
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  refer to the list version as @{text length}. *}
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abbreviation
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  length :: "'a list => nat" where
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  "length == size"
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primrec
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  "hd(x#xs) = x"
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primrec
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  "tl([]) = []"
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  "tl(x#xs) = xs"
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primrec
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  "last(x#xs) = (if xs=[] then x else last xs)"
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primrec
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  "butlast []= []"
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  "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
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primrec
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  "set [] = {}"
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  "set (x#xs) = insert x (set xs)"
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primrec
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  "map f [] = []"
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  "map f (x#xs) = f(x)#map f xs"
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function (*authentic syntax for append -- revert to primrec
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  as soon as "authentic" primrec is available*)
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  append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65)
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where
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  append_Nil: "[] @ ys = ys"
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  | append_Cons: "(x # xs) @ ys = x # (xs @ ys)"
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by (auto, case_tac a, auto)
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termination by (relation "measure (size o fst)") auto
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primrec
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  "rev([]) = []"
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  "rev(x#xs) = rev(xs) @ [x]"
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primrec
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  "filter P [] = []"
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  "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
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primrec
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  foldl_Nil:"foldl f a [] = a"
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  foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
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primrec
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  "foldr f [] a = a"
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  "foldr f (x#xs) a = f x (foldr f xs a)"
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primrec
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  "concat([]) = []"
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  "concat(x#xs) = x @ concat(xs)"
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primrec
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"listsum [] = 0"
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"listsum (x # xs) = x + listsum xs"
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primrec
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  drop_Nil:"drop n [] = []"
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  drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  take_Nil:"take n [] = []"
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  take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  "[][i:=v] = []"
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  "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])"
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primrec
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  "takeWhile P [] = []"
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  "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
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primrec
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  "dropWhile P [] = []"
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  "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
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primrec
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  "zip xs [] = []"
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  zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
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primrec
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  upt_0: "[i..<0] = []"
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  upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
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primrec
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  "distinct [] = True"
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  "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
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primrec
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  "remdups [] = []"
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  "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
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primrec
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  "remove1 x [] = []"
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  "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)"
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primrec
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  replicate_0: "replicate 0 x = []"
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  replicate_Suc: "replicate (Suc n) x = x # replicate n x"
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definition
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  rotate1 :: "'a list \<Rightarrow> 'a list" where
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  "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"
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definition
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  rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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  "rotate n = rotate1 ^ n"
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definition
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  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
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  "list_all2 P xs ys =
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    (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"
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definition
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  sublist :: "'a list => nat set => 'a list" where
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  "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"
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primrec
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  "splice [] ys = ys"
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  "splice (x#xs) ys = (if ys=[] then x#xs else x # hd ys # splice xs (tl ys))"
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    -- {*Warning: simpset does not contain the second eqn but a derived one. *}
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primrec
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"allpairs f [] ys = []"
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"allpairs f (x # xs) ys = map (f x) ys @ allpairs f xs ys"
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subsubsection {* List comprehension *}
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text{* Input syntax for Haskell-like list comprehension
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notation. Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"}, the list of all pairs of distinct elements from @{text xs} and @{text ys}.
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There are two differences to Haskell.  The general synatx is
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@{text"[e. p \<leftarrow> xs, \<dots>]"} rather than \verb![x| x <- xs, ...]!. Patterns in
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generators can only be tuples (at the moment).
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To avoid misunderstandings, the translation is not reversed upon
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output. You can add the inverse translations in your own theory if you
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desire.
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Hint: formulae containing complex list comprehensions may become quite
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unreadable after the simplifier has finished with them. It can be
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helpful to introduce definitions for such list comprehensions and
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treat them separately in suitable lemmas.
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*}
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(*
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Proper theorem proving support would be nice. For example, if
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@{text"set[f x y. x \<leftarrow> xs, y \<leftarrow> ys, P x y]"}
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produced something like
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@{term"{z. EX x: set xs. EX y:set ys. P x y \<and> z = f x y}"}.
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*)
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nonterminals lc_qual lc_quals
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syntax
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"_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
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"_lc_gen" :: "pttrn \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ <- _")
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"_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
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"_lc_end" :: "lc_quals" ("]")
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"_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals" (", __")
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translations
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"[e. p<-xs]" => "map (%p. e) xs"
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"_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
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 => "concat (map (%p. _listcompr e Q Qs) xs)"
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"[e. P]" => "if P then [e] else []"
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"_listcompr e (_lc_test P) (_lc_quals Q Qs)"
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 => "if P then (_listcompr e Q Qs) else []"
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syntax (xsymbols)
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"_lc_gen" :: "pttrn \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
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syntax (HTML output)
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"_lc_gen" :: "pttrn \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
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(*
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term "[(x,y,z). b]"
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term "[(x,y,z). x \<leftarrow> xs]"
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term "[(x,y,z). x<a, x>b]"
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term "[(x,y,z). x<a, x\<leftarrow>xs]"
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term "[(x,y,z). x\<leftarrow>xs, x>b]"
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term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys]"
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term "[(x,y,z). x<a, x>b, x=d]"
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term "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
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term "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
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term "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
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term "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
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term "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
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term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
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term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
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*)
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subsubsection {* @{const Nil} and @{const Cons} *}
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lemma not_Cons_self [simp]:
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  "xs \<noteq> x # xs"
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by (induct xs) auto
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lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
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lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
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by (induct xs) auto
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lemma length_induct:
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  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
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by (rule measure_induct [of length]) iprover
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subsubsection {* @{const length} *}
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text {*
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  Needs to come before @{text "@"} because of theorem @{text
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  append_eq_append_conv}.
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*}
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lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
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by (induct xs) auto
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lemma length_map [simp]: "length (map f xs) = length xs"
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by (induct xs) auto
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lemma length_rev [simp]: "length (rev xs) = length xs"
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by (induct xs) auto
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lemma length_tl [simp]: "length (tl xs) = length xs - 1"
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by (cases xs) auto
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lemma length_allpairs[simp]:
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 "length(allpairs f xs ys) = length xs * length ys"
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by(induct xs) auto
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lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
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   332
by (induct xs) auto
wenzelm@13114
   333
wenzelm@13142
   334
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
nipkow@13145
   335
by (induct xs) auto
wenzelm@13114
   336
nipkow@23479
   337
lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
nipkow@23479
   338
by auto
nipkow@23479
   339
wenzelm@13114
   340
lemma length_Suc_conv:
nipkow@13145
   341
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
nipkow@13145
   342
by (induct xs) auto
wenzelm@13142
   343
nipkow@14025
   344
lemma Suc_length_conv:
nipkow@14025
   345
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
paulson@14208
   346
apply (induct xs, simp, simp)
nipkow@14025
   347
apply blast
nipkow@14025
   348
done
nipkow@14025
   349
oheimb@14099
   350
lemma impossible_Cons [rule_format]: 
oheimb@14099
   351
  "length xs <= length ys --> xs = x # ys = False"
wenzelm@20503
   352
apply (induct xs)
wenzelm@20503
   353
apply auto
oheimb@14099
   354
done
oheimb@14099
   355
nipkow@14247
   356
lemma list_induct2[consumes 1]: "\<And>ys.
nipkow@14247
   357
 \<lbrakk> length xs = length ys;
nipkow@14247
   358
   P [] [];
nipkow@14247
   359
   \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
nipkow@14247
   360
 \<Longrightarrow> P xs ys"
nipkow@14247
   361
apply(induct xs)
nipkow@14247
   362
 apply simp
nipkow@14247
   363
apply(case_tac ys)
nipkow@14247
   364
 apply simp
nipkow@14247
   365
apply(simp)
nipkow@14247
   366
done
wenzelm@13114
   367
krauss@22493
   368
lemma list_induct2': 
krauss@22493
   369
  "\<lbrakk> P [] [];
krauss@22493
   370
  \<And>x xs. P (x#xs) [];
krauss@22493
   371
  \<And>y ys. P [] (y#ys);
krauss@22493
   372
   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
krauss@22493
   373
 \<Longrightarrow> P xs ys"
krauss@22493
   374
by (induct xs arbitrary: ys) (case_tac x, auto)+
krauss@22493
   375
nipkow@22143
   376
lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
nipkow@22143
   377
apply(rule Eq_FalseI)
nipkow@22143
   378
by auto
nipkow@22143
   379
nipkow@22143
   380
(*
nipkow@22143
   381
Reduces xs=ys to False if xs and ys cannot be of the same length.
nipkow@22143
   382
This is the case if the atomic sublists of one are a submultiset
nipkow@22143
   383
of those of the other list and there are fewer Cons's in one than the other.
nipkow@22143
   384
*)
nipkow@22143
   385
ML_setup {*
nipkow@22143
   386
local
nipkow@22143
   387
nipkow@22143
   388
fun len (Const("List.list.Nil",_)) acc = acc
nipkow@22143
   389
  | len (Const("List.list.Cons",_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
haftmann@23029
   390
  | len (Const("List.append",_) $ xs $ ys) acc = len xs (len ys acc)
nipkow@22143
   391
  | len (Const("List.rev",_) $ xs) acc = len xs acc
nipkow@22143
   392
  | len (Const("List.map",_) $ _ $ xs) acc = len xs acc
nipkow@22143
   393
  | len t (ts,n) = (t::ts,n);
nipkow@22143
   394
nipkow@22143
   395
fun list_eq ss (Const(_,eqT) $ lhs $ rhs) =
nipkow@22143
   396
  let
nipkow@22143
   397
    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
nipkow@22143
   398
    fun prove_neq() =
nipkow@22143
   399
      let
nipkow@22143
   400
        val Type(_,listT::_) = eqT;
haftmann@22994
   401
        val size = HOLogic.size_const listT;
nipkow@22143
   402
        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
nipkow@22143
   403
        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
nipkow@22143
   404
        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
haftmann@22633
   405
          (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
haftmann@22633
   406
      in SOME (thm RS @{thm neq_if_length_neq}) end
nipkow@22143
   407
  in
wenzelm@23214
   408
    if m < n andalso submultiset (op aconv) (ls,rs) orelse
wenzelm@23214
   409
       n < m andalso submultiset (op aconv) (rs,ls)
nipkow@22143
   410
    then prove_neq() else NONE
nipkow@22143
   411
  end;
nipkow@22143
   412
nipkow@22143
   413
in
nipkow@22143
   414
nipkow@22143
   415
val list_neq_simproc =
haftmann@22633
   416
  Simplifier.simproc @{theory} "list_neq" ["(xs::'a list) = ys"] (K list_eq);
nipkow@22143
   417
nipkow@22143
   418
end;
nipkow@22143
   419
nipkow@22143
   420
Addsimprocs [list_neq_simproc];
nipkow@22143
   421
*}
nipkow@22143
   422
nipkow@22143
   423
nipkow@15392
   424
subsubsection {* @{text "@"} -- append *}
wenzelm@13114
   425
wenzelm@13142
   426
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
nipkow@13145
   427
by (induct xs) auto
wenzelm@13114
   428
wenzelm@13142
   429
lemma append_Nil2 [simp]: "xs @ [] = xs"
nipkow@13145
   430
by (induct xs) auto
nipkow@3507
   431
wenzelm@13142
   432
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
nipkow@13145
   433
by (induct xs) auto
wenzelm@13114
   434
wenzelm@13142
   435
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
nipkow@13145
   436
by (induct xs) auto
wenzelm@13114
   437
wenzelm@13142
   438
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
nipkow@13145
   439
by (induct xs) auto
wenzelm@13114
   440
wenzelm@13142
   441
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
nipkow@13145
   442
by (induct xs) auto
wenzelm@13114
   443
berghofe@13883
   444
lemma append_eq_append_conv [simp]:
berghofe@13883
   445
 "!!ys. length xs = length ys \<or> length us = length vs
berghofe@13883
   446
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
berghofe@13883
   447
apply (induct xs)
paulson@14208
   448
 apply (case_tac ys, simp, force)
paulson@14208
   449
apply (case_tac ys, force, simp)
nipkow@13145
   450
done
wenzelm@13142
   451
nipkow@14495
   452
lemma append_eq_append_conv2: "!!ys zs ts.
nipkow@14495
   453
 (xs @ ys = zs @ ts) =
nipkow@14495
   454
 (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
nipkow@14495
   455
apply (induct xs)
nipkow@14495
   456
 apply fastsimp
nipkow@14495
   457
apply(case_tac zs)
nipkow@14495
   458
 apply simp
nipkow@14495
   459
apply fastsimp
nipkow@14495
   460
done
nipkow@14495
   461
wenzelm@13142
   462
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145
   463
by simp
wenzelm@13142
   464
wenzelm@13142
   465
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145
   466
by simp
wenzelm@13114
   467
wenzelm@13142
   468
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   469
by simp
wenzelm@13114
   470
wenzelm@13142
   471
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   472
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   473
wenzelm@13142
   474
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   475
using append_same_eq [of "[]"] by auto
wenzelm@13114
   476
wenzelm@13142
   477
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   478
by (induct xs) auto
wenzelm@13114
   479
wenzelm@13142
   480
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   481
by (induct xs) auto
wenzelm@13114
   482
wenzelm@13142
   483
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   484
by (simp add: hd_append split: list.split)
wenzelm@13114
   485
wenzelm@13142
   486
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   487
by (simp split: list.split)
wenzelm@13114
   488
wenzelm@13142
   489
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   490
by (simp add: tl_append split: list.split)
wenzelm@13114
   491
wenzelm@13114
   492
nipkow@14300
   493
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   494
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   495
by(cases ys) auto
nipkow@14300
   496
nipkow@15281
   497
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   498
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   499
by(cases ys) auto
nipkow@15281
   500
nipkow@14300
   501
wenzelm@13142
   502
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   503
wenzelm@13114
   504
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   505
by simp
wenzelm@13114
   506
wenzelm@13142
   507
lemma Cons_eq_appendI:
nipkow@13145
   508
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   509
by (drule sym) simp
wenzelm@13114
   510
wenzelm@13142
   511
lemma append_eq_appendI:
nipkow@13145
   512
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   513
by (drule sym) simp
wenzelm@13114
   514
wenzelm@13114
   515
wenzelm@13142
   516
text {*
nipkow@13145
   517
Simplification procedure for all list equalities.
