src/HOL/List.thy
author nipkow
Fri May 11 20:07:00 2007 +0200 (2007-05-11)
changeset 22940 42de50e78446
parent 22844 91c05f4b509e
child 22994 02440636214f
permissions -rw-r--r--
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(*  Title:      HOL/List.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow
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*)
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header {* The datatype of finite lists *}
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theory List
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imports PreList
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uses "Tools/string_syntax.ML"
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begin
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datatype 'a list =
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    Nil    ("[]")
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  | Cons 'a  "'a list"    (infixr "#" 65)
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subsection{*Basic list processing functions*}
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consts
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  "@" :: "'a list => 'a list => 'a list"    (infixr 65)
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  filter:: "('a => bool) => 'a list => 'a list"
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  concat:: "'a list list => 'a list"
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  foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"
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  foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"
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  hd:: "'a list => 'a"
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  tl:: "'a list => 'a list"
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  last:: "'a list => 'a"
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  butlast :: "'a list => 'a list"
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  set :: "'a list => 'a set"
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  map :: "('a=>'b) => ('a list => 'b list)"
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  nth :: "'a list => nat => 'a"    (infixl "!" 100)
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  list_update :: "'a list => nat => 'a => 'a list"
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  take:: "nat => 'a list => 'a list"
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  drop:: "nat => 'a list => 'a list"
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  takeWhile :: "('a => bool) => 'a list => 'a list"
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  dropWhile :: "('a => bool) => 'a list => 'a list"
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  rev :: "'a list => 'a list"
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  zip :: "'a list => 'b list => ('a * 'b) list"
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  upt :: "nat => nat => nat list" ("(1[_..</_'])")
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  remdups :: "'a list => 'a list"
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  remove1 :: "'a => 'a list => 'a list"
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  "distinct":: "'a list => bool"
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  replicate :: "nat => 'a => 'a list"
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  splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
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  allpairs :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'c list"
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abbreviation
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  upto:: "nat => nat => nat list"  ("(1[_../_])") where
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  "[i..j] == [i..<(Suc j)]"
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nonterminals lupdbinds lupdbind
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syntax
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  -- {* list Enumeration *}
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  "@list" :: "args => 'a list"    ("[(_)]")
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  -- {* Special syntax for filter *}
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_:_./ _])")
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  -- {* list update *}
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  "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
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  "" :: "lupdbind => lupdbinds"    ("_")
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  "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
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  "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
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translations
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  "[x, xs]" == "x#[xs]"
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  "[x]" == "x#[]"
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  "[x:xs . P]"== "filter (%x. P) xs"
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  "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"
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  "xs[i:=x]" == "list_update xs i x"
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syntax (xsymbols)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
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syntax (HTML output)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
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text {*
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  Function @{text size} is overloaded for all datatypes. Users may
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  refer to the list version as @{text length}. *}
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abbreviation
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  length :: "'a list => nat" where
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  "length == size"
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primrec
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  "hd(x#xs) = x"
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primrec
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  "tl([]) = []"
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  "tl(x#xs) = xs"
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primrec
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  "last(x#xs) = (if xs=[] then x else last xs)"
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primrec
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  "butlast []= []"
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  "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
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primrec
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  "set [] = {}"
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  "set (x#xs) = insert x (set xs)"
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primrec
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  "map f [] = []"
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  "map f (x#xs) = f(x)#map f xs"
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primrec
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  append_Nil: "[]@ys = ys"
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  append_Cons: "(x#xs)@ys = x#(xs@ys)"
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primrec
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  "rev([]) = []"
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  "rev(x#xs) = rev(xs) @ [x]"
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primrec
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  "filter P [] = []"
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  "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
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primrec
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  foldl_Nil:"foldl f a [] = a"
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  foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
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primrec
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  "foldr f [] a = a"
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  "foldr f (x#xs) a = f x (foldr f xs a)"
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primrec
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  "concat([]) = []"
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  "concat(x#xs) = x @ concat(xs)"
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primrec
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  drop_Nil:"drop n [] = []"
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  drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  take_Nil:"take n [] = []"
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  take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  "[][i:=v] = []"
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  "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])"
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primrec
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  "takeWhile P [] = []"
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  "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
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primrec
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  "dropWhile P [] = []"
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  "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
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primrec
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  "zip xs [] = []"
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  zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
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primrec
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  upt_0: "[i..<0] = []"
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  upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
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primrec
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  "distinct [] = True"
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  "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
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primrec
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  "remdups [] = []"
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  "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
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primrec
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  "remove1 x [] = []"
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  "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)"
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primrec
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  replicate_0: "replicate 0 x = []"
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  replicate_Suc: "replicate (Suc n) x = x # replicate n x"
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definition
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  rotate1 :: "'a list \<Rightarrow> 'a list" where
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  "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"
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definition
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  rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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  "rotate n = rotate1 ^ n"
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definition
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  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
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  "list_all2 P xs ys =
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    (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"
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definition
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  sublist :: "'a list => nat set => 'a list" where
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  "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"
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primrec
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  "splice [] ys = ys"
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  "splice (x#xs) ys = (if ys=[] then x#xs else x # hd ys # splice xs (tl ys))"
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    -- {*Warning: simpset does not contain the second eqn but a derived one. *}
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primrec
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"allpairs f [] ys = []"
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"allpairs f (x # xs) ys = map (f x) ys @ allpairs f xs ys"
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subsubsection {* @{const Nil} and @{const Cons} *}
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lemma not_Cons_self [simp]:
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  "xs \<noteq> x # xs"
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by (induct xs) auto
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lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
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lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
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by (induct xs) auto
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lemma length_induct:
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  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
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by (rule measure_induct [of length]) iprover
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subsubsection {* @{const length} *}
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text {*
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  Needs to come before @{text "@"} because of theorem @{text
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  append_eq_append_conv}.
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*}
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lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
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by (induct xs) auto
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lemma length_map [simp]: "length (map f xs) = length xs"
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by (induct xs) auto
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lemma length_rev [simp]: "length (rev xs) = length xs"
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by (induct xs) auto
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lemma length_tl [simp]: "length (tl xs) = length xs - 1"
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by (cases xs) auto
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lemma length_allpairs[simp]:
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 "length(allpairs f xs ys) = length xs * length ys"
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by(induct xs) auto
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lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
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by (induct xs) auto
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lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
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by (induct xs) auto
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lemma length_Suc_conv:
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"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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by (induct xs) auto
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lemma Suc_length_conv:
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"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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apply (induct xs, simp, simp)
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apply blast
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done
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lemma impossible_Cons [rule_format]: 
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  "length xs <= length ys --> xs = x # ys = False"
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apply (induct xs)
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apply auto
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done
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lemma list_induct2[consumes 1]: "\<And>ys.
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 \<lbrakk> length xs = length ys;
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   P [] [];
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   \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
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 \<Longrightarrow> P xs ys"
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apply(induct xs)
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 apply simp
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apply(case_tac ys)
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 apply simp
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apply(simp)
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done
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lemma list_induct2': 
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  "\<lbrakk> P [] [];
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  \<And>x xs. P (x#xs) [];
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  \<And>y ys. P [] (y#ys);
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   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
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 \<Longrightarrow> P xs ys"
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by (induct xs arbitrary: ys) (case_tac x, auto)+
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lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
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apply(rule Eq_FalseI)
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by auto
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(*
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Reduces xs=ys to False if xs and ys cannot be of the same length.
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This is the case if the atomic sublists of one are a submultiset
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of those of the other list and there are fewer Cons's in one than the other.
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*)
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ML_setup {*
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local
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fun len (Const("List.list.Nil",_)) acc = acc
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  | len (Const("List.list.Cons",_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
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  | len (Const("List.op @",_) $ xs $ ys) acc = len xs (len ys acc)
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  | len (Const("List.rev",_) $ xs) acc = len xs acc
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  | len (Const("List.map",_) $ _ $ xs) acc = len xs acc
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  | len t (ts,n) = (t::ts,n);
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fun list_eq ss (Const(_,eqT) $ lhs $ rhs) =
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  let
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    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
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    fun prove_neq() =
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      let
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        val Type(_,listT::_) = eqT;
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        val size = Const("Nat.size", listT --> HOLogic.natT);
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        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
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        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
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        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
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          (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
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      in SOME (thm RS @{thm neq_if_length_neq}) end
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  in
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    if m < n andalso gen_submultiset (op aconv) (ls,rs) orelse
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       n < m andalso gen_submultiset (op aconv) (rs,ls)
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    then prove_neq() else NONE
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  end;
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in
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nipkow@22143
   337
val list_neq_simproc =
haftmann@22633
   338
  Simplifier.simproc @{theory} "list_neq" ["(xs::'a list) = ys"] (K list_eq);
nipkow@22143
   339
nipkow@22143
   340
end;
nipkow@22143
   341
nipkow@22143
   342
Addsimprocs [list_neq_simproc];
nipkow@22143
   343
*}
nipkow@22143
   344
nipkow@22143
   345
nipkow@15392
   346
subsubsection {* @{text "@"} -- append *}
wenzelm@13114
   347
wenzelm@13142
   348
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
nipkow@13145
   349
by (induct xs) auto
wenzelm@13114
   350
wenzelm@13142
   351
lemma append_Nil2 [simp]: "xs @ [] = xs"
nipkow@13145
   352
by (induct xs) auto
nipkow@3507
   353
wenzelm@13142
   354
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
nipkow@13145
   355
by (induct xs) auto
wenzelm@13114
   356
wenzelm@13142
   357
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
nipkow@13145
   358
by (induct xs) auto
wenzelm@13114
   359
wenzelm@13142
   360
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
nipkow@13145
   361
by (induct xs) auto
wenzelm@13114
   362
wenzelm@13142
   363
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
nipkow@13145
   364
by (induct xs) auto
wenzelm@13114
   365
berghofe@13883
   366
lemma append_eq_append_conv [simp]:
berghofe@13883
   367
 "!!ys. length xs = length ys \<or> length us = length vs
berghofe@13883
   368
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
berghofe@13883
   369
apply (induct xs)
paulson@14208
   370
 apply (case_tac ys, simp, force)
paulson@14208
   371
apply (case_tac ys, force, simp)
nipkow@13145
   372
done
wenzelm@13142
   373
nipkow@14495
   374
lemma append_eq_append_conv2: "!!ys zs ts.
nipkow@14495
   375
 (xs @ ys = zs @ ts) =
nipkow@14495
   376
 (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
nipkow@14495
   377
apply (induct xs)
nipkow@14495
   378
 apply fastsimp
nipkow@14495
   379
apply(case_tac zs)
nipkow@14495
   380
 apply simp
nipkow@14495
   381
apply fastsimp
nipkow@14495
   382
done
nipkow@14495
   383
wenzelm@13142
   384
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145
   385
by simp
wenzelm@13142
   386
wenzelm@13142
   387
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145
   388
by simp
wenzelm@13114
   389
wenzelm@13142
   390
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   391
by simp
wenzelm@13114
   392
wenzelm@13142
   393
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   394
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   395
wenzelm@13142
   396
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   397
using append_same_eq [of "[]"] by auto
wenzelm@13114
   398
wenzelm@13142
   399
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   400
by (induct xs) auto
wenzelm@13114
   401
wenzelm@13142
   402
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   403
by (induct xs) auto
wenzelm@13114
   404
wenzelm@13142
   405
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   406
by (simp add: hd_append split: list.split)
wenzelm@13114
   407
wenzelm@13142
   408
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   409
by (simp split: list.split)
wenzelm@13114
   410
wenzelm@13142
   411
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   412
by (simp add: tl_append split: list.split)
wenzelm@13114
   413
wenzelm@13114
   414
nipkow@14300
   415
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   416
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   417
by(cases ys) auto
nipkow@14300
   418
nipkow@15281
   419
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   420
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   421
by(cases ys) auto
nipkow@15281
   422
nipkow@14300
   423
wenzelm@13142
   424
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   425
wenzelm@13114
   426
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   427
by simp
wenzelm@13114
   428
wenzelm@13142
   429
lemma Cons_eq_appendI:
nipkow@13145
   430
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   431
by (drule sym) simp
wenzelm@13114
   432
wenzelm@13142
   433
lemma append_eq_appendI:
nipkow@13145
   434
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   435
by (drule sym) simp
wenzelm@13114
   436
wenzelm@13114
   437
wenzelm@13142
   438
text {*
nipkow@13145
   439
Simplification procedure for all list equalities.
nipkow@13145
   440
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   441
- both lists end in a singleton list,
nipkow@13145
   442
- or both lists end in the same list.
