src/HOL/List.thy
author nipkow
Mon May 13 15:45:21 2002 +0200 (2002-05-13)
changeset 13147 491a48cf6023
parent 13146 f43153b63361
child 13187 e5434b822a96
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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   Copyright 1994 TU Muenchen
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*)
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header {* The datatype of finite lists *}
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theory List = PreList:
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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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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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list_all:: "('a => bool) => ('a list => bool)"
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list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool"
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map :: "('a=>'b) => ('a list => 'b list)"
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mem :: "'a => 'a list => bool"(infixl 55)
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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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null:: "'a list => bool"
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"distinct":: "'a list => bool"
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replicate :: "nat => 'a => 'a list"
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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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upto:: "nat => nat => nat list" ("(1[_../_])")
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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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"[i..j]" == "[i..(Suc j)(]"
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syntax (xsymbols)
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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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syntax length :: "'a list => nat"
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translations "length" => "size :: _ list => nat"
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typed_print_translation {*
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let
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fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] =
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Syntax.const "length" $ t
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| size_tr' _ _ _ = raise Match;
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in [("size", size_tr')] end
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*}
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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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"null([]) = True"
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"null(x#xs) = False"
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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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"x mem [] = False"
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"x mem (y#ys) = (if y=x then True else x mem ys)"
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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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list_all_Nil:"list_all P [] = True"
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list_all_Cons: "list_all P (x#xs) = (P(x) \<and> list_all P 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 *}
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-- {* but separate 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 *}
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-- {* but separate 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 *}
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-- {* but separate 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] =
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(case i of 0 => v # xs
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| 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 *}
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-- {* but separate 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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replicate_0: "replicate 0 x = []"
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replicate_Suc: "replicate (Suc n) x = x # replicate n x"
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defs
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 list_all2_def:
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 "list_all2 P xs ys == length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)"
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subsection {* Lexicographic orderings on lists *}
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consts
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lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
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primrec
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"lexn r 0 = {}"
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"lexn r (Suc n) =
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(prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
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{(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
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constdefs
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lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
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"lex r == \<Union>n. lexn r n"
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lexico :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
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"lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
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sublist :: "'a list => nat set => 'a list"
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"sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..size xs(]))"
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lemma not_Cons_self [simp]: "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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"(!!xs. \<forall>ys. length ys < length xs --> P ys ==> P xs) ==> P xs"
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by (rule measure_induct [of length]) rules
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subsection {* @{text lists}: the list-forming operator over sets *}
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consts lists :: "'a set => 'a list set"
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inductive "lists A"
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intros
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Nil [intro!]: "[]: lists A"
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Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A"
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inductive_cases listsE [elim!]: "x#l : lists A"
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lemma lists_mono: "A \<subseteq> B ==> lists A \<subseteq> lists B"
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by (unfold lists.defs) (blast intro!: lfp_mono)
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lemma lists_IntI [rule_format]:
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"l: lists A ==> l: lists B --> l: lists (A Int B)"
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apply (erule lists.induct)
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apply blast+
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done
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lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B"
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apply (rule mono_Int [THEN equalityI])
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apply (simp add: mono_def lists_mono)
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apply (blast intro!: lists_IntI)
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done
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lemma append_in_lists_conv [iff]:
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"(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"
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by (induct xs) auto
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subsection {* @{text 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_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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subsection {* @{text "@"} -- append *}
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lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
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by (induct xs) auto
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lemma append_Nil2 [simp]: "xs @ [] = xs"
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by (induct xs) auto
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lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
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by (induct xs) auto
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lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
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by (induct xs) auto
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lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
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by (induct xs) auto
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lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
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by (induct xs) auto
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lemma append_eq_append_conv [rule_format, simp]:
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 "\<forall>ys. length xs = length ys \<or> length us = length vs
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 --> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
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apply (induct_tac xs)
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 apply(rule allI)
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 apply (case_tac ys)
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apply simp
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 apply force
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apply (rule allI)
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apply (case_tac ys)
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 apply force
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apply simp
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done
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lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
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by simp
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lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
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by simp
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lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
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by simp
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lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
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using append_same_eq [of _ _ "[]"] by auto
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lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
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using append_same_eq [of "[]"] by auto
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lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
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by (induct xs) auto
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lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
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by (induct xs) auto
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lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
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by (simp add: hd_append split: list.split)
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lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
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by (simp split: list.split)
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lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
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by (simp add: tl_append split: list.split)
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   339
wenzelm@13142
   340
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   341
wenzelm@13114
   342
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
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   343
by simp
wenzelm@13114
   344
wenzelm@13142
   345
lemma Cons_eq_appendI:
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   346
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   347
by (drule sym) simp
wenzelm@13114
   348
wenzelm@13142
   349
lemma append_eq_appendI:
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   350
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   351
by (drule sym) simp
wenzelm@13114
   352
wenzelm@13114
   353
wenzelm@13142
   354
text {*
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   355
Simplification procedure for all list equalities.
nipkow@13145
   356
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   357
- both lists end in a singleton list,
nipkow@13145
   358
- or both lists end in the same list.