nipkow@13145
   518
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   519
- both lists end in a singleton list,
nipkow@13145
   520
- or both lists end in the same list.
wenzelm@13142
   521
*}
wenzelm@13142
   522
wenzelm@13142
   523
ML_setup {*
nipkow@3507
   524
local
nipkow@3507
   525
wenzelm@13114
   526
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462
   527
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
haftmann@23029
   528
  | last (Const("List.append",_) $ _ $ ys) = last ys
wenzelm@13462
   529
  | last t = t;
wenzelm@13114
   530
wenzelm@13114
   531
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462
   532
  | list1 _ = false;
wenzelm@13114
   533
wenzelm@13114
   534
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462
   535
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
haftmann@23029
   536
  | butlast ((app as Const("List.append",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462
   537
  | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114
   538
haftmann@22633
   539
val rearr_ss = HOL_basic_ss addsimps [@{thm append_assoc},
haftmann@22633
   540
  @{thm append_Nil}, @{thm append_Cons}];
wenzelm@16973
   541
wenzelm@20044
   542
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   543
  let
wenzelm@13462
   544
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   545
    fun rearr conv =
wenzelm@13462
   546
      let
wenzelm@13462
   547
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   548
        val Type(_,listT::_) = eqT
wenzelm@13462
   549
        val appT = [listT,listT] ---> listT
haftmann@23029
   550
        val app = Const("List.append",appT)
wenzelm@13462
   551
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   552
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@20044
   553
        val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
wenzelm@17877
   554
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
skalberg@15531
   555
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   556
wenzelm@13462
   557
  in
haftmann@22633
   558
    if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
haftmann@22633
   559
    else if lastl aconv lastr then rearr @{thm append_same_eq}
skalberg@15531
   560
    else NONE
wenzelm@13462
   561
  end;
wenzelm@13462
   562
wenzelm@13114
   563
in
wenzelm@13462
   564
wenzelm@13462
   565
val list_eq_simproc =
haftmann@22633
   566
  Simplifier.simproc @{theory} "list_eq" ["(xs::'a list) = ys"] (K list_eq);
wenzelm@13462
   567
wenzelm@13114
   568
end;
wenzelm@13114
   569
wenzelm@13114
   570
Addsimprocs [list_eq_simproc];
wenzelm@13114
   571
*}
wenzelm@13114
   572
wenzelm@13114
   573
nipkow@15392
   574
subsubsection {* @{text map} *}
wenzelm@13114
   575
wenzelm@13142
   576
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   577
by (induct xs) simp_all
wenzelm@13114
   578
wenzelm@13142
   579
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   580
by (rule ext, induct_tac xs) auto
wenzelm@13114
   581
wenzelm@13142
   582
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   583
by (induct xs) auto
wenzelm@13114
   584
wenzelm@13142
   585
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   586
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   587
wenzelm@13142
   588
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   589
by (induct xs) auto
wenzelm@13114
   590
nipkow@13737
   591
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   592
by (induct xs) auto
nipkow@13737
   593
krauss@19770
   594
lemma map_cong [fundef_cong, recdef_cong]:
nipkow@13145
   595
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   596
-- {* a congruence rule for @{text map} *}
nipkow@13737
   597
by simp
wenzelm@13114
   598
wenzelm@13142
   599
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   600
by (cases xs) auto
wenzelm@13114
   601
wenzelm@13142
   602
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   603
by (cases xs) auto
wenzelm@13114
   604
paulson@18447
   605
lemma map_eq_Cons_conv:
nipkow@14025
   606
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   607
by (cases xs) auto
wenzelm@13114
   608
paulson@18447
   609
lemma Cons_eq_map_conv:
nipkow@14025
   610
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   611
by (cases ys) auto
nipkow@14025
   612
paulson@18447
   613
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
paulson@18447
   614
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
paulson@18447
   615
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
paulson@18447
   616
nipkow@14111
   617
lemma ex_map_conv:
nipkow@14111
   618
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
paulson@18447
   619
by(induct ys, auto simp add: Cons_eq_map_conv)
nipkow@14111
   620
nipkow@15110
   621
lemma map_eq_imp_length_eq:
nipkow@15110
   622
  "!!xs. map f xs = map f ys ==> length xs = length ys"
nipkow@15110
   623
apply (induct ys)
nipkow@15110
   624
 apply simp
nipkow@15110
   625
apply(simp (no_asm_use))
nipkow@15110
   626
apply clarify
nipkow@15110
   627
apply(simp (no_asm_use))
nipkow@15110
   628
apply fast
nipkow@15110
   629
done
nipkow@15110
   630
nipkow@15110
   631
lemma map_inj_on:
nipkow@15110
   632
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   633
  ==> xs = ys"
nipkow@15110
   634
apply(frule map_eq_imp_length_eq)
nipkow@15110
   635
apply(rotate_tac -1)
nipkow@15110
   636
apply(induct rule:list_induct2)
nipkow@15110
   637
 apply simp
nipkow@15110
   638
apply(simp)
nipkow@15110
   639
apply (blast intro:sym)
nipkow@15110
   640
done
nipkow@15110
   641
nipkow@15110
   642
lemma inj_on_map_eq_map:
nipkow@15110
   643
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   644
by(blast dest:map_inj_on)
nipkow@15110
   645
wenzelm@13114
   646
lemma map_injective:
nipkow@14338
   647
 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@14338
   648
by (induct ys) (auto dest!:injD)
wenzelm@13114
   649
nipkow@14339
   650
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   651
by(blast dest:map_injective)
nipkow@14339
   652
wenzelm@13114
   653
lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@17589
   654
by (iprover dest: map_injective injD intro: inj_onI)
wenzelm@13114
   655
wenzelm@13114
   656
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   657
apply (unfold inj_on_def, clarify)
nipkow@13145
   658
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   659
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   660
apply blast
nipkow@13145
   661
done
wenzelm@13114
   662
nipkow@14339
   663
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   664
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   665
nipkow@15303
   666
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
   667
apply(rule inj_onI)
nipkow@15303
   668
apply(erule map_inj_on)
nipkow@15303
   669
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
   670
done
nipkow@15303
   671
kleing@14343
   672
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   673
by (induct xs, auto)
wenzelm@13114
   674
nipkow@14402
   675
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   676
by (induct xs) auto
nipkow@14402
   677
nipkow@15110
   678
lemma map_fst_zip[simp]:
nipkow@15110
   679
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   680
by (induct rule:list_induct2, simp_all)
nipkow@15110
   681
nipkow@15110
   682
lemma map_snd_zip[simp]:
nipkow@15110
   683
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   684
by (induct rule:list_induct2, simp_all)
nipkow@15110
   685
nipkow@15110
   686
nipkow@15392
   687
subsubsection {* @{text rev} *}
wenzelm@13114
   688
wenzelm@13142
   689
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   690
by (induct xs) auto
wenzelm@13114
   691
wenzelm@13142
   692
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   693
by (induct xs) auto
wenzelm@13114
   694
kleing@15870
   695
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870
   696
by auto
kleing@15870
   697
wenzelm@13142
   698
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   699
by (induct xs) auto
wenzelm@13114
   700
wenzelm@13142
   701
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   702
by (induct xs) auto
wenzelm@13114
   703
kleing@15870
   704
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870
   705
by (cases xs) auto
kleing@15870
   706
kleing@15870
   707
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870
   708
by (cases xs) auto
kleing@15870
   709
haftmann@21061
   710
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
haftmann@21061
   711
apply (induct xs arbitrary: ys, force)
paulson@14208
   712
apply (case_tac ys, simp, force)
nipkow@13145
   713
done
wenzelm@13114
   714
nipkow@15439
   715
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
   716
by(simp add:inj_on_def)
nipkow@15439
   717
wenzelm@13366
   718
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   719
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489
   720
apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145
   721
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   722
done
wenzelm@13114
   723
nipkow@13145
   724
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114
   725
wenzelm@13366
   726
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   727
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   728
by (induct xs rule: rev_induct) auto
wenzelm@13114
   729
wenzelm@13366
   730
lemmas rev_cases = rev_exhaust
wenzelm@13366
   731
nipkow@18423
   732
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
nipkow@18423
   733
by(rule rev_cases[of xs]) auto
nipkow@18423
   734
wenzelm@13114
   735
nipkow@15392
   736
subsubsection {* @{text set} *}
wenzelm@13114
   737
wenzelm@13142
   738
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   739
by (induct xs) auto
wenzelm@13114
   740
wenzelm@13142
   741
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   742
by (induct xs) auto
wenzelm@13114
   743
nipkow@17830
   744
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
nipkow@17830
   745
by(cases xs) auto
oheimb@14099
   746
wenzelm@13142
   747
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   748
by auto
wenzelm@13114
   749
oheimb@14099
   750
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   751
by auto
oheimb@14099
   752
wenzelm@13142
   753
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   754
by (induct xs) auto
wenzelm@13114
   755
nipkow@15245
   756
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   757
by(induct xs) auto
nipkow@15245
   758
wenzelm@13142
   759
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   760
by (induct xs) auto
wenzelm@13114
   761
wenzelm@13142
   762
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   763
by (induct xs) auto
wenzelm@13114
   764
nipkow@22828
   765
lemma set_allpairs[simp]:
nipkow@22830
   766
 "set(allpairs f xs ys) = {z. EX x : set xs. EX y : set ys. z = f x y}"
nipkow@22828
   767
by(induct xs) auto
nipkow@22828
   768
wenzelm@13142
   769
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   770
by (induct xs) auto
wenzelm@13114
   771
nipkow@15425
   772
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
paulson@14208
   773
apply (induct j, simp_all)
paulson@14208
   774
apply (erule ssubst, auto)
nipkow@13145
   775
done
wenzelm@13114
   776
wenzelm@13142
   777
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
paulson@15113
   778
proof (induct xs)
paulson@15113
   779
  case Nil show ?case by simp
paulson@15113
   780
  case (Cons a xs)
paulson@15113
   781
  show ?case
paulson@15113
   782
  proof 
paulson@15113
   783
    assume "x \<in> set (a # xs)"
paulson@15113
   784
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   785
      by (simp, blast intro: Cons_eq_appendI)
paulson@15113
   786
  next
paulson@15113
   787
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   788
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
paulson@15113
   789
    show "x \<in> set (a # xs)" 
paulson@15113
   790
      by (cases ys, auto simp add: eq)
paulson@15113
   791
  qed
paulson@15113
   792
qed
wenzelm@13142
   793
nipkow@18049
   794
lemma in_set_conv_decomp_first:
nipkow@18049
   795
 "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
nipkow@18049
   796
proof (induct xs)
nipkow@18049
   797
  case Nil show ?case by simp
nipkow@18049
   798
next
nipkow@18049
   799
  case (Cons a xs)
nipkow@18049
   800
  show ?case
nipkow@18049
   801
  proof cases
nipkow@18049
   802
    assume "x = a" thus ?case using Cons by force
nipkow@18049
   803
  next
nipkow@18049
   804
    assume "x \<noteq> a"
nipkow@18049
   805
    show ?case
nipkow@18049
   806
    proof
nipkow@18049
   807
      assume "x \<in> set (a # xs)"
nipkow@18049
   808
      from prems show "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@18049
   809
	by(fastsimp intro!: Cons_eq_appendI)
nipkow@18049
   810
    next
nipkow@18049
   811
      assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@18049
   812
      then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
nipkow@18049
   813
      show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq)
nipkow@18049
   814
    qed
nipkow@18049
   815
  qed
nipkow@18049
   816
qed
nipkow@18049
   817
nipkow@18049
   818
lemmas split_list       = in_set_conv_decomp[THEN iffD1, standard]
nipkow@18049
   819
lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard]
nipkow@18049
   820
nipkow@18049
   821
paulson@13508
   822
lemma finite_list: "finite A ==> EX l. set l = A"
paulson@13508
   823
apply (erule finite_induct, auto)
paulson@13508
   824
apply (rule_tac x="x#l" in exI, auto)
paulson@13508
   825
done
paulson@13508
   826
kleing@14388
   827
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
   828
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
   829
paulson@15168
   830
nipkow@15392
   831
subsubsection {* @{text filter} *}
wenzelm@13114
   832
wenzelm@13142
   833
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   834
by (induct xs) auto
wenzelm@13114
   835
nipkow@15305
   836
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
   837
by (induct xs) simp_all
nipkow@15305
   838
wenzelm@13142
   839
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   840
by (induct xs) auto
wenzelm@13114
   841
nipkow@16998
   842
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998
   843
by (induct xs) (auto simp add: le_SucI)
nipkow@16998
   844
nipkow@18423
   845
lemma sum_length_filter_compl:
nipkow@18423
   846
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
nipkow@18423
   847
by(induct xs) simp_all
nipkow@18423
   848
wenzelm@13142
   849
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   850
by (induct xs) auto
wenzelm@13114
   851
wenzelm@13142
   852
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   853
by (induct xs) auto
wenzelm@13114
   854
nipkow@16998
   855
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@16998
   856
  by (induct xs) simp_all
nipkow@16998
   857
nipkow@16998
   858
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998
   859
apply (induct xs)
nipkow@16998
   860
 apply auto
nipkow@16998
   861
apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998
   862
apply simp
nipkow@16998
   863
done
wenzelm@13114
   864
nipkow@16965
   865
lemma filter_map:
nipkow@16965
   866
  "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965
   867
by (induct xs) simp_all
nipkow@16965
   868
nipkow@16965
   869
lemma length_filter_map[simp]:
nipkow@16965
   870
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965
   871
by (simp add:filter_map)
nipkow@16965
   872
wenzelm@13142
   873
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   874
by auto
wenzelm@13114
   875
nipkow@15246
   876
lemma length_filter_less:
nipkow@15246
   877
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
   878
proof (induct xs)
nipkow@15246
   879
  case Nil thus ?case by simp
nipkow@15246
   880
next
nipkow@15246
   881
  case (Cons x xs) thus ?case
nipkow@15246
   882
    apply (auto split:split_if_asm)
nipkow@15246
   883
    using length_filter_le[of P xs] apply arith
nipkow@15246
   884
  done
nipkow@15246
   885
qed
wenzelm@13114
   886
nipkow@15281
   887
lemma length_filter_conv_card:
nipkow@15281
   888
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
   889
proof (induct xs)
nipkow@15281
   890
  case Nil thus ?case by simp
nipkow@15281
   891
next
nipkow@15281
   892
  case (Cons x xs)
nipkow@15281
   893
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
   894
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
   895
  show ?case (is "?l = card ?S'")
nipkow@15281
   896
  proof (cases)
nipkow@15281
   897
    assume "p x"
nipkow@15281
   898
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@15281
   899
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   900
    have "length (filter p (x # xs)) = Suc(card ?S)"
wenzelm@23388
   901
      using Cons `p x` by simp
nipkow@15281
   902
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
nipkow@15281
   903
      by (simp add: card_image inj_Suc)
nipkow@15281
   904
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   905
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
   906
    finally show ?thesis .
