wenzelm@13462
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(* Title: HOL/List.thy
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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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krauss@44884
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imports Plain Presburger Code_Numeral Quotient ATP
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bulwahn@41695
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uses
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bulwahn@41695
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("Tools/list_code.ML")
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bulwahn@41695
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("Tools/list_to_set_comprehension.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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clasohm@923
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syntax
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wenzelm@13366
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-- {* list Enumeration *}
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wenzelm@35118
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"_list" :: "args => 'a list" ("[(_)]")
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clasohm@923
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haftmann@34928
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translations
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haftmann@34928
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"[x, xs]" == "x#[xs]"
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haftmann@34928
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"[x]" == "x#[]"
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haftmann@34928
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wenzelm@35118
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wenzelm@35118
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subsection {* Basic list processing functions *}
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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hd :: "'a list \<Rightarrow> 'a" where
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haftmann@34928
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"hd (x # xs) = x"
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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tl :: "'a list \<Rightarrow> 'a list" where
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haftmann@34928
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"tl [] = []"
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haftmann@34928
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| "tl (x # xs) = xs"
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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last :: "'a list \<Rightarrow> 'a" where
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haftmann@34928
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"last (x # xs) = (if xs = [] then x else last xs)"
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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butlast :: "'a list \<Rightarrow> 'a list" where
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haftmann@34928
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"butlast []= []"
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haftmann@34928
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| "butlast (x # xs) = (if xs = [] then [] else x # butlast xs)"
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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set :: "'a list \<Rightarrow> 'a set" where
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haftmann@34928
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"set [] = {}"
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haftmann@34928
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| "set (x # xs) = insert x (set xs)"
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
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haftmann@34928
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"map f [] = []"
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haftmann@34928
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| "map f (x # xs) = f x # map f xs"
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65) where
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haftmann@34928
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append_Nil:"[] @ ys = ys"
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haftmann@34928
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| append_Cons: "(x#xs) @ ys = x # xs @ ys"
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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rev :: "'a list \<Rightarrow> 'a list" where
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haftmann@34928
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"rev [] = []"
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haftmann@34928
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| "rev (x # xs) = rev xs @ [x]"
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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filter:: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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haftmann@34928
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"filter P [] = []"
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haftmann@34928
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| "filter P (x # xs) = (if P x then x # filter P xs else filter P xs)"
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haftmann@34928
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haftmann@34928
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syntax
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wenzelm@13366
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-- {* Special syntax for filter *}
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wenzelm@35118
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"_filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_<-_./ _])")
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clasohm@923
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haftmann@34928
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translations
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haftmann@34928
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"[x<-xs . P]"== "CONST filter (%x. P) xs"
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haftmann@34928
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haftmann@34928
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syntax (xsymbols)
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wenzelm@35118
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"_filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
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haftmann@34928
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syntax (HTML output)
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wenzelm@35118
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"_filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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foldl :: "('b \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b" where
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haftmann@34928
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foldl_Nil: "foldl f a [] = a"
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haftmann@34928
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| foldl_Cons: "foldl f a (x # xs) = foldl f (f a x) xs"
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where
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haftmann@34928
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"foldr f [] a = a"
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haftmann@34928
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| "foldr f (x # xs) a = f x (foldr f xs a)"
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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concat:: "'a list list \<Rightarrow> 'a list" where
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haftmann@34928
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"concat [] = []"
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haftmann@34928
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| "concat (x # xs) = x @ concat xs"
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haftmann@34928
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haftmann@40007
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definition (in monoid_add)
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haftmann@34928
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listsum :: "'a list \<Rightarrow> 'a" where
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haftmann@40007
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"listsum xs = foldr plus xs 0"
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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drop:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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haftmann@34928
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drop_Nil: "drop n [] = []"
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haftmann@34928
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| drop_Cons: "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
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haftmann@34928
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-- {*Warning: simpset does not contain this definition, but separate
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haftmann@34928
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theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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take:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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haftmann@34928
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take_Nil:"take n [] = []"
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haftmann@34928
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| take_Cons: "take n (x # xs) = (case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # take m xs)"
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haftmann@34928
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-- {*Warning: simpset does not contain this definition, but separate
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haftmann@34928
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theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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nth :: "'a list => nat => 'a" (infixl "!" 100) where
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haftmann@34928
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nth_Cons: "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
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haftmann@34928
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-- {*Warning: simpset does not contain this definition, but separate
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haftmann@34928
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theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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list_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
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haftmann@34928
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"list_update [] i v = []"
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haftmann@34928
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| "list_update (x # xs) i v = (case i of 0 \<Rightarrow> v # xs | Suc j \<Rightarrow> x # list_update xs j v)"
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haftmann@34928
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wenzelm@41495
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nonterminal lupdbinds and lupdbind
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haftmann@34928
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haftmann@34928
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syntax
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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@35118
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"_LUpdate xs (_lupdbinds b bs)" == "_LUpdate (_LUpdate xs b) bs"
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haftmann@34928
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"xs[i:=x]" == "CONST list_update xs i x"
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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takeWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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haftmann@34928
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"takeWhile P [] = []"
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haftmann@34928
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| "takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])"
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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dropWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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haftmann@34928
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"dropWhile P [] = []"
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haftmann@34928
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| "dropWhile P (x # xs) = (if P x then dropWhile P xs else x # xs)"
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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zip :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
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haftmann@34928
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"zip xs [] = []"
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haftmann@34928
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| zip_Cons: "zip xs (y # ys) = (case xs of [] => [] | z # zs => (z, y) # zip zs ys)"
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haftmann@34928
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-- {*Warning: simpset does not contain this definition, but separate
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haftmann@34928
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theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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upt :: "nat \<Rightarrow> nat \<Rightarrow> nat list" ("(1[_..</_'])") where
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haftmann@34928
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upt_0: "[i..<0] = []"
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haftmann@34928
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| upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
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haftmann@34928
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haftmann@40096
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definition
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haftmann@40096
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insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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haftmann@40096
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"insert x xs = (if x \<in> set xs then xs else x # xs)"
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haftmann@40096
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haftmann@40096
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hide_const (open) insert
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haftmann@40096
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hide_fact (open) insert_def
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haftmann@40096
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haftmann@34928
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primrec
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haftmann@40096
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remove1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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haftmann@40096
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"remove1 x [] = []"
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haftmann@40096
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| "remove1 x (y # xs) = (if x = y then xs else y # remove1 x xs)"
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haftmann@40096
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haftmann@40096
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primrec
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haftmann@40096
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removeAll :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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haftmann@40096
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"removeAll x [] = []"
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haftmann@40096
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| "removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)"
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haftmann@40096
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haftmann@40366
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primrec
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haftmann@34928
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distinct :: "'a list \<Rightarrow> bool" where
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haftmann@40366
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"distinct [] \<longleftrightarrow> True"
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haftmann@40366
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| "distinct (x # xs) \<longleftrightarrow> x \<notin> set xs \<and> distinct xs"
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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remdups :: "'a list \<Rightarrow> 'a list" where
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haftmann@34928
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"remdups [] = []"
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haftmann@34928
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| "remdups (x # xs) = (if x \<in> set xs then remdups xs else x # remdups xs)"
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haftmann@34928
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haftmann@34928
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primrec
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haftmann@34928
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replicate :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
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haftmann@34928
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replicate_0: "replicate 0 x = []"
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haftmann@34928
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| replicate_Suc: "replicate (Suc n) x = x # replicate n x"
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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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haftmann@34928
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length :: "'a list \<Rightarrow> nat" where
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haftmann@34928
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"length \<equiv> size"
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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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haftmann@30971
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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@37767
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"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@40841
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fun splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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nipkow@40841
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"splice [] ys = ys" |
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nipkow@40841
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"splice xs [] = xs" |
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nipkow@40841
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"splice (x#xs) (y#ys) = x # y # splice xs ys"
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haftmann@21061
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nipkow@26771
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text{*
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nipkow@26771
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225 |
\begin{figure}[htbp]
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nipkow@26771
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226 |
\fbox{
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nipkow@26771
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227 |
\begin{tabular}{l}
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wenzelm@27381
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@{lemma "[a,b]@[c,d] = [a,b,c,d]" by simp}\\
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wenzelm@27381
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@{lemma "length [a,b,c] = 3" by simp}\\
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wenzelm@27381
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230 |
@{lemma "set [a,b,c] = {a,b,c}" by simp}\\
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wenzelm@27381
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@{lemma "map f [a,b,c] = [f a, f b, f c]" by simp}\\
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wenzelm@27381
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232 |
@{lemma "rev [a,b,c] = [c,b,a]" by simp}\\
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wenzelm@27381
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233 |
@{lemma "hd [a,b,c,d] = a" by simp}\\
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wenzelm@27381
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234 |
@{lemma "tl [a,b,c,d] = [b,c,d]" by simp}\\
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wenzelm@27381
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235 |
@{lemma "last [a,b,c,d] = d" by simp}\\
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wenzelm@27381
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236 |
@{lemma "butlast [a,b,c,d] = [a,b,c]" by simp}\\
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wenzelm@27381
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237 |
@{lemma[source] "filter (\<lambda>n::nat. n<2) [0,2,1] = [0,1]" by simp}\\
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wenzelm@27381
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238 |
@{lemma "concat [[a,b],[c,d,e],[],[f]] = [a,b,c,d,e,f]" by simp}\\
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wenzelm@27381
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239 |
@{lemma "foldl f x [a,b,c] = f (f (f x a) b) c" by simp}\\
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wenzelm@27381
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240 |
@{lemma "foldr f [a,b,c] x = f a (f b (f c x))" by simp}\\
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wenzelm@27381
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241 |
@{lemma "zip [a,b,c] [x,y,z] = [(a,x),(b,y),(c,z)]" by simp}\\
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wenzelm@27381
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242 |
@{lemma "zip [a,b] [x,y,z] = [(a,x),(b,y)]" by simp}\\
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wenzelm@27381
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243 |
@{lemma "splice [a,b,c] [x,y,z] = [a,x,b,y,c,z]" by simp}\\
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wenzelm@27381
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244 |
@{lemma "splice [a,b,c,d] [x,y] = [a,x,b,y,c,d]" by simp}\\
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wenzelm@27381
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245 |
@{lemma "take 2 [a,b,c,d] = [a,b]" by simp}\\
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wenzelm@27381
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246 |
@{lemma "take 6 [a,b,c,d] = [a,b,c,d]" by simp}\\
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wenzelm@27381
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247 |
@{lemma "drop 2 [a,b,c,d] = [c,d]" by simp}\\
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wenzelm@27381
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248 |
@{lemma "drop 6 [a,b,c,d] = []" by simp}\\
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wenzelm@27381
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249 |
@{lemma "takeWhile (%n::nat. n<3) [1,2,3,0] = [1,2]" by simp}\\
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wenzelm@27381
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250 |
@{lemma "dropWhile (%n::nat. n<3) [1,2,3,0] = [3,0]" by simp}\\
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wenzelm@27381
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251 |
@{lemma "distinct [2,0,1::nat]" by simp}\\
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wenzelm@27381
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252 |
@{lemma "remdups [2,0,2,1::nat,2] = [0,1,2]" by simp}\\
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haftmann@34965
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253 |
@{lemma "List.insert 2 [0::nat,1,2] = [0,1,2]" by (simp add: List.insert_def)}\\
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haftmann@35295
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254 |
@{lemma "List.insert 3 [0::nat,1,2] = [3,0,1,2]" by (simp add: List.insert_def)}\\
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wenzelm@27381
|
255 |
@{lemma "remove1 2 [2,0,2,1::nat,2] = [0,2,1,2]" by simp}\\
|
nipkow@27693
|
256 |
@{lemma "removeAll 2 [2,0,2,1::nat,2] = [0,1]" by simp}\\
|
wenzelm@27381
|
257 |
@{lemma "nth [a,b,c,d] 2 = c" by simp}\\
|
wenzelm@27381
|
258 |
@{lemma "[a,b,c,d][2 := x] = [a,b,x,d]" by simp}\\
|
wenzelm@27381
|
259 |
@{lemma "sublist [a,b,c,d,e] {0,2,3} = [a,c,d]" by (simp add:sublist_def)}\\
|
wenzelm@27381
|
260 |
@{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by (simp add:rotate1_def)}\\
|
nipkow@40258
|
261 |
@{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate1_def rotate_def eval_nat_numeral)}\\
|
nipkow@40258
|
262 |
@{lemma "replicate 4 a = [a,a,a,a]" by (simp add:eval_nat_numeral)}\\
|
nipkow@40258
|
263 |
@{lemma "[2..<5] = [2,3,4]" by (simp add:eval_nat_numeral)}\\
|
haftmann@40007
|
264 |
@{lemma "listsum [1,2,3::nat] = 6" by (simp add: listsum_def)}
|
nipkow@26771
|
265 |
\end{tabular}}
|
nipkow@26771
|
266 |
\caption{Characteristic examples}
|
nipkow@26771
|
267 |
\label{fig:Characteristic}
|
nipkow@26771
|
268 |
\end{figure}
|
blanchet@29864
|
269 |
Figure~\ref{fig:Characteristic} shows characteristic examples
|
nipkow@26771
|
270 |
that should give an intuitive understanding of the above functions.
|
nipkow@26771
|
271 |
*}
|
nipkow@26771
|
272 |
|
nipkow@24616
|
273 |
text{* The following simple sort functions are intended for proofs,
|
nipkow@24616
|
274 |
not for efficient implementations. *}
|
nipkow@24616
|
275 |
|
wenzelm@25221
|
276 |
context linorder
|
wenzelm@25221
|
277 |
begin
|
wenzelm@25221
|
278 |
|
haftmann@40096
|
279 |
inductive sorted :: "'a list \<Rightarrow> bool" where
|
haftmann@40096
|
280 |
Nil [iff]: "sorted []"
|
haftmann@40096
|
281 |
| Cons: "\<forall>y\<in>set xs. x \<le> y \<Longrightarrow> sorted xs \<Longrightarrow> sorted (x # xs)"
|
haftmann@40096
|
282 |
|
haftmann@40096
|
283 |
lemma sorted_single [iff]:
|
haftmann@40096
|
284 |
"sorted [x]"
|
haftmann@40096
|
285 |
by (rule sorted.Cons) auto
|
haftmann@40096
|
286 |
|
haftmann@40096
|
287 |
lemma sorted_many:
|
haftmann@40096
|
288 |
"x \<le> y \<Longrightarrow> sorted (y # zs) \<Longrightarrow> sorted (x # y # zs)"
|
haftmann@40096
|
289 |
by (rule sorted.Cons) (cases "y # zs" rule: sorted.cases, auto)
|
haftmann@40096
|
290 |
|
haftmann@40096
|
291 |
lemma sorted_many_eq [simp, code]:
|
haftmann@40096
|
292 |
"sorted (x # y # zs) \<longleftrightarrow> x \<le> y \<and> sorted (y # zs)"
|
haftmann@40096
|
293 |
by (auto intro: sorted_many elim: sorted.cases)
|
haftmann@40096
|
294 |
|
haftmann@40096
|
295 |
lemma [code]:
|
haftmann@40096
|
296 |
"sorted [] \<longleftrightarrow> True"
|
haftmann@40096
|
297 |
"sorted [x] \<longleftrightarrow> True"
|
haftmann@40096
|
298 |
by simp_all
|
nipkow@24697
|
299 |
|
hoelzl@33639
|
300 |
primrec insort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
|
hoelzl@33639
|
301 |
"insort_key f x [] = [x]" |
|
hoelzl@33639
|
302 |
"insort_key f x (y#ys) = (if f x \<le> f y then (x#y#ys) else y#(insort_key f x ys))"
|
hoelzl@33639
|
303 |
|
haftmann@35195
|
304 |
definition sort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b list \<Rightarrow> 'b list" where
|
haftmann@35195
|
305 |
"sort_key f xs = foldr (insort_key f) xs []"
|
hoelzl@33639
|
306 |
|
haftmann@40451
|
307 |
definition insort_insert_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
|
haftmann@40451
|
308 |
"insort_insert_key f x xs = (if f x \<in> f ` set xs then xs else insort_key f x xs)"
|
haftmann@40451
|
309 |
|
hoelzl@33639
|
310 |
abbreviation "sort \<equiv> sort_key (\<lambda>x. x)"
|
hoelzl@33639
|
311 |
abbreviation "insort \<equiv> insort_key (\<lambda>x. x)"
|
haftmann@40451
|
312 |
abbreviation "insort_insert \<equiv> insort_insert_key (\<lambda>x. x)"
|
haftmann@35608
|
313 |
|
wenzelm@25221
|
314 |
end
|
wenzelm@25221
|
315 |
|
nipkow@24616
|
316 |
|
wenzelm@23388
|
317 |
subsubsection {* List comprehension *}
|
nipkow@23192
|
318 |
|
nipkow@24349
|
319 |
text{* Input syntax for Haskell-like list comprehension notation.
|
nipkow@24349
|
320 |
Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
|
nipkow@24349
|
321 |
the list of all pairs of distinct elements from @{text xs} and @{text ys}.
|
nipkow@24349
|
322 |
The syntax is as in Haskell, except that @{text"|"} becomes a dot
|
nipkow@24349
|
323 |
(like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
|
nipkow@24349
|
324 |
\verb![e| x <- xs, ...]!.
|
nipkow@24349
|
325 |
|
nipkow@24349
|
326 |
The qualifiers after the dot are
|
nipkow@24349
|
327 |
\begin{description}
|
nipkow@24349
|
328 |
\item[generators] @{text"p \<leftarrow> xs"},
|
nipkow@24476
|
329 |
where @{text p} is a pattern and @{text xs} an expression of list type, or
|
nipkow@24476
|
330 |
\item[guards] @{text"b"}, where @{text b} is a boolean expression.
|
nipkow@24476
|
331 |
%\item[local bindings] @ {text"let x = e"}.
|
nipkow@24349
|
332 |
\end{description}
|
nipkow@23240
|
333 |
|
nipkow@24476
|
334 |
Just like in Haskell, list comprehension is just a shorthand. To avoid
|
nipkow@24476
|
335 |
misunderstandings, the translation into desugared form is not reversed
|
nipkow@24476
|
336 |
upon output. Note that the translation of @{text"[e. x \<leftarrow> xs]"} is
|
nipkow@24476
|
337 |
optmized to @{term"map (%x. e) xs"}.
|
nipkow@23240
|
338 |
|
nipkow@24349
|
339 |
It is easy to write short list comprehensions which stand for complex
|
nipkow@24349
|
340 |
expressions. During proofs, they may become unreadable (and
|
nipkow@24349
|
341 |
mangled). In such cases it can be advisable to introduce separate
|
nipkow@24349
|
342 |
definitions for the list comprehensions in question. *}
|
nipkow@24349
|
343 |
|
wenzelm@43015
|
344 |
nonterminal lc_gen and lc_qual and lc_quals
|
nipkow@23192
|
345 |
|
nipkow@23192
|
346 |
syntax
|
nipkow@23240
|
347 |
"_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list" ("[_ . __")
|
wenzelm@43015
|
348 |
"_lc_gen" :: "lc_gen \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ <- _")
|
nipkow@23240
|
349 |
"_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
|
nipkow@24476
|
350 |
(*"_lc_let" :: "letbinds => lc_qual" ("let _")*)
|
nipkow@23240
|
351 |
"_lc_end" :: "lc_quals" ("]")
|
nipkow@23240
|
352 |
"_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals" (", __")
|
nipkow@24349
|
353 |
"_lc_abs" :: "'a => 'b list => 'b list"
|
wenzelm@43015
|
354 |
"_strip_positions" :: "'a \<Rightarrow> lc_gen" ("_")
|
nipkow@23192
|
355 |
|
nipkow@24476
|
356 |
(* These are easier than ML code but cannot express the optimized
|
nipkow@24476
|
357 |
translation of [e. p<-xs]
|
nipkow@23192
|
358 |
translations
|
nipkow@24349
|
359 |
"[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
|
nipkow@23240
|
360 |
"_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
|
nipkow@24349
|
361 |
=> "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
|
nipkow@23240
|
362 |
"[e. P]" => "if P then [e] else []"
|
nipkow@23240
|
363 |
"_listcompr e (_lc_test P) (_lc_quals Q Qs)"
|
nipkow@23240
|
364 |
=> "if P then (_listcompr e Q Qs) else []"
|
nipkow@24349
|
365 |
"_listcompr e (_lc_let b) (_lc_quals Q Qs)"
|
nipkow@24349
|
366 |
=> "_Let b (_listcompr e Q Qs)"
|
nipkow@24476
|
367 |
*)
|
nipkow@23240
|
368 |
|
nipkow@23279
|
369 |
syntax (xsymbols)
|
wenzelm@43015
|
370 |
"_lc_gen" :: "lc_gen \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
|
nipkow@23279
|
371 |
syntax (HTML output)
|
wenzelm@43015
|
372 |
"_lc_gen" :: "lc_gen \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
|
nipkow@24349
|
373 |
|
nipkow@24349
|
374 |
parse_translation (advanced) {*
|
nipkow@24349
|
375 |
let
|
wenzelm@35256
|
376 |
val NilC = Syntax.const @{const_syntax Nil};
|
wenzelm@35256
|
377 |
val ConsC = Syntax.const @{const_syntax Cons};
|
wenzelm@35256
|
378 |
val mapC = Syntax.const @{const_syntax map};
|
wenzelm@35256
|
379 |
val concatC = Syntax.const @{const_syntax concat};
|
wenzelm@35256
|
380 |
val IfC = Syntax.const @{const_syntax If};
|
wenzelm@35118
|
381 |
|
nipkow@24476
|
382 |
fun singl x = ConsC $ x $ NilC;
|
nipkow@24476
|
383 |
|
wenzelm@35118
|
384 |
fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
|
nipkow@24349
|
385 |
let
|
wenzelm@44206
|
386 |
(* FIXME proper name context!? *)
|
wenzelm@44206
|
387 |
val x = Free (singleton (Name.variant_list (fold Term.add_free_names [p, e] [])) "x", dummyT);
|
nipkow@24476
|
388 |
val e = if opti then singl e else e;
|
wenzelm@43142
|
389 |
val case1 = Syntax.const @{syntax_const "_case1"} $ Term_Position.strip_positions p $ e;
|
wenzelm@35256
|
390 |
val case2 =
|
wenzelm@35256
|
391 |
Syntax.const @{syntax_const "_case1"} $
|
wenzelm@35256
|
392 |
Syntax.const @{const_syntax dummy_pattern} $ NilC;
|
wenzelm@35118
|
393 |
val cs = Syntax.const @{syntax_const "_case2"} $ case1 $ case2;
|
krauss@44442
|
394 |
val ft = Datatype_Case.case_tr false Datatype.info_of_constr_permissive ctxt [x, cs];
|
nipkow@24349
|
395 |
in lambda x ft end;
|
nipkow@24349
|
396 |
|
wenzelm@35256
|
397 |
fun abs_tr ctxt (p as Free (s, T)) e opti =
|
wenzelm@35118
|
398 |
let
|
wenzelm@43232
|
399 |
val thy = Proof_Context.theory_of ctxt;
|
wenzelm@43232
|
400 |
val s' = Proof_Context.intern_const ctxt s;
|
wenzelm@35118
|
401 |
in
|
wenzelm@35118
|
402 |
if Sign.declared_const thy s'
|
wenzelm@35118
|
403 |
then (pat_tr ctxt p e opti, false)
|
wenzelm@35118
|
404 |
else (lambda p e, true)
|
nipkow@24349
|
405 |
end
|
nipkow@24476
|
406 |
| abs_tr ctxt p e opti = (pat_tr ctxt p e opti, false);
|
nipkow@24476
|
407 |
|
wenzelm@35118
|
408 |
fun lc_tr ctxt [e, Const (@{syntax_const "_lc_test"}, _) $ b, qs] =
|
wenzelm@35118
|
409 |
let
|
wenzelm@35118
|
410 |
val res =
|
wenzelm@35118
|
411 |
(case qs of
|
wenzelm@35118
|
412 |
Const (@{syntax_const "_lc_end"}, _) => singl e
|
wenzelm@35118
|
413 |
| Const (@{syntax_const "_lc_quals"}, _) $ q $ qs => lc_tr ctxt [e, q, qs]);
|
nipkow@24476
|
414 |
in IfC $ b $ res $ NilC end
|
wenzelm@35118
|
415 |
| lc_tr ctxt
|
wenzelm@35118
|
416 |
[e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
|
wenzelm@35118
|
417 |
Const(@{syntax_const "_lc_end"}, _)] =
|
nipkow@24476
|
418 |
(case abs_tr ctxt p e true of
|
wenzelm@35118
|
419 |
(f, true) => mapC $ f $ es
|
wenzelm@35118
|
420 |
| (f, false) => concatC $ (mapC $ f $ es))
|
wenzelm@35118
|
421 |
| lc_tr ctxt
|
wenzelm@35118
|
422 |
[e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
|
wenzelm@35118
|
423 |
Const (@{syntax_const "_lc_quals"}, _) $ q $ qs] =
|
wenzelm@35118
|
424 |
let val e' = lc_tr ctxt [e, q, qs];
|
wenzelm@35118
|
425 |
in concatC $ (mapC $ (fst (abs_tr ctxt p e' false)) $ es) end;
|
wenzelm@35118
|
426 |
|
wenzelm@35118
|
427 |
in [(@{syntax_const "_listcompr"}, lc_tr)] end
|
nipkow@24349
|
428 |
*}
|
nipkow@23279
|
429 |
|
wenzelm@43038
|
430 |
ML {*
|
wenzelm@43038
|
431 |
let
|
wenzelm@43038
|
432 |
val read = Syntax.read_term @{context};
|
wenzelm@43038
|
433 |
fun check s1 s2 = read s1 aconv read s2 orelse error ("Check failed: " ^ quote s1);
|
wenzelm@43038
|
434 |
in
|
wenzelm@43038
|
435 |
check "[(x,y,z). b]" "if b then [(x, y, z)] else []";
|
wenzelm@43038
|
436 |
check "[(x,y,z). x\<leftarrow>xs]" "map (\<lambda>x. (x, y, z)) xs";
|
wenzelm@43038
|
437 |
check "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]" "concat (map (\<lambda>x. map (\<lambda>y. e x y) ys) xs)";
|
wenzelm@43038
|
438 |
check "[(x,y,z). x<a, x>b]" "if x < a then if b < x then [(x, y, z)] else [] else []";
|
wenzelm@43038
|
439 |
check "[(x,y,z). x\<leftarrow>xs, x>b]" "concat (map (\<lambda>x. if b < x then [(x, y, z)] else []) xs)";
|
wenzelm@43038
|
440 |
check "[(x,y,z). x<a, x\<leftarrow>xs]" "if x < a then map (\<lambda>x. (x, y, z)) xs else []";
|
wenzelm@43038
|
441 |
check "[(x,y). Cons True x \<leftarrow> xs]"
|
wenzelm@43038
|
442 |
"concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | True # x \<Rightarrow> [(x, y)] | False # x \<Rightarrow> []) xs)";
|
wenzelm@43038
|
443 |
check "[(x,y,z). Cons x [] \<leftarrow> xs]"
|
wenzelm@43038
|
444 |
"concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | [x] \<Rightarrow> [(x, y, z)] | x # aa # lista \<Rightarrow> []) xs)";
|
wenzelm@43038
|
445 |
check "[(x,y,z). x<a, x>b, x=d]"
|
wenzelm@43038
|
446 |
"if x < a then if b < x then if x = d then [(x, y, z)] else [] else [] else []";
|
wenzelm@43038
|
447 |
check "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
|
wenzelm@43038
|
448 |
"if x < a then if b < x then map (\<lambda>y. (x, y, z)) ys else [] else []";
|
wenzelm@43038
|
449 |
check "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
|
wenzelm@43038
|
450 |
"if x < a then concat (map (\<lambda>x. if b < y then [(x, y, z)] else []) xs) else []";
|
wenzelm@43038
|
451 |
check "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
|
wenzelm@43038
|
452 |
"if x < a then concat (map (\<lambda>x. map (\<lambda>y. (x, y, z)) ys) xs) else []";
|
wenzelm@43038
|
453 |
check "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
|
wenzelm@43038
|
454 |
"concat (map (\<lambda>x. if b < x then if y < a then [(x, y, z)] else [] else []) xs)";
|
wenzelm@43038
|
455 |
check "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
|
wenzelm@43038
|
456 |
"concat (map (\<lambda>x. if b < x then map (\<lambda>y. (x, y, z)) ys else []) xs)";
|
wenzelm@43038
|
457 |
check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
|
wenzelm@43038
|
458 |
"concat (map (\<lambda>x. concat (map (\<lambda>y. if x < y then [(x, y, z)] else []) ys)) xs)";
|
wenzelm@43038
|
459 |
check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
|
wenzelm@43038
|
460 |
"concat (map (\<lambda>x. concat (map (\<lambda>y. map (\<lambda>z. (x, y, z)) zs) ys)) xs)"
|
wenzelm@43038
|
461 |
end;
|
wenzelm@43038
|
462 |
*}
|
wenzelm@43038
|
463 |
|
wenzelm@35118
|
464 |
(*
|
nipkow@24349
|
465 |
term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
|
nipkow@23192
|
466 |
*)
|
nipkow@23192
|
467 |
|
wenzelm@43038
|
468 |
|
bulwahn@41695
|
469 |
use "Tools/list_to_set_comprehension.ML"
|
bulwahn@41695
|
470 |
|
bulwahn@41695
|
471 |
simproc_setup list_to_set_comprehension ("set xs") = {* K List_to_Set_Comprehension.simproc *}
|
bulwahn@41695
|
472 |
|
wenzelm@35118
|
473 |
|
haftmann@21061
|
474 |
subsubsection {* @{const Nil} and @{const Cons} *}
|
haftmann@21061
|
475 |
|
haftmann@21061
|
476 |
lemma not_Cons_self [simp]:
|
haftmann@21061
|
477 |
"xs \<noteq> x # xs"
|
nipkow@13145
|
478 |
by (induct xs) auto
|
nipkow@3507
|
479 |
|
wenzelm@42571
|
480 |
lemma not_Cons_self2 [simp]:
|
wenzelm@42571
|
481 |
"x # xs \<noteq> xs"
|
wenzelm@42571
|
482 |
by (rule not_Cons_self [symmetric])
|
wenzelm@13114
|
483 |
|
wenzelm@13142
|
484 |
lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
|
nipkow@13145
|
485 |
by (induct xs) auto
|
wenzelm@13114
|
486 |
|
wenzelm@13142
|
487 |
lemma length_induct:
|
haftmann@21061
|
488 |
"(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
|
nipkow@17589
|
489 |
by (rule measure_induct [of length]) iprover
|
wenzelm@13114
|
490 |
|
haftmann@37289
|
491 |
lemma list_nonempty_induct [consumes 1, case_names single cons]:
|
haftmann@37289
|
492 |
assumes "xs \<noteq> []"
|
haftmann@37289
|
493 |
assumes single: "\<And>x. P [x]"
|
haftmann@37289
|
494 |
assumes cons: "\<And>x xs. xs \<noteq> [] \<Longrightarrow> P xs \<Longrightarrow> P (x # xs)"
|
haftmann@37289
|
495 |
shows "P xs"
|
haftmann@37289
|
496 |
using `xs \<noteq> []` proof (induct xs)
|
haftmann@37289
|
497 |
case Nil then show ?case by simp
|
haftmann@37289
|
498 |
next
|
haftmann@37289
|
499 |
case (Cons x xs) show ?case proof (cases xs)
|
haftmann@37289
|
500 |
case Nil with single show ?thesis by simp
|
haftmann@37289
|
501 |
next
|
haftmann@37289
|
502 |
case Cons then have "xs \<noteq> []" by simp
|
haftmann@37289
|
503 |
moreover with Cons.hyps have "P xs" .