wenzelm@13142
   443
*}
wenzelm@13142
   444
wenzelm@13142
   445
ML_setup {*
nipkow@3507
   446
local
nipkow@3507
   447
wenzelm@13114
   448
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462
   449
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
wenzelm@13462
   450
  | last (Const("List.op @",_) $ _ $ ys) = last ys
wenzelm@13462
   451
  | last t = t;
wenzelm@13114
   452
wenzelm@13114
   453
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462
   454
  | list1 _ = false;
wenzelm@13114
   455
wenzelm@13114
   456
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462
   457
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
wenzelm@13462
   458
  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462
   459
  | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114
   460
haftmann@22633
   461
val rearr_ss = HOL_basic_ss addsimps [@{thm append_assoc},
haftmann@22633
   462
  @{thm append_Nil}, @{thm append_Cons}];
wenzelm@16973
   463
wenzelm@20044
   464
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   465
  let
wenzelm@13462
   466
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   467
    fun rearr conv =
wenzelm@13462
   468
      let
wenzelm@13462
   469
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   470
        val Type(_,listT::_) = eqT
wenzelm@13462
   471
        val appT = [listT,listT] ---> listT
wenzelm@13462
   472
        val app = Const("List.op @",appT)
wenzelm@13462
   473
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   474
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@20044
   475
        val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
wenzelm@17877
   476
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
skalberg@15531
   477
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   478
wenzelm@13462
   479
  in
haftmann@22633
   480
    if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
haftmann@22633
   481
    else if lastl aconv lastr then rearr @{thm append_same_eq}
skalberg@15531
   482
    else NONE
wenzelm@13462
   483
  end;
wenzelm@13462
   484
wenzelm@13114
   485
in
wenzelm@13462
   486
wenzelm@13462
   487
val list_eq_simproc =
haftmann@22633
   488
  Simplifier.simproc @{theory} "list_eq" ["(xs::'a list) = ys"] (K list_eq);
wenzelm@13462
   489
wenzelm@13114
   490
end;
wenzelm@13114
   491
wenzelm@13114
   492
Addsimprocs [list_eq_simproc];
wenzelm@13114
   493
*}
wenzelm@13114
   494
wenzelm@13114
   495
nipkow@15392
   496
subsubsection {* @{text map} *}
wenzelm@13114
   497
wenzelm@13142
   498
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   499
by (induct xs) simp_all
wenzelm@13114
   500
wenzelm@13142
   501
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   502
by (rule ext, induct_tac xs) auto
wenzelm@13114
   503
wenzelm@13142
   504
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   505
by (induct xs) auto
wenzelm@13114
   506
wenzelm@13142
   507
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   508
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   509
wenzelm@13142
   510
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   511
by (induct xs) auto
wenzelm@13114
   512
nipkow@13737
   513
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   514
by (induct xs) auto
nipkow@13737
   515
krauss@19770
   516
lemma map_cong [fundef_cong, recdef_cong]:
nipkow@13145
   517
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   518
-- {* a congruence rule for @{text map} *}
nipkow@13737
   519
by simp
wenzelm@13114
   520
wenzelm@13142
   521
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   522
by (cases xs) auto
wenzelm@13114
   523
wenzelm@13142
   524
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   525
by (cases xs) auto
wenzelm@13114
   526
paulson@18447
   527
lemma map_eq_Cons_conv:
nipkow@14025
   528
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   529
by (cases xs) auto
wenzelm@13114
   530
paulson@18447
   531
lemma Cons_eq_map_conv:
nipkow@14025
   532
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   533
by (cases ys) auto
nipkow@14025
   534
paulson@18447
   535
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
paulson@18447
   536
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
paulson@18447
   537
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
paulson@18447
   538
nipkow@14111
   539
lemma ex_map_conv:
nipkow@14111
   540
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
paulson@18447
   541
by(induct ys, auto simp add: Cons_eq_map_conv)
nipkow@14111
   542
nipkow@15110
   543
lemma map_eq_imp_length_eq:
nipkow@15110
   544
  "!!xs. map f xs = map f ys ==> length xs = length ys"
nipkow@15110
   545
apply (induct ys)
nipkow@15110
   546
 apply simp
nipkow@15110
   547
apply(simp (no_asm_use))
nipkow@15110
   548
apply clarify
nipkow@15110
   549
apply(simp (no_asm_use))
nipkow@15110
   550
apply fast
nipkow@15110
   551
done
nipkow@15110
   552
nipkow@15110
   553
lemma map_inj_on:
nipkow@15110
   554
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   555
  ==> xs = ys"
nipkow@15110
   556
apply(frule map_eq_imp_length_eq)
nipkow@15110
   557
apply(rotate_tac -1)
nipkow@15110
   558
apply(induct rule:list_induct2)
nipkow@15110
   559
 apply simp
nipkow@15110
   560
apply(simp)
nipkow@15110
   561
apply (blast intro:sym)
nipkow@15110
   562
done
nipkow@15110
   563
nipkow@15110
   564
lemma inj_on_map_eq_map:
nipkow@15110
   565
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   566
by(blast dest:map_inj_on)
nipkow@15110
   567
wenzelm@13114
   568
lemma map_injective:
nipkow@14338
   569
 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@14338
   570
by (induct ys) (auto dest!:injD)
wenzelm@13114
   571
nipkow@14339
   572
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   573
by(blast dest:map_injective)
nipkow@14339
   574
wenzelm@13114
   575
lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@17589
   576
by (iprover dest: map_injective injD intro: inj_onI)
wenzelm@13114
   577
wenzelm@13114
   578
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   579
apply (unfold inj_on_def, clarify)
nipkow@13145
   580
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   581
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   582
apply blast
nipkow@13145
   583
done
wenzelm@13114
   584
nipkow@14339
   585
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   586
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   587
nipkow@15303
   588
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
   589
apply(rule inj_onI)
nipkow@15303
   590
apply(erule map_inj_on)
nipkow@15303
   591
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
   592
done
nipkow@15303
   593
kleing@14343
   594
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   595
by (induct xs, auto)
wenzelm@13114
   596
nipkow@14402
   597
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   598
by (induct xs) auto
nipkow@14402
   599
nipkow@15110
   600
lemma map_fst_zip[simp]:
nipkow@15110
   601
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   602
by (induct rule:list_induct2, simp_all)
nipkow@15110
   603
nipkow@15110
   604
lemma map_snd_zip[simp]:
nipkow@15110
   605
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   606
by (induct rule:list_induct2, simp_all)
nipkow@15110
   607
nipkow@15110
   608
nipkow@15392
   609
subsubsection {* @{text rev} *}
wenzelm@13114
   610
wenzelm@13142
   611
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   612
by (induct xs) auto
wenzelm@13114
   613
wenzelm@13142
   614
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   615
by (induct xs) auto
wenzelm@13114
   616
kleing@15870
   617
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870
   618
by auto
kleing@15870
   619
wenzelm@13142
   620
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   621
by (induct xs) auto
wenzelm@13114
   622
wenzelm@13142
   623
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   624
by (induct xs) auto
wenzelm@13114
   625
kleing@15870
   626
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870
   627
by (cases xs) auto
kleing@15870
   628
kleing@15870
   629
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870
   630
by (cases xs) auto
kleing@15870
   631
haftmann@21061
   632
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
haftmann@21061
   633
apply (induct xs arbitrary: ys, force)
paulson@14208
   634
apply (case_tac ys, simp, force)
nipkow@13145
   635
done
wenzelm@13114
   636
nipkow@15439
   637
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
   638
by(simp add:inj_on_def)
nipkow@15439
   639
wenzelm@13366
   640
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   641
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489
   642
apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145
   643
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   644
done
wenzelm@13114
   645
nipkow@13145
   646
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114
   647
wenzelm@13366
   648
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   649
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   650
by (induct xs rule: rev_induct) auto
wenzelm@13114
   651
wenzelm@13366
   652
lemmas rev_cases = rev_exhaust
wenzelm@13366
   653
nipkow@18423
   654
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
nipkow@18423
   655
by(rule rev_cases[of xs]) auto
nipkow@18423
   656
wenzelm@13114
   657
nipkow@15392
   658
subsubsection {* @{text set} *}
wenzelm@13114
   659
wenzelm@13142
   660
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   661
by (induct xs) auto
wenzelm@13114
   662
wenzelm@13142
   663
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   664
by (induct xs) auto
wenzelm@13114
   665
nipkow@17830
   666
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
nipkow@17830
   667
by(cases xs) auto
oheimb@14099
   668
wenzelm@13142
   669
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   670
by auto
wenzelm@13114
   671
oheimb@14099
   672
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   673
by auto
oheimb@14099
   674
wenzelm@13142
   675
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   676
by (induct xs) auto
wenzelm@13114
   677
nipkow@15245
   678
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   679
by(induct xs) auto
nipkow@15245
   680
wenzelm@13142
   681
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   682
by (induct xs) auto
wenzelm@13114
   683
wenzelm@13142
   684
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   685
by (induct xs) auto
wenzelm@13114
   686
nipkow@22828
   687
lemma set_allpairs[simp]:
nipkow@22830
   688
 "set(allpairs f xs ys) = {z. EX x : set xs. EX y : set ys. z = f x y}"
nipkow@22828
   689
by(induct xs) auto
nipkow@22828
   690
wenzelm@13142
   691
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   692
by (induct xs) auto
wenzelm@13114
   693
nipkow@15425
   694
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
paulson@14208
   695
apply (induct j, simp_all)
paulson@14208
   696
apply (erule ssubst, auto)
nipkow@13145
   697
done
wenzelm@13114
   698
wenzelm@13142
   699
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
paulson@15113
   700
proof (induct xs)
paulson@15113
   701
  case Nil show ?case by simp
paulson@15113
   702
  case (Cons a xs)
paulson@15113
   703
  show ?case
paulson@15113
   704
  proof 
paulson@15113
   705
    assume "x \<in> set (a # xs)"
paulson@15113
   706
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   707
      by (simp, blast intro: Cons_eq_appendI)
paulson@15113
   708
  next
paulson@15113
   709
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   710
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
paulson@15113
   711
    show "x \<in> set (a # xs)" 
paulson@15113
   712
      by (cases ys, auto simp add: eq)
paulson@15113
   713
  qed
paulson@15113
   714
qed
wenzelm@13142
   715
nipkow@18049
   716
lemma in_set_conv_decomp_first:
nipkow@18049
   717
 "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
nipkow@18049
   718
proof (induct xs)
nipkow@18049
   719
  case Nil show ?case by simp
nipkow@18049
   720
next
nipkow@18049
   721
  case (Cons a xs)
nipkow@18049
   722
  show ?case
nipkow@18049
   723
  proof cases
nipkow@18049
   724
    assume "x = a" thus ?case using Cons by force
nipkow@18049
   725
  next
nipkow@18049
   726
    assume "x \<noteq> a"
nipkow@18049
   727
    show ?case
nipkow@18049
   728
    proof
nipkow@18049
   729
      assume "x \<in> set (a # xs)"
nipkow@18049
   730
      from prems show "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@18049
   731
	by(fastsimp intro!: Cons_eq_appendI)
nipkow@18049
   732
    next
nipkow@18049
   733
      assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@18049
   734
      then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
nipkow@18049
   735
      show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq)
nipkow@18049
   736
    qed
nipkow@18049
   737
  qed
nipkow@18049
   738
qed
nipkow@18049
   739
nipkow@18049
   740
lemmas split_list       = in_set_conv_decomp[THEN iffD1, standard]
nipkow@18049
   741
lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard]
nipkow@18049
   742
nipkow@18049
   743
paulson@13508
   744
lemma finite_list: "finite A ==> EX l. set l = A"
paulson@13508
   745
apply (erule finite_induct, auto)
paulson@13508
   746
apply (rule_tac x="x#l" in exI, auto)
paulson@13508
   747
done
paulson@13508
   748
kleing@14388
   749
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
   750
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
   751
paulson@15168
   752
nipkow@15392
   753
subsubsection {* @{text filter} *}
wenzelm@13114
   754
wenzelm@13142
   755
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   756
by (induct xs) auto
wenzelm@13114
   757
nipkow@15305
   758
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
   759
by (induct xs) simp_all
nipkow@15305
   760
wenzelm@13142
   761
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   762
by (induct xs) auto
wenzelm@13114
   763
nipkow@16998
   764
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998
   765
by (induct xs) (auto simp add: le_SucI)
nipkow@16998
   766
nipkow@18423
   767
lemma sum_length_filter_compl:
nipkow@18423
   768
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
nipkow@18423
   769
by(induct xs) simp_all
nipkow@18423
   770
wenzelm@13142
   771
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   772
by (induct xs) auto
wenzelm@13114
   773
wenzelm@13142
   774
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   775
by (induct xs) auto
wenzelm@13114
   776
nipkow@16998
   777
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@16998
   778
  by (induct xs) simp_all
nipkow@16998
   779
nipkow@16998
   780
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998
   781
apply (induct xs)
nipkow@16998
   782
 apply auto
nipkow@16998
   783
apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998
   784
apply simp
nipkow@16998
   785
done
wenzelm@13114
   786
nipkow@16965
   787
lemma filter_map:
nipkow@16965
   788
  "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965
   789
by (induct xs) simp_all
nipkow@16965
   790
nipkow@16965
   791
lemma length_filter_map[simp]:
nipkow@16965
   792
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965
   793
by (simp add:filter_map)
nipkow@16965
   794
wenzelm@13142
   795
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   796
by auto
wenzelm@13114
   797
nipkow@15246
   798
lemma length_filter_less:
nipkow@15246
   799
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
   800
proof (induct xs)
nipkow@15246
   801
  case Nil thus ?case by simp
nipkow@15246
   802
next
nipkow@15246
   803
  case (Cons x xs) thus ?case
nipkow@15246
   804
    apply (auto split:split_if_asm)
nipkow@15246
   805
    using length_filter_le[of P xs] apply arith
nipkow@15246
   806
  done
nipkow@15246
   807
qed
wenzelm@13114
   808
nipkow@15281
   809
lemma length_filter_conv_card:
nipkow@15281
   810
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
   811
proof (induct xs)
nipkow@15281
   812
  case Nil thus ?case by simp
nipkow@15281
   813
next
nipkow@15281
   814
  case (Cons x xs)
nipkow@15281
   815
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
   816
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
   817
  show ?case (is "?l = card ?S'")
nipkow@15281
   818
  proof (cases)
nipkow@15281
   819
    assume "p x"
nipkow@15281
   820
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@15281
   821
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   822
    have "length (filter p (x # xs)) = Suc(card ?S)"
nipkow@15281
   823
      using Cons by simp
nipkow@15281
   824
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
nipkow@15281
   825
      by (simp add: card_image inj_Suc)
nipkow@15281
   826
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   827
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
   828
    finally show ?thesis .
nipkow@15281
   829
  next
nipkow@15281
   830
    assume "\<not> p x"
nipkow@15281
   831
    hence eq: "?S' = Suc ` ?S"
nipkow@15281
   832
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   833
    have "length (filter p (x # xs)) = card ?S"
nipkow@15281
   834
      using Cons by simp
nipkow@15281
   835
    also have "\<dots> = card(Suc ` ?S)" using fin
nipkow@15281
   836
      by (simp add: card_image inj_Suc)
nipkow@15281
   837
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   838
      by (simp add:card_insert_if)
nipkow@15281
   839
    finally show ?thesis .