nipkow@15281
   907
  next
nipkow@15281
   908
    assume "\<not> p x"
nipkow@15281
   909
    hence eq: "?S' = Suc ` ?S"
nipkow@15281
   910
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   911
    have "length (filter p (x # xs)) = card ?S"
wenzelm@23388
   912
      using Cons `\<not> p x` by simp
nipkow@15281
   913
    also have "\<dots> = card(Suc ` ?S)" using fin
nipkow@15281
   914
      by (simp add: card_image inj_Suc)
nipkow@15281
   915
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   916
      by (simp add:card_insert_if)
nipkow@15281
   917
    finally show ?thesis .
nipkow@15281
   918
  qed
nipkow@15281
   919
qed
nipkow@15281
   920
nipkow@17629
   921
lemma Cons_eq_filterD:
nipkow@17629
   922
 "x#xs = filter P ys \<Longrightarrow>
nipkow@17629
   923
  \<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
   924
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
nipkow@17629
   925
proof(induct ys)
nipkow@17629
   926
  case Nil thus ?case by simp
nipkow@17629
   927
next
nipkow@17629
   928
  case (Cons y ys)
nipkow@17629
   929
  show ?case (is "\<exists>x. ?Q x")
nipkow@17629
   930
  proof cases
nipkow@17629
   931
    assume Py: "P y"
nipkow@17629
   932
    show ?thesis
nipkow@17629
   933
    proof cases
nipkow@17629
   934
      assume xy: "x = y"
nipkow@17629
   935
      show ?thesis
nipkow@17629
   936
      proof from Py xy Cons(2) show "?Q []" by simp qed
nipkow@17629
   937
    next
nipkow@17629
   938
      assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp
nipkow@17629
   939
    qed
nipkow@17629
   940
  next
nipkow@17629
   941
    assume Py: "\<not> P y"
nipkow@17629
   942
    with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp
nipkow@17629
   943
    show ?thesis (is "? us. ?Q us")
nipkow@17629
   944
    proof show "?Q (y#us)" using 1 by simp qed
nipkow@17629
   945
  qed
nipkow@17629
   946
qed
nipkow@17629
   947
nipkow@17629
   948
lemma filter_eq_ConsD:
nipkow@17629
   949
 "filter P ys = x#xs \<Longrightarrow>
nipkow@17629
   950
  \<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
   951
by(rule Cons_eq_filterD) simp
nipkow@17629
   952
nipkow@17629
   953
lemma filter_eq_Cons_iff:
nipkow@17629
   954
 "(filter P ys = x#xs) =
nipkow@17629
   955
  (\<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
   956
by(auto dest:filter_eq_ConsD)
nipkow@17629
   957
nipkow@17629
   958
lemma Cons_eq_filter_iff:
nipkow@17629
   959
 "(x#xs = filter P ys) =
nipkow@17629
   960
  (\<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
   961
by(auto dest:Cons_eq_filterD)
nipkow@17629
   962
krauss@19770
   963
lemma filter_cong[fundef_cong, recdef_cong]:
nipkow@17501
   964
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
nipkow@17501
   965
apply simp
nipkow@17501
   966
apply(erule thin_rl)
nipkow@17501
   967
by (induct ys) simp_all
nipkow@17501
   968
nipkow@15281
   969
nipkow@15392
   970
subsubsection {* @{text concat} *}
wenzelm@13114
   971
wenzelm@13142
   972
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
   973
by (induct xs) auto
wenzelm@13114
   974
paulson@18447
   975
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   976
by (induct xss) auto
wenzelm@13114
   977
paulson@18447
   978
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   979
by (induct xss) auto
wenzelm@13114
   980
wenzelm@13142
   981
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
nipkow@13145
   982
by (induct xs) auto
wenzelm@13114
   983
wenzelm@13142
   984
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
   985
by (induct xs) auto
wenzelm@13114
   986
wenzelm@13142
   987
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
   988
by (induct xs) auto
wenzelm@13114
   989
wenzelm@13142
   990
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
   991
by (induct xs) auto
wenzelm@13114
   992
wenzelm@13114
   993
nipkow@15392
   994
subsubsection {* @{text nth} *}
wenzelm@13114
   995
wenzelm@13142
   996
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
   997
by auto
wenzelm@13114
   998
wenzelm@13142
   999
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
  1000
by auto
wenzelm@13114
  1001
wenzelm@13142
  1002
declare nth.simps [simp del]
wenzelm@13114
  1003
wenzelm@13114
  1004
lemma nth_append:
nipkow@13145
  1005
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
paulson@14208
  1006
apply (induct "xs", simp)
paulson@14208
  1007
apply (case_tac n, auto)
nipkow@13145
  1008
done
wenzelm@13114
  1009
nipkow@14402
  1010
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
nipkow@14402
  1011
by (induct "xs") auto
nipkow@14402
  1012
nipkow@14402
  1013
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
nipkow@14402
  1014
by (induct "xs") auto
nipkow@14402
  1015
wenzelm@13142
  1016
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
paulson@14208
  1017
apply (induct xs, simp)
paulson@14208
  1018
apply (case_tac n, auto)
nipkow@13145
  1019
done
wenzelm@13114
  1020
nipkow@18423
  1021
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
nipkow@18423
  1022
by(cases xs) simp_all
nipkow@18423
  1023
nipkow@18049
  1024
nipkow@18049
  1025
lemma list_eq_iff_nth_eq:
nipkow@18049
  1026
 "!!ys. (xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
nipkow@18049
  1027
apply(induct xs)
nipkow@18049
  1028
 apply simp apply blast
nipkow@18049
  1029
apply(case_tac ys)
nipkow@18049
  1030
 apply simp
nipkow@18049
  1031
apply(simp add:nth_Cons split:nat.split)apply blast
nipkow@18049
  1032
done
nipkow@18049
  1033
wenzelm@13142
  1034
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
  1035
apply (induct xs, simp, simp)
nipkow@13145
  1036
apply safe
paulson@14208
  1037
apply (rule_tac x = 0 in exI, simp)
paulson@14208
  1038
 apply (rule_tac x = "Suc i" in exI, simp)
paulson@14208
  1039
apply (case_tac i, simp)
nipkow@13145
  1040
apply (rename_tac j)
paulson@14208
  1041
apply (rule_tac x = j in exI, simp)
nipkow@13145
  1042
done
wenzelm@13114
  1043
nipkow@17501
  1044
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
nipkow@17501
  1045
by(auto simp:set_conv_nth)
nipkow@17501
  1046
nipkow@13145
  1047
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
  1048
by (auto simp add: set_conv_nth)
wenzelm@13114
  1049
wenzelm@13142
  1050
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
  1051
by (auto simp add: set_conv_nth)
wenzelm@13114
  1052
wenzelm@13114
  1053
lemma all_nth_imp_all_set:
nipkow@13145
  1054
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
  1055
by (auto simp add: set_conv_nth)
wenzelm@13114
  1056
wenzelm@13114
  1057
lemma all_set_conv_all_nth:
nipkow@13145
  1058
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
  1059
by (auto simp add: set_conv_nth)
wenzelm@13114
  1060
wenzelm@13114
  1061
nipkow@15392
  1062
subsubsection {* @{text list_update} *}
wenzelm@13114
  1063
wenzelm@13142
  1064
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
nipkow@13145
  1065
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1066
wenzelm@13114
  1067
lemma nth_list_update:
nipkow@13145
  1068
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@13145
  1069
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1070
wenzelm@13142
  1071
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
  1072
by (simp add: nth_list_update)
wenzelm@13114
  1073
wenzelm@13142
  1074
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145
  1075
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1076
wenzelm@13142
  1077
lemma list_update_overwrite [simp]:
nipkow@13145
  1078
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145
  1079
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1080
nipkow@14402
  1081
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
paulson@14208
  1082
apply (induct xs, simp)
nipkow@14187
  1083
apply(simp split:nat.splits)
nipkow@14187
  1084
done
nipkow@14187
  1085
nipkow@17501
  1086
lemma list_update_beyond[simp]: "\<And>i. length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
nipkow@17501
  1087
apply (induct xs)
nipkow@17501
  1088
 apply simp
nipkow@17501
  1089
apply (case_tac i)
nipkow@17501
  1090
apply simp_all
nipkow@17501
  1091
done
nipkow@17501
  1092
wenzelm@13114
  1093
lemma list_update_same_conv:
nipkow@13145
  1094
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145
  1095
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1096
nipkow@14187
  1097
lemma list_update_append1:
nipkow@14187
  1098
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
paulson@14208
  1099
apply (induct xs, simp)
nipkow@14187
  1100
apply(simp split:nat.split)
nipkow@14187
  1101
done
nipkow@14187
  1102
kleing@15868
  1103
lemma list_update_append:
kleing@15868
  1104
  "!!n. (xs @ ys) [n:= x] = 
kleing@15868
  1105
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
kleing@15868
  1106
by (induct xs) (auto split:nat.splits)
kleing@15868
  1107
nipkow@14402
  1108
lemma list_update_length [simp]:
nipkow@14402
  1109
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
  1110
by (induct xs, auto)
nipkow@14402
  1111
wenzelm@13114
  1112
lemma update_zip:
nipkow@13145
  1113
"!!i xy xs. length xs = length ys ==>
nipkow@13145
  1114
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145
  1115
by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114
  1116
wenzelm@13114
  1117
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145
  1118
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1119
wenzelm@13114
  1120
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
  1121
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
  1122
kleing@15868
  1123
lemma set_update_memI: "!!n. n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
kleing@15868
  1124
by (induct xs) (auto split:nat.splits)
kleing@15868
  1125
wenzelm@13114
  1126
nipkow@15392
  1127
subsubsection {* @{text last} and @{text butlast} *}
wenzelm@13114
  1128
wenzelm@13142
  1129
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
  1130
by (induct xs) auto
wenzelm@13114
  1131
wenzelm@13142
  1132
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
  1133
by (induct xs) auto
wenzelm@13114
  1134
nipkow@14302
  1135
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
  1136
by(simp add:last.simps)
nipkow@14302
  1137
nipkow@14302
  1138
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
  1139
by(simp add:last.simps)
nipkow@14302
  1140
nipkow@14302
  1141
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
  1142
by (induct xs) (auto)
nipkow@14302
  1143
nipkow@14302
  1144
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
  1145
by(simp add:last_append)
nipkow@14302
  1146
nipkow@14302
  1147
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
  1148
by(simp add:last_append)
nipkow@14302
  1149
nipkow@17762
  1150
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
nipkow@17762
  1151
by(rule rev_exhaust[of xs]) simp_all
nipkow@17762
  1152
nipkow@17762
  1153
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
nipkow@17762
  1154
by(cases xs) simp_all
nipkow@17762
  1155
nipkow@17765
  1156
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
nipkow@17765
  1157
by (induct as) auto
nipkow@17762
  1158
wenzelm@13142
  1159
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
  1160
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1161
wenzelm@13114
  1162
lemma butlast_append:
nipkow@13145
  1163
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145
  1164
by (induct xs) auto
wenzelm@13114
  1165
wenzelm@13142
  1166
lemma append_butlast_last_id [simp]:
nipkow@13145
  1167
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
  1168
by (induct xs) auto
wenzelm@13114
  1169
wenzelm@13142
  1170
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
  1171
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1172
wenzelm@13114
  1173
lemma in_set_butlast_appendI:
nipkow@13145
  1174
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
  1175
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
  1176
nipkow@17501
  1177
lemma last_drop[simp]: "!!n. n < length xs \<Longrightarrow> last (drop n xs) = last xs"
nipkow@17501
  1178
apply (induct xs)
nipkow@17501
  1179
 apply simp
nipkow@17501
  1180
apply (auto split:nat.split)
nipkow@17501
  1181
done
nipkow@17501
  1182
nipkow@17589
  1183
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
nipkow@17589
  1184
by(induct xs)(auto simp:neq_Nil_conv)
nipkow@17589
  1185
nipkow@15392
  1186
subsubsection {* @{text take} and @{text drop} *}
wenzelm@13114
  1187
wenzelm@13142
  1188
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
  1189
by (induct xs) auto
wenzelm@13114
  1190
wenzelm@13142
  1191
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
  1192
by (induct xs) auto
wenzelm@13114
  1193
wenzelm@13142
  1194
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
  1195
by simp
wenzelm@13114
  1196
wenzelm@13142
  1197
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
  1198
by simp
wenzelm@13114
  1199
wenzelm@13142
  1200
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
  1201
nipkow@15110
  1202
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
  1203
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
  1204
nipkow@14187
  1205
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
  1206
by(cases xs, simp_all)
nipkow@14187
  1207
nipkow@14187
  1208
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
nipkow@14187
  1209
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@14187
  1210
nipkow@14187
  1211
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
paulson@14208
  1212
apply (induct xs, simp)
nipkow@14187
  1213
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
  1214
done
nipkow@14187
  1215
nipkow@13913
  1216
lemma take_Suc_conv_app_nth:
nipkow@13913
  1217
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
paulson@14208
  1218
apply (induct xs, simp)
paulson@14208
  1219
apply (case_tac i, auto)
nipkow@13913
  1220
done
nipkow@13913
  1221
mehta@14591
  1222
lemma drop_Suc_conv_tl:
mehta@14591
  1223
  "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
mehta@14591
  1224
apply (induct xs, simp)
mehta@14591
  1225
apply (case_tac i, auto)
mehta@14591
  1226
done
mehta@14591
  1227
wenzelm@13142
  1228
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
nipkow@13145
  1229
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1230
wenzelm@13142
  1231
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
nipkow@13145
  1232
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1233
wenzelm@13142
  1234
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
nipkow@13145
  1235
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1236
wenzelm@13142
  1237
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
nipkow@13145
  1238
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1239
wenzelm@13142
  1240
lemma take_append [simp]:
nipkow@13145
  1241
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@13145
  1242
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1243
wenzelm@13142
  1244
lemma drop_append [simp]:
nipkow@13145
  1245
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@13145
  1246
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1247
wenzelm@13142
  1248
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
paulson@14208
  1249
apply (induct m, auto)
paulson@14208