|
haftmann@37289
|
504 |
ultimately show ?thesis by (rule cons)
|
haftmann@37289
|
505 |
qed
|
haftmann@37289
|
506 |
qed
|
haftmann@37289
|
507 |
|
wenzelm@13114
|
508 |
|
haftmann@21061
|
509 |
subsubsection {* @{const length} *}
|
wenzelm@13114
|
510 |
|
wenzelm@13142
|
511 |
text {*
|
haftmann@21061
|
512 |
Needs to come before @{text "@"} because of theorem @{text
|
haftmann@21061
|
513 |
append_eq_append_conv}.
|
wenzelm@13142
|
514 |
*}
|
wenzelm@13114
|
515 |
|
wenzelm@13142
|
516 |
lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
|
nipkow@13145
|
517 |
by (induct xs) auto
|
wenzelm@13114
|
518 |
|
wenzelm@13142
|
519 |
lemma length_map [simp]: "length (map f xs) = length xs"
|
nipkow@13145
|
520 |
by (induct xs) auto
|
wenzelm@13114
|
521 |
|
wenzelm@13142
|
522 |
lemma length_rev [simp]: "length (rev xs) = length xs"
|
nipkow@13145
|
523 |
by (induct xs) auto
|
wenzelm@13114
|
524 |
|
wenzelm@13142
|
525 |
lemma length_tl [simp]: "length (tl xs) = length xs - 1"
|
nipkow@13145
|
526 |
by (cases xs) auto
|
wenzelm@13142
|
527 |
|
wenzelm@13142
|
528 |
lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
|
nipkow@13145
|
529 |
by (induct xs) auto
|
wenzelm@13142
|
530 |
|
wenzelm@13142
|
531 |
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
|
nipkow@13145
|
532 |
by (induct xs) auto
|
wenzelm@13114
|
533 |
|
nipkow@23479
|
534 |
lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
|
nipkow@23479
|
535 |
by auto
|
nipkow@23479
|
536 |
|
wenzelm@13114
|
537 |
lemma length_Suc_conv:
|
nipkow@13145
|
538 |
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
|
nipkow@13145
|
539 |
by (induct xs) auto
|
wenzelm@13114
|
540 |
|
nipkow@14025
|
541 |
lemma Suc_length_conv:
|
nipkow@14025
|
542 |
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
|
paulson@14208
|
543 |
apply (induct xs, simp, simp)
|
nipkow@14025
|
544 |
apply blast
|
nipkow@14025
|
545 |
done
|
nipkow@14025
|
546 |
|
wenzelm@25221
|
547 |
lemma impossible_Cons: "length xs <= length ys ==> xs = x # ys = False"
|
wenzelm@25221
|
548 |
by (induct xs) auto
|
wenzelm@25221
|
549 |
|
haftmann@26442
|
550 |
lemma list_induct2 [consumes 1, case_names Nil Cons]:
|
haftmann@26442
|
551 |
"length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
|
haftmann@26442
|
552 |
(\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
|
haftmann@26442
|
553 |
\<Longrightarrow> P xs ys"
|
haftmann@26442
|
554 |
proof (induct xs arbitrary: ys)
|
haftmann@26442
|
555 |
case Nil then show ?case by simp
|
haftmann@26442
|
556 |
next
|
haftmann@26442
|
557 |
case (Cons x xs ys) then show ?case by (cases ys) simp_all
|
haftmann@26442
|
558 |
qed
|
haftmann@26442
|
559 |
|
haftmann@26442
|
560 |
lemma list_induct3 [consumes 2, case_names Nil Cons]:
|
haftmann@26442
|
561 |
"length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
|
haftmann@26442
|
562 |
(\<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
|
563 |
\<Longrightarrow> P xs ys zs"
|
haftmann@26442
|
564 |
proof (induct xs arbitrary: ys zs)
|
haftmann@26442
|
565 |
case Nil then show ?case by simp
|
haftmann@26442
|
566 |
next
|
haftmann@26442
|
567 |
case (Cons x xs ys zs) then show ?case by (cases ys, simp_all)
|
haftmann@26442
|
568 |
(cases zs, simp_all)
|
haftmann@26442
|
569 |
qed
|
wenzelm@13114
|
570 |
|
kaliszyk@36148
|
571 |
lemma list_induct4 [consumes 3, case_names Nil Cons]:
|
kaliszyk@36148
|
572 |
"length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow>
|
kaliszyk@36148
|
573 |
P [] [] [] [] \<Longrightarrow> (\<And>x xs y ys z zs w ws. length xs = length ys \<Longrightarrow>
|
kaliszyk@36148
|
574 |
length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow> P xs ys zs ws \<Longrightarrow>
|
kaliszyk@36148
|
575 |
P (x#xs) (y#ys) (z#zs) (w#ws)) \<Longrightarrow> P xs ys zs ws"
|
kaliszyk@36148
|
576 |
proof (induct xs arbitrary: ys zs ws)
|
kaliszyk@36148
|
577 |
case Nil then show ?case by simp
|
kaliszyk@36148
|
578 |
next
|
kaliszyk@36148
|
579 |
case (Cons x xs ys zs ws) then show ?case by ((cases ys, simp_all), (cases zs,simp_all)) (cases ws, simp_all)
|
kaliszyk@36148
|
580 |
qed
|
kaliszyk@36148
|
581 |
|
krauss@22493
|
582 |
lemma list_induct2':
|
krauss@22493
|
583 |
"\<lbrakk> P [] [];
|
krauss@22493
|
584 |
\<And>x xs. P (x#xs) [];
|
krauss@22493
|
585 |
\<And>y ys. P [] (y#ys);
|
krauss@22493
|
586 |
\<And>x xs y ys. P xs ys \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
|
krauss@22493
|
587 |
\<Longrightarrow> P xs ys"
|
krauss@22493
|
588 |
by (induct xs arbitrary: ys) (case_tac x, auto)+
|
krauss@22493
|
589 |
|
nipkow@22143
|
590 |
lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
|
nipkow@24349
|
591 |
by (rule Eq_FalseI) auto
|
wenzelm@24037
|
592 |
|
wenzelm@24037
|
593 |
simproc_setup list_neq ("(xs::'a list) = ys") = {*
|
nipkow@22143
|
594 |
(*
|
nipkow@22143
|
595 |
Reduces xs=ys to False if xs and ys cannot be of the same length.
|
nipkow@22143
|
596 |
This is the case if the atomic sublists of one are a submultiset
|
nipkow@22143
|
597 |
of those of the other list and there are fewer Cons's in one than the other.
|
nipkow@22143
|
598 |
*)
|
wenzelm@24037
|
599 |
|
wenzelm@24037
|
600 |
let
|
nipkow@22143
|
601 |
|
huffman@29793
|
602 |
fun len (Const(@{const_name Nil},_)) acc = acc
|
huffman@29793
|
603 |
| len (Const(@{const_name Cons},_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
|
huffman@29793
|
604 |
| len (Const(@{const_name append},_) $ xs $ ys) acc = len xs (len ys acc)
|
huffman@29793
|
605 |
| len (Const(@{const_name rev},_) $ xs) acc = len xs acc
|
huffman@29793
|
606 |
| len (Const(@{const_name map},_) $ _ $ xs) acc = len xs acc
|
nipkow@22143
|
607 |
| len t (ts,n) = (t::ts,n);
|
nipkow@22143
|
608 |
|
wenzelm@24037
|
609 |
fun list_neq _ ss ct =
|
nipkow@22143
|
610 |
let
|
wenzelm@24037
|
611 |
val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
|
nipkow@22143
|
612 |
val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
|
nipkow@22143
|
613 |
fun prove_neq() =
|
nipkow@22143
|
614 |
let
|
nipkow@22143
|
615 |
val Type(_,listT::_) = eqT;
|
haftmann@22994
|
616 |
val size = HOLogic.size_const listT;
|
nipkow@22143
|
617 |
val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
|
nipkow@22143
|
618 |
val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
|
nipkow@22143
|
619 |
val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
|
haftmann@22633
|
620 |
(K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
|
haftmann@22633
|
621 |
in SOME (thm RS @{thm neq_if_length_neq}) end
|
nipkow@22143
|
622 |
in
|
wenzelm@23214
|
623 |
if m < n andalso submultiset (op aconv) (ls,rs) orelse
|
wenzelm@23214
|
624 |
n < m andalso submultiset (op aconv) (rs,ls)
|
nipkow@22143
|
625 |
then prove_neq() else NONE
|
nipkow@22143
|
626 |
end;
|
wenzelm@24037
|
627 |
in list_neq end;
|
nipkow@22143
|
628 |
*}
|
nipkow@22143
|
629 |
|
nipkow@22143
|
630 |
|
nipkow@15392
|
631 |
subsubsection {* @{text "@"} -- append *}
|
wenzelm@13114
|
632 |
|
wenzelm@13142
|
633 |
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
|
nipkow@13145
|
634 |
by (induct xs) auto
|
wenzelm@13114
|
635 |
|
wenzelm@13142
|
636 |
lemma append_Nil2 [simp]: "xs @ [] = xs"
|
nipkow@13145
|
637 |
by (induct xs) auto
|
wenzelm@13114
|
638 |
|
wenzelm@13142
|
639 |
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
|
nipkow@13145
|
640 |
by (induct xs) auto
|
wenzelm@13114
|
641 |
|
wenzelm@13142
|
642 |
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
|
nipkow@13145
|
643 |
by (induct xs) auto
|
wenzelm@13114
|
644 |
|
wenzelm@13142
|
645 |
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
|
nipkow@13145
|
646 |
by (induct xs) auto
|
wenzelm@13114
|
647 |
|
wenzelm@13142
|
648 |
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
|
nipkow@13145
|
649 |
by (induct xs) auto
|
wenzelm@13114
|
650 |
|
blanchet@35828
|
651 |
lemma append_eq_append_conv [simp, no_atp]:
|
nipkow@24526
|
652 |
"length xs = length ys \<or> length us = length vs
|
berghofe@13883
|
653 |
==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
|
nipkow@24526
|
654 |
apply (induct xs arbitrary: ys)
|
paulson@14208
|
655 |
apply (case_tac ys, simp, force)
|
paulson@14208
|
656 |
apply (case_tac ys, force, simp)
|
nipkow@13145
|
657 |
done
|
wenzelm@13114
|
658 |
|
nipkow@24526
|
659 |
lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
|
nipkow@24526
|
660 |
(EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
|
nipkow@24526
|
661 |
apply (induct xs arbitrary: ys zs ts)
|
nipkow@45761
|
662 |
apply fastforce
|
nipkow@14495
|
663 |
apply(case_tac zs)
|
nipkow@14495
|
664 |
apply simp
|
nipkow@45761
|
665 |
apply fastforce
|
nipkow@14495
|
666 |
done
|
nipkow@14495
|
667 |
|
berghofe@34910
|
668 |
lemma same_append_eq [iff, induct_simp]: "(xs @ ys = xs @ zs) = (ys = zs)"
|
nipkow@13145
|
669 |
by simp
|
wenzelm@13114
|
670 |
|
wenzelm@13142
|
671 |
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
|
nipkow@13145
|
672 |
by simp
|
wenzelm@13114
|
673 |
|
berghofe@34910
|
674 |
lemma append_same_eq [iff, induct_simp]: "(ys @ xs = zs @ xs) = (ys = zs)"
|
nipkow@13145
|
675 |
by simp
|
wenzelm@13114
|
676 |
|
wenzelm@13142
|
677 |
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
|
nipkow@13145
|
678 |
using append_same_eq [of _ _ "[]"] by auto
|
wenzelm@13114
|
679 |
|
wenzelm@13142
|
680 |
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
|
nipkow@13145
|
681 |
using append_same_eq [of "[]"] by auto
|
wenzelm@13114
|
682 |
|
blanchet@35828
|
683 |
lemma hd_Cons_tl [simp,no_atp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
|
nipkow@13145
|
684 |
by (induct xs) auto
|
wenzelm@13114
|
685 |
|
wenzelm@13142
|
686 |
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
|
nipkow@13145
|
687 |
by (induct xs) auto
|
wenzelm@13114
|
688 |
|
wenzelm@13142
|
689 |
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
|
nipkow@13145
|
690 |
by (simp add: hd_append split: list.split)
|
wenzelm@13114
|
691 |
|
wenzelm@13142
|
692 |
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
|
nipkow@13145
|
693 |
by (simp split: list.split)
|
wenzelm@13114
|
694 |
|
wenzelm@13142
|
695 |
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
|
nipkow@13145
|
696 |
by (simp add: tl_append split: list.split)
|
wenzelm@13114
|
697 |
|
wenzelm@13142
|
698 |
|
nipkow@14300
|
699 |
lemma Cons_eq_append_conv: "x#xs = ys@zs =
|
nipkow@14300
|
700 |
(ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
|
nipkow@14300
|
701 |
by(cases ys) auto
|
nipkow@14300
|
702 |
|
nipkow@15281
|
703 |
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
|
nipkow@15281
|
704 |
(ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
|
nipkow@15281
|
705 |
by(cases ys) auto
|
nipkow@15281
|
706 |
|
nipkow@14300
|
707 |
|
wenzelm@13142
|
708 |
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
|
wenzelm@13114
|
709 |
|
wenzelm@13114
|
710 |
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
|
nipkow@13145
|
711 |
by simp
|
wenzelm@13114
|
712 |
|
wenzelm@13142
|
713 |
lemma Cons_eq_appendI:
|
nipkow@13145
|
714 |
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
|
nipkow@13145
|
715 |
by (drule sym) simp
|
wenzelm@13114
|
716 |
|
wenzelm@13142
|
717 |
lemma append_eq_appendI:
|
nipkow@13145
|
718 |
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
|
nipkow@13145
|
719 |
by (drule sym) simp
|
wenzelm@13114
|
720 |
|
wenzelm@13114
|
721 |
|
wenzelm@13142
|
722 |
text {*
|
nipkow@13145
|
723 |
Simplification procedure for all list equalities.
|
nipkow@13145
|
724 |
Currently only tries to rearrange @{text "@"} to see if
|
nipkow@13145
|
725 |
- both lists end in a singleton list,
|
nipkow@13145
|
726 |
- or both lists end in the same list.
|
wenzelm@13142
|
727 |
*}
|
wenzelm@13142
|
728 |
|
wenzelm@44467
|
729 |
simproc_setup list_eq ("(xs::'a list) = ys") = {*
|
wenzelm@13462
|
730 |
let
|
wenzelm@44467
|
731 |
fun last (cons as Const (@{const_name Cons}, _) $ _ $ xs) =
|
wenzelm@44467
|
732 |
(case xs of Const (@{const_name Nil}, _) => cons | _ => last xs)
|
wenzelm@44467
|
733 |
| last (Const(@{const_name append},_) $ _ $ ys) = last ys
|
wenzelm@44467
|
734 |
| last t = t;
|
wenzelm@44467
|
735 |
|
wenzelm@44467
|
736 |
fun list1 (Const(@{const_name Cons},_) $ _ $ Const(@{const_name Nil},_)) = true
|
wenzelm@44467
|
737 |
| list1 _ = false;
|
wenzelm@44467
|
738 |
|
wenzelm@44467
|
739 |
fun butlast ((cons as Const(@{const_name Cons},_) $ x) $ xs) =
|
wenzelm@44467
|
740 |
(case xs of Const (@{const_name Nil}, _) => xs | _ => cons $ butlast xs)
|
wenzelm@44467
|
741 |
| butlast ((app as Const (@{const_name append}, _) $ xs) $ ys) = app $ butlast ys
|
wenzelm@44467
|
742 |
| butlast xs = Const(@{const_name Nil}, fastype_of xs);
|
wenzelm@44467
|
743 |
|
wenzelm@44467
|
744 |
val rearr_ss =
|
wenzelm@44467
|
745 |
HOL_basic_ss addsimps [@{thm append_assoc}, @{thm append_Nil}, @{thm append_Cons}];
|
wenzelm@44467
|
746 |
|
wenzelm@44467
|
747 |
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
|
wenzelm@13462
|
748 |
let
|
wenzelm@44467
|
749 |
val lastl = last lhs and lastr = last rhs;
|
wenzelm@44467
|
750 |
fun rearr conv =
|
wenzelm@44467
|
751 |
let
|
wenzelm@44467
|
752 |
val lhs1 = butlast lhs and rhs1 = butlast rhs;
|
wenzelm@44467
|
753 |
val Type(_,listT::_) = eqT
|
wenzelm@44467
|
754 |
val appT = [listT,listT] ---> listT
|
wenzelm@44467
|
755 |
val app = Const(@{const_name append},appT)
|
wenzelm@44467
|
756 |
val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
|
wenzelm@44467
|
757 |
val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
|
wenzelm@44467
|
758 |
val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
|
wenzelm@44467
|
759 |
(K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
|
wenzelm@44467
|
760 |
in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
|
wenzelm@44467
|
761 |
in
|
wenzelm@44467
|
762 |
if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
|
wenzelm@44467
|
763 |
else if lastl aconv lastr then rearr @{thm append_same_eq}
|
wenzelm@44467
|
764 |
else NONE
|
wenzelm@44467
|
765 |
end;
|
wenzelm@44467
|
766 |
in fn _ => fn ss => fn ct => list_eq ss (term_of ct) end;
|
wenzelm@13114
|
767 |
*}
|
wenzelm@13114
|
768 |
|
wenzelm@13114
|
769 |
|
nipkow@15392
|
770 |
subsubsection {* @{text map} *}
|
wenzelm@13114
|
771 |
|
haftmann@40451
|
772 |
lemma hd_map:
|
haftmann@40451
|
773 |
"xs \<noteq> [] \<Longrightarrow> hd (map f xs) = f (hd xs)"
|
haftmann@40451
|
774 |
by (cases xs) simp_all
|
haftmann@40451
|
775 |
|
haftmann@40451
|
776 |
lemma map_tl:
|
haftmann@40451
|
777 |
"map f (tl xs) = tl (map f xs)"
|
haftmann@40451
|
778 |
by (cases xs) simp_all
|
haftmann@40451
|
779 |
|
wenzelm@13142
|
780 |
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
|
nipkow@13145
|
781 |
by (induct xs) simp_all
|
wenzelm@13114
|
782 |
|
wenzelm@13142
|
783 |
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
|
nipkow@13145
|
784 |
by (rule ext, induct_tac xs) auto
|
wenzelm@13114
|
785 |
|
wenzelm@13142
|
786 |
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
|
nipkow@13145
|
787 |
by (induct xs) auto
|
wenzelm@13114
|
788 |
|
hoelzl@33639
|
789 |
lemma map_map [simp]: "map f (map g xs) = map (f \<circ> g) xs"
|
hoelzl@33639
|
790 |
by (induct xs) auto
|
hoelzl@33639
|
791 |
|
nipkow@35200
|
792 |
lemma map_comp_map[simp]: "((map f) o (map g)) = map(f o g)"
|
nipkow@35200
|
793 |
apply(rule ext)
|
nipkow@35200
|
794 |
apply(simp)
|
nipkow@35200
|
795 |
done
|
nipkow@35200
|
796 |
|
wenzelm@13142
|
797 |
lemma rev_map: "rev (map f xs) = map f (rev xs)"
|
nipkow@13145
|
798 |
by (induct xs) auto
|
wenzelm@13114
|
799 |
|
nipkow@13737
|
800 |
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
|
nipkow@13737
|
801 |
by (induct xs) auto
|
nipkow@13737
|
802 |
|
krauss@44884
|
803 |
lemma map_cong [fundef_cong]:
|
haftmann@40366
|
804 |
"xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> map f xs = map g ys"
|
haftmann@40366
|
805 |
by simp
|
wenzelm@13114
|
806 |
|
wenzelm@13142
|
807 |
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
|
nipkow@13145
|
808 |
by (cases xs) auto
|
wenzelm@13114
|
809 |
|
wenzelm@13142
|
810 |
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
|
nipkow@13145
|
811 |
by (cases xs) auto
|
wenzelm@13114
|
812 |
|
paulson@18447
|
813 |
lemma map_eq_Cons_conv:
|
nipkow@14025
|
814 |
"(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
|
nipkow@13145
|
815 |
by (cases xs) auto
|
wenzelm@13114
|
816 |
|
paulson@18447
|
817 |
lemma Cons_eq_map_conv:
|
nipkow@14025
|
818 |
"(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
|
nipkow@14025
|
819 |
by (cases ys) auto
|
nipkow@14025
|
820 |
|
paulson@18447
|
821 |
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
|
paulson@18447
|
822 |
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
|
paulson@18447
|
823 |
declare map_eq_Cons_D [dest!] Cons_eq_map_D [dest!]
