wenzelm@13462
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(* Title: HOL/List.thy
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wenzelm@13462
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ID: $Id$
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wenzelm@13462
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Author: Tobias Nipkow
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wenzelm@13462
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License: GPL (GNU GENERAL PUBLIC LICENSE)
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clasohm@923
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*)
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clasohm@923
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wenzelm@13114
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header {* The datatype of finite lists *}
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wenzelm@13122
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wenzelm@13122
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theory List = PreList:
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clasohm@923
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wenzelm@13142
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datatype 'a list =
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wenzelm@13366
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Nil ("[]")
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| Cons 'a "'a list" (infixr "#" 65)
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clasohm@923
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clasohm@923
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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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clasohm@923
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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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nipkow@5077
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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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kleing@14565
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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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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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berghofe@5183
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primrec
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"null([]) = True"
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"null(x#xs) = False"
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paulson@8972
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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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berghofe@5183
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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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berghofe@5183
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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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berghofe@5183
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primrec
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"remdups [] = []"
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nipkow@13145
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"remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
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berghofe@5183
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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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nipkow@8115
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paulson@3196
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wenzelm@13142
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subsection {* Lexicographic orderings on lists *}
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nipkow@5281
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consts
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lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
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nipkow@5281
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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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nipkow@13145
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{(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
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nipkow@5281
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nipkow@5281
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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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nipkow@5281
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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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paulson@9336
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nipkow@13145
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sublist :: "'a list => nat set => 'a list"
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nipkow@13145
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"sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..size xs(]))"
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nipkow@5281
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nipkow@3507
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wenzelm@13142
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lemma not_Cons_self [simp]: "xs \<noteq> x # xs"
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nipkow@13145
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by (induct xs) auto
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nipkow@3507
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lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
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wenzelm@13114
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wenzelm@13142
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lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
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nipkow@13145
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by (induct xs) auto
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wenzelm@13114
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wenzelm@13142
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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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nipkow@13145
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by (rule measure_induct [of length]) rules
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wenzelm@13114
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wenzelm@13114
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wenzelm@13142
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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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wenzelm@13142
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inductive "lists A"
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intros
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nipkow@13145
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Nil [intro!]: "[]: lists A"
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nipkow@13145
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Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A"
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wenzelm@13114
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wenzelm@13142
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inductive_cases listsE [elim!]: "x#l : lists A"
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wenzelm@13114
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wenzelm@13366
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lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B"
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nipkow@13145
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by (unfold lists.defs) (blast intro!: lfp_mono)
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wenzelm@13114
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berghofe@13883
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lemma lists_IntI:
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berghofe@13883
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assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l
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berghofe@13883
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by induct blast+
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wenzelm@13114
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wenzelm@13142
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lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B"
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nipkow@13145
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apply (rule mono_Int [THEN equalityI])
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nipkow@13145
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apply (simp add: mono_def lists_mono)
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nipkow@13145
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apply (blast intro!: lists_IntI)
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nipkow@13145
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done
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wenzelm@13114
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wenzelm@13142
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lemma append_in_lists_conv [iff]:
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nipkow@13145
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"(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"
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nipkow@13145
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by (induct xs) auto
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wenzelm@13114
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wenzelm@13114
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wenzelm@13142
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subsection {* @{text length} *}
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wenzelm@13114
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wenzelm@13142
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text {*
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nipkow@13145
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Needs to come before @{text "@"} because of theorem @{text
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nipkow@13145
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append_eq_append_conv}.
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wenzelm@13142
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*}
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wenzelm@13114
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wenzelm@13142
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lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
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nipkow@13145
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by (induct xs) auto
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wenzelm@13114
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wenzelm@13142
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lemma length_map [simp]: "length (map f xs) = length xs"
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nipkow@13145
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by (induct xs) auto
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wenzelm@13114
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wenzelm@13142
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lemma length_rev [simp]: "length (rev xs) = length xs"
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nipkow@13145
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by (induct xs) auto
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wenzelm@13114
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wenzelm@13142
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lemma length_tl [simp]: "length (tl xs) = length xs - 1"
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nipkow@13145
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by (cases xs) auto
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wenzelm@13142
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wenzelm@13142
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lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
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nipkow@13145
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by (induct xs) auto
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wenzelm@13142
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wenzelm@13142
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lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
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nipkow@13145
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by (induct xs) auto
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wenzelm@13114
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wenzelm@13114
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lemma length_Suc_conv:
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nipkow@13145
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271 |
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
|
nipkow@13145
|
272 |
by (induct xs) auto
|
wenzelm@13114
|
273 |
|
nipkow@14025
|
274 |
lemma Suc_length_conv:
|
nipkow@14025
|
275 |
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
|
paulson@14208
|
276 |
apply (induct xs, simp, simp)
|
nipkow@14025
|
277 |
apply blast
|
nipkow@14025
|
278 |
done
|
nipkow@14025
|
279 |
|
oheimb@14099
|
280 |
lemma impossible_Cons [rule_format]:
|
oheimb@14099
|
281 |
"length xs <= length ys --> xs = x # ys = False"
|
paulson@14208
|
282 |
apply (induct xs, auto)
|
oheimb@14099
|
283 |
done
|
oheimb@14099
|
284 |
|
nipkow@14247
|
285 |
lemma list_induct2[consumes 1]: "\<And>ys.
|
nipkow@14247
|
286 |
\<lbrakk> length xs = length ys;
|
nipkow@14247
|
287 |
P [] [];
|
nipkow@14247
|
288 |
\<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
|
nipkow@14247
|
289 |
\<Longrightarrow> P xs ys"
|
nipkow@14247
|
290 |
apply(induct xs)
|
nipkow@14247
|
291 |
apply simp
|
nipkow@14247
|
292 |
apply(case_tac ys)
|
nipkow@14247
|
293 |
apply simp
|
nipkow@14247
|
294 |
apply(simp)
|
nipkow@14247
|
295 |
done
|
wenzelm@13114
|
296 |
|
wenzelm@13142
|
297 |
subsection {* @{text "@"} -- append *}
|
wenzelm@13114
|
298 |
|
wenzelm@13142
|
299 |
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
|
nipkow@13145
|
300 |
by (induct xs) auto
|
wenzelm@13114
|
301 |
|
wenzelm@13142
|
302 |
lemma append_Nil2 [simp]: "xs @ [] = xs"
|
nipkow@13145
|
303 |
by (induct xs) auto
|
wenzelm@13114
|
304 |
|
wenzelm@13142
|
305 |
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
|
nipkow@13145
|
306 |
by (induct xs) auto
|
wenzelm@13114
|
307 |
|
wenzelm@13142
|
308 |
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
|
nipkow@13145
|
309 |
by (induct xs) auto
|
wenzelm@13114
|
310 |
|
wenzelm@13142
|
311 |
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
|
nipkow@13145
|
312 |
by (induct xs) auto
|
wenzelm@13114
|
313 |
|
wenzelm@13142
|
314 |
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
|
nipkow@13145
|
315 |
by (induct xs) auto
|
wenzelm@13114
|
316 |
|
berghofe@13883
|
317 |
lemma append_eq_append_conv [simp]:
|
berghofe@13883
|
318 |
"!!ys. length xs = length ys \<or> length us = length vs
|
berghofe@13883
|
319 |
==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
|
berghofe@13883
|
320 |
apply (induct xs)
|
paulson@14208
|
321 |
apply (case_tac ys, simp, force)
|
paulson@14208
|
322 |
apply (case_tac ys, force, simp)
|
nipkow@13145
|
323 |
done
|
wenzelm@13114
|
324 |
|
nipkow@14495
|
325 |
lemma append_eq_append_conv2: "!!ys zs ts.
|
nipkow@14495
|
326 |
(xs @ ys = zs @ ts) =
|
nipkow@14495
|
327 |
(EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
|
nipkow@14495
|
328 |
apply (induct xs)
|
nipkow@14495
|
329 |
apply fastsimp
|
nipkow@14495
|
330 |
apply(case_tac zs)
|
nipkow@14495
|
331 |
apply simp
|
nipkow@14495
|
332 |
apply fastsimp
|
nipkow@14495
|
333 |
done
|
nipkow@14495
|
334 |
|
wenzelm@13142
|
335 |
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
|
nipkow@13145
|
336 |
by simp
|
wenzelm@13114
|
337 |
|
wenzelm@13142
|
338 |
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
|
nipkow@13145
|
339 |
by simp
|
wenzelm@13114
|
340 |
|
wenzelm@13142
|
341 |
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
|
nipkow@13145
|
342 |
by simp
|
wenzelm@13114
|
343 |
|
wenzelm@13142
|
344 |
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
|
nipkow@13145
|
345 |
using append_same_eq [of _ _ "[]"] by auto
|
wenzelm@13114
|
346 |
|
wenzelm@13142
|
347 |
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
|
nipkow@13145
|
348 |
using append_same_eq [of "[]"] by auto
|
wenzelm@13114
|
349 |
|
wenzelm@13142
|
350 |
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
|
nipkow@13145
|
351 |
by (induct xs) auto
|
wenzelm@13114
|
352 |
|
wenzelm@13142
|
353 |
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
|
nipkow@13145
|
354 |
by (induct xs) auto
|
wenzelm@13114
|
355 |
|
wenzelm@13142
|
356 |
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
|
nipkow@13145
|
357 |
by (simp add: hd_append split: list.split)
|
wenzelm@13114
|
358 |
|
wenzelm@13142
|
359 |
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
|
nipkow@13145
|
360 |
by (simp split: list.split)
|
wenzelm@13114
|
361 |
|
wenzelm@13142
|
362 |
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
|
nipkow@13145
|
363 |
by (simp add: tl_append split: list.split)
|
wenzelm@13114
|
364 |
|
wenzelm@13142
|
365 |
|
nipkow@14300
|
366 |
lemma Cons_eq_append_conv: "x#xs = ys@zs =
|
nipkow@14300
|
367 |
(ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
|
nipkow@14300
|
368 |
by(cases ys) auto
|
nipkow@14300
|
369 |
|
nipkow@14300
|
370 |
|
wenzelm@13142
|
371 |
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
|
wenzelm@13114
|
372 |
|
wenzelm@13114
|
373 |
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
|
nipkow@13145
|
374 |
by simp
|
wenzelm@13114
|
375 |
|
wenzelm@13142
|
376 |
lemma Cons_eq_appendI:
|
nipkow@13145
|
377 |
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
|
nipkow@13145
|
378 |
by (drule sym) simp
|
wenzelm@13114
|
379 |
|
wenzelm@13142
|
380 |
lemma append_eq_appendI:
|
nipkow@13145
|
381 |
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
|
nipkow@13145
|
382 |
by (drule sym) simp
|
wenzelm@13114
|
383 |
|
wenzelm@13114
|
384 |
|
wenzelm@13142
|
385 |
text {*
|
nipkow@13145
|
386 |
Simplification procedure for all list equalities.
|
nipkow@13145
|
387 |
Currently only tries to rearrange @{text "@"} to see if
|
nipkow@13145
|
388 |
- both lists end in a singleton list,
|
nipkow@13145
|
389 |
- or both lists end in the same list.