  1250
apply (case_tac xs, auto)
nipkow@15236
  1251
apply (case_tac n, auto)
nipkow@13145
  1252
done
wenzelm@13114
  1253
wenzelm@13142
  1254
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
paulson@14208
  1255
apply (induct m, auto)
paulson@14208
  1256
apply (case_tac xs, auto)
nipkow@13145
  1257
done
wenzelm@13114
  1258
wenzelm@13114
  1259
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
paulson@14208
  1260
apply (induct m, auto)
paulson@14208
  1261
apply (case_tac xs, auto)
nipkow@13145
  1262
done
wenzelm@13114
  1263
nipkow@14802
  1264
lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@14802
  1265
apply(induct xs)
nipkow@14802
  1266
 apply simp
nipkow@14802
  1267
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  1268
done
nipkow@14802
  1269
wenzelm@13142
  1270
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
paulson@14208
  1271
apply (induct n, auto)
paulson@14208
  1272
apply (case_tac xs, auto)
nipkow@13145
  1273
done
wenzelm@13114
  1274
nipkow@15110
  1275
lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@15110
  1276
apply(induct xs)
nipkow@15110
  1277
 apply simp
nipkow@15110
  1278
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  1279
done
nipkow@15110
  1280
nipkow@15110
  1281
lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"
nipkow@15110
  1282
apply(induct xs)
nipkow@15110
  1283
apply simp
nipkow@15110
  1284
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1285
done
nipkow@15110
  1286
wenzelm@13114
  1287
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
paulson@14208
  1288
apply (induct n, auto)
paulson@14208
  1289
apply (case_tac xs, auto)
nipkow@13145
  1290
done
wenzelm@13114
  1291
wenzelm@13142
  1292
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
paulson@14208
  1293
apply (induct n, auto)
paulson@14208
  1294
apply (case_tac xs, auto)
nipkow@13145
  1295
done
wenzelm@13114
  1296
wenzelm@13114
  1297
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
paulson@14208
  1298
apply (induct xs, auto)
paulson@14208
  1299
apply (case_tac i, auto)
nipkow@13145
  1300
done
wenzelm@13114
  1301
wenzelm@13114
  1302
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
paulson@14208
  1303
apply (induct xs, auto)
paulson@14208
  1304
apply (case_tac i, auto)
nipkow@13145
  1305
done
wenzelm@13114
  1306
wenzelm@13142
  1307
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
paulson@14208
  1308
apply (induct xs, auto)
paulson@14208
  1309
apply (case_tac n, blast)
paulson@14208
  1310
apply (case_tac i, auto)
nipkow@13145
  1311
done
wenzelm@13114
  1312
wenzelm@13142
  1313
lemma nth_drop [simp]:
nipkow@13145
  1314
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
paulson@14208
  1315
apply (induct n, auto)
paulson@14208
  1316
apply (case_tac xs, auto)
nipkow@13145
  1317
done
nipkow@3507
  1318
nipkow@18423
  1319
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
nipkow@18423
  1320
by(simp add: hd_conv_nth)
nipkow@18423
  1321
nipkow@14025
  1322
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
nipkow@14025
  1323
by(induct xs)(auto simp:take_Cons split:nat.split)
nipkow@14025
  1324
nipkow@14025
  1325
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
nipkow@14025
  1326
by(induct xs)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1327
nipkow@14187
  1328
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1329
using set_take_subset by fast
nipkow@14187
  1330
nipkow@14187
  1331
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1332
using set_drop_subset by fast
nipkow@14187
  1333
wenzelm@13114
  1334
lemma append_eq_conv_conj:
nipkow@13145
  1335
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
paulson@14208
  1336
apply (induct xs, simp, clarsimp)
paulson@14208
  1337
apply (case_tac zs, auto)
nipkow@13145
  1338
done
wenzelm@13142
  1339
paulson@14050
  1340
lemma take_add [rule_format]: 
paulson@14050
  1341
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
paulson@14050
  1342
apply (induct xs, auto) 
paulson@14050
  1343
apply (case_tac i, simp_all) 
paulson@14050
  1344
done
paulson@14050
  1345
nipkow@14300
  1346
lemma append_eq_append_conv_if:
nipkow@14300
  1347
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1348
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1349
   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
  1350
   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
  1351
apply(induct xs\<^isub>1)
nipkow@14300
  1352
 apply simp
nipkow@14300
  1353
apply(case_tac ys\<^isub>1)
nipkow@14300
  1354
apply simp_all
nipkow@14300
  1355
done
nipkow@14300
  1356
nipkow@15110
  1357
lemma take_hd_drop:
nipkow@15110
  1358
  "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
nipkow@15110
  1359
apply(induct xs)
nipkow@15110
  1360
apply simp
nipkow@15110
  1361
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1362
done
nipkow@15110
  1363
nipkow@17501
  1364
lemma id_take_nth_drop:
nipkow@17501
  1365
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
nipkow@17501
  1366
proof -
nipkow@17501
  1367
  assume si: "i < length xs"
nipkow@17501
  1368
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
nipkow@17501
  1369
  moreover
nipkow@17501
  1370
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
nipkow@17501
  1371
    apply (rule_tac take_Suc_conv_app_nth) by arith
nipkow@17501
  1372
  ultimately show ?thesis by auto
nipkow@17501
  1373
qed
nipkow@17501
  1374
  
nipkow@17501
  1375
lemma upd_conv_take_nth_drop:
nipkow@17501
  1376
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1377
proof -
nipkow@17501
  1378
  assume i: "i < length xs"
nipkow@17501
  1379
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
nipkow@17501
  1380
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
nipkow@17501
  1381
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1382
    using i by (simp add: list_update_append)
nipkow@17501
  1383
  finally show ?thesis .
nipkow@17501
  1384
qed
nipkow@17501
  1385
wenzelm@13114
  1386
nipkow@15392
  1387
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1388
wenzelm@13142
  1389
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1390
by (induct xs) auto
wenzelm@13114
  1391
wenzelm@13142
  1392
lemma takeWhile_append1 [simp]:
nipkow@13145
  1393
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1394
by (induct xs) auto
wenzelm@13114
  1395
wenzelm@13142
  1396
lemma takeWhile_append2 [simp]:
nipkow@13145
  1397
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1398
by (induct xs) auto
wenzelm@13114
  1399
wenzelm@13142
  1400
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1401
by (induct xs) auto
wenzelm@13114
  1402
wenzelm@13142
  1403
lemma dropWhile_append1 [simp]:
nipkow@13145
  1404
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1405
by (induct xs) auto
wenzelm@13114
  1406
wenzelm@13142
  1407
lemma dropWhile_append2 [simp]:
nipkow@13145
  1408
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1409
by (induct xs) auto
wenzelm@13114
  1410
wenzelm@13142
  1411
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1412
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1413
nipkow@13913
  1414
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1415
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1416
by(induct xs, auto)
nipkow@13913
  1417
nipkow@13913
  1418
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1419
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1420
by(induct xs, auto)
nipkow@13913
  1421
nipkow@13913
  1422
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1423
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1424
by(induct xs, auto)
nipkow@13913
  1425
nipkow@17501
  1426
text{* The following two lemmmas could be generalized to an arbitrary
nipkow@17501
  1427
property. *}
nipkow@17501
  1428
nipkow@17501
  1429
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1430
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
nipkow@17501
  1431
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
nipkow@17501
  1432
nipkow@17501
  1433
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1434
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
nipkow@17501
  1435
apply(induct xs)
nipkow@17501
  1436
 apply simp
nipkow@17501
  1437
apply auto
nipkow@17501
  1438
apply(subst dropWhile_append2)
nipkow@17501
  1439
apply auto
nipkow@17501
  1440
done
nipkow@17501
  1441
nipkow@18423
  1442
lemma takeWhile_not_last:
nipkow@18423
  1443
 "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
nipkow@18423
  1444
apply(induct xs)
nipkow@18423
  1445
 apply simp
nipkow@18423
  1446
apply(case_tac xs)
nipkow@18423
  1447
apply(auto)
nipkow@18423
  1448
done
nipkow@18423
  1449
krauss@19770
  1450
lemma takeWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1451
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1452
  ==> takeWhile P l = takeWhile Q k"
wenzelm@20503
  1453
  by (induct k arbitrary: l) (simp_all)
krauss@18336
  1454
krauss@19770
  1455
lemma dropWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1456
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1457
  ==> dropWhile P l = dropWhile Q k"
wenzelm@20503
  1458
  by (induct k arbitrary: l, simp_all)
krauss@18336
  1459
wenzelm@13114
  1460
nipkow@15392
  1461
subsubsection {* @{text zip} *}
wenzelm@13114
  1462
wenzelm@13142
  1463
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1464
by (induct ys) auto
wenzelm@13114
  1465
wenzelm@13142
  1466
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1467
by simp
wenzelm@13114
  1468
wenzelm@13142
  1469
declare zip_Cons [simp del]
wenzelm@13114
  1470
nipkow@15281
  1471
lemma zip_Cons1:
nipkow@15281
  1472
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  1473
by(auto split:list.split)
nipkow@15281
  1474
wenzelm@13142
  1475
lemma length_zip [simp]:
krauss@22493
  1476
"length (zip xs ys) = min (length xs) (length ys)"
krauss@22493
  1477
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1478
wenzelm@13114
  1479
lemma zip_append1:
krauss@22493
  1480
"zip (xs @ ys) zs =
nipkow@13145
  1481
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
krauss@22493
  1482
by (induct xs zs rule:list_induct2') auto
wenzelm@13114
  1483
wenzelm@13114
  1484
lemma zip_append2:
krauss@22493
  1485
"zip xs (ys @ zs) =
nipkow@13145
  1486
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
krauss@22493
  1487
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1488
wenzelm@13142
  1489
lemma zip_append [simp]:
wenzelm@13142
  1490
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1491
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1492
by (simp add: zip_append1)
wenzelm@13114
  1493
wenzelm@13114
  1494
lemma zip_rev:
nipkow@14247
  1495
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1496
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1497
nipkow@23096
  1498
lemma map_zip_map:
nipkow@23096
  1499
 "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
nipkow@23096
  1500
apply(induct xs arbitrary:ys) apply simp
nipkow@23096
  1501
apply(case_tac ys)
nipkow@23096
  1502
apply simp_all
nipkow@23096
  1503
done
nipkow@23096
  1504
nipkow@23096
  1505
lemma map_zip_map2:
nipkow@23096
  1506
 "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
nipkow@23096
  1507
apply(induct xs arbitrary:ys) apply simp
nipkow@23096
  1508
apply(case_tac ys)
nipkow@23096
  1509
apply simp_all
nipkow@23096
  1510
done
nipkow@23096
  1511
wenzelm@13142
  1512
lemma nth_zip [simp]:
nipkow@13145
  1513
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
paulson@14208
  1514
apply (induct ys, simp)
nipkow@13145
  1515
apply (case_tac xs)
nipkow@13145
  1516
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1517
done
wenzelm@13114
  1518
wenzelm@13114
  1519
lemma set_zip:
nipkow@13145
  1520
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
  1521
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1522
wenzelm@13114
  1523
lemma zip_update:
nipkow@13145
  1524
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
  1525
by (rule sym, simp add: update_zip)
wenzelm@13114
  1526
wenzelm@13142
  1527
lemma zip_replicate [simp]:
nipkow@13145
  1528
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
paulson@14208
  1529
apply (induct i, auto)
paulson@14208
  1530
apply (case_tac j, auto)
nipkow@13145
  1531
done
wenzelm@13114
  1532
nipkow@19487
  1533
lemma take_zip:
nipkow@19487
  1534
 "!!xs ys. take n (zip xs ys) = zip (take n xs) (take n ys)"
nipkow@19487
  1535
apply (induct n)
nipkow@19487
  1536
 apply simp
nipkow@19487
  1537
apply (case_tac xs, simp)
nipkow@19487
  1538
apply (case_tac ys, simp_all)
nipkow@19487
  1539
done
nipkow@19487
  1540
nipkow@19487
  1541
lemma drop_zip:
nipkow@19487
  1542
 "!!xs ys. drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
nipkow@19487
  1543
apply (induct n)
nipkow@19487
  1544
 apply simp
nipkow@19487
  1545
apply (case_tac xs, simp)
nipkow@19487
  1546
apply (case_tac ys, simp_all)
nipkow@19487
  1547
done
nipkow@19487
  1548
krauss@22493
  1549
lemma set_zip_leftD:
krauss@22493
  1550
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
krauss@22493
  1551
by (induct xs ys rule:list_induct2') auto
krauss@22493
  1552
krauss@22493
  1553
lemma set_zip_rightD:
krauss@22493
  1554
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
krauss@22493
  1555
by (induct xs ys rule:list_induct2') auto
wenzelm@13142
  1556
nipkow@15392
  1557
subsubsection {* @{text list_all2} *}
wenzelm@13114
  1558
kleing@14316
  1559
lemma list_all2_lengthD [intro?]: 
kleing@14316
  1560
  "list_all2 P xs ys ==> length xs = length ys"
haftmann@19607
  1561
  by (simp add: list_all2_def)
haftmann@19607
  1562
haftmann@19787
  1563
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
haftmann@19607
  1564
  by (simp add: list_all2_def)
haftmann@19607
  1565
haftmann@19787
  1566
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
haftmann@19787
  1567
  by (simp add: list_all2_def)
haftmann@19607
  1568
haftmann@19607
  1569
lemma list_all2_Cons [iff, code]:
haftmann@19607
  1570
  "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
haftmann@19607
  1571
  by (auto simp add: list_all2_def)
wenzelm@13114
  1572
wenzelm@13114
  1573
lemma list_all2_Cons1:
nipkow@13145
  1574
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1575
by (cases ys) auto
wenzelm@13114
  1576
wenzelm@13114
  1577
lemma list_all2_Cons2:
nipkow@13145
  1578
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1579
by (cases xs) auto
wenzelm@13114
  1580
wenzelm@13142
  1581
lemma list_all2_rev [iff]:
nipkow@13145
  1582
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1583
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1584
kleing@13863
  1585
lemma list_all2_rev1:
kleing@13863
  1586
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1587
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1588
wenzelm@13114
  1589
lemma list_all2_append1:
nipkow@13145
  1590
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1591
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1592
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1593
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1594
apply (rule iffI)
nipkow@13145
  1595
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1596
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  1597