wenzelm@13142
   359
*}
wenzelm@13142
   360
wenzelm@13142
   361
ML_setup {*
nipkow@3507
   362
local
nipkow@3507
   363
wenzelm@13122
   364
val append_assoc = thm "append_assoc";
wenzelm@13122
   365
val append_Nil = thm "append_Nil";
wenzelm@13122
   366
val append_Cons = thm "append_Cons";
wenzelm@13122
   367
val append1_eq_conv = thm "append1_eq_conv";
wenzelm@13122
   368
val append_same_eq = thm "append_same_eq";
wenzelm@13122
   369
wenzelm@13114
   370
val list_eq_pattern =
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   371
Thm.read_cterm (Theory.sign_of (the_context ())) ("(xs::'a list) = ys",HOLogic.boolT)
wenzelm@13114
   372
wenzelm@13114
   373
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
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   374
(case xs of Const("List.list.Nil",_) => cons | _ => last xs)
nipkow@13145
   375
| last (Const("List.op @",_) $ _ $ ys) = last ys
nipkow@13145
   376
| last t = t
wenzelm@13114
   377
wenzelm@13114
   378
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
nipkow@13145
   379
| list1 _ = false
wenzelm@13114
   380
wenzelm@13114
   381
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
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   382
(case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
nipkow@13145
   383
| butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
nipkow@13145
   384
| butlast xs = Const("List.list.Nil",fastype_of xs)
wenzelm@13114
   385
wenzelm@13114
   386
val rearr_tac =
nipkow@13145
   387
simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons])
wenzelm@13114
   388
wenzelm@13114
   389
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
nipkow@13145
   390
let
nipkow@13145
   391
val lastl = last lhs and lastr = last rhs
nipkow@13145
   392
fun rearr conv =
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   393
let val lhs1 = butlast lhs and rhs1 = butlast rhs
nipkow@13145
   394
val Type(_,listT::_) = eqT
nipkow@13145
   395
val appT = [listT,listT] ---> listT
nipkow@13145
   396
val app = Const("List.op @",appT)
nipkow@13145
   397
val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
nipkow@13145
   398
val ct = cterm_of sg (HOLogic.mk_Trueprop(HOLogic.mk_eq(F,F2)))
nipkow@13145
   399
val thm = prove_goalw_cterm [] ct (K [rearr_tac 1])
nipkow@13145
   400
handle ERROR =>
nipkow@13145
   401
error("The error(s) above occurred while trying to prove " ^
nipkow@13145
   402
string_of_cterm ct)
nipkow@13145
   403
in Some((conv RS (thm RS trans)) RS eq_reflection) end
wenzelm@13114
   404
nipkow@13145
   405
in if list1 lastl andalso list1 lastr
nipkow@13145
   406
 then rearr append1_eq_conv
nipkow@13145
   407
 else
nipkow@13145
   408
 if lastl aconv lastr
nipkow@13145
   409
 then rearr append_same_eq
nipkow@13145
   410
 else None
nipkow@13145
   411
end
wenzelm@13114
   412
in
wenzelm@13114
   413
val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq
wenzelm@13114
   414
end;
wenzelm@13114
   415
wenzelm@13114
   416
Addsimprocs [list_eq_simproc];
wenzelm@13114
   417
*}
wenzelm@13114
   418
wenzelm@13114
   419
wenzelm@13142
   420
subsection {* @{text map} *}
wenzelm@13114
   421
wenzelm@13142
   422
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   423
by (induct xs) simp_all
wenzelm@13114
   424
wenzelm@13142
   425
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   426
by (rule ext, induct_tac xs) auto
wenzelm@13114
   427
wenzelm@13142
   428
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   429
by (induct xs) auto
wenzelm@13114
   430
wenzelm@13142
   431
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   432
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   433
wenzelm@13142
   434
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   435
by (induct xs) auto
wenzelm@13114
   436
wenzelm@13114
   437
lemma map_cong:
nipkow@13145
   438
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   439
-- {* a congruence rule for @{text map} *}
nipkow@13145
   440
by (clarify, induct ys) auto
wenzelm@13114
   441
wenzelm@13142
   442
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   443
by (cases xs) auto
wenzelm@13114
   444
wenzelm@13142
   445
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   446
by (cases xs) auto
wenzelm@13114
   447
wenzelm@13114
   448
lemma map_eq_Cons:
nipkow@13145
   449
"(map f xs = y # ys) = (\<exists>x xs'. xs = x # xs' \<and> f x = y \<and> map f xs' = ys)"
nipkow@13145
   450
by (cases xs) auto
wenzelm@13114
   451
wenzelm@13114
   452
lemma map_injective:
nipkow@13145
   453
"!!xs. map f xs = map f ys ==> (\<forall>x y. f x = f y --> x = y) ==> xs = ys"
nipkow@13145
   454
by (induct ys) (auto simp add: map_eq_Cons)
wenzelm@13114
   455
wenzelm@13114
   456
lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@13145
   457
by (rules dest: map_injective injD intro: injI)
wenzelm@13114
   458
wenzelm@13114
   459
lemma inj_mapD: "inj (map f) ==> inj f"
nipkow@13145
   460
apply (unfold inj_on_def)
nipkow@13145
   461
apply clarify
nipkow@13145
   462
apply (erule_tac x = "[x]" in ballE)
nipkow@13145
   463
 apply (erule_tac x = "[y]" in ballE)
nipkow@13145
   464
apply simp
nipkow@13145
   465
 apply blast
nipkow@13145
   466
apply blast
nipkow@13145
   467
done
wenzelm@13114
   468
wenzelm@13114
   469
lemma inj_map: "inj (map f) = inj f"
nipkow@13145
   470
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   471
wenzelm@13114
   472
wenzelm@13142
   473
subsection {* @{text rev} *}
wenzelm@13114
   474
wenzelm@13142
   475
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   476
by (induct xs) auto
wenzelm@13114
   477
wenzelm@13142
   478
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   479
by (induct xs) auto
wenzelm@13114
   480
wenzelm@13142
   481
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   482
by (induct xs) auto
wenzelm@13114
   483
wenzelm@13142
   484
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   485
by (induct xs) auto
wenzelm@13114
   486
wenzelm@13142
   487
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"
nipkow@13145
   488
apply (induct xs)
nipkow@13145
   489
 apply force
nipkow@13145
   490