nipkow@15281
   840
  qed
nipkow@15281
   841
qed
nipkow@15281
   842
nipkow@17629
   843
lemma Cons_eq_filterD:
nipkow@17629
   844
 "x#xs = filter P ys \<Longrightarrow>
nipkow@17629
   845
  \<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
   846
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
nipkow@17629
   847
proof(induct ys)
nipkow@17629
   848
  case Nil thus ?case by simp
nipkow@17629
   849
next
nipkow@17629
   850
  case (Cons y ys)
nipkow@17629
   851
  show ?case (is "\<exists>x. ?Q x")
nipkow@17629
   852
  proof cases
nipkow@17629
   853
    assume Py: "P y"
nipkow@17629
   854
    show ?thesis
nipkow@17629
   855
    proof cases
nipkow@17629
   856
      assume xy: "x = y"
nipkow@17629
   857
      show ?thesis
nipkow@17629
   858
      proof from Py xy Cons(2) show "?Q []" by simp qed
nipkow@17629
   859
    next
nipkow@17629
   860
      assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp
nipkow@17629
   861
    qed
nipkow@17629
   862
  next
nipkow@17629
   863
    assume Py: "\<not> P y"
nipkow@17629
   864
    with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp
nipkow@17629
   865
    show ?thesis (is "? us. ?Q us")
nipkow@17629
   866
    proof show "?Q (y#us)" using 1 by simp qed
nipkow@17629
   867
  qed
nipkow@17629
   868
qed
nipkow@17629
   869
nipkow@17629
   870
lemma filter_eq_ConsD:
nipkow@17629
   871
 "filter P ys = x#xs \<Longrightarrow>
nipkow@17629
   872
  \<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
   873
by(rule Cons_eq_filterD) simp
nipkow@17629
   874
nipkow@17629
   875
lemma filter_eq_Cons_iff:
nipkow@17629
   876
 "(filter P ys = x#xs) =
nipkow@17629
   877
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
nipkow@17629
   878
by(auto dest:filter_eq_ConsD)
nipkow@17629
   879
nipkow@17629
   880
lemma Cons_eq_filter_iff:
nipkow@17629
   881
 "(x#xs = filter P ys) =
nipkow@17629
   882
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
nipkow@17629
   883
by(auto dest:Cons_eq_filterD)
nipkow@17629
   884
krauss@19770
   885
lemma filter_cong[fundef_cong, recdef_cong]:
nipkow@17501
   886
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
nipkow@17501
   887
apply simp
nipkow@17501
   888
apply(erule thin_rl)
nipkow@17501
   889
by (induct ys) simp_all
nipkow@17501
   890
nipkow@15281
   891
nipkow@15392
   892
subsubsection {* @{text concat} *}
wenzelm@13114
   893
wenzelm@13142
   894
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
   895
by (induct xs) auto
wenzelm@13114
   896
paulson@18447
   897
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   898
by (induct xss) auto
wenzelm@13114
   899
paulson@18447
   900
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   901
by (induct xss) auto
wenzelm@13114
   902
wenzelm@13142
   903
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
nipkow@13145
   904
by (induct xs) auto
wenzelm@13114
   905
wenzelm@13142
   906
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
   907
by (induct xs) auto
wenzelm@13114
   908
wenzelm@13142
   909
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
   910
by (induct xs) auto
wenzelm@13114
   911
wenzelm@13142
   912
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
   913
by (induct xs) auto
wenzelm@13114
   914
wenzelm@13114
   915
nipkow@15392
   916
subsubsection {* @{text nth} *}
wenzelm@13114
   917
wenzelm@13142
   918
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
   919
by auto
wenzelm@13114
   920
wenzelm@13142
   921
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
   922
by auto
wenzelm@13114
   923
wenzelm@13142
   924
declare nth.simps [simp del]
wenzelm@13114
   925
wenzelm@13114
   926
lemma nth_append:
nipkow@13145
   927
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
paulson@14208
   928
apply (induct "xs", simp)
paulson@14208
   929
apply (case_tac n, auto)
nipkow@13145
   930
done
wenzelm@13114
   931
nipkow@14402
   932
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
nipkow@14402
   933
by (induct "xs") auto
nipkow@14402
   934
nipkow@14402
   935
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
nipkow@14402
   936
by (induct "xs") auto
nipkow@14402
   937
wenzelm@13142
   938
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
paulson@14208
   939
apply (induct xs, simp)
paulson@14208
   940
apply (case_tac n, auto)
nipkow@13145
   941
done
wenzelm@13114
   942
nipkow@18423
   943
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
nipkow@18423
   944
by(cases xs) simp_all
nipkow@18423
   945
nipkow@18049
   946
nipkow@18049
   947
lemma list_eq_iff_nth_eq:
nipkow@18049
   948
 "!!ys. (xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
nipkow@18049
   949
apply(induct xs)
nipkow@18049
   950
 apply simp apply blast
nipkow@18049
   951
apply(case_tac ys)
nipkow@18049
   952
 apply simp
nipkow@18049
   953
apply(simp add:nth_Cons split:nat.split)apply blast
nipkow@18049
   954
done
nipkow@18049
   955
wenzelm@13142
   956
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
   957
apply (induct xs, simp, simp)
nipkow@13145
   958
apply safe
paulson@14208
   959
apply (rule_tac x = 0 in exI, simp)
paulson@14208
   960
 apply (rule_tac x = "Suc i" in exI, simp)
paulson@14208
   961
apply (case_tac i, simp)
nipkow@13145
   962
apply (rename_tac j)
paulson@14208
   963
apply (rule_tac x = j in exI, simp)
nipkow@13145
   964
done
wenzelm@13114
   965
nipkow@17501
   966
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
nipkow@17501
   967
by(auto simp:set_conv_nth)
nipkow@17501
   968
nipkow@13145
   969
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
   970
by (auto simp add: set_conv_nth)
wenzelm@13114
   971
wenzelm@13142
   972
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
   973
by (auto simp add: set_conv_nth)
wenzelm@13114
   974
wenzelm@13114
   975
lemma all_nth_imp_all_set:
nipkow@13145
   976
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
   977
by (auto simp add: set_conv_nth)
wenzelm@13114
   978
wenzelm@13114
   979
lemma all_set_conv_all_nth:
nipkow@13145
   980
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
   981
by (auto simp add: set_conv_nth)
wenzelm@13114
   982
wenzelm@13114
   983
nipkow@15392
   984
subsubsection {* @{text list_update} *}
wenzelm@13114
   985
wenzelm@13142
   986
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
nipkow@13145
   987
by (induct xs) (auto split: nat.split)
wenzelm@13114
   988
wenzelm@13114
   989
lemma nth_list_update:
nipkow@13145
   990
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@13145
   991
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   992
wenzelm@13142
   993
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
   994
by (simp add: nth_list_update)
wenzelm@13114
   995
wenzelm@13142
   996
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145
   997
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   998
wenzelm@13142
   999
lemma list_update_overwrite [simp]:
nipkow@13145
  1000
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145
  1001
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1002
nipkow@14402
  1003
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
paulson@14208
  1004
apply (induct xs, simp)
nipkow@14187
  1005
apply(simp split:nat.splits)
nipkow@14187
  1006
done
nipkow@14187
  1007
nipkow@17501
  1008
lemma list_update_beyond[simp]: "\<And>i. length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
nipkow@17501
  1009
apply (induct xs)
nipkow@17501
  1010
 apply simp
nipkow@17501
  1011
apply (case_tac i)
nipkow@17501
  1012
apply simp_all
nipkow@17501
  1013
done
nipkow@17501
  1014
wenzelm@13114
  1015
lemma list_update_same_conv:
nipkow@13145
  1016
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145
  1017
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1018
nipkow@14187
  1019
lemma list_update_append1:
nipkow@14187
  1020
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
paulson@14208
  1021
apply (induct xs, simp)
nipkow@14187
  1022
apply(simp split:nat.split)
nipkow@14187
  1023
done
nipkow@14187
  1024
kleing@15868
  1025
lemma list_update_append:
kleing@15868
  1026
  "!!n. (xs @ ys) [n:= x] = 
kleing@15868
  1027
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
kleing@15868
  1028
by (induct xs) (auto split:nat.splits)
kleing@15868
  1029
nipkow@14402
  1030
lemma list_update_length [simp]:
nipkow@14402
  1031
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
  1032
by (induct xs, auto)
nipkow@14402
  1033
wenzelm@13114
  1034
lemma update_zip:
nipkow@13145
  1035
"!!i xy xs. length xs = length ys ==>
nipkow@13145
  1036
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145
  1037
by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114
  1038
wenzelm@13114
  1039
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145
  1040
by (induct xs) (auto split: nat.split)
wenzelm@13114
  1041
wenzelm@13114
  1042
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
  1043
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
  1044
kleing@15868
  1045
lemma set_update_memI: "!!n. n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
kleing@15868
  1046
by (induct xs) (auto split:nat.splits)
kleing@15868
  1047
wenzelm@13114
  1048
nipkow@15392
  1049
subsubsection {* @{text last} and @{text butlast} *}
wenzelm@13114
  1050
wenzelm@13142
  1051
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
  1052
by (induct xs) auto
wenzelm@13114
  1053
wenzelm@13142
  1054
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
  1055
by (induct xs) auto
wenzelm@13114
  1056
nipkow@14302
  1057
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
  1058
by(simp add:last.simps)
nipkow@14302
  1059
nipkow@14302
  1060
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
  1061
by(simp add:last.simps)
nipkow@14302
  1062
nipkow@14302
  1063
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
  1064
by (induct xs) (auto)
nipkow@14302
  1065
nipkow@14302
  1066
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
  1067
by(simp add:last_append)
nipkow@14302
  1068
nipkow@14302
  1069
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
  1070
by(simp add:last_append)
nipkow@14302
  1071
nipkow@17762
  1072
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
nipkow@17762
  1073
by(rule rev_exhaust[of xs]) simp_all
nipkow@17762
  1074
nipkow@17762
  1075
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
nipkow@17762
  1076
by(cases xs) simp_all
nipkow@17762
  1077
nipkow@17765
  1078
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
nipkow@17765
  1079
by (induct as) auto
nipkow@17762
  1080
wenzelm@13142
  1081
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
  1082
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1083
wenzelm@13114
  1084
lemma butlast_append:
nipkow@13145
  1085
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145
  1086
by (induct xs) auto
wenzelm@13114
  1087
wenzelm@13142
  1088
lemma append_butlast_last_id [simp]:
nipkow@13145
  1089
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
  1090
by (induct xs) auto
wenzelm@13114
  1091
wenzelm@13142
  1092
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
  1093
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1094
wenzelm@13114
  1095
lemma in_set_butlast_appendI:
nipkow@13145
  1096
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
  1097
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
  1098
nipkow@17501
  1099
lemma last_drop[simp]: "!!n. n < length xs \<Longrightarrow> last (drop n xs) = last xs"
nipkow@17501
  1100
apply (induct xs)
nipkow@17501
  1101
 apply simp
nipkow@17501
  1102
apply (auto split:nat.split)
nipkow@17501
  1103
done
nipkow@17501
  1104
nipkow@17589
  1105
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
nipkow@17589
  1106
by(induct xs)(auto simp:neq_Nil_conv)
nipkow@17589
  1107
nipkow@15392
  1108
subsubsection {* @{text take} and @{text drop} *}
wenzelm@13114
  1109
wenzelm@13142
  1110
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
  1111
by (induct xs) auto
wenzelm@13114
  1112
wenzelm@13142
  1113
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
  1114
by (induct xs) auto
wenzelm@13114
  1115
wenzelm@13142
  1116
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
  1117
by simp
wenzelm@13114
  1118
wenzelm@13142
  1119
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
  1120
by simp
wenzelm@13114
  1121
wenzelm@13142
  1122
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
  1123
nipkow@15110
  1124
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
  1125
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
  1126
nipkow@14187
  1127
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
  1128
by(cases xs, simp_all)
nipkow@14187
  1129
nipkow@14187
  1130
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
nipkow@14187
  1131
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@14187
  1132
nipkow@14187
  1133
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
paulson@14208
  1134
apply (induct xs, simp)
nipkow@14187
  1135
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
  1136
done
nipkow@14187
  1137
nipkow@13913
  1138
lemma take_Suc_conv_app_nth:
nipkow@13913
  1139
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
paulson@14208
  1140
apply (induct xs, simp)
paulson@14208
  1141
apply (case_tac i, auto)
nipkow@13913
  1142
done
nipkow@13913
  1143
mehta@14591
  1144
lemma drop_Suc_conv_tl:
mehta@14591
  1145
  "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
mehta@14591
  1146
apply (induct xs, simp)
mehta@14591
  1147
apply (case_tac i, auto)
mehta@14591
  1148
done
mehta@14591
  1149
wenzelm@13142
  1150
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
nipkow@13145
  1151
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1152
wenzelm@13142
  1153
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
nipkow@13145
  1154
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1155
wenzelm@13142
  1156
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
nipkow@13145
  1157
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1158
wenzelm@13142
  1159
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
nipkow@13145
  1160
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1161
wenzelm@13142
  1162
lemma take_append [simp]:
nipkow@13145
  1163
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@13145
  1164
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1165
wenzelm@13142
  1166
lemma drop_append [simp]:
nipkow@13145
  1167
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@13145
  1168
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1169
wenzelm@13142
  1170
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
paulson@14208
  1171
apply (induct m, auto)
paulson@14208
  1172
apply (case_tac xs, auto)
nipkow@15236
  1173
apply (case_tac n, auto)
nipkow@13145
  1174
done
wenzelm@13114
  1175
wenzelm@13142
  1176
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
paulson@14208
  1177
apply (induct m, auto)
paulson@14208
  1178
apply (case_tac xs, auto)
nipkow@13145
  1179
done
wenzelm@13114
  1180
wenzelm@13114
  1181
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
paulson@14208
  1182
apply (induct m, auto)
paulson@14208
  1183
apply (case_tac xs, auto)
nipkow@13145
  1184
done
wenzelm@13114
  1185
nipkow@14802
  1186
lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@14802
  1187
apply(induct xs)
nipkow@14802
  1188
 apply simp
nipkow@14802
  1189
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  1190
done
nipkow@14802
  1191
wenzelm@13142
  1192
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
paulson@14208
  1193
apply (induct n, auto)
paulson@14208
  1194
apply (case_tac xs, auto)
nipkow@13145
  1195
done
wenzelm@13114
  1196
nipkow@15110
  1197
lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@15110
  1198
apply(induct xs)
nipkow@15110
  1199
 apply simp
nipkow@15110
  1200
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  1201
done
nipkow@15110
  1202
nipkow@15110
  1203
lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"
nipkow@15110
  1204
apply(induct xs)
nipkow@15110
  1205
apply simp
nipkow@15110
  1206
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1207
done
nipkow@15110
  1208
wenzelm@13114
  1209
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
paulson@14208
  1210
apply (induct n, auto)
paulson@14208
  1211
apply (case_tac xs, auto)
nipkow@13145
  1212
done
wenzelm@13114
  1213
wenzelm@13142
  1214
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
paulson@14208
  1215
apply (induct n, auto)
paulson@14208
  1216
apply (case_tac xs, auto)
nipkow@13145
  1217
done
wenzelm@13114
  1218
wenzelm@13114
  1219
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
paulson@14208
  1220
apply (induct xs, auto)
paulson@14208
  1221
apply (case_tac i, auto)
nipkow@13145
  1222
done
wenzelm@13114
  1223
wenzelm@13114
  1224
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
paulson@14208
  1225
apply (induct xs, auto)
paulson@14208
  1226
apply (case_tac i, auto)
nipkow@13145
  1227
done
wenzelm@13114
  1228
wenzelm@13142
  1229
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
paulson@14208
  1230
apply (induct xs, auto)
paulson@14208
  1231
apply (case_tac n, blast)
paulson@14208
  1232
apply (case_tac i, auto)
nipkow@13145
  1233
done
wenzelm@13114
  1234
wenzelm@13142
  1235
lemma nth_drop [simp]:
nipkow@13145
  1236
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
paulson@14208
  1237
apply (induct n, auto)
paulson@14208
  1238
apply (case_tac xs, auto)
nipkow@13145
  1239
done
nipkow@3507
  1240
nipkow@18423
  1241
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
nipkow@18423
  1242
by(simp add: hd_conv_nth)
nipkow@18423
  1243
nipkow@14025
  1244
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
nipkow@14025
  1245
by(induct xs)(auto simp:take_Cons split:nat.split)
nipkow@14025
  1246
nipkow@14025
  1247
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
nipkow@14025
  1248
by(induct xs)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1249
nipkow@14187
  1250
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1251
using set_take_subset by fast
nipkow@14187
  1252
nipkow@14187
  1253
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1254
using set_drop_subset by fast
nipkow@14187
  1255
wenzelm@13114
  1256
lemma append_eq_conv_conj:
nipkow@13145
  1257
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
paulson@14208
  1258
apply (induct xs, simp, clarsimp)
paulson@14208
  1259
apply (case_tac zs, auto)
nipkow@13145
  1260
done
wenzelm@13142
  1261
paulson@14050
  1262
lemma take_add [rule_format]: 
paulson@14050
  1263
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
paulson@14050
  1264
apply (induct xs, auto) 
paulson@14050
  1265
apply (case_tac i, simp_all) 
paulson@14050
  1266
done
paulson@14050
  1267
nipkow@14300
  1268
lemma append_eq_append_conv_if:
nipkow@14300
  1269
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1270
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1271
   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
  1272
   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
  1273
apply(induct xs\<^isub>1)
nipkow@14300
  1274
 apply simp
nipkow@14300
  1275
apply(case_tac ys\<^isub>1)
nipkow@14300
  1276
apply simp_all
nipkow@14300
  1277
done
nipkow@14300
  1278
nipkow@15110
  1279
lemma take_hd_drop:
nipkow@15110
  1280
  "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
nipkow@15110
  1281
apply(induct xs)
nipkow@15110
  1282
apply simp
nipkow@15110
  1283
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1284
done
nipkow@15110
  1285
nipkow@17501
  1286
lemma id_take_nth_drop:
nipkow@17501
  1287
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
nipkow@17501
  1288
proof -
nipkow@17501
  1289
  assume si: "i < length xs"
nipkow@17501
  1290
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
nipkow@17501
  1291
  moreover
nipkow@17501
  1292
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
nipkow@17501
  1293
    apply (rule_tac take_Suc_conv_app_nth) by arith
nipkow@17501
  1294
  ultimately show ?thesis by auto
nipkow@17501
  1295
qed
nipkow@17501
  1296
  
nipkow@17501
  1297
lemma upd_conv_take_nth_drop:
nipkow@17501
  1298
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1299
proof -
nipkow@17501
  1300
  assume i: "i < length xs"
nipkow@17501
  1301
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
nipkow@17501
  1302
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
nipkow@17501
  1303
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1304
    using i by (simp add: list_update_append)
nipkow@17501
  1305
  finally show ?thesis .