|
paulson@18447
|
824 |
|
nipkow@14111
|
825 |
lemma ex_map_conv:
|
nipkow@14111
|
826 |
"(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
|
paulson@18447
|
827 |
by(induct ys, auto simp add: Cons_eq_map_conv)
|
nipkow@14111
|
828 |
|
nipkow@15110
|
829 |
lemma map_eq_imp_length_eq:
|
paulson@35496
|
830 |
assumes "map f xs = map g ys"
|
haftmann@26734
|
831 |
shows "length xs = length ys"
|
haftmann@26734
|
832 |
using assms proof (induct ys arbitrary: xs)
|
haftmann@26734
|
833 |
case Nil then show ?case by simp
|
haftmann@26734
|
834 |
next
|
haftmann@26734
|
835 |
case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto
|
paulson@35496
|
836 |
from Cons xs have "map f zs = map g ys" by simp
|
haftmann@26734
|
837 |
moreover with Cons have "length zs = length ys" by blast
|
haftmann@26734
|
838 |
with xs show ?case by simp
|
haftmann@26734
|
839 |
qed
|
haftmann@26734
|
840 |
|
nipkow@15110
|
841 |
lemma map_inj_on:
|
nipkow@15110
|
842 |
"[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
|
nipkow@15110
|
843 |
==> xs = ys"
|
nipkow@15110
|
844 |
apply(frule map_eq_imp_length_eq)
|
nipkow@15110
|
845 |
apply(rotate_tac -1)
|
nipkow@15110
|
846 |
apply(induct rule:list_induct2)
|
nipkow@15110
|
847 |
apply simp
|
nipkow@15110
|
848 |
apply(simp)
|
nipkow@15110
|
849 |
apply (blast intro:sym)
|
nipkow@15110
|
850 |
done
|
nipkow@15110
|
851 |
|
nipkow@15110
|
852 |
lemma inj_on_map_eq_map:
|
nipkow@15110
|
853 |
"inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
|
nipkow@15110
|
854 |
by(blast dest:map_inj_on)
|
nipkow@15110
|
855 |
|
wenzelm@13114
|
856 |
lemma map_injective:
|
nipkow@24526
|
857 |
"map f xs = map f ys ==> inj f ==> xs = ys"
|
nipkow@24526
|
858 |
by (induct ys arbitrary: xs) (auto dest!:injD)
|
wenzelm@13114
|
859 |
|
nipkow@14339
|
860 |
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
|
nipkow@14339
|
861 |
by(blast dest:map_injective)
|
nipkow@14339
|
862 |
|
wenzelm@13114
|
863 |
lemma inj_mapI: "inj f ==> inj (map f)"
|
nipkow@17589
|
864 |
by (iprover dest: map_injective injD intro: inj_onI)
|
wenzelm@13114
|
865 |
|
wenzelm@13114
|
866 |
lemma inj_mapD: "inj (map f) ==> inj f"
|
paulson@14208
|
867 |
apply (unfold inj_on_def, clarify)
|
nipkow@13145
|
868 |
apply (erule_tac x = "[x]" in ballE)
|
paulson@14208
|
869 |
apply (erule_tac x = "[y]" in ballE, simp, blast)
|
nipkow@13145
|
870 |
apply blast
|
nipkow@13145
|
871 |
done
|
wenzelm@13114
|
872 |
|
nipkow@14339
|
873 |
lemma inj_map[iff]: "inj (map f) = inj f"
|
nipkow@13145
|
874 |
by (blast dest: inj_mapD intro: inj_mapI)
|
wenzelm@13114
|
875 |
|
nipkow@15303
|
876 |
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
|
nipkow@15303
|
877 |
apply(rule inj_onI)
|
nipkow@15303
|
878 |
apply(erule map_inj_on)
|
nipkow@15303
|
879 |
apply(blast intro:inj_onI dest:inj_onD)
|
nipkow@15303
|
880 |
done
|
nipkow@15303
|
881 |
|
kleing@14343
|
882 |
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
|
kleing@14343
|
883 |
by (induct xs, auto)
|
wenzelm@13114
|
884 |
|
nipkow@14402
|
885 |
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
|
nipkow@14402
|
886 |
by (induct xs) auto
|
nipkow@14402
|
887 |
|
nipkow@15110
|
888 |
lemma map_fst_zip[simp]:
|
nipkow@15110
|
889 |
"length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
|
nipkow@15110
|
890 |
by (induct rule:list_induct2, simp_all)
|
nipkow@15110
|
891 |
|
nipkow@15110
|
892 |
lemma map_snd_zip[simp]:
|
nipkow@15110
|
893 |
"length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
|
nipkow@15110
|
894 |
by (induct rule:list_induct2, simp_all)
|
nipkow@15110
|
895 |
|
haftmann@41752
|
896 |
enriched_type map: map
|
haftmann@41620
|
897 |
by (simp_all add: fun_eq_iff id_def)
|
haftmann@40856
|
898 |
|
nipkow@15110
|
899 |
|
nipkow@15392
|
900 |
subsubsection {* @{text rev} *}
|
wenzelm@13114
|
901 |
|
wenzelm@13142
|
902 |
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
|
nipkow@13145
|
903 |
by (induct xs) auto
|
wenzelm@13114
|
904 |
|
wenzelm@13142
|
905 |
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
|
nipkow@13145
|
906 |
by (induct xs) auto
|
wenzelm@13114
|
907 |
|
kleing@15870
|
908 |
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
|
kleing@15870
|
909 |
by auto
|
kleing@15870
|
910 |
|
wenzelm@13142
|
911 |
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
|
nipkow@13145
|
912 |
by (induct xs) auto
|
wenzelm@13114
|
913 |
|
wenzelm@13142
|
914 |
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
|
nipkow@13145
|
915 |
by (induct xs) auto
|
wenzelm@13114
|
916 |
|
kleing@15870
|
917 |
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
|
kleing@15870
|
918 |
by (cases xs) auto
|
kleing@15870
|
919 |
|
kleing@15870
|
920 |
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
|
kleing@15870
|
921 |
by (cases xs) auto
|
kleing@15870
|
922 |
|
haftmann@21061
|
923 |
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
|
haftmann@21061
|
924 |
apply (induct xs arbitrary: ys, force)
|
paulson@14208
|
925 |
apply (case_tac ys, simp, force)
|
nipkow@13145
|
926 |
done
|
wenzelm@13114
|
927 |
|
nipkow@15439
|
928 |
lemma inj_on_rev[iff]: "inj_on rev A"
|
nipkow@15439
|
929 |
by(simp add:inj_on_def)
|
nipkow@15439
|
930 |
|
wenzelm@13366
|
931 |
lemma rev_induct [case_names Nil snoc]:
|
wenzelm@13366
|
932 |
"[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
|
berghofe@15489
|
933 |
apply(simplesubst rev_rev_ident[symmetric])
|
nipkow@13145
|
934 |
apply(rule_tac list = "rev xs" in list.induct, simp_all)
|
nipkow@13145
|
935 |
done
|
wenzelm@13114
|
936 |
|
wenzelm@13366
|
937 |
lemma rev_exhaust [case_names Nil snoc]:
|
wenzelm@13366
|
938 |
"(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
|
nipkow@13145
|
939 |
by (induct xs rule: rev_induct) auto
|
wenzelm@13114
|
940 |
|
wenzelm@13366
|
941 |
lemmas rev_cases = rev_exhaust
|
wenzelm@13366
|
942 |
|
nipkow@18423
|
943 |
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
|
nipkow@18423
|
944 |
by(rule rev_cases[of xs]) auto
|
nipkow@18423
|
945 |
|
wenzelm@13114
|
946 |
|
nipkow@15392
|
947 |
subsubsection {* @{text set} *}
|
wenzelm@13114
|
948 |
|
wenzelm@13142
|
949 |
lemma finite_set [iff]: "finite (set xs)"
|
nipkow@13145
|
950 |
by (induct xs) auto
|
wenzelm@13114
|
951 |
|
wenzelm@13142
|
952 |
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
|
nipkow@13145
|
953 |
by (induct xs) auto
|
wenzelm@13114
|
954 |
|
nipkow@17830
|
955 |
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
|
nipkow@17830
|
956 |
by(cases xs) auto
|
oheimb@14099
|
957 |
|
wenzelm@13142
|
958 |
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
|
nipkow@13145
|
959 |
by auto
|
wenzelm@13114
|
960 |
|
oheimb@14099
|
961 |
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs"
|
oheimb@14099
|
962 |
by auto
|
oheimb@14099
|
963 |
|
wenzelm@13142
|
964 |
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
|
nipkow@13145
|
965 |
by (induct xs) auto
|
wenzelm@13114
|
966 |
|
nipkow@15245
|
967 |
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
|
nipkow@15245
|
968 |
by(induct xs) auto
|
nipkow@15245
|
969 |
|
wenzelm@13142
|
970 |
lemma set_rev [simp]: "set (rev xs) = set xs"
|
nipkow@13145
|
971 |
by (induct xs) auto
|
wenzelm@13114
|
972 |
|
wenzelm@13142
|
973 |
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
|
nipkow@13145
|
974 |
by (induct xs) auto
|
wenzelm@13114
|
975 |
|
wenzelm@13142
|
976 |
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
|
nipkow@13145
|
977 |
by (induct xs) auto
|
wenzelm@13114
|
978 |
|
nipkow@32417
|
979 |
lemma set_upt [simp]: "set[i..<j] = {i..<j}"
|
bulwahn@41695
|
980 |
by (induct j) auto
|
wenzelm@13114
|
981 |
|
nipkow@26073
|
982 |
|
nipkow@26073
|
983 |
lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs"
|
nipkow@26073
|
984 |
proof (induct xs)
|
nipkow@26073
|
985 |
case Nil thus ?case by simp
|
nipkow@26073
|
986 |
next
|
nipkow@26073
|
987 |
case Cons thus ?case by (auto intro: Cons_eq_appendI)
|
nipkow@26073
|
988 |
qed
|
nipkow@26073
|
989 |
|
haftmann@26734
|
990 |
lemma in_set_conv_decomp: "x \<in> set xs \<longleftrightarrow> (\<exists>ys zs. xs = ys @ x # zs)"
|
haftmann@26734
|
991 |
by (auto elim: split_list)
|
nipkow@26073
|
992 |
|
nipkow@26073
|
993 |
lemma split_list_first: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
|
paulson@15113
|
994 |
proof (induct xs)
|
nipkow@26073
|
995 |
case Nil thus ?case by simp
|
nipkow@18049
|
996 |
next
|
nipkow@18049
|
997 |
case (Cons a xs)
|
nipkow@18049
|
998 |
show ?case
|
nipkow@18049
|
999 |
proof cases
|
nipkow@45761
|
1000 |
assume "x = a" thus ?case using Cons by fastforce
|
nipkow@18049
|
1001 |
next
|
nipkow@45761
|
1002 |
assume "x \<noteq> a" thus ?case using Cons by(fastforce intro!: Cons_eq_appendI)
|
nipkow@18049
|
1003 |
qed
|
nipkow@18049
|
1004 |
qed
|
nipkow@18049
|
1005 |
|
nipkow@26073
|
1006 |
lemma in_set_conv_decomp_first:
|
nipkow@26073
|
1007 |
"(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
|
haftmann@26734
|
1008 |
by (auto dest!: split_list_first)
|
nipkow@26073
|
1009 |
|
haftmann@40366
|
1010 |
lemma split_list_last: "x \<in> set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
|
haftmann@40366
|
1011 |
proof (induct xs rule: rev_induct)
|
nipkow@26073
|
1012 |
case Nil thus ?case by simp
|
nipkow@26073
|
1013 |
next
|
nipkow@26073
|
1014 |
case (snoc a xs)
|
nipkow@26073
|
1015 |
show ?case
|
nipkow@26073
|
1016 |
proof cases
|
haftmann@40366
|
1017 |
assume "x = a" thus ?case using snoc by (metis List.set.simps(1) emptyE)
|
nipkow@26073
|
1018 |
next
|
nipkow@45761
|
1019 |
assume "x \<noteq> a" thus ?case using snoc by fastforce
|
nipkow@26073
|
1020 |
qed
|
nipkow@26073
|
1021 |
qed
|
nipkow@26073
|
1022 |
|
nipkow@26073
|
1023 |
lemma in_set_conv_decomp_last:
|
nipkow@26073
|
1024 |
"(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
|
haftmann@26734
|
1025 |
by (auto dest!: split_list_last)
|
nipkow@26073
|
1026 |
|
nipkow@26073
|
1027 |
lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs & P x"
|
nipkow@26073
|
1028 |
proof (induct xs)
|
nipkow@26073
|
1029 |
case Nil thus ?case by simp
|
nipkow@26073
|
1030 |
next
|
nipkow@26073
|
1031 |
case Cons thus ?case
|
nipkow@26073
|
1032 |
by(simp add:Bex_def)(metis append_Cons append.simps(1))
|
nipkow@26073
|
1033 |
qed
|
nipkow@26073
|
1034 |
|
nipkow@26073
|
1035 |
lemma split_list_propE:
|
haftmann@26734
|
1036 |
assumes "\<exists>x \<in> set xs. P x"
|
haftmann@26734
|
1037 |
obtains ys x zs where "xs = ys @ x # zs" and "P x"
|
haftmann@26734
|
1038 |
using split_list_prop [OF assms] by blast
|
nipkow@26073
|
1039 |
|
nipkow@26073
|
1040 |
lemma split_list_first_prop:
|
nipkow@26073
|
1041 |
"\<exists>x \<in> set xs. P x \<Longrightarrow>
|
nipkow@26073
|
1042 |
\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)"
|
haftmann@26734
|
1043 |
proof (induct xs)
|
nipkow@26073
|
1044 |
case Nil thus ?case by simp
|
nipkow@26073
|
1045 |
next
|
nipkow@26073
|
1046 |
case (Cons x xs)
|
nipkow@26073
|
1047 |
show ?case
|
nipkow@26073
|
1048 |
proof cases
|
nipkow@26073
|
1049 |
assume "P x"
|
haftmann@40366
|
1050 |
thus ?thesis by simp (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
|
nipkow@26073
|
1051 |
next
|
nipkow@26073
|
1052 |
assume "\<not> P x"
|
nipkow@26073
|
1053 |
hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
|
nipkow@26073
|
1054 |
thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
|
nipkow@26073
|
1055 |
qed
|
nipkow@26073
|
1056 |
qed
|
nipkow@26073
|
1057 |
|
nipkow@26073
|
1058 |
lemma split_list_first_propE:
|
haftmann@26734
|
1059 |
assumes "\<exists>x \<in> set xs. P x"
|
haftmann@26734
|
1060 |
obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y"
|
haftmann@26734
|
1061 |
using split_list_first_prop [OF assms] by blast
|
nipkow@26073
|
1062 |
|
nipkow@26073
|
1063 |
lemma split_list_first_prop_iff:
|
nipkow@26073
|
1064 |
"(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
|
nipkow@26073
|
1065 |
(\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))"
|
haftmann@26734
|
1066 |
by (rule, erule split_list_first_prop) auto
|
nipkow@26073
|
1067 |
|
nipkow@26073
|
1068 |
lemma split_list_last_prop:
|
nipkow@26073
|
1069 |
"\<exists>x \<in> set xs. P x \<Longrightarrow>
|
nipkow@26073
|
1070 |
\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)"
|
nipkow@26073
|
1071 |
proof(induct xs rule:rev_induct)
|
nipkow@26073
|
1072 |
case Nil thus ?case by simp
|
nipkow@26073
|
1073 |
next
|
nipkow@26073
|
1074 |
case (snoc x xs)
|
nipkow@26073
|
1075 |
show ?case
|
nipkow@26073
|
1076 |
proof cases
|
nipkow@26073
|
1077 |
assume "P x" thus ?thesis by (metis emptyE set_empty)
|
nipkow@26073
|
1078 |
next
|
nipkow@26073
|
1079 |
assume "\<not> P x"
|
nipkow@26073
|
1080 |
hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
|
nipkow@45761
|
1081 |
thus ?thesis using `\<not> P x` snoc(1) by fastforce
|
nipkow@26073
|
1082 |
qed
|
nipkow@26073
|
1083 |
qed
|
nipkow@26073
|
1084 |
|
nipkow@26073
|
1085 |
lemma split_list_last_propE:
|
haftmann@26734
|
1086 |
assumes "\<exists>x \<in> set xs. P x"
|
haftmann@26734
|
1087 |
obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z"
|
haftmann@26734
|
1088 |
using split_list_last_prop [OF assms] by blast
|
nipkow@26073
|
1089 |
|
nipkow@26073
|
1090 |
lemma split_list_last_prop_iff:
|
nipkow@26073
|
1091 |
"(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
|
nipkow@26073
|
1092 |
(\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
|
haftmann@26734
|
1093 |
by (metis split_list_last_prop [where P=P] in_set_conv_decomp)
|
nipkow@26073
|
1094 |
|
nipkow@26073
|
1095 |
lemma finite_list: "finite A ==> EX xs. set xs = A"
|
haftmann@26734
|
1096 |
by (erule finite_induct)
|
haftmann@26734
|
1097 |
(auto simp add: set.simps(2) [symmetric] simp del: set.simps(2))
|
paulson@13508
|
1098 |
|
kleing@14388
|
1099 |
lemma card_length: "card (set xs) \<le> length xs"
|
kleing@14388
|
1100 |
by (induct xs) (auto simp add: card_insert_if)
|
wenzelm@13114
|
1101 |
|
haftmann@26442
|
1102 |
lemma set_minus_filter_out:
|
haftmann@26442
|
1103 |
"set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)"
|
haftmann@26442
|
1104 |
by (induct xs) auto
|
paulson@15168
|
1105 |
|
wenzelm@35118
|
1106 |
|
nipkow@15392
|
1107 |
subsubsection {* @{text filter} *}
|
wenzelm@13114
|
1108 |
|
wenzelm@13142
|
1109 |
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
|
nipkow@13145
|
1110 |
by (induct xs) auto
|
wenzelm@13114
|
1111 |
|
nipkow@15305
|
1112 |
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
|
nipkow@15305
|
1113 |
by (induct xs) simp_all
|
nipkow@15305
|
1114 |
|
wenzelm@13142
|
1115 |
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
|
nipkow@13145
|
1116 |
by (induct xs) auto
|
wenzelm@13114
|
1117 |
|
nipkow@16998
|
1118 |
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
|
nipkow@16998
|
1119 |
by (induct xs) (auto simp add: le_SucI)
|
nipkow@16998
|
1120 |
|
nipkow@18423
|
1121 |
lemma sum_length_filter_compl:
|
nipkow@18423
|
1122 |
"length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
|
nipkow@18423
|
1123 |
by(induct xs) simp_all
|
nipkow@18423
|
1124 |
|
wenzelm@13142
|
1125 |
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
|
nipkow@13145
|
1126 |
by (induct xs) auto
|
wenzelm@13114
|
1127 |
|
wenzelm@13142
|
1128 |
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
|
nipkow@13145
|
1129 |
by (induct xs) auto
|
wenzelm@13114
|
1130 |
|
nipkow@16998
|
1131 |
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)"
|
nipkow@24349
|
1132 |
by (induct xs) simp_all
|
nipkow@16998
|
1133 |
|
nipkow@16998
|
1134 |
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
|
nipkow@16998
|
1135 |
apply (induct xs)
|
nipkow@16998
|
1136 |
apply auto
|
nipkow@16998
|
1137 |
apply(cut_tac P=P and xs=xs in length_filter_le)
|
nipkow@16998
|
1138 |
apply simp
|
nipkow@16998
|
1139 |
done
|
wenzelm@13114
|
1140 |
|
nipkow@16965
|
1141 |
lemma filter_map:
|
nipkow@16965
|
1142 |
"filter P (map f xs) = map f (filter (P o f) xs)"
|
nipkow@16965
|
1143 |
by (induct xs) simp_all
|
nipkow@16965
|
1144 |
|
nipkow@16965
|
1145 |
lemma length_filter_map[simp]:
|
nipkow@16965
|
1146 |
"length (filter P (map f xs)) = length(filter (P o f) xs)"
|
nipkow@16965
|
1147 |
by (simp add:filter_map)
|
nipkow@16965
|
1148 |
|
wenzelm@13142
|
1149 |
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
|
nipkow@13145
|
1150 |
by auto
|
wenzelm@13114
|
1151 |
|
nipkow@15246
|
1152 |
lemma length_filter_less:
|
nipkow@15246
|
1153 |
"\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
|
nipkow@15246
|
1154 |
proof (induct xs)
|
nipkow@15246
|
1155 |
case Nil thus ?case by simp
|
nipkow@15246
|
1156 |
next
|
nipkow@15246
|
1157 |
case (Cons x xs) thus ?case
|
nipkow@15246
|
1158 |
apply (auto split:split_if_asm)
|
nipkow@15246
|
1159 |
using length_filter_le[of P xs] apply arith
|
nipkow@15246
|
1160 |
done
|
nipkow@15246
|
1161 |
qed
|
wenzelm@13114
|
1162 |
|
nipkow@15281
|
1163 |
lemma length_filter_conv_card:
|
nipkow@15281
|
1164 |
"length(filter p xs) = card{i. i < length xs & p(xs!i)}"
|
nipkow@15281
|
1165 |
proof (induct xs)
|
nipkow@15281
|
1166 |
case Nil thus ?case by simp
|
nipkow@15281
|
1167 |
next
|
nipkow@15281
|
1168 |
case (Cons x xs)
|
nipkow@15281
|
1169 |
let ?S = "{i. i < length xs & p(xs!i)}"
|
nipkow@15281
|
1170 |
have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
|
nipkow@15281
|
1171 |
show ?case (is "?l = card ?S'")
|
nipkow@15281
|
1172 |
proof (cases)
|
nipkow@15281
|
1173 |
assume "p x"
|
nipkow@15281
|
1174 |
hence eq: "?S' = insert 0 (Suc ` ?S)"
|
nipkow@25162
|
1175 |
by(auto simp: image_def split:nat.split dest:gr0_implies_Suc)
|
nipkow@15281
|
1176 |
have "length (filter p (x # xs)) = Suc(card ?S)"
|
wenzelm@23388
|
1177 |
using Cons `p x` by simp
|
nipkow@15281
|
1178 |
also have "\<dots> = Suc(card(Suc ` ?S))" using fin
|
huffman@45792
|
1179 |
by (simp add: card_image)
|
nipkow@15281
|
1180 |
also have "\<dots> = card ?S'" using eq fin
|
nipkow@15281
|
1181 |
by (simp add:card_insert_if) (simp add:image_def)
|
nipkow@15281
|
1182 |
finally show ?thesis .
|
nipkow@15281
|
1183 |
next
|
nipkow@15281
|
1184 |
assume "\<not> p x"
|
nipkow@15281
|
1185 |
hence eq: "?S' = Suc ` ?S"
|
nipkow@25162
|
1186 |
by(auto simp add: image_def split:nat.split elim:lessE)
|
nipkow@15281
|
1187 |
have "length (filter p (x # xs)) = card ?S"
|
wenzelm@23388
|
1188 |
using Cons `\<not> p x` by simp
|
nipkow@15281
|
1189 |
also have "\<dots> = card(Suc ` ?S)" using fin
|
huffman@45792
|
1190 |
by (simp add: card_image)
|
nipkow@15281
|
1191 |
also have "\<dots> = card ?S'" using eq fin
|
nipkow@15281
|
1192 |
by (simp add:card_insert_if)
|
nipkow@15281
|
1193 |
finally show ?thesis .