apply (case_tac ys)
nipkow@13145
   491
 apply simp
nipkow@13145
   492
apply force
nipkow@13145
   493
done
wenzelm@13114
   494
wenzelm@13142
   495
lemma rev_induct: "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
nipkow@13145
   496
apply(subst rev_rev_ident[symmetric])
nipkow@13145
   497
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   498
done
wenzelm@13114
   499
nipkow@13145
   500
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114
   501
nipkow@13145
   502
lemma rev_exhaust: "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   503
by (induct xs rule: rev_induct) auto
wenzelm@13114
   504
wenzelm@13114
   505
wenzelm@13142
   506
subsection {* @{text set} *}
wenzelm@13114
   507
wenzelm@13142
   508
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   509
by (induct xs) auto
wenzelm@13114
   510
wenzelm@13142
   511
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   512
by (induct xs) auto
wenzelm@13114
   513
wenzelm@13142
   514
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   515
by auto
wenzelm@13114
   516
wenzelm@13142
   517
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   518
by (induct xs) auto
wenzelm@13114
   519
wenzelm@13142
   520
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   521
by (induct xs) auto
wenzelm@13114
   522
wenzelm@13142
   523
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   524
by (induct xs) auto
wenzelm@13114
   525
wenzelm@13142
   526
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   527
by (induct xs) auto
wenzelm@13114
   528
wenzelm@13142
   529
lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> k < j}"
nipkow@13145
   530
apply (induct j)
nipkow@13145
   531
 apply simp_all
nipkow@13145
   532
apply(erule ssubst)
nipkow@13145
   533
apply auto
nipkow@13145
   534
apply arith
nipkow@13145
   535
done
wenzelm@13114
   536
wenzelm@13142
   537
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
nipkow@13145
   538
apply (induct xs)
nipkow@13145
   539
 apply simp
nipkow@13145
   540
apply simp
nipkow@13145
   541
apply (rule iffI)
nipkow@13145
   542
 apply (blast intro: eq_Nil_appendI Cons_eq_appendI)
nipkow@13145
   543
apply (erule exE)+
nipkow@13145
   544
apply (case_tac ys)
nipkow@13145
   545
apply auto
nipkow@13145
   546
done
wenzelm@13142
   547
wenzelm@13142
   548
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"
nipkow@13145
   549
-- {* eliminate @{text lists} in favour of @{text set} *}
nipkow@13145
   550
by (induct xs) auto
wenzelm@13142
   551
wenzelm@13142
   552
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"
nipkow@13145
   553
by (rule in_lists_conv_set [THEN iffD1])
wenzelm@13142
   554
wenzelm@13142
   555
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"
nipkow@13145
   556
by (rule in_lists_conv_set [THEN iffD2])
wenzelm@13114
   557
wenzelm@13114
   558
wenzelm@13142
   559
subsection {* @{text mem} *}
wenzelm@13114
   560
wenzelm@13114
   561
lemma set_mem_eq: "(x mem xs) = (x : set xs)"
nipkow@13145
   562
by (induct xs) auto
wenzelm@13114
   563
wenzelm@13114
   564
wenzelm@13142
   565
subsection {* @{text list_all} *}
wenzelm@13114
   566
wenzelm@13142
   567
lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)"
nipkow@13145
   568
by (induct xs) auto
wenzelm@13114
   569
wenzelm@13142
   570
lemma list_all_append [simp]:
nipkow@13145
   571
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
nipkow@13145
   572
by (induct xs) auto
wenzelm@13114
   573
wenzelm@13114
   574
wenzelm@13142
   575
subsection {* @{text filter} *}
wenzelm@13114
   576
wenzelm@13142
   577
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   578
by (induct xs) auto
wenzelm@13114
   579
wenzelm@13142
   580
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   581
by (induct xs) auto
wenzelm@13114
   582
wenzelm@13142
   583
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   584
by (induct xs) auto
wenzelm@13114
   585
wenzelm@13142
   586
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   587
by (induct xs) auto
wenzelm@13114
   588
wenzelm@13142
   589
lemma length_filter [simp]: "length (filter P xs) \<le> length xs"
nipkow@13145
   590
by (induct xs) (auto simp add: le_SucI)
wenzelm@13114
   591
wenzelm@13142
   592
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   593
by auto
wenzelm@13114
   594
wenzelm@13114
   595
wenzelm@13142
   596
subsection {* @{text concat} *}
wenzelm@13114
   597
wenzelm@13142
   598
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
   599
by (induct xs) auto
wenzelm@13114
   600
wenzelm@13142
   601
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   602
by (induct xss) auto
wenzelm@13114
   603
wenzelm@13142
   604
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   605
by (induct xss) auto
wenzelm@13114
   606
wenzelm@13142
   607
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
nipkow@13145
   608
by (induct xs) auto
wenzelm@13114
   609
wenzelm@13142
   610
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
   611
by (induct xs) auto
wenzelm@13114
   612
wenzelm@13142
   613
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
   614
by (induct xs) auto
wenzelm@13114
   615
wenzelm@13142
   616
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
   617
by (induct xs) auto
wenzelm@13114
   618
wenzelm@13114
   619
wenzelm@13142
   620
subsection {* @{text nth} *}
wenzelm@13114
   621
wenzelm@13142
   622
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
   623
by auto
wenzelm@13114
   624
wenzelm@13142
   625
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
   626
by auto
wenzelm@13114
   627
wenzelm@13142
   628
declare nth.simps [simp del]
wenzelm@13114
   629
wenzelm@13114
   630
lemma nth_append:
nipkow@13145
   631
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
nipkow@13145
   632
apply(induct "xs")
nipkow@13145
   633
 apply simp
nipkow@13145
   634
apply (case_tac n)
nipkow@13145
   635
 apply auto
nipkow@13145
   636
done
wenzelm@13114
   637
wenzelm@13142
   638
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
nipkow@13145
   639
apply(induct xs)
nipkow@13145
   640
 apply simp
nipkow@13145
   641
apply (case_tac n)
nipkow@13145