nipkow@17501
  1306
qed
nipkow@17501
  1307
wenzelm@13114
  1308
nipkow@15392
  1309
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1310
wenzelm@13142
  1311
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1312
by (induct xs) auto
wenzelm@13114
  1313
wenzelm@13142
  1314
lemma takeWhile_append1 [simp]:
nipkow@13145
  1315
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1316
by (induct xs) auto
wenzelm@13114
  1317
wenzelm@13142
  1318
lemma takeWhile_append2 [simp]:
nipkow@13145
  1319
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1320
by (induct xs) auto
wenzelm@13114
  1321
wenzelm@13142
  1322
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1323
by (induct xs) auto
wenzelm@13114
  1324
wenzelm@13142
  1325
lemma dropWhile_append1 [simp]:
nipkow@13145
  1326
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1327
by (induct xs) auto
wenzelm@13114
  1328
wenzelm@13142
  1329
lemma dropWhile_append2 [simp]:
nipkow@13145
  1330
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1331
by (induct xs) auto
wenzelm@13114
  1332
wenzelm@13142
  1333
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1334
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1335
nipkow@13913
  1336
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1337
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1338
by(induct xs, auto)
nipkow@13913
  1339
nipkow@13913
  1340
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1341
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1342
by(induct xs, auto)
nipkow@13913
  1343
nipkow@13913
  1344
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1345
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1346
by(induct xs, auto)
nipkow@13913
  1347
nipkow@17501
  1348
text{* The following two lemmmas could be generalized to an arbitrary
nipkow@17501
  1349
property. *}
nipkow@17501
  1350
nipkow@17501
  1351
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1352
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
nipkow@17501
  1353
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
nipkow@17501
  1354
nipkow@17501
  1355
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1356
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
nipkow@17501
  1357
apply(induct xs)
nipkow@17501
  1358
 apply simp
nipkow@17501
  1359
apply auto
nipkow@17501
  1360
apply(subst dropWhile_append2)
nipkow@17501
  1361
apply auto
nipkow@17501
  1362
done
nipkow@17501
  1363
nipkow@18423
  1364
lemma takeWhile_not_last:
nipkow@18423
  1365
 "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
nipkow@18423
  1366
apply(induct xs)
nipkow@18423
  1367
 apply simp
nipkow@18423
  1368
apply(case_tac xs)
nipkow@18423
  1369
apply(auto)
nipkow@18423
  1370
done
nipkow@18423
  1371
krauss@19770
  1372
lemma takeWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1373
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1374
  ==> takeWhile P l = takeWhile Q k"
wenzelm@20503
  1375
  by (induct k arbitrary: l) (simp_all)
krauss@18336
  1376
krauss@19770
  1377
lemma dropWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1378
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1379
  ==> dropWhile P l = dropWhile Q k"
wenzelm@20503
  1380
  by (induct k arbitrary: l, simp_all)
krauss@18336
  1381
wenzelm@13114
  1382
nipkow@15392
  1383
subsubsection {* @{text zip} *}
wenzelm@13114
  1384
wenzelm@13142
  1385
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1386
by (induct ys) auto
wenzelm@13114
  1387
wenzelm@13142
  1388
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1389
by simp
wenzelm@13114
  1390
wenzelm@13142
  1391
declare zip_Cons [simp del]
wenzelm@13114
  1392
nipkow@15281
  1393
lemma zip_Cons1:
nipkow@15281
  1394
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  1395
by(auto split:list.split)
nipkow@15281
  1396
wenzelm@13142
  1397
lemma length_zip [simp]:
krauss@22493
  1398
"length (zip xs ys) = min (length xs) (length ys)"
krauss@22493
  1399
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1400
wenzelm@13114
  1401
lemma zip_append1:
krauss@22493
  1402
"zip (xs @ ys) zs =
nipkow@13145
  1403
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
krauss@22493
  1404
by (induct xs zs rule:list_induct2') auto
wenzelm@13114
  1405
wenzelm@13114
  1406
lemma zip_append2:
krauss@22493
  1407
"zip xs (ys @ zs) =
nipkow@13145
  1408
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
krauss@22493
  1409
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1410
wenzelm@13142
  1411
lemma zip_append [simp]:
wenzelm@13142
  1412
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1413
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1414
by (simp add: zip_append1)
wenzelm@13114
  1415
wenzelm@13114
  1416
lemma zip_rev:
nipkow@14247
  1417
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1418
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1419
wenzelm@13142
  1420
lemma nth_zip [simp]:
nipkow@13145
  1421
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
paulson@14208
  1422
apply (induct ys, simp)
nipkow@13145
  1423
apply (case_tac xs)
nipkow@13145
  1424
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1425
done
wenzelm@13114
  1426
wenzelm@13114
  1427
lemma set_zip:
nipkow@13145
  1428
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
  1429
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1430
wenzelm@13114
  1431
lemma zip_update:
nipkow@13145
  1432
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
  1433
by (rule sym, simp add: update_zip)
wenzelm@13114
  1434
wenzelm@13142
  1435
lemma zip_replicate [simp]:
nipkow@13145
  1436
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
paulson@14208
  1437
apply (induct i, auto)
paulson@14208
  1438
apply (case_tac j, auto)
nipkow@13145
  1439
done
wenzelm@13114
  1440
nipkow@19487
  1441
lemma take_zip:
nipkow@19487
  1442
 "!!xs ys. take n (zip xs ys) = zip (take n xs) (take n ys)"
nipkow@19487
  1443
apply (induct n)
nipkow@19487
  1444
 apply simp
nipkow@19487
  1445
apply (case_tac xs, simp)
nipkow@19487
  1446
apply (case_tac ys, simp_all)
nipkow@19487
  1447
done
nipkow@19487
  1448
nipkow@19487
  1449
lemma drop_zip:
nipkow@19487
  1450
 "!!xs ys. drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
nipkow@19487
  1451
apply (induct n)
nipkow@19487
  1452
 apply simp
nipkow@19487
  1453
apply (case_tac xs, simp)
nipkow@19487
  1454
apply (case_tac ys, simp_all)
nipkow@19487
  1455
done
nipkow@19487
  1456
krauss@22493
  1457
lemma set_zip_leftD:
krauss@22493
  1458
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
krauss@22493
  1459
by (induct xs ys rule:list_induct2') auto
krauss@22493
  1460
krauss@22493
  1461
lemma set_zip_rightD:
krauss@22493
  1462
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
krauss@22493
  1463
by (induct xs ys rule:list_induct2') auto
wenzelm@13142
  1464
nipkow@15392
  1465
subsubsection {* @{text list_all2} *}
wenzelm@13114
  1466
kleing@14316
  1467
lemma list_all2_lengthD [intro?]: 
kleing@14316
  1468
  "list_all2 P xs ys ==> length xs = length ys"
haftmann@19607
  1469
  by (simp add: list_all2_def)
haftmann@19607
  1470
haftmann@19787
  1471
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
haftmann@19607
  1472
  by (simp add: list_all2_def)
haftmann@19607
  1473
haftmann@19787
  1474
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
haftmann@19787
  1475
  by (simp add: list_all2_def)
haftmann@19607
  1476
haftmann@19607
  1477
lemma list_all2_Cons [iff, code]:
haftmann@19607
  1478
  "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
haftmann@19607
  1479
  by (auto simp add: list_all2_def)
wenzelm@13114
  1480
wenzelm@13114
  1481
lemma list_all2_Cons1:
nipkow@13145
  1482
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1483
by (cases ys) auto
wenzelm@13114
  1484
wenzelm@13114
  1485
lemma list_all2_Cons2:
nipkow@13145
  1486
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1487
by (cases xs) auto
wenzelm@13114
  1488
wenzelm@13142
  1489
lemma list_all2_rev [iff]:
nipkow@13145
  1490
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1491
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1492
kleing@13863
  1493
lemma list_all2_rev1:
kleing@13863
  1494
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1495
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1496
wenzelm@13114
  1497
lemma list_all2_append1:
nipkow@13145
  1498
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1499
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1500
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1501
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1502
apply (rule iffI)
nipkow@13145
  1503
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1504
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  1505
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1506
apply (simp add: ball_Un)
nipkow@13145
  1507
done
wenzelm@13114
  1508
wenzelm@13114
  1509
lemma list_all2_append2:
nipkow@13145
  1510
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1511
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1512
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1513
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1514
apply (rule iffI)
nipkow@13145
  1515
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1516
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  1517
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1518
apply (simp add: ball_Un)
nipkow@13145
  1519
done
wenzelm@13114
  1520
kleing@13863
  1521
lemma list_all2_append:
nipkow@14247
  1522
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  1523
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  1524
by (induct rule:list_induct2, simp_all)
kleing@13863
  1525
kleing@13863
  1526
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1527
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
kleing@13863
  1528
  by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1529
wenzelm@13114
  1530
lemma list_all2_conv_all_nth:
nipkow@13145
  1531
"list_all2 P xs ys =
nipkow@13145
  1532
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1533
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1534
berghofe@13883
  1535
lemma list_all2_trans:
berghofe@13883
  1536
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1537
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1538
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1539
proof (induct as)
berghofe@13883
  1540
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1541
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1542
  proof (induct bs)
berghofe@13883
  1543
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1544
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1545
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1546
  qed simp
berghofe@13883
  1547
qed simp
berghofe@13883
  1548
kleing@13863
  1549
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1550
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
kleing@13863
  1551
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1552
paulson@14395
  1553
lemma list_all2I:
paulson@14395
  1554
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
paulson@14395
  1555
  by (simp add: list_all2_def)
paulson@14395
  1556
kleing@14328
  1557
lemma list_all2_nthD:
kleing@13863
  1558
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
kleing@13863
  1559
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1560
nipkow@14302
  1561
lemma list_all2_nthD2:
nipkow@14302
  1562
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@14302
  1563
  by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  1564
kleing@13863
  1565
lemma list_all2_map1: 
kleing@13863
  1566
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
kleing@13863
  1567
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1568
kleing@13863
  1569
lemma list_all2_map2: 
kleing@13863
  1570
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
kleing@13863
  1571
  by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1572
kleing@14316
  1573
lemma list_all2_refl [intro?]:
kleing@13863
  1574
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
kleing@13863
  1575
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1576
kleing@13863
  1577
lemma list_all2_update_cong:
kleing@13863
  1578
  "\<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
  1579
  by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1580
kleing@13863
  1581
lemma list_all2_update_cong2:
kleing@13863
  1582
  "\<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
  1583
  by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1584
nipkow@14302
  1585
lemma list_all2_takeI [simp,intro?]:
nipkow@14302
  1586
  "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@14302
  1587
  apply (induct xs)
nipkow@14302
  1588
   apply simp
nipkow@14302
  1589
  apply (clarsimp simp add: list_all2_Cons1)
nipkow@14302
  1590
  apply (case_tac n)
nipkow@14302
  1591
  apply auto
nipkow@14302
  1592
  done
nipkow@14302
  1593
nipkow@14302
  1594
lemma list_all2_dropI [simp,intro?]:
kleing@13863
  1595
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
paulson@14208
  1596
  apply (induct as, simp)
kleing@13863
  1597
  apply (clarsimp simp add: list_all2_Cons1)
paulson@14208
  1598
  apply (case_tac n, simp, simp)
kleing@13863
  1599
  done
kleing@13863
  1600
kleing@14327
  1601
lemma list_all2_mono [intro?]:
kleing@13863
  1602