|
nipkow@15281
|
1194 |
qed
|
nipkow@15281
|
1195 |
qed
|
nipkow@15281
|
1196 |
|
nipkow@17629
|
1197 |
lemma Cons_eq_filterD:
|
nipkow@17629
|
1198 |
"x#xs = filter P ys \<Longrightarrow>
|
nipkow@17629
|
1199 |
\<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
|
1200 |
(is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
|
nipkow@17629
|
1201 |
proof(induct ys)
|
nipkow@17629
|
1202 |
case Nil thus ?case by simp
|
nipkow@17629
|
1203 |
next
|
nipkow@17629
|
1204 |
case (Cons y ys)
|
nipkow@17629
|
1205 |
show ?case (is "\<exists>x. ?Q x")
|
nipkow@17629
|
1206 |
proof cases
|
nipkow@17629
|
1207 |
assume Py: "P y"
|
nipkow@17629
|
1208 |
show ?thesis
|
nipkow@17629
|
1209 |
proof cases
|
wenzelm@25221
|
1210 |
assume "x = y"
|
wenzelm@25221
|
1211 |
with Py Cons.prems have "?Q []" by simp
|
wenzelm@25221
|
1212 |
then show ?thesis ..
|
nipkow@17629
|
1213 |
next
|
wenzelm@25221
|
1214 |
assume "x \<noteq> y"
|
wenzelm@25221
|
1215 |
with Py Cons.prems show ?thesis by simp
|
nipkow@17629
|
1216 |
qed
|
nipkow@17629
|
1217 |
next
|
wenzelm@25221
|
1218 |
assume "\<not> P y"
|
nipkow@45761
|
1219 |
with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastforce
|
wenzelm@25221
|
1220 |
then have "?Q (y#us)" by simp
|
wenzelm@25221
|
1221 |
then show ?thesis ..
|
nipkow@17629
|
1222 |
qed
|
nipkow@17629
|
1223 |
qed
|
nipkow@17629
|
1224 |
|
nipkow@17629
|
1225 |
lemma filter_eq_ConsD:
|
nipkow@17629
|
1226 |
"filter P ys = x#xs \<Longrightarrow>
|
nipkow@17629
|
1227 |
\<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
|
1228 |
by(rule Cons_eq_filterD) simp
|
nipkow@17629
|
1229 |
|
nipkow@17629
|
1230 |
lemma filter_eq_Cons_iff:
|
nipkow@17629
|
1231 |
"(filter P ys = x#xs) =
|
nipkow@17629
|
1232 |
(\<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
|
1233 |
by(auto dest:filter_eq_ConsD)
|
nipkow@17629
|
1234 |
|
nipkow@17629
|
1235 |
lemma Cons_eq_filter_iff:
|
nipkow@17629
|
1236 |
"(x#xs = filter P ys) =
|
nipkow@17629
|
1237 |
(\<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
|
1238 |
by(auto dest:Cons_eq_filterD)
|
nipkow@17629
|
1239 |
|
krauss@44884
|
1240 |
lemma filter_cong[fundef_cong]:
|
nipkow@17501
|
1241 |
"xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
|
nipkow@17501
|
1242 |
apply simp
|
nipkow@17501
|
1243 |
apply(erule thin_rl)
|
nipkow@17501
|
1244 |
by (induct ys) simp_all
|
nipkow@17501
|
1245 |
|
nipkow@15281
|
1246 |
|
haftmann@26442
|
1247 |
subsubsection {* List partitioning *}
|
haftmann@26442
|
1248 |
|
haftmann@26442
|
1249 |
primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where
|
haftmann@26442
|
1250 |
"partition P [] = ([], [])"
|
haftmann@26442
|
1251 |
| "partition P (x # xs) =
|
haftmann@26442
|
1252 |
(let (yes, no) = partition P xs
|
haftmann@26442
|
1253 |
in if P x then (x # yes, no) else (yes, x # no))"
|
haftmann@26442
|
1254 |
|
haftmann@26442
|
1255 |
lemma partition_filter1:
|
haftmann@26442
|
1256 |
"fst (partition P xs) = filter P xs"
|
haftmann@26442
|
1257 |
by (induct xs) (auto simp add: Let_def split_def)
|
haftmann@26442
|
1258 |
|
haftmann@26442
|
1259 |
lemma partition_filter2:
|
haftmann@26442
|
1260 |
"snd (partition P xs) = filter (Not o P) xs"
|
haftmann@26442
|
1261 |
by (induct xs) (auto simp add: Let_def split_def)
|
haftmann@26442
|
1262 |
|
haftmann@26442
|
1263 |
lemma partition_P:
|
haftmann@26442
|
1264 |
assumes "partition P xs = (yes, no)"
|
haftmann@26442
|
1265 |
shows "(\<forall>p \<in> set yes. P p) \<and> (\<forall>p \<in> set no. \<not> P p)"
|
haftmann@26442
|
1266 |
proof -
|
haftmann@26442
|
1267 |
from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
|
haftmann@26442
|
1268 |
by simp_all
|
haftmann@26442
|
1269 |
then show ?thesis by (simp_all add: partition_filter1 partition_filter2)
|
haftmann@26442
|
1270 |
qed
|
haftmann@26442
|
1271 |
|
haftmann@26442
|
1272 |
lemma partition_set:
|
haftmann@26442
|
1273 |
assumes "partition P xs = (yes, no)"
|
haftmann@26442
|
1274 |
shows "set yes \<union> set no = set xs"
|
haftmann@26442
|
1275 |
proof -
|
haftmann@26442
|
1276 |
from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
|
haftmann@26442
|
1277 |
by simp_all
|
haftmann@26442
|
1278 |
then show ?thesis by (auto simp add: partition_filter1 partition_filter2)
|
haftmann@26442
|
1279 |
qed
|
haftmann@26442
|
1280 |
|
hoelzl@33639
|
1281 |
lemma partition_filter_conv[simp]:
|
hoelzl@33639
|
1282 |
"partition f xs = (filter f xs,filter (Not o f) xs)"
|
hoelzl@33639
|
1283 |
unfolding partition_filter2[symmetric]
|
hoelzl@33639
|
1284 |
unfolding partition_filter1[symmetric] by simp
|
hoelzl@33639
|
1285 |
|
hoelzl@33639
|
1286 |
declare partition.simps[simp del]
|
haftmann@26442
|
1287 |
|
wenzelm@35118
|
1288 |
|
nipkow@15392
|
1289 |
subsubsection {* @{text concat} *}
|
wenzelm@13114
|
1290 |
|
wenzelm@13142
|
1291 |
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
|
nipkow@13145
|
1292 |
by (induct xs) auto
|
wenzelm@13114
|
1293 |
|
paulson@18447
|
1294 |
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
|
nipkow@13145
|
1295 |
by (induct xss) auto
|
wenzelm@13114
|
1296 |
|
paulson@18447
|
1297 |
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
|
nipkow@13145
|
1298 |
by (induct xss) auto
|
wenzelm@13114
|
1299 |
|
nipkow@24308
|
1300 |
lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
|
nipkow@13145
|
1301 |
by (induct xs) auto
|
wenzelm@13114
|
1302 |
|
nipkow@24476
|
1303 |
lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs"
|
nipkow@24349
|
1304 |
by (induct xs) auto
|
nipkow@24349
|
1305 |
|
wenzelm@13142
|
1306 |
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
|
nipkow@13145
|
1307 |
by (induct xs) auto
|
wenzelm@13114
|
1308 |
|
wenzelm@13142
|
1309 |
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
|
nipkow@13145
|
1310 |
by (induct xs) auto
|
wenzelm@13114
|
1311 |
|
wenzelm@13142
|
1312 |
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
|
nipkow@13145
|
1313 |
by (induct xs) auto
|
wenzelm@13114
|
1314 |
|
bulwahn@40611
|
1315 |
lemma concat_eq_concat_iff: "\<forall>(x, y) \<in> set (zip xs ys). length x = length y ==> length xs = length ys ==> (concat xs = concat ys) = (xs = ys)"
|
bulwahn@40611
|
1316 |
proof (induct xs arbitrary: ys)
|
bulwahn@40611
|
1317 |
case (Cons x xs ys)
|
bulwahn@40611
|
1318 |
thus ?case by (cases ys) auto
|
bulwahn@40611
|
1319 |
qed (auto)
|
bulwahn@40611
|
1320 |
|
bulwahn@40611
|
1321 |
lemma concat_injective: "concat xs = concat ys ==> length xs = length ys ==> \<forall>(x, y) \<in> set (zip xs ys). length x = length y ==> xs = ys"
|
bulwahn@40611
|
1322 |
by (simp add: concat_eq_concat_iff)
|
bulwahn@40611
|
1323 |
|
wenzelm@13114
|
1324 |
|
nipkow@15392
|
1325 |
subsubsection {* @{text nth} *}
|
wenzelm@13114
|
1326 |
|
haftmann@29764
|
1327 |
lemma nth_Cons_0 [simp, code]: "(x # xs)!0 = x"
|
nipkow@13145
|
1328 |
by auto
|
wenzelm@13114
|
1329 |
|
haftmann@29764
|
1330 |
lemma nth_Cons_Suc [simp, code]: "(x # xs)!(Suc n) = xs!n"
|
nipkow@13145
|
1331 |
by auto
|
wenzelm@13114
|
1332 |
|
wenzelm@13142
|
1333 |
declare nth.simps [simp del]
|
wenzelm@13114
|
1334 |
|
nipkow@42713
|
1335 |
lemma nth_Cons_pos[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
|
nipkow@42713
|
1336 |
by(auto simp: Nat.gr0_conv_Suc)
|
nipkow@42713
|
1337 |
|
wenzelm@13114
|
1338 |
lemma nth_append:
|
nipkow@24526
|
1339 |
"(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
|
nipkow@24526
|
1340 |
apply (induct xs arbitrary: n, simp)
|
paulson@14208
|
1341 |
apply (case_tac n, auto)
|
nipkow@13145
|
1342 |
done
|
wenzelm@13114
|
1343 |
|
nipkow@14402
|
1344 |
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
|
wenzelm@25221
|
1345 |
by (induct xs) auto
|
nipkow@14402
|
1346 |
|
nipkow@14402
|
1347 |
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
|
wenzelm@25221
|
1348 |
by (induct xs) auto
|
nipkow@14402
|
1349 |
|
nipkow@24526
|
1350 |
lemma nth_map [simp]: "n < length xs ==> (map f xs)!n = f(xs!n)"
|
nipkow@24526
|
1351 |
apply (induct xs arbitrary: n, simp)
|
paulson@14208
|
1352 |
apply (case_tac n, auto)
|
nipkow@13145
|
1353 |
done
|
wenzelm@13114
|
1354 |
|
nipkow@18423
|
1355 |
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
|
nipkow@18423
|
1356 |
by(cases xs) simp_all
|
nipkow@18423
|
1357 |
|
nipkow@18049
|
1358 |
|
nipkow@18049
|
1359 |
lemma list_eq_iff_nth_eq:
|
nipkow@24526
|
1360 |
"(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
|
nipkow@24526
|
1361 |
apply(induct xs arbitrary: ys)
|
paulson@24632
|
1362 |
apply force
|
nipkow@18049
|
1363 |
apply(case_tac ys)
|
nipkow@18049
|
1364 |
apply simp
|
nipkow@18049
|
1365 |
apply(simp add:nth_Cons split:nat.split)apply blast
|
nipkow@18049
|
1366 |
done
|
nipkow@18049
|
1367 |
|
wenzelm@13142
|
1368 |
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
|
paulson@15251
|
1369 |
apply (induct xs, simp, simp)
|
nipkow@13145
|
1370 |
apply safe
|
paulson@24632
|
1371 |
apply (metis nat_case_0 nth.simps zero_less_Suc)
|
paulson@24632
|
1372 |
apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
|
paulson@14208
|
1373 |
apply (case_tac i, simp)
|
paulson@24632
|
1374 |
apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff)
|
nipkow@13145
|
1375 |
done
|
wenzelm@13114
|
1376 |
|
nipkow@17501
|
1377 |
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
|
nipkow@17501
|
1378 |
by(auto simp:set_conv_nth)
|
nipkow@17501
|
1379 |
|
nipkow@13145
|
1380 |
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
|
nipkow@13145
|
1381 |
by (auto simp add: set_conv_nth)
|
wenzelm@13114
|
1382 |
|
wenzelm@13142
|
1383 |
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
|
nipkow@13145
|
1384 |
by (auto simp add: set_conv_nth)
|
wenzelm@13114
|
1385 |
|
wenzelm@13114
|
1386 |
lemma all_nth_imp_all_set:
|
nipkow@13145
|
1387 |
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
|
nipkow@13145
|
1388 |
by (auto simp add: set_conv_nth)
|
wenzelm@13114
|
1389 |
|
wenzelm@13114
|
1390 |
lemma all_set_conv_all_nth:
|
nipkow@13145
|
1391 |
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
|
nipkow@13145
|
1392 |
by (auto simp add: set_conv_nth)
|
wenzelm@13114
|
1393 |
|
kleing@25296
|
1394 |
lemma rev_nth:
|
kleing@25296
|
1395 |
"n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)"
|
kleing@25296
|
1396 |
proof (induct xs arbitrary: n)
|
kleing@25296
|
1397 |
case Nil thus ?case by simp
|
kleing@25296
|
1398 |
next
|
kleing@25296
|
1399 |
case (Cons x xs)
|
kleing@25296
|
1400 |
hence n: "n < Suc (length xs)" by simp
|
kleing@25296
|
1401 |
moreover
|
kleing@25296
|
1402 |
{ assume "n < length xs"
|
kleing@25296
|
1403 |
with n obtain n' where "length xs - n = Suc n'"
|
kleing@25296
|
1404 |
by (cases "length xs - n", auto)
|
kleing@25296
|
1405 |
moreover
|
kleing@25296
|
1406 |
then have "length xs - Suc n = n'" by simp
|
kleing@25296
|
1407 |
ultimately
|
kleing@25296
|
1408 |
have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp
|
kleing@25296
|
1409 |
}
|
kleing@25296
|
1410 |
ultimately
|
kleing@25296
|
1411 |
show ?case by (clarsimp simp add: Cons nth_append)
|
kleing@25296
|
1412 |
qed
|
wenzelm@13114
|
1413 |
|
nipkow@31159
|
1414 |
lemma Skolem_list_nth:
|
nipkow@31159
|
1415 |
"(ALL i<k. EX x. P i x) = (EX xs. size xs = k & (ALL i<k. P i (xs!i)))"
|
nipkow@31159
|
1416 |
(is "_ = (EX xs. ?P k xs)")
|
nipkow@31159
|
1417 |
proof(induct k)
|
nipkow@31159
|
1418 |
case 0 show ?case by simp
|
nipkow@31159
|
1419 |
next
|
nipkow@31159
|
1420 |
case (Suc k)
|
nipkow@31159
|
1421 |
show ?case (is "?L = ?R" is "_ = (EX xs. ?P' xs)")
|
nipkow@31159
|
1422 |
proof
|
nipkow@31159
|
1423 |
assume "?R" thus "?L" using Suc by auto
|
nipkow@31159
|
1424 |
next
|
nipkow@31159
|
1425 |
assume "?L"
|
nipkow@31159
|
1426 |
with Suc obtain x xs where "?P k xs & P k x" by (metis less_Suc_eq)
|
nipkow@31159
|
1427 |
hence "?P'(xs@[x])" by(simp add:nth_append less_Suc_eq)
|
nipkow@31159
|
1428 |
thus "?R" ..
|
nipkow@31159
|
1429 |
qed
|
nipkow@31159
|
1430 |
qed
|
nipkow@31159
|
1431 |
|
nipkow@31159
|
1432 |
|
nipkow@15392
|
1433 |
subsubsection {* @{text list_update} *}
|
wenzelm@13114
|
1434 |
|
nipkow@24526
|
1435 |
lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
|
nipkow@24526
|
1436 |
by (induct xs arbitrary: i) (auto split: nat.split)
|
wenzelm@13114
|
1437 |
|
wenzelm@13114
|
1438 |
lemma nth_list_update:
|
nipkow@24526
|
1439 |
"i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
|
nipkow@24526
|
1440 |
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
|
wenzelm@13114
|
1441 |
|
wenzelm@13142
|
1442 |
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
|
nipkow@13145
|
1443 |
by (simp add: nth_list_update)
|
wenzelm@13114
|
1444 |
|
nipkow@24526
|
1445 |
lemma nth_list_update_neq [simp]: "i \<noteq> j ==> xs[i:=x]!j = xs!j"
|
nipkow@24526
|
1446 |
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
|
wenzelm@13114
|
1447 |
|
nipkow@24526
|
1448 |
lemma list_update_id[simp]: "xs[i := xs!i] = xs"
|
nipkow@24526
|
1449 |
by (induct xs arbitrary: i) (simp_all split:nat.splits)
|
nipkow@24526
|
1450 |
|
nipkow@24526
|
1451 |
lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
|
nipkow@24526
|
1452 |
apply (induct xs arbitrary: i)
|
nipkow@17501
|
1453 |
apply simp
|
nipkow@17501
|
1454 |
apply (case_tac i)
|
nipkow@17501
|
1455 |
apply simp_all
|
nipkow@17501
|
1456 |
done
|
nipkow@17501
|
1457 |
|
nipkow@31077
|
1458 |
lemma list_update_nonempty[simp]: "xs[k:=x] = [] \<longleftrightarrow> xs=[]"
|
nipkow@31077
|
1459 |
by(metis length_0_conv length_list_update)
|
nipkow@31077
|
1460 |
|
wenzelm@13114
|
1461 |
lemma list_update_same_conv:
|
nipkow@24526
|
1462 |
"i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
|
nipkow@24526
|
1463 |
by (induct xs arbitrary: i) (auto split: nat.split)
|
wenzelm@13114
|
1464 |
|
nipkow@14187
|
1465 |
lemma list_update_append1:
|
nipkow@24526
|
1466 |
"i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
|
nipkow@24526
|
1467 |
apply (induct xs arbitrary: i, simp)
|
nipkow@14187
|
1468 |
apply(simp split:nat.split)
|
nipkow@14187
|
1469 |
done
|
nipkow@14187
|
1470 |
|
kleing@15868
|
1471 |
lemma list_update_append:
|
nipkow@24526
|
1472 |
"(xs @ ys) [n:= x] =
|
kleing@15868
|
1473 |
(if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
|
nipkow@24526
|
1474 |
by (induct xs arbitrary: n) (auto split:nat.splits)
|
kleing@15868
|
1475 |
|
nipkow@14402
|
1476 |
lemma list_update_length [simp]:
|
nipkow@14402
|
1477 |
"(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
|
nipkow@14402
|
1478 |
by (induct xs, auto)
|
nipkow@14402
|
1479 |
|
nipkow@31258
|
1480 |
lemma map_update: "map f (xs[k:= y]) = (map f xs)[k := f y]"
|
nipkow@31258
|
1481 |
by(induct xs arbitrary: k)(auto split:nat.splits)
|
nipkow@31258
|
1482 |
|
nipkow@31258
|
1483 |
lemma rev_update:
|
nipkow@31258
|
1484 |
"k < length xs \<Longrightarrow> rev (xs[k:= y]) = (rev xs)[length xs - k - 1 := y]"
|
nipkow@31258
|
1485 |
by (induct xs arbitrary: k) (auto simp: list_update_append split:nat.splits)
|
nipkow@31258
|
1486 |
|
wenzelm@13114
|
1487 |
lemma update_zip:
|
nipkow@31080
|
1488 |
"(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
|
nipkow@24526
|
1489 |
by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split)
|
nipkow@24526
|
1490 |
|
nipkow@24526
|
1491 |
lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)"
|
nipkow@24526
|
1492 |
by (induct xs arbitrary: i) (auto split: nat.split)
|
wenzelm@13114
|
1493 |
|
wenzelm@13114
|
1494 |
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
|
nipkow@13145
|
1495 |
by (blast dest!: set_update_subset_insert [THEN subsetD])
|
wenzelm@13114
|
1496 |
|
nipkow@24526
|
1497 |
lemma set_update_memI: "n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
|
nipkow@24526
|
1498 |
by (induct xs arbitrary: n) (auto split:nat.splits)
|
kleing@15868
|
1499 |
|
nipkow@31077
|
1500 |
lemma list_update_overwrite[simp]:
|
haftmann@24796
|
1501 |
"xs [i := x, i := y] = xs [i := y]"
|
nipkow@31077
|
1502 |
apply (induct xs arbitrary: i) apply simp
|
nipkow@31077
|
1503 |
apply (case_tac i, simp_all)
|
haftmann@24796
|
1504 |
done
|
haftmann@24796
|
1505 |
|
haftmann@24796
|
1506 |
lemma list_update_swap:
|
haftmann@24796
|
1507 |
"i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]"
|
haftmann@24796
|
1508 |
apply (induct xs arbitrary: i i')
|
haftmann@24796
|
1509 |
apply simp
|
haftmann@24796
|
1510 |
apply (case_tac i, case_tac i')
|
haftmann@24796
|
1511 |
apply auto
|
haftmann@24796
|
1512 |
apply (case_tac i')
|
haftmann@24796
|
1513 |
apply auto
|
haftmann@24796
|
1514 |
done
|
haftmann@24796
|
1515 |
|
haftmann@29764
|
1516 |
lemma list_update_code [code]:
|
haftmann@29764
|
1517 |
"[][i := y] = []"
|
haftmann@29764
|
1518 |
"(x # xs)[0 := y] = y # xs"
|
haftmann@29764
|
1519 |
"(x # xs)[Suc i := y] = x # xs[i := y]"
|
haftmann@29764
|
1520 |
by simp_all
|
haftmann@29764
|
1521 |
|
wenzelm@13114
|
1522 |
|
nipkow@15392
|
1523 |
subsubsection {* @{text last} and @{text butlast} *}
|
wenzelm@13114
|
1524 |
|
wenzelm@13142
|
1525 |
lemma last_snoc [simp]: "last (xs @ [x]) = x"
|
nipkow@13145
|
1526 |
by (induct xs) auto
|
wenzelm@13114
|
1527 |
|
wenzelm@13142
|
1528 |
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
|
nipkow@13145
|
1529 |
by (induct xs) auto
|
wenzelm@13114
|
1530 |
|
nipkow@14302
|
1531 |
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
|
huffman@45792
|
1532 |
by simp
|
nipkow@14302
|
1533 |
|
nipkow@14302
|
1534 |
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
|
huffman@45792
|
1535 |
by simp
|
nipkow@14302
|
1536 |
|
nipkow@14302
|
1537 |
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
|
nipkow@14302
|
1538 |
by (induct xs) (auto)
|
nipkow@14302
|
1539 |
|
nipkow@14302
|
1540 |
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
|
nipkow@14302
|
1541 |
by(simp add:last_append)
|
nipkow@14302
|
1542 |
|
nipkow@14302
|
1543 |
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
|
nipkow@14302
|
1544 |
by(simp add:last_append)
|
nipkow@14302
|
1545 |
|
nipkow@17762
|
1546 |
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
|
nipkow@17762
|
1547 |
by(rule rev_exhaust[of xs]) simp_all
|
nipkow@17762
|
1548 |
|
nipkow@17762
|
1549 |
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
|
nipkow@17762
|
1550 |
by(cases xs) simp_all
|
nipkow@17762
|
1551 |
|
nipkow@17765
|
1552 |
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
|
nipkow@17765
|
1553 |
by (induct as) auto
|
nipkow@17762
|
1554 |
|
wenzelm@13142
|
1555 |
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
|
nipkow@13145
|
1556 |
by (induct xs rule: rev_induct) auto
|
wenzelm@13114
|
1557 |
|
wenzelm@13114
|
1558 |
lemma butlast_append:
|
nipkow@24526
|
1559 |
"butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
|
nipkow@24526
|
1560 |
by (induct xs arbitrary: ys) auto
|
wenzelm@13114
|
1561 |
|
wenzelm@13142
|
1562 |
lemma append_butlast_last_id [simp]:
|
nipkow@13145
|
1563 |
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
|
nipkow@13145
|
1564 |
by (induct xs) auto
|
wenzelm@13114
|
1565 |
|
wenzelm@13142
|
1566 |
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
|
nipkow@13145
|
1567 |
by (induct xs) (auto split: split_if_asm)
|
wenzelm@13114
|
1568 |
|
wenzelm@13114
|
1569 |
lemma in_set_butlast_appendI:
|
nipkow@13145
|
1570 |
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
|
nipkow@13145
|
1571 |
by (auto dest: in_set_butlastD simp add: butlast_append)
|
wenzelm@13114
|
1572 |
|
nipkow@24526
|
1573 |
lemma last_drop[simp]: "n < length xs \<Longrightarrow> last (drop n xs) = last xs"
|
nipkow@24526
|
1574 |
apply (induct xs arbitrary: n)
|
nipkow@17501
|
1575 |
apply simp
|
nipkow@17501
|
1576 |
apply (auto split:nat.split)
|
nipkow@17501
|
1577 |
done
|
nipkow@17501
|
1578 |
|
huffman@30128
|
1579 |