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1598
apply (simp add: ball_Un)
nipkow@13145
  1599
done
wenzelm@13114
  1600
wenzelm@13114
  1601
lemma list_all2_append2:
nipkow@13145
  1602
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1603
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1604
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1605
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1606
apply (rule iffI)
nipkow@13145
  1607
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1608
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  1609
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1610
apply (simp add: ball_Un)
nipkow@13145
  1611
done
wenzelm@13114
  1612
kleing@13863
  1613
lemma list_all2_append:
nipkow@14247
  1614
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  1615
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  1616
by (induct rule:list_induct2, simp_all)
kleing@13863
  1617
kleing@13863
  1618
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1619
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
kleing@13863
  1620
  by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1621
wenzelm@13114
  1622
lemma list_all2_conv_all_nth:
nipkow@13145
  1623
"list_all2 P xs ys =
nipkow@13145
  1624
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1625
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1626
berghofe@13883
  1627
lemma list_all2_trans:
berghofe@13883
  1628
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1629
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1630
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1631
proof (induct as)
berghofe@13883
  1632
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1633
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1634
  proof (induct bs)
berghofe@13883
  1635
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1636
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1637
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1638
  qed simp
berghofe@13883
  1639
qed simp
berghofe@13883
  1640
kleing@13863
  1641
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1642
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
kleing@13863
  1643
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1644
paulson@14395
  1645
lemma list_all2I:
paulson@14395
  1646
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
paulson@14395
  1647
  by (simp add: list_all2_def)
paulson@14395
  1648
kleing@14328
  1649
lemma list_all2_nthD:
kleing@13863
  1650
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
kleing@13863
  1651
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1652
nipkow@14302
  1653
lemma list_all2_nthD2:
nipkow@14302
  1654
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@14302
  1655
  by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  1656
kleing@13863
  1657
lemma list_all2_map1: 
kleing@13863
  1658
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
kleing@13863
  1659
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1660
kleing@13863
  1661
lemma list_all2_map2: 
kleing@13863
  1662
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
kleing@13863
  1663
  by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1664
kleing@14316
  1665
lemma list_all2_refl [intro?]:
kleing@13863
  1666
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
kleing@13863
  1667
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1668
kleing@13863
  1669
lemma list_all2_update_cong:
kleing@13863
  1670
  "\<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
  1671
  by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1672
kleing@13863
  1673
lemma list_all2_update_cong2:
kleing@13863
  1674
  "\<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
  1675
  by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1676
nipkow@14302
  1677
lemma list_all2_takeI [simp,intro?]:
nipkow@14302
  1678
  "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@14302
  1679
  apply (induct xs)
nipkow@14302
  1680
   apply simp
nipkow@14302
  1681
  apply (clarsimp simp add: list_all2_Cons1)
nipkow@14302
  1682
  apply (case_tac n)
nipkow@14302
  1683
  apply auto
nipkow@14302
  1684
  done
nipkow@14302
  1685
nipkow@14302
  1686
lemma list_all2_dropI [simp,intro?]:
kleing@13863
  1687
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
paulson@14208
  1688
  apply (induct as, simp)
kleing@13863
  1689
  apply (clarsimp simp add: list_all2_Cons1)
paulson@14208
  1690
  apply (case_tac n, simp, simp)
kleing@13863
  1691
  done
kleing@13863
  1692
kleing@14327
  1693
lemma list_all2_mono [intro?]:
kleing@13863
  1694
  "\<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
  1695
  apply (induct x, simp)
paulson@14208
  1696
  apply (case_tac y, auto)
kleing@13863
  1697
  done
kleing@13863
  1698
haftmann@22551
  1699
lemma list_all2_eq:
haftmann@22551
  1700
  "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
haftmann@22551
  1701
  by (induct xs ys rule: list_induct2') auto
haftmann@22551
  1702
wenzelm@13142
  1703
nipkow@15392
  1704
subsubsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  1705
wenzelm@13142
  1706
lemma foldl_append [simp]:
nipkow@13145
  1707
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145
  1708
by (induct xs) auto
wenzelm@13142
  1709
nipkow@14402
  1710
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  1711
by (induct xs) auto
nipkow@14402
  1712
nipkow@23096
  1713
lemma foldr_map: "foldr g (map f xs) a = foldr (g o f) xs a"
nipkow@23096
  1714
by(induct xs) simp_all
nipkow@23096
  1715
nipkow@23096
  1716
lemma foldl_map: "foldl g a (map f xs) = foldl (%a x. g a (f x)) a xs"
nipkow@23096
  1717
by(induct xs arbitrary:a) simp_all
nipkow@23096
  1718
krauss@19770
  1719
lemma foldl_cong [fundef_cong, recdef_cong]:
krauss@18336
  1720
  "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] 
krauss@18336
  1721
  ==> foldl f a l = foldl g b k"
wenzelm@20503
  1722
  by (induct k arbitrary: a b l) simp_all
krauss@18336
  1723
krauss@19770
  1724
lemma foldr_cong [fundef_cong, recdef_cong]:
krauss@18336
  1725
  "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] 
krauss@18336
  1726
  ==> foldr f l a = foldr g k b"
wenzelm@20503
  1727
  by (induct k arbitrary: a b l) simp_all
krauss@18336
  1728
nipkow@23096
  1729
text{* The ``First Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096
  1730
nipkow@23096
  1731
lemma foldl_foldr1_lemma:
nipkow@23096
  1732
 "foldl op + a xs = a + foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096
  1733
by (induct xs arbitrary: a) (auto simp:add_assoc)
nipkow@23096
  1734
nipkow@23096
  1735
corollary foldl_foldr1:
nipkow@23096
  1736
 "foldl op + 0 xs = foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096
  1737
by (simp add:foldl_foldr1_lemma)
nipkow@23096
  1738
nipkow@23096
  1739
nipkow@23096
  1740
text{* The ``Third Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096
  1741
nipkow@14402
  1742
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  1743
by (induct xs) auto
nipkow@14402
  1744
nipkow@14402
  1745
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  1746
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  1747
wenzelm@13142
  1748
text {*
nipkow@13145
  1749
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1750
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1751
*}
wenzelm@13142
  1752
wenzelm@13142
  1753
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
nipkow@13145
  1754
by (induct ns) auto
wenzelm@13142
  1755
wenzelm@13142
  1756
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1757
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1758
wenzelm@13142
  1759
lemma sum_eq_0_conv [iff]:
nipkow@13145
  1760
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145
  1761
by (induct ns) auto
wenzelm@13114
  1762
nipkow@23096
  1763
subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
nipkow@23096
  1764
nipkow@23096
  1765
lemma listsum_foldr:
nipkow@23096
  1766
 "listsum xs = foldr (op +) xs 0"
nipkow@23096
  1767
by(induct xs) auto
nipkow@23096
  1768
nipkow@23096
  1769
(* for efficient code generation *)
nipkow@23096
  1770
lemma listsum[code]: "listsum xs = foldl (op +) 0 xs"
nipkow@23096
  1771
by(simp add:listsum_foldr foldl_foldr1)
nipkow@23096
  1772
nipkow@23096
  1773
text{* Some syntactic sugar for summing a function over a list: *}
nipkow@23096
  1774
nipkow@23096
  1775
syntax
nipkow@23096
  1776
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
nipkow@23096
  1777
syntax (xsymbols)
nipkow@23096
  1778
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096
  1779
syntax (HTML output)
nipkow@23096
  1780
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096
  1781
nipkow@23096
  1782
translations -- {* Beware of argument permutation! *}
nipkow@23096
  1783
  "SUM x<-xs. b" == "CONST listsum (map (%x. b) xs)"
nipkow@23096
  1784
  "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (map (%x. b) xs)"
nipkow@23096
  1785
nipkow@23096
  1786
lemma listsum_0 [simp]: "(\<Sum>x\<leftarrow>xs. 0) = 0"
nipkow@23096
  1787
by (induct xs) simp_all
nipkow@23096
  1788
nipkow@23096
  1789
text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
nipkow@23096
  1790
lemma uminus_listsum_map:
nipkow@23096
  1791
 "- listsum (map f xs) = (listsum (map (uminus o f) xs) :: 'a::ab_group_add)"
nipkow@23096
  1792
by(induct xs) simp_all
nipkow@23096
  1793
wenzelm@13114
  1794
nipkow@15392
  1795
subsubsection {* @{text upto} *}
wenzelm@13114
  1796
nipkow@17090
  1797
lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
nipkow@17090
  1798
-- {* simp does not terminate! *}
nipkow@13145
  1799
by (induct j) auto
wenzelm@13142
  1800
nipkow@15425
  1801
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
nipkow@13145
  1802
by (subst upt_rec) simp
wenzelm@13114
  1803
nipkow@15425
  1804
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
nipkow@15281
  1805
by(induct j)simp_all
nipkow@15281
  1806
nipkow@15281
  1807
lemma upt_eq_Cons_conv:
nipkow@15425
  1808
 "!!x xs. ([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
nipkow@15281
  1809
apply(induct j)
nipkow@15281
  1810
 apply simp
nipkow@15281
  1811
apply(clarsimp simp add: append_eq_Cons_conv)
nipkow@15281
  1812
apply arith
nipkow@15281
  1813
done
nipkow@15281
  1814
nipkow@15425
  1815
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
nipkow@13145
  1816
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1817
by simp
wenzelm@13114
  1818
nipkow@15425
  1819
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
nipkow@13145
  1820
apply(rule trans)
nipkow@13145
  1821
apply(subst upt_rec)
paulson@14208
  1822
 prefer 2 apply (rule refl, simp)
nipkow@13145
  1823
done
wenzelm@13114
  1824
nipkow@15425
  1825
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
nipkow@13145
  1826
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1827
by (induct k) auto
wenzelm@13114
  1828
nipkow@15425
  1829
lemma length_upt [simp]: "length [i..<j] = j - i"
nipkow@13145
  1830
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1831
nipkow@15425
  1832
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
nipkow@13145
  1833
apply (induct j)
nipkow@13145
  1834
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1835
done
wenzelm@13114
  1836
nipkow@17906
  1837
nipkow@17906
  1838
lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
nipkow@17906
  1839
by(simp add:upt_conv_Cons)
nipkow@17906
  1840
nipkow@17906
  1841
lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
nipkow@17906
  1842
apply(cases j)
nipkow@17906
  1843
 apply simp
nipkow@17906
  1844
by(simp add:upt_Suc_append)
nipkow@17906
  1845
nipkow@15425
  1846
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..<n] = [i..<i+m]"
paulson@14208
  1847
apply (induct m, simp)
nipkow@13145
  1848
apply (subst upt_rec)
nipkow@13145
  1849
apply (rule sym)
nipkow@13145
  1850
apply (subst upt_rec)
nipkow@13145
  1851
apply (simp del: upt.simps)
nipkow@13145
  1852
done
nipkow@3507
  1853
nipkow@17501
  1854
lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
nipkow@17501
  1855
apply(induct j)
nipkow@17501
  1856
apply auto
nipkow@17501
  1857
done
nipkow@17501
  1858
nipkow@15425
  1859
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..n]"
nipkow@13145
  1860
by (induct n) auto
wenzelm@13114
  1861
nipkow@15425
  1862
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
nipkow@13145
  1863
apply (induct n m rule: diff_induct)
nipkow@13145
  1864
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1865
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1866
done
wenzelm@13114
  1867
berghofe@13883
  1868
lemma nth_take_lemma:
berghofe@13883
  1869
  "!!xs ys. k <= length xs ==> k <= length ys ==>
berghofe@13883
  1870
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
berghofe@13883
  1871
apply (atomize, induct k)
paulson@14208
  1872
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  1873
txt {* Both lists must be non-empty *}
paulson@14208
  1874
apply (case_tac xs, simp)
paulson@14208
  1875
apply (case_tac ys, clarify)
nipkow@13145
  1876
 apply (simp (no_asm_use))
nipkow@13145
  1877
apply clarify
nipkow@13145
  1878
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  1879
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  1880
apply blast
nipkow@13145
  1881
done
wenzelm@13114
  1882
wenzelm@13114
  1883
lemma nth_equalityI:
wenzelm@13114
  1884
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  1885
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  1886
apply (simp_all add: take_all)
nipkow@13145
  1887
done
wenzelm@13142
  1888
kleing@13863
  1889
(* needs nth_equalityI *)
kleing@13863
  1890
lemma list_all2_antisym:
kleing@13863
  1891
  "\<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
  1892
  \<Longrightarrow> xs = ys"
kleing@13863
  1893
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  1894
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  1895
  done
kleing@13863
  1896
wenzelm@13142
  1897
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  1898
-- {* The famous take-lemma. *}
nipkow@13145
  1899
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  1900
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  1901
done
wenzelm@13142
  1902
wenzelm@13142
  1903
nipkow@15302
  1904
lemma take_Cons':
nipkow@15302
  1905
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@15302
  1906
by (cases n) simp_all
nipkow@15302
  1907
nipkow@15302
  1908
lemma drop_Cons':
nipkow@15302
  1909
     "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@15302
  1910
by (cases n) simp_all
nipkow@15302
  1911
nipkow@15302
  1912
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@15302
  1913
by (cases n) simp_all
nipkow@15302
  1914
paulson@18622
  1915
lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard]
paulson@18622
  1916
lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard]
paulson@18622
  1917
lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard]
paulson@18622
  1918
paulson@18622
  1919
declare take_Cons_number_of [simp] 
paulson@18622
  1920
        drop_Cons_number_of [simp] 
paulson@18622
  1921
        nth_Cons_number_of [simp] 
nipkow@15302
  1922
nipkow@15302
  1923
nipkow@15392
  1924
subsubsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  1925
wenzelm@13142