  "\<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
  1603
  apply (induct x, simp)
paulson@14208
  1604
  apply (case_tac y, auto)
kleing@13863
  1605
  done
kleing@13863
  1606
haftmann@22551
  1607
lemma list_all2_eq:
haftmann@22551
  1608
  "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
haftmann@22551
  1609
  by (induct xs ys rule: list_induct2') auto
haftmann@22551
  1610
wenzelm@13142
  1611
nipkow@15392
  1612
subsubsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  1613
wenzelm@13142
  1614
lemma foldl_append [simp]:
nipkow@13145
  1615
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145
  1616
by (induct xs) auto
wenzelm@13142
  1617
nipkow@14402
  1618
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  1619
by (induct xs) auto
nipkow@14402
  1620
krauss@19770
  1621
lemma foldl_cong [fundef_cong, recdef_cong]:
krauss@18336
  1622
  "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] 
krauss@18336
  1623
  ==> foldl f a l = foldl g b k"
wenzelm@20503
  1624
  by (induct k arbitrary: a b l) simp_all
krauss@18336
  1625
krauss@19770
  1626
lemma foldr_cong [fundef_cong, recdef_cong]:
krauss@18336
  1627
  "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] 
krauss@18336
  1628
  ==> foldr f l a = foldr g k b"
wenzelm@20503
  1629
  by (induct k arbitrary: a b l) simp_all
krauss@18336
  1630
nipkow@14402
  1631
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  1632
by (induct xs) auto
nipkow@14402
  1633
nipkow@14402
  1634
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  1635
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  1636
wenzelm@13142
  1637
text {*
nipkow@13145
  1638
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1639
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1640
*}
wenzelm@13142
  1641
wenzelm@13142
  1642
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
nipkow@13145
  1643
by (induct ns) auto
wenzelm@13142
  1644
wenzelm@13142
  1645
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1646
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1647
wenzelm@13142
  1648
lemma sum_eq_0_conv [iff]:
nipkow@13145
  1649
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145
  1650
by (induct ns) auto
wenzelm@13114
  1651
wenzelm@13114
  1652
nipkow@15392
  1653
subsubsection {* @{text upto} *}
wenzelm@13114
  1654
nipkow@17090
  1655
lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
nipkow@17090
  1656
-- {* simp does not terminate! *}
nipkow@13145
  1657
by (induct j) auto
wenzelm@13142
  1658
nipkow@15425
  1659
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
nipkow@13145
  1660
by (subst upt_rec) simp
wenzelm@13114
  1661
nipkow@15425
  1662
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
nipkow@15281
  1663
by(induct j)simp_all
nipkow@15281
  1664
nipkow@15281
  1665
lemma upt_eq_Cons_conv:
nipkow@15425
  1666
 "!!x xs. ([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
nipkow@15281
  1667
apply(induct j)
nipkow@15281
  1668
 apply simp
nipkow@15281
  1669
apply(clarsimp simp add: append_eq_Cons_conv)
nipkow@15281
  1670
apply arith
nipkow@15281
  1671
done
nipkow@15281
  1672
nipkow@15425
  1673
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
nipkow@13145
  1674
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1675
by simp
wenzelm@13114
  1676
nipkow@15425
  1677
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
nipkow@13145
  1678
apply(rule trans)
nipkow@13145
  1679
apply(subst upt_rec)
paulson@14208
  1680
 prefer 2 apply (rule refl, simp)
nipkow@13145
  1681
done
wenzelm@13114
  1682
nipkow@15425
  1683
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
nipkow@13145
  1684
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1685
by (induct k) auto
wenzelm@13114
  1686
nipkow@15425
  1687
lemma length_upt [simp]: "length [i..<j] = j - i"
nipkow@13145
  1688
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1689
nipkow@15425
  1690
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
nipkow@13145
  1691
apply (induct j)
nipkow@13145
  1692
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1693
done
wenzelm@13114
  1694
nipkow@17906
  1695
nipkow@17906
  1696
lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
nipkow@17906
  1697
by(simp add:upt_conv_Cons)
nipkow@17906
  1698
nipkow@17906
  1699
lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
nipkow@17906
  1700
apply(cases j)
nipkow@17906
  1701
 apply simp
nipkow@17906
  1702
by(simp add:upt_Suc_append)
nipkow@17906
  1703
nipkow@15425
  1704
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..<n] = [i..<i+m]"
paulson@14208
  1705
apply (induct m, simp)
nipkow@13145
  1706
apply (subst upt_rec)
nipkow@13145
  1707
apply (rule sym)
nipkow@13145
  1708
apply (subst upt_rec)
nipkow@13145
  1709
apply (simp del: upt.simps)
nipkow@13145
  1710
done
nipkow@3507
  1711
nipkow@17501
  1712
lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
nipkow@17501
  1713
apply(induct j)
nipkow@17501
  1714
apply auto
nipkow@17501
  1715
done
nipkow@17501
  1716
nipkow@15425
  1717
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..n]"
nipkow@13145
  1718
by (induct n) auto
wenzelm@13114
  1719
nipkow@15425
  1720
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
nipkow@13145
  1721
apply (induct n m rule: diff_induct)
nipkow@13145
  1722
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1723
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1724
done
wenzelm@13114
  1725
berghofe@13883
  1726
lemma nth_take_lemma:
berghofe@13883
  1727
  "!!xs ys. k <= length xs ==> k <= length ys ==>
berghofe@13883
  1728
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
berghofe@13883
  1729
apply (atomize, induct k)
paulson@14208
  1730
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  1731
txt {* Both lists must be non-empty *}
paulson@14208
  1732
apply (case_tac xs, simp)
paulson@14208
  1733
apply (case_tac ys, clarify)
nipkow@13145
  1734
 apply (simp (no_asm_use))
nipkow@13145
  1735
apply clarify
nipkow@13145
  1736
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  1737
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  1738
apply blast
nipkow@13145
  1739
done
wenzelm@13114
  1740
wenzelm@13114
  1741
lemma nth_equalityI:
wenzelm@13114
  1742
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  1743
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  1744
apply (simp_all add: take_all)
nipkow@13145
  1745
done
wenzelm@13142
  1746
kleing@13863
  1747
(* needs nth_equalityI *)
kleing@13863
  1748
lemma list_all2_antisym:
kleing@13863
  1749
  "\<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
  1750
  \<Longrightarrow> xs = ys"
kleing@13863
  1751
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  1752
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  1753
  done
kleing@13863
  1754
wenzelm@13142
  1755
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  1756
-- {* The famous take-lemma. *}
nipkow@13145
  1757
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  1758
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  1759
done
wenzelm@13142
  1760
wenzelm@13142
  1761
nipkow@15302
  1762
lemma take_Cons':
nipkow@15302
  1763
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@15302
  1764
by (cases n) simp_all
nipkow@15302
  1765
nipkow@15302
  1766
lemma drop_Cons':
nipkow@15302
  1767
     "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@15302
  1768
by (cases n) simp_all
nipkow@15302
  1769
nipkow@15302
  1770
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@15302
  1771
by (cases n) simp_all
nipkow@15302
  1772
paulson@18622
  1773
lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard]
paulson@18622
  1774
lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard]
paulson@18622
  1775
lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard]
paulson@18622
  1776
paulson@18622
  1777
declare take_Cons_number_of [simp] 
paulson@18622
  1778
        drop_Cons_number_of [simp] 
paulson@18622
  1779
        nth_Cons_number_of [simp] 
nipkow@15302
  1780
nipkow@15302
  1781
nipkow@15392
  1782
subsubsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  1783
wenzelm@13142
  1784
lemma distinct_append [simp]:
nipkow@13145
  1785
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  1786
by (induct xs) auto
wenzelm@13142
  1787
nipkow@15305
  1788
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
nipkow@15305
  1789
by(induct xs) auto
nipkow@15305
  1790
wenzelm@13142
  1791
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  1792
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  1793
wenzelm@13142
  1794
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  1795
by (induct xs) auto
wenzelm@13142
  1796
paulson@15072
  1797
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
paulson@15251
  1798
  by (induct x, auto) 
paulson@15072
  1799
paulson@15072
  1800
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
paulson@15251
  1801
  by (induct x, auto)
paulson@15072
  1802
nipkow@15245
  1803
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245
  1804
by (induct xs) auto
nipkow@15245
  1805
nipkow@15245
  1806
lemma length_remdups_eq[iff]:
nipkow@15245
  1807
  "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245
  1808
apply(induct xs)
nipkow@15245
  1809
 apply auto
nipkow@15245
  1810
apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245
  1811
 apply arith
nipkow@15245
  1812
apply(rule length_remdups_leq)
nipkow@15245
  1813
done
nipkow@15245
  1814
nipkow@18490
  1815
nipkow@18490
  1816
lemma distinct_map:
nipkow@18490
  1817
  "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
nipkow@18490
  1818
by (induct xs) auto
nipkow@18490
  1819
nipkow@18490
  1820
wenzelm@13142
  1821
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  1822
by (induct xs) auto
wenzelm@13114
  1823
nipkow@17501
  1824
lemma distinct_upt[simp]: "distinct[i..<j]"
nipkow@17501
  1825
by (induct j) auto
nipkow@17501
  1826
nipkow@17501
  1827
lemma distinct_take[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (take i xs)"
nipkow@17501
  1828
apply(induct xs)
nipkow@17501
  1829
 apply simp
nipkow@17501
  1830
apply (case_tac i)
nipkow@17501
  1831
 apply simp_all
nipkow@17501
  1832
apply(blast dest:in_set_takeD)
nipkow@17501
  1833
done
nipkow@17501
  1834
nipkow@17501
  1835
lemma distinct_drop[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (drop i xs)"
nipkow@17501
  1836
apply(induct xs)
nipkow@17501
  1837
 apply simp
nipkow@17501
  1838
apply (case_tac i)
nipkow@17501
  1839
 apply simp_all
nipkow@17501
  1840
done
nipkow@17501
  1841
nipkow@17501
  1842
lemma distinct_list_update:
nipkow@17501
  1843
assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
nipkow@17501
  1844
shows "distinct (xs[i:=a])"
nipkow@17501
  1845
proof (cases "i < length xs")
nipkow@17501
  1846
  case True
nipkow@17501
  1847
  with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
nipkow@17501
  1848
    apply (drule_tac id_take_nth_drop) by simp
nipkow@17501
  1849
  with d True show ?thesis
nipkow@17501
  1850
    apply (simp add: upd_conv_take_nth_drop)
nipkow@17501
  1851
    apply (drule subst [OF id_take_nth_drop]) apply assumption
nipkow@17501
  1852
    apply simp apply (cases "a = xs!i") apply simp by blast
nipkow@17501
  1853
next
nipkow@17501
  1854
  case False with d show ?thesis by auto
nipkow@17501
  1855
qed
nipkow@17501
  1856
nipkow@17501
  1857
nipkow@17501
  1858
text {* It is best to avoid this indexed version of distinct, but
nipkow@17501
  1859
sometimes it is useful. *}
nipkow@17501
  1860
wenzelm@13142
  1861
lemma distinct_conv_nth:
nipkow@17501
  1862
"distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@15251
  1863
apply (induct xs, simp, simp)
paulson@14208
  1864
apply (rule iffI, clarsimp)
nipkow@13145
  1865
 apply (case_tac i)
paulson@14208
  1866
apply (case_tac j, simp)
nipkow@13145
  1867
apply (simp add: set_conv_nth)
nipkow@13145
  1868
 apply (case_tac j)
paulson@14208
  1869
apply (clarsimp simp add: set_conv_nth, simp)
nipkow@13145
  1870
apply (rule conjI)
nipkow@13145
  1871
 apply (clarsimp simp add: set_conv_nth)
nipkow@17501
  1872
 apply (erule_tac x = 0 in allE, simp)
paulson@14208
  1873
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
nipkow@17501
  1874
apply (erule_tac x = "Suc i" in allE, simp)
paulson@14208
  1875
apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145
  1876
done
wenzelm@13114
  1877
nipkow@18490
  1878
lemma nth_eq_iff_index_eq:
nipkow@18490
  1879
 "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
nipkow@18490
  1880
by(auto simp: distinct_conv_nth)
nipkow@18490
  1881
nipkow@15110
  1882
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
kleing@14388
  1883
  by (induct xs) auto
kleing@14388
  1884
nipkow@15110
  1885
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388
  1886
proof (induct xs)
kleing@14388
  1887
  case Nil thus ?case by simp
kleing@14388
  1888
next
kleing@14388
  1889
  case (Cons x xs)
kleing@14388
  1890
  show ?case
kleing@14388
  1891
  proof (cases "x \<in> set xs")
kleing@14388
  1892
    case False with Cons show ?thesis by simp
kleing@14388
  1893
  next
kleing@14388
  1894
    case True with Cons.prems
kleing@14388
  1895
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  1896
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  1897
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  1898
    ultimately have False by simp
kleing@14388
  1899
    thus ?thesis ..