|
wenzelm@13142
|
390 |
*}
|
wenzelm@13142
|
391 |
|
wenzelm@13142
|
392 |
ML_setup {*
|
nipkow@3507
|
393 |
local
|
nipkow@3507
|
394 |
|
wenzelm@13122
|
395 |
val append_assoc = thm "append_assoc";
|
wenzelm@13122
|
396 |
val append_Nil = thm "append_Nil";
|
wenzelm@13122
|
397 |
val append_Cons = thm "append_Cons";
|
wenzelm@13122
|
398 |
val append1_eq_conv = thm "append1_eq_conv";
|
wenzelm@13122
|
399 |
val append_same_eq = thm "append_same_eq";
|
wenzelm@13122
|
400 |
|
wenzelm@13114
|
401 |
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
|
wenzelm@13462
|
402 |
(case xs of Const("List.list.Nil",_) => cons | _ => last xs)
|
wenzelm@13462
|
403 |
| last (Const("List.op @",_) $ _ $ ys) = last ys
|
wenzelm@13462
|
404 |
| last t = t;
|
nipkow@3507
|
405 |
|
wenzelm@13114
|
406 |
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
|
wenzelm@13462
|
407 |
| list1 _ = false;
|
wenzelm@13114
|
408 |
|
wenzelm@13114
|
409 |
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
|
wenzelm@13462
|
410 |
(case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
|
wenzelm@13462
|
411 |
| butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
|
wenzelm@13462
|
412 |
| butlast xs = Const("List.list.Nil",fastype_of xs);
|
wenzelm@13114
|
413 |
|
wenzelm@13114
|
414 |
val rearr_tac =
|
wenzelm@13462
|
415 |
simp_tac (HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]);
|
wenzelm@13114
|
416 |
|
wenzelm@13114
|
417 |
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
|
wenzelm@13462
|
418 |
let
|
wenzelm@13462
|
419 |
val lastl = last lhs and lastr = last rhs;
|
wenzelm@13462
|
420 |
fun rearr conv =
|
wenzelm@13462
|
421 |
let
|
wenzelm@13462
|
422 |
val lhs1 = butlast lhs and rhs1 = butlast rhs;
|
wenzelm@13462
|
423 |
val Type(_,listT::_) = eqT
|
wenzelm@13462
|
424 |
val appT = [listT,listT] ---> listT
|
wenzelm@13462
|
425 |
val app = Const("List.op @",appT)
|
wenzelm@13462
|
426 |
val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
|
wenzelm@13480
|
427 |
val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
|
wenzelm@13480
|
428 |
val thm = Tactic.prove sg [] [] eq (K (rearr_tac 1));
|
wenzelm@13462
|
429 |
in Some ((conv RS (thm RS trans)) RS eq_reflection) end;
|
wenzelm@13114
|
430 |
|
wenzelm@13462
|
431 |
in
|
wenzelm@13462
|
432 |
if list1 lastl andalso list1 lastr then rearr append1_eq_conv
|
wenzelm@13462
|
433 |
else if lastl aconv lastr then rearr append_same_eq
|
wenzelm@13462
|
434 |
else None
|
wenzelm@13462
|
435 |
end;
|
wenzelm@13462
|
436 |
|
nipkow@3507
|
437 |
in
|
wenzelm@13462
|
438 |
|
wenzelm@13462
|
439 |
val list_eq_simproc =
|
wenzelm@13462
|
440 |
Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq;
|
wenzelm@13462
|
441 |
|
wenzelm@13114
|
442 |
end;
|
nipkow@3507
|
443 |
|
wenzelm@13114
|
444 |
Addsimprocs [list_eq_simproc];
|
wenzelm@13114
|
445 |
*}
|
wenzelm@13114
|
446 |
|
wenzelm@13114
|
447 |
|
wenzelm@13142
|
448 |
subsection {* @{text map} *}
|
wenzelm@13114
|
449 |
|
wenzelm@13142
|
450 |
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
|
nipkow@13145
|
451 |
by (induct xs) simp_all
|
wenzelm@13114
|
452 |
|
wenzelm@13142
|
453 |
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
|
nipkow@13145
|
454 |
by (rule ext, induct_tac xs) auto
|
wenzelm@13114
|
455 |
|
wenzelm@13142
|
456 |
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
|
nipkow@13145
|
457 |
by (induct xs) auto
|
wenzelm@13114
|
458 |
|
wenzelm@13142
|
459 |
lemma map_compose: "map (f o g) xs = map f (map g xs)"
|
nipkow@13145
|
460 |
by (induct xs) (auto simp add: o_def)
|
wenzelm@13114
|
461 |
|
wenzelm@13142
|
462 |
lemma rev_map: "rev (map f xs) = map f (rev xs)"
|
nipkow@13145
|
463 |
by (induct xs) auto
|
wenzelm@13114
|
464 |
|
nipkow@13737
|
465 |
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
|
nipkow@13737
|
466 |
by (induct xs) auto
|
nipkow@13737
|
467 |
|
wenzelm@13366
|
468 |
lemma map_cong [recdef_cong]:
|
nipkow@13145
|
469 |
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
|
nipkow@13145
|
470 |
-- {* a congruence rule for @{text map} *}
|
nipkow@13737
|
471 |
by simp
|
wenzelm@13114
|
472 |
|
wenzelm@13142
|
473 |
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
|
nipkow@13145
|
474 |
by (cases xs) auto
|
wenzelm@13114
|
475 |
|
wenzelm@13142
|
476 |
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
|
nipkow@13145
|
477 |
by (cases xs) auto
|
wenzelm@13114
|
478 |
|
nipkow@14025
|
479 |
lemma map_eq_Cons_conv[iff]:
|
nipkow@14025
|
480 |
"(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
|
nipkow@13145
|
481 |
by (cases xs) auto
|
wenzelm@13114
|
482 |
|
nipkow@14025
|
483 |
lemma Cons_eq_map_conv[iff]:
|
nipkow@14025
|
484 |
"(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
|
nipkow@14025
|
485 |
by (cases ys) auto
|
nipkow@14025
|
486 |
|
nipkow@14111
|
487 |
lemma ex_map_conv:
|
nipkow@14111
|
488 |
"(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
|
nipkow@14111
|
489 |
by(induct ys, auto)
|
nipkow@14111
|
490 |
|
wenzelm@13114
|
491 |
lemma map_injective:
|
nipkow@14338
|
492 |
"!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
|
nipkow@14338
|
493 |
by (induct ys) (auto dest!:injD)
|
wenzelm@13114
|
494 |
|
nipkow@14339
|
495 |
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
|
nipkow@14339
|
496 |
by(blast dest:map_injective)
|
nipkow@14339
|
497 |
|
wenzelm@13114
|
498 |
lemma inj_mapI: "inj f ==> inj (map f)"
|
paulson@13585
|
499 |
by (rules dest: map_injective injD intro: inj_onI)
|
wenzelm@13114
|
500 |
|
wenzelm@13114
|
501 |
lemma inj_mapD: "inj (map f) ==> inj f"
|
paulson@14208
|
502 |
apply (unfold inj_on_def, clarify)
|
nipkow@13145
|
503 |
apply (erule_tac x = "[x]" in ballE)
|
paulson@14208
|
504 |
apply (erule_tac x = "[y]" in ballE, simp, blast)
|
nipkow@13145
|
505 |
apply blast
|
nipkow@13145
|
506 |
done
|
wenzelm@13114
|
507 |
|
nipkow@14339
|
508 |
lemma inj_map[iff]: "inj (map f) = inj f"
|
nipkow@13145
|
509 |
by (blast dest: inj_mapD intro: inj_mapI)
|
wenzelm@13114
|
510 |
|
kleing@14343
|
511 |
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
|
kleing@14343
|
512 |
by (induct xs, auto)
|
wenzelm@13114
|
513 |
|
nipkow@14402
|
514 |
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
|
nipkow@14402
|
515 |
by (induct xs) auto
|
nipkow@14402
|
516 |
|
wenzelm@13142
|
517 |
subsection {* @{text rev} *}
|
wenzelm@13114
|
518 |
|
wenzelm@13142
|
519 |
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
|
nipkow@13145
|
520 |
by (induct xs) auto
|
wenzelm@13114
|
521 |
|
wenzelm@13142
|
522 |
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
|
nipkow@13145
|
523 |
by (induct xs) auto
|
wenzelm@13114
|
524 |
|
wenzelm@13142
|
525 |
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
|
nipkow@13145
|
526 |
by (induct xs) auto
|
wenzelm@13114
|
527 |
|
wenzelm@13142
|
528 |
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
|
nipkow@13145
|
529 |
by (induct xs) auto
|
wenzelm@13114
|
530 |
|
wenzelm@13142
|
531 |
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"
|
paulson@14208
|
532 |
apply (induct xs, force)
|
paulson@14208
|
533 |
apply (case_tac ys, simp, force)
|
nipkow@13145
|
534 |
done
|
wenzelm@13114
|
535 |
|
wenzelm@13366
|
536 |
lemma rev_induct [case_names Nil snoc]:
|
wenzelm@13366
|
537 |
"[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
|
nipkow@13145
|
538 |
apply(subst rev_rev_ident[symmetric])
|
nipkow@13145
|
539 |
apply(rule_tac list = "rev xs" in list.induct, simp_all)
|
nipkow@13145
|
540 |
done
|
wenzelm@13114
|
541 |
|
nipkow@13145
|
542 |
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
|
wenzelm@13114
|
543 |
|
wenzelm@13366
|
544 |
lemma rev_exhaust [case_names Nil snoc]:
|
wenzelm@13366
|
545 |
"(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
|
nipkow@13145
|
546 |
by (induct xs rule: rev_induct) auto
|
wenzelm@13114
|
547 |
|
wenzelm@13366
|
548 |
lemmas rev_cases = rev_exhaust
|
wenzelm@13366
|
549 |
|
wenzelm@13114
|
550 |
|
wenzelm@13142
|
551 |
subsection {* @{text set} *}
|
wenzelm@13114
|
552 |
|
wenzelm@13142
|
553 |
lemma finite_set [iff]: "finite (set xs)"
|
nipkow@13145
|
554 |
by (induct xs) auto
|
wenzelm@13114
|
555 |
|
wenzelm@13142
|
556 |
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
|
nipkow@13145
|
557 |
by (induct xs) auto
|
wenzelm@13114
|
558 |
|
oheimb@14099
|
559 |
lemma hd_in_set: "l = x#xs \<Longrightarrow> x\<in>set l"
|
paulson@14208
|
560 |
by (case_tac l, auto)
|
oheimb@14099
|
561 |
|
wenzelm@13142
|
562 |
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
|
nipkow@13145
|
563 |
by auto
|
wenzelm@13114
|
564 |
|
oheimb@14099
|
565 |
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs"
|
oheimb@14099
|
566 |
by auto
|
oheimb@14099
|
567 |
|
wenzelm@13142
|
568 |
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
|
nipkow@13145
|
569 |
by (induct xs) auto
|
wenzelm@13114
|
570 |
|
wenzelm@13142
|
571 |
lemma set_rev [simp]: "set (rev xs) = set xs"
|
nipkow@13145
|
572 |
by (induct xs) auto
|
wenzelm@13114