  1926
lemma distinct_append [simp]:
nipkow@13145
  1927
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  1928
by (induct xs) auto
wenzelm@13142
  1929
nipkow@15305
  1930
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
nipkow@15305
  1931
by(induct xs) auto
nipkow@15305
  1932
wenzelm@13142
  1933
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  1934
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  1935
wenzelm@13142
  1936
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  1937
by (induct xs) auto
wenzelm@13142
  1938
paulson@15072
  1939
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
paulson@15251
  1940
  by (induct x, auto) 
paulson@15072
  1941
paulson@15072
  1942
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
paulson@15251
  1943
  by (induct x, auto)
paulson@15072
  1944
nipkow@15245
  1945
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245
  1946
by (induct xs) auto
nipkow@15245
  1947
nipkow@15245
  1948
lemma length_remdups_eq[iff]:
nipkow@15245
  1949
  "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245
  1950
apply(induct xs)
nipkow@15245
  1951
 apply auto
nipkow@15245
  1952
apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245
  1953
 apply arith
nipkow@15245
  1954
apply(rule length_remdups_leq)
nipkow@15245
  1955
done
nipkow@15245
  1956
nipkow@18490
  1957
nipkow@18490
  1958
lemma distinct_map:
nipkow@18490
  1959
  "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
nipkow@18490
  1960
by (induct xs) auto
nipkow@18490
  1961
nipkow@18490
  1962
wenzelm@13142
  1963
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  1964
by (induct xs) auto
wenzelm@13114
  1965
nipkow@17501
  1966
lemma distinct_upt[simp]: "distinct[i..<j]"
nipkow@17501
  1967
by (induct j) auto
nipkow@17501
  1968
nipkow@17501
  1969
lemma distinct_take[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (take i xs)"
nipkow@17501
  1970
apply(induct xs)
nipkow@17501
  1971
 apply simp
nipkow@17501
  1972
apply (case_tac i)
nipkow@17501
  1973
 apply simp_all
nipkow@17501
  1974
apply(blast dest:in_set_takeD)
nipkow@17501
  1975
done
nipkow@17501
  1976
nipkow@17501
  1977
lemma distinct_drop[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (drop i xs)"
nipkow@17501
  1978
apply(induct xs)
nipkow@17501
  1979
 apply simp
nipkow@17501
  1980
apply (case_tac i)
nipkow@17501
  1981
 apply simp_all
nipkow@17501
  1982
done
nipkow@17501
  1983
nipkow@17501
  1984
lemma distinct_list_update:
nipkow@17501
  1985
assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
nipkow@17501
  1986
shows "distinct (xs[i:=a])"
nipkow@17501
  1987
proof (cases "i < length xs")
nipkow@17501
  1988
  case True
nipkow@17501
  1989
  with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
nipkow@17501
  1990
    apply (drule_tac id_take_nth_drop) by simp
nipkow@17501
  1991
  with d True show ?thesis
nipkow@17501
  1992
    apply (simp add: upd_conv_take_nth_drop)
nipkow@17501
  1993
    apply (drule subst [OF id_take_nth_drop]) apply assumption
nipkow@17501
  1994
    apply simp apply (cases "a = xs!i") apply simp by blast
nipkow@17501
  1995
next
nipkow@17501
  1996
  case False with d show ?thesis by auto
nipkow@17501
  1997
qed
nipkow@17501
  1998
nipkow@17501
  1999
nipkow@17501
  2000
text {* It is best to avoid this indexed version of distinct, but
nipkow@17501
  2001
sometimes it is useful. *}
nipkow@17501
  2002
wenzelm@13142
  2003
lemma distinct_conv_nth:
nipkow@17501
  2004
"distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@15251
  2005
apply (induct xs, simp, simp)
paulson@14208
  2006
apply (rule iffI, clarsimp)
nipkow@13145
  2007
 apply (case_tac i)
paulson@14208
  2008
apply (case_tac j, simp)
nipkow@13145
  2009
apply (simp add: set_conv_nth)
nipkow@13145
  2010
 apply (case_tac j)
paulson@14208
  2011
apply (clarsimp simp add: set_conv_nth, simp)
nipkow@13145
  2012
apply (rule conjI)
nipkow@13145
  2013
 apply (clarsimp simp add: set_conv_nth)
nipkow@17501
  2014
 apply (erule_tac x = 0 in allE, simp)
paulson@14208
  2015
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
nipkow@17501
  2016
apply (erule_tac x = "Suc i" in allE, simp)
paulson@14208
  2017
apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145
  2018
done
wenzelm@13114
  2019
nipkow@18490
  2020
lemma nth_eq_iff_index_eq:
nipkow@18490
  2021
 "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
nipkow@18490
  2022
by(auto simp: distinct_conv_nth)
nipkow@18490
  2023
nipkow@15110
  2024
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
kleing@14388
  2025
  by (induct xs) auto
kleing@14388
  2026
nipkow@15110
  2027
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388
  2028
proof (induct xs)
kleing@14388
  2029
  case Nil thus ?case by simp
kleing@14388
  2030
next
kleing@14388
  2031
  case (Cons x xs)
kleing@14388
  2032
  show ?case
kleing@14388
  2033
  proof (cases "x \<in> set xs")
kleing@14388
  2034
    case False with Cons show ?thesis by simp
kleing@14388
  2035
  next
kleing@14388
  2036
    case True with Cons.prems
kleing@14388
  2037
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  2038
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  2039
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  2040
    ultimately have False by simp
kleing@14388
  2041
    thus ?thesis ..
kleing@14388
  2042
  qed
kleing@14388
  2043
qed
kleing@14388
  2044
nipkow@18490
  2045
nipkow@18490
  2046
lemma length_remdups_concat:
nipkow@18490
  2047
 "length(remdups(concat xss)) = card(\<Union>xs \<in> set xss. set xs)"
nipkow@18490
  2048
by(simp add: distinct_card[symmetric])
nipkow@17906
  2049
nipkow@17906
  2050
nipkow@15392
  2051
subsubsection {* @{text remove1} *}
nipkow@15110
  2052
nipkow@18049
  2053
lemma remove1_append:
nipkow@18049
  2054
  "remove1 x (xs @ ys) =
nipkow@18049
  2055
  (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
nipkow@18049
  2056
by (induct xs) auto
nipkow@18049
  2057
nipkow@23479
  2058
lemma in_set_remove1[simp]:
nipkow@23479
  2059
  "a \<noteq> b \<Longrightarrow> a : set(remove1 b xs) = (a : set xs)"
nipkow@23479
  2060
apply (induct xs)
nipkow@23479
  2061
apply auto
nipkow@23479
  2062
done
nipkow@23479
  2063
nipkow@15110
  2064
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110
  2065
apply(induct xs)
nipkow@15110
  2066
 apply simp
nipkow@15110
  2067
apply simp
nipkow@15110
  2068
apply blast
nipkow@15110
  2069
done
nipkow@15110
  2070
paulson@17724
  2071
lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110
  2072
apply(induct xs)
nipkow@15110
  2073
 apply simp
nipkow@15110
  2074
apply simp
nipkow@15110
  2075
apply blast
nipkow@15110
  2076
done
nipkow@15110
  2077
nipkow@23479
  2078
lemma length_remove1:
nipkow@23479
  2079
  "length(remove1 x xs) = (if x : set xs then length xs - 1 else length xs)"
nipkow@23479
  2080
apply (induct xs)
nipkow@23479
  2081
 apply (auto dest!:length_pos_if_in_set)
nipkow@23479
  2082
done
nipkow@23479
  2083
nipkow@18049
  2084
lemma remove1_filter_not[simp]:
nipkow@18049
  2085
  "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
nipkow@18049
  2086
by(induct xs) auto
nipkow@18049
  2087
nipkow@15110
  2088
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110
  2089
apply(insert set_remove1_subset)
nipkow@15110
  2090
apply fast
nipkow@15110
  2091
done
nipkow@15110
  2092
nipkow@15110
  2093
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110
  2094
by (induct xs) simp_all
nipkow@15110
  2095
wenzelm@13114
  2096
nipkow@15392
  2097
subsubsection {* @{text replicate} *}
wenzelm@13114
  2098
wenzelm@13142
  2099
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  2100
by (induct n) auto
nipkow@13124
  2101
wenzelm@13142
  2102
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  2103
by (induct n) auto
wenzelm@13114
  2104
wenzelm@13114
  2105
lemma replicate_app_Cons_same:
nipkow@13145
  2106
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  2107
by (induct n) auto
wenzelm@13114
  2108
wenzelm@13142
  2109
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208
  2110
apply (induct n, simp)
nipkow@13145
  2111
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  2112
done
wenzelm@13114
  2113
wenzelm@13142
  2114
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  2115
by (induct n) auto
wenzelm@13114
  2116
nipkow@16397
  2117
text{* Courtesy of Matthias Daum: *}
nipkow@16397
  2118
lemma append_replicate_commute:
nipkow@16397
  2119
  "replicate n x @ replicate k x = replicate k x @ replicate n x"
nipkow@16397
  2120
apply (simp add: replicate_add [THEN sym])
nipkow@16397
  2121
apply (simp add: add_commute)
nipkow@16397
  2122
done
nipkow@16397
  2123
wenzelm@13142
  2124
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  2125
by (induct n) auto
wenzelm@13114
  2126
wenzelm@13142
  2127
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  2128
by (induct n) auto
wenzelm@13114
  2129
wenzelm@13142
  2130
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  2131
by (atomize (full), induct n) auto
wenzelm@13114
  2132
wenzelm@13142
  2133
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
paulson@14208
  2134
apply (induct n, simp)
nipkow@13145
  2135
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  2136
done
wenzelm@13114
  2137
nipkow@16397
  2138
text{* Courtesy of Matthias Daum (2 lemmas): *}
nipkow@16397
  2139
lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
nipkow@16397
  2140
apply (case_tac "k \<le> i")
nipkow@16397
  2141
 apply  (simp add: min_def)
nipkow@16397
  2142
apply (drule not_leE)
nipkow@16397
  2143
apply (simp add: min_def)
nipkow@16397
  2144
apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
nipkow@16397
  2145
 apply  simp
nipkow@16397
  2146
apply (simp add: replicate_add [symmetric])
nipkow@16397
  2147
done
nipkow@16397
  2148
nipkow@16397
  2149
lemma drop_replicate[simp]: "!!i. drop i (replicate k x) = replicate (k-i) x"
nipkow@16397
  2150
apply (induct k)
nipkow@16397
  2151
 apply simp
nipkow@16397
  2152
apply clarsimp
nipkow@16397
  2153
apply (case_tac i)
nipkow@16397
  2154
 apply simp
nipkow@16397
  2155
apply clarsimp
nipkow@16397
  2156
done
nipkow@16397
  2157
nipkow@16397
  2158
wenzelm@13142
  2159
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  2160
by (induct n) auto
wenzelm@13114
  2161
wenzelm@13142
  2162
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  2163
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  2164
wenzelm@13142
  2165
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  2166
by auto
wenzelm@13114
  2167
wenzelm@13142
  2168
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  2169
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  2170
wenzelm@13114
  2171
nipkow@15392
  2172
subsubsection{*@{text rotate1} and @{text rotate}*}
nipkow@15302
  2173
nipkow@15302
  2174
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
nipkow@15302
  2175
by(simp add:rotate1_def)
nipkow@15302
  2176
nipkow@15302
  2177
lemma rotate0[simp]: "rotate 0 = id"
nipkow@15302
  2178
by(simp add:rotate_def)
nipkow@15302
  2179
nipkow@15302
  2180
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
nipkow@15302
  2181
by(simp add:rotate_def)
nipkow@15302
  2182
nipkow@15302
  2183
lemma rotate_add:
nipkow@15302
  2184
  "rotate (m+n) = rotate m o rotate n"
nipkow@15302
  2185
by(simp add:rotate_def funpow_add)
nipkow@15302
  2186
nipkow@15302
  2187
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
nipkow@15302
  2188
by(simp add:rotate_add)
nipkow@15302
  2189
nipkow@18049
  2190
lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
nipkow@18049
  2191
by(simp add:rotate_def funpow_swap1)
nipkow@18049
  2192
nipkow@15302
  2193
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
nipkow@15302
  2194
by(cases xs) simp_all
nipkow@15302
  2195
nipkow@15302
  2196
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2197
apply(induct n)
nipkow@15302
  2198
 apply simp
nipkow@15302
  2199
apply (simp add:rotate_def)
nipkow@13145
  2200
done
wenzelm@13114
  2201
nipkow@15302
  2202
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
nipkow@15302
  2203
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2204
nipkow@15302
  2205
lemma rotate_drop_take:
nipkow@15302
  2206
  "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
nipkow@15302
  2207
apply(induct n)
nipkow@15302
  2208
 apply simp
nipkow@15302
  2209
apply(simp add:rotate_def)
nipkow@15302
  2210
apply(cases "xs = []")
nipkow@15302
  2211
 apply (simp)
nipkow@15302
  2212
apply(case_tac "n mod length xs = 0")
nipkow@15302
  2213
 apply(simp add:mod_Suc)
nipkow@15302
  2214
 apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
nipkow@15302
  2215
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
nipkow@15302
  2216
                take_hd_drop linorder_not_le)
nipkow@13145
  2217
done
wenzelm@13114
  2218
nipkow@15302
  2219
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
nipkow@15302
  2220
by(simp add:rotate_drop_take)
nipkow@15302
  2221
nipkow@15302
  2222
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2223
by(simp add:rotate_drop_take)
nipkow@15302
  2224
nipkow@15302
  2225
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
nipkow@15302
  2226
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2227
nipkow@15302
  2228
lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs"
nipkow@15302
  2229
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2230
nipkow@15302
  2231
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
nipkow@15302
  2232
by(simp add:rotate1_def split:list.split) blast
nipkow@15302
  2233
nipkow@15302
  2234
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
nipkow@15302
  2235
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2236
nipkow@15302
  2237
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
nipkow@15302
  2238
by(simp add:rotate_drop_take take_map drop_map)
nipkow@15302
  2239
nipkow@15302
  2240
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
nipkow@15302
  2241
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2242
nipkow@15302
  2243
lemma set_rotate[simp]: "set(rotate n xs) = set xs"
nipkow@15302
  2244
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2245
nipkow@15302
  2246
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
nipkow@15302
  2247
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2248
nipkow@15302
  2249
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
nipkow@15302
  2250
by (induct n) (simp_all add:rotate_def)
wenzelm@13114
  2251
nipkow@15439
  2252
lemma rotate_rev:
nipkow@15439
  2253
  "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
nipkow@15439
  2254
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2255
apply(cases "length xs = 0")
nipkow@15439
  2256
 apply simp
nipkow@15439
  2257
apply(cases "n mod length xs = 0")
nipkow@15439
  2258
 apply simp
nipkow@15439
  2259
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2260
done
nipkow@15439
  2261
nipkow@18423
  2262
lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
nipkow@18423
  2263
apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
nipkow@18423
  2264
apply(subgoal_tac "length xs \<noteq> 0")
nipkow@18423
  2265
 prefer 2 apply simp
nipkow@18423
  2266
using mod_less_divisor[of "length xs" n] by arith
nipkow@18423
  2267
wenzelm@13114
  2268
nipkow@15392
