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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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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nipkow@15131
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theory List
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haftmann@25591
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imports ATP_Linkup
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wenzelm@21754
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uses "Tools/string_syntax.ML"
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nipkow@15131
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begin
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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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wenzelm@13366
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| Cons 'a "'a list" (infixr "#" 65)
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clasohm@923
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nipkow@15392
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subsection{*Basic list processing functions*}
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nipkow@15302
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clasohm@923
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consts
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wenzelm@13366
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filter:: "('a => bool) => 'a list => 'a list"
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wenzelm@13366
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concat:: "'a list list => 'a list"
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wenzelm@13366
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foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"
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wenzelm@13366
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foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"
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wenzelm@13366
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hd:: "'a list => 'a"
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wenzelm@13366
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tl:: "'a list => 'a list"
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wenzelm@13366
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last:: "'a list => 'a"
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wenzelm@13366
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butlast :: "'a list => 'a list"
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wenzelm@13366
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set :: "'a list => 'a set"
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wenzelm@13366
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map :: "('a=>'b) => ('a list => 'b list)"
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nipkow@23096
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listsum :: "'a list => 'a::monoid_add"
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wenzelm@13366
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nth :: "'a list => nat => 'a" (infixl "!" 100)
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wenzelm@13366
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list_update :: "'a list => nat => 'a => 'a list"
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wenzelm@13366
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take:: "nat => 'a list => 'a list"
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wenzelm@13366
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drop:: "nat => 'a list => 'a list"
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wenzelm@13366
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takeWhile :: "('a => bool) => 'a list => 'a list"
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wenzelm@13366
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dropWhile :: "('a => bool) => 'a list => 'a list"
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wenzelm@13366
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rev :: "'a list => 'a list"
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wenzelm@13366
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zip :: "'a list => 'b list => ('a * 'b) list"
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nipkow@15425
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upt :: "nat => nat => nat list" ("(1[_..</_'])")
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wenzelm@13366
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remdups :: "'a list => 'a list"
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nipkow@15110
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remove1 :: "'a => 'a list => 'a list"
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wenzelm@13366
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"distinct":: "'a list => bool"
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wenzelm@13366
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replicate :: "nat => 'a => 'a list"
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nipkow@19390
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splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
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nipkow@15302
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clasohm@923
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nipkow@13146
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nonterminals lupdbinds lupdbind
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nipkow@5077
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clasohm@923
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syntax
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wenzelm@13366
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-- {* list Enumeration *}
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wenzelm@13366
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"@list" :: "args => 'a list" ("[(_)]")
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clasohm@923
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wenzelm@13366
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-- {* Special syntax for filter *}
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nipkow@23279
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"@filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_<-_./ _])")
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clasohm@923
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wenzelm@13366
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-- {* list update *}
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wenzelm@13366
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"_lupdbind":: "['a, 'a] => lupdbind" ("(2_ :=/ _)")
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wenzelm@13366
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"" :: "lupdbind => lupdbinds" ("_")
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wenzelm@13366
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"_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds" ("_,/ _")
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wenzelm@13366
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"_LUpdate" :: "['a, lupdbinds] => 'a" ("_/[(_)]" [900,0] 900)
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nipkow@5077
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clasohm@923
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translations
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wenzelm@13366
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"[x, xs]" == "x#[xs]"
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wenzelm@13366
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"[x]" == "x#[]"
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nipkow@23279
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"[x<-xs . P]"== "filter (%x. P) xs"
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clasohm@923
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wenzelm@13366
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"_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"
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wenzelm@13366
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"xs[i:=x]" == "list_update xs i x"
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nipkow@5077
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nipkow@5427
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wenzelm@12114
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syntax (xsymbols)
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nipkow@23279
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"@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
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kleing@14565
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syntax (HTML output)
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nipkow@23279
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"@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
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wenzelm@2262
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wenzelm@2262
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wenzelm@13142
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text {*
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wenzelm@14589
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Function @{text size} is overloaded for all datatypes. Users may
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wenzelm@13366
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refer to the list version as @{text length}. *}
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paulson@3342
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wenzelm@19363
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abbreviation
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wenzelm@21404
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length :: "'a list => nat" where
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wenzelm@19363
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"length == size"
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nipkow@15302
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berghofe@5183
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primrec
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paulson@15307
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"hd(x#xs) = x"
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paulson@15307
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berghofe@5183
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primrec
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paulson@15307
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"tl([]) = []"
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paulson@15307
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"tl(x#xs) = xs"
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paulson@15307
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berghofe@5183
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primrec
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paulson@15307
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"last(x#xs) = (if xs=[] then x else last xs)"
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paulson@15307
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berghofe@5183
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primrec
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paulson@15307
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"butlast []= []"
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paulson@15307
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"butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
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paulson@15307
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berghofe@5183
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primrec
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paulson@15307
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"set [] = {}"
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paulson@15307
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"set (x#xs) = insert x (set xs)"
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paulson@15307
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berghofe@5183
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primrec
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paulson@15307
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"map f [] = []"
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paulson@15307
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"map f (x#xs) = f(x)#map f xs"
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paulson@15307
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wenzelm@25221
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primrec
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haftmann@25559
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append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65)
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haftmann@25559
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where
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haftmann@25559
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append_Nil:"[] @ ys = ys"
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haftmann@25559
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| append_Cons: "(x#xs) @ ys = x # xs @ ys"
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paulson@15307
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berghofe@5183
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primrec
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paulson@15307
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"rev([]) = []"
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paulson@15307
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"rev(x#xs) = rev(xs) @ [x]"
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paulson@15307
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berghofe@5183
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primrec
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paulson@15307
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"filter P [] = []"
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paulson@15307
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"filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
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paulson@15307
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berghofe@5183
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primrec
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paulson@15307
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foldl_Nil:"foldl f a [] = a"
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paulson@15307
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foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
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paulson@15307
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berghofe@5183
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primrec
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paulson@15307
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"foldr f [] a = a"
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paulson@15307
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"foldr f (x#xs) a = f x (foldr f xs a)"
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paulson@15307
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paulson@8000
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primrec
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paulson@15307
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"concat([]) = []"
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paulson@15307
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"concat(x#xs) = x @ concat(xs)"
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paulson@15307
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berghofe@5183
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primrec
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nipkow@23096
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"listsum [] = 0"
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nipkow@23096
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"listsum (x # xs) = x + listsum xs"
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nipkow@23096
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nipkow@23096
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primrec
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paulson@15307
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drop_Nil:"drop n [] = []"
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paulson@15307
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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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paulson@15307
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-- {*Warning: simpset does not contain this definition, but separate
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paulson@15307
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theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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paulson@15307
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berghofe@5183
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primrec
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paulson@15307
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take_Nil:"take n [] = []"
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paulson@15307
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take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
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paulson@15307
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-- {*Warning: simpset does not contain this definition, but separate
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paulson@15307
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theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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paulson@15307
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berghofe@5183
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primrec
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paulson@15307
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nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
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paulson@15307
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-- {*Warning: simpset does not contain this definition, but separate
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paulson@15307
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theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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paulson@15307
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wenzelm@13142
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primrec
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paulson@15307
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"[][i:=v] = []"
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paulson@15307
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"(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])"
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paulson@15307
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berghofe@5183
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primrec
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paulson@15307
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"takeWhile P [] = []"
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paulson@15307
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"takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
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paulson@15307
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berghofe@5183
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primrec
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paulson@15307
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"dropWhile P [] = []"
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paulson@15307
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"dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
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paulson@15307
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berghofe@5183
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primrec
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paulson@15307
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"zip xs [] = []"
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paulson@15307
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zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
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paulson@15307
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-- {*Warning: simpset does not contain this definition, but separate
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paulson@15307
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theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
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paulson@15307
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nipkow@5427
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primrec
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nipkow@15425
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upt_0: "[i..<0] = []"
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nipkow@15425
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upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
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paulson@15307
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berghofe@5183
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primrec
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paulson@15307
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"distinct [] = True"
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paulson@15307
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"distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
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paulson@15307
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berghofe@5183
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primrec
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paulson@15307
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"remdups [] = []"
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paulson@15307
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"remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
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paulson@15307
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berghofe@5183
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primrec
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paulson@15307
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"remove1 x [] = []"
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paulson@15307
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"remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)"
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paulson@15307
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nipkow@15110
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primrec
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paulson@15307
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replicate_0: "replicate 0 x = []"
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paulson@15307
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replicate_Suc: "replicate (Suc n) x = x # replicate n x"
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paulson@15307
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haftmann@21061
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definition
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wenzelm@21404
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rotate1 :: "'a list \<Rightarrow> 'a list" where
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wenzelm@21404
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"rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"
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wenzelm@21404
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wenzelm@21404
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definition
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wenzelm@21404
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rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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wenzelm@21404
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"rotate n = rotate1 ^ n"
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wenzelm@21404
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wenzelm@21404
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definition
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wenzelm@21404
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list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
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haftmann@25966
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[code func del]: "list_all2 P xs ys =
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haftmann@21061
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(length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"
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wenzelm@21404
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wenzelm@21404
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definition
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wenzelm@21404
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sublist :: "'a list => nat set => 'a list" where
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wenzelm@21404
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"sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"
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nipkow@5281
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nipkow@17086
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primrec
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haftmann@21061
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"splice [] ys = ys"
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haftmann@21061
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"splice (x#xs) ys = (if ys=[] then x#xs else x # hd ys # splice xs (tl ys))"
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haftmann@21061
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-- {*Warning: simpset does not contain the second eqn but a derived one. *}
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haftmann@21061
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nipkow@24616
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text{* The following simple sort functions are intended for proofs,
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nipkow@24616
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not for efficient implementations. *}
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nipkow@24616
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wenzelm@25221
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context linorder
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wenzelm@25221
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begin
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wenzelm@25221
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wenzelm@25221
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fun sorted :: "'a list \<Rightarrow> bool" where
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nipkow@24697
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"sorted [] \<longleftrightarrow> True" |
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nipkow@24697
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"sorted [x] \<longleftrightarrow> True" |
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haftmann@25062
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"sorted (x#y#zs) \<longleftrightarrow> x <= y \<and> sorted (y#zs)"
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nipkow@24697
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haftmann@25559
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primrec insort :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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nipkow@24697
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"insort x [] = [x]" |
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haftmann@25062
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"insort x (y#ys) = (if x <= y then (x#y#ys) else y#(insort x ys))"
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nipkow@24697
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haftmann@25559
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primrec sort :: "'a list \<Rightarrow> 'a list" where
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nipkow@24697
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"sort [] = []" |
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nipkow@24697
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"sort (x#xs) = insort x (sort xs)"
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nipkow@24616
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wenzelm@25221
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end
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wenzelm@25221
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nipkow@24616
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wenzelm@23388
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subsubsection {* List comprehension *}
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nipkow@23192
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nipkow@24349
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text{* Input syntax for Haskell-like list comprehension notation.
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nipkow@24349
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Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
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nipkow@24349
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the list of all pairs of distinct elements from @{text xs} and @{text ys}.
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nipkow@24349
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The syntax is as in Haskell, except that @{text"|"} becomes a dot
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nipkow@24349
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(like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
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nipkow@24349
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\verb![e| x <- xs, ...]!.
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nipkow@24349
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nipkow@24349
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The qualifiers after the dot are
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nipkow@24349
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\begin{description}
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nipkow@24349
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\item[generators] @{text"p \<leftarrow> xs"},
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nipkow@24476
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where @{text p} is a pattern and @{text xs} an expression of list type, or
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nipkow@24476
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\item[guards] @{text"b"}, where @{text b} is a boolean expression.
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nipkow@24476
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%\item[local bindings] @ {text"let x = e"}.
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nipkow@24349
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\end{description}
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nipkow@23240
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nipkow@24476
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Just like in Haskell, list comprehension is just a shorthand. To avoid
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nipkow@24476
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misunderstandings, the translation into desugared form is not reversed
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nipkow@24476
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upon output. Note that the translation of @{text"[e. x \<leftarrow> xs]"} is
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nipkow@24476
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optmized to @{term"map (%x. e) xs"}.
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nipkow@23240
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nipkow@24349
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It is easy to write short list comprehensions which stand for complex
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nipkow@24349
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expressions. During proofs, they may become unreadable (and
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nipkow@24349
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mangled). In such cases it can be advisable to introduce separate
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nipkow@24349
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definitions for the list comprehensions in question. *}
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nipkow@24349
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nipkow@23209
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(*
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nipkow@23240
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Proper theorem proving support would be nice. For example, if
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nipkow@23192
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@{text"set[f x y. x \<leftarrow> xs, y \<leftarrow> ys, P x y]"}
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nipkow@23192
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produced something like
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nipkow@23209
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@{term"{z. EX x: set xs. EX y:set ys. P x y \<and> z = f x y}"}.
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nipkow@23209
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*)
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nipkow@23209
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nipkow@23240
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nonterminals lc_qual lc_quals
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nipkow@23192
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nipkow@23192
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syntax
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nipkow@23240
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"_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list" ("[_ . __")
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nipkow@24349
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"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ <- _")
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nipkow@23240
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"_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
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nipkow@24476
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(*"_lc_let" :: "letbinds => lc_qual" ("let _")*)
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nipkow@23240
|
277 |
"_lc_end" :: "lc_quals" ("]")
|
nipkow@23240
|
278 |
"_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals" (", __")
|
nipkow@24349
|
279 |
"_lc_abs" :: "'a => 'b list => 'b list"
|
nipkow@23192
|
280 |
|
nipkow@24476
|
281 |
(* These are easier than ML code but cannot express the optimized
|
nipkow@24476
|
282 |
translation of [e. p<-xs]
|
nipkow@23192
|
283 |
translations
|
nipkow@24349
|
284 |
"[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
|
nipkow@23240
|
285 |
"_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
|
nipkow@24349
|
286 |
=> "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
|
nipkow@23240
|
287 |
"[e. P]" => "if P then [e] else []"
|
nipkow@23240
|
288 |
"_listcompr e (_lc_test P) (_lc_quals Q Qs)"
|
nipkow@23240
|
289 |
=> "if P then (_listcompr e Q Qs) else []"
|
nipkow@24349
|
290 |
"_listcompr e (_lc_let b) (_lc_quals Q Qs)"
|
nipkow@24349
|
291 |
=> "_Let b (_listcompr e Q Qs)"
|
nipkow@24476
|
292 |
*)
|
nipkow@23240
|
293 |
|
nipkow@23279
|
294 |
syntax (xsymbols)
|
nipkow@24349
|
295 |
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
|
nipkow@23279
|
296 |
syntax (HTML output)
|
nipkow@24349
|
297 |
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
|
nipkow@24349
|
298 |
|
nipkow@24349
|
299 |
parse_translation (advanced) {*
|
nipkow@24349
|
300 |
let
|
nipkow@24476
|
301 |
val NilC = Syntax.const @{const_name Nil};
|
nipkow@24476
|
302 |
val ConsC = Syntax.const @{const_name Cons};
|
nipkow@24476
|
303 |
val mapC = Syntax.const @{const_name map};
|
nipkow@24476
|
304 |
val concatC = Syntax.const @{const_name concat};
|
nipkow@24476
|
305 |
val IfC = Syntax.const @{const_name If};
|
nipkow@24476
|
306 |
fun singl x = ConsC $ x $ NilC;
|
nipkow@24476
|
307 |
|
nipkow@24476
|
308 |
fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
|
nipkow@24349
|
309 |
let
|
nipkow@24476
|
310 |
val x = Free (Name.variant (add_term_free_names (p$e, [])) "x", dummyT);
|
nipkow@24476
|
311 |
val e = if opti then singl e else e;
|
nipkow@24476
|
312 |
val case1 = Syntax.const "_case1" $ p $ e;
|
nipkow@24349
|
313 |
val case2 = Syntax.const "_case1" $ Syntax.const Term.dummy_patternN
|
nipkow@24476
|
314 |
$ NilC;
|
nipkow@24349
|
315 |
val cs = Syntax.const "_case2" $ case1 $ case2
|
nipkow@24349
|
316 |
val ft = DatatypeCase.case_tr false DatatypePackage.datatype_of_constr
|
nipkow@24349
|
317 |
ctxt [x, cs]
|
nipkow@24349
|
318 |
in lambda x ft end;
|
nipkow@24349
|
319 |
|
nipkow@24476
|
320 |
fun abs_tr ctxt (p as Free(s,T)) e opti =
|
nipkow@24349
|
321 |
let val thy = ProofContext.theory_of ctxt;
|
nipkow@24349
|
322 |
val s' = Sign.intern_const thy s
|
nipkow@24476
|
323 |
in if Sign.declared_const thy s'
|
nipkow@24476
|
324 |
then (pat_tr ctxt p e opti, false)
|
nipkow@24476
|
325 |
else (lambda p e, true)
|
nipkow@24349
|
326 |
end
|
nipkow@24476
|
327 |
| abs_tr ctxt p e opti = (pat_tr ctxt p e opti, false);
|
nipkow@24476
|
328 |
|
nipkow@24476
|
329 |
fun lc_tr ctxt [e, Const("_lc_test",_)$b, qs] =
|
nipkow@24476
|
330 |
let val res = case qs of Const("_lc_end",_) => singl e
|
nipkow@24476
|
331 |
| Const("_lc_quals",_)$q$qs => lc_tr ctxt [e,q,qs];
|
nipkow@24476
|
332 |
in IfC $ b $ res $ NilC end
|
nipkow@24476
|
333 |
| lc_tr ctxt [e, Const("_lc_gen",_) $ p $ es, Const("_lc_end",_)] =
|
nipkow@24476
|
334 |
(case abs_tr ctxt p e true of
|
nipkow@24476
|
335 |
(f,true) => mapC $ f $ es
|
nipkow@24476
|
336 |
| (f, false) => concatC $ (mapC $ f $ es))
|
nipkow@24476
|
337 |
| lc_tr ctxt [e, Const("_lc_gen",_) $ p $ es, Const("_lc_quals",_)$q$qs] =
|
nipkow@24476
|
338 |
let val e' = lc_tr ctxt [e,q,qs];
|
nipkow@24476
|
339 |
in concatC $ (mapC $ (fst(abs_tr ctxt p e' false)) $ es) end
|
nipkow@24476
|
340 |
|
nipkow@24476
|
341 |
in [("_listcompr", lc_tr)] end
|
nipkow@24349
|
342 |
*}
|
nipkow@23279
|
343 |
|
nipkow@23240
|
344 |
(*
|
nipkow@23240
|
345 |
term "[(x,y,z). b]"
|
nipkow@24476
|
346 |
term "[(x,y,z). x\<leftarrow>xs]"
|
nipkow@24476
|
347 |
term "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]"
|
nipkow@24476
|
348 |
term "[(x,y,z). x<a, x>b]"
|
nipkow@24476
|
349 |
term "[(x,y,z). x\<leftarrow>xs, x>b]"
|
nipkow@24476
|
350 |
term "[(x,y,z). x<a, x\<leftarrow>xs]"
|
nipkow@24349
|
351 |
term "[(x,y). Cons True x \<leftarrow> xs]"
|
nipkow@24349
|
352 |
term "[(x,y,z). Cons x [] \<leftarrow> xs]"
|
nipkow@23240
|
353 |
term "[(x,y,z). x<a, x>b, x=d]"
|
nipkow@23240
|
354 |
term "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
|
nipkow@23240
|
355 |
term "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
|
nipkow@23240
|
356 |
term "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
|
nipkow@23240
|
357 |
term "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
|
nipkow@23240
|
358 |
term "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
|
nipkow@23240
|
359 |
term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
|
nipkow@23240
|
360 |
term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
|
nipkow@24349
|
361 |
term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
|
nipkow@23192
|
362 |
*)
|
nipkow@23192
|
363 |
|
haftmann@21061
|
364 |
subsubsection {* @{const Nil} and @{const Cons} *}
|
haftmann@21061
|
365 |
|
haftmann@21061
|
366 |
lemma not_Cons_self [simp]:
|
haftmann@21061
|
367 |
"xs \<noteq> x # xs"
|
nipkow@13145
|
368 |
by (induct xs) auto
|
nipkow@3507
|
369 |
|
wenzelm@13142
|
370 |
lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
|
wenzelm@13114
|
371 |
|
wenzelm@13142
|
372 |
lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
|
nipkow@13145
|
373 |
by (induct xs) auto
|
wenzelm@13114
|
374 |
|
wenzelm@13142
|
375 |
lemma length_induct:
|
haftmann@21061
|
376 |
"(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
|
nipkow@17589
|
377 |
by (rule measure_induct [of length]) iprover
|
wenzelm@13114
|
378 |
|
wenzelm@13114
|
379 |
|
haftmann@21061
|
380 |
subsubsection {* @{const length} *}
|
wenzelm@13114
|
381 |
|
wenzelm@13142
|
382 |
text {*
|
haftmann@21061
|
383 |
Needs to come before @{text "@"} because of theorem @{text
|
haftmann@21061
|
384 |
append_eq_append_conv}.
|
wenzelm@13142
|
385 |
*}
|
wenzelm@13114
|
386 |
|
wenzelm@13142
|
387 |
lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
|
nipkow@13145
|
388 |
by (induct xs) auto
|
wenzelm@13114
|
389 |
|
wenzelm@13142
|
390 |
lemma length_map [simp]: "length (map f xs) = length xs"
|
nipkow@13145
|
391 |
by (induct xs) auto
|
wenzelm@13114
|
392 |
|
wenzelm@13142
|
393 |
lemma length_rev [simp]: "length (rev xs) = length xs"
|
nipkow@13145
|
394 |
by (induct xs) auto
|
wenzelm@13114
|
395 |
|
wenzelm@13142
|
396 |
lemma length_tl [simp]: "length (tl xs) = length xs - 1"
|
nipkow@13145
|
397 |
by (cases xs) auto
|
wenzelm@13142
|
398 |
|
wenzelm@13142
|
399 |
lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
|
nipkow@13145
|
400 |
by (induct xs) auto
|
wenzelm@13142
|
401 |
|
wenzelm@13142
|
402 |
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
|
nipkow@13145
|
403 |
by (induct xs) auto
|
wenzelm@13114
|
404 |
|
nipkow@23479
|
405 |
lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
|
nipkow@23479
|
406 |
by auto
|
nipkow@23479
|
407 |
|
wenzelm@13114
|
408 |
lemma length_Suc_conv:
|
nipkow@13145
|
409 |
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
|
nipkow@13145
|
410 |
by (induct xs) auto
|
wenzelm@13114
|
411 |
|
nipkow@14025
|
412 |
lemma Suc_length_conv:
|
nipkow@14025
|
413 |
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
|
paulson@14208
|
414 |
apply (induct xs, simp, simp)
|
nipkow@14025
|
415 |
apply blast
|
nipkow@14025
|
416 |
done
|
nipkow@14025
|
417 |
|
wenzelm@25221
|
418 |
lemma impossible_Cons: "length xs <= length ys ==> xs = x # ys = False"
|
wenzelm@25221
|
419 |
by (induct xs) auto
|
wenzelm@25221
|
420 |
|
haftmann@26442
|
421 |
lemma list_induct2 [consumes 1, case_names Nil Cons]:
|
haftmann@26442
|
422 |
"length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
|
haftmann@26442
|
423 |
(\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
|
haftmann@26442
|
424 |
\<Longrightarrow> P xs ys"
|
haftmann@26442
|
425 |
proof (induct xs arbitrary: ys)
|
haftmann@26442
|
426 |
case Nil then show ?case by simp
|
haftmann@26442
|
427 |
next
|
haftmann@26442
|
428 |
case (Cons x xs ys) then show ?case by (cases ys) simp_all
|
haftmann@26442
|
429 |
qed
|
haftmann@26442
|
430 |
|
haftmann@26442
|
431 |
lemma list_induct3 [consumes 2, case_names Nil Cons]:
|
haftmann@26442
|
432 |
"length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
|
haftmann@26442
|
433 |
(\<And>x xs y ys z zs. length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P xs ys zs \<Longrightarrow> P (x#xs) (y#ys) (z#zs))
|
haftmann@26442
|
434 |
\<Longrightarrow> P xs ys zs"
|
haftmann@26442
|
435 |
proof (induct xs arbitrary: ys zs)
|
haftmann@26442
|
436 |
case Nil then show ?case by simp
|
haftmann@26442
|
437 |
next
|
haftmann@26442
|
438 |
case (Cons x xs ys zs) then show ?case by (cases ys, simp_all)
|
haftmann@26442
|
439 |
(cases zs, simp_all)
|
haftmann@26442
|
440 |
qed
|
wenzelm@13114
|
441 |
|
krauss@22493
|
442 |
lemma list_induct2':
|
krauss@22493
|
443 |
"\<lbrakk> P [] [];
|
krauss@22493
|
444 |
\<And>x xs. P (x#xs) [];
|
krauss@22493
|
445 |
\<And>y ys. P [] (y#ys);
|
krauss@22493
|
446 |
\<And>x xs y ys. P xs ys \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
|
krauss@22493
|
447 |
\<Longrightarrow> P xs ys"
|
krauss@22493
|
448 |
by (induct xs arbitrary: ys) (case_tac x, auto)+
|
krauss@22493
|
449 |
|
nipkow@22143
|
450 |
lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
|
nipkow@24349
|
451 |
by (rule Eq_FalseI) auto
|
wenzelm@24037
|
452 |
|
wenzelm@24037
|
453 |
simproc_setup list_neq ("(xs::'a list) = ys") = {*
|
nipkow@22143
|
454 |
(*
|
nipkow@22143
|
455 |
Reduces xs=ys to False if xs and ys cannot be of the same length.
|
nipkow@22143
|
456 |
This is the case if the atomic sublists of one are a submultiset
|
nipkow@22143
|
457 |
of those of the other list and there are fewer Cons's in one than the other.