kleing@14388
  1900
  qed
kleing@14388
  1901
qed
kleing@14388
  1902
nipkow@18490
  1903
nipkow@18490
  1904
lemma length_remdups_concat:
nipkow@18490
  1905
 "length(remdups(concat xss)) = card(\<Union>xs \<in> set xss. set xs)"
nipkow@18490
  1906
by(simp add: distinct_card[symmetric])
nipkow@17906
  1907
nipkow@17906
  1908
nipkow@15392
  1909
subsubsection {* @{text remove1} *}
nipkow@15110
  1910
nipkow@18049
  1911
lemma remove1_append:
nipkow@18049
  1912
  "remove1 x (xs @ ys) =
nipkow@18049
  1913
  (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
nipkow@18049
  1914
by (induct xs) auto
nipkow@18049
  1915
nipkow@15110
  1916
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110
  1917
apply(induct xs)
nipkow@15110
  1918
 apply simp
nipkow@15110
  1919
apply simp
nipkow@15110
  1920
apply blast
nipkow@15110
  1921
done
nipkow@15110
  1922
paulson@17724
  1923
lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110
  1924
apply(induct xs)
nipkow@15110
  1925
 apply simp
nipkow@15110
  1926
apply simp
nipkow@15110
  1927
apply blast
nipkow@15110
  1928
done
nipkow@15110
  1929
nipkow@18049
  1930
lemma remove1_filter_not[simp]:
nipkow@18049
  1931
  "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
nipkow@18049
  1932
by(induct xs) auto
nipkow@18049
  1933
nipkow@15110
  1934
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110
  1935
apply(insert set_remove1_subset)
nipkow@15110
  1936
apply fast
nipkow@15110
  1937
done
nipkow@15110
  1938
nipkow@15110
  1939
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110
  1940
by (induct xs) simp_all
nipkow@15110
  1941
wenzelm@13114
  1942
nipkow@15392
  1943
subsubsection {* @{text replicate} *}
wenzelm@13114
  1944
wenzelm@13142
  1945
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  1946
by (induct n) auto
nipkow@13124
  1947
wenzelm@13142
  1948
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  1949
by (induct n) auto
wenzelm@13114
  1950
wenzelm@13114
  1951
lemma replicate_app_Cons_same:
nipkow@13145
  1952
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  1953
by (induct n) auto
wenzelm@13114
  1954
wenzelm@13142
  1955
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208
  1956
apply (induct n, simp)
nipkow@13145
  1957
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  1958
done
wenzelm@13114
  1959
wenzelm@13142
  1960
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  1961
by (induct n) auto
wenzelm@13114
  1962
nipkow@16397
  1963
text{* Courtesy of Matthias Daum: *}
nipkow@16397
  1964
lemma append_replicate_commute:
nipkow@16397
  1965
  "replicate n x @ replicate k x = replicate k x @ replicate n x"
nipkow@16397
  1966
apply (simp add: replicate_add [THEN sym])
nipkow@16397
  1967
apply (simp add: add_commute)
nipkow@16397
  1968
done
nipkow@16397
  1969
wenzelm@13142
  1970
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  1971
by (induct n) auto
wenzelm@13114
  1972
wenzelm@13142
  1973
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  1974
by (induct n) auto
wenzelm@13114
  1975
wenzelm@13142
  1976
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  1977
by (atomize (full), induct n) auto
wenzelm@13114
  1978
wenzelm@13142
  1979
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
paulson@14208
  1980
apply (induct n, simp)
nipkow@13145
  1981
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  1982
done
wenzelm@13114
  1983
nipkow@16397
  1984
text{* Courtesy of Matthias Daum (2 lemmas): *}
nipkow@16397
  1985
lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
nipkow@16397
  1986
apply (case_tac "k \<le> i")
nipkow@16397
  1987
 apply  (simp add: min_def)
nipkow@16397
  1988
apply (drule not_leE)
nipkow@16397
  1989
apply (simp add: min_def)
nipkow@16397
  1990
apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
nipkow@16397
  1991
 apply  simp
nipkow@16397
  1992
apply (simp add: replicate_add [symmetric])
nipkow@16397
  1993
done
nipkow@16397
  1994
nipkow@16397
  1995
lemma drop_replicate[simp]: "!!i. drop i (replicate k x) = replicate (k-i) x"
nipkow@16397
  1996
apply (induct k)
nipkow@16397
  1997
 apply simp
nipkow@16397
  1998
apply clarsimp
nipkow@16397
  1999
apply (case_tac i)
nipkow@16397
  2000
 apply simp
nipkow@16397
  2001
apply clarsimp
nipkow@16397
  2002
done
nipkow@16397
  2003
nipkow@16397
  2004
wenzelm@13142
  2005
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  2006
by (induct n) auto
wenzelm@13114
  2007
wenzelm@13142
  2008
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  2009
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  2010
wenzelm@13142
  2011
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  2012
by auto
wenzelm@13114
  2013
wenzelm@13142
  2014
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  2015
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  2016
wenzelm@13114
  2017
nipkow@15392
  2018
subsubsection{*@{text rotate1} and @{text rotate}*}
nipkow@15302
  2019
nipkow@15302
  2020
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
nipkow@15302
  2021
by(simp add:rotate1_def)
nipkow@15302
  2022
nipkow@15302
  2023
lemma rotate0[simp]: "rotate 0 = id"
nipkow@15302
  2024
by(simp add:rotate_def)
nipkow@15302
  2025
nipkow@15302
  2026
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
nipkow@15302
  2027
by(simp add:rotate_def)
nipkow@15302
  2028
nipkow@15302
  2029
lemma rotate_add:
nipkow@15302
  2030
  "rotate (m+n) = rotate m o rotate n"
nipkow@15302
  2031
by(simp add:rotate_def funpow_add)
nipkow@15302
  2032
nipkow@15302
  2033
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
nipkow@15302
  2034
by(simp add:rotate_add)
nipkow@15302
  2035
nipkow@18049
  2036
lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
nipkow@18049
  2037
by(simp add:rotate_def funpow_swap1)
nipkow@18049
  2038
nipkow@15302
  2039
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
nipkow@15302
  2040
by(cases xs) simp_all
nipkow@15302
  2041
nipkow@15302
  2042
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2043
apply(induct n)
nipkow@15302
  2044
 apply simp
nipkow@15302
  2045
apply (simp add:rotate_def)
nipkow@13145
  2046
done
wenzelm@13114
  2047
nipkow@15302
  2048
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
nipkow@15302
  2049
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2050
nipkow@15302
  2051
lemma rotate_drop_take:
nipkow@15302
  2052
  "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
nipkow@15302
  2053
apply(induct n)
nipkow@15302
  2054
 apply simp
nipkow@15302
  2055
apply(simp add:rotate_def)
nipkow@15302
  2056
apply(cases "xs = []")
nipkow@15302
  2057
 apply (simp)
nipkow@15302
  2058
apply(case_tac "n mod length xs = 0")
nipkow@15302
  2059
 apply(simp add:mod_Suc)
nipkow@15302
  2060
 apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
nipkow@15302
  2061
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
nipkow@15302
  2062
                take_hd_drop linorder_not_le)
nipkow@13145
  2063
done
wenzelm@13114
  2064
nipkow@15302
  2065
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
nipkow@15302
  2066
by(simp add:rotate_drop_take)
nipkow@15302
  2067
nipkow@15302
  2068
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2069
by(simp add:rotate_drop_take)
nipkow@15302
  2070
nipkow@15302
  2071
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
nipkow@15302
  2072
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2073
nipkow@15302
  2074
lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs"
nipkow@15302
  2075
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2076
nipkow@15302
  2077
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
nipkow@15302
  2078
by(simp add:rotate1_def split:list.split) blast
nipkow@15302
  2079
nipkow@15302
  2080
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
nipkow@15302
  2081
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2082
nipkow@15302
  2083
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
nipkow@15302
  2084
by(simp add:rotate_drop_take take_map drop_map)
nipkow@15302
  2085
nipkow@15302
  2086
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
nipkow@15302
  2087
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2088
nipkow@15302
  2089
lemma set_rotate[simp]: "set(rotate n xs) = set xs"
nipkow@15302
  2090
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2091
nipkow@15302
  2092
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
nipkow@15302
  2093
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2094
nipkow@15302
  2095
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
nipkow@15302
  2096
by (induct n) (simp_all add:rotate_def)
wenzelm@13114
  2097
nipkow@15439
  2098
lemma rotate_rev:
nipkow@15439
  2099
  "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
nipkow@15439
  2100
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2101
apply(cases "length xs = 0")
nipkow@15439
  2102
 apply simp
nipkow@15439
  2103
apply(cases "n mod length xs = 0")
nipkow@15439
  2104
 apply simp
nipkow@15439
  2105
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2106
done
nipkow@15439
  2107
nipkow@18423
  2108
lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
nipkow@18423
  2109
apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
nipkow@18423
  2110
apply(subgoal_tac "length xs \<noteq> 0")
nipkow@18423
  2111
 prefer 2 apply simp
nipkow@18423
  2112
using mod_less_divisor[of "length xs" n] by arith
nipkow@18423
  2113
wenzelm@13114
  2114
nipkow@15392
  2115
subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  2116
wenzelm@13142
  2117
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  2118
by (auto simp add: sublist_def)
wenzelm@13114
  2119
wenzelm@13142
  2120
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  2121
by (auto simp add: sublist_def)
wenzelm@13114
  2122
nipkow@15281
  2123
lemma length_sublist:
nipkow@15281
  2124
  "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
nipkow@15281
  2125
by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
nipkow@15281
  2126
nipkow@15281
  2127
lemma sublist_shift_lemma_Suc:
nipkow@15281
  2128
  "!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
nipkow@15281
  2129
         map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
nipkow@15281
  2130
apply(induct xs)
nipkow@15281
  2131
 apply simp
nipkow@15281
  2132
apply (case_tac "is")
nipkow@15281
  2133
 apply simp
nipkow@15281
  2134
apply simp
nipkow@15281
  2135
done
nipkow@15281
  2136
wenzelm@13114
  2137
lemma sublist_shift_lemma:
nipkow@15425
  2138
     "map fst [p:zip xs [i..<i + length xs] . snd p : A] =
nipkow@15425
  2139
      map fst [p:zip xs [0..<length xs] . snd p + i : A]"
nipkow@13145
  2140
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  2141
wenzelm@13114
  2142
lemma sublist_append:
paulson@15168
  2143
     "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  2144
apply (unfold sublist_def)
paulson@14208
  2145
apply (induct l' rule: rev_induct, simp)
nipkow@13145
  2146
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  2147
apply (simp add: add_commute)
nipkow@13145
  2148
done
wenzelm@13114
  2149
wenzelm@13114
  2150
lemma sublist_Cons:
nipkow@13145
  2151
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  2152
apply (induct l rule: rev_induct)
nipkow@13145
  2153
 apply (simp add: sublist_def)
nipkow@13145
  2154
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  2155
done
wenzelm@13114
  2156
nipkow@15281
  2157
lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
nipkow@15281
  2158
apply(induct xs)
nipkow@15281
  2159
 apply simp
nipkow@15281
  2160
apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE)
nipkow@15281
  2161
 apply(erule lessE)
nipkow@15281
  2162
  apply auto
nipkow@15281
  2163
apply(erule lessE)
nipkow@15281
  2164
apply auto
nipkow@15281
  2165
done
nipkow@15281
  2166
nipkow@15281
  2167
lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
nipkow@15281
  2168
by(auto simp add:set_sublist)
nipkow@15281
  2169
nipkow@15281
  2170
lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
nipkow@15281
  2171
by(auto simp add:set_sublist)
nipkow@15281
  2172
nipkow@15281
  2173
lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
nipkow@15281
  2174
by(auto simp add:set_sublist)
nipkow@15281
  2175
wenzelm@13142
  2176
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  2177
by (simp add: sublist_Cons)
wenzelm@13114
  2178
nipkow@15281
  2179
nipkow@15281
  2180