|
573 |
|
wenzelm@13142
|
574 |
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
|
nipkow@13145
|
575 |
by (induct xs) auto
|
wenzelm@13114
|
576 |
|
wenzelm@13142
|
577 |
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
|
nipkow@13145
|
578 |
by (induct xs) auto
|
wenzelm@13114
|
579 |
|
wenzelm@13142
|
580 |
lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> k < j}"
|
paulson@14208
|
581 |
apply (induct j, simp_all)
|
paulson@14208
|
582 |
apply (erule ssubst, auto)
|
nipkow@13145
|
583 |
done
|
wenzelm@13114
|
584 |
|
wenzelm@13142
|
585 |
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
|
paulson@14208
|
586 |
apply (induct xs, simp, simp)
|
nipkow@13145
|
587 |
apply (rule iffI)
|
nipkow@13145
|
588 |
apply (blast intro: eq_Nil_appendI Cons_eq_appendI)
|
nipkow@13145
|
589 |
apply (erule exE)+
|
paulson@14208
|
590 |
apply (case_tac ys, auto)
|
nipkow@13145
|
591 |
done
|
wenzelm@13114
|
592 |
|
wenzelm@13142
|
593 |
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"
|
nipkow@13145
|
594 |
-- {* eliminate @{text lists} in favour of @{text set} *}
|
nipkow@13145
|
595 |
by (induct xs) auto
|
wenzelm@13114
|
596 |
|
wenzelm@13142
|
597 |
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"
|
nipkow@13145
|
598 |
by (rule in_lists_conv_set [THEN iffD1])
|
wenzelm@13114
|
599 |
|
wenzelm@13142
|
600 |
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"
|
nipkow@13145
|
601 |
by (rule in_lists_conv_set [THEN iffD2])
|
wenzelm@13114
|
602 |
|
paulson@13508
|
603 |
lemma finite_list: "finite A ==> EX l. set l = A"
|
paulson@13508
|
604 |
apply (erule finite_induct, auto)
|
paulson@13508
|
605 |
apply (rule_tac x="x#l" in exI, auto)
|
paulson@13508
|
606 |
done
|
paulson@13508
|
607 |
|
kleing@14388
|
608 |
lemma card_length: "card (set xs) \<le> length xs"
|
kleing@14388
|
609 |
by (induct xs) (auto simp add: card_insert_if)
|
wenzelm@13114
|
610 |
|
wenzelm@13142
|
611 |
subsection {* @{text mem} *}
|
wenzelm@13114
|
612 |
|
wenzelm@13114
|
613 |
lemma set_mem_eq: "(x mem xs) = (x : set xs)"
|
nipkow@13145
|
614 |
by (induct xs) auto
|
wenzelm@13114
|
615 |
|
wenzelm@13114
|
616 |
|
wenzelm@13142
|
617 |
subsection {* @{text list_all} *}
|
wenzelm@13114
|
618 |
|
wenzelm@13142
|
619 |
lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)"
|
nipkow@13145
|
620 |
by (induct xs) auto
|
wenzelm@13114
|
621 |
|
wenzelm@13142
|
622 |
lemma list_all_append [simp]:
|
nipkow@13145
|
623 |
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
|
nipkow@13145
|
624 |
by (induct xs) auto
|
wenzelm@13114
|
625 |
|
wenzelm@13114
|
626 |
|
wenzelm@13142
|
627 |
subsection {* @{text filter} *}
|
wenzelm@13114
|
628 |
|
wenzelm@13142
|
629 |
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
|
nipkow@13145
|
630 |
by (induct xs) auto
|
wenzelm@13114
|
631 |
|
wenzelm@13142
|
632 |
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
|
nipkow@13145
|
633 |
by (induct xs) auto
|
wenzelm@13114
|
634 |
|
wenzelm@13142
|
635 |
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
|
nipkow@13145
|
636 |
by (induct xs) auto
|
wenzelm@13114
|
637 |
|
wenzelm@13142
|
638 |
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
|
nipkow@13145
|
639 |
by (induct xs) auto
|
wenzelm@13114
|
640 |
|
wenzelm@13142
|
641 |
lemma length_filter [simp]: "length (filter P xs) \<le> length xs"
|
nipkow@13145
|
642 |
by (induct xs) (auto simp add: le_SucI)
|
wenzelm@13114
|
643 |
|
wenzelm@13142
|
644 |
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
|
nipkow@13145
|
645 |
by auto
|
wenzelm@13114
|
646 |
|
wenzelm@13114
|
647 |
|
wenzelm@13142
|
648 |
subsection {* @{text concat} *}
|
wenzelm@13114
|
649 |
|
wenzelm@13142
|
650 |
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
|
nipkow@13145
|
651 |
by (induct xs) auto
|
wenzelm@13114
|
652 |
|
wenzelm@13142
|
653 |
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
|
nipkow@13145
|
654 |
by (induct xss) auto
|
wenzelm@13114
|
655 |
|
wenzelm@13142
|
656 |
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
|
nipkow@13145
|
657 |
by (induct xss) auto
|
wenzelm@13114
|
658 |
|
wenzelm@13142
|
659 |
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
|
nipkow@13145
|
660 |
by (induct xs) auto
|
wenzelm@13114
|
661 |
|
wenzelm@13142
|
662 |
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
|
nipkow@13145
|
663 |
by (induct xs) auto
|
wenzelm@13114
|
664 |
|
wenzelm@13142
|
665 |
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
|
nipkow@13145
|
666 |
by (induct xs) auto
|
wenzelm@13114
|
667 |
|
wenzelm@13142
|
668 |
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
|
nipkow@13145
|
669 |
by (induct xs) auto
|
wenzelm@13114
|
670 |
|
wenzelm@13114
|
671 |
|
wenzelm@13142
|
672 |
subsection {* @{text nth} *}
|
wenzelm@13114
|
673 |
|
wenzelm@13142
|
674 |
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
|
nipkow@13145
|
675 |
by auto
|
wenzelm@13114
|
676 |
|
wenzelm@13142
|
677 |
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
|
nipkow@13145
|
678 |
by auto
|
wenzelm@13114
|
679 |
|
wenzelm@13142
|
680 |
declare nth.simps [simp del]
|
wenzelm@13114
|
681 |
|
wenzelm@13114
|
682 |
lemma nth_append:
|
nipkow@13145
|
683 |
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
|
paulson@14208
|
684 |
apply (induct "xs", simp)
|
paulson@14208
|
685 |
apply (case_tac n, auto)
|
nipkow@13145
|
686 |
done
|
wenzelm@13114
|
687 |
|
nipkow@14402
|
688 |
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
|
nipkow@14402
|
689 |
by (induct "xs") auto
|
nipkow@14402
|
690 |
|
nipkow@14402
|
691 |
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
|
nipkow@14402
|
692 |
by (induct "xs") auto
|
nipkow@14402
|
693 |
|
wenzelm@13142
|
694 |
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
|
paulson@14208
|
695 |
apply (induct xs, simp)
|
paulson@14208
|
696 |
apply (case_tac n, auto)
|
nipkow@13145
|
697 |
done
|
wenzelm@13114
|
698 |
|
wenzelm@13142
|
699 |
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
|
paulson@14208
|
700 |
apply (induct_tac xs, simp, simp)
|
nipkow@13145
|
701 |
apply safe
|
paulson@14208
|
702 |
apply (rule_tac x = 0 in exI, simp)
|
paulson@14208
|
703 |
apply (rule_tac x = "Suc i" in exI, simp)
|
paulson@14208
|
704 |
apply (case_tac i, simp)
|
nipkow@13145
|
705 |
apply (rename_tac j)
|
paulson@14208
|
706 |
apply (rule_tac x = j in exI, simp)
|
nipkow@13145
|
707 |
done
|
wenzelm@13114
|
708 |
|
nipkow@13145
|
709 |
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
|
nipkow@13145
|
710 |
by (auto simp add: set_conv_nth)
|
wenzelm@13114
|
711 |
|
wenzelm@13142
|
712 |
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
|
nipkow@13145
|
713 |
by (auto simp add: set_conv_nth)
|
wenzelm@13114
|
714 |
|
wenzelm@13114
|
715 |
lemma all_nth_imp_all_set:
|
nipkow@13145
|
716 |
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
|
nipkow@13145
|
717 |
by (auto simp add: set_conv_nth)
|
wenzelm@13114
|
718 |
|
wenzelm@13114
|
719 |
lemma all_set_conv_all_nth:
|
nipkow@13145
|
720 |
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
|
nipkow@13145
|
721 |
by (auto simp add: set_conv_nth)
|
wenzelm@13114
|
722 |
|
wenzelm@13114
|
723 |
|
wenzelm@13142
|
724 |
subsection {* @{text list_update} *}
|
wenzelm@13114
|
725 |
|
wenzelm@13142
|
726 |
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
|
nipkow@13145
|
727 |
by (induct xs) (auto split: nat.split)
|
wenzelm@13114
|
728 |
|
wenzelm@13114
|
729 |
lemma nth_list_update:
|
nipkow@13145
|
730 |
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
|
nipkow@13145
|
731 |
by (induct xs) (auto simp add: nth_Cons split: nat.split)
|
wenzelm@13114
|
732 |
|
wenzelm@13142
|
733 |
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
|
nipkow@13145
|
734 |
by (simp add: nth_list_update)
|
wenzelm@13114
|
735 |
|
wenzelm@13142
|
736 |
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
|
nipkow@13145
|
737 |
by (induct xs) (auto simp add: nth_Cons split: nat.split)
|
wenzelm@13114
|
738 |
|
wenzelm@13142
|
739 |
lemma list_update_overwrite [simp]:
|
nipkow@13145
|
740 |
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
|
nipkow@13145
|
741 |
by (induct xs) (auto split: nat.split)
|
wenzelm@13114
|
742 |
|
nipkow@14402
|
743 |
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
|
paulson@14208
|
744 |
apply (induct xs, simp)
|
nipkow@14187
|
745 |
apply(simp split:nat.splits)
|
nipkow@14187
|
746 |
done
|
nipkow@14187
|
747 |
|
wenzelm@13114
|
748 |
lemma list_update_same_conv:
|
nipkow@13145
|
749 |
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
|
nipkow@13145
|
750 |
by (induct xs) (auto split: nat.split)
|
wenzelm@13114
|
751 |
|
nipkow@14187
|
752 |
lemma list_update_append1:
|
nipkow@14187
|
753 |
"!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
|
paulson@14208
|
754 |
apply (induct xs, simp)
|
nipkow@14187
|
755 |
apply(simp split:nat.split)
|
nipkow@14187
|
756 |
done
|
nipkow@14187
|
757 |
|
nipkow@14402
|
758 |
lemma list_update_length [simp]:
|
nipkow@14402
|