|
nipkow@22143
|
458 |
*)
|
wenzelm@24037
|
459 |
|
wenzelm@24037
|
460 |
let
|
nipkow@22143
|
461 |
|
nipkow@22143
|
462 |
fun len (Const("List.list.Nil",_)) acc = acc
|
nipkow@22143
|
463 |
| len (Const("List.list.Cons",_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
|
haftmann@23029
|
464 |
| len (Const("List.append",_) $ xs $ ys) acc = len xs (len ys acc)
|
nipkow@22143
|
465 |
| len (Const("List.rev",_) $ xs) acc = len xs acc
|
nipkow@22143
|
466 |
| len (Const("List.map",_) $ _ $ xs) acc = len xs acc
|
nipkow@22143
|
467 |
| len t (ts,n) = (t::ts,n);
|
nipkow@22143
|
468 |
|
wenzelm@24037
|
469 |
fun list_neq _ ss ct =
|
nipkow@22143
|
470 |
let
|
wenzelm@24037
|
471 |
val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
|
nipkow@22143
|
472 |
val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
|
nipkow@22143
|
473 |
fun prove_neq() =
|
nipkow@22143
|
474 |
let
|
nipkow@22143
|
475 |
val Type(_,listT::_) = eqT;
|
haftmann@22994
|
476 |
val size = HOLogic.size_const listT;
|
nipkow@22143
|
477 |
val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
|
nipkow@22143
|
478 |
val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
|
nipkow@22143
|
479 |
val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
|
haftmann@22633
|
480 |
(K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
|
haftmann@22633
|
481 |
in SOME (thm RS @{thm neq_if_length_neq}) end
|
nipkow@22143
|
482 |
in
|
wenzelm@23214
|
483 |
if m < n andalso submultiset (op aconv) (ls,rs) orelse
|
wenzelm@23214
|
484 |
n < m andalso submultiset (op aconv) (rs,ls)
|
nipkow@22143
|
485 |
then prove_neq() else NONE
|
nipkow@22143
|
486 |
end;
|
wenzelm@24037
|
487 |
in list_neq end;
|
nipkow@22143
|
488 |
*}
|
nipkow@22143
|
489 |
|
nipkow@22143
|
490 |
|
nipkow@15392
|
491 |
subsubsection {* @{text "@"} -- append *}
|
wenzelm@13114
|
492 |
|
wenzelm@13142
|
493 |
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
|
nipkow@13145
|
494 |
by (induct xs) auto
|
wenzelm@13114
|
495 |
|
wenzelm@13142
|
496 |
lemma append_Nil2 [simp]: "xs @ [] = xs"
|
nipkow@13145
|
497 |
by (induct xs) auto
|
wenzelm@13114
|
498 |
|
nipkow@24449
|
499 |
interpretation semigroup_append: semigroup_add ["op @"]
|
nipkow@24449
|
500 |
by unfold_locales simp
|
nipkow@24449
|
501 |
interpretation monoid_append: monoid_add ["[]" "op @"]
|
nipkow@24449
|
502 |
by unfold_locales (simp+)
|
nipkow@24449
|
503 |
|
wenzelm@13142
|
504 |
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
|
nipkow@13145
|
505 |
by (induct xs) auto
|
wenzelm@13114
|
506 |
|
wenzelm@13142
|
507 |
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
|
nipkow@13145
|
508 |
by (induct xs) auto
|
wenzelm@13114
|
509 |
|
wenzelm@13142
|
510 |
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
|
nipkow@13145
|
511 |
by (induct xs) auto
|
wenzelm@13114
|
512 |
|
wenzelm@13142
|
513 |
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
|
nipkow@13145
|
514 |
by (induct xs) auto
|
wenzelm@13114
|
515 |
|
wenzelm@25221
|
516 |
lemma append_eq_append_conv [simp, noatp]:
|
nipkow@24526
|
517 |
"length xs = length ys \<or> length us = length vs
|
berghofe@13883
|
518 |
==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
|
nipkow@24526
|
519 |
apply (induct xs arbitrary: ys)
|
paulson@14208
|
520 |
apply (case_tac ys, simp, force)
|
paulson@14208
|
521 |
apply (case_tac ys, force, simp)
|
nipkow@13145
|
522 |
done
|
wenzelm@13114
|
523 |
|
nipkow@24526
|
524 |
lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
|
nipkow@24526
|
525 |
(EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
|
nipkow@24526
|
526 |
apply (induct xs arbitrary: ys zs ts)
|
nipkow@14495
|
527 |
apply fastsimp
|
nipkow@14495
|
528 |
apply(case_tac zs)
|
nipkow@14495
|
529 |
apply simp
|
nipkow@14495
|
530 |
apply fastsimp
|
nipkow@14495
|
531 |
done
|
nipkow@14495
|
532 |
|
wenzelm@13142
|
533 |
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
|
nipkow@13145
|
534 |
by simp
|
wenzelm@13114
|
535 |
|
wenzelm@13142
|
536 |
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
|
nipkow@13145
|
537 |
by simp
|
wenzelm@13114
|
538 |
|
wenzelm@13142
|
539 |
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
|
nipkow@13145
|
540 |
by simp
|
wenzelm@13114
|
541 |
|
wenzelm@13142
|
542 |
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
|
nipkow@13145
|
543 |
using append_same_eq [of _ _ "[]"] by auto
|
wenzelm@13114
|
544 |
|
wenzelm@13142
|
545 |
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
|
nipkow@13145
|
546 |
using append_same_eq [of "[]"] by auto
|
wenzelm@13114
|
547 |
|
paulson@24286
|
548 |
lemma hd_Cons_tl [simp,noatp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
|
nipkow@13145
|
549 |
by (induct xs) auto
|
wenzelm@13114
|
550 |
|
wenzelm@13142
|
551 |
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
|
nipkow@13145
|
552 |
by (induct xs) auto
|
wenzelm@13114
|
553 |
|
wenzelm@13142
|
554 |
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
|
nipkow@13145
|
555 |
by (simp add: hd_append split: list.split)
|
wenzelm@13114
|
556 |
|
wenzelm@13142
|
557 |
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
|
nipkow@13145
|
558 |
by (simp split: list.split)
|
wenzelm@13114
|
559 |
|
wenzelm@13142
|
560 |
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
|
nipkow@13145
|
561 |
by (simp add: tl_append split: list.split)
|
wenzelm@13114
|
562 |
|
wenzelm@13142
|
563 |
|
nipkow@14300
|
564 |
lemma Cons_eq_append_conv: "x#xs = ys@zs =
|
nipkow@14300
|
565 |
(ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
|
nipkow@14300
|
566 |
by(cases ys) auto
|
nipkow@14300
|
567 |
|
nipkow@15281
|
568 |
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
|
nipkow@15281
|
569 |
(ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
|
nipkow@15281
|
570 |
by(cases ys) auto
|
nipkow@15281
|
571 |
|
nipkow@14300
|
572 |
|
wenzelm@13142
|
573 |
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
|
wenzelm@13114
|
574 |
|
wenzelm@13114
|
575 |
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
|
nipkow@13145
|
576 |
by simp
|
wenzelm@13114
|
577 |
|
wenzelm@13142
|
578 |
lemma Cons_eq_appendI:
|
nipkow@13145
|
579 |
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
|
nipkow@13145
|
580 |
by (drule sym) simp
|
wenzelm@13114
|
581 |
|
wenzelm@13142
|
582 |
lemma append_eq_appendI:
|
nipkow@13145
|
583 |
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
|
nipkow@13145
|
584 |
by (drule sym) simp
|
wenzelm@13114
|
585 |
|
wenzelm@13114
|
586 |
|
wenzelm@13142
|
587 |
text {*
|
nipkow@13145
|
588 |
Simplification procedure for all list equalities.
|
nipkow@13145
|
589 |
Currently only tries to rearrange @{text "@"} to see if
|
nipkow@13145
|
590 |
- both lists end in a singleton list,
|
nipkow@13145
|
591 |
- or both lists end in the same list.
|
wenzelm@13142
|
592 |
*}
|
wenzelm@13142
|
593 |
|
wenzelm@26480
|
594 |
ML {*
|
nipkow@3507
|
595 |
local
|
nipkow@3507
|
596 |
|
wenzelm@13114
|
597 |
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
|
wenzelm@13462
|
598 |
(case xs of Const("List.list.Nil",_) => cons | _ => last xs)
|
haftmann@23029
|
599 |
| last (Const("List.append",_) $ _ $ ys) = last ys
|
wenzelm@13462
|
600 |
| last t = t;
|
nipkow@3507
|
601 |
|
wenzelm@13114
|
602 |
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
|
wenzelm@13462
|
603 |
| list1 _ = false;
|
wenzelm@13114
|
604 |
|
wenzelm@13114
|
605 |
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
|
wenzelm@13462
|
606 |
(case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
|
haftmann@23029
|
607 |
| butlast ((app as Const("List.append",_) $ xs) $ ys) = app $ butlast ys
|
wenzelm@13462
|
608 |
| butlast xs = Const("List.list.Nil",fastype_of xs);
|
wenzelm@13114
|
609 |
|
haftmann@22633
|
610 |
val rearr_ss = HOL_basic_ss addsimps [@{thm append_assoc},
|
haftmann@22633
|
611 |
@{thm append_Nil}, @{thm append_Cons}];
|
wenzelm@16973
|
612 |
|
wenzelm@20044
|
613 |
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
|
wenzelm@13462
|
614 |
let
|
wenzelm@13462
|
615 |
val lastl = last lhs and lastr = last rhs;
|
wenzelm@13462
|
616 |
fun rearr conv =
|
wenzelm@13462
|
617 |
let
|
wenzelm@13462
|
618 |
val lhs1 = butlast lhs and rhs1 = butlast rhs;
|
wenzelm@13462
|
619 |
val Type(_,listT::_) = eqT
|
wenzelm@13462
|
620 |
val appT = [listT,listT] ---> listT
|
haftmann@23029
|
621 |
val app = Const("List.append",appT)
|
wenzelm@13462
|
622 |
val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
|
wenzelm@13480
|
623 |
val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
|
wenzelm@20044
|
624 |
val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
|
wenzelm@17877
|
625 |
(K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
|
skalberg@15531
|
626 |
in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
|
wenzelm@13114
|
627 |
|
wenzelm@13462
|
628 |
in
|
haftmann@22633
|
629 |
if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
|
haftmann@22633
|
630 |
else if lastl aconv lastr then rearr @{thm append_same_eq}
|
skalberg@15531
|
631 |
else NONE
|
wenzelm@13462
|
632 |
end;
|
wenzelm@13462
|
633 |
|
nipkow@3507
|
634 |
in
|
wenzelm@13462
|
635 |
|
wenzelm@13462
|
636 |
val list_eq_simproc =
|
haftmann@22633
|
637 |
Simplifier.simproc @{theory} "list_eq" ["(xs::'a list) = ys"] (K list_eq);
|
wenzelm@13462
|
638 |
|
wenzelm@13114
|
639 |
end;
|
nipkow@3507
|
640 |
|
wenzelm@13114
|
641 |
Addsimprocs [list_eq_simproc];
|
wenzelm@13114
|
642 |
*}
|
wenzelm@13114
|
643 |
|
wenzelm@13114
|
644 |
|
nipkow@15392
|
645 |
subsubsection {* @{text map} *}
|
wenzelm@13114
|
646 |
|
wenzelm@13142
|
647 |
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
|
nipkow@13145
|
648 |
by (induct xs) simp_all
|
wenzelm@13114
|
649 |
|
wenzelm@13142
|
650 |
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
|
nipkow@13145
|
651 |
by (rule ext, induct_tac xs) auto
|
wenzelm@13114
|
652 |
|
wenzelm@13142
|
653 |
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
|
nipkow@13145
|
654 |
by (induct xs) auto
|
wenzelm@13114
|
655 |
|
wenzelm@13142
|
656 |
lemma map_compose: "map (f o g) xs = map f (map g xs)"
|
nipkow@13145
|
657 |
by (induct xs) (auto simp add: o_def)
|
wenzelm@13114
|
658 |
|
wenzelm@13142
|
659 |
lemma rev_map: "rev (map f xs) = map f (rev xs)"
|
nipkow@13145
|
660 |
by (induct xs) auto
|
wenzelm@13114
|
661 |
|
nipkow@13737
|
662 |
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
|
nipkow@13737
|
663 |
by (induct xs) auto
|
nipkow@13737
|
664 |
|
krauss@19770
|
665 |
lemma map_cong [fundef_cong, recdef_cong]:
|
nipkow@13145
|
666 |
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
|
nipkow@13145
|
667 |
-- {* a congruence rule for @{text map} *}
|
nipkow@13737
|
668 |
by simp
|
wenzelm@13114
|
669 |
|
wenzelm@13142
|
670 |
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
|
nipkow@13145
|
671 |
by (cases xs) auto
|
wenzelm@13114
|
672 |
|
wenzelm@13142
|
673 |
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
|
nipkow@13145
|
674 |
by (cases xs) auto
|
wenzelm@13114
|
675 |
|
paulson@18447
|
676 |
lemma map_eq_Cons_conv:
|
nipkow@14025
|
677 |
"(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
|
nipkow@13145
|
678 |
by (cases xs) auto
|
wenzelm@13114
|
679 |
|
paulson@18447
|
680 |
lemma Cons_eq_map_conv:
|
nipkow@14025
|
681 |
"(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
|
nipkow@14025
|
682 |
by (cases ys) auto
|
nipkow@14025
|
683 |
|
paulson@18447
|
684 |
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
|
paulson@18447
|
685 |
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
|
paulson@18447
|
686 |
declare map_eq_Cons_D [dest!] Cons_eq_map_D [dest!]
|
paulson@18447
|
687 |
|
nipkow@14111
|
688 |
lemma ex_map_conv:
|
nipkow@14111
|
689 |
"(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
|
paulson@18447
|
690 |
by(induct ys, auto simp add: Cons_eq_map_conv)
|
nipkow@14111
|
691 |
|
nipkow@15110
|
692 |
lemma map_eq_imp_length_eq:
|
nipkow@24526
|
693 |
"map f xs = map f ys ==> length xs = length ys"
|
nipkow@24526
|
694 |
apply (induct ys arbitrary: xs)
|
nipkow@15110
|
695 |
apply simp
|
paulson@24632
|
696 |
apply (metis Suc_length_conv length_map)
|
nipkow@15110
|
697 |
done
|
nipkow@15110
|
698 |
|
nipkow@15110
|
699 |
lemma map_inj_on:
|
nipkow@15110
|
700 |
"[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
|
nipkow@15110
|
701 |
==> xs = ys"
|
nipkow@15110
|
702 |
apply(frule map_eq_imp_length_eq)
|
nipkow@15110
|
703 |
apply(rotate_tac -1)
|
nipkow@15110
|
704 |
apply(induct rule:list_induct2)
|
nipkow@15110
|
705 |
apply simp
|
nipkow@15110
|
706 |
apply(simp)
|
nipkow@15110
|
707 |
apply (blast intro:sym)
|
nipkow@15110
|
708 |
done
|
nipkow@15110
|
709 |
|
nipkow@15110
|
710 |
lemma inj_on_map_eq_map:
|
nipkow@15110
|
711 |
"inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
|
nipkow@15110
|
712 |
by(blast dest:map_inj_on)
|
nipkow@15110
|
713 |
|
wenzelm@13114
|
714 |
lemma map_injective:
|
nipkow@24526
|
715 |
"map f xs = map f ys ==> inj f ==> xs = ys"
|
nipkow@24526
|
716 |
by (induct ys arbitrary: xs) (auto dest!:injD)
|
wenzelm@13114
|
717 |
|
nipkow@14339
|
718 |
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
|
nipkow@14339
|
719 |
by(blast dest:map_injective)
|
nipkow@14339
|
720 |
|
wenzelm@13114
|
721 |
lemma inj_mapI: "inj f ==> inj (map f)"
|
nipkow@17589
|
722 |
by (iprover dest: map_injective injD intro: inj_onI)
|
wenzelm@13114
|
723 |
|
wenzelm@13114
|
724 |
lemma inj_mapD: "inj (map f) ==> inj f"
|
paulson@14208
|
725 |
apply (unfold inj_on_def, clarify)
|
nipkow@13145
|
726 |
apply (erule_tac x = "[x]" in ballE)
|
paulson@14208
|
727 |
apply (erule_tac x = "[y]" in ballE, simp, blast)
|
nipkow@13145
|
728 |
apply blast
|
nipkow@13145
|
729 |
done
|
wenzelm@13114
|
730 |
|
nipkow@14339
|
731 |
lemma inj_map[iff]: "inj (map f) = inj f"
|
nipkow@13145
|
732 |
by (blast dest: inj_mapD intro: inj_mapI)
|
wenzelm@13114
|
733 |
|
nipkow@15303
|
734 |
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
|
nipkow@15303
|
735 |
apply(rule inj_onI)
|
nipkow@15303
|
736 |
apply(erule map_inj_on)
|
nipkow@15303
|
737 |
apply(blast intro:inj_onI dest:inj_onD)
|
nipkow@15303
|
738 |
done
|
nipkow@15303
|
739 |
|
kleing@14343
|
740 |
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
|
kleing@14343
|
741 |
by (induct xs, auto)
|
wenzelm@13114
|
742 |
|
nipkow@14402
|
743 |
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
|
nipkow@14402
|
744 |
by (induct xs) auto
|
nipkow@14402
|
745 |
|
nipkow@15110
|
746 |
lemma map_fst_zip[simp]:
|
nipkow@15110
|
747 |
"length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
|
nipkow@15110
|
748 |
by (induct rule:list_induct2, simp_all)
|
nipkow@15110
|
749 |
|
nipkow@15110
|
750 |
lemma map_snd_zip[simp]:
|
nipkow@15110
|
751 |
"length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
|
nipkow@15110
|
752 |
by (induct rule:list_induct2, simp_all)
|
nipkow@15110
|
753 |
|
nipkow@15110
|
754 |
|
nipkow@15392
|
755 |
subsubsection {* @{text rev} *}
|
wenzelm@13114
|
756 |
|
wenzelm@13142
|
757 |
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
|
nipkow@13145
|
758 |
by (induct xs) auto
|
wenzelm@13114
|
759 |
|
wenzelm@13142
|
760 |
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
|
nipkow@13145
|
761 |
by (induct xs) auto
|
wenzelm@13114
|
762 |
|
kleing@15870
|
763 |
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
|
kleing@15870
|
764 |
by auto
|
kleing@15870
|
765 |
|
wenzelm@13142
|
766 |
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
|
nipkow@13145
|
767 |
by (induct xs) auto
|
wenzelm@13114
|
768 |
|
wenzelm@13142
|
769 |
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
|
nipkow@13145
|
770 |
by (induct xs) auto
|
wenzelm@13114
|
771 |
|
kleing@15870
|
772 |
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
|
kleing@15870
|
773 |
by (cases xs) auto
|
kleing@15870
|
774 |
|
kleing@15870
|
775 |
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
|
kleing@15870
|
776 |
by (cases xs) auto
|
kleing@15870
|
777 |
|
haftmann@21061
|
778 |
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
|
haftmann@21061
|
779 |
apply (induct xs arbitrary: ys, force)
|
paulson@14208
|
780 |
apply (case_tac ys, simp, force)
|
nipkow@13145
|
781 |
done
|
wenzelm@13114
|
782 |
|
nipkow@15439
|
783 |
lemma inj_on_rev[iff]: "inj_on rev A"
|
nipkow@15439
|
784 |
by(simp add:inj_on_def)
|
nipkow@15439
|
785 |
|
wenzelm@13366
|
786 |
lemma rev_induct [case_names Nil snoc]:
|
wenzelm@13366
|
787 |
"[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
|
berghofe@15489
|
788 |
apply(simplesubst rev_rev_ident[symmetric])
|
nipkow@13145
|
789 |
apply(rule_tac list = "rev xs" in list.induct, simp_all)
|
nipkow@13145
|
790 |
done
|
wenzelm@13114
|
791 |
|
wenzelm@13366
|
792 |
lemma rev_exhaust [case_names Nil snoc]:
|
wenzelm@13366
|
793 |
"(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
|
nipkow@13145
|
794 |
by (induct xs rule: rev_induct) auto
|
wenzelm@13114
|
795 |
|
wenzelm@13366
|
796 |
lemmas rev_cases = rev_exhaust
|
wenzelm@13366
|
797 |
|
nipkow@18423
|
798 |
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
|
nipkow@18423
|
799 |
by(rule rev_cases[of xs]) auto
|
nipkow@18423
|
800 |
|
wenzelm@13114
|
801 |
|
nipkow@15392
|
802 |
subsubsection {* @{text set} *}
|
wenzelm@13114
|
803 |
|
wenzelm@13142
|
804 |
lemma finite_set [iff]: "finite (set xs)"
|
nipkow@13145
|
805 |
by (induct xs) auto
|
wenzelm@13114
|
806 |
|
wenzelm@13142
|
807 |
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
|
nipkow@13145
|
808 |
by (induct xs) auto
|
wenzelm@13114
|
809 |
|
nipkow@17830
|
810 |
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
|
nipkow@17830
|
811 |
by(cases xs) auto
|
oheimb@14099
|
812 |
|
wenzelm@13142
|
813 |
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
|
nipkow@13145
|
814 |
by auto
|
wenzelm@13114
|
815 |
|
oheimb@14099
|
816 |
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs"
|
oheimb@14099
|
817 |
by auto
|
oheimb@14099
|
818 |
|
wenzelm@13142
|
819 |
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
|
nipkow@13145
|
820 |
by (induct xs) auto
|
wenzelm@13114
|
821 |
|
nipkow@15245
|
822 |
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
|
nipkow@15245
|
823 |
by(induct xs) auto
|
nipkow@15245
|
824 |
|
wenzelm@13142
|
825 |
lemma set_rev [simp]: "set (rev xs) = set xs"
|
nipkow@13145
|
826 |
by (induct xs) auto
|
wenzelm@13114
|
827 |
|
wenzelm@13142
|
828 |
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
|
nipkow@13145
|
829 |
by (induct xs) auto
|
wenzelm@13114
|
830 |
|
wenzelm@13142
|
831 |
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
|
nipkow@13145
|
832 |
by (induct xs) auto
|
wenzelm@13114
|
833 |
|
nipkow@15425
|
834 |
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
|
paulson@14208
|
835 |
apply (induct j, simp_all)
|
paulson@14208
|
836 |
apply (erule ssubst, auto)
|
nipkow@13145
|
837 |
done
|
wenzelm@13114
|
838 |
|
nipkow@26073
|
839 |
|
nipkow@26073
|
840 |
lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs"
|
nipkow@26073
|
841 |
proof (induct xs)
|
nipkow@26073
|
842 |
case Nil thus ?case by simp
|
nipkow@26073
|
843 |
next
|
nipkow@26073
|
844 |
case Cons thus ?case by (auto intro: Cons_eq_appendI)
|
nipkow@26073
|
845 |
qed
|
nipkow@26073
|
846 |
|
wenzelm@13142
|
847 |
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
|
nipkow@26073
|
848 |
by (metis Un_upper2 insert_subset set.simps(2) set_append split_list)
|
nipkow@26073
|
849 |
|
nipkow@26073
|
850 |
lemma split_list_first: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
|
paulson@15113
|
851 |
proof (induct xs)
|
nipkow@26073
|
852 |
case Nil thus ?case by simp
|
nipkow@18049
|
853 |
next
|
nipkow@18049
|
854 |
case (Cons a xs)
|
nipkow@18049
|
855 |
show ?case
|
nipkow@18049
|
856 |
proof cases
|
wenzelm@25221
|
857 |
assume "x = a" thus ?case using Cons by fastsimp
|
nipkow@18049
|
858 |
next
|
nipkow@26073
|
859 |
assume "x \<noteq> a" thus ?case using Cons by(fastsimp intro!: Cons_eq_appendI)
|
nipkow@18049
|
860 |
qed
|
nipkow@18049
|
861 |
qed
|
nipkow@18049
|
862 |
|
nipkow@26073
|
863 |
lemma in_set_conv_decomp_first:
|
nipkow@26073
|
864 |
"(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
|
nipkow@26073
|
865 |
by (metis in_set_conv_decomp split_list_first)
|
nipkow@26073
|
866 |
|
nipkow@26073
|
867 |
lemma split_list_last: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
|
nipkow@26073
|
868 |
proof (induct xs rule:rev_induct)
|
nipkow@26073
|
869 |
case Nil thus ?case by simp
|
nipkow@26073
|
870 |
next
|
nipkow@26073
|
871 |
case (snoc a xs)
|
nipkow@26073
|
872 |
show ?case
|
nipkow@26073
|
873 |
proof cases
|
nipkow@26073
|
874 |
assume "x = a" thus ?case using snoc by simp (metis ex_in_conv set_empty2)
|
nipkow@26073
|
875 |
next
|
nipkow@26073
|
876 |
assume "x \<noteq> a" thus ?case using snoc by fastsimp
|
nipkow@26073
|
877 |
qed
|
nipkow@26073
|
878 |
qed
|
nipkow@26073
|
879 |
|
nipkow@26073
|
880 |
lemma in_set_conv_decomp_last:
|
nipkow@26073
|
881 |
"(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
|
nipkow@26073
|
882 |
by (metis in_set_conv_decomp split_list_last)
|
nipkow@26073
|
883 |
|
nipkow@26073
|
884 |
lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs & P x"
|
nipkow@26073
|
885 |
proof (induct xs)
|
nipkow@26073
|
886 |
case Nil thus ?case by simp
|
nipkow@26073
|
887 |
next
|
nipkow@26073
|
888 |
case Cons thus ?case
|
nipkow@26073
|
889 |
by(simp add:Bex_def)(metis append_Cons append.simps(1))
|
nipkow@26073
|
890 |
qed
|
nipkow@26073
|
891 |
|
nipkow@26073
|
892 |
lemma split_list_propE:
|
nipkow@26073
|
893 |
assumes "\<exists>x \<in> set xs. P x"
|
nipkow@26073
|
894 |
obtains ys x zs where "xs = ys @ x # zs" and "P x"
|
nipkow@26073
|
895 |
by(metis split_list_prop[OF assms])
|
nipkow@26073
|
896 |
|
nipkow@26073
|
897 |
|
nipkow@26073
|
898 |
lemma split_list_first_prop:
|
nipkow@26073
|
899 |
"\<exists>x \<in> set xs. P x \<Longrightarrow>
|
nipkow@26073
|
900 |
\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)"
|
nipkow@26073
|
901 |
proof(induct xs)
|
nipkow@26073
|
902 |
case Nil thus ?case by simp
|
nipkow@26073
|
903 |
next
|
nipkow@26073
|
904 |
case (Cons x xs)
|
nipkow@26073
|
905 |
show ?case
|
nipkow@26073
|
906 |
proof cases
|
nipkow@26073
|
907 |
assume "P x"
|
nipkow@26073
|
908 |
thus ?thesis by simp (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
|
nipkow@26073
|
909 |
next
|
nipkow@26073
|
910 |
assume "\<not> P x"
|
nipkow@26073
|
911 |
hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
|
nipkow@26073
|
912 |
thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
|
nipkow@26073
|
913 |
qed
|
nipkow@26073
|
914 |
qed
|
nipkow@26073
|
915 |
|
nipkow@26073
|
916 |
lemma split_list_first_propE:
|
nipkow@26073
|
917 |
assumes "\<exists>x \<in> set xs. P x"
|
nipkow@26073
|
918 |
obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y"
|
nipkow@26073
|
919 |
by(metis split_list_first_prop[OF assms])
|
nipkow@26073
|
920 |
|
nipkow@26073
|
921 |
lemma split_list_first_prop_iff:
|
nipkow@26073
|
922 |
"(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
|
nipkow@26073
|
923 |
(\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))"
|
nipkow@26073
|
924 |
by(metis split_list_first_prop[where P=P] in_set_conv_decomp)
|
nipkow@26073
|
925 |
|
nipkow@26073
|
926 |
|
nipkow@26073
|
927 |
lemma split_list_last_prop:
|
nipkow@26073
|
928 |
"\<exists>x \<in> set xs. P x \<Longrightarrow>
|
nipkow@26073
|
929 |
\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)"
|
nipkow@26073
|
930 |
proof(induct xs rule:rev_induct)
|
nipkow@26073
|
931 |
case Nil thus ?case by simp
|
nipkow@26073
|
932 |
next
|
nipkow@26073
|
933 |
case (snoc x xs)
|
nipkow@26073
|
934 |
show ?case
|
nipkow@26073
|
935 |
proof cases
|
nipkow@26073
|
936 |
assume "P x" thus ?thesis by (metis emptyE set_empty)
|
nipkow@26073
|
937 |
next
|
nipkow@26073
|
938 |
assume "\<not> P x"
|
nipkow@26073
|
939 |
hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
|
nipkow@26073
|
940 |
thus ?thesis using `\<not> P x` snoc(1) by fastsimp
|
nipkow@26073
|
941 |
qed
|
nipkow@26073
|
942 |
qed
|
nipkow@26073
|
943 |
|
nipkow@26073
|
944 |
lemma split_list_last_propE:
|
nipkow@26073
|
945 |
assumes "\<exists>x \<in> set xs. P x"
|
nipkow@26073
|
946 |
obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z"
|
nipkow@26073
|
947 |
by(metis split_list_last_prop[OF assms])
|
nipkow@26073
|
948 |
|
nipkow@26073
|
949 |
lemma split_list_last_prop_iff:
|
nipkow@26073
|
950 |
"(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
|
nipkow@26073
|
951 |
(\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
|
nipkow@26073
|
952 |
by(metis split_list_last_prop[where P=P] in_set_conv_decomp)
|
nipkow@26073
|
953 |
|
nipkow@26073
|
954 |
|
nipkow@26073
|
955 |
lemma finite_list: "finite A ==> EX xs. set xs = A"
|
paulson@13508
|
956 |
apply (erule finite_induct, auto)
|
nipkow@26073
|
957 |
apply (metis set.simps(2))
|
paulson@13508
|
958 |
done
|
paulson@13508
|
959 |
|
kleing@14388
|
960 |
lemma card_length: "card (set xs) \<le> length xs"
|
kleing@14388
|
961 |
by (induct xs) (auto simp add: card_insert_if)
|
wenzelm@13114
|
962 |
|
haftmann@26442
|
963 |
lemma set_minus_filter_out:
|
haftmann@26442
|
964 |
"set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)"
|
haftmann@26442
|
965 |
by (induct xs) auto
|
paulson@15168
|
966 |
|
nipkow@15392
|
967 |
subsubsection {* @{text filter} *}
|
wenzelm@13114
|
968 |
|
wenzelm@13142
|
969 |
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
|
nipkow@13145
|
970 |
by (induct xs) auto
|
wenzelm@13114
|
971 |
|
nipkow@15305
|
972 |
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
|
nipkow@15305
|
973 |
by (induct xs) simp_all
|
nipkow@15305
|
974 |
|
wenzelm@13142
|
975 |
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
|
nipkow@13145
|
976 |
by (induct xs) auto
|
wenzelm@13114
|
977 |
|
nipkow@16998
|
978 |
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
|
nipkow@16998
|
979 |
by (induct xs) (auto simp add: le_SucI)
|
nipkow@16998
|
980 |
|
nipkow@18423
|
981 |
lemma sum_length_filter_compl:
|
nipkow@18423
|
982 |
"length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
|
nipkow@18423
|
983 |
by(induct xs) simp_all
|
nipkow@18423
|
984 |
|
wenzelm@13142
|
985 |
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
|
nipkow@13145
|
986 |
by (induct xs) auto
|
wenzelm@13114
|
987 |
|
wenzelm@13142
|
988 |
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
|
nipkow@13145
|
989 |
by (induct xs) auto
|
wenzelm@13114
|
990 |
|
nipkow@16998
|
991 |
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)"
|
nipkow@24349
|
992 |
by (induct xs) simp_all
|
nipkow@16998
|
993 |
|
nipkow@16998
|
994 |
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
|
nipkow@16998
|
995 |
apply (induct xs)
|
nipkow@16998
|
996 |
apply auto
|
nipkow@16998
|
997 |
apply(cut_tac P=P and xs=xs in length_filter_le)
|
nipkow@16998
|
998 |
apply simp
|
nipkow@16998
|
999 |
done
|
wenzelm@13114
|
1000 |
|
nipkow@16965
|
1001 |
lemma filter_map:
|
nipkow@16965
|
1002 |
"filter P (map f xs) = map f (filter (P o f) xs)"
|
nipkow@16965
|
1003 |
by (induct xs) simp_all
|
nipkow@16965
|
1004 |
|
nipkow@16965
|
1005 |
lemma length_filter_map[simp]:
|
nipkow@16965
|
1006 |
"length (filter P (map f xs)) = length(filter (P o f) xs)"
|
nipkow@16965
|
1007 |
by (simp add:filter_map)
|
nipkow@16965
|
1008 |
|
wenzelm@13142
|
1009 |
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
|
nipkow@13145
|
1010 |
by auto
|
wenzelm@13114
|
1011 |
|
nipkow@15246
|
1012 |
lemma length_filter_less:
|
nipkow@15246
|
1013 |
"\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
|
nipkow@15246
|
1014 |
proof (induct xs)
|
nipkow@15246
|
1015 |
case Nil thus ?case by simp
|
nipkow@15246
|
1016 |
next
|
nipkow@15246
|
1017 |
case (Cons x xs) thus ?case
|
nipkow@15246
|
1018 |
apply (auto split:split_if_asm)
|
nipkow@15246
|
1019 |
using length_filter_le[of P xs] apply arith
|
nipkow@15246
|
1020 |
done
|
nipkow@15246
|
1021 |
qed
|
wenzelm@13114
|
1022 |
|
nipkow@15281
|
1023 |
lemma length_filter_conv_card:
|
nipkow@15281
|
1024 |
"length(filter p xs) = card{i. i < length xs & p(xs!i)}"
|
nipkow@15281
|
1025 |
proof (induct xs)
|
nipkow@15281
|
1026 |
case Nil thus ?case by simp
|
nipkow@15281
|
1027 |
next
|
nipkow@15281
|
1028 |
case (Cons x xs)
|
nipkow@15281
|
1029 |
let ?S = "{i. i < length xs & p(xs!i)}"
|
nipkow@15281
|
1030 |
have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
|
nipkow@15281
|
1031 |
show ?case (is "?l = card ?S'")
|
nipkow@15281
|
1032 |
proof (cases)
|
nipkow@15281
|
1033 |
assume "p x"
|
nipkow@15281
|
1034 |
hence eq: "?S' = insert 0 (Suc ` ?S)"
|
nipkow@25162
|
1035 |
by(auto simp: image_def split:nat.split dest:gr0_implies_Suc)
|
nipkow@15281
|
1036 |
have "length (filter p (x # xs)) = Suc(card ?S)"
|
wenzelm@23388
|
1037 |
using Cons `p x` by simp
|
nipkow@15281
|
1038 |
also have "\<dots> = Suc(card(Suc ` ?S))" using fin
|
nipkow@15281
|
1039 |
by (simp add: card_image inj_Suc)
|
nipkow@15281
|
1040 |
also have "\<dots> = card ?S'" using eq fin
|
nipkow@15281
|
1041 |
by (simp add:card_insert_if) (simp add:image_def)
|
nipkow@15281
|
1042 |
finally show ?thesis .