  2269
subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  2270
wenzelm@13142
  2271
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  2272
by (auto simp add: sublist_def)
wenzelm@13114
  2273
wenzelm@13142
  2274
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  2275
by (auto simp add: sublist_def)
wenzelm@13114
  2276
nipkow@15281
  2277
lemma length_sublist:
nipkow@15281
  2278
  "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
nipkow@15281
  2279
by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
nipkow@15281
  2280
nipkow@15281
  2281
lemma sublist_shift_lemma_Suc:
nipkow@15281
  2282
  "!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
nipkow@15281
  2283
         map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
nipkow@15281
  2284
apply(induct xs)
nipkow@15281
  2285
 apply simp
nipkow@15281
  2286
apply (case_tac "is")
nipkow@15281
  2287
 apply simp
nipkow@15281
  2288
apply simp
nipkow@15281
  2289
done
nipkow@15281
  2290
wenzelm@13114
  2291
lemma sublist_shift_lemma:
nipkow@23279
  2292
     "map fst [p<-zip xs [i..<i + length xs] . snd p : A] =
nipkow@23279
  2293
      map fst [p<-zip xs [0..<length xs] . snd p + i : A]"
nipkow@13145
  2294
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  2295
wenzelm@13114
  2296
lemma sublist_append:
paulson@15168
  2297
     "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  2298
apply (unfold sublist_def)
paulson@14208
  2299
apply (induct l' rule: rev_induct, simp)
nipkow@13145
  2300
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  2301
apply (simp add: add_commute)
nipkow@13145
  2302
done
wenzelm@13114
  2303
wenzelm@13114
  2304
lemma sublist_Cons:
nipkow@13145
  2305
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  2306
apply (induct l rule: rev_induct)
nipkow@13145
  2307
 apply (simp add: sublist_def)
nipkow@13145
  2308
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  2309
done
wenzelm@13114
  2310
nipkow@15281
  2311
lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
nipkow@15281
  2312
apply(induct xs)
nipkow@15281
  2313
 apply simp
nipkow@15281
  2314
apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE)
nipkow@15281
  2315
 apply(erule lessE)
nipkow@15281
  2316
  apply auto
nipkow@15281
  2317
apply(erule lessE)
nipkow@15281
  2318
apply auto
nipkow@15281
  2319
done
nipkow@15281
  2320
nipkow@15281
  2321
lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
nipkow@15281
  2322
by(auto simp add:set_sublist)
nipkow@15281
  2323
nipkow@15281
  2324
lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
nipkow@15281
  2325
by(auto simp add:set_sublist)
nipkow@15281
  2326
nipkow@15281
  2327
lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
nipkow@15281
  2328
by(auto simp add:set_sublist)
nipkow@15281
  2329
wenzelm@13142
  2330
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  2331
by (simp add: sublist_Cons)
wenzelm@13114
  2332
nipkow@15281
  2333
nipkow@15281
  2334
lemma distinct_sublistI[simp]: "!!I. distinct xs \<Longrightarrow> distinct(sublist xs I)"
nipkow@15281
  2335
apply(induct xs)
nipkow@15281
  2336
 apply simp
nipkow@15281
  2337
apply(auto simp add:sublist_Cons)
nipkow@15281
  2338
done
nipkow@15281
  2339
nipkow@15281
  2340
nipkow@15045
  2341
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
paulson@14208
  2342
apply (induct l rule: rev_induct, simp)
nipkow@13145
  2343
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  2344
done
wenzelm@13114
  2345
nipkow@17501
  2346
lemma filter_in_sublist: "\<And>s. distinct xs \<Longrightarrow>
nipkow@17501
  2347
  filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
nipkow@17501
  2348
proof (induct xs)
nipkow@17501
  2349
  case Nil thus ?case by simp
nipkow@17501
  2350
next
nipkow@17501
  2351
  case (Cons a xs)
nipkow@17501
  2352
  moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
nipkow@17501
  2353
  ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
nipkow@17501
  2354
qed
nipkow@17501
  2355
wenzelm@13114
  2356
nipkow@19390
  2357
subsubsection {* @{const splice} *}
nipkow@19390
  2358
haftmann@19607
  2359
lemma splice_Nil2 [simp, code]:
nipkow@19390
  2360
 "splice xs [] = xs"
nipkow@19390
  2361
by (cases xs) simp_all
nipkow@19390
  2362
haftmann@19607
  2363
lemma splice_Cons_Cons [simp, code]:
nipkow@19390
  2364
 "splice (x#xs) (y#ys) = x # y # splice xs ys"
nipkow@19390
  2365
by simp
nipkow@19390
  2366
haftmann@19607
  2367
declare splice.simps(2) [simp del, code del]
nipkow@19390
  2368
nipkow@22793
  2369
lemma length_splice[simp]: "!!ys. length(splice xs ys) = length xs + length ys"
nipkow@22793
  2370
apply(induct xs) apply simp
nipkow@22793
  2371
apply(case_tac ys)
nipkow@22793
  2372
 apply auto
nipkow@22793
  2373
done
nipkow@22793
  2374
nipkow@22828
  2375
subsubsection {* @{const allpairs} *}
nipkow@22828
  2376
nipkow@22940
  2377
lemma allpairs_conv_concat:
nipkow@22940
  2378
 "allpairs f xs ys = concat(map (%x. map (f x) ys) xs)"
nipkow@22940
  2379
by(induct xs) auto
nipkow@22940
  2380
nipkow@22828
  2381
lemma allpairs_append:
nipkow@22830
  2382
 "allpairs f (xs @ ys) zs = allpairs f xs zs @ allpairs f ys zs"
nipkow@22828
  2383
by(induct xs) auto
nipkow@22828
  2384
nipkow@15392
  2385
subsubsection {* @{text lists}: the list-forming operator over sets *}
nipkow@15302
  2386
berghofe@22262
  2387
inductive2
berghofe@22262
  2388
  listsp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
berghofe@22262
  2389
  for A :: "'a \<Rightarrow> bool"
berghofe@22262
  2390
where
berghofe@22262
  2391
    Nil [intro!]: "listsp A []"
berghofe@22262
  2392
  | Cons [intro!]: "[| A a; listsp A l |] ==> listsp A (a # l)"
berghofe@22262
  2393
berghofe@22262
  2394
constdefs
berghofe@22262
  2395
  lists :: "'a set => 'a list set"
berghofe@22262
  2396
  "lists A == Collect (listsp (member A))"
berghofe@22262
  2397
berghofe@22262
  2398
lemma listsp_lists_eq [pred_set_conv]: "listsp (member A) = member (lists A)"
berghofe@22262
  2399
  by (simp add: lists_def)
berghofe@22262
  2400
berghofe@22262
  2401
lemmas lists_intros [intro!] = listsp.intros [to_set]
berghofe@22262
  2402
berghofe@22262
  2403
lemmas lists_induct [consumes 1, case_names Nil Cons, induct set: lists] =
berghofe@22262
  2404
  listsp.induct [to_set]
berghofe@22262
  2405
berghofe@22262
  2406
inductive_cases2 listspE [elim!]: "listsp A (x # l)"
berghofe@22262
  2407
berghofe@22262
  2408
lemmas listsE [elim!] = listspE [to_set]
berghofe@22262
  2409
berghofe@22262
  2410
lemma listsp_mono [mono2]: "A \<le> B ==> listsp A \<le> listsp B"
berghofe@22262
  2411
  by (clarify, erule listsp.induct, blast+)
berghofe@22262
  2412
berghofe@22262
  2413
lemmas lists_mono [mono] = listsp_mono [to_set]
berghofe@22262
  2414
haftmann@22422
  2415
lemma listsp_infI:
haftmann@22422
  2416
  assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
nipkow@15302
  2417
  by induct blast+
nipkow@15302
  2418
haftmann@22422
  2419
lemmas lists_IntI = listsp_infI [to_set]
haftmann@22422
  2420
haftmann@22422
  2421
lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
haftmann@22422
  2422
proof (rule mono_inf [where f=listsp, THEN order_antisym])
berghofe@22262
  2423
  show "mono listsp" by (simp add: mono_def listsp_mono)
haftmann@22422
  2424
  show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro: listsp_infI)
kleing@14388
  2425
qed
kleing@14388
  2426
haftmann@22422
  2427
lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_eq inf_bool_eq]
haftmann@22422
  2428
haftmann@22422
  2429
lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set]
berghofe@22262
  2430
berghofe@22262
  2431
lemma append_in_listsp_conv [iff]:
berghofe@22262
  2432
     "(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
nipkow@15302
  2433
by (induct xs) auto
nipkow@15302
  2434
berghofe@22262
  2435
lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]
berghofe@22262
  2436
berghofe@22262
  2437
lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
berghofe@22262
  2438
-- {* eliminate @{text listsp} in favour of @{text set} *}
nipkow@15302
  2439
by (induct xs) auto
nipkow@15302
  2440
berghofe@22262
  2441
lemmas in_lists_conv_set = in_listsp_conv_set [to_set]
berghofe@22262
  2442
berghofe@22262
  2443
lemma in_listspD [dest!]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
berghofe@22262
  2444
by (rule in_listsp_conv_set [THEN iffD1])
berghofe@22262
  2445
berghofe@22262
  2446
lemmas in_listsD [dest!] = in_listspD [to_set]
berghofe@22262
  2447
berghofe@22262
  2448
lemma in_listspI [intro!]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
berghofe@22262
  2449
by (rule in_listsp_conv_set [THEN iffD2])
berghofe@22262
  2450
berghofe@22262
  2451
lemmas in_listsI [intro!] = in_listspI [to_set]
nipkow@15302
  2452
nipkow@15302
  2453
lemma lists_UNIV [simp]: "lists UNIV = UNIV"
nipkow@15302
  2454
by auto
nipkow@15302
  2455
nipkow@17086
  2456
nipkow@17086
  2457
nipkow@17086
  2458
subsubsection{* Inductive definition for membership *}
nipkow@17086
  2459
berghofe@22262
  2460
inductive2 ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
berghofe@22262
  2461
where
berghofe@22262
  2462
    elem:  "ListMem x (x # xs)"
berghofe@22262
  2463
  | insert:  "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"
berghofe@22262
  2464
berghofe@22262
  2465
lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
nipkow@17086
  2466
apply (rule iffI)
nipkow@17086
  2467
 apply (induct set: ListMem)
nipkow@17086
  2468
  apply auto
nipkow@17086
  2469
apply (induct xs)
nipkow@17086
  2470
 apply (auto intro: ListMem.intros)
nipkow@17086
  2471
done
nipkow@17086
  2472
nipkow@17086
  2473
nipkow@17086
  2474
nipkow@15392
  2475
subsubsection{*Lists as Cartesian products*}
nipkow@15302
  2476
nipkow@15302
  2477
text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
nipkow@15302
  2478
@{term A} and tail drawn from @{term Xs}.*}
nipkow@15302
  2479
nipkow@15302
  2480
constdefs
nipkow@15302
  2481
  set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
nipkow@15302
  2482
  "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"
nipkow@15302
  2483
paulson@17724
  2484
lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
nipkow@15302
  2485
by (auto simp add: set_Cons_def)
nipkow@15302
  2486
nipkow@15302
  2487
text{*Yields the set of lists, all of the same length as the argument and
nipkow@15302
  2488
with elements drawn from the corresponding element of the argument.*}
nipkow@15302
  2489
nipkow@15302
  2490
consts  listset :: "'a set list \<Rightarrow> 'a list set"
nipkow@15302
  2491
primrec
nipkow@15302
  2492
   "listset []    = {[]}"
nipkow@15302
  2493
   "listset(A#As) = set_Cons A (listset As)"
nipkow@15302
  2494
nipkow@15302
  2495
paulson@15656
  2496
subsection{*Relations on Lists*}
paulson@15656
  2497
paulson@15656
  2498
subsubsection {* Length Lexicographic Ordering *}
paulson@15656
  2499
paulson@15656
  2500
text{*These orderings preserve well-foundedness: shorter lists 
paulson@15656
  2501
  precede longer lists. These ordering are not used in dictionaries.*}
paulson@15656
  2502
paulson@15656
  2503
consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
paulson@15656
  2504
        --{*The lexicographic ordering for lists of the specified length*}
nipkow@15302
  2505
primrec
paulson@15656
  2506
  "lexn r 0 = {}"
paulson@15656
  2507
  "lexn r (Suc n) =
paulson@15656
  2508
    (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
paulson@15656
  2509
    {(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
nipkow@15302
  2510
nipkow@15302
  2511
constdefs
paulson@15656
  2512
  lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
paulson@15656
  2513
    "lex r == \<Union>n. lexn r n"
paulson@15656
  2514
        --{*Holds only between lists of the same length*}
paulson@15656
  2515
nipkow@15693
  2516
  lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@15693
  2517
    "lenlex r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
paulson@15656
  2518
        --{*Compares lists by their length and then lexicographically*}
nipkow@15302
  2519
nipkow@15302
  2520
nipkow@15302
  2521
lemma wf_lexn: "wf r ==> wf (lexn r n)"
nipkow@15302
  2522
apply (induct n, simp, simp)
nipkow@15302
  2523
apply(rule wf_subset)
nipkow@15302
  2524
 prefer 2 apply (rule Int_lower1)
nipkow@15302
  2525
apply(rule wf_prod_fun_image)
nipkow@15302
  2526
 prefer 2 apply (rule inj_onI, auto)
nipkow@15302
  2527
done
nipkow@15302
  2528
nipkow@15302
  2529
lemma lexn_length:
nipkow@15302
  2530
     "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@15302
  2531
by (induct n) auto
nipkow@15302
  2532
nipkow@15302
  2533
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@15302
  2534
apply (unfold lex_def)
nipkow@15302
  2535
apply (rule wf_UN)
nipkow@15302
  2536
apply (blast intro: wf_lexn, clarify)
nipkow@15302
  2537
apply (rename_tac m n)
nipkow@15302
  2538
apply (subgoal_tac "m \<noteq> n")
nipkow@15302
  2539
 prefer 2 apply blast
nipkow@15302
  2540
apply (blast dest: lexn_length not_sym)
nipkow@15302
  2541
done
nipkow@15302
  2542
nipkow@15302
  2543
lemma lexn_conv:
paulson@15656
  2544
  "lexn r n =
paulson@15656
  2545
    {(xs,ys). length xs = n \<and> length ys = n \<and>
paulson@15656
  2546
    (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
nipkow@18423
  2547
apply (induct n, simp)
nipkow@15302
  2548
apply (simp add: image_Collect lex_prod_def, safe, blast)
nipkow@15302
  2549
 apply (rule_tac x = "ab # xys" in exI, simp)
nipkow@15302
  2550
apply (case_tac xys, simp_all, blast)
nipkow@15302
  2551
done
nipkow@15302
  2552
nipkow@15302
  2553
lemma lex_conv:
paulson@15656
  2554
  "lex r =
paulson@15656
  2555
    {(xs,ys). length xs = length ys \<and>
paulson@15656
  2556
    (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@15302
  2557
by (force simp add: lex_def lexn_conv)
nipkow@15302
  2558
nipkow@15693
  2559
lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
nipkow@15693
  2560
by (unfold lenlex_def) blast
nipkow@15693
  2561
nipkow@15693
  2562
lemma lenlex_conv:
nipkow@15693
  2563
    "lenlex r = {(xs,ys). length xs < length ys |
paulson@15656
  2564
                 length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@19623
  2565
by (simp add: lenlex_def diag_def lex_prod_def inv_image_def)
nipkow@15302
  2566
nipkow@15302
  2567
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@15302
  2568
by (simp add: lex_conv)
nipkow@15302
  2569
nipkow@15302
  2570
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@15302
  2571
by (simp add:lex_conv)
nipkow@15302
  2572
paulson@18447
  2573
lemma Cons_in_lex [simp]:
paulson@15656
  2574
    "((x # xs, y # ys) : lex r) =
paulson@15656
  2575
      ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@15302
  2576
apply (simp add: lex_conv)
nipkow@15302
  2577
apply (rule iffI)
nipkow@15302
  2578
 prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
nipkow@15302
  2579
apply (case_tac xys, simp, simp)
nipkow@15302
  2580
apply blast
nipkow@15302
  2581
done
nipkow@15302
  2582
nipkow@15302
  2583
paulson@15656
  2584
subsubsection {* Lexicographic Ordering *}
paulson@15656
  2585
paulson@15656
  2586
text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
paulson@15656
  2587
    This ordering does \emph{not} preserve well-foundedness.