759 |
"(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
|
nipkow@14402
|
760 |
by (induct xs, auto)
|
nipkow@14402
|
761 |
|
wenzelm@13114
|
762 |
lemma update_zip:
|
nipkow@13145
|
763 |
"!!i xy xs. length xs = length ys ==>
|
nipkow@13145
|
764 |
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
|
nipkow@13145
|
765 |
by (induct ys) (auto, case_tac xs, auto split: nat.split)
|
wenzelm@13114
|
766 |
|
wenzelm@13114
|
767 |
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
|
nipkow@13145
|
768 |
by (induct xs) (auto split: nat.split)
|
wenzelm@13114
|
769 |
|
wenzelm@13114
|
770 |
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
|
nipkow@13145
|
771 |
by (blast dest!: set_update_subset_insert [THEN subsetD])
|
wenzelm@13114
|
772 |
|
wenzelm@13114
|
773 |
|
wenzelm@13142
|
774 |
subsection {* @{text last} and @{text butlast} *}
|
wenzelm@13114
|
775 |
|
wenzelm@13142
|
776 |
lemma last_snoc [simp]: "last (xs @ [x]) = x"
|
nipkow@13145
|
777 |
by (induct xs) auto
|
wenzelm@13114
|
778 |
|
wenzelm@13142
|
779 |
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
|
nipkow@13145
|
780 |
by (induct xs) auto
|
wenzelm@13114
|
781 |
|
nipkow@14302
|
782 |
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
|
nipkow@14302
|
783 |
by(simp add:last.simps)
|
nipkow@14302
|
784 |
|
nipkow@14302
|
785 |
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
|
nipkow@14302
|
786 |
by(simp add:last.simps)
|
nipkow@14302
|
787 |
|
nipkow@14302
|
788 |
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
|
nipkow@14302
|
789 |
by (induct xs) (auto)
|
nipkow@14302
|
790 |
|
nipkow@14302
|
791 |
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
|
nipkow@14302
|
792 |
by(simp add:last_append)
|
nipkow@14302
|
793 |
|
nipkow@14302
|
794 |
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
|
nipkow@14302
|
795 |
by(simp add:last_append)
|
nipkow@14302
|
796 |
|
nipkow@14302
|
797 |
|
nipkow@14302
|
798 |
|
wenzelm@13142
|
799 |
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
|
nipkow@13145
|
800 |
by (induct xs rule: rev_induct) auto
|
wenzelm@13114
|
801 |
|
wenzelm@13114
|
802 |
lemma butlast_append:
|
nipkow@13145
|
803 |
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
|
nipkow@13145
|
804 |
by (induct xs) auto
|
wenzelm@13114
|
805 |
|
wenzelm@13142
|
806 |
lemma append_butlast_last_id [simp]:
|
nipkow@13145
|
807 |
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
|
nipkow@13145
|
808 |
by (induct xs) auto
|
wenzelm@13114
|
809 |
|
wenzelm@13142
|
810 |
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
|
nipkow@13145
|
811 |
by (induct xs) (auto split: split_if_asm)
|
wenzelm@13114
|
812 |
|
wenzelm@13114
|
813 |
lemma in_set_butlast_appendI:
|
nipkow@13145
|
814 |
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
|
nipkow@13145
|
815 |
by (auto dest: in_set_butlastD simp add: butlast_append)
|
wenzelm@13114
|
816 |
|
wenzelm@13114
|
817 |
|
wenzelm@13142
|
818 |
subsection {* @{text take} and @{text drop} *}
|
wenzelm@13114
|
819 |
|
wenzelm@13142
|
820 |
lemma take_0 [simp]: "take 0 xs = []"
|
nipkow@13145
|
821 |
by (induct xs) auto
|
wenzelm@13114
|
822 |
|
wenzelm@13142
|
823 |
lemma drop_0 [simp]: "drop 0 xs = xs"
|
nipkow@13145
|
824 |
by (induct xs) auto
|
wenzelm@13114
|
825 |
|
wenzelm@13142
|
826 |
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
|
nipkow@13145
|
827 |
by simp
|
wenzelm@13114
|
828 |
|
wenzelm@13142
|
829 |
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
|
nipkow@13145
|
830 |
by simp
|
wenzelm@13114
|
831 |
|
wenzelm@13142
|
832 |
declare take_Cons [simp del] and drop_Cons [simp del]
|
wenzelm@13114
|
833 |
|
nipkow@14187
|
834 |
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
|
nipkow@14187
|
835 |
by(cases xs, simp_all)
|
nipkow@14187
|
836 |
|
nipkow@14187
|
837 |
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
|
nipkow@14187
|
838 |
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
|
nipkow@14187
|
839 |
|
nipkow@14187
|
840 |
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
|
paulson@14208
|
841 |
apply (induct xs, simp)
|
nipkow@14187
|
842 |
apply(simp add:drop_Cons nth_Cons split:nat.splits)
|
nipkow@14187
|
843 |
done
|
nipkow@14187
|
844 |
|
nipkow@13913
|
845 |
lemma take_Suc_conv_app_nth:
|
nipkow@13913
|
846 |
"!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
|
paulson@14208
|
847 |
apply (induct xs, simp)
|
paulson@14208
|
848 |
apply (case_tac i, auto)
|
nipkow@13913
|
849 |
done
|
nipkow@13913
|
850 |
|
wenzelm@13142
|
851 |
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
|
nipkow@13145
|
852 |
by (induct n) (auto, case_tac xs, auto)
|
wenzelm@13114
|
853 |
|
wenzelm@13142
|
854 |
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
|
nipkow@13145
|
855 |
by (induct n) (auto, case_tac xs, auto)
|
wenzelm@13114
|
856 |
|
wenzelm@13142
|
857 |
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
|
nipkow@13145
|
858 |
by (induct n) (auto, case_tac xs, auto)
|
wenzelm@13114
|
859 |
|
wenzelm@13142
|
860 |
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
|
nipkow@13145
|
861 |
by (induct n) (auto, case_tac xs, auto)
|
wenzelm@13114
|
862 |
|
wenzelm@13142
|
863 |
lemma take_append [simp]:
|
nipkow@13145
|
864 |
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
|
nipkow@13145
|
865 |
by (induct n) (auto, case_tac xs, auto)
|
wenzelm@13114
|
866 |
|
wenzelm@13142
|
867 |
lemma drop_append [simp]:
|
nipkow@13145
|
868 |
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
|
nipkow@13145
|
869 |
by (induct n) (auto, case_tac xs, auto)
|
wenzelm@13114
|
870 |
|
wenzelm@13142
|
871 |
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
|
paulson@14208
|
872 |
apply (induct m, auto)
|
paulson@14208
|
873 |
apply (case_tac xs, auto)
|
paulson@14208
|
874 |
apply (case_tac na, auto)
|
nipkow@13145
|
875 |
done
|
wenzelm@13142
|
876 |
|
wenzelm@13142
|
877 |
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
|
paulson@14208
|
878 |
apply (induct m, auto)
|
paulson@14208
|
879 |
apply (case_tac xs, auto)
|
nipkow@13145
|
880 |
done
|
wenzelm@13114
|
881 |
|
wenzelm@13114
|
882 |
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
|
paulson@14208
|
883 |
apply (induct m, auto)
|
paulson@14208
|
884 |
apply (case_tac xs, auto)
|
nipkow@13145
|
885 |
done
|
wenzelm@13114
|
886 |
|
wenzelm@13142
|
887 |
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
|
paulson@14208
|
888 |
apply (induct n, auto)
|
paulson@14208
|
889 |
apply (case_tac xs, auto)
|
nipkow@13145
|
890 |
done
|
wenzelm@13114
|
891 |
|
wenzelm@13114
|
892 |
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
|
paulson@14208
|
893 |
apply (induct n, auto)
|
paulson@14208
|
894 |
apply (case_tac xs, auto)
|
nipkow@13145
|
895 |
done
|
wenzelm@13114
|
896 |
|
wenzelm@13142
|
897 |
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
|
paulson@14208
|
898 |
apply (induct n, auto)
|
paulson@14208
|
899 |
apply (case_tac xs, auto)
|
nipkow@13145
|
900 |
done
|
wenzelm@13114
|
901 |
|
wenzelm@13114
|
902 |
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
|
paulson@14208
|
903 |
apply (induct xs, auto)
|
paulson@14208
|
904 |
apply (case_tac i, auto)
|
nipkow@13145
|
905 |
done
|
wenzelm@13114
|
906 |
|
wenzelm@13114
|
907 |
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
|
paulson@14208
|
908 |
apply (induct xs, auto)
|
paulson@14208
|
909 |
apply (case_tac i, auto)
|
nipkow@13145
|
910 |
done
|
wenzelm@13114
|
911 |
|
wenzelm@13142
|
912 |
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
|
paulson@14208
|
913 |
apply (induct xs, auto)
|
paulson@14208
|
914 |
apply (case_tac n, blast)
|
paulson@14208
|
915 |
apply (case_tac i, auto)
|
nipkow@13145
|
916 |
done
|
wenzelm@13114
|
917 |
|
wenzelm@13142
|
918 |
lemma nth_drop [simp]:
|
nipkow@13145
|
919 |
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
|
paulson@14208
|
920 |
apply (induct n, auto)
|
paulson@14208
|
921 |
apply (case_tac xs, auto)
|
nipkow@13145
|
922 |
done
|
wenzelm@13114
|
923 |
|
nipkow@14025
|
924 |
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
|
nipkow@14025
|
925 |
by(induct xs)(auto simp:take_Cons split:nat.split)
|
nipkow@14025
|
926 |
|
nipkow@14025
|
927 |
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
|
nipkow@14025
|
928 |
by(induct xs)(auto simp:drop_Cons split:nat.split)
|
nipkow@14025
|
929 |
|
nipkow@14187
|
930 |
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
|
nipkow@14187
|
931 |
using set_take_subset by fast
|
nipkow@14187
|
932 |
|
nipkow@14187
|
933 |
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
|
nipkow@14187
|
934 |
using set_drop_subset by fast
|
nipkow@14187
|
935 |
|
wenzelm@13114
|
936 |
lemma append_eq_conv_conj:
|
nipkow@13145
|
937 |
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
|
paulson@14208
|
938 |
apply (induct xs, simp, clarsimp)
|
paulson@14208
|
939 |
apply (case_tac zs, auto)
|
nipkow@13145
|
940 |
done
|
wenzelm@13114
|
941 |
|
paulson@14050
|
942 |
lemma take_add [rule_format]:
|
paulson@14050