|
nipkow@15281
|
1043 |
next
|
nipkow@15281
|
1044 |
assume "\<not> p x"
|
nipkow@15281
|
1045 |
hence eq: "?S' = Suc ` ?S"
|
nipkow@25162
|
1046 |
by(auto simp add: image_def split:nat.split elim:lessE)
|
nipkow@15281
|
1047 |
have "length (filter p (x # xs)) = card ?S"
|
wenzelm@23388
|
1048 |
using Cons `\<not> p x` by simp
|
nipkow@15281
|
1049 |
also have "\<dots> = card(Suc ` ?S)" using fin
|
nipkow@15281
|
1050 |
by (simp add: card_image inj_Suc)
|
nipkow@15281
|
1051 |
also have "\<dots> = card ?S'" using eq fin
|
nipkow@15281
|
1052 |
by (simp add:card_insert_if)
|
nipkow@15281
|
1053 |
finally show ?thesis .
|
nipkow@15281
|
1054 |
qed
|
nipkow@15281
|
1055 |
qed
|
nipkow@15281
|
1056 |
|
nipkow@17629
|
1057 |
lemma Cons_eq_filterD:
|
nipkow@17629
|
1058 |
"x#xs = filter P ys \<Longrightarrow>
|
nipkow@17629
|
1059 |
\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
|
wenzelm@19585
|
1060 |
(is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
|
nipkow@17629
|
1061 |
proof(induct ys)
|
nipkow@17629
|
1062 |
case Nil thus ?case by simp
|
nipkow@17629
|
1063 |
next
|
nipkow@17629
|
1064 |
case (Cons y ys)
|
nipkow@17629
|
1065 |
show ?case (is "\<exists>x. ?Q x")
|
nipkow@17629
|
1066 |
proof cases
|
nipkow@17629
|
1067 |
assume Py: "P y"
|
nipkow@17629
|
1068 |
show ?thesis
|
nipkow@17629
|
1069 |
proof cases
|
wenzelm@25221
|
1070 |
assume "x = y"
|
wenzelm@25221
|
1071 |
with Py Cons.prems have "?Q []" by simp
|
wenzelm@25221
|
1072 |
then show ?thesis ..
|
nipkow@17629
|
1073 |
next
|
wenzelm@25221
|
1074 |
assume "x \<noteq> y"
|
wenzelm@25221
|
1075 |
with Py Cons.prems show ?thesis by simp
|
nipkow@17629
|
1076 |
qed
|
nipkow@17629
|
1077 |
next
|
wenzelm@25221
|
1078 |
assume "\<not> P y"
|
wenzelm@25221
|
1079 |
with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastsimp
|
wenzelm@25221
|
1080 |
then have "?Q (y#us)" by simp
|
wenzelm@25221
|
1081 |
then show ?thesis ..
|
nipkow@17629
|
1082 |
qed
|
nipkow@17629
|
1083 |
qed
|
nipkow@17629
|
1084 |
|
nipkow@17629
|
1085 |
lemma filter_eq_ConsD:
|
nipkow@17629
|
1086 |
"filter P ys = x#xs \<Longrightarrow>
|
nipkow@17629
|
1087 |
\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
|
nipkow@17629
|
1088 |
by(rule Cons_eq_filterD) simp
|
nipkow@17629
|
1089 |
|
nipkow@17629
|
1090 |
lemma filter_eq_Cons_iff:
|
nipkow@17629
|
1091 |
"(filter P ys = x#xs) =
|
nipkow@17629
|
1092 |
(\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
|
nipkow@17629
|
1093 |
by(auto dest:filter_eq_ConsD)
|
nipkow@17629
|
1094 |
|
nipkow@17629
|
1095 |
lemma Cons_eq_filter_iff:
|
nipkow@17629
|
1096 |
"(x#xs = filter P ys) =
|
nipkow@17629
|
1097 |
(\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
|
nipkow@17629
|
1098 |
by(auto dest:Cons_eq_filterD)
|
nipkow@17629
|
1099 |
|
krauss@19770
|
1100 |
lemma filter_cong[fundef_cong, recdef_cong]:
|
nipkow@17501
|
1101 |
"xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
|
nipkow@17501
|
1102 |
apply simp
|
nipkow@17501
|
1103 |
apply(erule thin_rl)
|
nipkow@17501
|
1104 |
by (induct ys) simp_all
|
nipkow@17501
|
1105 |
|
nipkow@15281
|
1106 |
|
haftmann@26442
|
1107 |
subsubsection {* List partitioning *}
|
haftmann@26442
|
1108 |
|
haftmann@26442
|
1109 |
primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where
|
haftmann@26442
|
1110 |
"partition P [] = ([], [])"
|
haftmann@26442
|
1111 |
| "partition P (x # xs) =
|
haftmann@26442
|
1112 |
(let (yes, no) = partition P xs
|
haftmann@26442
|
1113 |
in if P x then (x # yes, no) else (yes, x # no))"
|
haftmann@26442
|
1114 |
|
haftmann@26442
|
1115 |
lemma partition_filter1:
|
haftmann@26442
|
1116 |
"fst (partition P xs) = filter P xs"
|
haftmann@26442
|
1117 |
by (induct xs) (auto simp add: Let_def split_def)
|
haftmann@26442
|
1118 |
|
haftmann@26442
|
1119 |
lemma partition_filter2:
|
haftmann@26442
|
1120 |
"snd (partition P xs) = filter (Not o P) xs"
|
haftmann@26442
|
1121 |
by (induct xs) (auto simp add: Let_def split_def)
|
haftmann@26442
|
1122 |
|
haftmann@26442
|
1123 |
lemma partition_P:
|
haftmann@26442
|
1124 |
assumes "partition P xs = (yes, no)"
|
haftmann@26442
|
1125 |
shows "(\<forall>p \<in> set yes. P p) \<and> (\<forall>p \<in> set no. \<not> P p)"
|
haftmann@26442
|
1126 |
proof -
|
haftmann@26442
|
1127 |
from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
|
haftmann@26442
|
1128 |
by simp_all
|
haftmann@26442
|
1129 |
then show ?thesis by (simp_all add: partition_filter1 partition_filter2)
|
haftmann@26442
|
1130 |
qed
|
haftmann@26442
|
1131 |
|
haftmann@26442
|
1132 |
lemma partition_set:
|
haftmann@26442
|
1133 |
assumes "partition P xs = (yes, no)"
|
haftmann@26442
|
1134 |
shows "set yes \<union> set no = set xs"
|
haftmann@26442
|
1135 |
proof -
|
haftmann@26442
|
1136 |
from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
|
haftmann@26442
|
1137 |
by simp_all
|
haftmann@26442
|
1138 |
then show ?thesis by (auto simp add: partition_filter1 partition_filter2)
|
haftmann@26442
|
1139 |
qed
|
haftmann@26442
|
1140 |
|
haftmann@26442
|
1141 |
|
nipkow@15392
|
1142 |
subsubsection {* @{text concat} *}
|
wenzelm@13114
|
1143 |
|
wenzelm@13142
|
1144 |
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
|
nipkow@13145
|
1145 |
by (induct xs) auto
|
wenzelm@13114
|
1146 |
|
paulson@18447
|
1147 |
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
|
nipkow@13145
|
1148 |
by (induct xss) auto
|
wenzelm@13114
|
1149 |
|
paulson@18447
|
1150 |
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
|
nipkow@13145
|
1151 |
by (induct xss) auto
|
wenzelm@13114
|
1152 |
|
nipkow@24308
|
1153 |
lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
|
nipkow@13145
|
1154 |
by (induct xs) auto
|
wenzelm@13114
|
1155 |
|
nipkow@24476
|
1156 |
lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs"
|
nipkow@24349
|
1157 |
by (induct xs) auto
|
nipkow@24349
|
1158 |
|
wenzelm@13142
|
1159 |
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
|
nipkow@13145
|
1160 |
by (induct xs) auto
|
wenzelm@13114
|
1161 |
|
wenzelm@13142
|
1162 |
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
|
nipkow@13145
|
1163 |
by (induct xs) auto
|
wenzelm@13114
|
1164 |
|
wenzelm@13142
|
1165 |
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
|
nipkow@13145
|
1166 |
by (induct xs) auto
|
wenzelm@13114
|
1167 |
|
wenzelm@13114
|
1168 |
|
nipkow@15392
|
1169 |
subsubsection {* @{text nth} *}
|
wenzelm@13114
|
1170 |
|
wenzelm@13142
|
1171 |
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
|
nipkow@13145
|
1172 |
by auto
|
wenzelm@13114
|
1173 |
|
wenzelm@13142
|
1174 |
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
|
nipkow@13145
|
1175 |
by auto
|
wenzelm@13114
|
1176 |
|
wenzelm@13142
|
1177 |
declare nth.simps [simp del]
|
wenzelm@13114
|
1178 |
|
wenzelm@13114
|
1179 |
lemma nth_append:
|
nipkow@24526
|
1180 |
"(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
|
nipkow@24526
|
1181 |
apply (induct xs arbitrary: n, simp)
|
paulson@14208
|
1182 |
apply (case_tac n, auto)
|
nipkow@13145
|
1183 |
done
|
wenzelm@13114
|
1184 |
|
nipkow@14402
|
1185 |
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
|
wenzelm@25221
|
1186 |
by (induct xs) auto
|
nipkow@14402
|
1187 |
|
nipkow@14402
|
1188 |
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
|
wenzelm@25221
|
1189 |
by (induct xs) auto
|
nipkow@14402
|
1190 |
|
nipkow@24526
|
1191 |
lemma nth_map [simp]: "n < length xs ==> (map f xs)!n = f(xs!n)"
|
nipkow@24526
|
1192 |
apply (induct xs arbitrary: n, simp)
|
paulson@14208
|
1193 |
apply (case_tac n, auto)
|
nipkow@13145
|
1194 |
done
|
wenzelm@13114
|
1195 |
|
nipkow@18423
|
1196 |
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
|
nipkow@18423
|
1197 |
by(cases xs) simp_all
|
nipkow@18423
|
1198 |
|
nipkow@18049
|
1199 |
|
nipkow@18049
|
1200 |
lemma list_eq_iff_nth_eq:
|
nipkow@24526
|
1201 |
"(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
|
nipkow@24526
|
1202 |
apply(induct xs arbitrary: ys)
|
paulson@24632
|
1203 |
apply force
|
nipkow@18049
|
1204 |
apply(case_tac ys)
|
nipkow@18049
|
1205 |
apply simp
|
nipkow@18049
|
1206 |
apply(simp add:nth_Cons split:nat.split)apply blast
|
nipkow@18049
|
1207 |
done
|
nipkow@18049
|
1208 |
|
wenzelm@13142
|
1209 |
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
|
paulson@15251
|
1210 |
apply (induct xs, simp, simp)
|
nipkow@13145
|
1211 |
apply safe
|
paulson@24632
|
1212 |
apply (metis nat_case_0 nth.simps zero_less_Suc)
|
paulson@24632
|
1213 |
apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
|
paulson@14208
|
1214 |
apply (case_tac i, simp)
|
paulson@24632
|
1215 |
apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff)
|
nipkow@13145
|
1216 |
done
|
wenzelm@13114
|
1217 |
|
nipkow@17501
|
1218 |
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
|
nipkow@17501
|
1219 |
by(auto simp:set_conv_nth)
|
nipkow@17501
|
1220 |
|
nipkow@13145
|
1221 |
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
|
nipkow@13145
|
1222 |
by (auto simp add: set_conv_nth)
|
wenzelm@13114
|
1223 |
|
wenzelm@13142
|
1224 |
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
|
nipkow@13145
|
1225 |
by (auto simp add: set_conv_nth)
|
wenzelm@13114
|
1226 |
|
wenzelm@13114
|
1227 |
lemma all_nth_imp_all_set:
|
nipkow@13145
|
1228 |
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
|
nipkow@13145
|
1229 |
by (auto simp add: set_conv_nth)
|
wenzelm@13114
|
1230 |
|
wenzelm@13114
|
1231 |
lemma all_set_conv_all_nth:
|
nipkow@13145
|
1232 |
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
|
nipkow@13145
|
1233 |
by (auto simp add: set_conv_nth)
|
wenzelm@13114
|
1234 |
|
kleing@25296
|
1235 |
lemma rev_nth:
|
kleing@25296
|
1236 |
"n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)"
|
kleing@25296
|
1237 |
proof (induct xs arbitrary: n)
|
kleing@25296
|
1238 |
case Nil thus ?case by simp
|
kleing@25296
|
1239 |
next
|
kleing@25296
|
1240 |
case (Cons x xs)
|
kleing@25296
|
1241 |
hence n: "n < Suc (length xs)" by simp
|
kleing@25296
|
1242 |
moreover
|
kleing@25296
|
1243 |
{ assume "n < length xs"
|
kleing@25296
|
1244 |
with n obtain n' where "length xs - n = Suc n'"
|
kleing@25296
|
1245 |
by (cases "length xs - n", auto)
|
kleing@25296
|
1246 |
moreover
|
kleing@25296
|
1247 |
then have "length xs - Suc n = n'" by simp
|
kleing@25296
|
1248 |
ultimately
|
kleing@25296
|
1249 |
have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp
|
kleing@25296
|
1250 |
}
|
kleing@25296
|
1251 |
ultimately
|
kleing@25296
|
1252 |
show ?case by (clarsimp simp add: Cons nth_append)
|
kleing@25296
|
1253 |
qed
|
wenzelm@13114
|
1254 |
|
nipkow@15392
|
1255 |
subsubsection {* @{text list_update} *}
|
wenzelm@13114
|
1256 |
|
nipkow@24526
|
1257 |
lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
|
nipkow@24526
|
1258 |
by (induct xs arbitrary: i) (auto split: nat.split)
|
wenzelm@13114
|
1259 |
|
wenzelm@13114
|
1260 |
lemma nth_list_update:
|
nipkow@24526
|
1261 |
"i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
|
nipkow@24526
|
1262 |
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
|
wenzelm@13114
|
1263 |
|
wenzelm@13142
|
1264 |
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
|
nipkow@13145
|
1265 |
by (simp add: nth_list_update)
|
wenzelm@13114
|
1266 |
|
nipkow@24526
|
1267 |
lemma nth_list_update_neq [simp]: "i \<noteq> j ==> xs[i:=x]!j = xs!j"
|
nipkow@24526
|
1268 |
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
|
wenzelm@13114
|
1269 |
|
nipkow@24526
|
1270 |
lemma list_update_id[simp]: "xs[i := xs!i] = xs"
|
nipkow@24526
|
1271 |
by (induct xs arbitrary: i) (simp_all split:nat.splits)
|
nipkow@24526
|
1272 |
|
nipkow@24526
|
1273 |
lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
|
nipkow@24526
|
1274 |
apply (induct xs arbitrary: i)
|
nipkow@17501
|
1275 |
apply simp
|
nipkow@17501
|
1276 |
apply (case_tac i)
|
nipkow@17501
|
1277 |
apply simp_all
|
nipkow@17501
|
1278 |
done
|
nipkow@17501
|
1279 |
|
wenzelm@13114
|
1280 |
lemma list_update_same_conv:
|
nipkow@24526
|
1281 |
"i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
|
nipkow@24526
|
1282 |
by (induct xs arbitrary: i) (auto split: nat.split)
|
wenzelm@13114
|
1283 |
|
nipkow@14187
|
1284 |
lemma list_update_append1:
|
nipkow@24526
|
1285 |
"i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
|
nipkow@24526
|
1286 |
apply (induct xs arbitrary: i, simp)
|
nipkow@14187
|
1287 |
apply(simp split:nat.split)
|
nipkow@14187
|
1288 |
done
|
nipkow@14187
|
1289 |
|
kleing@15868
|
1290 |
lemma list_update_append:
|
nipkow@24526
|
1291 |
"(xs @ ys) [n:= x] =
|
kleing@15868
|
1292 |
(if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
|
nipkow@24526
|
1293 |
by (induct xs arbitrary: n) (auto split:nat.splits)
|
kleing@15868
|
1294 |
|
nipkow@14402
|
1295 |
lemma list_update_length [simp]:
|
nipkow@14402
|
1296 |
"(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
|
nipkow@14402
|
1297 |
by (induct xs, auto)
|
nipkow@14402
|
1298 |
|
wenzelm@13114
|
1299 |
lemma update_zip:
|
nipkow@24526
|
1300 |
"length xs = length ys ==>
|
nipkow@24526
|
1301 |
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
|
nipkow@24526
|
1302 |
by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split)
|
nipkow@24526
|
1303 |
|
nipkow@24526
|
1304 |
lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)"
|
nipkow@24526
|
1305 |
by (induct xs arbitrary: i) (auto split: nat.split)
|
wenzelm@13114
|
1306 |
|
wenzelm@13114
|
1307 |
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
|
nipkow@13145
|
1308 |
by (blast dest!: set_update_subset_insert [THEN subsetD])
|
wenzelm@13114
|
1309 |
|
nipkow@24526
|
1310 |
lemma set_update_memI: "n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
|
nipkow@24526
|
1311 |
by (induct xs arbitrary: n) (auto split:nat.splits)
|
kleing@15868
|
1312 |
|
haftmann@24796
|
1313 |
lemma list_update_overwrite:
|
haftmann@24796
|
1314 |
"xs [i := x, i := y] = xs [i := y]"
|
haftmann@24796
|
1315 |
apply (induct xs arbitrary: i)
|
haftmann@24796
|
1316 |
apply simp
|
haftmann@24796
|
1317 |
apply (case_tac i)
|
haftmann@24796
|
1318 |
apply simp_all
|
haftmann@24796
|
1319 |
done
|
haftmann@24796
|
1320 |
|
haftmann@24796
|
1321 |
lemma list_update_swap:
|
haftmann@24796
|
1322 |
"i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]"
|
haftmann@24796
|
1323 |
apply (induct xs arbitrary: i i')
|
haftmann@24796
|
1324 |
apply simp
|
haftmann@24796
|
1325 |
apply (case_tac i, case_tac i')
|
haftmann@24796
|
1326 |
apply auto
|
haftmann@24796
|
1327 |
apply (case_tac i')
|
haftmann@24796
|
1328 |
apply auto
|
haftmann@24796
|
1329 |
done
|
haftmann@24796
|
1330 |
|
wenzelm@13114
|
1331 |
|
nipkow@15392
|
1332 |
subsubsection {* @{text last} and @{text butlast} *}
|
wenzelm@13114
|
1333 |
|
wenzelm@13142
|
1334 |
lemma last_snoc [simp]: "last (xs @ [x]) = x"
|
nipkow@13145
|
1335 |
by (induct xs) auto
|
wenzelm@13114
|
1336 |
|
wenzelm@13142
|
1337 |
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
|
nipkow@13145
|
1338 |
by (induct xs) auto
|
wenzelm@13114
|
1339 |
|
nipkow@14302
|
1340 |
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
|
nipkow@14302
|
1341 |
by(simp add:last.simps)
|
nipkow@14302
|
1342 |
|
nipkow@14302
|
1343 |
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
|
nipkow@14302
|
1344 |
by(simp add:last.simps)
|
nipkow@14302
|
1345 |
|
nipkow@14302
|
1346 |
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
|
nipkow@14302
|
1347 |
by (induct xs) (auto)
|
nipkow@14302
|
1348 |
|
nipkow@14302
|
1349 |
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
|
nipkow@14302
|
1350 |
by(simp add:last_append)
|
nipkow@14302
|
1351 |
|
nipkow@14302
|
1352 |
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
|
nipkow@14302
|
1353 |
by(simp add:last_append)
|
nipkow@14302
|
1354 |
|
nipkow@17762
|
1355 |
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
|
nipkow@17762
|
1356 |
by(rule rev_exhaust[of xs]) simp_all
|
nipkow@17762
|
1357 |
|
nipkow@17762
|
1358 |
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
|
nipkow@17762
|
1359 |
by(cases xs) simp_all
|
nipkow@17762
|
1360 |
|
nipkow@17765
|
1361 |
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
|
nipkow@17765
|
1362 |
by (induct as) auto
|
nipkow@17762
|
1363 |
|
wenzelm@13142
|
1364 |
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
|
nipkow@13145
|
1365 |
by (induct xs rule: rev_induct) auto
|
wenzelm@13114
|
1366 |
|
wenzelm@13114
|
1367 |
lemma butlast_append:
|
nipkow@24526
|
1368 |
"butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
|
nipkow@24526
|
1369 |
by (induct xs arbitrary: ys) auto
|
wenzelm@13114
|
1370 |
|
wenzelm@13142
|
1371 |
lemma append_butlast_last_id [simp]:
|
nipkow@13145
|
1372 |
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
|
nipkow@13145
|
1373 |
by (induct xs) auto
|
wenzelm@13114
|
1374 |
|
wenzelm@13142
|
1375 |
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
|
nipkow@13145
|
1376 |
by (induct xs) (auto split: split_if_asm)
|
wenzelm@13114
|
1377 |
|
wenzelm@13114
|
1378 |
lemma in_set_butlast_appendI:
|
nipkow@13145
|
1379 |
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
|
nipkow@13145
|
1380 |
by (auto dest: in_set_butlastD simp add: butlast_append)
|
wenzelm@13114
|
1381 |
|
nipkow@24526
|
1382 |
lemma last_drop[simp]: "n < length xs \<Longrightarrow> last (drop n xs) = last xs"
|
nipkow@24526
|
1383 |
apply (induct xs arbitrary: n)
|
nipkow@17501
|
1384 |
apply simp
|
nipkow@17501
|
1385 |
apply (auto split:nat.split)
|
nipkow@17501
|
1386 |
done
|
nipkow@17501
|
1387 |
|
nipkow@17589
|
1388 |
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
|
nipkow@17589
|
1389 |
by(induct xs)(auto simp:neq_Nil_conv)
|
nipkow@17589
|
1390 |
|
haftmann@24796
|
1391 |
|
nipkow@15392
|
1392 |
subsubsection {* @{text take} and @{text drop} *}
|
wenzelm@13114
|
1393 |
|
wenzelm@13142
|
1394 |
lemma take_0 [simp]: "take 0 xs = []"
|
nipkow@13145
|
1395 |
by (induct xs) auto
|
wenzelm@13114
|
1396 |
|
wenzelm@13142
|
1397 |
lemma drop_0 [simp]: "drop 0 xs = xs"
|
nipkow@13145
|
1398 |
by (induct xs) auto
|
wenzelm@13114
|
1399 |
|
wenzelm@13142
|
1400 |
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
|
nipkow@13145
|
1401 |
by simp
|
wenzelm@13114
|
1402 |
|
wenzelm@13142
|
1403 |
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
|
nipkow@13145
|
1404 |
by simp
|
wenzelm@13114
|
1405 |
|
wenzelm@13142
|
1406 |
declare take_Cons [simp del] and drop_Cons [simp del]
|
wenzelm@13114
|
1407 |
|
nipkow@15110
|
1408 |
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
|
nipkow@15110
|
1409 |
by(clarsimp simp add:neq_Nil_conv)
|
nipkow@15110
|
1410 |
|
nipkow@14187
|
1411 |
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
|
nipkow@14187
|
1412 |
by(cases xs, simp_all)
|
nipkow@14187
|
1413 |
|
nipkow@24526
|
1414 |
lemma drop_tl: "drop n (tl xs) = tl(drop n xs)"
|
nipkow@24526
|
1415 |
by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split)
|
nipkow@24526
|
1416 |
|
nipkow@24526
|
1417 |
lemma nth_via_drop: "drop n xs = y#ys \<Longrightarrow> xs!n = y"
|
nipkow@24526
|
1418 |
apply (induct xs arbitrary: n, simp)
|
nipkow@14187
|
1419 |
apply(simp add:drop_Cons nth_Cons split:nat.splits)
|
nipkow@14187
|
1420 |
done
|
nipkow@14187
|
1421 |
|
nipkow@13913
|
1422 |
lemma take_Suc_conv_app_nth:
|
nipkow@24526
|
1423 |
"i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
|
nipkow@24526
|
1424 |
apply (induct xs arbitrary: i, simp)
|
paulson@14208
|
1425 |
apply (case_tac i, auto)
|
nipkow@13913
|
1426 |
done
|
nipkow@13913
|
1427 |
|
mehta@14591
|
1428 |
lemma drop_Suc_conv_tl:
|
nipkow@24526
|
1429 |
"i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
|
nipkow@24526
|
1430 |
apply (induct xs arbitrary: i, simp)
|
mehta@14591
|
1431 |
apply (case_tac i, auto)
|
mehta@14591
|
1432 |
done
|
mehta@14591
|
1433 |
|
nipkow@24526
|
1434 |
lemma length_take [simp]: "length (take n xs) = min (length xs) n"
|
nipkow@24526
|
1435 |
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
|
nipkow@24526
|
1436 |
|
nipkow@24526
|
1437 |
lemma length_drop [simp]: "length (drop n xs) = (length xs - n)"
|
nipkow@24526
|
1438 |
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
|
nipkow@24526
|
1439 |
|
nipkow@24526
|
1440 |
lemma take_all [simp]: "length xs <= n ==> take n xs = xs"
|
nipkow@24526
|
1441 |
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
|
nipkow@24526
|
1442 |
|
nipkow@24526
|
1443 |
lemma drop_all [simp]: "length xs <= n ==> drop n xs = []"
|
nipkow@24526
|
1444 |
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
|
wenzelm@13114
|
1445 |
|
wenzelm@13142
|
1446 |
lemma take_append [simp]:
|
nipkow@24526
|
1447 |
"take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
|
nipkow@24526
|
1448 |
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
|
wenzelm@13114
|
1449 |
|
wenzelm@13142
|
1450 |
lemma drop_append [simp]:
|
nipkow@24526
|
1451 |
"drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
|
nipkow@24526
|
1452 |
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
|
nipkow@24526
|
1453 |
|
nipkow@24526
|
1454 |
lemma take_take [simp]: "take n (take m xs) = take (min n m) xs"
|
nipkow@24526
|
1455 |
apply (induct m arbitrary: xs n, auto)
|
paulson@14208
|
1456 |
apply (case_tac xs, auto)
|
nipkow@15236
|
1457 |
apply (case_tac n, auto)
|
nipkow@13145
|
1458 |
done
|
wenzelm@13142
|
1459 |
|
nipkow@24526
|
1460 |
lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs"
|
nipkow@24526
|
1461 |
apply (induct m arbitrary: xs, auto)
|
paulson@14208
|
1462 |
apply (case_tac xs, auto)
|
nipkow@13145
|
1463 |
done
|
wenzelm@13114
|
1464 |
|
nipkow@24526
|
1465 |
lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)"
|
nipkow@24526
|
1466 |
apply (induct m arbitrary: xs n, auto)
|
paulson@14208
|
1467 |
apply (case_tac xs, auto)
|
nipkow@13145
|
1468 |
done
|
wenzelm@13114
|
1469 |
|
nipkow@24526
|
1470 |
lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)"
|
nipkow@24526
|
1471 |
apply(induct xs arbitrary: m n)
|
nipkow@14802
|
1472 |
apply simp
|
nipkow@14802
|
1473 |
apply(simp add: take_Cons drop_Cons split:nat.split)
|
nipkow@14802
|
1474 |
done
|
nipkow@14802
|
1475 |
|
nipkow@24526
|
1476 |
lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs"
|
nipkow@24526
|
1477 |
apply (induct n arbitrary: xs, auto)
|
paulson@14208
|