   642
 apply auto
nipkow@13145
   643
done
wenzelm@13114
   644
wenzelm@13142
   645
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
nipkow@13145
   646
apply (induct_tac xs)
nipkow@13145
   647
 apply simp
nipkow@13145
   648
apply simp
nipkow@13145
   649
apply safe
nipkow@13145
   650
apply (rule_tac x = 0 in exI)
nipkow@13145
   651
apply simp
nipkow@13145
   652
 apply (rule_tac x = "Suc i" in exI)
nipkow@13145
   653
 apply simp
nipkow@13145
   654
apply (case_tac i)
nipkow@13145
   655
 apply simp
nipkow@13145
   656
apply (rename_tac j)
nipkow@13145
   657
apply (rule_tac x = j in exI)
nipkow@13145
   658
apply simp
nipkow@13145
   659
done
wenzelm@13114
   660
nipkow@13145
   661
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
   662
by (auto simp add: set_conv_nth)
wenzelm@13114
   663
wenzelm@13142
   664
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
   665
by (auto simp add: set_conv_nth)
wenzelm@13114
   666
wenzelm@13114
   667
lemma all_nth_imp_all_set:
nipkow@13145
   668
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
   669
by (auto simp add: set_conv_nth)
wenzelm@13114
   670
wenzelm@13114
   671
lemma all_set_conv_all_nth:
nipkow@13145
   672
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
   673
by (auto simp add: set_conv_nth)
wenzelm@13114
   674
wenzelm@13114
   675
wenzelm@13142
   676
subsection {* @{text list_update} *}
wenzelm@13114
   677
wenzelm@13142
   678
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
nipkow@13145
   679
by (induct xs) (auto split: nat.split)
wenzelm@13114
   680
wenzelm@13114
   681
lemma nth_list_update:
nipkow@13145
   682
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@13145
   683
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   684
wenzelm@13142
   685
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
   686
by (simp add: nth_list_update)
wenzelm@13114
   687
wenzelm@13142
   688
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145
   689
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   690
wenzelm@13142
   691
lemma list_update_overwrite [simp]:
nipkow@13145
   692
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145
   693
by (induct xs) (auto split: nat.split)
wenzelm@13114
   694
wenzelm@13114
   695
lemma list_update_same_conv:
nipkow@13145
   696
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145
   697
by (induct xs) (auto split: nat.split)
wenzelm@13114
   698
wenzelm@13114
   699
lemma update_zip:
nipkow@13145
   700
"!!i xy xs. length xs = length ys ==>
nipkow@13145
   701
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145
   702
by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114
   703
wenzelm@13114
   704
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145
   705
by (induct xs) (auto split: nat.split)
wenzelm@13114
   706
wenzelm@13114
   707
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
   708
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
   709
wenzelm@13114
   710
wenzelm@13142
   711
subsection {* @{text last} and @{text butlast} *}
wenzelm@13114
   712
wenzelm@13142
   713
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
   714
by (induct xs) auto
wenzelm@13114
   715
wenzelm@13142
   716
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
   717
by (induct xs) auto
wenzelm@13114
   718
wenzelm@13142
   719
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
   720
by (induct xs rule: rev_induct) auto
wenzelm@13114
   721
wenzelm@13114
   722
lemma butlast_append:
nipkow@13145
   723
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145
   724
by (induct xs) auto
wenzelm@13114
   725
wenzelm@13142
   726
lemma append_butlast_last_id [simp]:
nipkow@13145
   727
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
   728
by (induct xs) auto
wenzelm@13114
   729
wenzelm@13142
   730
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
   731
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
   732
wenzelm@13114
   733
lemma in_set_butlast_appendI:
nipkow@13145
   734
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
   735
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
   736
wenzelm@13142
   737
wenzelm@13142
   738
subsection {* @{text take} and @{text drop} *}
wenzelm@13114
   739
wenzelm@13142
   740
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
   741
by (induct xs) auto
wenzelm@13114
   742
wenzelm@13142
   743
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
   744
by (induct xs) auto
wenzelm@13114
   745
wenzelm@13142
   746
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
   747
by simp
wenzelm@13114
   748
wenzelm@13142
   749
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
   750
by simp
wenzelm@13114
   751
wenzelm@13142
   752
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
   753
wenzelm@13142
   754
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
nipkow@13145
   755
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   756
wenzelm@13142
   757
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
nipkow@13145
   758
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   759
wenzelm@13142
   760
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
nipkow@13145
   761
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   762
wenzelm@13142
   763
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
nipkow@13145
   764
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   765
wenzelm@13142
   766
lemma take_append [simp]:
nipkow@13145
   767
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@13145
   768
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   769
wenzelm@13142
   770
lemma drop_append [simp]:
nipkow@13145
   771
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@13145
   772
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   773
wenzelm@13142
   774
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
nipkow@13145
   775
apply (induct m)
nipkow@13145
   776
 apply auto
nipkow@13145
   777
apply (case_tac xs)
nipkow@13145
   778
 apply auto
nipkow@13145
   779
apply (case_tac na)
nipkow@13145
   780
 apply auto
nipkow@13145
   781
done
wenzelm@13114
   782
wenzelm@13142
   783