|
943 |
"\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
|
paulson@14050
|
944 |
apply (induct xs, auto)
|
paulson@14050
|
945 |
apply (case_tac i, simp_all)
|
paulson@14050
|
946 |
done
|
paulson@14050
|
947 |
|
nipkow@14300
|
948 |
lemma append_eq_append_conv_if:
|
nipkow@14300
|
949 |
"!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
|
nipkow@14300
|
950 |
(if size xs\<^isub>1 \<le> size ys\<^isub>1
|
nipkow@14300
|
951 |
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
|
952 |
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
|
953 |
apply(induct xs\<^isub>1)
|
nipkow@14300
|
954 |
apply simp
|
nipkow@14300
|
955 |
apply(case_tac ys\<^isub>1)
|
nipkow@14300
|
956 |
apply simp_all
|
nipkow@14300
|
957 |
done
|
nipkow@14300
|
958 |
|
wenzelm@13114
|
959 |
|
wenzelm@13142
|
960 |
subsection {* @{text takeWhile} and @{text dropWhile} *}
|
wenzelm@13114
|
961 |
|
wenzelm@13142
|
962 |
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
|
nipkow@13145
|
963 |
by (induct xs) auto
|
wenzelm@13114
|
964 |
|
wenzelm@13142
|
965 |
lemma takeWhile_append1 [simp]:
|
nipkow@13145
|
966 |
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
|
nipkow@13145
|
967 |
by (induct xs) auto
|
wenzelm@13114
|
968 |
|
wenzelm@13142
|
969 |
lemma takeWhile_append2 [simp]:
|
nipkow@13145
|
970 |
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
|
nipkow@13145
|
971 |
by (induct xs) auto
|
wenzelm@13114
|
972 |
|
wenzelm@13142
|
973 |
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
|
nipkow@13145
|
974 |
by (induct xs) auto
|
wenzelm@13114
|
975 |
|
wenzelm@13142
|
976 |
lemma dropWhile_append1 [simp]:
|
nipkow@13145
|
977 |
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
|
nipkow@13145
|
978 |
by (induct xs) auto
|
wenzelm@13114
|
979 |
|
wenzelm@13142
|
980 |
lemma dropWhile_append2 [simp]:
|
nipkow@13145
|
981 |
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
|
nipkow@13145
|
982 |
by (induct xs) auto
|
wenzelm@13114
|
983 |
|
wenzelm@13142
|
984 |
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
|
nipkow@13145
|
985 |
by (induct xs) (auto split: split_if_asm)
|
wenzelm@13114
|
986 |
|
nipkow@13913
|
987 |
lemma takeWhile_eq_all_conv[simp]:
|
nipkow@13913
|
988 |
"(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
|
nipkow@13913
|
989 |
by(induct xs, auto)
|
nipkow@13913
|
990 |
|
nipkow@13913
|
991 |
lemma dropWhile_eq_Nil_conv[simp]:
|
nipkow@13913
|
992 |
"(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
|
nipkow@13913
|
993 |
by(induct xs, auto)
|
nipkow@13913
|
994 |
|
nipkow@13913
|
995 |
lemma dropWhile_eq_Cons_conv:
|
nipkow@13913
|
996 |
"(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
|
nipkow@13913
|
997 |
by(induct xs, auto)
|
nipkow@13913
|
998 |
|
wenzelm@13114
|
999 |
|
wenzelm@13142
|
1000 |
subsection {* @{text zip} *}
|
wenzelm@13114
|
1001 |
|
wenzelm@13142
|
1002 |
lemma zip_Nil [simp]: "zip [] ys = []"
|
nipkow@13145
|
1003 |
by (induct ys) auto
|
wenzelm@13114
|
1004 |
|
wenzelm@13142
|
1005 |
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
|
nipkow@13145
|
1006 |
by simp
|
wenzelm@13114
|
1007 |
|
wenzelm@13142
|
1008 |
declare zip_Cons [simp del]
|
wenzelm@13114
|
1009 |
|
wenzelm@13142
|
1010 |
lemma length_zip [simp]:
|
nipkow@13145
|
1011 |
"!!xs. length (zip xs ys) = min (length xs) (length ys)"
|
paulson@14208
|
1012 |
apply (induct ys, simp)
|
paulson@14208
|
1013 |
apply (case_tac xs, auto)
|
nipkow@13145
|
1014 |
done
|
wenzelm@13114
|
1015 |
|
wenzelm@13114
|
1016 |
lemma zip_append1:
|
nipkow@13145
|
1017 |
"!!xs. zip (xs @ ys) zs =
|
nipkow@13145
|
1018 |
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
|
paulson@14208
|
1019 |
apply (induct zs, simp)
|
paulson@14208
|
1020 |
apply (case_tac xs, simp_all)
|
nipkow@13145
|
1021 |
done
|
wenzelm@13114
|
1022 |
|
wenzelm@13114
|
1023 |
lemma zip_append2:
|
nipkow@13145
|
1024 |
"!!ys. zip xs (ys @ zs) =
|
nipkow@13145
|
1025 |
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
|
paulson@14208
|
1026 |
apply (induct xs, simp)
|
paulson@14208
|
1027 |
apply (case_tac ys, simp_all)
|
nipkow@13145
|
1028 |
done
|
wenzelm@13114
|
1029 |
|
wenzelm@13142
|
1030 |
lemma zip_append [simp]:
|
wenzelm@13142
|
1031 |
"[| length xs = length us; length ys = length vs |] ==>
|
nipkow@13145
|
1032 |
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
|
nipkow@13145
|
1033 |
by (simp add: zip_append1)
|
wenzelm@13114
|
1034 |
|
wenzelm@13114
|
1035 |
lemma zip_rev:
|
nipkow@14247
|
1036 |
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
|
nipkow@14247
|
1037 |
by (induct rule:list_induct2, simp_all)
|
wenzelm@13114
|
1038 |
|
wenzelm@13142
|
1039 |
lemma nth_zip [simp]:
|
nipkow@13145
|
1040 |
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
|
paulson@14208
|
1041 |
apply (induct ys, simp)
|
nipkow@13145
|
1042 |
apply (case_tac xs)
|
nipkow@13145
|
1043 |
apply (simp_all add: nth.simps split: nat.split)
|
nipkow@13145
|
1044 |
done
|
wenzelm@13114
|
1045 |
|
wenzelm@13114
|
1046 |
lemma set_zip:
|
nipkow@13145
|
1047 |
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
|
nipkow@13145
|
1048 |
by (simp add: set_conv_nth cong: rev_conj_cong)
|
wenzelm@13114
|
1049 |
|
wenzelm@13114
|
1050 |
lemma zip_update:
|
nipkow@13145
|
1051 |
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
|
nipkow@13145
|
1052 |
by (rule sym, simp add: update_zip)
|
wenzelm@13114
|
1053 |
|
wenzelm@13142
|
1054 |
lemma zip_replicate [simp]:
|
nipkow@13145
|
1055 |
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
|
paulson@14208
|
1056 |
apply (induct i, auto)
|
paulson@14208
|
1057 |
apply (case_tac j, auto)
|
nipkow@13145
|
1058 |
done
|
wenzelm@13114
|
1059 |
|
wenzelm@13142
|
1060 |
|
wenzelm@13142
|
1061 |
subsection {* @{text list_all2} *}
|
wenzelm@13114
|
1062 |
|
kleing@14316
|
1063 |
lemma list_all2_lengthD [intro?]:
|
kleing@14316
|
1064 |
"list_all2 P xs ys ==> length xs = length ys"
|
nipkow@13145
|
1065 |
by (simp add: list_all2_def)
|
wenzelm@13114
|
1066 |
|
wenzelm@13142
|
1067 |
lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])"
|
nipkow@13145
|
1068 |
by (simp add: list_all2_def)
|
wenzelm@13114
|
1069 |
|
wenzelm@13142
|
1070 |
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"
|
nipkow@13145
|
1071 |
by (simp add: list_all2_def)
|
wenzelm@13114
|
1072 |
|
wenzelm@13142
|
1073 |
lemma list_all2_Cons [iff]:
|
nipkow@13145
|
1074 |
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
|
nipkow@13145
|
1075 |
by (auto simp add: list_all2_def)
|
wenzelm@13114
|
1076 |
|
wenzelm@13114
|
1077 |
lemma list_all2_Cons1:
|
nipkow@13145
|
1078 |
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
|
nipkow@13145
|
1079 |
by (cases ys) auto
|
wenzelm@13114
|
1080 |
|
wenzelm@13114
|
1081 |
lemma list_all2_Cons2:
|
nipkow@13145
|
1082 |
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
|
nipkow@13145
|
1083 |
by (cases xs) auto
|
wenzelm@13114
|
1084 |
|
wenzelm@13142
|
1085 |
lemma list_all2_rev [iff]:
|
nipkow@13145
|
1086 |
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
|
nipkow@13145
|
1087 |
by (simp add: list_all2_def zip_rev cong: conj_cong)
|
wenzelm@13114
|
1088 |
|
kleing@13863
|
1089 |
lemma list_all2_rev1:
|
kleing@13863
|
1090 |
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
|
kleing@13863
|
1091 |
by (subst list_all2_rev [symmetric]) simp
|
kleing@13863
|
1092 |
|
wenzelm@13114
|
1093 |
lemma list_all2_append1:
|
nipkow@13145
|
1094 |
"list_all2 P (xs @ ys) zs =
|
nipkow@13145
|
1095 |
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
|
nipkow@13145
|
1096 |
list_all2 P xs us \<and> list_all2 P ys vs)"
|
nipkow@13145
|
1097 |
apply (simp add: list_all2_def zip_append1)
|
nipkow@13145
|
1098 |
apply (rule iffI)
|
nipkow@13145
|
1099 |
apply (rule_tac x = "take (length xs) zs" in exI)
|
nipkow@13145
|
1100 |
apply (rule_tac x = "drop (length xs) zs" in exI)
|
paulson@14208
|
1101 |
apply (force split: nat_diff_split simp add: min_def, clarify)
|
nipkow@13145
|
1102 |
apply (simp add: ball_Un)
|
nipkow@13145
|
1103 |
done
|
wenzelm@13114
|
1104 |
|
wenzelm@13114
|
1105 |
lemma list_all2_append2:
|
nipkow@13145
|
1106 |
"list_all2 P xs (ys @ zs) =
|
nipkow@13145
|
1107 |
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
|
nipkow@13145
|
1108 |
list_all2 P us ys \<and> list_all2 P vs zs)"
|
nipkow@13145
|
1109 |
apply (simp add: list_all2_def zip_append2)
|
nipkow@13145
|
1110 |
apply (rule iffI)
|
nipkow@13145
|
1111 |
apply (rule_tac x = "take (length ys) xs" in exI)
|
nipkow@13145
|
1112 |
apply (rule_tac x = "drop (length ys) xs" in exI)
|
paulson@14208
|
1113 |
apply (force split: nat_diff_split simp add: min_def, clarify)
|
nipkow@13145
|
1114 |
apply (simp add: ball_Un)
|
nipkow@13145
|
1115 |
done
|
wenzelm@13114
|
1116 |
|
kleing@13863
|
1117 |
lemma list_all2_append:
|
nipkow@14247
|
1118 |
"length xs = length ys \<Longrightarrow>
|
nipkow@14247
|
1119 |
list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
|
nipkow@14247
|
1120 |