1478 |
apply (case_tac xs, auto)
|
nipkow@13145
|
1479 |
done
|
wenzelm@13114
|
1480 |
|
nipkow@24526
|
1481 |
lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])"
|
nipkow@24526
|
1482 |
apply(induct xs arbitrary: n)
|
nipkow@15110
|
1483 |
apply simp
|
nipkow@15110
|
1484 |
apply(simp add:take_Cons split:nat.split)
|
nipkow@15110
|
1485 |
done
|
nipkow@15110
|
1486 |
|
nipkow@24526
|
1487 |
lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)"
|
nipkow@24526
|
1488 |
apply(induct xs arbitrary: n)
|
nipkow@15110
|
1489 |
apply simp
|
nipkow@15110
|
1490 |
apply(simp add:drop_Cons split:nat.split)
|
nipkow@15110
|
1491 |
done
|
nipkow@15110
|
1492 |
|
nipkow@24526
|
1493 |
lemma take_map: "take n (map f xs) = map f (take n xs)"
|
nipkow@24526
|
1494 |
apply (induct n arbitrary: xs, auto)
|
paulson@14208
|
1495 |
apply (case_tac xs, auto)
|
nipkow@13145
|
1496 |
done
|
wenzelm@13114
|
1497 |
|
nipkow@24526
|
1498 |
lemma drop_map: "drop n (map f xs) = map f (drop n xs)"
|
nipkow@24526
|
1499 |
apply (induct n arbitrary: xs, auto)
|
paulson@14208
|
1500 |
apply (case_tac xs, auto)
|
nipkow@13145
|
1501 |
done
|
wenzelm@13114
|
1502 |
|
nipkow@24526
|
1503 |
lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)"
|
nipkow@24526
|
1504 |
apply (induct xs arbitrary: i, auto)
|
paulson@14208
|
1505 |
apply (case_tac i, auto)
|
nipkow@13145
|
1506 |
done
|
wenzelm@13114
|
1507 |
|
nipkow@24526
|
1508 |
lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)"
|
nipkow@24526
|
1509 |
apply (induct xs arbitrary: i, auto)
|
paulson@14208
|
1510 |
apply (case_tac i, auto)
|
nipkow@13145
|
1511 |
done
|
wenzelm@13114
|
1512 |
|
nipkow@24526
|
1513 |
lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i"
|
nipkow@24526
|
1514 |
apply (induct xs arbitrary: i n, auto)
|
paulson@14208
|
1515 |
apply (case_tac n, blast)
|
paulson@14208
|
1516 |
apply (case_tac i, auto)
|
nipkow@13145
|
1517 |
done
|
wenzelm@13114
|
1518 |
|
wenzelm@13142
|
1519 |
lemma nth_drop [simp]:
|
nipkow@24526
|
1520 |
"n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
|
nipkow@24526
|
1521 |
apply (induct n arbitrary: xs i, auto)
|
paulson@14208
|
1522 |
apply (case_tac xs, auto)
|
nipkow@13145
|
1523 |
done
|
wenzelm@13114
|
1524 |
|
nipkow@18423
|
1525 |
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
|
nipkow@18423
|
1526 |
by(simp add: hd_conv_nth)
|
nipkow@18423
|
1527 |
|
nipkow@24526
|
1528 |
lemma set_take_subset: "set(take n xs) \<subseteq> set xs"
|
nipkow@24526
|
1529 |
by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split)
|
nipkow@24526
|
1530 |
|
nipkow@24526
|
1531 |
lemma set_drop_subset: "set(drop n xs) \<subseteq> set xs"
|
nipkow@24526
|
1532 |
by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split)
|
nipkow@14025
|
1533 |
|
nipkow@14187
|
1534 |
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
|
nipkow@14187
|
1535 |
using set_take_subset by fast
|
nipkow@14187
|
1536 |
|
nipkow@14187
|
1537 |
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
|
nipkow@14187
|
1538 |
using set_drop_subset by fast
|
nipkow@14187
|
1539 |
|
wenzelm@13114
|
1540 |
lemma append_eq_conv_conj:
|
nipkow@24526
|
1541 |
"(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
|
nipkow@24526
|
1542 |
apply (induct xs arbitrary: zs, simp, clarsimp)
|
paulson@14208
|
1543 |
apply (case_tac zs, auto)
|
nipkow@13145
|
1544 |
done
|
wenzelm@13114
|
1545 |
|
nipkow@24526
|
1546 |
lemma take_add:
|
nipkow@24526
|
1547 |
"i+j \<le> length(xs) \<Longrightarrow> take (i+j) xs = take i xs @ take j (drop i xs)"
|
nipkow@24526
|
1548 |
apply (induct xs arbitrary: i, auto)
|
nipkow@24526
|
1549 |
apply (case_tac i, simp_all)
|
paulson@14050
|
1550 |
done
|
paulson@14050
|
1551 |
|
nipkow@14300
|
1552 |
lemma append_eq_append_conv_if:
|
nipkow@24526
|
1553 |
"(xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
|
nipkow@14300
|
1554 |
(if size xs\<^isub>1 \<le> size ys\<^isub>1
|
nipkow@14300
|
1555 |
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
|
1556 |
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@24526
|
1557 |
apply(induct xs\<^isub>1 arbitrary: ys\<^isub>1)
|
nipkow@14300
|
1558 |
apply simp
|
nipkow@14300
|
1559 |
apply(case_tac ys\<^isub>1)
|
nipkow@14300
|
1560 |
apply simp_all
|
nipkow@14300
|
1561 |
done
|
nipkow@14300
|
1562 |
|
nipkow@15110
|
1563 |
lemma take_hd_drop:
|
nipkow@24526
|
1564 |
"n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
|
nipkow@24526
|
1565 |
apply(induct xs arbitrary: n)
|
nipkow@15110
|
1566 |
apply simp
|
nipkow@15110
|
1567 |
apply(simp add:drop_Cons split:nat.split)
|
nipkow@15110
|
1568 |
done
|
nipkow@15110
|
1569 |
|
nipkow@17501
|
1570 |
lemma id_take_nth_drop:
|
nipkow@17501
|
1571 |
"i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs"
|
nipkow@17501
|
1572 |
proof -
|
nipkow@17501
|
1573 |
assume si: "i < length xs"
|
nipkow@17501
|
1574 |
hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
|
nipkow@17501
|
1575 |
moreover
|
nipkow@17501
|
1576 |
from si have "take (Suc i) xs = take i xs @ [xs!i]"
|
nipkow@17501
|
1577 |
apply (rule_tac take_Suc_conv_app_nth) by arith
|
nipkow@17501
|
1578 |
ultimately show ?thesis by auto
|
nipkow@17501
|
1579 |
qed
|
nipkow@17501
|
1580 |
|
nipkow@17501
|
1581 |
lemma upd_conv_take_nth_drop:
|
nipkow@17501
|
1582 |
"i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
|
nipkow@17501
|
1583 |
proof -
|
nipkow@17501
|
1584 |
assume i: "i < length xs"
|
nipkow@17501
|
1585 |
have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
|
nipkow@17501
|
1586 |
by(rule arg_cong[OF id_take_nth_drop[OF i]])
|
nipkow@17501
|
1587 |
also have "\<dots> = take i xs @ a # drop (Suc i) xs"
|
nipkow@17501
|
1588 |
using i by (simp add: list_update_append)
|
nipkow@17501
|
1589 |
finally show ?thesis .
|
nipkow@17501
|
1590 |
qed
|
nipkow@17501
|
1591 |
|
haftmann@24796
|
1592 |
lemma nth_drop':
|
haftmann@24796
|
1593 |
"i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs"
|
haftmann@24796
|
1594 |
apply (induct i arbitrary: xs)
|
haftmann@24796
|
1595 |
apply (simp add: neq_Nil_conv)
|
haftmann@24796
|
1596 |
apply (erule exE)+
|
haftmann@24796
|
1597 |
apply simp
|
haftmann@24796
|
1598 |
apply (case_tac xs)
|
haftmann@24796
|
1599 |
apply simp_all
|
haftmann@24796
|
1600 |
done
|
haftmann@24796
|
1601 |
|
wenzelm@13114
|
1602 |
|
nipkow@15392
|
1603 |
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
|
wenzelm@13114
|
1604 |
|
wenzelm@13142
|
1605 |
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
|
nipkow@13145
|
1606 |
by (induct xs) auto
|
wenzelm@13114
|
1607 |
|
wenzelm@13142
|
1608 |
lemma takeWhile_append1 [simp]:
|
nipkow@13145
|
1609 |
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
|
nipkow@13145
|
1610 |
by (induct xs) auto
|
wenzelm@13114
|
1611 |
|
wenzelm@13142
|
1612 |
lemma takeWhile_append2 [simp]:
|
nipkow@13145
|
1613 |
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
|
nipkow@13145
|
1614 |
by (induct xs) auto
|
wenzelm@13114
|
1615 |
|
wenzelm@13142
|
1616 |
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
|
nipkow@13145
|
1617 |
by (induct xs) auto
|
wenzelm@13114
|
1618 |
|
wenzelm@13142
|
1619 |
lemma dropWhile_append1 [simp]:
|
nipkow@13145
|
1620 |
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
|
nipkow@13145
|
1621 |
by (induct xs) auto
|
wenzelm@13114
|
1622 |
|
wenzelm@13142
|
1623 |
lemma dropWhile_append2 [simp]:
|
nipkow@13145
|
1624 |
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
|
nipkow@13145
|
1625 |
by (induct xs) auto
|
wenzelm@13114
|
1626 |
|
krauss@23971
|
1627 |
lemma set_takeWhileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
|
nipkow@13145
|
1628 |
by (induct xs) (auto split: split_if_asm)
|
wenzelm@13114
|
1629 |
|
nipkow@13913
|
1630 |
lemma takeWhile_eq_all_conv[simp]:
|
nipkow@13913
|
1631 |
"(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
|
nipkow@13913
|
1632 |
by(induct xs, auto)
|
nipkow@13913
|
1633 |
|
nipkow@13913
|
1634 |
lemma dropWhile_eq_Nil_conv[simp]:
|
nipkow@13913
|
1635 |
"(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
|
nipkow@13913
|
1636 |
by(induct xs, auto)
|
nipkow@13913
|
1637 |
|
nipkow@13913
|
1638 |
lemma dropWhile_eq_Cons_conv:
|
nipkow@13913
|
1639 |
"(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
|
nipkow@13913
|
1640 |
by(induct xs, auto)
|
nipkow@13913
|
1641 |
|
nipkow@17501
|
1642 |
text{* The following two lemmmas could be generalized to an arbitrary
|
nipkow@17501
|
1643 |
property. *}
|
nipkow@17501
|
1644 |
|
nipkow@17501
|
1645 |
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
|
nipkow@17501
|
1646 |
takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
|
nipkow@17501
|
1647 |
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
|
nipkow@17501
|
1648 |
|
nipkow@17501
|
1649 |
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
|
nipkow@17501
|
1650 |
dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
|
nipkow@17501
|
1651 |
apply(induct xs)
|
nipkow@17501
|
1652 |
apply simp
|
nipkow@17501
|
1653 |
apply auto
|
nipkow@17501
|
1654 |
apply(subst dropWhile_append2)
|
nipkow@17501
|
1655 |
apply auto
|
nipkow@17501
|
1656 |
done
|
nipkow@17501
|
1657 |
|
nipkow@18423
|
1658 |
lemma takeWhile_not_last:
|
nipkow@18423
|
1659 |
"\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
|
nipkow@18423
|
1660 |
apply(induct xs)
|
nipkow@18423
|
1661 |
apply simp
|
nipkow@18423
|
1662 |
apply(case_tac xs)
|
nipkow@18423
|
1663 |
apply(auto)
|
nipkow@18423
|
1664 |
done
|
nipkow@18423
|
1665 |
|
krauss@19770
|
1666 |
lemma takeWhile_cong [fundef_cong, recdef_cong]:
|
krauss@18336
|
1667 |
"[| l = k; !!x. x : set l ==> P x = Q x |]
|
krauss@18336
|
1668 |
==> takeWhile P l = takeWhile Q k"
|
nipkow@24349
|
1669 |
by (induct k arbitrary: l) (simp_all)
|
krauss@18336
|
1670 |
|
krauss@19770
|
1671 |
lemma dropWhile_cong [fundef_cong, recdef_cong]:
|
krauss@18336
|
1672 |
"[| l = k; !!x. x : set l ==> P x = Q x |]
|
krauss@18336
|
1673 |
==> dropWhile P l = dropWhile Q k"
|
nipkow@24349
|
1674 |
by (induct k arbitrary: l, simp_all)
|
krauss@18336
|
1675 |
|
wenzelm@13114
|
1676 |
|
nipkow@15392
|
1677 |
subsubsection {* @{text zip} *}
|
wenzelm@13114
|
1678 |
|
wenzelm@13142
|
1679 |
lemma zip_Nil [simp]: "zip [] ys = []"
|
nipkow@13145
|
1680 |
by (induct ys) auto
|
wenzelm@13114
|
1681 |
|
wenzelm@13142
|
1682 |
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
|
nipkow@13145
|
1683 |
by simp
|
wenzelm@13114
|
1684 |
|
wenzelm@13142
|
1685 |
declare zip_Cons [simp del]
|
wenzelm@13114
|
1686 |
|
nipkow@15281
|
1687 |
lemma zip_Cons1:
|
nipkow@15281
|
1688 |
"zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
|
nipkow@15281
|
1689 |
by(auto split:list.split)
|
nipkow@15281
|
1690 |
|
wenzelm@13142
|
1691 |
lemma length_zip [simp]:
|
krauss@22493
|
1692 |
"length (zip xs ys) = min (length xs) (length ys)"
|
krauss@22493
|
1693 |
by (induct xs ys rule:list_induct2') auto
|
wenzelm@13114
|
1694 |
|
wenzelm@13114
|
1695 |
lemma zip_append1:
|
krauss@22493
|
1696 |
"zip (xs @ ys) zs =
|
nipkow@13145
|
1697 |
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
|
krauss@22493
|
1698 |
by (induct xs zs rule:list_induct2') auto
|
wenzelm@13114
|
1699 |
|
wenzelm@13114
|
1700 |
lemma zip_append2:
|
krauss@22493
|
1701 |
"zip xs (ys @ zs) =
|
nipkow@13145
|
1702 |
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
|
krauss@22493
|
1703 |
by (induct xs ys rule:list_induct2') auto
|
wenzelm@13114
|
1704 |
|
wenzelm@13142
|
1705 |
lemma zip_append [simp]:
|
wenzelm@13142
|
1706 |
"[| length xs = length us; length ys = length vs |] ==>
|
nipkow@13145
|
1707 |
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
|
nipkow@13145
|
1708 |
by (simp add: zip_append1)
|
wenzelm@13114
|
1709 |
|
wenzelm@13114
|
1710 |
lemma zip_rev:
|
nipkow@14247
|
1711 |
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
|
nipkow@14247
|
1712 |
by (induct rule:list_induct2, simp_all)
|
wenzelm@13114
|
1713 |
|
nipkow@23096
|
1714 |
lemma map_zip_map:
|
nipkow@23096
|
1715 |
"map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
|
nipkow@23096
|
1716 |
apply(induct xs arbitrary:ys) apply simp
|
nipkow@23096
|
1717 |
apply(case_tac ys)
|
nipkow@23096
|
1718 |
apply simp_all
|
nipkow@23096
|
1719 |
done
|
nipkow@23096
|
1720 |
|
nipkow@23096
|
1721 |
lemma map_zip_map2:
|
nipkow@23096
|
1722 |
"map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
|
nipkow@23096
|
1723 |
apply(induct xs arbitrary:ys) apply simp
|
nipkow@23096
|
1724 |
apply(case_tac ys)
|
nipkow@23096
|
1725 |
apply simp_all
|
nipkow@23096
|
1726 |
done
|
nipkow@23096
|
1727 |
|
wenzelm@13142
|
1728 |
lemma nth_zip [simp]:
|
nipkow@24526
|
1729 |
"[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
|
nipkow@24526
|
1730 |
apply (induct ys arbitrary: i xs, simp)
|
nipkow@13145
|
1731 |
apply (case_tac xs)
|
nipkow@13145
|
1732 |
apply (simp_all add: nth.simps split: nat.split)
|
nipkow@13145
|
1733 |
done
|
wenzelm@13114
|
1734 |
|
wenzelm@13114
|
1735 |
lemma set_zip:
|
nipkow@13145
|
1736 |
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
|
nipkow@13145
|
1737 |
by (simp add: set_conv_nth cong: rev_conj_cong)
|
wenzelm@13114
|
1738 |
|
wenzelm@13114
|
1739 |
lemma zip_update:
|
nipkow@13145
|
1740 |
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
|
nipkow@13145
|
1741 |
by (rule sym, simp add: update_zip)
|
wenzelm@13114
|
1742 |
|
wenzelm@13142
|
1743 |
lemma zip_replicate [simp]:
|
nipkow@24526
|
1744 |
"zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
|
nipkow@24526
|
1745 |
apply (induct i arbitrary: j, auto)
|
paulson@14208
|
1746 |
apply (case_tac j, auto)
|
nipkow@13145
|
1747 |
done
|
wenzelm@13114
|
1748 |
|
nipkow@19487
|
1749 |
lemma take_zip:
|
nipkow@24526
|
1750 |
"take n (zip xs ys) = zip (take n xs) (take n ys)"
|
nipkow@24526
|
1751 |
apply (induct n arbitrary: xs ys)
|
nipkow@19487
|
1752 |
apply simp
|
nipkow@19487
|
1753 |
apply (case_tac xs, simp)
|
nipkow@19487
|
1754 |
apply (case_tac ys, simp_all)
|
nipkow@19487
|
1755 |
done
|
nipkow@19487
|
1756 |
|
nipkow@19487
|
1757 |
lemma drop_zip:
|
nipkow@24526
|
1758 |
"drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
|
nipkow@24526
|
1759 |
apply (induct n arbitrary: xs ys)
|
nipkow@19487
|
1760 |
apply simp
|
nipkow@19487
|
1761 |
apply (case_tac xs, simp)
|
nipkow@19487
|
1762 |
apply (case_tac ys, simp_all)
|
nipkow@19487
|
1763 |
done
|
nipkow@19487
|
1764 |
|
krauss@22493
|
1765 |
lemma set_zip_leftD:
|
krauss@22493
|
1766 |
"(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
|
krauss@22493
|
1767 |
by (induct xs ys rule:list_induct2') auto
|
krauss@22493
|
1768 |
|
krauss@22493
|
1769 |
lemma set_zip_rightD:
|
krauss@22493
|
1770 |
"(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
|
krauss@22493
|
1771 |
by (induct xs ys rule:list_induct2') auto
|
wenzelm@13142
|
1772 |
|
nipkow@23983
|
1773 |
lemma in_set_zipE:
|
nipkow@23983
|
1774 |
"(x,y) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
|
nipkow@23983
|
1775 |
by(blast dest: set_zip_leftD set_zip_rightD)
|
nipkow@23983
|
1776 |
|
nipkow@15392
|
1777 |
subsubsection {* @{text list_all2} *}
|
wenzelm@13114
|
1778 |
|
kleing@14316
|
1779 |
lemma list_all2_lengthD [intro?]:
|
kleing@14316
|
1780 |
"list_all2 P xs ys ==> length xs = length ys"
|
nipkow@24349
|
1781 |
by (simp add: list_all2_def)
|
haftmann@19607
|
1782 |
|
haftmann@19787
|
1783 |
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
|
nipkow@24349
|
1784 |
by (simp add: list_all2_def)
|
haftmann@19607
|
1785 |
|
haftmann@19787
|
1786 |
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
|
nipkow@24349
|
1787 |
by (simp add: list_all2_def)
|
haftmann@19607
|
1788 |
|
haftmann@19607
|
1789 |
lemma list_all2_Cons [iff, code]:
|
haftmann@19607
|
1790 |
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
|
nipkow@24349
|
1791 |
by (auto simp add: list_all2_def)
|
wenzelm@13114
|
1792 |
|
wenzelm@13114
|
1793 |
lemma list_all2_Cons1:
|
nipkow@13145
|
1794 |
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
|
nipkow@13145
|
1795 |
by (cases ys) auto
|
wenzelm@13114
|
1796 |
|
wenzelm@13114
|
1797 |
lemma list_all2_Cons2:
|
nipkow@13145
|
1798 |
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
|
nipkow@13145
|
1799 |
by (cases xs) auto
|
wenzelm@13114
|
1800 |
|
wenzelm@13142
|
1801 |
lemma list_all2_rev [iff]:
|
nipkow@13145
|
1802 |
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
|
nipkow@13145
|
1803 |
by (simp add: list_all2_def zip_rev cong: conj_cong)
|
wenzelm@13114
|
1804 |
|
kleing@13863
|
1805 |
lemma list_all2_rev1:
|
kleing@13863
|
1806 |
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
|
kleing@13863
|
1807 |
by (subst list_all2_rev [symmetric]) simp
|
kleing@13863
|
1808 |
|
wenzelm@13114
|
1809 |
lemma list_all2_append1:
|
nipkow@13145
|
1810 |
"list_all2 P (xs @ ys) zs =
|
nipkow@13145
|
1811 |
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
|
nipkow@13145
|
1812 |
list_all2 P xs us \<and> list_all2 P ys vs)"
|
nipkow@13145
|
1813 |
apply (simp add: list_all2_def zip_append1)
|
nipkow@13145
|
1814 |
apply (rule iffI)
|
nipkow@13145
|
1815 |
apply (rule_tac x = "take (length xs) zs" in exI)
|
nipkow@13145
|
1816 |
apply (rule_tac x = "drop (length xs) zs" in exI)
|
paulson@14208
|
1817 |
apply (force split: nat_diff_split simp add: min_def, clarify)
|
nipkow@13145
|
1818 |
apply (simp add: ball_Un)
|
nipkow@13145
|
1819 |
done
|
wenzelm@13114
|
1820 |
|
wenzelm@13114
|
1821 |
lemma list_all2_append2:
|
nipkow@13145
|
1822 |
"list_all2 P xs (ys @ zs) =
|
nipkow@13145
|
1823 |
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
|
nipkow@13145
|
1824 |
list_all2 P us ys \<and> list_all2 P vs zs)"
|
nipkow@13145
|
1825 |
apply (simp add: list_all2_def zip_append2)
|
nipkow@13145
|
1826 |
apply (rule iffI)
|
nipkow@13145
|
1827 |
apply (rule_tac x = "take (length ys) xs" in exI)
|
nipkow@13145
|
1828 |
apply (rule_tac x = "drop (length ys) xs" in exI)
|
paulson@14208
|
1829 |
apply (force split: nat_diff_split simp add: min_def, clarify)
|
nipkow@13145
|
1830 |
apply (simp add: ball_Un)
|
nipkow@13145
|
1831 |
done
|
wenzelm@13114
|
1832 |
|
kleing@13863
|
1833 |
lemma list_all2_append:
|
nipkow@14247
|
1834 |
"length xs = length ys \<Longrightarrow>
|
nipkow@14247
|
1835 |
list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
|
nipkow@14247
|
1836 |
by (induct rule:list_induct2, simp_all)
|
kleing@13863
|
1837 |
|
kleing@13863
|
1838 |
lemma list_all2_appendI [intro?, trans]:
|
kleing@13863
|
1839 |
"\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
|
nipkow@24349
|
1840 |
by (simp add: list_all2_append list_all2_lengthD)
|
kleing@13863
|
1841 |
|
wenzelm@13114
|
1842 |
lemma list_all2_conv_all_nth:
|
nipkow@13145
|
1843 |
"list_all2 P xs ys =
|
nipkow@13145
|
1844 |
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
|
nipkow@13145
|
1845 |
by (force simp add: list_all2_def set_zip)
|
wenzelm@13114
|
1846 |
|
berghofe@13883
|
1847 |
lemma list_all2_trans:
|
berghofe@13883
|
1848 |
assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
|
berghofe@13883
|
1849 |
shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
|
berghofe@13883
|
1850 |
(is "!!bs cs. PROP ?Q as bs cs")
|
berghofe@13883
|
1851 |
proof (induct as)
|
berghofe@13883
|
1852 |
fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
|
berghofe@13883
|
1853 |
show "!!cs. PROP ?Q (x # xs) bs cs"
|
berghofe@13883
|
1854 |
proof (induct bs)
|
berghofe@13883
|
1855 |
fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
|
berghofe@13883
|
1856 |
show "PROP ?Q (x # xs) (y # ys) cs"
|
berghofe@13883
|
1857 |
by (induct cs) (auto intro: tr I1 I2)
|
berghofe@13883
|
1858 |
qed simp
|
berghofe@13883
|
1859 |
qed simp
|
berghofe@13883
|
1860 |
|
kleing@13863
|
1861 |
lemma list_all2_all_nthI [intro?]:
|
kleing@13863
|
1862 |
"length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
|
nipkow@24349
|
1863 |
by (simp add: list_all2_conv_all_nth)
|
kleing@13863
|
1864 |
|
paulson@14395
|
1865 |
lemma list_all2I:
|
paulson@14395
|
1866 |
"\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
|
nipkow@24349
|
1867 |
by (simp add: list_all2_def)
|
paulson@14395
|
1868 |
|
kleing@14328
|
1869 |
lemma list_all2_nthD:
|
kleing@13863
|
1870 |
"\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
|
nipkow@24349
|
1871 |
by (simp add: list_all2_conv_all_nth)
|
kleing@13863
|
1872 |
|
nipkow@14302
|
1873 |
lemma list_all2_nthD2:
|
nipkow@14302
|
1874 |
"\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
|
nipkow@24349
|
1875 |
by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
|
nipkow@14302
|
1876 |
|
kleing@13863
|
1877 |
lemma list_all2_map1:
|
kleing@13863
|
1878 |
"list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
|
nipkow@24349
|
1879 |
by (simp add: list_all2_conv_all_nth)
|
kleing@13863
|
1880 |
|
kleing@13863
|
1881 |
lemma list_all2_map2:
|
kleing@13863
|
1882 |
"list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
|
nipkow@24349
|
1883 |
by (auto simp add: list_all2_conv_all_nth)
|
kleing@13863
|
1884 |
|
kleing@14316
|
1885 |
lemma list_all2_refl [intro?]:
|
kleing@13863
|
1886 |
"(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
|
nipkow@24349
|
1887 |
by (simp add: list_all2_conv_all_nth)
|
kleing@13863
|
1888 |
|
kleing@13863
|
1889 |
lemma list_all2_update_cong:
|
kleing@13863
|
1890 |
"\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
|
nipkow@24349
|
1891 |
by (simp add: list_all2_conv_all_nth nth_list_update)
|
kleing@13863
|
1892 |
|
kleing@13863
|
1893 |
lemma list_all2_update_cong2:
|
kleing@13863
|
1894 |
"\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
|
nipkow@24349
|
1895 |
by (simp add: list_all2_lengthD list_all2_update_cong)
|
kleing@13863
|
1896 |
|
nipkow@14302
|
1897 |
lemma list_all2_takeI [simp,intro?]:
|
nipkow@24526
|
1898 |
"list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
|
nipkow@24526
|
1899 |
apply (induct xs arbitrary: n ys)
|
nipkow@24526
|
1900 |
apply simp
|
nipkow@24526
|
1901 |
apply (clarsimp simp add: list_all2_Cons1)
|
nipkow@24526
|
1902 |
apply (case_tac n)
|
nipkow@24526
|
1903 |
apply auto
|
nipkow@24526
|
1904 |
done
|
nipkow@14302
|
1905 |
|
nipkow@14302
|
1906 |
lemma list_all2_dropI [simp,intro?]:
|
nipkow@24526
|
1907 |
"list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
|