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
nipkow@13145
   784
apply (induct m)
nipkow@13145
   785
 apply auto
nipkow@13145
   786
apply (case_tac xs)
nipkow@13145
   787
 apply auto
nipkow@13145
   788
done
wenzelm@13114
   789
wenzelm@13114
   790
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
nipkow@13145
   791
apply (induct m)
nipkow@13145
   792
 apply auto
nipkow@13145
   793
apply (case_tac xs)
nipkow@13145
   794
 apply auto
nipkow@13145
   795
done
wenzelm@13114
   796
wenzelm@13142
   797
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
nipkow@13145
   798
apply (induct n)
nipkow@13145
   799
 apply auto
nipkow@13145
   800
apply (case_tac xs)
nipkow@13145
   801
 apply auto
nipkow@13145
   802
done
wenzelm@13114
   803
wenzelm@13114
   804
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
nipkow@13145
   805
apply (induct n)
nipkow@13145
   806
 apply auto
nipkow@13145
   807
apply (case_tac xs)
nipkow@13145
   808
 apply auto
nipkow@13145
   809
done
wenzelm@13114
   810
wenzelm@13142
   811
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
nipkow@13145
   812
apply (induct n)
nipkow@13145
   813
 apply auto
nipkow@13145
   814
apply (case_tac xs)
nipkow@13145
   815
 apply auto
nipkow@13145
   816
done
wenzelm@13114
   817
wenzelm@13114
   818
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
nipkow@13145
   819
apply (induct xs)
nipkow@13145
   820
 apply auto
nipkow@13145
   821
apply (case_tac i)
nipkow@13145
   822
 apply auto
nipkow@13145
   823
done
wenzelm@13114
   824
wenzelm@13114
   825
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
nipkow@13145
   826
apply (induct xs)
nipkow@13145
   827
 apply auto
nipkow@13145
   828
apply (case_tac i)
nipkow@13145
   829
 apply auto
nipkow@13145
   830
done
wenzelm@13114
   831
wenzelm@13142
   832
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
nipkow@13145
   833
apply (induct xs)
nipkow@13145
   834
 apply auto
nipkow@13145
   835
apply (case_tac n)
nipkow@13145
   836
 apply(blast )
nipkow@13145
   837
apply (case_tac i)
nipkow@13145
   838
 apply auto
nipkow@13145
   839
done
wenzelm@13114
   840
wenzelm@13142
   841
lemma nth_drop [simp]:
nipkow@13145
   842
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
nipkow@13145
   843
apply (induct n)
nipkow@13145
   844
 apply auto
nipkow@13145
   845
apply (case_tac xs)
nipkow@13145
   846
 apply auto
nipkow@13145
   847
done
nipkow@3507
   848
wenzelm@13114
   849
lemma append_eq_conv_conj:
nipkow@13145
   850
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
nipkow@13145
   851
apply(induct xs)
nipkow@13145
   852
 apply simp
nipkow@13145
   853
apply clarsimp
nipkow@13145
   854
apply (case_tac zs)
nipkow@13145
   855
apply auto
nipkow@13145
   856
done
wenzelm@13142
   857
wenzelm@13114
   858
wenzelm@13142
   859
subsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
   860
wenzelm@13142
   861
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
   862
by (induct xs) auto
wenzelm@13114
   863
wenzelm@13142
   864
lemma takeWhile_append1 [simp]:
nipkow@13145
   865
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
   866
by (induct xs) auto
wenzelm@13114
   867
wenzelm@13142
   868
lemma takeWhile_append2 [simp]:
nipkow@13145
   869
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
   870
by (induct xs) auto
wenzelm@13114
   871
wenzelm@13142
   872
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
   873
by (induct xs) auto
wenzelm@13114
   874
wenzelm@13142
   875
lemma dropWhile_append1 [simp]:
nipkow@13145
   876
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
   877
by (induct xs) auto
wenzelm@13114
   878
wenzelm@13142
   879
lemma dropWhile_append2 [simp]:
nipkow@13145
   880
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
   881
by (induct xs) auto
wenzelm@13114
   882
wenzelm@13142
   883
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
   884
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
   885
wenzelm@13114
   886
wenzelm@13142
   887
subsection {* @{text zip} *}
wenzelm@13114
   888
wenzelm@13142
   889
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
   890
by (induct ys) auto
wenzelm@13114
   891
wenzelm@13142
   892
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
   893
by simp
wenzelm@13114
   894
wenzelm@13142
   895
declare zip_Cons [simp del]
wenzelm@13114
   896
wenzelm@13142
   897
lemma length_zip [simp]:
nipkow@13145
   898
"!!xs. length (zip xs ys) = min (length xs) (length ys)"
nipkow@13145
   899
apply(induct ys)
nipkow@13145
   900
 apply simp
nipkow@13145
   901
apply (case_tac xs)
nipkow@13145
   902
 apply auto
nipkow@13145
   903
done
wenzelm@13114
   904
wenzelm@13114
   905
lemma zip_append1:
nipkow@13145
   906
"!!xs. zip (xs @ ys) zs =
nipkow@13145
   907
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
nipkow@13145
   908
apply (induct zs)
nipkow@13145
   909
 apply simp
nipkow@13145
   910
apply (case_tac xs)
nipkow@13145
   911
 apply simp_all
nipkow@13145
   912
done
wenzelm@13114
   913
wenzelm@13114
   914
lemma zip_append2:
nipkow@13145
   915
"!!ys. zip xs (ys @ zs) =
nipkow@13145
   916
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
nipkow@13145
   917
apply (induct xs)
nipkow@13145
   918
 apply simp
nipkow@13145
   919
apply (case_tac ys)
nipkow@13145
   920
 apply simp_all
nipkow@13145
   921
done
wenzelm@13114
   922
wenzelm@13142
   923
lemma zip_append [simp]:
wenzelm@13142
   924
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
   925
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
   926
by (simp add: zip_append1)
wenzelm@13114
   927
wenzelm@13114
   928
lemma zip_rev:
nipkow@13145
   929
"!!xs. length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@13145
   930
apply(induct ys)
nipkow@13145
   931
 apply simp
nipkow@13145
   932
apply (case_tac xs)
nipkow@13145
   933
 apply simp_all
nipkow@13145
   934
done
wenzelm@13114
   935
wenzelm@13142
   936
lemma nth_zip [simp]:
nipkow@13145
   937
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
nipkow@13145
   938
apply (induct ys)
nipkow@13145
   939
 apply simp
nipkow@13145
   940
apply (case_tac xs)
nipkow@13145
   941