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
|
nipkow@17589
|
1580 |
by(induct xs)(auto simp:neq_Nil_conv)
|
nipkow@17589
|
1581 |
|
huffman@30128
|
1582 |
lemma butlast_conv_take: "butlast xs = take (length xs - 1) xs"
|
huffman@26584
|
1583 |
by (induct xs, simp, case_tac xs, simp_all)
|
huffman@26584
|
1584 |
|
nipkow@31077
|
1585 |
lemma last_list_update:
|
nipkow@31077
|
1586 |
"xs \<noteq> [] \<Longrightarrow> last(xs[k:=x]) = (if k = size xs - 1 then x else last xs)"
|
nipkow@31077
|
1587 |
by (auto simp: last_conv_nth)
|
nipkow@31077
|
1588 |
|
nipkow@31077
|
1589 |
lemma butlast_list_update:
|
nipkow@31077
|
1590 |
"butlast(xs[k:=x]) =
|
nipkow@31077
|
1591 |
(if k = size xs - 1 then butlast xs else (butlast xs)[k:=x])"
|
nipkow@31077
|
1592 |
apply(cases xs rule:rev_cases)
|
nipkow@31077
|
1593 |
apply simp
|
nipkow@31077
|
1594 |
apply(simp add:list_update_append split:nat.splits)
|
nipkow@31077
|
1595 |
done
|
nipkow@31077
|
1596 |
|
haftmann@36846
|
1597 |
lemma last_map:
|
haftmann@36846
|
1598 |
"xs \<noteq> [] \<Longrightarrow> last (map f xs) = f (last xs)"
|
haftmann@36846
|
1599 |
by (cases xs rule: rev_cases) simp_all
|
haftmann@36846
|
1600 |
|
haftmann@36846
|
1601 |
lemma map_butlast:
|
haftmann@36846
|
1602 |
"map f (butlast xs) = butlast (map f xs)"
|
haftmann@36846
|
1603 |
by (induct xs) simp_all
|
haftmann@36846
|
1604 |
|
nipkow@40476
|
1605 |
lemma snoc_eq_iff_butlast:
|
nipkow@40476
|
1606 |
"xs @ [x] = ys \<longleftrightarrow> (ys \<noteq> [] & butlast ys = xs & last ys = x)"
|
nipkow@40476
|
1607 |
by (metis append_butlast_last_id append_is_Nil_conv butlast_snoc last_snoc not_Cons_self)
|
nipkow@40476
|
1608 |
|
haftmann@24796
|
1609 |
|
nipkow@15392
|
1610 |
subsubsection {* @{text take} and @{text drop} *}
|
wenzelm@13114
|
1611 |
|
wenzelm@13142
|
1612 |
lemma take_0 [simp]: "take 0 xs = []"
|
nipkow@13145
|
1613 |
by (induct xs) auto
|
wenzelm@13114
|
1614 |
|
wenzelm@13142
|
1615 |
lemma drop_0 [simp]: "drop 0 xs = xs"
|
nipkow@13145
|
1616 |
by (induct xs) auto
|
wenzelm@13114
|
1617 |
|
wenzelm@13142
|
1618 |
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
|
nipkow@13145
|
1619 |
by simp
|
wenzelm@13114
|
1620 |
|
wenzelm@13142
|
1621 |
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
|
nipkow@13145
|
1622 |
by simp
|
wenzelm@13114
|
1623 |
|
wenzelm@13142
|
1624 |
declare take_Cons [simp del] and drop_Cons [simp del]
|
wenzelm@13114
|
1625 |
|
huffman@30128
|
1626 |
lemma take_1_Cons [simp]: "take 1 (x # xs) = [x]"
|
huffman@30128
|
1627 |
unfolding One_nat_def by simp
|
huffman@30128
|
1628 |
|
huffman@30128
|
1629 |
lemma drop_1_Cons [simp]: "drop 1 (x # xs) = xs"
|
huffman@30128
|
1630 |
unfolding One_nat_def by simp
|
huffman@30128
|
1631 |
|
nipkow@15110
|
1632 |
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
|
nipkow@15110
|
1633 |
by(clarsimp simp add:neq_Nil_conv)
|
nipkow@15110
|
1634 |
|
nipkow@14187
|
1635 |
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
|
nipkow@14187
|
1636 |
by(cases xs, simp_all)
|
nipkow@14187
|
1637 |
|
huffman@26584
|
1638 |
lemma take_tl: "take n (tl xs) = tl (take (Suc n) xs)"
|
huffman@26584
|
1639 |
by (induct xs arbitrary: n) simp_all
|
huffman@26584
|
1640 |
|
nipkow@24526
|
1641 |
lemma drop_tl: "drop n (tl xs) = tl(drop n xs)"
|
nipkow@24526
|
1642 |
by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split)
|
nipkow@24526
|
1643 |
|
huffman@26584
|
1644 |
lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)"
|
huffman@26584
|
1645 |
by (cases n, simp, cases xs, auto)
|
huffman@26584
|
1646 |
|
huffman@26584
|
1647 |
lemma tl_drop: "tl (drop n xs) = drop n (tl xs)"
|
huffman@26584
|
1648 |
by (simp only: drop_tl)
|
huffman@26584
|
1649 |
|
nipkow@24526
|
1650 |
lemma nth_via_drop: "drop n xs = y#ys \<Longrightarrow> xs!n = y"
|
nipkow@24526
|
1651 |
apply (induct xs arbitrary: n, simp)
|
nipkow@14187
|
1652 |
apply(simp add:drop_Cons nth_Cons split:nat.splits)
|
nipkow@14187
|
1653 |
done
|
nipkow@14187
|
1654 |
|
nipkow@13913
|
1655 |
lemma take_Suc_conv_app_nth:
|
nipkow@24526
|
1656 |
"i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
|
nipkow@24526
|
1657 |
apply (induct xs arbitrary: i, simp)
|
paulson@14208
|
1658 |
apply (case_tac i, auto)
|
nipkow@13913
|
1659 |
done
|
nipkow@13913
|
1660 |
|
mehta@14591
|
1661 |
lemma drop_Suc_conv_tl:
|
nipkow@24526
|
1662 |
"i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
|
nipkow@24526
|
1663 |
apply (induct xs arbitrary: i, simp)
|
mehta@14591
|
1664 |
apply (case_tac i, auto)
|
mehta@14591
|
1665 |
done
|
mehta@14591
|
1666 |
|
nipkow@24526
|
1667 |
lemma length_take [simp]: "length (take n xs) = min (length xs) n"
|
nipkow@24526
|
1668 |
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
|
nipkow@24526
|
1669 |
|
nipkow@24526
|
1670 |
lemma length_drop [simp]: "length (drop n xs) = (length xs - n)"
|
nipkow@24526
|
1671 |
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
|
nipkow@24526
|
1672 |
|
nipkow@24526
|
1673 |
lemma take_all [simp]: "length xs <= n ==> take n xs = xs"
|
nipkow@24526
|
1674 |
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
|
nipkow@24526
|
1675 |
|
nipkow@24526
|
1676 |
lemma drop_all [simp]: "length xs <= n ==> drop n xs = []"
|
nipkow@24526
|
1677 |
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
|
wenzelm@13114
|
1678 |
|
wenzelm@13142
|
1679 |
lemma take_append [simp]:
|
nipkow@24526
|
1680 |
"take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
|
nipkow@24526
|
1681 |
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
|
wenzelm@13114
|
1682 |
|
wenzelm@13142
|
1683 |
lemma drop_append [simp]:
|
nipkow@24526
|
1684 |
"drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
|
nipkow@24526
|
1685 |
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
|
nipkow@24526
|
1686 |
|
nipkow@24526
|
1687 |
lemma take_take [simp]: "take n (take m xs) = take (min n m) xs"
|
nipkow@24526
|
1688 |
apply (induct m arbitrary: xs n, auto)
|
paulson@14208
|
1689 |
apply (case_tac xs, auto)
|
nipkow@15236
|
1690 |
apply (case_tac n, auto)
|
nipkow@13145
|
1691 |
done
|
wenzelm@13142
|
1692 |
|
nipkow@24526
|
1693 |
lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs"
|
nipkow@24526
|
1694 |
apply (induct m arbitrary: xs, auto)
|
paulson@14208
|
1695 |
apply (case_tac xs, auto)
|
nipkow@13145
|
1696 |
done
|
wenzelm@13114
|
1697 |
|
nipkow@24526
|
1698 |
lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)"
|
nipkow@24526
|
1699 |
apply (induct m arbitrary: xs n, auto)
|
paulson@14208
|
1700 |
apply (case_tac xs, auto)
|
nipkow@13145
|
1701 |
done
|
wenzelm@13114
|
1702 |
|
nipkow@24526
|
1703 |
lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)"
|
nipkow@24526
|
1704 |
apply(induct xs arbitrary: m n)
|
nipkow@14802
|
1705 |
apply simp
|
nipkow@14802
|
1706 |
apply(simp add: take_Cons drop_Cons split:nat.split)
|
nipkow@14802
|
1707 |
done
|
nipkow@14802
|
1708 |
|
nipkow@24526
|
1709 |
lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs"
|
nipkow@24526
|
1710 |
apply (induct n arbitrary: xs, auto)
|
paulson@14208
|
1711 |
apply (case_tac xs, auto)
|
nipkow@13145
|
1712 |
done
|
wenzelm@13114
|
1713 |
|
nipkow@24526
|
1714 |
lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])"
|
nipkow@24526
|
1715 |
apply(induct xs arbitrary: n)
|
nipkow@15110
|
1716 |
apply simp
|
nipkow@15110
|
1717 |
apply(simp add:take_Cons split:nat.split)
|
nipkow@15110
|
1718 |
done
|
nipkow@15110
|
1719 |
|
nipkow@24526
|
1720 |
lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)"
|
nipkow@24526
|
1721 |
apply(induct xs arbitrary: n)
|
nipkow@15110
|
1722 |
apply simp
|
nipkow@15110
|
1723 |
apply(simp add:drop_Cons split:nat.split)
|
nipkow@15110
|
1724 |
done
|
nipkow@15110
|
1725 |
|
nipkow@24526
|
1726 |
lemma take_map: "take n (map f xs) = map f (take n xs)"
|
nipkow@24526
|
1727 |
apply (induct n arbitrary: xs, auto)
|
paulson@14208
|
1728 |
apply (case_tac xs, auto)
|
nipkow@13145
|
1729 |
done
|
wenzelm@13114
|
1730 |
|
nipkow@24526
|
1731 |
lemma drop_map: "drop n (map f xs) = map f (drop n xs)"
|
nipkow@24526
|
1732 |
apply (induct n arbitrary: xs, auto)
|
paulson@14208
|
1733 |
apply (case_tac xs, auto)
|
nipkow@13145
|
1734 |
done
|
wenzelm@13114
|
1735 |
|
nipkow@24526
|
1736 |
lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)"
|
nipkow@24526
|
1737 |
apply (induct xs arbitrary: i, auto)
|
paulson@14208
|
1738 |
apply (case_tac i, auto)
|
nipkow@13145
|
1739 |
done
|
wenzelm@13114
|
1740 |
|
nipkow@24526
|
1741 |
lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)"
|
nipkow@24526
|
1742 |
apply (induct xs arbitrary: i, auto)
|
paulson@14208
|
1743 |
apply (case_tac i, auto)
|
nipkow@13145
|
1744 |
done
|
wenzelm@13114
|
1745 |
|
nipkow@24526
|
1746 |
lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i"
|
nipkow@24526
|
1747 |
apply (induct xs arbitrary: i n, auto)
|
paulson@14208
|
1748 |
apply (case_tac n, blast)
|
paulson@14208
|
1749 |
apply (case_tac i, auto)
|
nipkow@13145
|
1750 |
done
|
wenzelm@13114
|
1751 |
|
wenzelm@13142
|
1752 |
lemma nth_drop [simp]:
|
nipkow@24526
|
1753 |
"n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
|
nipkow@24526
|
1754 |
apply (induct n arbitrary: xs i, auto)
|
paulson@14208
|
1755 |
apply (case_tac xs, auto)
|
nipkow@13145
|
1756 |
done
|
wenzelm@13114
|
1757 |
|
huffman@26584
|
1758 |
lemma butlast_take:
|
huffman@30128
|
1759 |
"n <= length xs ==> butlast (take n xs) = take (n - 1) xs"
|
huffman@26584
|
1760 |
by (simp add: butlast_conv_take min_max.inf_absorb1 min_max.inf_absorb2)
|
huffman@26584
|
1761 |
|
huffman@26584
|
1762 |
lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)"
|
huffman@30128
|
1763 |
by (simp add: butlast_conv_take drop_take add_ac)
|
huffman@26584
|
1764 |
|
huffman@26584
|
1765 |
lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs"
|
huffman@26584
|
1766 |
by (simp add: butlast_conv_take min_max.inf_absorb1)
|
huffman@26584
|
1767 |
|
huffman@26584
|
1768 |
lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)"
|
huffman@30128
|
1769 |
by (simp add: butlast_conv_take drop_take add_ac)
|
huffman@26584
|
1770 |
|
nipkow@18423
|
1771 |
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
|
nipkow@18423
|
1772 |
by(simp add: hd_conv_nth)
|
nipkow@18423
|
1773 |
|
nipkow@35248
|
1774 |
lemma set_take_subset_set_take:
|
nipkow@35248
|
1775 |
"m <= n \<Longrightarrow> set(take m xs) <= set(take n xs)"
|
bulwahn@41695
|
1776 |
apply (induct xs arbitrary: m n)
|
bulwahn@41695
|
1777 |
apply simp
|
bulwahn@41695
|
1778 |
apply (case_tac n)
|
bulwahn@41695
|
1779 |
apply (auto simp: take_Cons)
|
bulwahn@41695
|
1780 |
done
|
nipkow@35248
|
1781 |
|
nipkow@24526
|
1782 |
lemma set_take_subset: "set(take n xs) \<subseteq> set xs"
|
nipkow@24526
|
1783 |
by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split)
|
nipkow@24526
|
1784 |
|
nipkow@24526
|
1785 |
lemma set_drop_subset: "set(drop n xs) \<subseteq> set xs"
|
nipkow@24526
|
1786 |
by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split)
|
nipkow@14025
|
1787 |
|
nipkow@35248
|
1788 |
lemma set_drop_subset_set_drop:
|
nipkow@35248
|
1789 |
"m >= n \<Longrightarrow> set(drop m xs) <= set(drop n xs)"
|
nipkow@35248
|
1790 |
apply(induct xs arbitrary: m n)
|
nipkow@35248
|
1791 |
apply(auto simp:drop_Cons split:nat.split)
|
nipkow@35248
|
1792 |
apply (metis set_drop_subset subset_iff)
|
nipkow@35248
|
1793 |
done
|
nipkow@35248
|
1794 |
|
nipkow@14187
|
1795 |
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
|
nipkow@14187
|
1796 |
using set_take_subset by fast
|
nipkow@14187
|
1797 |
|
nipkow@14187
|
1798 |
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
|
nipkow@14187
|
1799 |
using set_drop_subset by fast
|
nipkow@14187
|
1800 |
|
wenzelm@13114
|
1801 |
lemma append_eq_conv_conj:
|
nipkow@24526
|
1802 |
"(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
|
nipkow@24526
|
1803 |
apply (induct xs arbitrary: zs, simp, clarsimp)
|
paulson@14208
|
1804 |
apply (case_tac zs, auto)
|
nipkow@13145
|
1805 |
done
|
wenzelm@13114
|
1806 |
|
nipkow@24526
|
1807 |
lemma take_add:
|
noschinl@43584
|
1808 |
"take (i+j) xs = take i xs @ take j (drop i xs)"
|
nipkow@24526
|
1809 |
apply (induct xs arbitrary: i, auto)
|
nipkow@24526
|
1810 |
apply (case_tac i, simp_all)
|
paulson@14050
|
1811 |
done
|
paulson@14050
|
1812 |
|
nipkow@14300
|
1813 |
lemma append_eq_append_conv_if:
|
nipkow@24526
|
1814 |
"(xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
|
nipkow@14300
|
1815 |
(if size xs\<^isub>1 \<le> size ys\<^isub>1
|
nipkow@14300
|
1816 |
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
|
1817 |
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
|
1818 |
apply(induct xs\<^isub>1 arbitrary: ys\<^isub>1)
|
nipkow@14300
|
1819 |
apply simp
|
nipkow@14300
|
1820 |
apply(case_tac ys\<^isub>1)
|
nipkow@14300
|
1821 |
apply simp_all
|
nipkow@14300
|
1822 |
done
|
nipkow@14300
|
1823 |
|
nipkow@15110
|
1824 |
lemma take_hd_drop:
|
huffman@30016
|
1825 |
"n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (Suc n) xs"
|
nipkow@24526
|
1826 |
apply(induct xs arbitrary: n)
|
nipkow@15110
|
1827 |
apply simp
|
nipkow@15110
|
1828 |
apply(simp add:drop_Cons split:nat.split)
|
nipkow@15110
|
1829 |
done
|
nipkow@15110
|
1830 |
|
nipkow@17501
|
1831 |
lemma id_take_nth_drop:
|
nipkow@17501
|
1832 |
"i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs"
|
nipkow@17501
|
1833 |
proof -
|
nipkow@17501
|
1834 |
assume si: "i < length xs"
|
nipkow@17501
|
1835 |
hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
|
nipkow@17501
|
1836 |
moreover
|
nipkow@17501
|
1837 |
from si have "take (Suc i) xs = take i xs @ [xs!i]"
|
nipkow@17501
|
1838 |
apply (rule_tac take_Suc_conv_app_nth) by arith
|
nipkow@17501
|
1839 |
ultimately show ?thesis by auto
|
nipkow@17501
|
1840 |
qed
|
nipkow@17501
|
1841 |
|
nipkow@17501
|
1842 |
lemma upd_conv_take_nth_drop:
|
nipkow@17501
|
1843 |
"i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
|
nipkow@17501
|
1844 |
proof -
|
nipkow@17501
|
1845 |
assume i: "i < length xs"
|
nipkow@17501
|
1846 |
have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
|
nipkow@17501
|
1847 |
by(rule arg_cong[OF id_take_nth_drop[OF i]])
|
nipkow@17501
|
1848 |
also have "\<dots> = take i xs @ a # drop (Suc i) xs"
|
nipkow@17501
|
1849 |
using i by (simp add: list_update_append)
|
nipkow@17501
|
1850 |
finally show ?thesis .
|
nipkow@17501
|
1851 |
qed
|
nipkow@17501
|
1852 |
|
haftmann@24796
|
1853 |
lemma nth_drop':
|
haftmann@24796
|
1854 |
"i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs"
|
haftmann@24796
|
1855 |
apply (induct i arbitrary: xs)
|
haftmann@24796
|
1856 |
apply (simp add: neq_Nil_conv)
|
haftmann@24796
|
1857 |
apply (erule exE)+
|
haftmann@24796
|
1858 |
apply simp
|
haftmann@24796
|
1859 |
apply (case_tac xs)
|
haftmann@24796
|
1860 |
apply simp_all
|
haftmann@24796
|
1861 |
done
|
haftmann@24796
|
1862 |
|
wenzelm@13114
|
1863 |
|
nipkow@15392
|
1864 |
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
|
wenzelm@13114
|
1865 |
|
hoelzl@33639
|
1866 |
lemma length_takeWhile_le: "length (takeWhile P xs) \<le> length xs"
|
hoelzl@33639
|
1867 |
by (induct xs) auto
|
hoelzl@33639
|
1868 |
|
wenzelm@13142
|
1869 |
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
|
nipkow@13145
|
1870 |
by (induct xs) auto
|
wenzelm@13114
|
1871 |
|
wenzelm@13142
|
1872 |
lemma takeWhile_append1 [simp]:
|
nipkow@13145
|
1873 |
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
|
nipkow@13145
|
1874 |
by (induct xs) auto
|
wenzelm@13114
|
1875 |
|
wenzelm@13142
|
1876 |
lemma takeWhile_append2 [simp]:
|
nipkow@13145
|
1877 |
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
|
nipkow@13145
|
1878 |
by (induct xs) auto
|
wenzelm@13114
|
1879 |
|
wenzelm@13142
|
1880 |
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
|
nipkow@13145
|
1881 |
by (induct xs) auto
|
wenzelm@13114
|
1882 |
|
hoelzl@33639
|
1883 |
lemma takeWhile_nth: "j < length (takeWhile P xs) \<Longrightarrow> takeWhile P xs ! j = xs ! j"
|
hoelzl@33639
|
1884 |
apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
|
hoelzl@33639
|
1885 |
|
hoelzl@33639
|
1886 |
lemma dropWhile_nth: "j < length (dropWhile P xs) \<Longrightarrow> dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))"
|
hoelzl@33639
|
1887 |
apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
|
hoelzl@33639
|
1888 |
|
hoelzl@33639
|
1889 |
lemma length_dropWhile_le: "length (dropWhile P xs) \<le> length xs"
|
hoelzl@33639
|
1890 |
by (induct xs) auto
|
hoelzl@33639
|
1891 |
|
wenzelm@13142
|
1892 |
lemma dropWhile_append1 [simp]:
|
nipkow@13145
|
1893 |
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
|
nipkow@13145
|
1894 |
by (induct xs) auto
|
wenzelm@13114
|
1895 |
|
wenzelm@13142
|
1896 |
lemma dropWhile_append2 [simp]:
|
nipkow@13145
|
1897 |
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
|
nipkow@13145
|
1898 |
by (induct xs) auto
|
wenzelm@13114
|
1899 |
|
krauss@23971
|
1900 |
lemma set_takeWhileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
|
nipkow@13145
|
1901 |
by (induct xs) (auto split: split_if_asm)
|
wenzelm@13114
|
1902 |
|
nipkow@13913
|
1903 |
lemma takeWhile_eq_all_conv[simp]:
|
nipkow@13913
|
1904 |
"(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
|
nipkow@13913
|
1905 |
by(induct xs, auto)
|
nipkow@13913
|
1906 |
|
nipkow@13913
|
1907 |
lemma dropWhile_eq_Nil_conv[simp]:
|
nipkow@13913
|
1908 |
"(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
|
nipkow@13913
|
1909 |
by(induct xs, auto)
|
nipkow@13913
|
1910 |
|
nipkow@13913
|
1911 |
lemma dropWhile_eq_Cons_conv:
|
nipkow@13913
|
1912 |
"(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
|
nipkow@13913
|
1913 |
by(induct xs, auto)
|
nipkow@13913
|
1914 |
|
nipkow@31077
|
1915 |
lemma distinct_takeWhile[simp]: "distinct xs ==> distinct (takeWhile P xs)"
|
nipkow@31077
|
1916 |
by (induct xs) (auto dest: set_takeWhileD)
|
nipkow@31077
|
1917 |
|
nipkow@31077
|
1918 |
lemma distinct_dropWhile[simp]: "distinct xs ==> distinct (dropWhile P xs)"
|
nipkow@31077
|
1919 |
by (induct xs) auto
|
nipkow@31077
|
1920 |
|
hoelzl@33639
|
1921 |
lemma takeWhile_map: "takeWhile P (map f xs) = map f (takeWhile (P \<circ> f) xs)"
|
hoelzl@33639
|
1922 |
by (induct xs) auto
|
hoelzl@33639
|
1923 |
|
hoelzl@33639
|
1924 |
lemma dropWhile_map: "dropWhile P (map f xs) = map f (dropWhile (P \<circ> f) xs)"
|
hoelzl@33639
|
1925 |
by (induct xs) auto
|
hoelzl@33639
|
1926 |
|
hoelzl@33639
|
1927 |
lemma takeWhile_eq_take: "takeWhile P xs = take (length (takeWhile P xs)) xs"
|
hoelzl@33639
|
1928 |
by (induct xs) auto
|
hoelzl@33639
|
1929 |
|
hoelzl@33639
|
1930 |
lemma dropWhile_eq_drop: "dropWhile P xs = drop (length (takeWhile P xs)) xs"
|
hoelzl@33639
|
1931 |
by (induct xs) auto
|
hoelzl@33639
|
1932 |
|
hoelzl@33639
|
1933 |
lemma hd_dropWhile:
|
hoelzl@33639
|
1934 |
"dropWhile P xs \<noteq> [] \<Longrightarrow> \<not> P (hd (dropWhile P xs))"
|
hoelzl@33639
|
1935 |
using assms by (induct xs) auto
|
hoelzl@33639
|
1936 |
|
hoelzl@33639
|
1937 |
lemma takeWhile_eq_filter:
|
hoelzl@33639
|
1938 |
assumes "\<And> x. x \<in> set (dropWhile P xs) \<Longrightarrow> \<not> P x"
|
hoelzl@33639
|
1939 |
shows "takeWhile P xs = filter P xs"
|
hoelzl@33639
|
1940 |
proof -
|
hoelzl@33639
|
1941 |
have A: "filter P xs = filter P (takeWhile P xs @ dropWhile P xs)"
|
hoelzl@33639
|
1942 |
by simp
|
hoelzl@33639
|
1943 |
have B: "filter P (dropWhile P xs) = []"
|
hoelzl@33639
|
1944 |
unfolding filter_empty_conv using assms by blast
|
hoelzl@33639
|
1945 |
have "filter P xs = takeWhile P xs"
|
hoelzl@33639
|
1946 |
unfolding A filter_append B
|
hoelzl@33639
|
1947 |
by (auto simp add: filter_id_conv dest: set_takeWhileD)
|
hoelzl@33639
|
1948 |
thus ?thesis ..