nipkow@24526
|
1908 |
apply (induct as arbitrary: n bs, simp)
|
nipkow@24526
|
1909 |
apply (clarsimp simp add: list_all2_Cons1)
|
nipkow@24526
|
1910 |
apply (case_tac n, simp, simp)
|
nipkow@24526
|
1911 |
done
|
kleing@13863
|
1912 |
|
kleing@14327
|
1913 |
lemma list_all2_mono [intro?]:
|
nipkow@24526
|
1914 |
"list_all2 P xs ys \<Longrightarrow> (\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys) \<Longrightarrow> list_all2 Q xs ys"
|
nipkow@24526
|
1915 |
apply (induct xs arbitrary: ys, simp)
|
nipkow@24526
|
1916 |
apply (case_tac ys, auto)
|
nipkow@24526
|
1917 |
done
|
kleing@13863
|
1918 |
|
haftmann@22551
|
1919 |
lemma list_all2_eq:
|
haftmann@22551
|
1920 |
"xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
|
nipkow@24349
|
1921 |
by (induct xs ys rule: list_induct2') auto
|
haftmann@22551
|
1922 |
|
wenzelm@13114
|
1923 |
|
nipkow@15392
|
1924 |
subsubsection {* @{text foldl} and @{text foldr} *}
|
wenzelm@13114
|
1925 |
|
wenzelm@13142
|
1926 |
lemma foldl_append [simp]:
|
nipkow@24526
|
1927 |
"foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
|
nipkow@24526
|
1928 |
by (induct xs arbitrary: a) auto
|
wenzelm@13114
|
1929 |
|
nipkow@14402
|
1930 |
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
|
nipkow@14402
|
1931 |
by (induct xs) auto
|
nipkow@14402
|
1932 |
|
nipkow@23096
|
1933 |
lemma foldr_map: "foldr g (map f xs) a = foldr (g o f) xs a"
|
nipkow@23096
|
1934 |
by(induct xs) simp_all
|
nipkow@23096
|
1935 |
|
nipkow@24449
|
1936 |
text{* For efficient code generation: avoid intermediate list. *}
|
nipkow@24449
|
1937 |
lemma foldl_map[code unfold]:
|
nipkow@24449
|
1938 |
"foldl g a (map f xs) = foldl (%a x. g a (f x)) a xs"
|
nipkow@23096
|
1939 |
by(induct xs arbitrary:a) simp_all
|
nipkow@23096
|
1940 |
|
krauss@19770
|
1941 |
lemma foldl_cong [fundef_cong, recdef_cong]:
|
krauss@18336
|
1942 |
"[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |]
|
krauss@18336
|
1943 |
==> foldl f a l = foldl g b k"
|
nipkow@24349
|
1944 |
by (induct k arbitrary: a b l) simp_all
|
krauss@18336
|
1945 |
|
krauss@19770
|
1946 |
lemma foldr_cong [fundef_cong, recdef_cong]:
|
krauss@18336
|
1947 |
"[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |]
|
krauss@18336
|
1948 |
==> foldr f l a = foldr g k b"
|
nipkow@24349
|
1949 |
by (induct k arbitrary: a b l) simp_all
|
krauss@18336
|
1950 |
|
nipkow@24449
|
1951 |
lemma (in semigroup_add) foldl_assoc:
|
haftmann@25062
|
1952 |
shows "foldl op+ (x+y) zs = x + (foldl op+ y zs)"
|
nipkow@24449
|
1953 |
by (induct zs arbitrary: y) (simp_all add:add_assoc)
|
nipkow@24449
|
1954 |
|
nipkow@24449
|
1955 |
lemma (in monoid_add) foldl_absorb0:
|
haftmann@25062
|
1956 |
shows "x + (foldl op+ 0 zs) = foldl op+ x zs"
|
nipkow@24449
|
1957 |
by (induct zs) (simp_all add:foldl_assoc)
|
nipkow@24449
|
1958 |
|
nipkow@24449
|
1959 |
|
nipkow@23096
|
1960 |
text{* The ``First Duality Theorem'' in Bird \& Wadler: *}
|
nipkow@23096
|
1961 |
|
nipkow@23096
|
1962 |
lemma foldl_foldr1_lemma:
|
nipkow@23096
|
1963 |
"foldl op + a xs = a + foldr op + xs (0\<Colon>'a::monoid_add)"
|
nipkow@23096
|
1964 |
by (induct xs arbitrary: a) (auto simp:add_assoc)
|
nipkow@23096
|
1965 |
|
nipkow@23096
|
1966 |
corollary foldl_foldr1:
|
nipkow@23096
|
1967 |
"foldl op + 0 xs = foldr op + xs (0\<Colon>'a::monoid_add)"
|
nipkow@23096
|
1968 |
by (simp add:foldl_foldr1_lemma)
|
nipkow@23096
|
1969 |
|
nipkow@23096
|
1970 |
|
nipkow@23096
|
1971 |
text{* The ``Third Duality Theorem'' in Bird \& Wadler: *}
|
nipkow@23096
|
1972 |
|
nipkow@14402
|
1973 |
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
|
nipkow@14402
|
1974 |
by (induct xs) auto
|
nipkow@14402
|
1975 |
|
nipkow@14402
|
1976 |
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
|
nipkow@14402
|
1977 |
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
|
nipkow@14402
|
1978 |
|
haftmann@25062
|
1979 |
lemma (in ab_semigroup_add) foldr_conv_foldl: "foldr op + xs a = foldl op + a xs"
|
chaieb@24471
|
1980 |
by (induct xs, auto simp add: foldl_assoc add_commute)
|
chaieb@24471
|
1981 |
|
wenzelm@13142
|
1982 |
text {*
|
nipkow@13145
|
1983 |
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
|
nipkow@13145
|
1984 |
difficult to use because it requires an additional transitivity step.
|
wenzelm@13142
|
1985 |
*}
|
wenzelm@13114
|
1986 |
|
nipkow@24526
|
1987 |
lemma start_le_sum: "(m::nat) <= n ==> m <= foldl (op +) n ns"
|
nipkow@24526
|
1988 |
by (induct ns arbitrary: n) auto
|
nipkow@24526
|
1989 |
|
nipkow@24526
|
1990 |
lemma elem_le_sum: "(n::nat) : set ns ==> n <= foldl (op +) 0 ns"
|
nipkow@13145
|
1991 |
by (force intro: start_le_sum simp add: in_set_conv_decomp)
|
wenzelm@13114
|
1992 |
|
wenzelm@13142
|
1993 |
lemma sum_eq_0_conv [iff]:
|
nipkow@24526
|
1994 |
"(foldl (op +) (m::nat) ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
|
nipkow@24526
|
1995 |
by (induct ns arbitrary: m) auto
|
wenzelm@13114
|
1996 |
|
chaieb@24471
|
1997 |
lemma foldr_invariant:
|
chaieb@24471
|
1998 |
"\<lbrakk>Q x ; \<forall> x\<in> set xs. P x; \<forall> x y. P x \<and> Q y \<longrightarrow> Q (f x y) \<rbrakk> \<Longrightarrow> Q (foldr f xs x)"
|
chaieb@24471
|
1999 |
by (induct xs, simp_all)
|
chaieb@24471
|
2000 |
|
chaieb@24471
|
2001 |
lemma foldl_invariant:
|
chaieb@24471
|
2002 |
"\<lbrakk>Q x ; \<forall> x\<in> set xs. P x; \<forall> x y. P x \<and> Q y \<longrightarrow> Q (f y x) \<rbrakk> \<Longrightarrow> Q (foldl f x xs)"
|
chaieb@24471
|
2003 |
by (induct xs arbitrary: x, simp_all)
|
chaieb@24471
|
2004 |
|
nipkow@24449
|
2005 |
text{* @{const foldl} and @{text concat} *}
|
nipkow@24449
|
2006 |
|
nipkow@24449
|
2007 |
lemma concat_conv_foldl: "concat xss = foldl op@ [] xss"
|
nipkow@24449
|
2008 |
by (induct xss) (simp_all add:monoid_append.foldl_absorb0)
|
nipkow@24449
|
2009 |
|
nipkow@24449
|
2010 |
lemma foldl_conv_concat:
|
nipkow@24449
|
2011 |
"foldl (op @) xs xxs = xs @ (concat xxs)"
|
nipkow@24449
|
2012 |
by(simp add:concat_conv_foldl monoid_append.foldl_absorb0)
|
nipkow@24449
|
2013 |
|
nipkow@23096
|
2014 |
subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
|
nipkow@23096
|
2015 |
|
haftmann@26442
|
2016 |
lemma listsum_append [simp]: "listsum (xs @ ys) = listsum xs + listsum ys"
|
nipkow@24449
|
2017 |
by (induct xs) (simp_all add:add_assoc)
|
nipkow@24449
|
2018 |
|
haftmann@26442
|
2019 |
lemma listsum_rev [simp]:
|
haftmann@26442
|
2020 |
fixes xs :: "'a\<Colon>comm_monoid_add list"
|
haftmann@26442
|
2021 |
shows "listsum (rev xs) = listsum xs"
|
nipkow@24449
|
2022 |
by (induct xs) (simp_all add:add_ac)
|
nipkow@24449
|
2023 |
|
haftmann@26442
|
2024 |
lemma listsum_foldr: "listsum xs = foldr (op +) xs 0"
|
haftmann@26442
|
2025 |
by (induct xs) auto
|
haftmann@26442
|
2026 |
|
haftmann@26442
|
2027 |
lemma length_concat: "length (concat xss) = listsum (map length xss)"
|
haftmann@26442
|
2028 |
by (induct xss) simp_all
|
nipkow@23096
|
2029 |
|
nipkow@24449
|
2030 |
text{* For efficient code generation ---
|
nipkow@24449
|
2031 |
@{const listsum} is not tail recursive but @{const foldl} is. *}
|
nipkow@24449
|
2032 |
lemma listsum[code unfold]: "listsum xs = foldl (op +) 0 xs"
|
nipkow@23096
|
2033 |
by(simp add:listsum_foldr foldl_foldr1)
|
nipkow@23096
|
2034 |
|
nipkow@24449
|
2035 |
|
nipkow@23096
|
2036 |
text{* Some syntactic sugar for summing a function over a list: *}
|
nipkow@23096
|
2037 |
|
nipkow@23096
|
2038 |
syntax
|
nipkow@23096
|
2039 |
"_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3SUM _<-_. _)" [0, 51, 10] 10)
|
nipkow@23096
|
2040 |
syntax (xsymbols)
|
nipkow@23096
|
2041 |
"_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
|
nipkow@23096
|
2042 |
syntax (HTML output)
|
nipkow@23096
|
2043 |
"_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
|
nipkow@23096
|
2044 |
|
nipkow@23096
|
2045 |
translations -- {* Beware of argument permutation! *}
|
nipkow@23096
|
2046 |
"SUM x<-xs. b" == "CONST listsum (map (%x. b) xs)"
|
nipkow@23096
|
2047 |
"\<Sum>x\<leftarrow>xs. b" == "CONST listsum (map (%x. b) xs)"
|
nipkow@23096
|
2048 |
|
haftmann@26442
|
2049 |
lemma listsum_triv: "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
|
haftmann@26442
|
2050 |
by (induct xs) (simp_all add: left_distrib)
|
haftmann@26442
|
2051 |
|
nipkow@23096
|
2052 |
lemma listsum_0 [simp]: "(\<Sum>x\<leftarrow>xs. 0) = 0"
|
haftmann@26442
|
2053 |
by (induct xs) (simp_all add: left_distrib)
|
nipkow@23096
|
2054 |
|
nipkow@23096
|
2055 |
text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
|
nipkow@23096
|
2056 |
lemma uminus_listsum_map:
|
haftmann@26442
|
2057 |
fixes f :: "'a \<Rightarrow> 'b\<Colon>ab_group_add"
|
haftmann@26442
|
2058 |
shows "- listsum (map f xs) = (listsum (map (uminus o f) xs))"
|
haftmann@26442
|
2059 |
by (induct xs) simp_all
|
nipkow@23096
|
2060 |
|
wenzelm@13142
|
2061 |
|
nipkow@24645
|
2062 |
subsubsection {* @{text upt} *}
|
wenzelm@13142
|
2063 |
|
nipkow@17090
|
2064 |
lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
|
nipkow@17090
|
2065 |
-- {* simp does not terminate! *}
|
nipkow@13145
|
2066 |
by (induct j) auto
|
wenzelm@13114
|
2067 |
|
nipkow@15425
|
2068 |
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
|
nipkow@13145
|
2069 |
by (subst upt_rec) simp
|
wenzelm@13114
|
2070 |
|
nipkow@15425
|
2071 |
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
|
nipkow@15281
|
2072 |
by(induct j)simp_all
|
nipkow@15281
|
2073 |
|
nipkow@15281
|
2074 |
lemma upt_eq_Cons_conv:
|
nipkow@24526
|
2075 |
"([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
|
nipkow@24526
|
2076 |
apply(induct j arbitrary: x xs)
|
nipkow@15281
|
2077 |
apply simp
|
nipkow@15281
|
2078 |
apply(clarsimp simp add: append_eq_Cons_conv)
|
nipkow@15281
|
2079 |
apply arith
|
nipkow@15281
|
2080 |
done
|
nipkow@15281
|
2081 |
|
nipkow@15425
|
2082 |
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
|
nipkow@13145
|
2083 |
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
|
nipkow@13145
|
2084 |
by simp
|
wenzelm@13114
|
2085 |
|
nipkow@15425
|
2086 |
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
|
paulson@24632
|
2087 |
by (metis upt_rec)
|
wenzelm@13114
|
2088 |
|
nipkow@15425
|
2089 |
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
|
nipkow@13145
|
2090 |
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
|
nipkow@13145
|
2091 |
by (induct k) auto
|
wenzelm@13114
|
2092 |
|
nipkow@15425
|
2093 |
lemma length_upt [simp]: "length [i..<j] = j - i"
|
nipkow@13145
|
2094 |
by (induct j) (auto simp add: Suc_diff_le)
|
wenzelm@13114
|
2095 |
|
nipkow@15425
|
2096 |
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
|
nipkow@13145
|
2097 |
apply (induct j)
|
nipkow@13145
|
2098 |
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
|
nipkow@13145
|
2099 |
done
|
wenzelm@13114
|
2100 |
|
nipkow@17906
|
2101 |
|
nipkow@17906
|
2102 |
lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
|
nipkow@17906
|
2103 |
by(simp add:upt_conv_Cons)
|
nipkow@17906
|
2104 |
|
nipkow@17906
|
2105 |
lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
|
nipkow@17906
|
2106 |
apply(cases j)
|
nipkow@17906
|
2107 |
apply simp
|
nipkow@17906
|
2108 |
by(simp add:upt_Suc_append)
|
nipkow@17906
|
2109 |
|
nipkow@24526
|
2110 |
lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]"
|
nipkow@24526
|
2111 |
apply (induct m arbitrary: i, simp)
|
nipkow@13145
|
2112 |
apply (subst upt_rec)
|
nipkow@13145
|
2113 |
apply (rule sym)
|
nipkow@13145
|
2114 |
apply (subst upt_rec)
|
nipkow@13145
|
2115 |
apply (simp del: upt.simps)
|
nipkow@13145
|
2116 |
done
|
wenzelm@13114
|
2117 |
|
nipkow@17501
|
2118 |
lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
|
nipkow@17501
|
2119 |
apply(induct j)
|
nipkow@17501
|
2120 |
apply auto
|
nipkow@17501
|
2121 |
done
|
nipkow@17501
|
2122 |
|
nipkow@24645
|
2123 |
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]"
|
nipkow@13145
|
2124 |
by (induct n) auto
|
wenzelm@13114
|
2125 |
|
nipkow@24526
|
2126 |
lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
|
nipkow@24526
|
2127 |
apply (induct n m arbitrary: i rule: diff_induct)
|
nipkow@13145
|
2128 |
prefer 3 apply (subst map_Suc_upt[symmetric])
|
nipkow@13145
|
2129 |
apply (auto simp add: less_diff_conv nth_upt)
|
nipkow@13145
|
2130 |
done
|
wenzelm@13114
|
2131 |
|
berghofe@13883
|
2132 |
lemma nth_take_lemma:
|
nipkow@24526
|
2133 |
"k <= length xs ==> k <= length ys ==>
|
berghofe@13883
|
2134 |
(!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
|
nipkow@24526
|
2135 |
apply (atomize, induct k arbitrary: xs ys)
|
paulson@14208
|
2136 |
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
|
nipkow@13145
|
2137 |
txt {* Both lists must be non-empty *}
|
paulson@14208
|
2138 |
apply (case_tac xs, simp)
|
paulson@14208
|
2139 |
apply (case_tac ys, clarify)
|
nipkow@13145
|
2140 |
apply (simp (no_asm_use))
|
nipkow@13145
|
2141 |
apply clarify
|
nipkow@13145
|
2142 |
txt {* prenexing's needed, not miniscoping *}
|
nipkow@13145
|
2143 |
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
|
nipkow@13145
|
2144 |
apply blast
|
nipkow@13145
|
2145 |
done
|
wenzelm@13114
|
2146 |
|
wenzelm@13114
|
2147 |
lemma nth_equalityI:
|
wenzelm@13114
|
2148 |
"[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
|
nipkow@13145
|
2149 |
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
|
nipkow@13145
|
2150 |
apply (simp_all add: take_all)
|
nipkow@13145
|
2151 |
done
|
wenzelm@13114
|
2152 |
|
haftmann@24796
|
2153 |
lemma map_nth:
|
haftmann@24796
|
2154 |
"map (\<lambda>i. xs ! i) [0..<length xs] = xs"
|
haftmann@24796
|
2155 |
by (rule nth_equalityI, auto)
|
haftmann@24796
|
2156 |
|
kleing@13863
|
2157 |
(* needs nth_equalityI *)
|
kleing@13863
|
2158 |
lemma list_all2_antisym:
|
kleing@13863
|
2159 |
"\<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
|
2160 |
\<Longrightarrow> xs = ys"
|
kleing@13863
|
2161 |
apply (simp add: list_all2_conv_all_nth)
|
paulson@14208
|
2162 |
apply (rule nth_equalityI, blast, simp)
|
kleing@13863
|
2163 |
done
|
kleing@13863
|
2164 |
|
wenzelm@13142
|
2165 |
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
|
nipkow@13145
|
2166 |
-- {* The famous take-lemma. *}
|
nipkow@13145
|
2167 |
apply (drule_tac x = "max (length xs) (length ys)" in spec)
|
nipkow@13145
|
2168 |
apply (simp add: le_max_iff_disj take_all)
|
nipkow@13145
|
2169 |
done
|
wenzelm@13114
|
2170 |
|
wenzelm@13114
|
2171 |
|
nipkow@15302
|
2172 |
lemma take_Cons':
|
nipkow@15302
|
2173 |
"take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
|
nipkow@15302
|
2174 |
by (cases n) simp_all
|
nipkow@15302
|
2175 |
|
nipkow@15302
|
2176 |
lemma drop_Cons':
|
nipkow@15302
|
2177 |
"drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
|
nipkow@15302
|
2178 |
by (cases n) simp_all
|
nipkow@15302
|
2179 |
|
nipkow@15302
|
2180 |
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
|
nipkow@15302
|
2181 |
by (cases n) simp_all
|
nipkow@15302
|
2182 |
|
paulson@18622
|
2183 |
lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard]
|
paulson@18622
|
2184 |
lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard]
|
paulson@18622
|
2185 |
lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard]
|
paulson@18622
|
2186 |
|
paulson@18622
|
2187 |
declare take_Cons_number_of [simp]
|
paulson@18622
|
2188 |
drop_Cons_number_of [simp]
|
paulson@18622
|
2189 |
nth_Cons_number_of [simp]
|
nipkow@15302
|
2190 |
|
nipkow@15302
|
2191 |
|
nipkow@15392
|
2192 |
subsubsection {* @{text "distinct"} and @{text remdups} *}
|
wenzelm@13114
|
2193 |
|
wenzelm@13142
|
2194 |
lemma distinct_append [simp]:
|
nipkow@13145
|
2195 |
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
|
nipkow@13145
|
2196 |
by (induct xs) auto
|
wenzelm@13114
|
2197 |
|
nipkow@15305
|
2198 |
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
|
nipkow@15305
|
2199 |
by(induct xs) auto
|
nipkow@15305
|
2200 |
|
wenzelm@13142
|
2201 |
lemma set_remdups [simp]: "set (remdups xs) = set xs"
|
nipkow@13145
|
2202 |
by (induct xs) (auto simp add: insert_absorb)
|
wenzelm@13114
|
2203 |
|
wenzelm@13142
|
2204 |
lemma distinct_remdups [iff]: "distinct (remdups xs)"
|
nipkow@13145
|
2205 |
by (induct xs) auto
|
wenzelm@13114
|
2206 |
|
nipkow@25287
|
2207 |
lemma distinct_remdups_id: "distinct xs ==> remdups xs = xs"
|
nipkow@25287
|
2208 |
by (induct xs, auto)
|
nipkow@25287
|
2209 |
|
nipkow@25287
|
2210 |
lemma remdups_id_iff_distinct[simp]: "(remdups xs = xs) = distinct xs"
|
nipkow@25287
|
2211 |
by(metis distinct_remdups distinct_remdups_id)
|
nipkow@25287
|
2212 |
|
nipkow@24566
|
2213 |
lemma finite_distinct_list: "finite A \<Longrightarrow> EX xs. set xs = A & distinct xs"
|
paulson@24632
|
2214 |
by (metis distinct_remdups finite_list set_remdups)
|
nipkow@24566
|
2215 |
|
paulson@15072
|
2216 |
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
|
nipkow@24349
|
2217 |
by (induct x, auto)
|
paulson@15072
|
2218 |
|
paulson@15072
|
2219 |
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
|
nipkow@24349
|
2220 |
by (induct x, auto)
|
paulson@15072
|
2221 |
|
nipkow@15245
|
2222 |
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
|
nipkow@15245
|
2223 |
by (induct xs) auto
|
nipkow@15245
|
2224 |
|
nipkow@15245
|
2225 |
lemma length_remdups_eq[iff]:
|
nipkow@15245
|
2226 |
"(length (remdups xs) = length xs) = (remdups xs = xs)"
|
nipkow@15245
|
2227 |
apply(induct xs)
|
nipkow@15245
|
2228 |
apply auto
|
nipkow@15245
|
2229 |
apply(subgoal_tac "length (remdups xs) <= length xs")
|
nipkow@15245
|
2230 |
apply arith
|
nipkow@15245
|
2231 |
apply(rule length_remdups_leq)
|
nipkow@15245
|
2232 |
done
|
nipkow@15245
|
2233 |
|
nipkow@18490
|
2234 |
|
nipkow@18490
|
2235 |
lemma distinct_map:
|
nipkow@18490
|
2236 |
"distinct(map f xs) = (distinct xs & inj_on f (set xs))"
|
nipkow@18490
|
2237 |
by (induct xs) auto
|
nipkow@18490
|
2238 |
|
nipkow@18490
|
2239 |
|
wenzelm@13142
|
2240 |
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
|
nipkow@13145
|
2241 |
by (induct xs) auto
|
wenzelm@13114
|
2242 |
|
nipkow@17501
|
2243 |
lemma distinct_upt[simp]: "distinct[i..<j]"
|
nipkow@17501
|
2244 |
by (induct j) auto
|
nipkow@17501
|
2245 |
|
nipkow@24526
|
2246 |
lemma distinct_take[simp]: "distinct xs \<Longrightarrow> distinct (take i xs)"
|
nipkow@24526
|
2247 |
apply(induct xs arbitrary: i)
|
nipkow@17501
|
2248 |
apply simp
|
nipkow@17501
|
2249 |
apply (case_tac i)
|
nipkow@17501
|
2250 |
apply simp_all
|
nipkow@17501
|
2251 |
apply(blast dest:in_set_takeD)
|
nipkow@17501
|
2252 |
done
|
nipkow@17501
|
2253 |
|
nipkow@24526
|
2254 |
lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)"
|
nipkow@24526
|
2255 |
apply(induct xs arbitrary: i)
|
nipkow@17501
|
2256 |
apply simp
|
nipkow@17501
|
2257 |
apply (case_tac i)
|
nipkow@17501
|
2258 |
apply simp_all
|
nipkow@17501
|
2259 |
done
|
nipkow@17501
|
2260 |
|
nipkow@17501
|
2261 |
lemma distinct_list_update:
|
nipkow@17501
|
2262 |
assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
|
nipkow@17501
|
2263 |
shows "distinct (xs[i:=a])"
|
nipkow@17501
|
2264 |
proof (cases "i < length xs")
|
nipkow@17501
|
2265 |
case True
|
nipkow@17501
|
2266 |
with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
|
nipkow@17501
|
2267 |
apply (drule_tac id_take_nth_drop) by simp
|
nipkow@17501
|
2268 |
with d True show ?thesis
|
nipkow@17501
|
2269 |
apply (simp add: upd_conv_take_nth_drop)
|
nipkow@17501
|
2270 |
apply (drule subst [OF id_take_nth_drop]) apply assumption
|
nipkow@17501
|
2271 |
apply simp apply (cases "a = xs!i") apply simp by blast
|
nipkow@17501
|
2272 |
next
|
nipkow@17501
|
2273 |
case False with d show ?thesis by auto
|
nipkow@17501
|
2274 |
qed
|
nipkow@17501
|
2275 |
|
nipkow@17501
|
2276 |
|
nipkow@17501
|
2277 |
text {* It is best to avoid this indexed version of distinct, but
|
nipkow@17501
|
2278 |
sometimes it is useful. *}
|
nipkow@17501
|
2279 |
|
nipkow@13124
|
2280 |
lemma distinct_conv_nth:
|
nipkow@17501
|
2281 |
"distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
|
paulson@15251
|
2282 |
apply (induct xs, simp, simp)
|
paulson@14208
|
2283 |
apply (rule iffI, clarsimp)
|
nipkow@13145
|
2284 |
apply (case_tac i)
|
paulson@14208
|
2285 |
apply (case_tac j, simp)
|
nipkow@13145
|
2286 |
apply (simp add: set_conv_nth)
|
nipkow@13145
|
2287 |
apply (case_tac j)
|
paulson@24648
|
2288 |
apply (clarsimp simp add: set_conv_nth, simp)
|
nipkow@13145
|
2289 |
apply (rule conjI)
|
paulson@24648
|
2290 |
(*TOO SLOW
|
paulson@24632
|
2291 |
apply (metis Zero_neq_Suc gr0_conv_Suc in_set_conv_nth lessI less_trans_Suc nth_Cons' nth_Cons_Suc)
|
paulson@24648
|
2292 |
*)
|
paulson@24648
|
2293 |
apply (clarsimp simp add: set_conv_nth)
|
paulson@24648
|
2294 |
apply (erule_tac x = 0 in allE, simp)
|
paulson@24648
|
2295 |
apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
|
wenzelm@25130
|
2296 |
(*TOO SLOW
|
paulson@24632
|
2297 |
apply (metis Suc_Suc_eq lessI less_trans_Suc nth_Cons_Suc)
|
wenzelm@25130
|
2298 |
*)
|
wenzelm@25130
|
2299 |
apply (erule_tac x = "Suc i" in allE, simp)
|
wenzelm@25130
|
2300 |
apply (erule_tac x = "Suc j" in allE, simp)
|
nipkow@13145
|
2301 |
done
|
nipkow@13124
|
2302 |
|
nipkow@18490
|
2303 |
lemma nth_eq_iff_index_eq:
|
nipkow@18490
|
2304 |
"\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
|
nipkow@18490
|
2305 |
by(auto simp: distinct_conv_nth)
|
nipkow@18490
|
2306 |
|
nipkow@15110
|
2307 |
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
|
nipkow@24349
|
2308 |
by (induct xs) auto
|
kleing@14388
|
2309 |
|
nipkow@15110
|
2310 |
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
|
kleing@14388
|
2311 |
proof (induct xs)
|
kleing@14388
|
2312 |
case Nil thus ?case by simp
|
kleing@14388
|
2313 |
next
|
kleing@14388
|
2314 |
case (Cons x xs)
|
kleing@14388
|
2315 |
show ?case
|
kleing@14388
|
2316 |
proof (cases "x \<in> set xs")
|
kleing@14388
|
2317 |
case False with Cons show ?thesis by simp
|
kleing@14388
|
2318 |
next
|
kleing@14388
|
2319 |
case True with Cons.prems
|
kleing@14388
|
2320 |
have "card (set xs) = Suc (length xs)"
|
kleing@14388
|
2321 |
by (simp add: card_insert_if split: split_if_asm)
|
kleing@14388
|
2322 |
moreover have "card (set xs) \<le> length xs" by (rule card_length)
|
kleing@14388
|
2323 |
ultimately have False by simp
|
kleing@14388
|
2324 |
thus ?thesis ..