lemma distinct_sublistI[simp]: "!!I. distinct xs \<Longrightarrow> distinct(sublist xs I)"
nipkow@15281
  2181
apply(induct xs)
nipkow@15281
  2182
 apply simp
nipkow@15281
  2183
apply(auto simp add:sublist_Cons)
nipkow@15281
  2184
done
nipkow@15281
  2185
nipkow@15281
  2186
nipkow@15045
  2187
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
paulson@14208
  2188
apply (induct l rule: rev_induct, simp)
nipkow@13145
  2189
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  2190
done
wenzelm@13114
  2191
nipkow@17501
  2192
lemma filter_in_sublist: "\<And>s. distinct xs \<Longrightarrow>
nipkow@17501
  2193
  filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
nipkow@17501
  2194
proof (induct xs)
nipkow@17501
  2195
  case Nil thus ?case by simp
nipkow@17501
  2196
next
nipkow@17501
  2197
  case (Cons a xs)
nipkow@17501
  2198
  moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
nipkow@17501
  2199
  ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
nipkow@17501
  2200
qed
nipkow@17501
  2201
wenzelm@13114
  2202
nipkow@19390
  2203
subsubsection {* @{const splice} *}
nipkow@19390
  2204
haftmann@19607
  2205
lemma splice_Nil2 [simp, code]:
nipkow@19390
  2206
 "splice xs [] = xs"
nipkow@19390
  2207
by (cases xs) simp_all
nipkow@19390
  2208
haftmann@19607
  2209
lemma splice_Cons_Cons [simp, code]:
nipkow@19390
  2210
 "splice (x#xs) (y#ys) = x # y # splice xs ys"
nipkow@19390
  2211
by simp
nipkow@19390
  2212
haftmann@19607
  2213
declare splice.simps(2) [simp del, code del]
nipkow@19390
  2214
nipkow@22793
  2215
lemma length_splice[simp]: "!!ys. length(splice xs ys) = length xs + length ys"
nipkow@22793
  2216
apply(induct xs) apply simp
nipkow@22793
  2217
apply(case_tac ys)
nipkow@22793
  2218
 apply auto
nipkow@22793
  2219
done
nipkow@22793
  2220
nipkow@22828
  2221
nipkow@22828
  2222
subsubsection {* @{const allpairs} *}
nipkow@22828
  2223
nipkow@22940
  2224
lemma allpairs_conv_concat:
nipkow@22940
  2225
 "allpairs f xs ys = concat(map (%x. map (f x) ys) xs)"
nipkow@22940
  2226
by(induct xs) auto
nipkow@22940
  2227
nipkow@22828
  2228
lemma allpairs_append:
nipkow@22830
  2229
 "allpairs f (xs @ ys) zs = allpairs f xs zs @ allpairs f ys zs"
nipkow@22828
  2230
by(induct xs) auto
nipkow@22828
  2231
nipkow@22828
  2232
nipkow@15392
  2233
subsubsection{*Sets of Lists*}
nipkow@15392
  2234
nipkow@15392
  2235
subsubsection {* @{text lists}: the list-forming operator over sets *}
nipkow@15302
  2236
berghofe@22262
  2237
inductive2
berghofe@22262
  2238
  listsp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
berghofe@22262
  2239
  for A :: "'a \<Rightarrow> bool"
berghofe@22262
  2240
where
berghofe@22262
  2241
    Nil [intro!]: "listsp A []"
berghofe@22262
  2242
  | Cons [intro!]: "[| A a; listsp A l |] ==> listsp A (a # l)"
berghofe@22262
  2243
berghofe@22262
  2244
constdefs
berghofe@22262
  2245
  lists :: "'a set => 'a list set"
berghofe@22262
  2246
  "lists A == Collect (listsp (member A))"
berghofe@22262
  2247
berghofe@22262
  2248
lemma listsp_lists_eq [pred_set_conv]: "listsp (member A) = member (lists A)"
berghofe@22262
  2249
  by (simp add: lists_def)
berghofe@22262
  2250
berghofe@22262
  2251
lemmas lists_intros [intro!] = listsp.intros [to_set]
berghofe@22262
  2252
berghofe@22262
  2253
lemmas lists_induct [consumes 1, case_names Nil Cons, induct set: lists] =
berghofe@22262
  2254
  listsp.induct [to_set]
berghofe@22262
  2255
berghofe@22262
  2256
inductive_cases2 listspE [elim!]: "listsp A (x # l)"
berghofe@22262
  2257
berghofe@22262
  2258
lemmas listsE [elim!] = listspE [to_set]
berghofe@22262
  2259
berghofe@22262
  2260
lemma listsp_mono [mono2]: "A \<le> B ==> listsp A \<le> listsp B"
berghofe@22262
  2261
  by (clarify, erule listsp.induct, blast+)
berghofe@22262
  2262
berghofe@22262
  2263
lemmas lists_mono [mono] = listsp_mono [to_set]
berghofe@22262
  2264
haftmann@22422
  2265
lemma listsp_infI:
haftmann@22422
  2266
  assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
nipkow@15302
  2267
  by induct blast+
nipkow@15302
  2268
haftmann@22422
  2269
lemmas lists_IntI = listsp_infI [to_set]
haftmann@22422
  2270
haftmann@22422
  2271
lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
haftmann@22422
  2272
proof (rule mono_inf [where f=listsp, THEN order_antisym])
berghofe@22262
  2273
  show "mono listsp" by (simp add: mono_def listsp_mono)
haftmann@22422
  2274
  show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro: listsp_infI)
kleing@14388
  2275
qed
kleing@14388
  2276
haftmann@22422
  2277
lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_eq inf_bool_eq]
haftmann@22422
  2278
haftmann@22422
  2279
lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set]
berghofe@22262
  2280
berghofe@22262
  2281
lemma append_in_listsp_conv [iff]:
berghofe@22262
  2282
     "(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
nipkow@15302
  2283
by (induct xs) auto
nipkow@15302
  2284
berghofe@22262
  2285
lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]
berghofe@22262
  2286
berghofe@22262
  2287
lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
berghofe@22262
  2288
-- {* eliminate @{text listsp} in favour of @{text set} *}
nipkow@15302
  2289
by (induct xs) auto
nipkow@15302
  2290
berghofe@22262
  2291
lemmas in_lists_conv_set = in_listsp_conv_set [to_set]
berghofe@22262
  2292
berghofe@22262
  2293
lemma in_listspD [dest!]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
berghofe@22262
  2294
by (rule in_listsp_conv_set [THEN iffD1])
berghofe@22262
  2295
berghofe@22262
  2296
lemmas in_listsD [dest!] = in_listspD [to_set]
berghofe@22262
  2297
berghofe@22262
  2298
lemma in_listspI [intro!]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
berghofe@22262
  2299
by (rule in_listsp_conv_set [THEN iffD2])
berghofe@22262
  2300
berghofe@22262
  2301
lemmas in_listsI [intro!] = in_listspI [to_set]
nipkow@15302
  2302
nipkow@15302
  2303
lemma lists_UNIV [simp]: "lists UNIV = UNIV"
nipkow@15302
  2304
by auto
nipkow@15302
  2305
nipkow@17086
  2306
nipkow@17086
  2307
nipkow@17086
  2308
subsubsection{* Inductive definition for membership *}
nipkow@17086
  2309
berghofe@22262
  2310
inductive2 ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
berghofe@22262
  2311
where
berghofe@22262
  2312
    elem:  "ListMem x (x # xs)"
berghofe@22262
  2313
  | insert:  "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"
berghofe@22262
  2314
berghofe@22262
  2315
lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
nipkow@17086
  2316
apply (rule iffI)
nipkow@17086
  2317
 apply (induct set: ListMem)
nipkow@17086
  2318
  apply auto
nipkow@17086
  2319
apply (induct xs)
nipkow@17086
  2320
 apply (auto intro: ListMem.intros)
nipkow@17086
  2321
done
nipkow@17086
  2322
nipkow@17086
  2323
nipkow@17086
  2324
nipkow@15392
  2325
subsubsection{*Lists as Cartesian products*}
nipkow@15302
  2326
nipkow@15302
  2327
text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
nipkow@15302
  2328
@{term A} and tail drawn from @{term Xs}.*}
nipkow@15302
  2329
nipkow@15302
  2330
constdefs
nipkow@15302
  2331
  set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
nipkow@15302
  2332
  "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"
nipkow@15302
  2333
paulson@17724
  2334
lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
nipkow@15302
  2335
by (auto simp add: set_Cons_def)
nipkow@15302
  2336
nipkow@15302
  2337
text{*Yields the set of lists, all of the same length as the argument and
nipkow@15302
  2338
with elements drawn from the corresponding element of the argument.*}
nipkow@15302
  2339
nipkow@15302
  2340
consts  listset :: "'a set list \<Rightarrow> 'a list set"
nipkow@15302
  2341
primrec
nipkow@15302
  2342
   "listset []    = {[]}"
nipkow@15302
  2343
   "listset(A#As) = set_Cons A (listset As)"
nipkow@15302
  2344
nipkow@15302
  2345
paulson@15656
  2346
subsection{*Relations on Lists*}
paulson@15656
  2347
paulson@15656
  2348
subsubsection {* Length Lexicographic Ordering *}
paulson@15656
  2349
paulson@15656
  2350
text{*These orderings preserve well-foundedness: shorter lists 
paulson@15656
  2351
  precede longer lists. These ordering are not used in dictionaries.*}
paulson@15656
  2352
paulson@15656
  2353
consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
paulson@15656
  2354
        --{*The lexicographic ordering for lists of the specified length*}
nipkow@15302
  2355
primrec
paulson@15656
  2356
  "lexn r 0 = {}"
paulson@15656
  2357
  "lexn r (Suc n) =
paulson@15656
  2358
    (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
paulson@15656
  2359
    {(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
nipkow@15302
  2360
nipkow@15302
  2361
constdefs
paulson@15656
  2362
  lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
paulson@15656
  2363
    "lex r == \<Union>n. lexn r n"
paulson@15656
  2364
        --{*Holds only between lists of the same length*}
paulson@15656
  2365
nipkow@15693
  2366
  lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@15693
  2367
    "lenlex r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
paulson@15656
  2368
        --{*Compares lists by their length and then lexicographically*}
nipkow@15302
  2369
nipkow@15302
  2370
nipkow@15302
  2371
lemma wf_lexn: "wf r ==> wf (lexn r n)"
nipkow@15302
  2372
apply (induct n, simp, simp)
nipkow@15302
  2373
apply(rule wf_subset)
nipkow@15302
  2374
 prefer 2 apply (rule Int_lower1)
nipkow@15302
  2375
apply(rule wf_prod_fun_image)
nipkow@15302
  2376
 prefer 2 apply (rule inj_onI, auto)
nipkow@15302
  2377
done
nipkow@15302
  2378
nipkow@15302
  2379
lemma lexn_length:
nipkow@15302
  2380
     "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@15302
  2381
by (induct n) auto
nipkow@15302
  2382
nipkow@15302
  2383
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@15302
  2384
apply (unfold lex_def)
nipkow@15302
  2385
apply (rule wf_UN)
nipkow@15302
  2386
apply (blast intro: wf_lexn, clarify)
nipkow@15302
  2387
apply (rename_tac m n)
nipkow@15302
  2388
apply (subgoal_tac "m \<noteq> n")
nipkow@15302
  2389
 prefer 2 apply blast
nipkow@15302
  2390
apply (blast dest: lexn_length not_sym)
nipkow@15302
  2391
done
nipkow@15302
  2392
nipkow@15302
  2393
lemma lexn_conv:
paulson@15656
  2394
  "lexn r n =
paulson@15656
  2395
    {(xs,ys). length xs = n \<and> length ys = n \<and>
paulson@15656
  2396
    (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
nipkow@18423
  2397
apply (induct n, simp)
nipkow@15302
  2398
apply (simp add: image_Collect lex_prod_def, safe, blast)
nipkow@15302
  2399
 apply (rule_tac x = "ab # xys" in exI, simp)
nipkow@15302
  2400
apply (case_tac xys, simp_all, blast)
nipkow@15302
  2401
done
nipkow@15302
  2402
nipkow@15302
  2403
lemma lex_conv:
paulson@15656
  2404
  "lex r =
paulson@15656
  2405
    {(xs,ys). length xs = length ys \<and>
paulson@15656
  2406
    (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@15302
  2407
by (force simp add: lex_def lexn_conv)
nipkow@15302
  2408
nipkow@15693
  2409
lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
nipkow@15693
  2410
by (unfold lenlex_def) blast
nipkow@15693
  2411
nipkow@15693
  2412
lemma lenlex_conv:
nipkow@15693
  2413
    "lenlex r = {(xs,ys). length xs < length ys |
paulson@15656
  2414
                 length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@19623
  2415
by (simp add: lenlex_def diag_def lex_prod_def inv_image_def)
nipkow@15302
  2416
nipkow@15302
  2417
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@15302
  2418
by (simp add: lex_conv)
nipkow@15302
  2419
nipkow@15302
  2420
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@15302
  2421
by (simp add:lex_conv)
nipkow@15302
  2422
paulson@18447
  2423
lemma Cons_in_lex [simp]:
paulson@15656
  2424
    "((x # xs, y # ys) : lex r) =
paulson@15656
  2425
      ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@15302
  2426
apply (simp add: lex_conv)
nipkow@15302
  2427
apply (rule iffI)
nipkow@15302
  2428
 prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
nipkow@15302
  2429
apply (case_tac xys, simp, simp)
nipkow@15302
  2430
apply blast
nipkow@15302
  2431
done
nipkow@15302
  2432
nipkow@15302
  2433
paulson@15656
  2434
subsubsection {* Lexicographic Ordering *}
paulson@15656
  2435
paulson@15656
  2436
text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
paulson@15656
  2437
    This ordering does \emph{not} preserve well-foundedness.