|
hoelzl@33639
|
1949 |
qed
|
hoelzl@33639
|
1950 |
|
hoelzl@33639
|
1951 |
lemma takeWhile_eq_take_P_nth:
|
hoelzl@33639
|
1952 |
"\<lbrakk> \<And> i. \<lbrakk> i < n ; i < length xs \<rbrakk> \<Longrightarrow> P (xs ! i) ; n < length xs \<Longrightarrow> \<not> P (xs ! n) \<rbrakk> \<Longrightarrow>
|
hoelzl@33639
|
1953 |
takeWhile P xs = take n xs"
|
hoelzl@33639
|
1954 |
proof (induct xs arbitrary: n)
|
hoelzl@33639
|
1955 |
case (Cons x xs)
|
hoelzl@33639
|
1956 |
thus ?case
|
hoelzl@33639
|
1957 |
proof (cases n)
|
hoelzl@33639
|
1958 |
case (Suc n') note this[simp]
|
hoelzl@33639
|
1959 |
have "P x" using Cons.prems(1)[of 0] by simp
|
hoelzl@33639
|
1960 |
moreover have "takeWhile P xs = take n' xs"
|
hoelzl@33639
|
1961 |
proof (rule Cons.hyps)
|
hoelzl@33639
|
1962 |
case goal1 thus "P (xs ! i)" using Cons.prems(1)[of "Suc i"] by simp
|
hoelzl@33639
|
1963 |
next case goal2 thus ?case using Cons by auto
|
hoelzl@33639
|
1964 |
qed
|
hoelzl@33639
|
1965 |
ultimately show ?thesis by simp
|
hoelzl@33639
|
1966 |
qed simp
|
hoelzl@33639
|
1967 |
qed simp
|
hoelzl@33639
|
1968 |
|
hoelzl@33639
|
1969 |
lemma nth_length_takeWhile:
|
hoelzl@33639
|
1970 |
"length (takeWhile P xs) < length xs \<Longrightarrow> \<not> P (xs ! length (takeWhile P xs))"
|
hoelzl@33639
|
1971 |
by (induct xs) auto
|
hoelzl@33639
|
1972 |
|
hoelzl@33639
|
1973 |
lemma length_takeWhile_less_P_nth:
|
hoelzl@33639
|
1974 |
assumes all: "\<And> i. i < j \<Longrightarrow> P (xs ! i)" and "j \<le> length xs"
|
hoelzl@33639
|
1975 |
shows "j \<le> length (takeWhile P xs)"
|
hoelzl@33639
|
1976 |
proof (rule classical)
|
hoelzl@33639
|
1977 |
assume "\<not> ?thesis"
|
hoelzl@33639
|
1978 |
hence "length (takeWhile P xs) < length xs" using assms by simp
|
hoelzl@33639
|
1979 |
thus ?thesis using all `\<not> ?thesis` nth_length_takeWhile[of P xs] by auto
|
hoelzl@33639
|
1980 |
qed
|
nipkow@31077
|
1981 |
|
nipkow@17501
|
1982 |
text{* The following two lemmmas could be generalized to an arbitrary
|
nipkow@17501
|
1983 |
property. *}
|
nipkow@17501
|
1984 |
|
nipkow@17501
|
1985 |
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
|
nipkow@17501
|
1986 |
takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
|
nipkow@17501
|
1987 |
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
|
nipkow@17501
|
1988 |
|
nipkow@17501
|
1989 |
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
|
nipkow@17501
|
1990 |
dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
|
nipkow@17501
|
1991 |
apply(induct xs)
|
nipkow@17501
|
1992 |
apply simp
|
nipkow@17501
|
1993 |
apply auto
|
nipkow@17501
|
1994 |
apply(subst dropWhile_append2)
|
nipkow@17501
|
1995 |
apply auto
|
nipkow@17501
|
1996 |
done
|
nipkow@17501
|
1997 |
|
nipkow@18423
|
1998 |
lemma takeWhile_not_last:
|
nipkow@18423
|
1999 |
"\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
|
nipkow@18423
|
2000 |
apply(induct xs)
|
nipkow@18423
|
2001 |
apply simp
|
nipkow@18423
|
2002 |
apply(case_tac xs)
|
nipkow@18423
|
2003 |
apply(auto)
|
nipkow@18423
|
2004 |
done
|
nipkow@18423
|
2005 |
|
krauss@44884
|
2006 |
lemma takeWhile_cong [fundef_cong]:
|
krauss@18336
|
2007 |
"[| l = k; !!x. x : set l ==> P x = Q x |]
|
krauss@18336
|
2008 |
==> takeWhile P l = takeWhile Q k"
|
nipkow@24349
|
2009 |
by (induct k arbitrary: l) (simp_all)
|
krauss@18336
|
2010 |
|
krauss@44884
|
2011 |
lemma dropWhile_cong [fundef_cong]:
|
krauss@18336
|
2012 |
"[| l = k; !!x. x : set l ==> P x = Q x |]
|
krauss@18336
|
2013 |
==> dropWhile P l = dropWhile Q k"
|
nipkow@24349
|
2014 |
by (induct k arbitrary: l, simp_all)
|
krauss@18336
|
2015 |
|
wenzelm@13114
|
2016 |
|
nipkow@15392
|
2017 |
subsubsection {* @{text zip} *}
|
wenzelm@13114
|
2018 |
|
wenzelm@13142
|
2019 |
lemma zip_Nil [simp]: "zip [] ys = []"
|
nipkow@13145
|
2020 |
by (induct ys) auto
|
wenzelm@13114
|
2021 |
|
wenzelm@13142
|
2022 |
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
|
nipkow@13145
|
2023 |
by simp
|
wenzelm@13114
|
2024 |
|
wenzelm@13142
|
2025 |
declare zip_Cons [simp del]
|
wenzelm@13114
|
2026 |
|
haftmann@36198
|
2027 |
lemma [code]:
|
haftmann@36198
|
2028 |
"zip [] ys = []"
|
haftmann@36198
|
2029 |
"zip xs [] = []"
|
haftmann@36198
|
2030 |
"zip (x # xs) (y # ys) = (x, y) # zip xs ys"
|
haftmann@36198
|
2031 |
by (fact zip_Nil zip.simps(1) zip_Cons_Cons)+
|
haftmann@36198
|
2032 |
|
nipkow@15281
|
2033 |
lemma zip_Cons1:
|
nipkow@15281
|
2034 |
"zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
|
nipkow@15281
|
2035 |
by(auto split:list.split)
|
nipkow@15281
|
2036 |
|
wenzelm@13142
|
2037 |
lemma length_zip [simp]:
|
krauss@22493
|
2038 |
"length (zip xs ys) = min (length xs) (length ys)"
|
krauss@22493
|
2039 |
by (induct xs ys rule:list_induct2') auto
|
wenzelm@13114
|
2040 |
|
haftmann@34965
|
2041 |
lemma zip_obtain_same_length:
|
haftmann@34965
|
2042 |
assumes "\<And>zs ws n. length zs = length ws \<Longrightarrow> n = min (length xs) (length ys)
|
haftmann@34965
|
2043 |
\<Longrightarrow> zs = take n xs \<Longrightarrow> ws = take n ys \<Longrightarrow> P (zip zs ws)"
|
haftmann@34965
|
2044 |
shows "P (zip xs ys)"
|
haftmann@34965
|
2045 |
proof -
|
haftmann@34965
|
2046 |
let ?n = "min (length xs) (length ys)"
|
haftmann@34965
|
2047 |
have "P (zip (take ?n xs) (take ?n ys))"
|
haftmann@34965
|
2048 |
by (rule assms) simp_all
|
haftmann@34965
|
2049 |
moreover have "zip xs ys = zip (take ?n xs) (take ?n ys)"
|
haftmann@34965
|
2050 |
proof (induct xs arbitrary: ys)
|
haftmann@34965
|
2051 |
case Nil then show ?case by simp
|
haftmann@34965
|
2052 |
next
|
haftmann@34965
|
2053 |
case (Cons x xs) then show ?case by (cases ys) simp_all
|
haftmann@34965
|
2054 |
qed
|
haftmann@34965
|
2055 |
ultimately show ?thesis by simp
|
haftmann@34965
|
2056 |
qed
|
haftmann@34965
|
2057 |
|
wenzelm@13114
|
2058 |
lemma zip_append1:
|
krauss@22493
|
2059 |
"zip (xs @ ys) zs =
|
nipkow@13145
|
2060 |
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
|
krauss@22493
|
2061 |
by (induct xs zs rule:list_induct2') auto
|
wenzelm@13114
|
2062 |
|
wenzelm@13114
|
2063 |
lemma zip_append2:
|
krauss@22493
|
2064 |
"zip xs (ys @ zs) =
|
nipkow@13145
|
2065 |
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
|
krauss@22493
|
2066 |
by (induct xs ys rule:list_induct2') auto
|
wenzelm@13114
|
2067 |
|
wenzelm@13142
|
2068 |
lemma zip_append [simp]:
|
wenzelm@13142
|
2069 |
"[| length xs = length us; length ys = length vs |] ==>
|
nipkow@13145
|
2070 |
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
|
nipkow@13145
|
2071 |
by (simp add: zip_append1)
|
wenzelm@13114
|
2072 |
|
wenzelm@13114
|
2073 |
lemma zip_rev:
|
nipkow@14247
|
2074 |
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
|
nipkow@14247
|
2075 |
by (induct rule:list_induct2, simp_all)
|
wenzelm@13114
|
2076 |
|
hoelzl@33639
|
2077 |
lemma zip_map_map:
|
hoelzl@33639
|
2078 |
"zip (map f xs) (map g ys) = map (\<lambda> (x, y). (f x, g y)) (zip xs ys)"
|
hoelzl@33639
|
2079 |
proof (induct xs arbitrary: ys)
|
hoelzl@33639
|
2080 |
case (Cons x xs) note Cons_x_xs = Cons.hyps
|
hoelzl@33639
|
2081 |
show ?case
|
hoelzl@33639
|
2082 |
proof (cases ys)
|
hoelzl@33639
|
2083 |
case (Cons y ys')
|
hoelzl@33639
|
2084 |
show ?thesis unfolding Cons using Cons_x_xs by simp
|
hoelzl@33639
|
2085 |
qed simp
|
hoelzl@33639
|
2086 |
qed simp
|
hoelzl@33639
|
2087 |
|
hoelzl@33639
|
2088 |
lemma zip_map1:
|
hoelzl@33639
|
2089 |
"zip (map f xs) ys = map (\<lambda>(x, y). (f x, y)) (zip xs ys)"
|
hoelzl@33639
|
2090 |
using zip_map_map[of f xs "\<lambda>x. x" ys] by simp
|
hoelzl@33639
|
2091 |
|
hoelzl@33639
|
2092 |
lemma zip_map2:
|
hoelzl@33639
|
2093 |
"zip xs (map f ys) = map (\<lambda>(x, y). (x, f y)) (zip xs ys)"
|
hoelzl@33639
|
2094 |
using zip_map_map[of "\<lambda>x. x" xs f ys] by simp
|
hoelzl@33639
|
2095 |
|
nipkow@23096
|
2096 |
lemma map_zip_map:
|
hoelzl@33639
|
2097 |
"map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
|
hoelzl@33639
|
2098 |
unfolding zip_map1 by auto
|
nipkow@23096
|
2099 |
|
nipkow@23096
|
2100 |
lemma map_zip_map2:
|
hoelzl@33639
|
2101 |
"map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
|
hoelzl@33639
|
2102 |
unfolding zip_map2 by auto
|
nipkow@23096
|
2103 |
|
nipkow@31080
|
2104 |
text{* Courtesy of Andreas Lochbihler: *}
|
nipkow@31080
|
2105 |
lemma zip_same_conv_map: "zip xs xs = map (\<lambda>x. (x, x)) xs"
|
nipkow@31080
|
2106 |
by(induct xs) auto
|
nipkow@31080
|
2107 |
|
wenzelm@13142
|
2108 |
lemma nth_zip [simp]:
|
nipkow@24526
|
2109 |
"[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
|
nipkow@24526
|
2110 |
apply (induct ys arbitrary: i xs, simp)
|
nipkow@13145
|
2111 |
apply (case_tac xs)
|
nipkow@13145
|
2112 |
apply (simp_all add: nth.simps split: nat.split)
|
nipkow@13145
|
2113 |
done
|
wenzelm@13114
|
2114 |
|
wenzelm@13114
|
2115 |
lemma set_zip:
|
nipkow@13145
|
2116 |
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
|
nipkow@31080
|
2117 |
by(simp add: set_conv_nth cong: rev_conj_cong)
|
wenzelm@13114
|
2118 |
|
hoelzl@33639
|
2119 |
lemma zip_same: "((a,b) \<in> set (zip xs xs)) = (a \<in> set xs \<and> a = b)"
|
hoelzl@33639
|
2120 |
by(induct xs) auto
|
hoelzl@33639
|
2121 |
|
wenzelm@13114
|
2122 |
lemma zip_update:
|
nipkow@31080
|
2123 |
"zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
|
nipkow@31080
|
2124 |
by(rule sym, simp add: update_zip)
|
wenzelm@13114
|
2125 |
|
wenzelm@13142
|
2126 |
lemma zip_replicate [simp]:
|
nipkow@24526
|
2127 |
"zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
|
nipkow@24526
|
2128 |
apply (induct i arbitrary: j, auto)
|
paulson@14208
|
2129 |
apply (case_tac j, auto)
|
nipkow@13145
|
2130 |
done
|
wenzelm@13114
|
2131 |
|
nipkow@19487
|
2132 |
lemma take_zip:
|
nipkow@24526
|
2133 |
"take n (zip xs ys) = zip (take n xs) (take n ys)"
|
nipkow@24526
|
2134 |
apply (induct n arbitrary: xs ys)
|
nipkow@19487
|
2135 |
apply simp
|
nipkow@19487
|
2136 |
apply (case_tac xs, simp)
|
nipkow@19487
|
2137 |
apply (case_tac ys, simp_all)
|
nipkow@19487
|
2138 |
done
|
nipkow@19487
|
2139 |
|
nipkow@19487
|
2140 |
lemma drop_zip:
|
nipkow@24526
|
2141 |
"drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
|
nipkow@24526
|
2142 |
apply (induct n arbitrary: xs ys)
|
nipkow@19487
|
2143 |
apply simp
|
nipkow@19487
|
2144 |
apply (case_tac xs, simp)
|
nipkow@19487
|
2145 |
apply (case_tac ys, simp_all)
|
nipkow@19487
|
2146 |
done
|
nipkow@19487
|
2147 |
|
hoelzl@33639
|
2148 |
lemma zip_takeWhile_fst: "zip (takeWhile P xs) ys = takeWhile (P \<circ> fst) (zip xs ys)"
|
hoelzl@33639
|
2149 |
proof (induct xs arbitrary: ys)
|
hoelzl@33639
|
2150 |
case (Cons x xs) thus ?case by (cases ys) auto
|
hoelzl@33639
|
2151 |
qed simp
|
hoelzl@33639
|
2152 |
|
hoelzl@33639
|
2153 |
lemma zip_takeWhile_snd: "zip xs (takeWhile P ys) = takeWhile (P \<circ> snd) (zip xs ys)"
|
hoelzl@33639
|
2154 |
proof (induct xs arbitrary: ys)
|
hoelzl@33639
|
2155 |
case (Cons x xs) thus ?case by (cases ys) auto
|
hoelzl@33639
|
2156 |
qed simp
|
hoelzl@33639
|
2157 |
|
krauss@22493
|
2158 |
lemma set_zip_leftD:
|
krauss@22493
|
2159 |
"(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
|
krauss@22493
|
2160 |
by (induct xs ys rule:list_induct2') auto
|
krauss@22493
|
2161 |
|
krauss@22493
|
2162 |
lemma set_zip_rightD:
|
krauss@22493
|
2163 |
"(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
|
krauss@22493
|
2164 |
by (induct xs ys rule:list_induct2') auto
|
wenzelm@13142
|
2165 |
|
nipkow@23983
|
2166 |
lemma in_set_zipE:
|
nipkow@23983
|
2167 |
"(x,y) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
|
nipkow@23983
|
2168 |
by(blast dest: set_zip_leftD set_zip_rightD)
|
nipkow@23983
|
2169 |
|
haftmann@29766
|
2170 |
lemma zip_map_fst_snd:
|
haftmann@29766
|
2171 |
"zip (map fst zs) (map snd zs) = zs"
|
haftmann@29766
|
2172 |
by (induct zs) simp_all
|
haftmann@29766
|
2173 |
|
haftmann@29766
|
2174 |
lemma zip_eq_conv:
|
haftmann@29766
|
2175 |
"length xs = length ys \<Longrightarrow> zip xs ys = zs \<longleftrightarrow> map fst zs = xs \<and> map snd zs = ys"
|
haftmann@29766
|
2176 |
by (auto simp add: zip_map_fst_snd)
|
haftmann@29766
|
2177 |
|
wenzelm@35118
|
2178 |
|
nipkow@15392
|
2179 |
subsubsection {* @{text list_all2} *}
|
wenzelm@13114
|
2180 |
|
kleing@14316
|
2181 |
lemma list_all2_lengthD [intro?]:
|
kleing@14316
|
2182 |
"list_all2 P xs ys ==> length xs = length ys"
|
nipkow@24349
|
2183 |
by (simp add: list_all2_def)
|
haftmann@19607
|
2184 |
|
haftmann@19787
|
2185 |
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
|
nipkow@24349
|
2186 |
by (simp add: list_all2_def)
|
haftmann@19607
|
2187 |
|
haftmann@19787
|
2188 |
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
|
nipkow@24349
|
2189 |
by (simp add: list_all2_def)
|
haftmann@19607
|
2190 |
|
haftmann@19607
|
2191 |
lemma list_all2_Cons [iff, code]:
|
haftmann@19607
|
2192 |
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
|
nipkow@24349
|
2193 |
by (auto simp add: list_all2_def)
|
wenzelm@13114
|
2194 |
|
wenzelm@13114
|
2195 |
lemma list_all2_Cons1:
|
nipkow@13145
|
2196 |
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
|
nipkow@13145
|
2197 |
by (cases ys) auto
|
wenzelm@13114
|
2198 |
|
wenzelm@13114
|
2199 |
lemma list_all2_Cons2:
|
nipkow@13145
|
2200 |
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
|
nipkow@13145
|
2201 |
by (cases xs) auto
|
wenzelm@13114
|
2202 |
|
wenzelm@13142
|
2203 |
lemma list_all2_rev [iff]:
|
nipkow@13145
|
2204 |
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
|
nipkow@13145
|
2205 |
by (simp add: list_all2_def zip_rev cong: conj_cong)
|
wenzelm@13114
|
2206 |
|
kleing@13863
|
2207 |
lemma list_all2_rev1:
|
kleing@13863
|
2208 |
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
|
kleing@13863
|
2209 |
by (subst list_all2_rev [symmetric]) simp
|
kleing@13863
|
2210 |
|
wenzelm@13114
|
2211 |
lemma list_all2_append1:
|
nipkow@13145
|
2212 |
"list_all2 P (xs @ ys) zs =
|
nipkow@13145
|
2213 |
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
|
nipkow@13145
|
2214 |
list_all2 P xs us \<and> list_all2 P ys vs)"
|
nipkow@13145
|
2215 |
apply (simp add: list_all2_def zip_append1)
|
nipkow@13145
|
2216 |
apply (rule iffI)
|
nipkow@13145
|
2217 |
apply (rule_tac x = "take (length xs) zs" in exI)
|
nipkow@13145
|
2218 |
apply (rule_tac x = "drop (length xs) zs" in exI)
|
paulson@14208
|
2219 |
apply (force split: nat_diff_split simp add: min_def, clarify)
|
nipkow@13145
|
2220 |
apply (simp add: ball_Un)
|
nipkow@13145
|
2221 |
done
|
wenzelm@13114
|
2222 |
|
wenzelm@13114
|
2223 |
lemma list_all2_append2:
|
nipkow@13145
|
2224 |
"list_all2 P xs (ys @ zs) =
|
nipkow@13145
|
2225 |
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
|
nipkow@13145
|
2226 |
list_all2 P us ys \<and> list_all2 P vs zs)"
|
nipkow@13145
|
2227 |
apply (simp add: list_all2_def zip_append2)
|
nipkow@13145
|
2228 |
apply (rule iffI)
|
nipkow@13145
|
2229 |
apply (rule_tac x = "take (length ys) xs" in exI)
|
nipkow@13145
|
2230 |
apply (rule_tac x = "drop (length ys) xs" in exI)
|
paulson@14208
|
2231 |
apply (force split: nat_diff_split simp add: min_def, clarify)
|
nipkow@13145
|
2232 |
apply (simp add: ball_Un)
|
nipkow@13145
|
2233 |
done
|
wenzelm@13114
|
2234 |
|
kleing@13863
|
2235 |
lemma list_all2_append:
|
nipkow@14247
|
2236 |
"length xs = length ys \<Longrightarrow>
|
nipkow@14247
|
2237 |
list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
|
nipkow@14247
|
2238 |
by (induct rule:list_induct2, simp_all)
|
kleing@13863
|
2239 |
|
kleing@13863
|
2240 |
lemma list_all2_appendI [intro?, trans]:
|
kleing@13863
|
2241 |
"\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
|
nipkow@24349
|
2242 |
by (simp add: list_all2_append list_all2_lengthD)
|
kleing@13863
|
2243 |
|
wenzelm@13114
|
2244 |
lemma list_all2_conv_all_nth:
|
nipkow@13145
|
2245 |
"list_all2 P xs ys =
|
nipkow@13145
|
2246 |
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
|
nipkow@13145
|
2247 |
by (force simp add: list_all2_def set_zip)
|
wenzelm@13114
|
2248 |
|
berghofe@13883
|
2249 |
lemma list_all2_trans:
|
berghofe@13883
|
2250 |
assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
|
berghofe@13883
|
2251 |
shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
|
berghofe@13883
|
2252 |
(is "!!bs cs. PROP ?Q as bs cs")
|
berghofe@13883
|
2253 |
proof (induct as)
|
berghofe@13883
|
2254 |
fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
|
berghofe@13883
|
2255 |
show "!!cs. PROP ?Q (x # xs) bs cs"
|
berghofe@13883
|
2256 |
proof (induct bs)
|
berghofe@13883
|
2257 |
fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
|
berghofe@13883
|
2258 |
show "PROP ?Q (x # xs) (y # ys) cs"
|
berghofe@13883
|
2259 |
by (induct cs) (auto intro: tr I1 I2)
|
berghofe@13883
|
2260 |
qed simp
|
berghofe@13883
|
2261 |
qed simp
|
berghofe@13883
|
2262 |
|
kleing@13863
|
2263 |
lemma list_all2_all_nthI [intro?]:
|
kleing@13863
|
2264 |
"length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
|
nipkow@24349
|
2265 |
by (simp add: list_all2_conv_all_nth)
|
kleing@13863
|
2266 |
|
paulson@14395
|
2267 |
lemma list_all2I:
|
paulson@14395
|
2268 |
"\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
|
nipkow@24349
|
2269 |
by (simp add: list_all2_def)
|
paulson@14395
|
2270 |
|
kleing@14328
|
2271 |
lemma list_all2_nthD:
|
kleing@13863
|
2272 |
"\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
|
nipkow@24349
|
2273 |
by (simp add: list_all2_conv_all_nth)
|
kleing@13863
|
2274 |
|
nipkow@14302
|
2275 |
lemma list_all2_nthD2:
|
nipkow@14302
|
2276 |
"\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
|
nipkow@24349
|
2277 |
by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
|
nipkow@14302
|
2278 |
|
kleing@13863
|
2279 |
lemma list_all2_map1:
|
kleing@13863
|
2280 |
"list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
|
nipkow@24349
|
2281 |
by (simp add: list_all2_conv_all_nth)
|
kleing@13863
|
2282 |
|
kleing@13863
|
2283 |
lemma list_all2_map2:
|
kleing@13863
|
2284 |
"list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
|
nipkow@24349
|
2285 |
by (auto simp add: list_all2_conv_all_nth)
|
kleing@13863
|
2286 |
|
kleing@14316
|
2287 |
lemma list_all2_refl [intro?]:
|
kleing@13863
|
2288 |
"(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
|
nipkow@24349
|
2289 |
by (simp add: list_all2_conv_all_nth)
|
kleing@13863
|
2290 |
|
kleing@13863
|
2291 |
lemma list_all2_update_cong:
|
kleing@13863
|
2292 |
"\<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
|
2293 |
by (simp add: list_all2_conv_all_nth nth_list_update)
|
kleing@13863
|
2294 |
|
kleing@13863
|
2295 |
lemma list_all2_update_cong2:
|
kleing@13863
|
2296 |
"\<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
|
2297 |
by (simp add: list_all2_lengthD list_all2_update_cong)
|
kleing@13863
|
2298 |
|
nipkow@14302
|
2299 |
lemma list_all2_takeI [simp,intro?]:
|
nipkow@24526
|
2300 |
"list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
|
nipkow@24526
|
2301 |
apply (induct xs arbitrary: n ys)
|
nipkow@24526
|
2302 |
apply simp
|
nipkow@24526
|
2303 |
apply (clarsimp simp add: list_all2_Cons1)
|
nipkow@24526
|