|
kleing@14388
|
2325 |
qed
|
kleing@14388
|
2326 |
qed
|
kleing@14388
|
2327 |
|
nipkow@25287
|
2328 |
lemma not_distinct_decomp: "~ distinct ws ==> EX xs ys zs y. ws = xs@[y]@ys@[y]@zs"
|
nipkow@25287
|
2329 |
apply (induct n == "length ws" arbitrary:ws) apply simp
|
nipkow@25287
|
2330 |
apply(case_tac ws) apply simp
|
nipkow@25287
|
2331 |
apply (simp split:split_if_asm)
|
nipkow@25287
|
2332 |
apply (metis Cons_eq_appendI eq_Nil_appendI split_list)
|
nipkow@25287
|
2333 |
done
|
nipkow@18490
|
2334 |
|
nipkow@18490
|
2335 |
lemma length_remdups_concat:
|
nipkow@18490
|
2336 |
"length(remdups(concat xss)) = card(\<Union>xs \<in> set xss. set xs)"
|
nipkow@24308
|
2337 |
by(simp add: set_concat distinct_card[symmetric])
|
nipkow@17906
|
2338 |
|
nipkow@17906
|
2339 |
|
nipkow@15392
|
2340 |
subsubsection {* @{text remove1} *}
|
nipkow@15110
|
2341 |
|
nipkow@18049
|
2342 |
lemma remove1_append:
|
nipkow@18049
|
2343 |
"remove1 x (xs @ ys) =
|
nipkow@18049
|
2344 |
(if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
|
nipkow@18049
|
2345 |
by (induct xs) auto
|
nipkow@18049
|
2346 |
|
nipkow@23479
|
2347 |
lemma in_set_remove1[simp]:
|
nipkow@23479
|
2348 |
"a \<noteq> b \<Longrightarrow> a : set(remove1 b xs) = (a : set xs)"
|
nipkow@23479
|
2349 |
apply (induct xs)
|
nipkow@23479
|
2350 |
apply auto
|
nipkow@23479
|
2351 |
done
|
nipkow@23479
|
2352 |
|
nipkow@15110
|
2353 |
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
|
nipkow@15110
|
2354 |
apply(induct xs)
|
nipkow@15110
|
2355 |
apply simp
|
nipkow@15110
|
2356 |
apply simp
|
nipkow@15110
|
2357 |
apply blast
|
nipkow@15110
|
2358 |
done
|
nipkow@15110
|
2359 |
|
paulson@17724
|
2360 |
lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
|
nipkow@15110
|
2361 |
apply(induct xs)
|
nipkow@15110
|
2362 |
apply simp
|
nipkow@15110
|
2363 |
apply simp
|
nipkow@15110
|
2364 |
apply blast
|
nipkow@15110
|
2365 |
done
|
nipkow@15110
|
2366 |
|
nipkow@23479
|
2367 |
lemma length_remove1:
|
nipkow@23479
|
2368 |
"length(remove1 x xs) = (if x : set xs then length xs - 1 else length xs)"
|
nipkow@23479
|
2369 |
apply (induct xs)
|
nipkow@23479
|
2370 |
apply (auto dest!:length_pos_if_in_set)
|
nipkow@23479
|
2371 |
done
|
nipkow@23479
|
2372 |
|
nipkow@18049
|
2373 |
lemma remove1_filter_not[simp]:
|
nipkow@18049
|
2374 |
"\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
|
nipkow@18049
|
2375 |
by(induct xs) auto
|
nipkow@18049
|
2376 |
|
nipkow@15110
|
2377 |
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
|
nipkow@15110
|
2378 |
apply(insert set_remove1_subset)
|
nipkow@15110
|
2379 |
apply fast
|
nipkow@15110
|
2380 |
done
|
nipkow@15110
|
2381 |
|
nipkow@15110
|
2382 |
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
|
nipkow@15110
|
2383 |
by (induct xs) simp_all
|
nipkow@15110
|
2384 |
|
wenzelm@13114
|
2385 |
|
nipkow@15392
|
2386 |
subsubsection {* @{text replicate} *}
|
wenzelm@13114
|
2387 |
|
wenzelm@13142
|
2388 |
lemma length_replicate [simp]: "length (replicate n x) = n"
|
nipkow@13145
|
2389 |
by (induct n) auto
|
wenzelm@13142
|
2390 |
|
wenzelm@13142
|
2391 |
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
|
nipkow@13145
|
2392 |
by (induct n) auto
|
wenzelm@13114
|
2393 |
|
wenzelm@13114
|
2394 |
lemma replicate_app_Cons_same:
|
nipkow@13145
|
2395 |
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
|
nipkow@13145
|
2396 |
by (induct n) auto
|
wenzelm@13114
|
2397 |
|
wenzelm@13142
|
2398 |
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
|
paulson@14208
|
2399 |
apply (induct n, simp)
|
nipkow@13145
|
2400 |
apply (simp add: replicate_app_Cons_same)
|
nipkow@13145
|
2401 |
done
|
wenzelm@13114
|
2402 |
|
wenzelm@13142
|
2403 |
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
|
nipkow@13145
|
2404 |
by (induct n) auto
|
wenzelm@13114
|
2405 |
|
nipkow@16397
|
2406 |
text{* Courtesy of Matthias Daum: *}
|
nipkow@16397
|
2407 |
lemma append_replicate_commute:
|
nipkow@16397
|
2408 |
"replicate n x @ replicate k x = replicate k x @ replicate n x"
|
nipkow@16397
|
2409 |
apply (simp add: replicate_add [THEN sym])
|
nipkow@16397
|
2410 |
apply (simp add: add_commute)
|
nipkow@16397
|
2411 |
done
|
nipkow@16397
|
2412 |
|
wenzelm@13142
|
2413 |
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
|
nipkow@13145
|
2414 |
by (induct n) auto
|
wenzelm@13114
|
2415 |
|
wenzelm@13142
|
2416 |
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
|
nipkow@13145
|
2417 |
by (induct n) auto
|
wenzelm@13114
|
2418 |
|
wenzelm@13142
|
2419 |
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
|
nipkow@13145
|
2420 |
by (atomize (full), induct n) auto
|
wenzelm@13114
|
2421 |
|
nipkow@24526
|
2422 |
lemma nth_replicate[simp]: "i < n ==> (replicate n x)!i = x"
|
nipkow@24526
|
2423 |
apply (induct n arbitrary: i, simp)
|
nipkow@13145
|
2424 |
apply (simp add: nth_Cons split: nat.split)
|
nipkow@13145
|
2425 |
done
|
wenzelm@13114
|
2426 |
|
nipkow@16397
|
2427 |
text{* Courtesy of Matthias Daum (2 lemmas): *}
|
nipkow@16397
|
2428 |
lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
|
nipkow@16397
|
2429 |
apply (case_tac "k \<le> i")
|
nipkow@16397
|
2430 |
apply (simp add: min_def)
|
nipkow@16397
|
2431 |
apply (drule not_leE)
|
nipkow@16397
|
2432 |
apply (simp add: min_def)
|
nipkow@16397
|
2433 |
apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
|
nipkow@16397
|
2434 |
apply simp
|
nipkow@16397
|
2435 |
apply (simp add: replicate_add [symmetric])
|
nipkow@16397
|
2436 |
done
|
nipkow@16397
|
2437 |
|
nipkow@24526
|
2438 |
lemma drop_replicate[simp]: "drop i (replicate k x) = replicate (k-i) x"
|
nipkow@24526
|
2439 |
apply (induct k arbitrary: i)
|
nipkow@16397
|
2440 |
apply simp
|
nipkow@16397
|
2441 |
apply clarsimp
|
nipkow@16397
|
2442 |
apply (case_tac i)
|
nipkow@16397
|
2443 |
apply simp
|
nipkow@16397
|
2444 |
apply clarsimp
|
nipkow@16397
|
2445 |
done
|
nipkow@16397
|
2446 |
|
nipkow@16397
|
2447 |
|
wenzelm@13142
|
2448 |
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
|
nipkow@13145
|
2449 |
by (induct n) auto
|
wenzelm@13114
|
2450 |
|
wenzelm@13142
|
2451 |
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
|
nipkow@13145
|
2452 |
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
|
wenzelm@13114
|
2453 |
|
wenzelm@13142
|
2454 |
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
|
nipkow@13145
|
2455 |
by auto
|
wenzelm@13114
|
2456 |
|
wenzelm@13142
|
2457 |
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
|
nipkow@13145
|
2458 |
by (simp add: set_replicate_conv_if split: split_if_asm)
|
wenzelm@13114
|
2459 |
|
haftmann@24796
|
2460 |
lemma replicate_append_same:
|
haftmann@24796
|
2461 |
"replicate i x @ [x] = x # replicate i x"
|
haftmann@24796
|
2462 |
by (induct i) simp_all
|
haftmann@24796
|
2463 |
|
haftmann@24796
|
2464 |
lemma map_replicate_trivial:
|
haftmann@24796
|
2465 |
"map (\<lambda>i. x) [0..<i] = replicate i x"
|
haftmann@24796
|
2466 |
by (induct i) (simp_all add: replicate_append_same)
|
haftmann@24796
|
2467 |
|
wenzelm@13114
|
2468 |
|
nipkow@15392
|
2469 |
subsubsection{*@{text rotate1} and @{text rotate}*}
|
nipkow@15302
|
2470 |
|
nipkow@15302
|
2471 |
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
|
nipkow@15302
|
2472 |
by(simp add:rotate1_def)
|
nipkow@15302
|
2473 |
|
nipkow@15302
|
2474 |
lemma rotate0[simp]: "rotate 0 = id"
|
nipkow@15302
|
2475 |
by(simp add:rotate_def)
|
nipkow@15302
|
2476 |
|
nipkow@15302
|
2477 |
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
|
nipkow@15302
|
2478 |
by(simp add:rotate_def)
|
nipkow@15302
|
2479 |
|
nipkow@15302
|
2480 |
lemma rotate_add:
|
nipkow@15302
|
2481 |
"rotate (m+n) = rotate m o rotate n"
|
nipkow@15302
|
2482 |
by(simp add:rotate_def funpow_add)
|
nipkow@15302
|
2483 |
|
nipkow@15302
|
2484 |
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
|
nipkow@15302
|
2485 |
by(simp add:rotate_add)
|
nipkow@15302
|
2486 |
|
nipkow@18049
|
2487 |
lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
|
nipkow@18049
|
2488 |
by(simp add:rotate_def funpow_swap1)
|
nipkow@18049
|
2489 |
|
nipkow@15302
|
2490 |
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
|
nipkow@15302
|
2491 |
by(cases xs) simp_all
|
nipkow@15302
|
2492 |
|
nipkow@15302
|
2493 |
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
|
nipkow@15302
|
2494 |
apply(induct n)
|
nipkow@15302
|
2495 |
apply simp
|
nipkow@15302
|
2496 |
apply (simp add:rotate_def)
|
nipkow@15302
|
2497 |
done
|
nipkow@15302
|
2498 |
|
nipkow@15302
|
2499 |
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
|
nipkow@15302
|
2500 |
by(simp add:rotate1_def split:list.split)
|
nipkow@15302
|
2501 |
|
nipkow@15302
|
2502 |
lemma rotate_drop_take:
|
nipkow@15302
|
2503 |
"rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
|
nipkow@15302
|
2504 |
apply(induct n)
|
nipkow@15302
|
2505 |
apply simp
|
nipkow@15302
|
2506 |
apply(simp add:rotate_def)
|
nipkow@15302
|
2507 |
apply(cases "xs = []")
|
nipkow@15302
|
2508 |
apply (simp)
|
nipkow@15302
|
2509 |
apply(case_tac "n mod length xs = 0")
|
nipkow@15302
|
2510 |
apply(simp add:mod_Suc)
|
nipkow@15302
|
2511 |
apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
|
nipkow@15302
|
2512 |
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
|
nipkow@15302
|
2513 |
take_hd_drop linorder_not_le)
|
nipkow@15302
|
2514 |
done
|
nipkow@15302
|
2515 |
|
nipkow@15302
|
2516 |
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
|
nipkow@15302
|
2517 |
by(simp add:rotate_drop_take)
|
nipkow@15302
|
2518 |
|
nipkow@15302
|
2519 |
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
|
nipkow@15302
|
2520 |
by(simp add:rotate_drop_take)
|
nipkow@15302
|
2521 |
|
nipkow@15302
|
2522 |
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
|
nipkow@15302
|
2523 |
by(simp add:rotate1_def split:list.split)
|
nipkow@15302
|
2524 |
|
nipkow@24526
|
2525 |
lemma length_rotate[simp]: "length(rotate n xs) = length xs"
|
nipkow@24526
|
2526 |
by (induct n arbitrary: xs) (simp_all add:rotate_def)
|
nipkow@15302
|
2527 |
|
nipkow@15302
|
2528 |
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
|
nipkow@15302
|
2529 |
by(simp add:rotate1_def split:list.split) blast
|
nipkow@15302
|
2530 |
|
nipkow@15302
|
2531 |
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
|
nipkow@15302
|
2532 |
by (induct n) (simp_all add:rotate_def)
|
nipkow@15302
|
2533 |
|
nipkow@15302
|
2534 |
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
|
nipkow@15302
|
2535 |
by(simp add:rotate_drop_take take_map drop_map)
|
nipkow@15302
|
2536 |
|
nipkow@15302
|
2537 |
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
|
nipkow@15302
|
2538 |
by(simp add:rotate1_def split:list.split)
|
nipkow@15302
|
2539 |
|
nipkow@15302
|
2540 |
lemma set_rotate[simp]: "set(rotate n xs) = set xs"
|
nipkow@15302
|
2541 |
by (induct n) (simp_all add:rotate_def)
|
nipkow@15302
|
2542 |
|
nipkow@15302
|
2543 |
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
|
nipkow@15302
|
2544 |
by(simp add:rotate1_def split:list.split)
|
nipkow@15302
|
2545 |
|
nipkow@15302
|
2546 |
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
|
nipkow@15302
|
2547 |
by (induct n) (simp_all add:rotate_def)
|
nipkow@15302
|
2548 |
|
nipkow@15439
|
2549 |
lemma rotate_rev:
|
nipkow@15439
|
2550 |
"rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
|
nipkow@15439
|
2551 |
apply(simp add:rotate_drop_take rev_drop rev_take)
|
nipkow@15439
|
2552 |
apply(cases "length xs = 0")
|
nipkow@15439
|
2553 |
apply simp
|
nipkow@15439
|
2554 |
apply(cases "n mod length xs = 0")
|
nipkow@15439
|
2555 |
apply simp
|
nipkow@15439
|
2556 |
apply(simp add:rotate_drop_take rev_drop rev_take)
|
nipkow@15439
|
2557 |
done
|
nipkow@15439
|
2558 |
|
nipkow@18423
|
2559 |
lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
|
nipkow@18423
|
2560 |
apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
|
nipkow@18423
|
2561 |
apply(subgoal_tac "length xs \<noteq> 0")
|
nipkow@18423
|
2562 |
prefer 2 apply simp
|
nipkow@18423
|
2563 |
using mod_less_divisor[of "length xs" n] by arith
|
nipkow@18423
|
2564 |
|
nipkow@15302
|
2565 |
|
nipkow@15392
|
2566 |
subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
|
nipkow@15302
|
2567 |
|
nipkow@15302
|
2568 |
lemma sublist_empty [simp]: "sublist xs {} = []"
|
nipkow@15302
|
2569 |
by (auto simp add: sublist_def)
|
nipkow@15302
|
2570 |
|
nipkow@15302
|
2571 |
lemma sublist_nil [simp]: "sublist [] A = []"
|
nipkow@15302
|
2572 |
by (auto simp add: sublist_def)
|
nipkow@15302
|
2573 |
|
nipkow@15302
|
2574 |
lemma length_sublist:
|
nipkow@15302
|
2575 |
"length(sublist xs I) = card{i. i < length xs \<and> i : I}"
|
nipkow@15302
|
2576 |
by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
|
nipkow@15302
|
2577 |
|
nipkow@15302
|
2578 |
lemma sublist_shift_lemma_Suc:
|
nipkow@24526
|
2579 |
"map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
|
nipkow@24526
|
2580 |
map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
|
nipkow@24526
|
2581 |
apply(induct xs arbitrary: "is")
|
nipkow@15302
|
2582 |
apply simp
|
nipkow@15302
|
2583 |
apply (case_tac "is")
|
nipkow@15302
|
2584 |
apply simp
|
nipkow@15302
|
2585 |
apply simp
|
nipkow@15302
|
2586 |
done
|
nipkow@15302
|
2587 |
|
nipkow@15302
|
2588 |
lemma sublist_shift_lemma:
|
nipkow@23279
|
2589 |
"map fst [p<-zip xs [i..<i + length xs] . snd p : A] =
|
nipkow@23279
|
2590 |
map fst [p<-zip xs [0..<length xs] . snd p + i : A]"
|
nipkow@15302
|
2591 |
by (induct xs rule: rev_induct) (simp_all add: add_commute)
|
nipkow@15302
|
2592 |
|
nipkow@15302
|
2593 |
lemma sublist_append:
|
nipkow@15302
|
2594 |
"sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
|
nipkow@15302
|
2595 |
apply (unfold sublist_def)
|
nipkow@15302
|
2596 |
apply (induct l' rule: rev_induct, simp)
|
nipkow@15302
|
2597 |
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
|
nipkow@15302
|
2598 |
apply (simp add: add_commute)
|
nipkow@15302
|
2599 |
done
|
nipkow@15302
|
2600 |
|
nipkow@15302
|
2601 |
lemma sublist_Cons:
|
nipkow@15302
|
2602 |
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
|
nipkow@15302
|
2603 |
apply (induct l rule: rev_induct)
|
nipkow@15302
|
2604 |
apply (simp add: sublist_def)
|
nipkow@15302
|
2605 |
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
|
nipkow@15302
|
2606 |
done
|
nipkow@15302
|
2607 |
|
nipkow@24526
|
2608 |
lemma set_sublist: "set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
|
nipkow@24526
|
2609 |
apply(induct xs arbitrary: I)
|
nipkow@25162
|
2610 |
apply(auto simp: sublist_Cons nth_Cons split:nat.split dest!: gr0_implies_Suc)
|
nipkow@15302
|
2611 |
done
|
nipkow@15302
|
2612 |
|
nipkow@15302
|
2613 |
lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
|
nipkow@15302
|
2614 |
by(auto simp add:set_sublist)
|
nipkow@15302
|
2615 |
|
nipkow@15302
|
2616 |
lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
|
nipkow@15302
|
2617 |
by(auto simp add:set_sublist)
|
nipkow@15302
|
2618 |
|
nipkow@15302
|
2619 |
lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
|
nipkow@15302
|
2620 |
by(auto simp add:set_sublist)
|
nipkow@15302
|
2621 |
|
nipkow@15302
|
2622 |
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
|
nipkow@15302
|
2623 |
by (simp add: sublist_Cons)
|
nipkow@15302
|
2624 |
|
nipkow@15302
|
2625 |
|
nipkow@24526
|
2626 |
lemma distinct_sublistI[simp]: "distinct xs \<Longrightarrow> distinct(sublist xs I)"
|
nipkow@24526
|
2627 |
apply(induct xs arbitrary: I)
|
nipkow@15302
|
2628 |
apply simp
|
nipkow@15302
|
2629 |
apply(auto simp add:sublist_Cons)
|
nipkow@15302
|
2630 |
done
|
nipkow@15302
|
2631 |
|
nipkow@15302
|
2632 |
|
nipkow@15302
|
2633 |
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
|
nipkow@15302
|
2634 |
apply (induct l rule: rev_induct, simp)
|
nipkow@15302
|
2635 |
apply (simp split: nat_diff_split add: sublist_append)
|
nipkow@15302
|
2636 |
done
|
nipkow@15302
|
2637 |
|
nipkow@24526
|
2638 |
lemma filter_in_sublist:
|
nipkow@24526
|
2639 |
"distinct xs \<Longrightarrow> filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
|
nipkow@24526
|
2640 |
proof (induct xs arbitrary: s)
|
nipkow@17501
|
2641 |
case Nil thus ?case by simp
|
nipkow@17501
|
2642 |
next
|
nipkow@17501
|
2643 |
case (Cons a xs)
|
nipkow@17501
|
2644 |
moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
|
nipkow@17501
|
2645 |
ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
|
nipkow@17501
|
2646 |
qed
|
nipkow@17501
|
2647 |
|
nipkow@15302
|
2648 |
|
nipkow@19390
|
2649 |
subsubsection {* @{const splice} *}
|
nipkow@19390
|
2650 |
|
haftmann@19607
|
2651 |
lemma splice_Nil2 [simp, code]:
|
nipkow@19390
|
2652 |
"splice xs [] = xs"
|
nipkow@19390
|
2653 |
by (cases xs) simp_all
|
nipkow@19390
|
2654 |
|
haftmann@19607
|
2655 |
lemma splice_Cons_Cons [simp, code]:
|
nipkow@19390
|
2656 |
"splice (x#xs) (y#ys) = x # y # splice xs ys"
|
nipkow@19390
|
2657 |
by simp
|
nipkow@19390
|
2658 |
|
haftmann@19607
|
2659 |
declare splice.simps(2) [simp del, code del]
|
nipkow@19390
|
2660 |
|
nipkow@24526
|
2661 |
lemma length_splice[simp]: "length(splice xs ys) = length xs + length ys"
|
nipkow@24526
|
2662 |
apply(induct xs arbitrary: ys) apply simp
|
nipkow@22793
|
2663 |
apply(case_tac ys)
|
nipkow@22793
|
2664 |
apply auto
|
nipkow@22793
|
2665 |
done
|
nipkow@22793
|
2666 |
|
nipkow@24616
|
2667 |
|
nipkow@24616
|
2668 |
subsection {*Sorting*}
|
nipkow@24616
|
2669 |
|
nipkow@24617
|
2670 |
text{* Currently it is not shown that @{const sort} returns a
|
nipkow@24617
|
2671 |
permutation of its input because the nicest proof is via multisets,
|
nipkow@24617
|
2672 |
which are not yet available. Alternatively one could define a function
|
nipkow@24617
|
2673 |
that counts the number of occurrences of an element in a list and use
|
nipkow@24617
|
2674 |
that instead of multisets to state the correctness property. *}
|
nipkow@24617
|
2675 |
|
nipkow@24616
|
2676 |
context linorder
|
nipkow@24616
|
2677 |
begin
|
nipkow@24616
|
2678 |
|
haftmann@25062
|
2679 |
lemma sorted_Cons: "sorted (x#xs) = (sorted xs & (ALL y:set xs. x <= y))"
|
nipkow@24616
|
2680 |
apply(induct xs arbitrary: x) apply simp
|
nipkow@24616
|
2681 |
by simp (blast intro: order_trans)
|
nipkow@24616
|
2682 |
|
nipkow@24616
|
2683 |
lemma sorted_append:
|
haftmann@25062
|
2684 |
"sorted (xs@ys) = (sorted xs & sorted ys & (\<forall>x \<in> set xs. \<forall>y \<in> set ys. x\<le>y))"