nipkow@17090
  2438
     Author: N. Voelker, March 2005. *} 
paulson@15656
  2439
paulson@15656
  2440
constdefs 
paulson@15656
  2441
  lexord :: "('a * 'a)set \<Rightarrow> ('a list * 'a list) set" 
paulson@15656
  2442
  "lexord  r == {(x,y). \<exists> a v. y = x @ a # v \<or> 
paulson@15656
  2443
            (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
paulson@15656
  2444
paulson@15656
  2445
lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
paulson@15656
  2446
  by (unfold lexord_def, induct_tac y, auto) 
paulson@15656
  2447
paulson@15656
  2448
lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
paulson@15656
  2449
  by (unfold lexord_def, induct_tac x, auto)
paulson@15656
  2450
paulson@15656
  2451
lemma lexord_cons_cons[simp]:
paulson@15656
  2452
     "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
paulson@15656
  2453
  apply (unfold lexord_def, safe, simp_all)
paulson@15656
  2454
  apply (case_tac u, simp, simp)
paulson@15656
  2455
  apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
paulson@15656
  2456
  apply (erule_tac x="b # u" in allE)
paulson@15656
  2457
  by force
paulson@15656
  2458
paulson@15656
  2459
lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
paulson@15656
  2460
paulson@15656
  2461
lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
paulson@15656
  2462
  by (induct_tac x, auto)  
paulson@15656
  2463
paulson@15656
  2464
lemma lexord_append_left_rightI:
paulson@15656
  2465
     "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
paulson@15656
  2466
  by (induct_tac u, auto)
paulson@15656
  2467
paulson@15656
  2468
lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
paulson@15656
  2469
  by (induct x, auto)
paulson@15656
  2470
paulson@15656
  2471
lemma lexord_append_leftD:
paulson@15656
  2472
     "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
paulson@15656
  2473
  by (erule rev_mp, induct_tac x, auto)
paulson@15656
  2474
paulson@15656
  2475
lemma lexord_take_index_conv: 
paulson@15656
  2476
   "((x,y) : lexord r) = 
paulson@15656
  2477
    ((length x < length y \<and> take (length x) y = x) \<or> 
paulson@15656
  2478
     (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
paulson@15656
  2479
  apply (unfold lexord_def Let_def, clarsimp) 
paulson@15656
  2480
  apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
paulson@15656
  2481
  apply auto 
paulson@15656
  2482
  apply (rule_tac x="hd (drop (length x) y)" in exI)
paulson@15656
  2483
  apply (rule_tac x="tl (drop (length x) y)" in exI)
paulson@15656
  2484
  apply (erule subst, simp add: min_def) 
paulson@15656
  2485
  apply (rule_tac x ="length u" in exI, simp) 
paulson@15656
  2486
  apply (rule_tac x ="take i x" in exI) 
paulson@15656
  2487
  apply (rule_tac x ="x ! i" in exI) 
paulson@15656
  2488
  apply (rule_tac x ="y ! i" in exI, safe) 
paulson@15656
  2489
  apply (rule_tac x="drop (Suc i) x" in exI)
paulson@15656
  2490
  apply (drule sym, simp add: drop_Suc_conv_tl) 
paulson@15656
  2491
  apply (rule_tac x="drop (Suc i) y" in exI)
paulson@15656
  2492
  by (simp add: drop_Suc_conv_tl) 
paulson@15656
  2493
paulson@15656
  2494
-- {* lexord is extension of partial ordering List.lex *} 
paulson@15656
  2495
lemma lexord_lex: " (x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
paulson@15656
  2496
  apply (rule_tac x = y in spec) 
paulson@15656
  2497
  apply (induct_tac x, clarsimp) 
paulson@15656
  2498
  by (clarify, case_tac x, simp, force)
paulson@15656
  2499
paulson@15656
  2500
lemma lexord_irreflexive: "(! x. (x,x) \<notin> r) \<Longrightarrow> (y,y) \<notin> lexord r"
paulson@15656
  2501
  by (induct y, auto)
paulson@15656
  2502
paulson@15656
  2503
lemma lexord_trans: 
paulson@15656
  2504
    "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
paulson@15656
  2505
   apply (erule rev_mp)+
paulson@15656
  2506
   apply (rule_tac x = x in spec) 
paulson@15656
  2507
  apply (rule_tac x = z in spec) 
paulson@15656
  2508
  apply ( induct_tac y, simp, clarify)
paulson@15656
  2509
  apply (case_tac xa, erule ssubst) 
paulson@15656
  2510
  apply (erule allE, erule allE) -- {* avoid simp recursion *} 
paulson@15656
  2511
  apply (case_tac x, simp, simp) 
paulson@15656
  2512
  apply (case_tac x, erule allE, erule allE, simp) 
paulson@15656
  2513
  apply (erule_tac x = listb in allE) 
paulson@15656
  2514
  apply (erule_tac x = lista in allE, simp)
paulson@15656
  2515
  apply (unfold trans_def)
paulson@15656
  2516
  by blast
paulson@15656
  2517
paulson@15656
  2518
lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
paulson@15656
  2519
  by (rule transI, drule lexord_trans, blast) 
paulson@15656
  2520
paulson@15656
  2521
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
  2522
  apply (rule_tac x = y in spec) 
paulson@15656
  2523
  apply (induct_tac x, rule allI) 
paulson@15656
  2524
  apply (case_tac x, simp, simp) 
paulson@15656
  2525
  apply (rule allI, case_tac x, simp, simp) 
paulson@15656
  2526
  by blast
paulson@15656
  2527
paulson@15656
  2528
krauss@21103
  2529
subsection {* Lexicographic combination of measure functions *}
krauss@21103
  2530
krauss@21103
  2531
text {* These are useful for termination proofs *}
krauss@21103
  2532
krauss@21103
  2533
definition
krauss@21103
  2534
  "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"
krauss@21103
  2535
krauss@21106
  2536
lemma wf_measures[recdef_wf, simp]: "wf (measures fs)"
krauss@21103
  2537
  unfolding measures_def
krauss@21103
  2538
  by blast
krauss@21103
  2539
krauss@21103
  2540
lemma in_measures[simp]: 
krauss@21103
  2541
  "(x, y) \<in> measures [] = False"
krauss@21103
  2542
  "(x, y) \<in> measures (f # fs)
krauss@21103
  2543
         = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"  
krauss@21103
  2544
  unfolding measures_def
krauss@21103
  2545
  by auto
krauss@21103
  2546
krauss@21103
  2547
lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
krauss@21103
  2548
  by simp
krauss@21103
  2549
krauss@21103
  2550
lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
krauss@21103
  2551
  by auto
krauss@21103
  2552
krauss@21211
  2553
(* install the lexicographic_order method and the "fun" command *)
bulwahn@21131
  2554
use "Tools/function_package/lexicographic_order.ML"
krauss@21211
  2555
use "Tools/function_package/fundef_datatype.ML"
krauss@21211
  2556
setup LexicographicOrder.setup
krauss@21211
  2557
setup FundefDatatype.setup
krauss@21211
  2558
krauss@21103
  2559
nipkow@15392
  2560
subsubsection{*Lifting a Relation on List Elements to the Lists*}
nipkow@15302
  2561
berghofe@22262
  2562
inductive2
berghofe@22262
  2563
  list_all2' :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool"
berghofe@22262
  2564
  for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
berghofe@22262
  2565
where
berghofe@22262
  2566
    Nil:  "list_all2' r [] []"
berghofe@22262
  2567
  | Cons: "[| r x y; list_all2' r xs ys |] ==> list_all2' r (x#xs) (y#ys)"
berghofe@22262
  2568
berghofe@22262
  2569
constdefs
berghofe@22262
  2570
  listrel :: "('a * 'b) set => ('a list * 'b list) set"
berghofe@22262
  2571
  "listrel r == Collect2 (list_all2' (member2 r))"
berghofe@22262
  2572
berghofe@22262
  2573
lemma list_all2_listrel_eq [pred_set_conv]:
berghofe@22262
  2574
  "list_all2' (member2 r) = member2 (listrel r)"
berghofe@22262
  2575
  by (simp add: listrel_def)
berghofe@22262
  2576
berghofe@22262
  2577
lemmas listrel_induct [consumes 1, case_names Nil Cons, induct set: listrel] =
berghofe@22262
  2578
  list_all2'.induct [to_set]
berghofe@22262
  2579
berghofe@22262
  2580
lemmas listrel_intros = list_all2'.intros [to_set]
berghofe@22262
  2581
berghofe@22262
  2582
inductive_cases2 listrel_Nil1 [to_set, elim!]: "list_all2' r [] xs"
berghofe@22262
  2583
inductive_cases2 listrel_Nil2 [to_set, elim!]: "list_all2' r xs []"
berghofe@22262
  2584
inductive_cases2 listrel_Cons1 [to_set, elim!]: "list_all2' r  (y#ys) xs"
berghofe@22262
  2585
inductive_cases2 listrel_Cons2 [to_set, elim!]: "list_all2' r xs (y#ys)"
nipkow@15302
  2586
nipkow@15302
  2587
nipkow@15302
  2588
lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
nipkow@15302
  2589
apply clarify  
berghofe@22262
  2590
apply (erule listrel_induct)
berghofe@22262
  2591
apply (blast intro: listrel_intros)+
nipkow@15302
  2592
done
nipkow@15302
  2593
nipkow@15302
  2594
lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
nipkow@15302
  2595
apply clarify 
berghofe@22262
  2596
apply (erule listrel_induct, auto) 
nipkow@15302
  2597
done
nipkow@15302
  2598
nipkow@15302
  2599
lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" 
nipkow@15302
  2600
apply (simp add: refl_def listrel_subset Ball_def)
nipkow@15302
  2601
apply (rule allI) 
nipkow@15302
  2602
apply (induct_tac x) 
berghofe@22262
  2603
apply (auto intro: listrel_intros)
nipkow@15302
  2604
done
nipkow@15302
  2605
nipkow@15302
  2606
lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
nipkow@15302
  2607
apply (auto simp add: sym_def)
berghofe@22262
  2608
apply (erule listrel_induct) 
berghofe@22262
  2609
apply (blast intro: listrel_intros)+
nipkow@15302
  2610
done
nipkow@15302
  2611
nipkow@15302
  2612
lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
nipkow@15302
  2613
apply (simp add: trans_def)
nipkow@15302
  2614
apply (intro allI) 
nipkow@15302
  2615
apply (rule impI) 
berghofe@22262
  2616
apply (erule listrel_induct) 
berghofe@22262
  2617
apply (blast intro: listrel_intros)+
nipkow@15302
  2618
done
nipkow@15302
  2619
nipkow@15302
  2620
theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
nipkow@15302
  2621
by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) 
nipkow@15302
  2622
nipkow@15302
  2623
lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
berghofe@22262
  2624
by (blast intro: listrel_intros)
nipkow@15302
  2625
nipkow@15302
  2626
lemma listrel_Cons:
nipkow@15302
  2627
     "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})";
berghofe@22262
  2628
by (auto simp add: set_Cons_def intro: listrel_intros) 
nipkow@15302
  2629
nipkow@15302
  2630
nipkow@15392
  2631
subsection{*Miscellany*}
nipkow@15392
  2632
nipkow@15392
  2633
subsubsection {* Characters and strings *}
wenzelm@13366
  2634
wenzelm@13366
  2635
datatype nibble =
wenzelm@13366
  2636
    Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
wenzelm@13366
  2637
  | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
wenzelm@13366
  2638
wenzelm@13366
  2639
datatype char = Char nibble nibble
wenzelm@13366
  2640
  -- "Note: canonical order of character encoding coincides with standard term ordering"
wenzelm@13366
  2641
wenzelm@13366
  2642
types string = "char list"
wenzelm@13366
  2643
wenzelm@13366
  2644
syntax
wenzelm@13366
  2645
  "_Char" :: "xstr => char"    ("CHR _")
wenzelm@13366
  2646
  "_String" :: "xstr => string"    ("_")
wenzelm@13366
  2647
wenzelm@21754
  2648
setup StringSyntax.setup
wenzelm@13366
  2649
haftmann@20453
  2650
haftmann@21061
  2651
subsection {* Code generator *}
haftmann@21061
  2652
haftmann@21061
  2653
subsubsection {* Setup *}
berghofe@15064
  2654
berghofe@16770
  2655
types_code
berghofe@16770
  2656
  "list" ("_ list")
berghofe@16770
  2657
attach (term_of) {*
wenzelm@21760
  2658
fun term_of_list f T = HOLogic.mk_list T o map f;
berghofe@16770
  2659
*}
berghofe@16770
  2660
attach (test) {*
berghofe@15064
  2661
fun gen_list' aG i j = frequency
berghofe@15064
  2662
  [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] ()
berghofe@15064
  2663
and gen_list aG i = gen_list' aG i i;
berghofe@16770
  2664
*}
berghofe@16770
  2665
  "char" ("string")
berghofe@16770
  2666
attach (term_of) {*
haftmann@21455
  2667
val term_of_char = HOLogic.mk_char;
berghofe@16770
  2668
*}
berghofe@16770
  2669
attach (test) {*
berghofe@15064
  2670
fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z")));
berghofe@15064
  2671
*}
berghofe@15064
  2672
berghofe@15064
  2673
consts_code "Cons" ("(_ ::/ _)")
berghofe@15064
  2674
haftmann@20453
  2675
code_type list
haftmann@20453
  2676
  (SML "_ list")
haftmann@21911
  2677
  (OCaml "_ list")
haftmann@21113
  2678
  (Haskell "![_]")
haftmann@20453
  2679
haftmann@22799
  2680
code_reserved SML
haftmann@22799
  2681
  list
haftmann@22799
  2682
haftmann@22799
  2683
code_reserved OCaml
haftmann@22799
  2684
  list
haftmann@22799
  2685
haftmann@20453
  2686
code_const Nil
haftmann@21113
  2687
  (SML "[]")
haftmann@21911
  2688
  (OCaml "[]")
haftmann@21113
  2689
  (Haskell "[]")
haftmann@20453
  2690
haftmann@21911
  2691
setup {*
haftmann@22799
  2692
  fold (fn target => CodegenSerializer.add_pretty_list target
haftmann@22799
  2693
    @{const_name Nil} @{const_name Cons}
haftmann@22799
  2694
  ) ["SML", "OCaml", "Haskell"]
haftmann@21911
  2695
*}
haftmann@21911
  2696
haftmann@22799
  2697
code_instance list :: eq
haftmann@22799
  2698
  (Haskell -)
haftmann@20588
  2699
haftmann@21455
  2700
code_const "op = \<Colon> 'a\<Colon>eq list \<Rightarrow> 'a list \<Rightarrow> bool"
haftmann@20588
  2701
  (Haskell infixl 4 "==")
haftmann@20588
  2702
haftmann@20453
  2703
setup {*
haftmann@20453
  2704
let
haftmann@20453
  2705
haftmann@20453
  2706
fun list_codegen thy defs gr dep thyname b t =
haftmann@20453
  2707
  let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy defs dep thyname false)
haftmann@20453
  2708
    (gr, HOLogic.dest_list t)
haftmann@20453
  2709
  in SOME (gr', Pretty.list "[" "]" ps) end handle TERM _ => NONE;
haftmann@20453
  2710
haftmann@20453
  2711
fun char_codegen thy defs gr dep thyname b t =
berghofe@22539
  2712
  case Option.map chr (try HOLogic.dest_char t) of
berghofe@22539
  2713
      SOME c => SOME (gr, Pretty.quote (Pretty.str (ML_Syntax.print_char c)))
haftmann@20453
  2714
    | NONE => NONE;
haftmann@20453
  2715
haftmann@20453
  2716
in
haftmann@20453
  2717
  Codegen.add_codegen "list_codegen" list_codegen
haftmann@20453
  2718
  #> Codegen.add_codegen "char_codegen" char_codegen
haftmann@20453
  2719
end;
haftmann@20453
  2720
*}
berghofe@15064
  2721
haftmann@21061
  2722
haftmann@21061
  2723
subsubsection {* Generation of efficient code *}
haftmann@21061
  2724
haftmann@21061
  2725
consts
haftmann@21079
  2726
  memberl :: "'a \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "mem" 55)
haftmann@21061
  2727
  null:: "'a list \<Rightarrow> bool"
haftmann@21061
  2728
  list_inter :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
haftmann@21061
  2729
  list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
haftmann@21061
  2730
  list_all :: "('a \<Rightarrow> bool) \<Rightarrow> ('a list \<Rightarrow> bool)"
haftmann@21061
  2731
  itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
haftmann@21061
  2732
  filtermap :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list"
haftmann@21061
  2733
  map_filter :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list"
haftmann@21061
  2734
haftmann@21061
  2735
primrec
haftmann@21061
  2736
  "x mem [] = False"
haftmann@21079
  2737
  "x mem (y#ys) = (x = y \<or> x mem ys)"
haftmann@21061
  2738
haftmann@21061
  2739
primrec
haftmann@21061
  2740
  "null [] = True"
haftmann@21061
  2741
  "null (x#xs) = False"
haftmann@21061
  2742
haftmann@21061
  2743
primrec
haftmann@21061
  2744
  "list_inter [] bs = []"
haftmann@21061
  2745
  "list_inter (a#as) bs =
haftmann@21061
  2746
     (if a \<in> set bs then a # list_inter as bs else list_inter as bs)"
haftmann@21061
  2747
haftmann@21061
  2748
primrec
haftmann@21061
  2749
  "list_all P [] = True"
haftmann@21061
  2750
  "list_all P (x#xs) = (P x \<and> list_all P xs)"
haftmann@21061
  2751</