nipkow@17090
  2588
     Author: N. Voelker, March 2005. *} 
paulson@15656
  2589
paulson@15656
  2590
constdefs 
paulson@15656
  2591
  lexord :: "('a * 'a)set \<Rightarrow> ('a list * 'a list) set" 
paulson@15656
  2592
  "lexord  r == {(x,y). \<exists> a v. y = x @ a # v \<or> 
paulson@15656
  2593
            (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
paulson@15656
  2594
paulson@15656
  2595
lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
paulson@15656
  2596
  by (unfold lexord_def, induct_tac y, auto) 
paulson@15656
  2597
paulson@15656
  2598
lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
paulson@15656
  2599
  by (unfold lexord_def, induct_tac x, auto)
paulson@15656
  2600
paulson@15656
  2601
lemma lexord_cons_cons[simp]:
paulson@15656
  2602
     "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
paulson@15656
  2603
  apply (unfold lexord_def, safe, simp_all)
paulson@15656
  2604
  apply (case_tac u, simp, simp)
paulson@15656
  2605
  apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
paulson@15656
  2606
  apply (erule_tac x="b # u" in allE)
paulson@15656
  2607
  by force
paulson@15656
  2608
paulson@15656
  2609
lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
paulson@15656
  2610
paulson@15656
  2611
lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
paulson@15656
  2612
  by (induct_tac x, auto)  
paulson@15656
  2613
paulson@15656
  2614
lemma lexord_append_left_rightI:
paulson@15656
  2615
     "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
paulson@15656
  2616
  by (induct_tac u, auto)
paulson@15656
  2617
paulson@15656
  2618
lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
paulson@15656
  2619
  by (induct x, auto)
paulson@15656
  2620
paulson@15656
  2621
lemma lexord_append_leftD:
paulson@15656
  2622
     "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
paulson@15656
  2623
  by (erule rev_mp, induct_tac x, auto)
paulson@15656
  2624
paulson@15656
  2625
lemma lexord_take_index_conv: 
paulson@15656
  2626
   "((x,y) : lexord r) = 
paulson@15656
  2627
    ((length x < length y \<and> take (length x) y = x) \<or> 
paulson@15656
  2628
     (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
paulson@15656
  2629
  apply (unfold lexord_def Let_def, clarsimp) 
paulson@15656
  2630
  apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
paulson@15656
  2631
  apply auto 
paulson@15656
  2632
  apply (rule_tac x="hd (drop (length x) y)" in exI)
paulson@15656
  2633
  apply (rule_tac x="tl (drop (length x) y)" in exI)
paulson@15656
  2634
  apply (erule subst, simp add: min_def) 
paulson@15656
  2635
  apply (rule_tac x ="length u" in exI, simp) 
paulson@15656
  2636
  apply (rule_tac x ="take i x" in exI) 
paulson@15656
  2637
  apply (rule_tac x ="x ! i" in exI) 
paulson@15656
  2638
  apply (rule_tac x ="y ! i" in exI, safe) 
paulson@15656
  2639
  apply (rule_tac x="drop (Suc i) x" in exI)
paulson@15656
  2640
  apply (drule sym, simp add: drop_Suc_conv_tl) 
paulson@15656
  2641
  apply (rule_tac x="drop (Suc i) y" in exI)
paulson@15656
  2642
  by (simp add: drop_Suc_conv_tl) 
paulson@15656
  2643
paulson@15656
  2644
-- {* lexord is extension of partial ordering List.lex *} 
paulson@15656
  2645
lemma lexord_lex: " (x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
paulson@15656
  2646
  apply (rule_tac x = y in spec) 
paulson@15656
  2647
  apply (induct_tac x, clarsimp) 
paulson@15656
  2648
  by (clarify, case_tac x, simp, force)
paulson@15656
  2649
paulson@15656
  2650
lemma lexord_irreflexive: "(! x. (x,x) \<notin> r) \<Longrightarrow> (y,y) \<notin> lexord r"
paulson@15656
  2651
  by (induct y, auto)
paulson@15656
  2652
paulson@15656
  2653
lemma lexord_trans: 
paulson@15656
  2654
    "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
paulson@15656
  2655
   apply (erule rev_mp)+
paulson@15656
  2656
   apply (rule_tac x = x in spec) 
paulson@15656
  2657
  apply (rule_tac x = z in spec) 
paulson@15656
  2658
  apply ( induct_tac y, simp, clarify)
paulson@15656
  2659
  apply (case_tac xa, erule ssubst) 
paulson@15656
  2660
  apply (erule allE, erule allE) -- {* avoid simp recursion *} 
paulson@15656
  2661
  apply (case_tac x, simp, simp) 
paulson@15656
  2662
  apply (case_tac x, erule allE, erule allE, simp) 
paulson@15656
  2663
  apply (erule_tac x = listb in allE) 
paulson@15656
  2664
  apply (erule_tac x = lista in allE, simp)
paulson@15656
  2665
  apply (unfold trans_def)
paulson@15656
  2666
  by blast
paulson@15656
  2667
paulson@15656
  2668
lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
paulson@15656
  2669
  by (rule transI, drule lexord_trans, blast) 
paulson@15656
  2670
paulson@15656
  2671
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
  2672
  apply (rule_tac x = y in spec) 
paulson@15656
  2673
  apply (induct_tac x, rule allI) 
paulson@15656
  2674
  apply (case_tac x, simp, simp) 
paulson@15656
  2675
  apply (rule allI, case_tac x, simp, simp) 
paulson@15656
  2676
  by blast
paulson@15656
  2677
paulson@15656
  2678
krauss@21103
  2679
subsection {* Lexicographic combination of measure functions *}
krauss@21103
  2680
krauss@21103
  2681
text {* These are useful for termination proofs *}
krauss@21103
  2682
krauss@21103
  2683
definition
krauss@21103
  2684
  "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"
krauss@21103
  2685
krauss@21106
  2686
lemma wf_measures[recdef_wf, simp]: "wf (measures fs)"
krauss@21103
  2687
  unfolding measures_def
krauss@21103
  2688
  by blast
krauss@21103
  2689
krauss@21103
  2690
lemma in_measures[simp]: 
krauss@21103
  2691
  "(x, y) \<in> measures [] = False"
krauss@21103
  2692
  "(x, y) \<in> measures (f # fs)
krauss@21103
  2693
         = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"  
krauss@21103
  2694
  unfolding measures_def
krauss@21103
  2695
  by auto
krauss@21103
  2696
krauss@21103
  2697
lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
krauss@21103
  2698
  by simp
krauss@21103
  2699
krauss@21103
  2700
lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
krauss@21103
  2701
  by auto
krauss@21103
  2702
krauss@21211
  2703
(* install the lexicographic_order method and the "fun" command *)
bulwahn@21131
  2704
use "Tools/function_package/lexicographic_order.ML"
krauss@21211
  2705
use "Tools/function_package/fundef_datatype.ML"
krauss@21211
  2706
setup LexicographicOrder.setup
krauss@21211
  2707
setup FundefDatatype.setup
krauss@21211
  2708
krauss@21103
  2709
nipkow@15392
  2710
subsubsection{*Lifting a Relation on List Elements to the Lists*}
nipkow@15302
  2711
berghofe@22262
  2712
inductive2
berghofe@22262
  2713
  list_all2' :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool"
berghofe@22262
  2714
  for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
berghofe@22262
  2715
where
berghofe@22262
  2716
    Nil:  "list_all2' r [] []"
berghofe@22262
  2717
  | Cons: "[| r x y; list_all2' r xs ys |] ==> list_all2' r (x#xs) (y#ys)"
berghofe@22262
  2718
berghofe@22262
  2719
constdefs
berghofe@22262
  2720
  listrel :: "('a * 'b) set => ('a list * 'b list) set"
berghofe@22262
  2721
  "listrel r == Collect2 (list_all2' (member2 r))"
berghofe@22262
  2722
berghofe@22262
  2723
lemma list_all2_listrel_eq [pred_set_conv]:
berghofe@22262
  2724
  "list_all2' (member2 r) = member2 (listrel r)"
berghofe@22262
  2725
  by (simp add: listrel_def)
berghofe@22262
  2726
berghofe@22262
  2727
lemmas listrel_induct [consumes 1, case_names Nil Cons, induct set: listrel] =
berghofe@22262
  2728
  list_all2'.induct [to_set]
berghofe@22262
  2729
berghofe@22262
  2730
lemmas listrel_intros = list_all2'.intros [to_set]
berghofe@22262
  2731
berghofe@22262
  2732
inductive_cases2 listrel_Nil1 [to_set, elim!]: "list_all2' r [] xs"
berghofe@22262
  2733
inductive_cases2 listrel_Nil2 [to_set, elim!]: "list_all2' r xs []"
berghofe@22262
  2734
inductive_cases2 listrel_Cons1 [to_set, elim!]: "list_all2' r  (y#ys) xs"
berghofe@22262
  2735
inductive_cases2 listrel_Cons2 [to_set, elim!]: "list_all2' r xs (y#ys)"
nipkow@15302
  2736
nipkow@15302
  2737
nipkow@15302
  2738
lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
nipkow@15302
  2739
apply clarify  
berghofe@22262
  2740
apply (erule listrel_induct)
berghofe@22262
  2741
apply (blast intro: listrel_intros)+
nipkow@15302
  2742
done
nipkow@15302
  2743
nipkow@15302
  2744
lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
nipkow@15302
  2745
apply clarify 
berghofe@22262
  2746
apply (erule listrel_induct, auto) 
nipkow@15302
  2747
done
nipkow@15302
  2748
nipkow@15302
  2749
lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" 
nipkow@15302
  2750
apply (simp add: refl_def listrel_subset Ball_def)
nipkow@15302
  2751
apply (rule allI) 
nipkow@15302
  2752
apply (induct_tac x) 
berghofe@22262
  2753
apply (auto intro: listrel_intros)
nipkow@15302
  2754
done
nipkow@15302
  2755
nipkow@15302
  2756
lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
nipkow@15302
  2757
apply (auto simp add: sym_def)
berghofe@22262
  2758
apply (erule listrel_induct) 
berghofe@22262
  2759
apply (blast intro: listrel_intros)+
nipkow@15302
  2760
done