by (induct rule:list_induct2, simp_all)
|
kleing@13863
|
1121 |
|
kleing@13863
|
1122 |
lemma list_all2_appendI [intro?, trans]:
|
kleing@13863
|
1123 |
"\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
|
kleing@13863
|
1124 |
by (simp add: list_all2_append list_all2_lengthD)
|
kleing@13863
|
1125 |
|
wenzelm@13114
|
1126 |
lemma list_all2_conv_all_nth:
|
nipkow@13145
|
1127 |
"list_all2 P xs ys =
|
nipkow@13145
|
1128 |
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
|
nipkow@13145
|
1129 |
by (force simp add: list_all2_def set_zip)
|
wenzelm@13114
|
1130 |
|
berghofe@13883
|
1131 |
lemma list_all2_trans:
|
berghofe@13883
|
1132 |
assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
|
berghofe@13883
|
1133 |
shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
|
berghofe@13883
|
1134 |
(is "!!bs cs. PROP ?Q as bs cs")
|
berghofe@13883
|
1135 |
proof (induct as)
|
berghofe@13883
|
1136 |
fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
|
berghofe@13883
|
1137 |
show "!!cs. PROP ?Q (x # xs) bs cs"
|
berghofe@13883
|
1138 |
proof (induct bs)
|
berghofe@13883
|
1139 |
fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
|
berghofe@13883
|
1140 |
show "PROP ?Q (x # xs) (y # ys) cs"
|
berghofe@13883
|
1141 |
by (induct cs) (auto intro: tr I1 I2)
|
berghofe@13883
|
1142 |
qed simp
|
berghofe@13883
|
1143 |
qed simp
|
berghofe@13883
|
1144 |
|
kleing@13863
|
1145 |
lemma list_all2_all_nthI [intro?]:
|
kleing@13863
|
1146 |
"length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
|
kleing@13863
|
1147 |
by (simp add: list_all2_conv_all_nth)
|
kleing@13863
|
1148 |
|
paulson@14395
|
1149 |
lemma list_all2I:
|
paulson@14395
|
1150 |
"\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
|
paulson@14395
|
1151 |
by (simp add: list_all2_def)
|
paulson@14395
|
1152 |
|
kleing@14328
|
1153 |
lemma list_all2_nthD:
|
kleing@13863
|
1154 |
"\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
|
kleing@13863
|
1155 |
by (simp add: list_all2_conv_all_nth)
|
kleing@13863
|
1156 |
|
nipkow@14302
|
1157 |
lemma list_all2_nthD2:
|
nipkow@14302
|
1158 |
"\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
|
nipkow@14302
|
1159 |
by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
|
nipkow@14302
|
1160 |
|
kleing@13863
|
1161 |
lemma list_all2_map1:
|
kleing@13863
|
1162 |
"list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
|
kleing@13863
|
1163 |
by (simp add: list_all2_conv_all_nth)
|
kleing@13863
|
1164 |
|
kleing@13863
|
1165 |
lemma list_all2_map2:
|
kleing@13863
|
1166 |
"list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
|
kleing@13863
|
1167 |
by (auto simp add: list_all2_conv_all_nth)
|
kleing@13863
|
1168 |
|
kleing@14316
|
1169 |
lemma list_all2_refl [intro?]:
|
kleing@13863
|
1170 |
"(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
|
kleing@13863
|
1171 |
by (simp add: list_all2_conv_all_nth)
|
kleing@13863
|
1172 |
|
kleing@13863
|
1173 |
lemma list_all2_update_cong:
|
kleing@13863
|
1174 |
"\<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
|
1175 |
by (simp add: list_all2_conv_all_nth nth_list_update)
|
kleing@13863
|
1176 |
|
kleing@13863
|
1177 |
lemma list_all2_update_cong2:
|
kleing@13863
|
1178 |
"\<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
|
1179 |
by (simp add: list_all2_lengthD list_all2_update_cong)
|
kleing@13863
|
1180 |
|
nipkow@14302
|
1181 |
lemma list_all2_takeI [simp,intro?]:
|
nipkow@14302
|
1182 |
"\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
|
nipkow@14302
|
1183 |
apply (induct xs)
|
nipkow@14302
|
1184 |
apply simp
|
nipkow@14302
|
1185 |
apply (clarsimp simp add: list_all2_Cons1)
|
nipkow@14302
|
1186 |
apply (case_tac n)
|
nipkow@14302
|
1187 |
apply auto
|
nipkow@14302
|
1188 |
done
|
nipkow@14302
|
1189 |
|
nipkow@14302
|
1190 |
lemma list_all2_dropI [simp,intro?]:
|
kleing@13863
|
1191 |
"\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
|
paulson@14208
|
1192 |
apply (induct as, simp)
|
kleing@13863
|
1193 |
apply (clarsimp simp add: list_all2_Cons1)
|
paulson@14208
|
1194 |
apply (case_tac n, simp, simp)
|
kleing@13863
|
1195 |
done
|
kleing@13863
|
1196 |
|
kleing@14327
|
1197 |
lemma list_all2_mono [intro?]:
|
kleing@13863
|
1198 |
"\<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
|
1199 |
apply (induct x, simp)
|
paulson@14208
|
1200 |
apply (case_tac y, auto)
|
kleing@13863
|
1201 |
done
|
kleing@13863
|
1202 |
|
wenzelm@13114
|
1203 |
|
nipkow@14402
|
1204 |
subsection {* @{text foldl} and @{text foldr} *}
|
wenzelm@13114
|
1205 |
|
wenzelm@13142
|
1206 |
lemma foldl_append [simp]:
|
nipkow@13145
|
1207 |
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
|
nipkow@13145
|
1208 |
by (induct xs) auto
|
wenzelm@13114
|
1209 |
|
nipkow@14402
|
1210 |
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
|
nipkow@14402
|
1211 |
by (induct xs) auto
|
nipkow@14402
|
1212 |
|
nipkow@14402
|
1213 |
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
|
nipkow@14402
|
1214 |
by (induct xs) auto
|
nipkow@14402
|
1215 |
|
nipkow@14402
|
1216 |
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
|
nipkow@14402
|
1217 |
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
|
nipkow@14402
|
1218 |
|
wenzelm@13142
|
1219 |
text {*
|
nipkow@13145
|
1220 |
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
|
nipkow@13145
|
1221 |
difficult to use because it requires an additional transitivity step.
|
wenzelm@13142
|
1222 |
*}
|
wenzelm@13114
|
1223 |
|
wenzelm@13142
|
1224 |
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
|
nipkow@13145
|
1225 |
by (induct ns) auto
|
wenzelm@13114
|
1226 |
|
wenzelm@13142
|
1227 |
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
|
nipkow@13145
|
1228 |
by (force intro: start_le_sum simp add: in_set_conv_decomp)
|
wenzelm@13114
|
1229 |
|
wenzelm@13142
|
1230 |
lemma sum_eq_0_conv [iff]:
|
nipkow@13145
|
1231 |
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
|
nipkow@13145
|
1232 |
by (induct ns) auto
|
wenzelm@13114
|
1233 |
|
wenzelm@13142
|
1234 |
|
wenzelm@13142
|
1235 |
subsection {* @{text upto} *}
|
wenzelm@13142
|
1236 |
|
wenzelm@13114
|
1237 |
lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])"
|
nipkow@13145
|
1238 |
-- {* Does not terminate! *}
|
nipkow@13145
|
1239 |
by (induct j) auto
|
wenzelm@13114
|
1240 |
|
wenzelm@13142
|
1241 |
lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []"
|
nipkow@13145
|
1242 |
by (subst upt_rec) simp
|
wenzelm@13114
|
1243 |
|
wenzelm@13142
|
1244 |
lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]"
|
nipkow@13145
|
1245 |
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
|
nipkow@13145
|
1246 |
by simp
|
wenzelm@13114
|
1247 |
|
wenzelm@13142
|
1248 |
lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]"
|
nipkow@13145
|
1249 |
apply(rule trans)
|
nipkow@13145
|
1250 |
apply(subst upt_rec)
|
paulson@14208
|
1251 |
prefer 2 apply (rule refl, simp)
|
nipkow@13145
|
1252 |
done
|
wenzelm@13114
|
1253 |
|
wenzelm@13114
|
1254 |
lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]"
|
nipkow@13145
|
1255 |
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
|
nipkow@13145
|
1256 |
by (induct k) auto
|
wenzelm@13114
|
1257 |
|
wenzelm@13142
|
1258 |
lemma length_upt [simp]: "length [i..j(] = j - i"
|
nipkow@13145
|
1259 |
by (induct j) (auto simp add: Suc_diff_le)
|
wenzelm@13114
|
1260 |
|
wenzelm@13142
|
1261 |
lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k"
|
nipkow@13145
|
1262 |
apply (induct j)
|
nipkow@13145
|
1263 |
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
|
nipkow@13145
|
1264 |
done
|
wenzelm@13114
|
1265 |
|
wenzelm@13142
|
1266 |
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]"
|
paulson@14208
|
1267 |
apply (induct m, simp)
|
nipkow@13145
|
1268 |
apply (subst upt_rec)
|
nipkow@13145
|
1269 |
apply (rule sym)
|
nipkow@13145
|
1270 |
apply (subst upt_rec)
|
nipkow@13145
|
1271 |
apply (simp del: upt.simps)
|
nipkow@13145
|
1272 |
done
|
wenzelm@13114
|
1273 |
|
wenzelm@13114
|
1274 |
lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]"
|
nipkow@13145
|
1275 |
by (induct n) auto
|
wenzelm@13114
|
1276 |
|
wenzelm@13114
|
1277 |
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..n(]) ! i = f(m+i)"
|
nipkow@13145
|
1278 |
apply (induct n m rule: diff_induct)
|
nipkow@13145
|
1279 |
prefer 3 apply (subst map_Suc_upt[symmetric])
|
nipkow@13145
|
1280 |
apply (auto simp add: less_diff_conv nth_upt)
|
nipkow@13145
|
1281 |
done
|
wenzelm@13114
|
1282 |
|
berghofe@13883
|
1283 |
lemma nth_take_lemma:
|
berghofe@13883
|
1284 |
"!!xs ys. k <= length xs ==> k <= length ys ==>
|
berghofe@13883
|
1285 |
(!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
|
berghofe@13883
|
1286 |
apply (atomize, induct k)
|
paulson@14208
|
1287 |
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
|
nipkow@13145
|
1288 |
txt {* Both lists must be non-empty *}
|
paulson@14208
|
1289 |
apply (case_tac xs, simp)
|
paulson@14208
|
1290 |
apply (case_tac ys, clarify)
|
nipkow@13145
|
1291 |
apply (simp (no_asm_use))