2304 |
apply (case_tac n)
|
nipkow@24526
|
2305 |
apply auto
|
nipkow@24526
|
2306 |
done
|
nipkow@14302
|
2307 |
|
nipkow@14302
|
2308 |
lemma list_all2_dropI [simp,intro?]:
|
nipkow@24526
|
2309 |
"list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
|
nipkow@24526
|
2310 |
apply (induct as arbitrary: n bs, simp)
|
nipkow@24526
|
2311 |
apply (clarsimp simp add: list_all2_Cons1)
|
nipkow@24526
|
2312 |
apply (case_tac n, simp, simp)
|
nipkow@24526
|
2313 |
done
|
kleing@13863
|
2314 |
|
kleing@14327
|
2315 |
lemma list_all2_mono [intro?]:
|
nipkow@24526
|
2316 |
"list_all2 P xs ys \<Longrightarrow> (\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys) \<Longrightarrow> list_all2 Q xs ys"
|
nipkow@24526
|
2317 |
apply (induct xs arbitrary: ys, simp)
|
nipkow@24526
|
2318 |
apply (case_tac ys, auto)
|
nipkow@24526
|
2319 |
done
|
kleing@13863
|
2320 |
|
haftmann@22551
|
2321 |
lemma list_all2_eq:
|
haftmann@22551
|
2322 |
"xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
|
nipkow@24349
|
2323 |
by (induct xs ys rule: list_induct2') auto
|
haftmann@22551
|
2324 |
|
nipkow@40476
|
2325 |
lemma list_eq_iff_zip_eq:
|
nipkow@40476
|
2326 |
"xs = ys \<longleftrightarrow> length xs = length ys \<and> (\<forall>(x,y) \<in> set (zip xs ys). x = y)"
|
nipkow@40476
|
2327 |
by(auto simp add: set_zip list_all2_eq list_all2_conv_all_nth cong: conj_cong)
|
nipkow@40476
|
2328 |
|
wenzelm@13114
|
2329 |
|
nipkow@15392
|
2330 |
subsubsection {* @{text foldl} and @{text foldr} *}
|
wenzelm@13114
|
2331 |
|
wenzelm@13142
|
2332 |
lemma foldl_append [simp]:
|
nipkow@24526
|
2333 |
"foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
|
nipkow@24526
|
2334 |
by (induct xs arbitrary: a) auto
|
wenzelm@13114
|
2335 |
|
nipkow@14402
|
2336 |
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
|
nipkow@14402
|
2337 |
by (induct xs) auto
|
nipkow@14402
|
2338 |
|
nipkow@23096
|
2339 |
lemma foldr_map: "foldr g (map f xs) a = foldr (g o f) xs a"
|
nipkow@23096
|
2340 |
by(induct xs) simp_all
|
nipkow@23096
|
2341 |
|
nipkow@24449
|
2342 |
text{* For efficient code generation: avoid intermediate list. *}
|
haftmann@31998
|
2343 |
lemma foldl_map[code_unfold]:
|
nipkow@24449
|
2344 |
"foldl g a (map f xs) = foldl (%a x. g a (f x)) a xs"
|
nipkow@23096
|
2345 |
by(induct xs arbitrary:a) simp_all
|
nipkow@23096
|
2346 |
|
haftmann@34965
|
2347 |
lemma foldl_apply:
|
haftmann@34965
|
2348 |
assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x \<circ> h = h \<circ> g x"
|
haftmann@34965
|
2349 |
shows "foldl (\<lambda>s x. f x s) (h s) xs = h (foldl (\<lambda>s x. g x s) s xs)"
|
nipkow@39535
|
2350 |
by (rule sym, insert assms, induct xs arbitrary: s) (simp_all add: fun_eq_iff)
|
haftmann@31929
|
2351 |
|
krauss@44884
|
2352 |
lemma foldl_cong [fundef_cong]:
|
krauss@18336
|
2353 |
"[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |]
|
krauss@18336
|
2354 |
==> foldl f a l = foldl g b k"
|
nipkow@24349
|
2355 |
by (induct k arbitrary: a b l) simp_all
|
krauss@18336
|
2356 |
|
krauss@44884
|
2357 |
lemma foldr_cong [fundef_cong]:
|
krauss@18336
|
2358 |
"[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |]
|
krauss@18336
|
2359 |
==> foldr f l a = foldr g k b"
|
nipkow@24349
|
2360 |
by (induct k arbitrary: a b l) simp_all
|
krauss@18336
|
2361 |
|
haftmann@35195
|
2362 |
lemma foldl_fun_comm:
|
haftmann@35195
|
2363 |
assumes "\<And>x y s. f (f s x) y = f (f s y) x"
|
haftmann@35195
|
2364 |
shows "f (foldl f s xs) x = foldl f (f s x) xs"
|
haftmann@35195
|
2365 |
by (induct xs arbitrary: s)
|
haftmann@35195
|
2366 |
(simp_all add: assms)
|
haftmann@35195
|
2367 |
|
nipkow@24449
|
2368 |
lemma (in semigroup_add) foldl_assoc:
|
haftmann@25062
|
2369 |
shows "foldl op+ (x+y) zs = x + (foldl op+ y zs)"
|
nipkow@24449
|
2370 |
by (induct zs arbitrary: y) (simp_all add:add_assoc)
|
nipkow@24449
|
2371 |
|
nipkow@24449
|
2372 |
lemma (in monoid_add) foldl_absorb0:
|
haftmann@25062
|
2373 |
shows "x + (foldl op+ 0 zs) = foldl op+ x zs"
|
nipkow@24449
|
2374 |
by (induct zs) (simp_all add:foldl_assoc)
|
nipkow@24449
|
2375 |
|
haftmann@35195
|
2376 |
lemma foldl_rev:
|
haftmann@35195
|
2377 |
assumes "\<And>x y s. f (f s x) y = f (f s y) x"
|
haftmann@35195
|
2378 |
shows "foldl f s (rev xs) = foldl f s xs"
|
haftmann@35195
|
2379 |
proof (induct xs arbitrary: s)
|
haftmann@35195
|
2380 |
case Nil then show ?case by simp
|
haftmann@35195
|
2381 |
next
|
haftmann@35195
|
2382 |
case (Cons x xs) with assms show ?case by (simp add: foldl_fun_comm)
|
haftmann@35195
|
2383 |
qed
|
haftmann@35195
|
2384 |
|
haftmann@37605
|
2385 |
lemma rev_foldl_cons [code]:
|
haftmann@37605
|
2386 |
"rev xs = foldl (\<lambda>xs x. x # xs) [] xs"
|
haftmann@37605
|
2387 |
proof (induct xs)
|
haftmann@37605
|
2388 |
case Nil then show ?case by simp
|
haftmann@37605
|
2389 |
next
|
haftmann@37605
|
2390 |
case Cons
|
haftmann@37605
|
2391 |
{
|
haftmann@37605
|
2392 |
fix x xs ys
|
haftmann@37605
|
2393 |
have "foldl (\<lambda>xs x. x # xs) ys xs @ [x]
|
haftmann@37605
|
2394 |
= foldl (\<lambda>xs x. x # xs) (ys @ [x]) xs"
|
haftmann@37605
|
2395 |
by (induct xs arbitrary: ys) auto
|
haftmann@37605
|
2396 |
}
|
haftmann@37605
|
2397 |
note aux = this
|
haftmann@37605
|
2398 |
show ?case by (induct xs) (auto simp add: Cons aux)
|
haftmann@37605
|
2399 |
qed
|
haftmann@37605
|
2400 |
|
nipkow@24449
|
2401 |
|
haftmann@40007
|
2402 |
text{* The ``Third Duality Theorem'' in Bird \& Wadler: *}
|
haftmann@40007
|
2403 |
|
haftmann@40007
|
2404 |
lemma foldr_foldl:
|
haftmann@40007
|
2405 |
"foldr f xs a = foldl (%x y. f y x) a (rev xs)"
|
haftmann@40007
|
2406 |
by (induct xs) auto
|
haftmann@40007
|
2407 |
|
haftmann@40007
|
2408 |
lemma foldl_foldr:
|
haftmann@40007
|
2409 |
"foldl f a xs = foldr (%x y. f y x) (rev xs) a"
|
haftmann@40007
|
2410 |
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
|
haftmann@40007
|
2411 |
|
haftmann@40007
|
2412 |
|
nipkow@23096
|
2413 |
text{* The ``First Duality Theorem'' in Bird \& Wadler: *}
|
nipkow@23096
|
2414 |
|
haftmann@40007
|
2415 |
lemma (in monoid_add) foldl_foldr1_lemma:
|
haftmann@40007
|
2416 |
"foldl op + a xs = a + foldr op + xs 0"
|
haftmann@40007
|
2417 |
by (induct xs arbitrary: a) (auto simp: add_assoc)
|
haftmann@40007
|
2418 |
|
haftmann@40007
|
2419 |
corollary (in monoid_add) foldl_foldr1:
|
haftmann@40007
|
2420 |
"foldl op + 0 xs = foldr op + xs 0"
|
haftmann@40007
|
2421 |
by (simp add: foldl_foldr1_lemma)
|
haftmann@40007
|
2422 |
|
haftmann@40007
|
2423 |
lemma (in ab_semigroup_add) foldr_conv_foldl:
|
haftmann@40007
|
2424 |
"foldr op + xs a = foldl op + a xs"
|
haftmann@40007
|
2425 |
by (induct xs) (simp_all add: foldl_assoc add.commute)
|
chaieb@24471
|
2426 |
|
wenzelm@13142
|
2427 |
text {*
|
nipkow@13145
|
2428 |
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
|
nipkow@13145
|
2429 |
difficult to use because it requires an additional transitivity step.
|
wenzelm@13142
|
2430 |
*}
|
wenzelm@13114
|
2431 |
|
nipkow@24526
|
2432 |
lemma start_le_sum: "(m::nat) <= n ==> m <= foldl (op +) n ns"
|
nipkow@24526
|
2433 |
by (induct ns arbitrary: n) auto
|
nipkow@24526
|
2434 |
|
nipkow@24526
|
2435 |
lemma elem_le_sum: "(n::nat) : set ns ==> n <= foldl (op +) 0 ns"
|
nipkow@13145
|
2436 |
by (force intro: start_le_sum simp add: in_set_conv_decomp)
|
wenzelm@13114
|
2437 |
|
wenzelm@13142
|
2438 |
lemma sum_eq_0_conv [iff]:
|
nipkow@24526
|
2439 |
"(foldl (op +) (m::nat) ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
|
nipkow@24526
|
2440 |
by (induct ns arbitrary: m) auto
|
wenzelm@13114
|
2441 |
|
chaieb@24471
|
2442 |
lemma foldr_invariant:
|
chaieb@24471
|
2443 |
"\<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
|
2444 |
by (induct xs, simp_all)
|
chaieb@24471
|
2445 |
|
chaieb@24471
|
2446 |
lemma foldl_invariant:
|
chaieb@24471
|
2447 |
"\<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
|
2448 |
by (induct xs arbitrary: x, simp_all)
|
chaieb@24471
|
2449 |
|
haftmann@34965
|
2450 |
lemma foldl_weak_invariant:
|
haftmann@34965
|
2451 |
assumes "P s"
|
haftmann@34965
|
2452 |
and "\<And>s x. x \<in> set xs \<Longrightarrow> P s \<Longrightarrow> P (f s x)"
|
haftmann@34965
|
2453 |
shows "P (foldl f s xs)"
|
haftmann@34965
|
2454 |
using assms by (induct xs arbitrary: s) simp_all
|
haftmann@34965
|
2455 |
|
haftmann@31455
|
2456 |
text {* @{const foldl} and @{const concat} *}
|
nipkow@24449
|
2457 |
|
nipkow@24449
|
2458 |
lemma foldl_conv_concat:
|
haftmann@29719
|
2459 |
"foldl (op @) xs xss = xs @ concat xss"
|
haftmann@29719
|
2460 |
proof (induct xss arbitrary: xs)
|
haftmann@29719
|
2461 |
case Nil show ?case by simp
|
haftmann@29719
|
2462 |
next
|
haftmann@35267
|
2463 |
interpret monoid_add "op @" "[]" proof qed simp_all
|
haftmann@29719
|
2464 |
case Cons then show ?case by (simp add: foldl_absorb0)
|
haftmann@29719
|
2465 |
qed
|
haftmann@29719
|
2466 |
|
haftmann@29719
|
2467 |
lemma concat_conv_foldl: "concat xss = foldl (op @) [] xss"
|
haftmann@29719
|
2468 |
by (simp add: foldl_conv_concat)
|
haftmann@29719
|
2469 |
|
haftmann@31455
|
2470 |
text {* @{const Finite_Set.fold} and @{const foldl} *}
|
haftmann@31455
|
2471 |
|
haftmann@43740
|
2472 |
lemma (in comp_fun_commute) fold_set_remdups:
|
haftmann@35195
|
2473 |
"fold f y (set xs) = foldl (\<lambda>y x. f x y) y (remdups xs)"
|
haftmann@35195
|
2474 |
by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm insert_absorb)
|
haftmann@35195
|
2475 |
|
haftmann@43740
|
2476 |
lemma (in comp_fun_idem) fold_set:
|
haftmann@31455
|
2477 |
"fold f y (set xs) = foldl (\<lambda>y x. f x y) y xs"
|
haftmann@31455
|
2478 |
by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm)
|
haftmann@31455
|
2479 |
|
haftmann@32681
|
2480 |
lemma (in ab_semigroup_idem_mult) fold1_set:
|
haftmann@32681
|
2481 |
assumes "xs \<noteq> []"
|
haftmann@32681
|
2482 |
shows "fold1 times (set xs) = foldl times (hd xs) (tl xs)"
|
haftmann@32681
|
2483 |
proof -
|
haftmann@43740
|
2484 |
interpret comp_fun_idem times by (fact comp_fun_idem)
|
haftmann@32681
|
2485 |
from assms obtain y ys where xs: "xs = y # ys"
|
haftmann@32681
|
2486 |
by (cases xs) auto
|
haftmann@32681
|
2487 |
show ?thesis
|
haftmann@32681
|
2488 |
proof (cases "set ys = {}")
|
haftmann@32681
|
2489 |
case True with xs show ?thesis by simp
|
haftmann@32681
|
2490 |
next
|
haftmann@32681
|
2491 |
case False
|
haftmann@32681
|
2492 |
then have "fold1 times (insert y (set ys)) = fold times y (set ys)"
|
haftmann@32681
|
2493 |
by (simp only: finite_set fold1_eq_fold_idem)
|
haftmann@32681
|
2494 |
with xs show ?thesis by (simp add: fold_set mult_commute)
|
haftmann@32681
|
2495 |
qed
|
haftmann@32681
|
2496 |
qed
|
haftmann@32681
|
2497 |
|
haftmann@32681
|
2498 |
lemma (in lattice) Inf_fin_set_fold [code_unfold]:
|
haftmann@32681
|
2499 |
"Inf_fin (set (x # xs)) = foldl inf x xs"
|
haftmann@32681
|
2500 |
proof -
|
haftmann@32681
|
2501 |
interpret ab_semigroup_idem_mult "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
|
haftmann@32681
|
2502 |
by (fact ab_semigroup_idem_mult_inf)
|
haftmann@32681
|
2503 |
show ?thesis
|
haftmann@32681
|
2504 |
by (simp add: Inf_fin_def fold1_set del: set.simps)
|
haftmann@32681
|
2505 |
qed
|
haftmann@32681
|
2506 |
|
haftmann@32681
|
2507 |
lemma (in lattice) Sup_fin_set_fold [code_unfold]:
|
haftmann@32681
|
2508 |
"Sup_fin (set (x # xs)) = foldl sup x xs"
|
haftmann@32681
|
2509 |
proof -
|
haftmann@32681
|
2510 |
interpret ab_semigroup_idem_mult "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
|
haftmann@32681
|
2511 |
by (fact ab_semigroup_idem_mult_sup)
|
haftmann@32681
|
2512 |
show ?thesis
|
haftmann@32681
|
2513 |
by (simp add: Sup_fin_def fold1_set del: set.simps)
|
haftmann@32681
|
2514 |
qed
|
haftmann@32681
|
2515 |
|
haftmann@32681
|
2516 |
lemma (in linorder) Min_fin_set_fold [code_unfold]:
|
haftmann@32681
|
2517 |
"Min (set (x # xs)) = foldl min x xs"
|
haftmann@32681
|
2518 |
proof -
|
haftmann@32681
|
2519 |
interpret ab_semigroup_idem_mult "min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
|
haftmann@32681
|
2520 |
by (fact ab_semigroup_idem_mult_min)
|
haftmann@32681
|
2521 |
show ?thesis
|
haftmann@32681
|
2522 |
by (simp add: Min_def fold1_set del: set.simps)
|
haftmann@32681
|
2523 |
qed
|
haftmann@32681
|
2524 |
|
haftmann@32681
|
2525 |
lemma (in linorder) Max_fin_set_fold [code_unfold]:
|
haftmann@32681
|
2526 |
"Max (set (x # xs)) = foldl max x xs"
|
haftmann@32681
|
2527 |
proof -
|
haftmann@32681
|
2528 |
interpret ab_semigroup_idem_mult "max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
|
haftmann@32681
|
2529 |
by (fact ab_semigroup_idem_mult_max)
|
haftmann@32681
|
2530 |
show ?thesis
|
haftmann@32681
|
2531 |
by (simp add: Max_def fold1_set del: set.simps)
|
haftmann@32681
|
2532 |
qed
|
haftmann@32681
|
2533 |
|
haftmann@32681
|
2534 |
lemma (in complete_lattice) Inf_set_fold [code_unfold]:
|
haftmann@32681
|
2535 |
"Inf (set xs) = foldl inf top xs"
|
haftmann@33998
|
2536 |
proof -
|
haftmann@43740
|
2537 |
interpret comp_fun_idem "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
|
haftmann@43740
|
2538 |
by (fact comp_fun_idem_inf)
|
haftmann@33998
|
2539 |
show ?thesis by (simp add: Inf_fold_inf fold_set inf_commute)
|
haftmann@33998
|
2540 |
qed
|
haftmann@32681
|
2541 |
|
haftmann@32681
|
2542 |
lemma (in complete_lattice) Sup_set_fold [code_unfold]:
|
haftmann@32681
|
2543 |
"Sup (set xs) = foldl sup bot xs"
|
haftmann@33998
|
2544 |
proof -
|
haftmann@43740
|
2545 |
interpret comp_fun_idem "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
|
haftmann@43740
|
2546 |
by (fact comp_fun_idem_sup)
|
haftmann@33998
|
2547 |
show ?thesis by (simp add: Sup_fold_sup fold_set sup_commute)
|
haftmann@33998
|
2548 |
qed
|
haftmann@33998
|
2549 |
|
haftmann@33998
|
2550 |
lemma (in complete_lattice) INFI_set_fold:
|
haftmann@33998
|
2551 |
"INFI (set xs) f = foldl (\<lambda>y x. inf (f x) y) top xs"
|
hoelzl@45799
|
2552 |
unfolding INF_def set_map [symmetric] Inf_set_fold foldl_map
|
haftmann@33998
|
2553 |
by (simp add: inf_commute)
|
haftmann@33998
|
2554 |
|
haftmann@33998
|
2555 |
lemma (in complete_lattice) SUPR_set_fold:
|
haftmann@33998
|
2556 |
"SUPR (set xs) f = foldl (\<lambda>y x. sup (f x) y) bot xs"
|
hoelzl@45799
|
2557 |
unfolding SUP_def set_map [symmetric] Sup_set_fold foldl_map
|
haftmann@33998
|
2558 |
by (simp add: sup_commute)
|
haftmann@31455
|
2559 |
|
wenzelm@35118
|
2560 |
|
nipkow@24645
|
2561 |
subsubsection {* @{text upt} *}
|
wenzelm@13142
|
2562 |
|
nipkow@17090
|
2563 |
lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
|
nipkow@17090
|
2564 |
-- {* simp does not terminate! *}
|
nipkow@13145
|
2565 |
by (induct j) auto
|
wenzelm@13114
|
2566 |
|
nipkow@32005
|
2567 |
lemmas upt_rec_number_of[simp] = upt_rec[of "number_of m" "number_of n", standard]
|
nipkow@32005
|
2568 |
|
nipkow@15425
|
2569 |
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
|
nipkow@13145
|
2570 |
by (subst upt_rec) simp
|
wenzelm@13114
|
2571 |
|
nipkow@15425
|
2572 |
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
|
nipkow@15281
|
2573 |
by(induct j)simp_all
|
nipkow@15281
|
2574 |
|
nipkow@15281
|
2575 |
lemma upt_eq_Cons_conv:
|
nipkow@24526
|
2576 |
"([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
|
nipkow@24526
|
2577 |
apply(induct j arbitrary: x xs)
|
nipkow@15281
|
2578 |
apply simp
|
nipkow@15281
|
2579 |
apply(clarsimp simp add: append_eq_Cons_conv)
|
nipkow@15281
|
2580 |
apply arith
|
nipkow@15281
|
2581 |
done
|
nipkow@15281
|
2582 |
|
nipkow@15425
|
2583 |
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
|
nipkow@13145
|
2584 |
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
|
nipkow@13145
|
2585 |
by simp
|
wenzelm@13114
|
2586 |
|
nipkow@15425
|
2587 |
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
|
haftmann@26734
|
2588 |
by (simp add: upt_rec)
|
wenzelm@13114
|
2589 |
|
nipkow@15425
|
2590 |
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
|
nipkow@13145
|
2591 |
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
|
nipkow@13145
|
2592 |
by (induct k) auto
|
wenzelm@13114
|
2593 |
|
nipkow@15425
|
2594 |
lemma length_upt [simp]: "length [i..<j] = j - i"
|
nipkow@13145
|
2595 |
by (induct j) (auto simp add: Suc_diff_le)
|
wenzelm@13114
|
2596 |
|
nipkow@15425
|
2597 |
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
|
nipkow@13145
|
2598 |
apply (induct j)
|
nipkow@13145
|
2599 |
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
|
nipkow@13145
|
2600 |
done
|
wenzelm@13114
|
2601 |
|
nipkow@17906
|
2602 |
|
nipkow@17906
|
2603 |
lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
|
nipkow@17906
|
2604 |
by(simp add:upt_conv_Cons)
|
nipkow@17906
|
2605 |
|
nipkow@17906
|
2606 |
lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
|
nipkow@17906
|
2607 |
apply(cases j)
|
nipkow@17906
|
2608 |
apply simp
|
nipkow@17906
|
2609 |
by(simp add:upt_Suc_append)
|
nipkow@17906
|
2610 |
|
nipkow@24526
|
2611 |
lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]"
|
nipkow@24526
|
2612 |
apply (induct m arbitrary: i, simp)
|
nipkow@13145
|
2613 |
apply (subst upt_rec)
|
nipkow@13145
|
2614 |
apply (rule sym)
|
nipkow@13145
|
2615 |
apply (subst upt_rec)
|
nipkow@13145
|
2616 |
apply (simp del: upt.simps)
|
nipkow@13145
|
2617 |
done
|
wenzelm@13114
|
2618 |
|
nipkow@17501
|
2619 |
lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
|
nipkow@17501
|
2620 |
apply(induct j)
|
nipkow@17501
|
2621 |
apply auto
|
nipkow@17501
|
2622 |
done
|
nipkow@17501
|
2623 |
|
nipkow@24645
|
2624 |
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]"
|
nipkow@13145
|
2625 |
by (induct n) auto
|
wenzelm@13114
|
2626 |
|
nipkow@24526
|
2627 |
lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
|
nipkow@24526
|
2628 |
apply (induct n m arbitrary: i rule: diff_induct)
|
nipkow@13145
|
2629 |
prefer 3 apply (subst map_Suc_upt[symmetric])
|
huffman@45792
|
2630 |
apply (auto simp add: less_diff_conv)
|
nipkow@13145
|
2631 |
done
|
wenzelm@13114
|
2632 |
|
berghofe@13883
|
2633 |
lemma nth_take_lemma:
|
nipkow@24526
|
2634 |
"k <= length xs ==> k <= length ys ==>
|
berghofe@13883
|
2635 |
(!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
|
nipkow@24526
|
2636 |
apply (atomize, induct k arbitrary: xs ys)
|
paulson@14208
|
2637 |
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
|
nipkow@13145
|
2638 |
txt {* Both lists must be non-empty *}
|
paulson@14208
|
2639 |
apply (case_tac xs, simp)
|
paulson@14208
|
2640 |
apply (case_tac ys, clarify)
|
nipkow@13145
|
2641 |
apply (simp (no_asm_use))
|
nipkow@13145
|
2642 |
apply clarify
|
nipkow@13145
|
2643 |
txt {* prenexing's needed, not miniscoping *}
|
nipkow@13145
|
2644 |
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
|
nipkow@13145
|
2645 |
apply blast
|
nipkow@13145
|
2646 |
done
|
wenzelm@13114
|
2647 |
|
wenzelm@13114
|
2648 |
lemma nth_equalityI:
|
wenzelm@13114
|
2649 |
"[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
|
huffman@45792
|
2650 |
by (frule nth_take_lemma [OF le_refl eq_imp_le]) simp_all
|
wenzelm@13114
|
2651 |
|
haftmann@24796
|
2652 |
lemma map_nth:
|