|
nipkow@24616
|
2685 |
by (induct xs) (auto simp add:sorted_Cons)
|
nipkow@24616
|
2686 |
|
nipkow@24616
|
2687 |
lemma set_insort: "set(insort x xs) = insert x (set xs)"
|
nipkow@24616
|
2688 |
by (induct xs) auto
|
nipkow@24616
|
2689 |
|
nipkow@24617
|
2690 |
lemma set_sort[simp]: "set(sort xs) = set xs"
|
nipkow@24616
|
2691 |
by (induct xs) (simp_all add:set_insort)
|
nipkow@24616
|
2692 |
|
nipkow@24616
|
2693 |
lemma distinct_insort: "distinct (insort x xs) = (x \<notin> set xs \<and> distinct xs)"
|
nipkow@24616
|
2694 |
by(induct xs)(auto simp:set_insort)
|
nipkow@24616
|
2695 |
|
nipkow@24617
|
2696 |
lemma distinct_sort[simp]: "distinct (sort xs) = distinct xs"
|
nipkow@24616
|
2697 |
by(induct xs)(simp_all add:distinct_insort set_sort)
|
nipkow@24616
|
2698 |
|
nipkow@24616
|
2699 |
lemma sorted_insort: "sorted (insort x xs) = sorted xs"
|
nipkow@24616
|
2700 |
apply (induct xs)
|
nipkow@24650
|
2701 |
apply(auto simp:sorted_Cons set_insort)
|
nipkow@24616
|
2702 |
done
|
nipkow@24616
|
2703 |
|
nipkow@24616
|
2704 |
theorem sorted_sort[simp]: "sorted (sort xs)"
|
nipkow@24616
|
2705 |
by (induct xs) (auto simp:sorted_insort)
|
nipkow@24616
|
2706 |
|
bulwahn@26143
|
2707 |
lemma insort_is_Cons: "\<forall>x\<in>set xs. a \<le> x \<Longrightarrow> insort a xs = a # xs"
|
bulwahn@26143
|
2708 |
by (cases xs) auto
|
bulwahn@26143
|
2709 |
|
bulwahn@26143
|
2710 |
lemma sorted_remove1: "sorted xs \<Longrightarrow> sorted (remove1 a xs)"
|
bulwahn@26143
|
2711 |
by (induct xs, auto simp add: sorted_Cons)
|
bulwahn@26143
|
2712 |
|
bulwahn@26143
|
2713 |
lemma insort_remove1: "\<lbrakk> a \<in> set xs; sorted xs \<rbrakk> \<Longrightarrow> insort a (remove1 a xs) = xs"
|
bulwahn@26143
|
2714 |
by (induct xs, auto simp add: sorted_Cons insort_is_Cons)
|
bulwahn@26143
|
2715 |
|
bulwahn@26143
|
2716 |
lemma sorted_remdups[simp]:
|
bulwahn@26143
|
2717 |
"sorted l \<Longrightarrow> sorted (remdups l)"
|
bulwahn@26143
|
2718 |
by (induct l) (auto simp: sorted_Cons)
|
bulwahn@26143
|
2719 |
|
nipkow@24645
|
2720 |
lemma sorted_distinct_set_unique:
|
nipkow@24645
|
2721 |
assumes "sorted xs" "distinct xs" "sorted ys" "distinct ys" "set xs = set ys"
|
nipkow@24645
|
2722 |
shows "xs = ys"
|
nipkow@24645
|
2723 |
proof -
|
nipkow@24645
|
2724 |
from assms have 1: "length xs = length ys" by (metis distinct_card)
|
nipkow@24645
|
2725 |
from assms show ?thesis
|
nipkow@24645
|
2726 |
proof(induct rule:list_induct2[OF 1])
|
nipkow@24645
|
2727 |
case 1 show ?case by simp
|
nipkow@24645
|
2728 |
next
|
nipkow@24645
|
2729 |
case 2 thus ?case by (simp add:sorted_Cons)
|
nipkow@24645
|
2730 |
(metis Diff_insert_absorb antisym insertE insert_iff)
|
nipkow@24645
|
2731 |
qed
|
nipkow@24645
|
2732 |
qed
|
nipkow@24645
|
2733 |
|
nipkow@24645
|
2734 |
lemma finite_sorted_distinct_unique:
|
nipkow@24645
|
2735 |
shows "finite A \<Longrightarrow> EX! xs. set xs = A & sorted xs & distinct xs"
|
nipkow@24645
|
2736 |
apply(drule finite_distinct_list)
|
nipkow@24645
|
2737 |
apply clarify
|
nipkow@24645
|
2738 |
apply(rule_tac a="sort xs" in ex1I)
|
nipkow@24645
|
2739 |
apply (auto simp: sorted_distinct_set_unique)
|
nipkow@24645
|
2740 |
done
|
nipkow@24645
|
2741 |
|
nipkow@24616
|
2742 |
end
|
nipkow@24616
|
2743 |
|
nipkow@25277
|
2744 |
lemma sorted_upt[simp]: "sorted[i..<j]"
|
nipkow@25277
|
2745 |
by (induct j) (simp_all add:sorted_append)
|
nipkow@25277
|
2746 |
|
nipkow@24616
|
2747 |
|
nipkow@25069
|
2748 |
subsubsection {* @{text sorted_list_of_set} *}
|
nipkow@25069
|
2749 |
|
nipkow@25069
|
2750 |
text{* This function maps (finite) linearly ordered sets to sorted
|
nipkow@25069
|
2751 |
lists. Warning: in most cases it is not a good idea to convert from
|
nipkow@25069
|
2752 |
sets to lists but one should convert in the other direction (via
|
nipkow@25069
|
2753 |
@{const set}). *}
|
nipkow@25069
|
2754 |
|
nipkow@25069
|
2755 |
|
nipkow@25069
|
2756 |
context linorder
|
nipkow@25069
|
2757 |
begin
|
nipkow@25069
|
2758 |
|
nipkow@25069
|
2759 |
definition
|
nipkow@25069
|
2760 |
sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where
|
nipkow@25069
|
2761 |
"sorted_list_of_set A == THE xs. set xs = A & sorted xs & distinct xs"
|
nipkow@25069
|
2762 |
|
nipkow@25069
|
2763 |
lemma sorted_list_of_set[simp]: "finite A \<Longrightarrow>
|
nipkow@25069
|
2764 |
set(sorted_list_of_set A) = A &
|
nipkow@25069
|
2765 |
sorted(sorted_list_of_set A) & distinct(sorted_list_of_set A)"
|
nipkow@25069
|
2766 |
apply(simp add:sorted_list_of_set_def)
|
nipkow@25069
|
2767 |
apply(rule the1I2)
|
nipkow@25069
|
2768 |
apply(simp_all add: finite_sorted_distinct_unique)
|
nipkow@25069
|
2769 |
done
|
nipkow@25069
|
2770 |
|
nipkow@25069
|
2771 |
lemma sorted_list_of_empty[simp]: "sorted_list_of_set {} = []"
|
nipkow@25069
|
2772 |
unfolding sorted_list_of_set_def
|
nipkow@25069
|
2773 |
apply(subst the_equality[of _ "[]"])
|
nipkow@25069
|
2774 |
apply simp_all
|
nipkow@25069
|
2775 |
done
|
nipkow@25069
|
2776 |
|
nipkow@25069
|
2777 |
end
|
nipkow@25069
|
2778 |
|
nipkow@25069
|
2779 |
|
nipkow@24645
|
2780 |
subsubsection {* @{text upto}: the generic interval-list *}
|
nipkow@24645
|
2781 |
|
nipkow@24697
|
2782 |
class finite_intvl_succ = linorder +
|
nipkow@24697
|
2783 |
fixes successor :: "'a \<Rightarrow> 'a"
|
nipkow@25069
|
2784 |
assumes finite_intvl: "finite{a..b}"
|
haftmann@25062
|
2785 |
and successor_incr: "a < successor a"
|
haftmann@25062
|
2786 |
and ord_discrete: "\<not>(\<exists>x. a < x & x < successor a)"
|
nipkow@24697
|
2787 |
|
nipkow@24697
|
2788 |
context finite_intvl_succ
|
nipkow@24697
|
2789 |
begin
|
nipkow@24697
|
2790 |
|
nipkow@24697
|
2791 |
definition
|
haftmann@25062
|
2792 |
upto :: "'a \<Rightarrow> 'a \<Rightarrow> 'a list" ("(1[_../_])") where
|
nipkow@25069
|
2793 |
"upto i j == sorted_list_of_set {i..j}"
|
nipkow@25069
|
2794 |
|
nipkow@25069
|
2795 |
lemma upto[simp]: "set[a..b] = {a..b} & sorted[a..b] & distinct[a..b]"
|
nipkow@25069
|
2796 |
by(simp add:upto_def finite_intvl)
|
nipkow@24645
|
2797 |
|
haftmann@25062
|
2798 |
lemma insert_intvl: "i \<le> j \<Longrightarrow> insert i {successor i..j} = {i..j}"
|
nipkow@24697
|
2799 |
apply(insert successor_incr[of i])
|
nipkow@24697
|
2800 |
apply(auto simp: atLeastAtMost_def atLeast_def atMost_def)
|
nipkow@24697
|
2801 |
apply (metis ord_discrete less_le not_le)
|
nipkow@24645
|
2802 |
done
|
nipkow@24645
|
2803 |
|
nipkow@25069
|
2804 |
lemma sorted_list_of_set_rec: "i \<le> j \<Longrightarrow>
|
nipkow@25069
|
2805 |
sorted_list_of_set {i..j} = i # sorted_list_of_set {successor i..j}"
|
nipkow@25069
|
2806 |
apply(simp add:sorted_list_of_set_def upto_def)
|
nipkow@25069
|
2807 |
apply (rule the1_equality[OF finite_sorted_distinct_unique])
|
nipkow@25069
|
2808 |
apply (simp add:finite_intvl)
|
nipkow@25069
|
2809 |
apply(rule the1I2[OF finite_sorted_distinct_unique])
|
nipkow@25069
|
2810 |
apply (simp add:finite_intvl)
|
nipkow@25069
|
2811 |
apply (simp add: sorted_Cons insert_intvl Ball_def)
|
nipkow@25069
|
2812 |
apply (metis successor_incr leD less_imp_le order_trans)
|
nipkow@25069
|
2813 |
done
|
nipkow@25069
|
2814 |
|
haftmann@25062
|
2815 |
lemma upto_rec[code]: "[i..j] = (if i \<le> j then i # [successor i..j] else [])"
|
nipkow@25069
|
2816 |
by(simp add: upto_def sorted_list_of_set_rec)
|
nipkow@24697
|
2817 |
|
nipkow@24697
|
2818 |
end
|
nipkow@24697
|
2819 |
|
nipkow@24697
|
2820 |
text{* The integers are an instance of the above class: *}
|
nipkow@24697
|
2821 |
|
haftmann@25571
|
2822 |
instantiation int:: finite_intvl_succ
|
haftmann@25571
|
2823 |
begin
|
haftmann@25571
|
2824 |
|
haftmann@25571
|
2825 |
definition
|
haftmann@25571
|
2826 |
successor_int_def: "successor = (%i\<Colon>int. i+1)"
|
haftmann@25571
|
2827 |
|
haftmann@25571
|
2828 |
instance
|
haftmann@25571
|
2829 |
by intro_classes (simp_all add: successor_int_def)
|
haftmann@25571
|
2830 |
|
haftmann@25571
|
2831 |
end
|
nipkow@24697
|
2832 |
|
nipkow@24697
|
2833 |
text{* Now @{term"[i..j::int]"} is defined for integers. *}
|
nipkow@24697
|
2834 |
|
nipkow@24698
|
2835 |
hide (open) const successor
|
nipkow@24698
|
2836 |
|
nipkow@24645
|
2837 |
|
nipkow@15392
|
2838 |
subsubsection {* @{text lists}: the list-forming operator over sets *}
|
nipkow@15302
|
2839 |
|
berghofe@23740
|
2840 |
inductive_set
|
berghofe@23740
|
2841 |
lists :: "'a set => 'a list set"
|
berghofe@23740
|
2842 |
for A :: "'a set"
|
berghofe@22262
|
2843 |
where
|
berghofe@23740
|
2844 |
Nil [intro!]: "[]: lists A"
|
paulson@24286
|
2845 |
| Cons [intro!,noatp]: "[| a: A;l: lists A|] ==> a#l : lists A"
|
paulson@24286
|
2846 |
|
paulson@24286
|
2847 |
inductive_cases listsE [elim!,noatp]: "x#l : lists A"
|
paulson@24286
|
2848 |
inductive_cases listspE [elim!,noatp]: "listsp A (x # l)"
|
berghofe@23740
|
2849 |
|
berghofe@23740
|
2850 |
lemma listsp_mono [mono]: "A \<le> B ==> listsp A \<le> listsp B"
|
nipkow@24349
|
2851 |
by (clarify, erule listsp.induct, blast+)
|
berghofe@22262
|
2852 |
|
berghofe@23740
|
2853 |
lemmas lists_mono = listsp_mono [to_set]
|
berghofe@22262
|
2854 |
|
haftmann@22422
|
2855 |
lemma listsp_infI:
|
haftmann@22422
|
2856 |
assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
|
nipkow@24349
|
2857 |
by induct blast+
|
nipkow@15302
|
2858 |
|
haftmann@22422
|
2859 |
lemmas lists_IntI = listsp_infI [to_set]
|
haftmann@22422
|
2860 |
|
haftmann@22422
|
2861 |
lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
|
haftmann@22422
|
2862 |
proof (rule mono_inf [where f=listsp, THEN order_antisym])
|
berghofe@22262
|
2863 |
show "mono listsp" by (simp add: mono_def listsp_mono)
|
haftmann@22422
|
2864 |
show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro: listsp_infI)
|
nipkow@15302
|
2865 |
qed
|
nipkow@15302
|
2866 |
|
haftmann@22422
|
2867 |
lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_eq inf_bool_eq]
|
haftmann@22422
|
2868 |
|
haftmann@22422
|
2869 |
lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set]
|
berghofe@22262
|
2870 |
|
berghofe@22262
|
2871 |
lemma append_in_listsp_conv [iff]:
|
berghofe@22262
|
2872 |
"(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
|
nipkow@15302
|
2873 |
by (induct xs) auto
|
nipkow@15302
|
2874 |
|
berghofe@22262
|
2875 |
lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]
|
berghofe@22262
|
2876 |
|
berghofe@22262
|
2877 |
lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
|
berghofe@22262
|
2878 |
-- {* eliminate @{text listsp} in favour of @{text set} *}
|
nipkow@15302
|
2879 |
by (induct xs) auto
|
nipkow@15302
|
2880 |
|
berghofe@22262
|
2881 |
lemmas in_lists_conv_set = in_listsp_conv_set [to_set]
|
berghofe@22262
|
2882 |
|
paulson@24286
|
2883 |
lemma in_listspD [dest!,noatp]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
|
berghofe@22262
|
2884 |
by (rule in_listsp_conv_set [THEN iffD1])
|
berghofe@22262
|
2885 |
|
paulson@24286
|
2886 |
lemmas in_listsD [dest!,noatp] = in_listspD [to_set]
|
paulson@24286
|
2887 |
|
paulson@24286
|
2888 |
lemma in_listspI [intro!,noatp]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
|
berghofe@22262
|
2889 |
by (rule in_listsp_conv_set [THEN iffD2])
|
berghofe@22262
|
2890 |
|
paulson@24286
|
2891 |
lemmas in_listsI [intro!,noatp] = in_listspI [to_set]
|
nipkow@15302
|
2892 |
|
nipkow@15302
|
2893 |
lemma lists_UNIV [simp]: "lists UNIV = UNIV"
|
nipkow@15302
|
2894 |
by auto
|
nipkow@15302
|
2895 |
|
nipkow@17086
|
2896 |
|
nipkow@17086
|
2897 |
|
nipkow@17086
|
2898 |
subsubsection{* Inductive definition for membership *}
|
nipkow@17086
|
2899 |
|
berghofe@23740
|
2900 |
inductive ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
|
berghofe@22262
|
2901 |
where
|
berghofe@22262
|
2902 |
elem: "ListMem x (x # xs)"
|
berghofe@22262
|
2903 |
| insert: "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"
|
berghofe@22262
|
2904 |
|
berghofe@22262
|
2905 |
lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
|
nipkow@17086
|
2906 |
apply (rule iffI)
|
nipkow@17086
|
2907 |
apply (induct set: ListMem)
|
nipkow@17086
|
2908 |
apply auto
|
nipkow@17086
|
2909 |
apply (induct xs)
|
nipkow@17086
|
2910 |
apply (auto intro: ListMem.intros)
|
nipkow@17086
|
2911 |
done
|
nipkow@17086
|
2912 |
|
nipkow@17086
|
2913 |
|
nipkow@17086
|
2914 |
|
nipkow@15392
|
2915 |
subsubsection{*Lists as Cartesian products*}
|
nipkow@15302
|
2916 |
|
nipkow@15302
|
2917 |
text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
|
nipkow@15302
|
2918 |
@{term A} and tail drawn from @{term Xs}.*}
|
nipkow@15302
|
2919 |
|
nipkow@15302
|
2920 |
constdefs
|
nipkow@15302
|
2921 |
set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
|
nipkow@15302
|
2922 |
"set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"
|
nipkow@15302
|
2923 |
|
paulson@17724
|
2924 |
lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
|
nipkow@15302
|
2925 |
by (auto simp add: set_Cons_def)
|
nipkow@15302
|
2926 |
|
nipkow@15302
|
2927 |
text{*Yields the set of lists, all of the same length as the argument and
|
nipkow@15302
|
2928 |
with elements drawn from the corresponding element of the argument.*}
|
nipkow@15302
|
2929 |
|
nipkow@15302
|
2930 |
consts listset :: "'a set list \<Rightarrow> 'a list set"
|
nipkow@15302
|
2931 |
primrec
|
nipkow@15302
|
2932 |
"listset [] = {[]}"
|
nipkow@15302
|
2933 |
"listset(A#As) = set_Cons A (listset As)"
|
nipkow@15302
|
2934 |
|
nipkow@15302
|
2935 |
|
paulson@15656
|
2936 |
subsection{*Relations on Lists*}
|
paulson@15656
|
2937 |
|
paulson@15656
|
2938 |
subsubsection {* Length Lexicographic Ordering *}
|
paulson@15656
|
2939 |
|
paulson@15656
|
2940 |
text{*These orderings preserve well-foundedness: shorter lists
|
paulson@15656
|
2941 |
precede longer lists. These ordering are not used in dictionaries.*}
|
paulson@15656
|
2942 |
|
paulson@15656
|
2943 |
consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
|
paulson@15656
|
2944 |
--{*The lexicographic ordering for lists of the specified length*}
|
nipkow@15302
|
2945 |
primrec
|
paulson@15656
|
2946 |
"lexn r 0 = {}"
|
paulson@15656
|
2947 |
"lexn r (Suc n) =
|
paulson@15656
|
2948 |
(prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
|
paulson@15656
|
2949 |
{(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
|
nipkow@15302
|
2950 |
|
nipkow@15302
|
2951 |
constdefs
|
paulson@15656
|
2952 |
lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
|
paulson@15656
|
2953 |
"lex r == \<Union>n. lexn r n"
|
paulson@15656
|
2954 |
--{*Holds only between lists of the same length*}
|
paulson@15656
|
2955 |
|
nipkow@15693
|
2956 |
lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
|
nipkow@15693
|
2957 |
"lenlex r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
|
paulson@15656
|
2958 |
--{*Compares lists by their length and then lexicographically*}
|
nipkow@15302
|
2959 |
|
nipkow@15302
|
2960 |
|
wenzelm@13142
|
2961 |
lemma wf_lexn: "wf r ==> wf (lexn r n)"
|
paulson@15251
|
2962 |
apply (induct n, simp, simp)
|
nipkow@13145
|
2963 |
apply(rule wf_subset)
|
nipkow@13145
|
2964 |
prefer 2 apply (rule Int_lower1)
|
nipkow@13145
|
2965 |
apply(rule wf_prod_fun_image)
|
paulson@14208
|
2966 |
prefer 2 apply (rule inj_onI, auto)
|
nipkow@13145
|
2967 |
done
|
wenzelm@13114
|
2968 |
|
wenzelm@13114
|
2969 |
lemma lexn_length:
|
nipkow@24526
|
2970 |
"(xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
|
nipkow@24526
|
2971 |
by (induct n arbitrary: xs ys) auto
|
wenzelm@13114
|
2972 |
|
wenzelm@13142
|
2973 |
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
|
nipkow@13145
|
2974 |
apply (unfold lex_def)
|
nipkow@13145
|
2975 |
apply (rule wf_UN)
|
paulson@14208
|
2976 |
apply (blast intro: wf_lexn, clarify)
|
nipkow@13145
|
2977 |
apply (rename_tac m n)
|
nipkow@13145
|
2978 |
apply (subgoal_tac "m \<noteq> n")
|
nipkow@13145
|
2979 |
prefer 2 apply blast
|
nipkow@13145
|
2980 |
apply (blast dest: lexn_length not_sym)
|
nipkow@13145
|
2981 |
done
|
wenzelm@13114
|
2982 |
|
wenzelm@13114
|
2983 |
lemma lexn_conv:
|
paulson@15656
|
2984 |
"lexn r n =
|
paulson@15656
|
2985 |
{(xs,ys). length xs = n \<and> length ys = n \<and>
|
paulson@15656
|
2986 |
(\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
|
nipkow@18423
|
2987 |
apply (induct n, simp)
|
paulson@14208
|
2988 |
apply (simp add: image_Collect lex_prod_def, safe, blast)
|
paulson@14208
|
2989 |
apply (rule_tac x = "ab # xys" in exI, simp)
|
paulson@14208
|
2990 |
apply (case_tac xys, simp_all, blast)
|
nipkow@13145
|
2991 |
done
|
wenzelm@13114
|
2992 |
|
wenzelm@13114
|
2993 |
lemma lex_conv:
|
paulson@15656
|
2994 |
"lex r =
|
paulson@15656
|
2995 |
{(xs,ys). length xs = length ys \<and>
|
paulson@15656
|
2996 |
(\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
|
nipkow@13145
|
2997 |
by (force simp add: lex_def lexn_conv)
|
wenzelm@13114
|
2998 |
|
nipkow@15693
|
2999 |
lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
|
nipkow@15693
|
3000 |
by (unfold lenlex_def) blast
|
nipkow@15693
|
3001 |
|
nipkow@15693
|
3002 |
lemma lenlex_conv:
|
nipkow@15693
|
3003 |
"lenlex r = {(xs,ys). length xs < length ys |
|
paulson@15656
|
3004 |
length xs = length ys \<and> (xs, ys) : lex r}"
|
nipkow@19623
|
3005 |
by (simp add: lenlex_def diag_def lex_prod_def inv_image_def)
|
wenzelm@13114
|
3006 |
|
wenzelm@13142
|
3007 |
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
|
nipkow@13145
|
3008 |
by (simp add: lex_conv)
|
wenzelm@13114
|
3009 |
|
wenzelm@13142
|
3010 |
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
|
nipkow@13145
|
3011 |
by (simp add:lex_conv)
|
wenzelm@13114
|
3012 |
|
paulson@18447
|
3013 |
lemma Cons_in_lex [simp]:
|
paulson@15656
|
3014 |
"((x # xs, y # ys) : lex r) =
|
paulson@15656
|
3015 |
((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
|
nipkow@13145
|
3016 |
apply (simp add: lex_conv)
|
nipkow@13145
|
3017 |
apply (rule iffI)
|
paulson@14208
|
3018 |
prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
|
paulson@14208
|
3019 |
apply (case_tac xys, simp, simp)
|
nipkow@13145
|
3020 |
apply blast
|
nipkow@13145
|
3021 |
done
|
wenzelm@13114
|
3022 |
|
wenzelm@13114
|
3023 |
|
paulson@15656
|
3024 |
subsubsection {* Lexicographic Ordering *}
|
paulson@15656
|
3025 |
|
paulson@15656
|
3026 |
text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
|
paulson@15656
|
3027 |
This ordering does \emph{not} preserve well-foundedness.