|
nipkow@13145
|
1292 |
apply clarify
|
nipkow@13145
|
1293 |
txt {* prenexing's needed, not miniscoping *}
|
nipkow@13145
|
1294 |
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
|
nipkow@13145
|
1295 |
apply blast
|
nipkow@13145
|
1296 |
done
|
wenzelm@13114
|
1297 |
|
wenzelm@13114
|
1298 |
lemma nth_equalityI:
|
wenzelm@13114
|
1299 |
"[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
|
nipkow@13145
|
1300 |
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
|
nipkow@13145
|
1301 |
apply (simp_all add: take_all)
|
nipkow@13145
|
1302 |
done
|
wenzelm@13114
|
1303 |
|
kleing@13863
|
1304 |
(* needs nth_equalityI *)
|
kleing@13863
|
1305 |
lemma list_all2_antisym:
|
kleing@13863
|
1306 |
"\<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
|
1307 |
\<Longrightarrow> xs = ys"
|
kleing@13863
|
1308 |
apply (simp add: list_all2_conv_all_nth)
|
paulson@14208
|
1309 |
apply (rule nth_equalityI, blast, simp)
|
kleing@13863
|
1310 |
done
|
kleing@13863
|
1311 |
|
wenzelm@13142
|
1312 |
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
|
nipkow@13145
|
1313 |
-- {* The famous take-lemma. *}
|
nipkow@13145
|
1314 |
apply (drule_tac x = "max (length xs) (length ys)" in spec)
|
nipkow@13145
|
1315 |
apply (simp add: le_max_iff_disj take_all)
|
nipkow@13145
|
1316 |
done
|
wenzelm@13114
|
1317 |
|
wenzelm@13114
|
1318 |
|
wenzelm@13142
|
1319 |
subsection {* @{text "distinct"} and @{text remdups} *}
|
wenzelm@13114
|
1320 |
|
wenzelm@13142
|
1321 |
lemma distinct_append [simp]:
|
nipkow@13145
|
1322 |
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
|
nipkow@13145
|
1323 |
by (induct xs) auto
|
wenzelm@13114
|
1324 |
|
wenzelm@13142
|
1325 |
lemma set_remdups [simp]: "set (remdups xs) = set xs"
|
nipkow@13145
|
1326 |
by (induct xs) (auto simp add: insert_absorb)
|
wenzelm@13114
|
1327 |
|
wenzelm@13142
|
1328 |
lemma distinct_remdups [iff]: "distinct (remdups xs)"
|
nipkow@13145
|
1329 |
by (induct xs) auto
|
wenzelm@13114
|
1330 |
|
wenzelm@13142
|
1331 |
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
|
nipkow@13145
|
1332 |
by (induct xs) auto
|
wenzelm@13114
|
1333 |
|
wenzelm@13142
|
1334 |
text {*
|
nipkow@13145
|
1335 |
It is best to avoid this indexed version of distinct, but sometimes
|
nipkow@13145
|
1336 |
it is useful. *}
|
nipkow@13124
|
1337 |
lemma distinct_conv_nth:
|
nipkow@13145
|
1338 |
"distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)"
|
paulson@14208
|
1339 |
apply (induct_tac xs, simp, simp)
|
paulson@14208
|
1340 |
apply (rule iffI, clarsimp)
|
nipkow@13145
|
1341 |
apply (case_tac i)
|
paulson@14208
|
1342 |
apply (case_tac j, simp)
|
nipkow@13145
|
1343 |
apply (simp add: set_conv_nth)
|
nipkow@13145
|
1344 |
apply (case_tac j)
|
paulson@14208
|
1345 |
apply (clarsimp simp add: set_conv_nth, simp)
|
nipkow@13145
|
1346 |
apply (rule conjI)
|
nipkow@13145
|
1347 |
apply (clarsimp simp add: set_conv_nth)
|
nipkow@13145
|
1348 |
apply (erule_tac x = 0 in allE)
|
paulson@14208
|
1349 |
apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
|
nipkow@13145
|
1350 |
apply (erule_tac x = "Suc i" in allE)
|
paulson@14208
|
1351 |
apply (erule_tac x = "Suc j" in allE, simp)
|
nipkow@13145
|
1352 |
done
|
nipkow@13124
|
1353 |
|
kleing@14388
|
1354 |
lemma distinct_card: "distinct xs \<Longrightarrow> card (set xs) = size xs"
|
kleing@14388
|
1355 |
by (induct xs) auto
|
kleing@14388
|
1356 |
|
kleing@14388
|
1357 |
lemma card_distinct: "card (set xs) = size xs \<Longrightarrow> distinct xs"
|
kleing@14388
|
1358 |
proof (induct xs)
|
kleing@14388
|
1359 |
case Nil thus ?case by simp
|
kleing@14388
|
1360 |
next
|
kleing@14388
|
1361 |
case (Cons x xs)
|
kleing@14388
|
1362 |
show ?case
|
kleing@14388
|
1363 |
proof (cases "x \<in> set xs")
|
kleing@14388
|
1364 |
case False with Cons show ?thesis by simp
|
kleing@14388
|
1365 |
next
|
kleing@14388
|
1366 |
case True with Cons.prems
|
kleing@14388
|
1367 |
have "card (set xs) = Suc (length xs)"
|
kleing@14388
|
1368 |
by (simp add: card_insert_if split: split_if_asm)
|
kleing@14388
|
1369 |
moreover have "card (set xs) \<le> length xs" by (rule card_length)
|
kleing@14388
|
1370 |
ultimately have False by simp
|
kleing@14388
|
1371 |
thus ?thesis ..
|
kleing@14388
|
1372 |
qed
|
kleing@14388
|
1373 |
qed
|
kleing@14388
|
1374 |
|
wenzelm@13114
|
1375 |
|
wenzelm@13142
|
1376 |
subsection {* @{text replicate} *}
|
wenzelm@13114
|
1377 |
|
wenzelm@13142
|
1378 |
lemma length_replicate [simp]: "length (replicate n x) = n"
|
nipkow@13145
|
1379 |
by (induct n) auto
|
wenzelm@13142
|
1380 |
|
wenzelm@13142
|
1381 |
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
|
nipkow@13145
|
1382 |
by (induct n) auto
|
wenzelm@13114
|
1383 |
|
wenzelm@13114
|
1384 |
lemma replicate_app_Cons_same:
|
nipkow@13145
|
1385 |
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
|
nipkow@13145
|
1386 |
by (induct n) auto
|
wenzelm@13114
|
1387 |
|
wenzelm@13142
|
1388 |
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
|
paulson@14208
|
1389 |
apply (induct n, simp)
|
nipkow@13145
|
1390 |
apply (simp add: replicate_app_Cons_same)
|
nipkow@13145
|
1391 |
done
|
wenzelm@13114
|
1392 |
|
wenzelm@13142
|
1393 |
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
|
nipkow@13145
|
1394 |
by (induct n) auto
|
wenzelm@13114
|
1395 |
|
wenzelm@13142
|
1396 |
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
|
nipkow@13145
|
1397 |
by (induct n) auto
|
wenzelm@13114
|
1398 |
|
wenzelm@13142
|
1399 |
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
|
nipkow@13145
|
1400 |
by (induct n) auto
|
wenzelm@13114
|
1401 |
|
wenzelm@13142
|
1402 |
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
|
nipkow@13145
|
1403 |
by (atomize (full), induct n) auto
|
wenzelm@13114
|
1404 |
|
wenzelm@13142
|
1405 |
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
|
paulson@14208
|
1406 |
apply (induct n, simp)
|
nipkow@13145
|
1407 |
apply (simp add: nth_Cons split: nat.split)
|
nipkow@13145
|
1408 |
done
|
wenzelm@13114
|
1409 |
|
wenzelm@13142
|
1410 |
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
|
nipkow@13145
|
1411 |
by (induct n) auto
|
wenzelm@13114
|
1412 |
|
wenzelm@13142
|
1413 |
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
|
nipkow@13145
|
1414 |
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
|
wenzelm@13114
|
1415 |
|
wenzelm@13142
|
1416 |
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
|
nipkow@13145
|
1417 |
by auto
|
wenzelm@13114
|
1418 |
|
wenzelm@13142
|
1419 |
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
|
nipkow@13145
|
1420 |
by (simp add: set_replicate_conv_if split: split_if_asm)
|
wenzelm@13114
|
1421 |
|
wenzelm@13114
|
1422 |
|
oheimb@14099
|
1423 |
subsection {* Lexicographic orderings on lists *}
|
wenzelm@13114
|
1424 |
|
wenzelm@13142
|
1425 |
lemma wf_lexn: "wf r ==> wf (lexn r n)"
|
paulson@14208
|
1426 |
apply (induct_tac n, simp, simp)
|
nipkow@13145
|
1427 |
apply(rule wf_subset)
|
nipkow@13145
|
1428 |
prefer 2 apply (rule Int_lower1)
|
nipkow@13145
|
1429 |
apply(rule wf_prod_fun_image)
|
paulson@14208
|
1430 |
prefer 2 apply (rule inj_onI, auto)
|
nipkow@13145
|
1431 |
done
|
wenzelm@13114
|
1432 |
|
wenzelm@13114
|
1433 |
lemma lexn_length:
|
nipkow@13145
|
1434 |
"!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
|
nipkow@13145
|
1435 |
by (induct n) auto
|
wenzelm@13114
|
1436 |
|
wenzelm@13142
|
1437 |
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
|
nipkow@13145
|
1438 |
apply (unfold lex_def)
|
nipkow@13145
|
1439 |
apply (rule wf_UN)
|
paulson@14208
|
1440 |
apply (blast intro: wf_lexn, clarify)
|
nipkow@13145
|
1441 |
apply (rename_tac m n)
|
nipkow@13145
|
1442 |
apply (subgoal_tac "m \<noteq> n")
|
nipkow@13145
|
1443 |
prefer 2 apply blast
|
nipkow@13145
|
1444 |
apply (blast dest: lexn_length not_sym)
|
nipkow@13145
|
1445 |
done
|
wenzelm@13114
|
1446 |
|
wenzelm@13114
|
1447 |
lemma lexn_conv:
|
nipkow@13145
|
1448 |
"lexn r n =
|
nipkow@13145
|
1449 |
{(xs,ys). length xs = n \<and> length ys = n \<and>
|
nipkow@13145
|
1450 |
(\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
|
paulson@14208
|
1451 |
apply (induct_tac n, simp, blast)
|
paulson@14208
|
1452 |
apply (simp add: image_Collect lex_prod_def, safe, blast)
|
paulson@14208
|
1453 |
apply (rule_tac x = "ab # xys" in exI, simp)
|
paulson@14208
|
1454 |
apply (case_tac xys, simp_all, blast)
|
nipkow@13145
|
1455 |
done
|
wenzelm@13114
|
1456 |
|
wenzelm@13114
|
1457 |
lemma lex_conv:
|
nipkow@13145
|
1458 |
"lex r =
|
nipkow@13145
|
1459 |
{(xs,ys). length xs = length ys \<and>
|
nipkow@13145
|
1460 |
(\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
|
nipkow@13145
|
1461 |
by (force simp add: lex_def lexn_conv)
|
wenzelm@13114
|
1462 |
|
wenzelm@13142
|
1463 |
lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)"
|
nipkow@13145
|
1464 |
by (unfold lexico_def) blast
|
wenzelm@13114
|
1465 |
|
wenzelm@13114
|
1466 |
lemma lexico_conv:
|
nipkow@13145
|
1467 |
"lexico r = {(xs,ys). length xs < length ys |