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
   942
done
wenzelm@13114
   943
wenzelm@13114
   944
lemma set_zip:
nipkow@13145
   945
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
   946
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
   947
wenzelm@13114
   948
lemma zip_update:
nipkow@13145
   949
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
   950
by (rule sym, simp add: update_zip)
wenzelm@13114
   951
wenzelm@13142
   952
lemma zip_replicate [simp]:
nipkow@13145
   953
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
nipkow@13145
   954
apply (induct i)
nipkow@13145
   955
 apply auto
nipkow@13145
   956
apply (case_tac j)
nipkow@13145
   957
 apply auto
nipkow@13145
   958
done
wenzelm@13114
   959
wenzelm@13142
   960
wenzelm@13142
   961
subsection {* @{text list_all2} *}
wenzelm@13114
   962
wenzelm@13114
   963
lemma list_all2_lengthD: "list_all2 P xs ys ==> length xs = length ys"
nipkow@13145
   964
by (simp add: list_all2_def)
wenzelm@13114
   965
wenzelm@13142
   966
lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])"
nipkow@13145
   967
by (simp add: list_all2_def)
wenzelm@13114
   968
wenzelm@13142
   969
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"
nipkow@13145
   970
by (simp add: list_all2_def)
wenzelm@13114
   971
wenzelm@13142
   972
lemma list_all2_Cons [iff]:
nipkow@13145
   973
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@13145
   974
by (auto simp add: list_all2_def)
wenzelm@13114
   975
wenzelm@13114
   976
lemma list_all2_Cons1:
nipkow@13145
   977
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
   978
by (cases ys) auto
wenzelm@13114
   979
wenzelm@13114
   980
lemma list_all2_Cons2:
nipkow@13145
   981
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
   982
by (cases xs) auto
wenzelm@13114
   983
wenzelm@13142
   984
lemma list_all2_rev [iff]:
nipkow@13145
   985
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
   986
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
   987
wenzelm@13114
   988
lemma list_all2_append1:
nipkow@13145
   989
"list_all2 P (xs @ ys) zs =
nipkow@13145
   990
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
   991
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
   992
apply (simp add: list_all2_def zip_append1)
nipkow@13145
   993
apply (rule iffI)
nipkow@13145
   994
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
   995
 apply (rule_tac x = "drop (length xs) zs" in exI)
nipkow@13145
   996
 apply (force split: nat_diff_split simp add: min_def)
nipkow@13145
   997
apply clarify
nipkow@13145
   998
apply (simp add: ball_Un)
nipkow@13145
   999
done
wenzelm@13114
  1000
wenzelm@13114
  1001
lemma list_all2_append2:
nipkow@13145
  1002
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1003
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1004
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1005
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1006
apply (rule iffI)
nipkow@13145
  1007
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1008
 apply (rule_tac x = "drop (length ys) xs" in exI)
nipkow@13145
  1009
 apply (force split: nat_diff_split simp add: min_def)
nipkow@13145
  1010
apply clarify
nipkow@13145
  1011
apply (simp add: ball_Un)
nipkow@13145
  1012
done
wenzelm@13114
  1013
wenzelm@13114
  1014
lemma list_all2_conv_all_nth:
nipkow@13145
  1015
"list_all2 P xs ys =
nipkow@13145
  1016
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1017
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1018
wenzelm@13114
  1019
lemma list_all2_trans[rule_format]:
nipkow@13145
  1020
"\<forall>a b c. P1 a b --> P2 b c --> P3 a c ==>
nipkow@13145
  1021
\<forall>bs cs. list_all2 P1 as bs --> list_all2 P2 bs cs --> list_all2 P3 as cs"
nipkow@13145
  1022
apply(induct_tac as)
nipkow@13145
  1023
 apply simp
nipkow@13145
  1024
apply(rule allI)
nipkow@13145
  1025
apply(induct_tac bs)
nipkow@13145
  1026
 apply simp
nipkow@13145
  1027
apply(rule allI)
nipkow@13145
  1028
apply(induct_tac cs)
nipkow@13145
  1029
 apply auto
nipkow@13145
  1030
done
wenzelm@13142
  1031
wenzelm@13142
  1032
wenzelm@13142
  1033
subsection {* @{text foldl} *}
wenzelm@13142
  1034
wenzelm@13142
  1035
lemma foldl_append [simp]:
nipkow@13145
  1036
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145
  1037
by (induct xs) auto
wenzelm@13142
  1038
wenzelm@13142
  1039
text {*
nipkow@13145
  1040
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1041
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1042
*}
wenzelm@13142
  1043
wenzelm@13142
  1044
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
nipkow@13145
  1045
by (induct ns) auto
wenzelm@13142
  1046
wenzelm@13142
  1047
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1048
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1049
wenzelm@13142
  1050
lemma sum_eq_0_conv [iff]:
nipkow@13145
  1051
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145
  1052
by (induct ns) auto
wenzelm@13114
  1053
wenzelm@13114
  1054
wenzelm@13142
  1055
subsection {* @{text upto} *}
wenzelm@13114
  1056
wenzelm@13142
  1057
lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])"
nipkow@13145
  1058
-- {* Does not terminate! *}
nipkow@13145
  1059
by (induct j) auto
wenzelm@13142
  1060
wenzelm@13142
  1061
lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []"
nipkow@13145
  1062
by (subst upt_rec) simp
wenzelm@13114
  1063
wenzelm@13142
  1064
lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]"
nipkow@13145
  1065
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1066
by simp
wenzelm@13114
  1067
wenzelm@13142
  1068
lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]"
nipkow@13145
  1069
apply(rule trans)
nipkow@13145
  1070
apply(subst upt_rec)
nipkow@13145
  1071
 prefer 2 apply(rule refl)
nipkow@13145
  1072
apply simp
nipkow@13145
  1073
done
wenzelm@13114
  1074
wenzelm@13142
  1075
lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]"
nipkow@13145
  1076
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1077
by (induct k) auto
wenzelm@13114
  1078
wenzelm@13142
  1079
lemma length_upt [simp]: "length [i..j(] = j - i"
nipkow@13145
  1080