|
nipkow@17090
|
3028 |
Author: N. Voelker, March 2005. *}
|
paulson@15656
|
3029 |
|
paulson@15656
|
3030 |
constdefs
|
paulson@15656
|
3031 |
lexord :: "('a * 'a)set \<Rightarrow> ('a list * 'a list) set"
|
paulson@15656
|
3032 |
"lexord r == {(x,y). \<exists> a v. y = x @ a # v \<or>
|
paulson@15656
|
3033 |
(\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
|
paulson@15656
|
3034 |
|
paulson@15656
|
3035 |
lemma lexord_Nil_left[simp]: "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
|
nipkow@24349
|
3036 |
by (unfold lexord_def, induct_tac y, auto)
|
paulson@15656
|
3037 |
|
paulson@15656
|
3038 |
lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
|
nipkow@24349
|
3039 |
by (unfold lexord_def, induct_tac x, auto)
|
paulson@15656
|
3040 |
|
paulson@15656
|
3041 |
lemma lexord_cons_cons[simp]:
|
paulson@15656
|
3042 |
"((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
|
paulson@15656
|
3043 |
apply (unfold lexord_def, safe, simp_all)
|
paulson@15656
|
3044 |
apply (case_tac u, simp, simp)
|
paulson@15656
|
3045 |
apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
|
paulson@15656
|
3046 |
apply (erule_tac x="b # u" in allE)
|
paulson@15656
|
3047 |
by force
|
paulson@15656
|
3048 |
|
paulson@15656
|
3049 |
lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
|
paulson@15656
|
3050 |
|
paulson@15656
|
3051 |
lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
|
nipkow@24349
|
3052 |
by (induct_tac x, auto)
|
paulson@15656
|
3053 |
|
paulson@15656
|
3054 |
lemma lexord_append_left_rightI:
|
paulson@15656
|
3055 |
"(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
|
nipkow@24349
|
3056 |
by (induct_tac u, auto)
|
paulson@15656
|
3057 |
|
paulson@15656
|
3058 |
lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
|
nipkow@24349
|
3059 |
by (induct x, auto)
|
paulson@15656
|
3060 |
|
paulson@15656
|
3061 |
lemma lexord_append_leftD:
|
paulson@15656
|
3062 |
"\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
|
nipkow@24349
|
3063 |
by (erule rev_mp, induct_tac x, auto)
|
paulson@15656
|
3064 |
|
paulson@15656
|
3065 |
lemma lexord_take_index_conv:
|
paulson@15656
|
3066 |
"((x,y) : lexord r) =
|
paulson@15656
|
3067 |
((length x < length y \<and> take (length x) y = x) \<or>
|
paulson@15656
|
3068 |
(\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
|
paulson@15656
|
3069 |
apply (unfold lexord_def Let_def, clarsimp)
|
paulson@15656
|
3070 |
apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
|
paulson@15656
|
3071 |
apply auto
|
paulson@15656
|
3072 |
apply (rule_tac x="hd (drop (length x) y)" in exI)
|
paulson@15656
|
3073 |
apply (rule_tac x="tl (drop (length x) y)" in exI)
|
paulson@15656
|
3074 |
apply (erule subst, simp add: min_def)
|
paulson@15656
|
3075 |
apply (rule_tac x ="length u" in exI, simp)
|
paulson@15656
|
3076 |
apply (rule_tac x ="take i x" in exI)
|
paulson@15656
|
3077 |
apply (rule_tac x ="x ! i" in exI)
|
paulson@15656
|
3078 |
apply (rule_tac x ="y ! i" in exI, safe)
|
paulson@15656
|
3079 |
apply (rule_tac x="drop (Suc i) x" in exI)
|
paulson@15656
|
3080 |
apply (drule sym, simp add: drop_Suc_conv_tl)
|
paulson@15656
|
3081 |
apply (rule_tac x="drop (Suc i) y" in exI)
|
paulson@15656
|
3082 |
by (simp add: drop_Suc_conv_tl)
|
paulson@15656
|
3083 |
|
paulson@15656
|
3084 |
-- {* lexord is extension of partial ordering List.lex *}
|
paulson@15656
|
3085 |
lemma lexord_lex: " (x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
|
paulson@15656
|
3086 |
apply (rule_tac x = y in spec)
|
paulson@15656
|
3087 |
apply (induct_tac x, clarsimp)
|
paulson@15656
|
3088 |
by (clarify, case_tac x, simp, force)
|
paulson@15656
|
3089 |
|
paulson@15656
|
3090 |
lemma lexord_irreflexive: "(! x. (x,x) \<notin> r) \<Longrightarrow> (y,y) \<notin> lexord r"
|
paulson@15656
|
3091 |
by (induct y, auto)
|
paulson@15656
|
3092 |
|
paulson@15656
|
3093 |
lemma lexord_trans:
|
paulson@15656
|
3094 |
"\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
|
paulson@15656
|
3095 |
apply (erule rev_mp)+
|
paulson@15656
|
3096 |
apply (rule_tac x = x in spec)
|
paulson@15656
|
3097 |
apply (rule_tac x = z in spec)
|
paulson@15656
|
3098 |
apply ( induct_tac y, simp, clarify)
|
paulson@15656
|
3099 |
apply (case_tac xa, erule ssubst)
|
paulson@15656
|
3100 |
apply (erule allE, erule allE) -- {* avoid simp recursion *}
|
paulson@15656
|
3101 |
apply (case_tac x, simp, simp)
|
paulson@24632
|
3102 |
apply (case_tac x, erule allE, erule allE, simp)
|
paulson@15656
|
3103 |
apply (erule_tac x = listb in allE)
|
paulson@15656
|
3104 |
apply (erule_tac x = lista in allE, simp)
|
paulson@15656
|
3105 |
apply (unfold trans_def)
|
paulson@15656
|
3106 |
by blast
|
paulson@15656
|
3107 |
|
paulson@15656
|
3108 |
lemma lexord_transI: "trans r \<Longrightarrow> trans (lexord r)"
|
nipkow@24349
|
3109 |
by (rule transI, drule lexord_trans, blast)
|
paulson@15656
|
3110 |
|
paulson@15656
|
3111 |
lemma lexord_linear: "(! a b. (a,b)\<in> r | a = b | (b,a) \<in> r) \<Longrightarrow> (x,y) : lexord r | x = y | (y,x) : lexord r"
|
paulson@15656
|
3112 |
apply (rule_tac x = y in spec)
|
paulson@15656
|
3113 |
apply (induct_tac x, rule allI)
|
paulson@15656
|
3114 |
apply (case_tac x, simp, simp)
|
paulson@15656
|
3115 |
apply (rule allI, case_tac x, simp, simp)
|
paulson@15656
|
3116 |
by blast
|
paulson@15656
|
3117 |
|
paulson@15656
|
3118 |
|
krauss@21103
|
3119 |
subsection {* Lexicographic combination of measure functions *}
|
krauss@21103
|
3120 |
|
krauss@21103
|
3121 |
text {* These are useful for termination proofs *}
|
krauss@21103
|
3122 |
|
krauss@21103
|
3123 |
definition
|
krauss@21103
|
3124 |
"measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"
|
krauss@21103
|
3125 |
|
krauss@21106
|
3126 |
lemma wf_measures[recdef_wf, simp]: "wf (measures fs)"
|
nipkow@24349
|
3127 |
unfolding measures_def
|
nipkow@24349
|
3128 |
by blast
|
krauss@21103
|
3129 |
|
krauss@21103
|
3130 |
lemma in_measures[simp]:
|
krauss@21103
|
3131 |
"(x, y) \<in> measures [] = False"
|
krauss@21103
|
3132 |
"(x, y) \<in> measures (f # fs)
|
krauss@21103
|
3133 |
= (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"
|
nipkow@24349
|
3134 |
unfolding measures_def
|
nipkow@24349
|
3135 |
by auto
|
krauss@21103
|
3136 |
|
krauss@21103
|
3137 |
lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
|
nipkow@24349
|
3138 |
by simp
|
krauss@21103
|
3139 |
|
krauss@21103
|
3140 |
lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
|
nipkow@24349
|
3141 |
by auto
|
krauss@21103
|
3142 |
|
krauss@21103
|
3143 |
|
nipkow@15392
|
3144 |
subsubsection{*Lifting a Relation on List Elements to the Lists*}
|
nipkow@15302
|
3145 |
|
berghofe@23740
|
3146 |
inductive_set
|
berghofe@23740
|
3147 |
listrel :: "('a * 'a)set => ('a list * 'a list)set"
|
berghofe@23740
|
3148 |
for r :: "('a * 'a)set"
|
berghofe@22262
|
3149 |
where
|
berghofe@23740
|
3150 |
Nil: "([],[]) \<in> listrel r"
|
berghofe@23740
|
3151 |
| Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
|
berghofe@23740
|
3152 |
|
berghofe@23740
|
3153 |
inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
|
berghofe@23740
|
3154 |
inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
|
berghofe@23740
|
3155 |
inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
|
berghofe@23740
|
3156 |
inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
|
nipkow@15302
|
3157 |
|
nipkow@15302
|
3158 |
|
nipkow@15302
|
3159 |
lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
|
nipkow@15302
|
3160 |
apply clarify
|
berghofe@23740
|
3161 |
apply (erule listrel.induct)
|
berghofe@23740
|
3162 |
apply (blast intro: listrel.intros)+
|
nipkow@15281
|
3163 |
done
|
nipkow@15281
|
3164 |
|
nipkow@15302
|
3165 |
lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
|
nipkow@15302
|
3166 |
apply clarify
|
berghofe@23740
|
3167 |
apply (erule listrel.induct, auto)
|
nipkow@13145
|
3168 |
done
|
wenzelm@13114
|
3169 |
|
nipkow@15302
|
3170 |
lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)"
|
nipkow@15302
|
3171 |
apply (simp add: refl_def listrel_subset Ball_def)
|
nipkow@15302
|
3172 |
apply (rule allI)
|
nipkow@15302
|
3173 |
apply (induct_tac x)
|
berghofe@23740
|
3174 |
apply (auto intro: listrel.intros)
|
nipkow@13145
|
3175 |
done
|
wenzelm@13114
|
3176 |
|
nipkow@15302
|
3177 |
lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)"
|
nipkow@15302
|
3178 |
apply (auto simp add: sym_def)
|
berghofe@23740
|
3179 |
apply (erule listrel.induct)
|
berghofe@23740
|
3180 |
apply (blast intro: listrel.intros)+
|
nipkow@15281
|
3181 |
done
|
nipkow@15281
|
3182 |
|
nipkow@15302
|
3183 |
lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)"
|
nipkow@15302
|
3184 |
apply (simp add: trans_def)
|
nipkow@15302
|
3185 |
apply (intro allI)
|
nipkow@15302
|
3186 |
apply (rule impI)
|
berghofe@23740
|
3187 |
apply (erule listrel.induct)
|
berghofe@23740
|
3188 |
apply (blast intro: listrel.intros)+
|
nipkow@15281
|
3189 |
done
|
nipkow@15281
|
3190 |
|
nipkow@15302
|
3191 |
theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
|
nipkow@15302
|
3192 |
by (simp add: equiv_def listrel_refl listrel_sym listrel_trans)
|
nipkow@15302
|
3193 |
|
nipkow@15302
|
3194 |
lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
|
berghofe@23740
|
3195 |
by (blast intro: listrel.intros)
|
nipkow@15302
|
3196 |
|
nipkow@15302
|
3197 |
lemma listrel_Cons:
|
nipkow@15302
|
3198 |
"listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})";
|
berghofe@23740
|
3199 |
by (auto simp add: set_Cons_def intro: listrel.intros)
|
nipkow@15302
|
3200 |
|
nipkow@15302
|
3201 |
|
nipkow@15392
|
3202 |
subsection{*Miscellany*}
|
nipkow@15392
|
3203 |
|
nipkow@15392
|
3204 |
subsubsection {* Characters and strings *}
|
wenzelm@13366
|
3205 |
|
wenzelm@13366
|
3206 |
datatype nibble =
|
wenzelm@13366
|
3207 |
Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
|
wenzelm@13366
|
3208 |
| Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
|
wenzelm@13366
|
3209 |
|
haftmann@26148
|
3210 |
lemma UNIV_nibble:
|
haftmann@26148
|
3211 |
"UNIV = {Nibble0, Nibble1, Nibble2, Nibble3, Nibble4, Nibble5, Nibble6, Nibble7,
|
haftmann@26148
|
3212 |
Nibble8, Nibble9, NibbleA, NibbleB, NibbleC, NibbleD, NibbleE, NibbleF}" (is "_ = ?A")
|
haftmann@26148
|
3213 |
proof (rule UNIV_eq_I)
|
haftmann@26148
|
3214 |
fix x show "x \<in> ?A" by (cases x) simp_all
|
haftmann@26148
|
3215 |
qed
|
haftmann@26148
|
3216 |
|
haftmann@26148
|
3217 |
instance nibble :: finite
|
haftmann@26148
|
3218 |
by default (simp add: UNIV_nibble)
|
haftmann@26148
|
3219 |
|
wenzelm@13366
|
3220 |
datatype char = Char nibble nibble
|
wenzelm@13366
|
3221 |
-- "Note: canonical order of character encoding coincides with standard term ordering"
|
wenzelm@13366
|
3222 |
|
haftmann@26148
|
3223 |
lemma UNIV_char:
|
haftmann@26148
|
3224 |
"UNIV = image (split Char) (UNIV \<times> UNIV)"
|
haftmann@26148
|
3225 |
proof (rule UNIV_eq_I)
|
haftmann@26148
|
3226 |
fix x show "x \<in> image (split Char) (UNIV \<times> UNIV)" by (cases x) auto
|
haftmann@26148
|
3227 |
qed
|
haftmann@26148
|
3228 |
|
haftmann@26148
|
3229 |
instance char :: finite
|
haftmann@26148
|
3230 |
by default (simp add: UNIV_char)
|
haftmann@26148
|
3231 |
|
wenzelm@13366
|
3232 |
types string = "char list"
|
wenzelm@13366
|
3233 |
|
wenzelm@13366
|
3234 |
syntax
|
wenzelm@13366
|
3235 |
"_Char" :: "xstr => char" ("CHR _")
|
wenzelm@13366
|
3236 |
"_String" :: "xstr => string" ("_")
|
wenzelm@13366
|
3237 |
|
wenzelm@21754
|
3238 |
setup StringSyntax.setup
|
wenzelm@13366
|
3239 |
|
haftmann@20453
|
3240 |
|
haftmann@21061
|
3241 |
subsection {* Code generator *}
|
haftmann@21061
|
3242 |
|
haftmann@21061
|
3243 |
subsubsection {* Setup *}
|
berghofe@15064
|
3244 |
|
berghofe@16770
|
3245 |
types_code
|
berghofe@16770
|
3246 |
"list" ("_ list")
|
berghofe@16770
|
3247 |
attach (term_of) {*
|
wenzelm@21760
|
3248 |
fun term_of_list f T = HOLogic.mk_list T o map f;
|
berghofe@16770
|
3249 |
*}
|
berghofe@16770
|
3250 |
attach (test) {*
|
berghofe@25885
|
3251 |
fun gen_list' aG aT i j = frequency
|
berghofe@25885
|
3252 |
[(i, fn () =>
|
berghofe@25885
|
3253 |
let
|
berghofe@25885
|
3254 |
val (x, t) = aG j;
|
berghofe@25885
|
3255 |
val (xs, ts) = gen_list' aG aT (i-1) j
|
berghofe@25885
|
3256 |
in (x :: xs, fn () => HOLogic.cons_const aT $ t () $ ts ()) end),
|
berghofe@25885
|
3257 |
(1, fn () => ([], fn () => HOLogic.nil_const aT))] ()
|
berghofe@25885
|
3258 |
and gen_list aG aT i = gen_list' aG aT i i;
|
berghofe@16770
|
3259 |
*}
|
berghofe@16770
|
3260 |
"char" ("string")
|
berghofe@16770
|
3261 |
attach (term_of) {*
|
berghofe@24130
|
3262 |
val term_of_char = HOLogic.mk_char o ord;
|
berghofe@16770
|
3263 |
*}
|
berghofe@16770
|
3264 |
attach (test) {*
|
berghofe@25885
|
3265 |
fun gen_char i =
|
berghofe@25885
|
3266 |
let val j = random_range (ord "a") (Int.min (ord "a" + i, ord "z"))
|
berghofe@25885
|
3267 |
in (chr j, fn () => HOLogic.mk_char j) end;
|
berghofe@15064
|
3268 |
*}
|
berghofe@15064
|
3269 |
|
berghofe@15064
|
3270 |
consts_code "Cons" ("(_ ::/ _)")
|
berghofe@15064
|
3271 |
|
haftmann@20453
|
3272 |
code_type list
|
haftmann@20453
|
3273 |
(SML "_ list")
|
haftmann@21911
|
3274 |
(OCaml "_ list")
|
haftmann@21113
|
3275 |
(Haskell "![_]")
|
haftmann@20453
|
3276 |
|
haftmann@22799
|
3277 |
code_reserved SML
|
haftmann@22799
|
3278 |
list
|
haftmann@22799
|
3279 |
|
haftmann@22799
|
3280 |
code_reserved OCaml
|
haftmann@22799
|
3281 |
list
|
haftmann@22799
|
3282 |
|
haftmann@20453
|
3283 |
code_const Nil
|
haftmann@21113
|
3284 |
(SML "[]")
|
haftmann@21911
|
3285 |
(OCaml "[]")
|
haftmann@21113
|
3286 |
(Haskell "[]")
|
haftmann@20453
|
3287 |
|
haftmann@21911
|
3288 |
setup {*
|
haftmann@24219
|
3289 |
fold (fn target => CodeTarget.add_pretty_list target
|
haftmann@22799
|
3290 |
@{const_name Nil} @{const_name Cons}
|
haftmann@22799
|
3291 |
) ["SML", "OCaml", "Haskell"]
|
haftmann@21911
|
3292 |
*}
|
haftmann@21911
|
3293 |
|
haftmann@22799
|
3294 |
code_instance list :: eq
|
haftmann@22799
|
3295 |
(Haskell -)
|
haftmann@20588
|
3296 |
|
haftmann@21455
|
3297 |
code_const "op = \<Colon> 'a\<Colon>eq list \<Rightarrow> 'a list \<Rightarrow> bool"
|
haftmann@20588
|
3298 |
(Haskell infixl 4 "==")
|
haftmann@20588
|
3299 |
|
haftmann@20453
|
3300 |
setup {*
|
haftmann@20453
|
3301 |
let
|
haftmann@20453
|
3302 |
|
haftmann@20453
|
3303 |
fun list_codegen thy defs gr dep thyname b t =
|
berghofe@24902
|
3304 |
let
|
berghofe@24902
|
3305 |
val ts = HOLogic.dest_list t;
|
berghofe@24902
|
3306 |
val (gr', _) = Codegen.invoke_tycodegen thy defs dep thyname false
|
berghofe@24902
|
3307 |
(gr, fastype_of t);
|
berghofe@24902
|
3308 |
val (gr'', ps) = foldl_map
|
berghofe@24902
|
3309 |
(Codegen.invoke_codegen thy defs dep thyname false) (gr', ts)
|
berghofe@24902
|
3310 |
in SOME (gr'', Pretty.list "[" "]" ps) end handle TERM _ => NONE;
|
haftmann@20453
|
3311 |
|
haftmann@20453
|
3312 |
fun char_codegen thy defs gr dep thyname b t =
|
berghofe@24902
|
3313 |
let
|
berghofe@24902
|
3314 |
val i = HOLogic.dest_char t;
|
berghofe@24902
|
3315 |
val (gr', _) = Codegen.invoke_tycodegen thy defs dep thyname false
|
berghofe@24902
|
3316 |
(gr, fastype_of t)
|
berghofe@24902
|
3317 |
in SOME (gr', Pretty.str (ML_Syntax.print_string (chr i)))
|
berghofe@24902
|
3318 |
end handle TERM _ => NONE;
|
haftmann@20453
|
3319 |
|
haftmann@20453
|
3320 |
in
|
haftmann@20453
|
3321 |
Codegen.add_codegen "list_codegen" list_codegen
|
haftmann@20453
|
3322 |
#> Codegen.add_codegen "char_codegen" char_codegen
|
haftmann@20453
|
3323 |
end;
|
haftmann@20453
|
3324 |
*}
|
berghofe@15064
|
3325 |
|
haftmann@21061
|
3326 |
|
haftmann@21061
|
3327 |
subsubsection {* Generation of efficient code *}
|
haftmann@21061
|
3328 |
|
wenzelm@25221
|
3329 |
primrec
|
haftmann@25559
|
3330 |
member :: "'a \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "mem" 55)
|
haftmann@25559
|
3331 |
where
|
haftmann@25559
|
3332 |
"x mem [] \<longleftrightarrow> False"
|
haftmann@25559
|
3333 |
| "x mem (y#ys) \<longleftrightarrow> (if y = x then True else x mem ys)"
|
haftmann@21061
|
3334 |
|
haftmann@21061
|
3335 |
primrec
|
haftmann@26442
|
3336 |
null:: "'a list \<Rightarrow> bool"
|
haftmann@26442
|
3337 |
where
|
haftmann@21061
|
3338 |
"null [] = True"
|
haftmann@26442
|
3339 |
| "null (x#xs) = False"
|
haftmann@21061
|
3340 |
|
haftmann@21061
|
3341 |
primrec
|
haftmann@26442
|
3342 |
list_inter :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
|
haftmann@26442
|
3343 |
where
|
haftmann@21061
|
3344 |
"list_inter [] bs = []"
|
haftmann@26442
|
3345 |
| "list_inter (a#as) bs =
|
haftmann@21061
|
3346 |
(if a \<in> set bs then a # list_inter as bs else list_inter as bs)"
|
haftmann@21061
|
3347 |
|
haftmann@21061
|
3348 |
primrec
|
haftmann@26442
|
3349 |
list_all :: "('a \<Rightarrow> bool) \<Rightarrow> ('a list \<Rightarrow> bool)"
|
haftmann@26442
|
3350 |
where
|
haftmann@21061
|
3351 |
"list_all P [] = True"
|
haftmann@26442
|
3352 |
| "list_all P (x#xs) = (P x \<and> list_all P xs)"
|
haftmann@21061
|
3353 |
|
haftmann@21061
|
3354 |
primrec
|
haftmann@26442
|
3355 |
list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
|
haftmann@26442
|
3356 |
where
|
haftmann@21061
|
3357 |
"list_ex P [] = False"
|
haftmann@26442
|
3358 |
| "list_ex P (x#xs) = (P x \<or> list_ex P xs)"
|
haftmann@21061
|
3359 |
|
haftmann@21061
|
3360 |
primrec
|
haftmann@26442
|
3361 |
filtermap :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list"
|
haftmann@26442
|
3362 |
where
|
haftmann@21061
|
3363 |
"filtermap f [] = []"
|
haftmann@26442
|
3364 |
| "filtermap f (x#xs) =
|
haftmann@21061
|
3365 |
(case f x of None \<Rightarrow> filtermap f xs
|
haftmann@21061
|
3366 |
| Some y \<Rightarrow> y # filtermap f xs)"
|
haftmann@21061
|
3367 |
|
haftmann@21061
|
3368 |
primrec
|
haftmann@26442
|
3369 |
map_filter :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list"
|
haftmann@26442
|
3370 |
where
|
haftmann@21061
|
3371 |
"map_filter f P [] = []"
|
haftmann@26442
|
3372 |
| "map_filter f P (x#xs) =
|
haftmann@21061
|
3373 |
(if P x then f x # map_filter f P xs else map_filter f P xs)"
|
haftmann@21061
|
3374 |
|
haftmann@21061
|
3375 |
text {*
|
wenzelm@21754
|
3376 |
Only use @{text mem} for generating executable code. Otherwise use
|
wenzelm@21754
|
3377 |
@{prop "x : set xs"} instead --- it is much easier to reason about.
|
haftmann@21061
|
3378 |
The same is true for @{const list_all} and @{const list_ex}: write
|
haftmann@21061
|
3379 |
@{text "\<forall>x\<in>set xs"} and @{text "\<exists>x\<in>set xs"} instead because the HOL
|
wenzelm@21754
|
3380 |
quantifiers are aleady known to the automatic provers. In fact, the
|
wenzelm@21754
|
3381 |
declarations in the code subsection make sure that @{text "\<in>"},
|
wenzelm@21754
|
3382 |
@{text "\<forall>x\<in>set xs"} and @{text "\<exists>x\<in>set xs"} are implemented
|
wenzelm@21754
|
3383 |
efficiently.
|
haftmann@21061
|
3384 |
|
haftmann@21061
|
3385 |
Efficient emptyness check is implemented by @{const null}.
|
haftmann@21061
|
3386 |
|
haftmann@23060
|
3387 |
The functions @{const filtermap} and @{const map_filter} are just
|
haftmann@23060
|
3388 |
there to generate efficient code. Do not use
|
wenzelm@21754
|
3389 |
them for modelling and proving.
|
haftmann@21061
|
3390 |
*}
|
haftmann@21061
|
3391 |
|
haftmann@23060
|
3392 |
lemma rev_foldl_cons [code]:
|
haftmann@23060
|
3393 |
"rev xs = foldl (\<lambda>xs x. x # xs) [] xs"
|
haftmann@23060
|
3394 |
proof (induct xs)
|
haftmann@23060
|
3395 |
case Nil then show ?case by simp
|
haftmann@23060
|
3396 |
next
|
haftmann@23060
|
3397 |
case Cons
|
haftmann@23060
|
3398 |
{
|
haftmann@23060
|
3399 |
fix x xs ys
|
haftmann@23060
|
3400 |
have "foldl (\<lambda>xs x. x # xs) ys xs @ [x]
|
haftmann@23060
|
3401 |
= foldl (\<lambda>xs x. x # xs) (ys @ [x]) xs"
|
haftmann@23060
|
3402 |
by (induct xs arbitrary: ys) auto
|
haftmann@23060
|
3403 |
}
|
haftmann@23060
|
3404 |
note aux = this
|
haftmann@23060
|
3405 |
show ?case by (induct xs) (auto simp add: Cons aux)
|
haftmann@23060
|
3406 |
qed
|
haftmann@23060
|
3407 |
|
haftmann@24166
|
3408 |
lemma mem_iff [code post]:
|
haftmann@22422
|
3409 |
"x mem xs \<longleftrightarrow> x \<in> set xs"
|
nipkow@24349
|
3410 |
by (induct xs) auto
|
haftmann@21061
|
3411 |
|
haftmann@22799
|
3412 |
lemmas in_set_code [code unfold] = mem_iff [symmetric]
|
haftmann@21061
|
3413 |
|
haftmann@21061
|
3414 |
lemma empty_null [code inline]:
|
haftmann@22422
|
3415 |
"xs = [] \<longleftrightarrow> null xs"
|
nipkow@24349
|
3416 |
by (cases xs) simp_all
|
haftmann@21061
|
3417 |
|
haftmann@24166
|
3418 |
lemmas null_empty [code post] =
|
haftmann@21061
|
3419 |
empty_null [symmetric]
|
haftmann@21061
|
3420 |
|
haftmann@21061
|
3421 |
lemma list_inter_conv:
|
haftmann@21061
|
3422 |
"set (list_inter xs ys) = set xs \<inter> set ys"
|
nipkow@24349
|
3423 |
by (induct xs) auto
|
haftmann@21061
|
3424 |
|
haftmann@24166
|
3425 |
lemma list_all_iff [code post]:
|
haftmann@22422
|
3426 |
"list_all P xs \<longleftrightarrow> (\<forall>x \<in> set xs. P x)"
|
nipkow@24349
|
3427 |
by (induct xs) auto
|
haftmann@21061
|
3428 |
|
haftmann@22799
|
3429 |
lemmas list_ball_code [code unfold] = list_all_iff [symmetric]
|
haftmann@21061
|
3430 |
|
haftmann@21061
|
3431 |
lemma list_all_append [simp]:
|
haftmann@22422
|
3432 |
"list_all P (xs @ ys) \<longleftrightarrow> (list_all P xs \<and> list_all P ys)"
|
nipkow@24349
|
3433 |
by (induct xs) auto
|
haftmann@21061
|
3434 |
|
haftmann@21061
|
3435 |
lemma list_all_rev [simp]:
|
haftmann@22422
|
3436 |
"list_all P (rev xs) \<longleftrightarrow> list_all P xs"
|
nipkow@24349
|
3437 |
by (simp add: list_all_iff)
|
haftmann@21061
|
3438 |
|
haftmann@22506
|
3439 |
lemma list_all_length:
|
haftmann@22506
|
3440 |
"list_all P xs \<longleftrightarrow> (\<forall>n < length xs. P (xs ! n))"
|
haftmann@22506
|
3441 |
unfolding list_all_iff by (auto intro: all_nth_imp_all_set)
|
haftmann@22506
|
3442 |
|
haftmann@24166
|
3443 |
lemma list_ex_iff [code post]:
|
haftmann@22422
|
3444 |
"list_ex P xs \<longleftrightarrow> (\<exists>x \<in> set xs. P x)"
|
nipkow@24349
|
3445 |
by (induct xs) simp_all
|
haftmann@21061
|
3446 |
|
haftmann@21061
|
3447 |
lemmas list_bex_code [code unfold] =
|
haftmann@22799
|
3448 |
list_ex_iff [symmetric]
|
haftmann@21061
|
3449 |
|
haftmann@22506
|
3450 |
lemma list_ex_length:
|
haftmann@22506
|
3451 |
"list_ex P xs \<longleftrightarrow> (\<exists>n < length xs. P (xs ! n))"
|
haftmann@22506
|
3452 |
unfolding list_ex_iff set_conv_nth by auto
|
haftmann@22506
|
3453 |
|
haftmann@21061
|
3454 |
lemma filtermap_conv:
|
haftmann@21061
|
3455 |
"filtermap f xs = map (\<lambda>x. the (f x)) (filter (\<lambda>x. f x \<noteq> None) xs)"
|
nipkow@24349
|
3456 |
by (induct xs) (simp_all split: option.split)
|
haftmann@21061
|
3457 |
|
haftmann@21061
|
3458 |
lemma map_filter_conv [simp]:
|
haftmann@21061
|
3459 |
"map_filter f P xs = map f (filter P xs)"
|
nipkow@24349
|
3460 |
by (induct xs) auto
|
haftmann@21061
|
3461 |
|
nipkow@24449
|
3462 |
|
nipkow@24449
|
3463 |
text {* Code for bounded quantification and summation over nats. *}
|
haftmann@21891
|
3464 |
|
haftmann@22799
|
3465 |
lemma atMost_upto [code unfold]:
|
nipkow@24645
|
3466 |
"{..n} = set [0..<Suc n]"
|
nipkow@24349
|
3467 |
by auto
|
haftmann@22799
|
3468 |
|
haftmann@22799
|
3469 |
lemma atLeast_upt [code unfold]:
|
haftmann@21891
|
3470 |
"{..<n} = set [0..<n]"
|
nipkow@24349
|
3471 |
by auto
|
haftmann@22799
|
3472 |
|
nipkow@24449
|
3473 |
lemma greaterThanLessThan_upt [code unfold]:
|
haftmann@21891
|
3474 |
"{n<..<m} = set [Suc n..<m]"
|
nipkow@24349
|
3475 |
by auto
|
haftmann@22799
|
3476 |
|
nipkow@24449
|
3477 |
lemma atLeastLessThan_upt [code unfold]:
|
haftmann@21891
|
3478 |
"{n..<m} = set [n..<m]"
|
nipkow@24349
|
3479 |
by auto
|
haftmann@22799
|
3480 |
|
haftmann@22799
|
3481 |
lemma greaterThanAtMost_upto [code unfold]:
|
nipkow@24645
|
3482 |
"{n<..m} = set [Suc n..<Suc m]"
|
nipkow@24349
|
3483 |
by auto
|
haftmann@22799
|
3484 |
|
haftmann@22799
|
3485 |
lemma atLeastAtMost_upto [code unfold]:
|
nipkow@24645
|
3486 |
"{n..m} = set [n..<Suc m]"
|
nipkow@24349
|
3487 |
by auto
|
haftmann@22799
|
3488 |
|
haftmann@22799
|
3489 |
lemma all_nat_less_eq [code unfold]:
|
haftmann@21891
|
3490 |
"(\<forall>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..<n}. P m)"
|
nipkow@24349
|
3491 |
by auto
|
haftmann@22799
|
3492 |
|
haftmann@22799
|
3493 |
lemma ex_nat_less_eq [code unfold]:
|
haftmann@21891
|
3494 |
"(\<exists>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..<n}. P m)"
|
nipkow@24349
|
3495 |
by auto
|
haftmann@22799
|
3496 |
|
haftmann@22799
|
3497 |
lemma all_nat_less [code unfold]:
|
haftmann@21891
|
3498 |
"(\<forall>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..n}. P m)"
|
nipkow@24349
|
3499 |
by auto
|
haftmann@22799
|
3500 |
|
haftmann@22799
|
3501 |
lemma ex_nat_less [code unfold]:
|
haftmann@21891
|
3502 |
"(\<exists>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..n}. P m)"
|
nipkow@24349
|
3503 |
by auto
|
haftmann@22799
|
3504 |
|
haftmann@26442
|
3505 |
lemma setsum_set_upt_conv_listsum [code unfold]:
|
haftmann@26442
|
3506 |
"setsum f (set [k..<n]) = listsum (map f [k..<n])"
|
nipkow@24449
|
3507 |
apply(subst atLeastLessThan_upt[symmetric])
|
nipkow@24449
|
3508 |
by (induct n) simp_all
|
nipkow@24449
|
3509 |
|
wenzelm@23388
|
3510 |
end
|