|
nipkow@13145
|
1468 |
length xs = length ys \<and> (xs, ys) : lex r}"
|
nipkow@13145
|
1469 |
by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def)
|
wenzelm@13114
|
1470 |
|
wenzelm@13142
|
1471 |
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
|
nipkow@13145
|
1472 |
by (simp add: lex_conv)
|
wenzelm@13114
|
1473 |
|
wenzelm@13142
|
1474 |
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
|
nipkow@13145
|
1475 |
by (simp add:lex_conv)
|
wenzelm@13114
|
1476 |
|
wenzelm@13142
|
1477 |
lemma Cons_in_lex [iff]:
|
nipkow@13145
|
1478 |
"((x # xs, y # ys) : lex r) =
|
nipkow@13145
|
1479 |
((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
|
nipkow@13145
|
1480 |
apply (simp add: lex_conv)
|
nipkow@13145
|
1481 |
apply (rule iffI)
|
paulson@14208
|
1482 |
prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
|
paulson@14208
|
1483 |
apply (case_tac xys, simp, simp)
|
nipkow@13145
|
1484 |
apply blast
|
nipkow@13145
|
1485 |
done
|
wenzelm@13114
|
1486 |
|
wenzelm@13114
|
1487 |
|
wenzelm@13142
|
1488 |
subsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
|
wenzelm@13114
|
1489 |
|
wenzelm@13142
|
1490 |
lemma sublist_empty [simp]: "sublist xs {} = []"
|
nipkow@13145
|
1491 |
by (auto simp add: sublist_def)
|
wenzelm@13114
|
1492 |
|
wenzelm@13142
|
1493 |
lemma sublist_nil [simp]: "sublist [] A = []"
|
nipkow@13145
|
1494 |
by (auto simp add: sublist_def)
|
wenzelm@13114
|
1495 |
|
wenzelm@13114
|
1496 |
lemma sublist_shift_lemma:
|
nipkow@13145
|
1497 |
"map fst [p:zip xs [i..i + length xs(] . snd p : A] =
|
nipkow@13145
|
1498 |
map fst [p:zip xs [0..length xs(] . snd p + i : A]"
|
nipkow@13145
|
1499 |
by (induct xs rule: rev_induct) (simp_all add: add_commute)
|
wenzelm@13114
|
1500 |
|
wenzelm@13114
|
1501 |
lemma sublist_append:
|
nipkow@13145
|
1502 |
"sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
|
nipkow@13145
|
1503 |
apply (unfold sublist_def)
|
paulson@14208
|
1504 |
apply (induct l' rule: rev_induct, simp)
|
nipkow@13145
|
1505 |
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
|
nipkow@13145
|
1506 |
apply (simp add: add_commute)
|
nipkow@13145
|
1507 |
done
|
wenzelm@13114
|
1508 |
|
wenzelm@13114
|
1509 |
lemma sublist_Cons:
|
nipkow@13145
|
1510 |
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
|
nipkow@13145
|
1511 |
apply (induct l rule: rev_induct)
|
nipkow@13145
|
1512 |
apply (simp add: sublist_def)
|
nipkow@13145
|
1513 |
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
|
nipkow@13145
|
1514 |
done
|
wenzelm@13114
|
1515 |
|
wenzelm@13142
|
1516 |
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
|
nipkow@13145
|
1517 |
by (simp add: sublist_Cons)
|
wenzelm@13114
|
1518 |
|
wenzelm@13142
|
1519 |
lemma sublist_upt_eq_take [simp]: "sublist l {..n(} = take n l"
|
paulson@14208
|
1520 |
apply (induct l rule: rev_induct, simp)
|
nipkow@13145
|
1521 |
apply (simp split: nat_diff_split add: sublist_append)
|
nipkow@13145
|
1522 |
done
|
wenzelm@13114
|
1523 |
|
wenzelm@13114
|
1524 |
|
wenzelm@13142
|
1525 |
lemma take_Cons':
|
nipkow@13145
|
1526 |
"take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
|
nipkow@13145
|
1527 |
by (cases n) simp_all
|
wenzelm@13114
|
1528 |
|
wenzelm@13142
|
1529 |
lemma drop_Cons':
|
nipkow@13145
|
1530 |
"drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
|
nipkow@13145
|
1531 |
by (cases n) simp_all
|
wenzelm@13114
|
1532 |
|
wenzelm@13142
|
1533 |
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
|
nipkow@13145
|
1534 |
by (cases n) simp_all
|
wenzelm@13114
|
1535 |
|
nipkow@13145
|
1536 |
lemmas [simp] = take_Cons'[of "number_of v",standard]
|
nipkow@13145
|
1537 |
drop_Cons'[of "number_of v",standard]
|
nipkow@13145
|
1538 |
nth_Cons'[of _ _ "number_of v",standard]
|
nipkow@3507
|
1539 |
|
wenzelm@13462
|
1540 |
|
kleing@14388
|
1541 |
lemma distinct_card: "distinct xs \<Longrightarrow> card (set xs) = size xs"
|
kleing@14388
|
1542 |
by (induct xs) auto
|
kleing@14388
|
1543 |
|
kleing@14388
|
1544 |
lemma card_length: "card (set xs) \<le> length xs"
|
kleing@14388
|
1545 |
by (induct xs) (auto simp add: card_insert_if)
|
kleing@14388
|
1546 |
|
kleing@14388
|
1547 |
lemma "card (set xs) = size xs \<Longrightarrow> distinct xs"
|
kleing@14388
|
1548 |
proof (induct xs)
|
kleing@14388
|
1549 |
case Nil thus ?case by simp
|
kleing@14388
|
1550 |
next
|
kleing@14388
|
1551 |
case (Cons x xs)
|
kleing@14388
|
1552 |
show ?case
|
kleing@14388
|
1553 |
proof (cases "x \<in> set xs")
|
kleing@14388
|
1554 |
case False with Cons show ?thesis by simp
|
kleing@14388
|
1555 |
next
|
kleing@14388
|
1556 |
case True with Cons.prems
|
kleing@14388
|
1557 |
have "card (set xs) = Suc (length xs)"
|
kleing@14388
|
1558 |
by (simp add: card_insert_if split: split_if_asm)
|
kleing@14388
|
1559 |
moreover have "card (set xs) \<le> length xs" by (rule card_length)
|
kleing@14388
|
1560 |
ultimately have False by simp
|
kleing@14388
|
1561 |
thus ?thesis ..
|
kleing@14388
|
1562 |
qed
|
kleing@14388
|
1563 |
qed
|
kleing@14388
|
1564 |
|
wenzelm@13366
|
1565 |
subsection {* Characters and strings *}
|
wenzelm@13366
|
1566 |
|
wenzelm@13366
|
1567 |
datatype nibble =
|
wenzelm@13366
|
1568 |
Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
|
wenzelm@13366
|
1569 |
| Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
|
wenzelm@13366
|
1570 |
|
wenzelm@13366
|
1571 |
datatype char = Char nibble nibble
|
wenzelm@13366
|
1572 |
-- "Note: canonical order of character encoding coincides with standard term ordering"
|
wenzelm@13366
|
1573 |
|
wenzelm@13366
|
1574 |
types string = "char list"
|
wenzelm@13366
|
1575 |
|
wenzelm@13366
|
1576 |
syntax
|
wenzelm@13366
|
1577 |
"_Char" :: "xstr => char" ("CHR _")
|
wenzelm@13366
|
1578 |
"_String" :: "xstr => string" ("_")
|
wenzelm@13366
|
1579 |
|
wenzelm@13366
|
1580 |
parse_ast_translation {*
|
wenzelm@13366
|
1581 |
let
|
wenzelm@13366
|
1582 |
val constants = Syntax.Appl o map Syntax.Constant;
|
wenzelm@13366
|
1583 |
|
wenzelm@13366
|
1584 |
fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10));
|
wenzelm@13366
|
1585 |
fun mk_char c =
|
wenzelm@13366
|
1586 |
if Symbol.is_ascii c andalso Symbol.is_printable c then
|
wenzelm@13366
|
1587 |
constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)]
|
wenzelm@13366
|
1588 |
else error ("Printable ASCII character expected: " ^ quote c);
|
wenzelm@13366
|
1589 |
|
wenzelm@13366
|
1590 |
fun mk_string [] = Syntax.Constant "Nil"
|
wenzelm@13366
|
1591 |
| mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs];
|
wenzelm@13366
|
1592 |
|
wenzelm@13366
|
1593 |
fun char_ast_tr [Syntax.Variable xstr] =
|
wenzelm@13366
|
1594 |
(case Syntax.explode_xstr xstr of
|
wenzelm@13366
|
1595 |
[c] => mk_char c
|
wenzelm@13366
|
1596 |
| _ => error ("Single character expected: " ^ xstr))
|
wenzelm@13366
|
1597 |
| char_ast_tr asts = raise AST ("char_ast_tr", asts);
|
wenzelm@13366
|
1598 |
|
wenzelm@13366
|
1599 |
fun string_ast_tr [Syntax.Variable xstr] =
|
wenzelm@13366
|
1600 |
(case Syntax.explode_xstr xstr of
|
wenzelm@13366
|
1601 |
[] => constants [Syntax.constrainC, "Nil", "string"]
|
wenzelm@13366
|
1602 |
| cs => mk_string cs)
|
wenzelm@13366
|
1603 |
| string_ast_tr asts = raise AST ("string_tr", asts);
|
wenzelm@13366
|
1604 |
in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end;
|
wenzelm@13366
|
1605 |
*}
|
wenzelm@13366
|
1606 |
|
wenzelm@13366
|
1607 |
print_ast_translation {*
|
wenzelm@13366
|
1608 |
let
|
wenzelm@13366
|
1609 |
fun dest_nib (Syntax.Constant c) =
|
wenzelm@13366
|
1610 |
(case explode c of
|
wenzelm@13366
|
1611 |
["N", "i", "b", "b", "l", "e", h] =>
|
wenzelm@13366
|
1612 |
if "0" <= h andalso h <= "9" then ord h - ord "0"
|
wenzelm@13366
|
1613 |
else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10
|
wenzelm@13366
|
1614 |
else raise Match
|
wenzelm@13366
|
1615 |
| _ => raise Match)
|
wenzelm@13366
|
1616 |
| dest_nib _ = raise Match;
|
wenzelm@13366
|
1617 |
|
wenzelm@13366
|
1618 |
fun dest_chr c1 c2 =
|
wenzelm@13366
|
1619 |
let val c = chr (dest_nib c1 * 16 + dest_nib c2)
|
wenzelm@13366
|
1620 |
in if Symbol.is_printable c then c else raise Match end;
|
wenzelm@13366
|
1621 |
|
wenzelm@13366
|
1622 |
fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2
|
wenzelm@13366
|
1623 |
| dest_char _ = raise Match;
|
wenzelm@13366
|
1624 |
|
wenzelm@13366
|
1625 |
fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)];
|
wenzelm@13366
|
1626 |
|
wenzelm@13366
|
1627 |
fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]]
|
wenzelm@13366
|
1628 |
| char_ast_tr' _ = raise Match;
|
wenzelm@13366
|
1629 |
|
wenzelm@13366
|
1630 |
fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String",
|
wenzelm@13366
|
1631 |
xstr (map dest_char (Syntax.unfold_ast "_args" args))]
|
wenzelm@13366
|
1632 |
| list_ast_tr' ts = raise Match;
|
wenzelm@13366
|
1633 |
in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end;
|
wenzelm@13366
|
1634 |
*}
|
wenzelm@13366
|
1635 |
|
wenzelm@13122
|
1636 |
end
|