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1081
wenzelm@13142
  1082
lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k"
nipkow@13145
  1083
apply (induct j)
nipkow@13145
  1084
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1085
done
wenzelm@13114
  1086
wenzelm@13142
  1087
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]"
nipkow@13145
  1088
apply (induct m)
nipkow@13145
  1089
 apply simp
nipkow@13145
  1090
apply (subst upt_rec)
nipkow@13145
  1091
apply (rule sym)
nipkow@13145
  1092
apply (subst upt_rec)
nipkow@13145
  1093
apply (simp del: upt.simps)
nipkow@13145
  1094
done
nipkow@3507
  1095
wenzelm@13114
  1096
lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]"
nipkow@13145
  1097
by (induct n) auto
wenzelm@13114
  1098
wenzelm@13114
  1099
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..n(]) ! i = f(m+i)"
nipkow@13145
  1100
apply (induct n m rule: diff_induct)
nipkow@13145
  1101
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1102
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1103
done
wenzelm@13114
  1104
wenzelm@13142
  1105
lemma nth_take_lemma [rule_format]:
nipkow@13145
  1106
"ALL xs ys. k <= length xs --> k <= length ys
nipkow@13145
  1107
--> (ALL i. i < k --> xs!i = ys!i)
nipkow@13145
  1108
--> take k xs = take k ys"
nipkow@13145
  1109
apply (induct k)
nipkow@13145
  1110
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib)
nipkow@13145
  1111
apply clarify
nipkow@13145
  1112
txt {* Both lists must be non-empty *}
nipkow@13145
  1113
apply (case_tac xs)
nipkow@13145
  1114
 apply simp
nipkow@13145
  1115
apply (case_tac ys)
nipkow@13145
  1116
 apply clarify
nipkow@13145
  1117
 apply (simp (no_asm_use))
nipkow@13145
  1118
apply clarify
nipkow@13145
  1119
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  1120
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  1121
apply blast
nipkow@13145
  1122
done
wenzelm@13114
  1123
wenzelm@13114
  1124
lemma nth_equalityI:
wenzelm@13114
  1125
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  1126
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  1127
apply (simp_all add: take_all)
nipkow@13145
  1128
done
wenzelm@13142
  1129
wenzelm@13142
  1130
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  1131
-- {* The famous take-lemma. *}
nipkow@13145
  1132
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  1133
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  1134
done
wenzelm@13142
  1135
wenzelm@13142
  1136
wenzelm@13142
  1137
subsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  1138
wenzelm@13142
  1139
lemma distinct_append [simp]:
nipkow@13145
  1140
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  1141
by (induct xs) auto
wenzelm@13142
  1142
wenzelm@13142
  1143
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  1144
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  1145
wenzelm@13142
  1146
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  1147
by (induct xs) auto
wenzelm@13142
  1148
wenzelm@13142
  1149
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  1150
by (induct xs) auto
wenzelm@13114
  1151
wenzelm@13142
  1152
text {*
nipkow@13145
  1153
It is best to avoid this indexed version of distinct, but sometimes
nipkow@13145
  1154
it is useful. *}
wenzelm@13142
  1155
lemma distinct_conv_nth:
nipkow@13145
  1156
"distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)"
nipkow@13145
  1157
apply (induct_tac xs)
nipkow@13145
  1158
 apply simp
nipkow@13145
  1159
apply simp
nipkow@13145
  1160
apply (rule iffI)
nipkow@13145
  1161
 apply clarsimp
nipkow@13145
  1162
 apply (case_tac i)
nipkow@13145
  1163
apply (case_tac j)
nipkow@13145
  1164
 apply simp
nipkow@13145
  1165
apply (simp add: set_conv_nth)
nipkow@13145
  1166
 apply (case_tac j)
nipkow@13145
  1167
apply (clarsimp simp add: set_conv_nth)
nipkow@13145
  1168
 apply simp
nipkow@13145
  1169
apply (rule conjI)
nipkow@13145
  1170
 apply (clarsimp simp add: set_conv_nth)
nipkow@13145
  1171
 apply (erule_tac x = 0 in allE)
nipkow@13145
  1172
 apply (erule_tac x = "Suc i" in allE)
nipkow@13145
  1173
 apply simp
nipkow@13145
  1174
apply clarsimp
nipkow@13145
  1175
apply (erule_tac x = "Suc i" in allE)
nipkow@13145
  1176
apply (erule_tac x = "Suc j" in allE)
nipkow@13145
  1177
apply simp
nipkow@13145
  1178
done
wenzelm@13114
  1179
wenzelm@13114
  1180
wenzelm@13142
  1181
subsection {* @{text replicate} *}
wenzelm@13114
  1182
wenzelm@13142
  1183
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  1184
by (induct n) auto
nipkow@13124
  1185
wenzelm@13142
  1186
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  1187
by (induct n) auto
wenzelm@13114
  1188
wenzelm@13114
  1189
lemma replicate_app_Cons_same:
nipkow@13145
  1190
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  1191
by (induct n) auto
wenzelm@13114
  1192
wenzelm@13142
  1193
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
nipkow@13145
  1194
apply(induct n)
nipkow@13145
  1195
 apply simp
nipkow@13145
  1196
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  1197
done
wenzelm@13114
  1198
wenzelm@13142
  1199
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  1200
by (induct n) auto
wenzelm@13114
  1201
wenzelm@13142
  1202
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  1203
by (induct n) auto
wenzelm@13114
  1204
wenzelm@13142
  1205
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  1206
by (induct n) auto
wenzelm@13114
  1207
wenzelm@13142
  1208
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  1209
by (atomize (full), induct n) auto
wenzelm@13114
  1210
wenzelm@13142
  1211
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
nipkow@13145
  1212
apply(induct n)
nipkow@13145
  1213
 apply simp
nipkow@13145
  1214
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  1215
done
wenzelm@13114
  1216
wenzelm@13142
  1217
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  1218
by (induct n) auto
wenzelm@13114
  1219
wenzelm@13142
  1220
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  1221
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  1222
wenzelm@13142
  1223
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  1224
by auto
wenzelm@13114
  1225
wenzelm@13142
  1226
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  1227
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  1228
wenzelm@13114
  1229
wenzelm@13142
  1230
subsection {* Lexcicographic orderings on lists *}
nipkow@3507
  1231
wenzelm@13142
  1232
lemma wf_lexn: "wf r ==> wf (lexn r n)"
nipkow@13145
  1233
apply (induct_tac n)
nipkow@13145
  1234
 apply simp
nipkow@13145
  1235
apply simp
nipkow@13145
  1236
apply(rule wf_subset)
nipkow@13145
  1237
 prefer 2 apply (rule Int_lower1)
nipkow@13145
  1238
apply(rule wf_prod_fun_image)
nipkow@13145
  1239
 prefer 2 apply (rule injI)
nipkow@13145
  1240
apply auto
nipkow@13145
  1241
done
wenzelm@13114
  1242
wenzelm@13114
  1243
lemma lexn_length:
nipkow@13145
  1244
"!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@13145
  1245
by (induct n) auto
wenzelm@13114
  1246
wenzelm@13142
  1247
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@13145
  1248
apply (unfold lex_def)
nipkow@13145
  1249
apply (rule wf_UN)
nipkow@13145
  1250
apply (blast intro: wf_lexn)
nipkow@13145
  1251
apply clarify
nipkow@13145
  1252
apply (rename_tac m n)
nipkow@13145
  1253
apply (subgoal_tac "m \<noteq> n")
nipkow@13145
  1254
 prefer 2 apply blast
nipkow@13145
  1255
apply (blast dest: lexn_length not_sym)
nipkow@13145
  1256
done
wenzelm@13114
  1257
wenzelm@13114
  1258
lemma lexn_conv:
nipkow@13145
  1259
"lexn r n =
nipkow@13145
  1260
{(xs,ys). length xs = n \<and> length ys = n \<and>
nipkow@13145
  1261
(\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
nipkow@13145
  1262
apply (induct_tac n)
nipkow@13145
  1263
 apply simp
nipkow@13145
  1264
 apply blast
nipkow@13145
  1265
apply (simp add: image_Collect lex_prod_def)
nipkow@13145
  1266
apply auto
nipkow@13145
  1267
apply blast
nipkow@13145
  1268
 apply (rename_tac a xys x xs' y ys')
nipkow@13145
  1269
 apply (rule_tac x = "a # xys" in exI)
nipkow@13145
  1270
 apply simp
nipkow@13145
  1271
apply (case_tac xys)
nipkow@13145
  1272
 apply simp_all
nipkow@13145
  1273
apply blast
nipkow@13145
  1274
done
wenzelm@13114
  1275
wenzelm@13114
  1276
lemma lex_conv:
nipkow@13145
  1277
"lex r =
nipkow@13145
  1278
{(xs,ys). length xs = length ys \<and>
nipkow@13145
  1279
(\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@13145
  1280
by (force simp add: lex_def lexn_conv)
wenzelm@13114
  1281
wenzelm@13142
  1282
lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)"
nipkow@13145
  1283
by (unfold lexico_def) blast
wenzelm@13114
  1284
wenzelm@13114
  1285
lemma lexico_conv:
nipkow@13145
  1286
"lexico r = {(xs,ys). length xs < length ys |
nipkow@13145
  1287
length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@13145
  1288
by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def)
wenzelm@13114
  1289
wenzelm@13142
  1290
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@13145
  1291
by (simp add: lex_conv)
wenzelm@13114
  1292
wenzelm@13142
  1293
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@13145
  1294
by (simp add:lex_conv)
wenzelm@13114
  1295
wenzelm@13142
  1296
lemma Cons_in_lex [iff]:
nipkow@13145
  1297
"((x # xs, y # ys) : lex r) =
nipkow@13145
  1298
((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@13145
  1299
apply (simp add: lex_conv)
nipkow@13145
  1300
apply (rule iffI)
nipkow@13145
  1301
 prefer 2 apply (blast intro: Cons_eq_appendI)
nipkow@13145
  1302
apply clarify
nipkow@13145
  1303
apply (case_tac xys)
nipkow@13145
  1304
 apply simp
nipkow@13145
  1305
apply simp
nipkow@13145
  1306
apply blast
nipkow@13145
  1307
done
wenzelm@13114
  1308
wenzelm@13114
  1309
wenzelm@13142
  1310
subsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  1311
wenzelm@13142
  1312
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  1313
by (auto simp add: sublist_def)
wenzelm@13114
  1314
wenzelm@13142
  1315
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  1316
by (auto simp add: sublist_def)
wenzelm@13114
  1317
wenzelm@13114
  1318
lemma sublist_shift_lemma:
nipkow@13145
  1319
"map fst [p:zip xs [i..i + length xs(] . snd p : A] =
nipkow@13145
  1320
map fst [p:zip xs [0..length xs(] . snd p + i : A]"
nipkow@13145
  1321
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  1322
wenzelm@13114
  1323
lemma sublist_append:
nipkow@13145
  1324
"sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  1325
apply (unfold sublist_def)
nipkow@13145
  1326
apply (induct l' rule: rev_induct)
nipkow@13145
  1327
 apply simp
nipkow@13145
  1328
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  1329
apply (simp add: add_commute)
nipkow@13145
  1330
done
wenzelm@13114
  1331
wenzelm@13114
  1332
lemma sublist_Cons:
nipkow@13145
  1333
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  1334
apply (induct l rule: rev_induct)
nipkow@13145
  1335
 apply (simp add: sublist_def)
nipkow@13145
  1336
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  1337
done
wenzelm@13114
  1338
wenzelm@13142
  1339
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  1340
by (simp add: sublist_Cons)
wenzelm@13114
  1341
wenzelm@13142
  1342
lemma sublist_upt_eq_take [simp]: "sublist l {..n(} = take n l"
nipkow@13145
  1343
apply (induct l rule: rev_induct)
nipkow@13145
  1344
 apply simp
nipkow@13145
  1345
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  1346
done
wenzelm@13114
  1347
wenzelm@13114
  1348
wenzelm@13142
  1349
lemma take_Cons':
nipkow@13145
  1350
"take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@13145
  1351
by (cases n) simp_all
wenzelm@13114
  1352
wenzelm@13142
  1353
lemma drop_Cons':
nipkow@13145
  1354
"drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@13145
  1355
by (cases n) simp_all
wenzelm@13114
  1356
wenzelm@13142
  1357
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@13145
  1358
by (cases n) simp_all
wenzelm@13142
  1359
nipkow@13145
  1360
lemmas [simp] = take_Cons'[of "number_of v",standard]
nipkow@13145
  1361
                drop_Cons'[of "number_of v",standard]
nipkow@13145
  1362
                nth_Cons'[of _ _ "number_of v",standard]
nipkow@3507
  1363
wenzelm@13122
  1364
end