haftmann@24796
|
2653 |
"map (\<lambda>i. xs ! i) [0..<length xs] = xs"
|
haftmann@24796
|
2654 |
by (rule nth_equalityI, auto)
|
haftmann@24796
|
2655 |
|
kleing@13863
|
2656 |
(* needs nth_equalityI *)
|
kleing@13863
|
2657 |
lemma list_all2_antisym:
|
kleing@13863
|
2658 |
"\<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
|
2659 |
\<Longrightarrow> xs = ys"
|
kleing@13863
|
2660 |
apply (simp add: list_all2_conv_all_nth)
|
paulson@14208
|
2661 |
apply (rule nth_equalityI, blast, simp)
|
kleing@13863
|
2662 |
done
|
kleing@13863
|
2663 |
|
wenzelm@13142
|
2664 |
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
|
nipkow@13145
|
2665 |
-- {* The famous take-lemma. *}
|
nipkow@13145
|
2666 |
apply (drule_tac x = "max (length xs) (length ys)" in spec)
|
huffman@45792
|
2667 |
apply (simp add: le_max_iff_disj)
|
nipkow@13145
|
2668 |
done
|
wenzelm@13114
|
2669 |
|
wenzelm@13114
|
2670 |
|
nipkow@15302
|
2671 |
lemma take_Cons':
|
nipkow@15302
|
2672 |
"take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
|
nipkow@15302
|
2673 |
by (cases n) simp_all
|
nipkow@15302
|
2674 |
|
nipkow@15302
|
2675 |
lemma drop_Cons':
|
nipkow@15302
|
2676 |
"drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
|
nipkow@15302
|
2677 |
by (cases n) simp_all
|
nipkow@15302
|
2678 |
|
nipkow@15302
|
2679 |
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
|
nipkow@15302
|
2680 |
by (cases n) simp_all
|
nipkow@15302
|
2681 |
|
paulson@18622
|
2682 |
lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard]
|
paulson@18622
|
2683 |
lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard]
|
paulson@18622
|
2684 |
lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard]
|
paulson@18622
|
2685 |
|
paulson@18622
|
2686 |
declare take_Cons_number_of [simp]
|
paulson@18622
|
2687 |
drop_Cons_number_of [simp]
|
paulson@18622
|
2688 |
nth_Cons_number_of [simp]
|
nipkow@15302
|
2689 |
|
nipkow@15302
|
2690 |
|
nipkow@32415
|
2691 |
subsubsection {* @{text upto}: interval-list on @{typ int} *}
|
nipkow@32415
|
2692 |
|
nipkow@32415
|
2693 |
(* FIXME make upto tail recursive? *)
|
nipkow@32415
|
2694 |
|
nipkow@32415
|
2695 |
function upto :: "int \<Rightarrow> int \<Rightarrow> int list" ("(1[_../_])") where
|
nipkow@32415
|
2696 |
"upto i j = (if i \<le> j then i # [i+1..j] else [])"
|
nipkow@32415
|
2697 |
by auto
|
nipkow@32415
|
2698 |
termination
|
nipkow@32415
|
2699 |
by(relation "measure(%(i::int,j). nat(j - i + 1))") auto
|
nipkow@32415
|
2700 |
|
nipkow@32415
|
2701 |
declare upto.simps[code, simp del]
|
nipkow@32415
|
2702 |
|
nipkow@32415
|
2703 |
lemmas upto_rec_number_of[simp] =
|
nipkow@32415
|
2704 |
upto.simps[of "number_of m" "number_of n", standard]
|
nipkow@32415
|
2705 |
|
nipkow@32415
|
2706 |
lemma upto_empty[simp]: "j < i \<Longrightarrow> [i..j] = []"
|
nipkow@32415
|
2707 |
by(simp add: upto.simps)
|
nipkow@32415
|
2708 |
|
nipkow@32415
|
2709 |
lemma set_upto[simp]: "set[i..j] = {i..j}"
|
bulwahn@41695
|
2710 |
proof(induct i j rule:upto.induct)
|
bulwahn@41695
|
2711 |
case (1 i j)
|
bulwahn@41695
|
2712 |
from this show ?case
|
bulwahn@41695
|
2713 |
unfolding upto.simps[of i j] simp_from_to[of i j] by auto
|
bulwahn@41695
|
2714 |
qed
|
nipkow@32415
|
2715 |
|
nipkow@32415
|
2716 |
|
nipkow@15392
|
2717 |
subsubsection {* @{text "distinct"} and @{text remdups} *}
|
wenzelm@13114
|
2718 |
|
haftmann@40451
|
2719 |
lemma distinct_tl:
|
haftmann@40451
|
2720 |
"distinct xs \<Longrightarrow> distinct (tl xs)"
|
haftmann@40451
|
2721 |
by (cases xs) simp_all
|
haftmann@40451
|
2722 |
|
wenzelm@13142
|
2723 |
lemma distinct_append [simp]:
|
nipkow@13145
|
2724 |
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
|
nipkow@13145
|
2725 |
by (induct xs) auto
|
wenzelm@13114
|
2726 |
|
nipkow@15305
|
2727 |
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
|
nipkow@15305
|
2728 |
by(induct xs) auto
|
nipkow@15305
|
2729 |
|
wenzelm@13142
|
2730 |
lemma set_remdups [simp]: "set (remdups xs) = set xs"
|
nipkow@13145
|
2731 |
by (induct xs) (auto simp add: insert_absorb)
|
wenzelm@13114
|
2732 |
|
wenzelm@13142
|
2733 |
lemma distinct_remdups [iff]: "distinct (remdups xs)"
|
nipkow@13145
|
2734 |
by (induct xs) auto
|
wenzelm@13114
|
2735 |
|
nipkow@25287
|
2736 |
lemma distinct_remdups_id: "distinct xs ==> remdups xs = xs"
|
nipkow@25287
|
2737 |
by (induct xs, auto)
|
nipkow@25287
|
2738 |
|
haftmann@26734
|
2739 |
lemma remdups_id_iff_distinct [simp]: "remdups xs = xs \<longleftrightarrow> distinct xs"
|
haftmann@26734
|
2740 |
by (metis distinct_remdups distinct_remdups_id)
|
nipkow@25287
|
2741 |
|
nipkow@24566
|
2742 |
lemma finite_distinct_list: "finite A \<Longrightarrow> EX xs. set xs = A & distinct xs"
|
paulson@24632
|
2743 |
by (metis distinct_remdups finite_list set_remdups)
|
nipkow@24566
|
2744 |
|
paulson@15072
|
2745 |
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
|
nipkow@24349
|
2746 |
by (induct x, auto)
|
paulson@15072
|
2747 |
|
paulson@15072
|
2748 |
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
|
nipkow@24349
|
2749 |
by (induct x, auto)
|
paulson@15072
|
2750 |
|
nipkow@15245
|
2751 |
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
|
nipkow@15245
|
2752 |
by (induct xs) auto
|
nipkow@15245
|
2753 |
|
nipkow@15245
|
2754 |
lemma length_remdups_eq[iff]:
|
nipkow@15245
|
2755 |
"(length (remdups xs) = length xs) = (remdups xs = xs)"
|
nipkow@15245
|
2756 |
apply(induct xs)
|
nipkow@15245
|
2757 |
apply auto
|
nipkow@15245
|
2758 |
apply(subgoal_tac "length (remdups xs) <= length xs")
|
nipkow@15245
|
2759 |
apply arith
|
nipkow@15245
|
2760 |
apply(rule length_remdups_leq)
|
nipkow@15245
|
2761 |
done
|
nipkow@15245
|
2762 |
|
nipkow@33911
|
2763 |
lemma remdups_filter: "remdups(filter P xs) = filter P (remdups xs)"
|
nipkow@33911
|
2764 |
apply(induct xs)
|
nipkow@33911
|
2765 |
apply auto
|
nipkow@33911
|
2766 |
done
|
nipkow@18490
|
2767 |
|
nipkow@18490
|
2768 |
lemma distinct_map:
|
nipkow@18490
|
2769 |
"distinct(map f xs) = (distinct xs & inj_on f (set xs))"
|
nipkow@18490
|
2770 |
by (induct xs) auto
|
nipkow@18490
|
2771 |
|
nipkow@18490
|
2772 |
|
wenzelm@13142
|
2773 |
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
|
nipkow@13145
|
2774 |
by (induct xs) auto
|
wenzelm@13114
|
2775 |
|
nipkow@17501
|
2776 |
lemma distinct_upt[simp]: "distinct[i..<j]"
|
nipkow@17501
|
2777 |
by (induct j) auto
|
nipkow@17501
|
2778 |
|
nipkow@32415
|
2779 |
lemma distinct_upto[simp]: "distinct[i..j]"
|
nipkow@32415
|
2780 |
apply(induct i j rule:upto.induct)
|
nipkow@32415
|
2781 |
apply(subst upto.simps)
|
nipkow@32415
|
2782 |
apply(simp)
|
nipkow@32415
|
2783 |
done
|
nipkow@32415
|
2784 |
|
nipkow@24526
|
2785 |
lemma distinct_take[simp]: "distinct xs \<Longrightarrow> distinct (take i xs)"
|
nipkow@24526
|
2786 |
apply(induct xs arbitrary: i)
|
nipkow@17501
|
2787 |
apply simp
|
nipkow@17501
|
2788 |
apply (case_tac i)
|
nipkow@17501
|
2789 |
apply simp_all
|
nipkow@17501
|
2790 |
apply(blast dest:in_set_takeD)
|
nipkow@17501
|
2791 |
done
|
nipkow@17501
|
2792 |
|
nipkow@24526
|
2793 |
lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)"
|
nipkow@24526
|
2794 |
apply(induct xs arbitrary: i)
|
nipkow@17501
|
2795 |
apply simp
|
nipkow@17501
|
2796 |
apply (case_tac i)
|
nipkow@17501
|
2797 |
apply simp_all
|
nipkow@17501
|
2798 |
done
|
nipkow@17501
|
2799 |
|
nipkow@17501
|
2800 |
lemma distinct_list_update:
|
nipkow@17501
|
2801 |
assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
|
nipkow@17501
|
2802 |
shows "distinct (xs[i:=a])"
|
nipkow@17501
|
2803 |
proof (cases "i < length xs")
|
nipkow@17501
|
2804 |
case True
|
nipkow@17501
|
2805 |
with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
|
nipkow@17501
|
2806 |
apply (drule_tac id_take_nth_drop) by simp
|
nipkow@17501
|
2807 |
with d True show ?thesis
|
nipkow@17501
|
2808 |
apply (simp add: upd_conv_take_nth_drop)
|
nipkow@17501
|
2809 |
apply (drule subst [OF id_take_nth_drop]) apply assumption
|
nipkow@17501
|
2810 |
apply simp apply (cases "a = xs!i") apply simp by blast
|
nipkow@17501
|
2811 |
next
|
nipkow@17501
|
2812 |
case False with d show ?thesis by auto
|
nipkow@17501
|
2813 |
qed
|
nipkow@17501
|
2814 |
|
hoelzl@31350
|
2815 |
lemma distinct_concat:
|
hoelzl@31350
|
2816 |
assumes "distinct xs"
|
hoelzl@31350
|
2817 |
and "\<And> ys. ys \<in> set xs \<Longrightarrow> distinct ys"
|
hoelzl@31350
|
2818 |
and "\<And> ys zs. \<lbrakk> ys \<in> set xs ; zs \<in> set xs ; ys \<noteq> zs \<rbrakk> \<Longrightarrow> set ys \<inter> set zs = {}"
|
hoelzl@31350
|
2819 |
shows "distinct (concat xs)"
|
hoelzl@31350
|
2820 |
using assms by (induct xs) auto
|
nipkow@17501
|
2821 |
|
nipkow@17501
|
2822 |
text {* It is best to avoid this indexed version of distinct, but
|
nipkow@17501
|
2823 |
sometimes it is useful. *}
|
nipkow@17501
|
2824 |
|
nipkow@13124
|
2825 |
lemma distinct_conv_nth:
|
nipkow@17501
|
2826 |
"distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
|
paulson@15251
|
2827 |
apply (induct xs, simp, simp)
|
paulson@14208
|
2828 |
apply (rule iffI, clarsimp)
|
nipkow@13145
|
2829 |
apply (case_tac i)
|
paulson@14208
|
2830 |
apply (case_tac j, simp)
|
nipkow@13145
|
2831 |
apply (simp add: set_conv_nth)
|
nipkow@13145
|
2832 |
apply (case_tac j)
|
paulson@24648
|
2833 |
apply (clarsimp simp add: set_conv_nth, simp)
|
nipkow@13145
|
2834 |
apply (rule conjI)
|
paulson@24648
|
2835 |
(*TOO SLOW
|
paulson@24632
|
2836 |
apply (metis Zero_neq_Suc gr0_conv_Suc in_set_conv_nth lessI less_trans_Suc nth_Cons' nth_Cons_Suc)
|
paulson@24648
|
2837 |
*)
|
paulson@24648
|
2838 |
apply (clarsimp simp add: set_conv_nth)
|
paulson@24648
|
2839 |
apply (erule_tac x = 0 in allE, simp)
|
paulson@24648
|
2840 |
apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
|
wenzelm@25130
|
2841 |
(*TOO SLOW
|
paulson@24632
|
2842 |
apply (metis Suc_Suc_eq lessI less_trans_Suc nth_Cons_Suc)
|
wenzelm@25130
|
2843 |
*)
|
wenzelm@25130
|
2844 |
apply (erule_tac x = "Suc i" in allE, simp)
|
wenzelm@25130
|
2845 |
apply (erule_tac x = "Suc j" in allE, simp)
|
nipkow@13145
|
2846 |
done
|
nipkow@13124
|
2847 |
|
nipkow@18490
|
2848 |
lemma nth_eq_iff_index_eq:
|
nipkow@18490
|
2849 |
"\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
|
nipkow@18490
|
2850 |
by(auto simp: distinct_conv_nth)
|
nipkow@18490
|
2851 |
|
nipkow@15110
|
2852 |
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
|
nipkow@24349
|
2853 |
by (induct xs) auto
|
kleing@14388
|
2854 |
|
nipkow@15110
|
2855 |
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
|
kleing@14388
|
2856 |
proof (induct xs)
|
kleing@14388
|
2857 |
case Nil thus ?case by simp
|
kleing@14388
|
2858 |
next
|
kleing@14388
|
2859 |
case (Cons x xs)
|
kleing@14388
|
2860 |
show ?case
|
kleing@14388
|
2861 |
proof (cases "x \<in> set xs")
|
kleing@14388
|
2862 |
case False with Cons show ?thesis by simp
|
kleing@14388
|
2863 |
next
|
kleing@14388
|
2864 |
case True with Cons.prems
|
kleing@14388
|
2865 |
have "card (set xs) = Suc (length xs)"
|
kleing@14388
|
2866 |
by (simp add: card_insert_if split: split_if_asm)
|
kleing@14388
|
2867 |
moreover have "card (set xs) \<le> length xs" by (rule card_length)
|
kleing@14388
|
2868 |
ultimately have False by simp
|
kleing@14388
|
2869 |
thus ?thesis ..
|
kleing@14388
|
2870 |
qed
|
kleing@14388
|
2871 |
qed
|
kleing@14388
|
2872 |
|
bulwahn@45967
|
2873 |
lemma distinct_length_filter: "distinct xs \<Longrightarrow> length (filter P xs) = card ({x. P x} Int set xs)"
|
bulwahn@45967
|
2874 |
by (induct xs) (auto)
|
bulwahn@45967
|
2875 |
|
nipkow@25287
|
2876 |
lemma not_distinct_decomp: "~ distinct ws ==> EX xs ys zs y. ws = xs@[y]@ys@[y]@zs"
|
nipkow@25287
|
2877 |
apply (induct n == "length ws" arbitrary:ws) apply simp
|
nipkow@25287
|
2878 |
apply(case_tac ws) apply simp
|
nipkow@25287
|
2879 |
apply (simp split:split_if_asm)
|
nipkow@25287
|
2880 |
apply (metis Cons_eq_appendI eq_Nil_appendI split_list)
|
nipkow@25287
|
2881 |
done
|
nipkow@18490
|
2882 |
|
nipkow@18490
|
2883 |
lemma length_remdups_concat:
|
huffman@45792
|
2884 |
"length (remdups (concat xss)) = card (\<Union>xs\<in>set xss. set xs)"
|
huffman@45792
|
2885 |
by (simp add: distinct_card [symmetric])
|
nipkow@17906
|
2886 |
|
hoelzl@33639
|
2887 |
lemma length_remdups_card_conv: "length(remdups xs) = card(set xs)"
|
hoelzl@33639
|
2888 |
proof -
|
hoelzl@33639
|
2889 |
have xs: "concat[xs] = xs" by simp
|
hoelzl@33639
|
2890 |
from length_remdups_concat[of "[xs]"] show ?thesis unfolding xs by simp
|
hoelzl@33639
|
2891 |
qed
|
nipkow@17906
|
2892 |
|
haftmann@36275
|
2893 |
lemma remdups_remdups:
|
haftmann@36275
|
2894 |
"remdups (remdups xs) = remdups xs"
|
haftmann@36275
|
2895 |
by (induct xs) simp_all
|
haftmann@36275
|
2896 |
|
haftmann@36846
|
2897 |
lemma distinct_butlast:
|
haftmann@36846
|
2898 |
assumes "xs \<noteq> []" and "distinct xs"
|
haftmann@36846
|
2899 |
shows "distinct (butlast xs)"
|
haftmann@36846
|
2900 |
proof -
|
haftmann@36846
|
2901 |
from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
|
haftmann@36846
|
2902 |
with `distinct xs` show ?thesis by simp
|
haftmann@36846
|
2903 |
qed
|
haftmann@36846
|
2904 |
|
haftmann@39960
|
2905 |
lemma remdups_map_remdups:
|
haftmann@39960
|
2906 |
"remdups (map f (remdups xs)) = remdups (map f xs)"
|
haftmann@39960
|
2907 |
by (induct xs) simp_all
|
haftmann@39960
|
2908 |
|
haftmann@40096
|
2909 |
lemma distinct_zipI1:
|
haftmann@40096
|
2910 |
assumes "distinct xs"
|
haftmann@40096
|
2911 |
shows "distinct (zip xs ys)"
|
haftmann@40096
|
2912 |
proof (rule zip_obtain_same_length)
|
haftmann@40096
|
2913 |
fix xs' :: "'a list" and ys' :: "'b list" and n
|
haftmann@40096
|
2914 |
assume "length xs' = length ys'"
|
haftmann@40096
|
2915 |
assume "xs' = take n xs"
|
haftmann@40096
|
2916 |
with assms have "distinct xs'" by simp
|
haftmann@40096
|
2917 |
with `length xs' = length ys'` show "distinct (zip xs' ys')"
|
haftmann@40096
|
2918 |
by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
|
haftmann@40096
|
2919 |
qed
|
haftmann@40096
|
2920 |
|
haftmann@40096
|
2921 |
lemma distinct_zipI2:
|
haftmann@40096
|
2922 |
assumes "distinct ys"
|
haftmann@40096
|
2923 |
shows "distinct (zip xs ys)"
|
haftmann@40096
|
2924 |
proof (rule zip_obtain_same_length)
|
haftmann@40096
|
2925 |
fix xs' :: "'b list" and ys' :: "'a list" and n
|
haftmann@40096
|
2926 |
assume "length xs' = length ys'"
|
haftmann@40096
|
2927 |
assume "ys' = take n ys"
|
haftmann@40096
|
2928 |
with assms have "distinct ys'" by simp
|
haftmann@40096
|
2929 |
with `length xs' = length ys'` show "distinct (zip xs' ys')"
|
haftmann@40096
|
2930 |
by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
|
haftmann@40096
|
2931 |
qed
|
haftmann@40096
|
2932 |
|
blanchet@45493
|
2933 |
(* The next two lemmas help Sledgehammer. *)
|
blanchet@45493
|
2934 |
|
blanchet@45493
|
2935 |
lemma distinct_singleton: "distinct [x]" by simp
|
blanchet@45493
|
2936 |
|
blanchet@45493
|
2937 |
lemma distinct_length_2_or_more:
|
blanchet@45493
|
2938 |
"distinct (a # b # xs) \<longleftrightarrow> (a \<noteq> b \<and> distinct (a # xs) \<and> distinct (b # xs))"
|
blanchet@45493
|
2939 |
by (metis distinct.simps(2) hd.simps hd_in_set list.simps(2) set_ConsD set_rev_mp set_subset_Cons)
|
blanchet@45493
|
2940 |
|
wenzelm@35118
|
2941 |
|
haftmann@37605
|
2942 |
subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
|
haftmann@37605
|
2943 |
|
haftmann@40007
|
2944 |
lemma (in monoid_add) listsum_foldl [code]:
|
haftmann@40007
|
2945 |
"listsum = foldl (op +) 0"
|
haftmann@40007
|
2946 |
by (simp add: listsum_def foldl_foldr1 fun_eq_iff)
|
haftmann@40007
|
2947 |
|
haftmann@40007
|
2948 |
lemma (in monoid_add) listsum_simps [simp]:
|
haftmann@40007
|
2949 |
"listsum [] = 0"
|
haftmann@40007
|
2950 |
"listsum (x#xs) = x + listsum xs"
|
haftmann@40007
|
2951 |
by (simp_all add: listsum_def)
|
haftmann@40007
|
2952 |
|
haftmann@40007
|
2953 |
lemma (in monoid_add) listsum_append [simp]:
|
haftmann@40007
|
2954 |
"listsum (xs @ ys) = listsum xs + listsum ys"
|
haftmann@40007
|
2955 |
by (induct xs) (simp_all add: add.assoc)
|
haftmann@40007
|
2956 |
|
haftmann@40007
|
2957 |
lemma (in comm_monoid_add) listsum_rev [simp]:
|
haftmann@40007
|
2958 |
"listsum (rev xs) = listsum xs"
|
haftmann@40007
|
2959 |
by (simp add: listsum_def [of "rev xs"]) (simp add: listsum_foldl foldr_foldl add.commute)
|
haftmann@40007
|
2960 |
|
haftmann@40007
|
2961 |
lemma (in comm_monoid_add) listsum_map_remove1:
|
haftmann@40007
|
2962 |
"x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))"
|
haftmann@40007
|
2963 |
by (induct xs) (auto simp add: ac_simps)
|
haftmann@40007
|
2964 |
|
haftmann@40007
|
2965 |
lemma (in monoid_add) list_size_conv_listsum:
|
haftmann@37605
|
2966 |
"list_size f xs = listsum (map f xs) + size xs"
|
haftmann@40007
|
2967 |
by (induct xs) auto
|
haftmann@40007
|
2968 |
|
haftmann@40007
|
2969 |
lemma (in monoid_add) length_concat:
|
haftmann@40007
|
2970 |
"length (concat xss) = listsum (map length xss)"
|
haftmann@40007
|
2971 |
by (induct xss) simp_all
|
haftmann@40007
|
2972 |
|
haftmann@40007
|
2973 |
lemma (in monoid_add) listsum_map_filter:
|
haftmann@40007
|
2974 |
assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
|
haftmann@37605
|
2975 |
shows "listsum (map f (filter P xs)) = listsum (map f xs)"
|
haftmann@40007
|
2976 |
using assms by (induct xs) auto
|
haftmann@40007
|
2977 |
|
haftmann@40007
|
2978 |
lemma (in monoid_add) distinct_listsum_conv_Setsum:
|
haftmann@40007
|
2979 |
"distinct xs \<Longrightarrow> listsum xs = Setsum (set xs)"
|
haftmann@40007
|
2980 |
by (induct xs) simp_all
|
haftmann@40007
|
2981 |
|
haftmann@40007
|
2982 |
lemma listsum_eq_0_nat_iff_nat [simp]:
|
haftmann@40007
|
2983 |
"listsum ns = (0::nat) \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
|
haftmann@40007
|
2984 |
by (simp add: listsum_foldl)
|
haftmann@40007
|
2985 |
|
haftmann@40007
|
2986 |
lemma elem_le_listsum_nat:
|
haftmann@40007
|
2987 |
"k < size ns \<Longrightarrow> ns ! k \<le> listsum (ns::nat list)"
|
haftmann@37605
|
2988 |
apply(induct ns arbitrary: k)
|
haftmann@37605
|
2989 |
apply simp
|
nipkow@45761
|
2990 |
apply(fastforce simp add:nth_Cons split: nat.split)
|
haftmann@37605
|
2991 |
done
|
haftmann@37605
|
2992 |
|
haftmann@40007
|
2993 |
lemma listsum_update_nat:
|
haftmann@40007
|
2994 |
"k<size ns \<Longrightarrow> listsum (ns[k := (n::nat)]) = listsum ns + n - ns ! k"
|
haftmann@37605
|
2995 |
apply(induct ns arbitrary:k)
|
haftmann@37605
|
2996 |
apply (auto split:nat.split)
|
haftmann@37605
|
2997 |
apply(drule elem_le_listsum_nat)
|
haftmann@37605
|
2998 |
apply arith
|
haftmann@37605
|
2999 |
done
|
haftmann@37605
|
3000 |
|
haftmann@37605
|
3001 |
text{* Some syntactic sugar for summing a function over a list: *}
|
haftmann@37605
|
3002 |
|
haftmann@37605
|
3003 |
syntax
|
haftmann@37605
|
3004 |
"_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3SUM _<-_. _)" [0, 51, 10] 10)
|
haftmann@37605
|
3005 |
syntax (xsymbols)
|
haftmann@37605
|
3006 |
"_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
|
haftmann@37605
|
3007 |
syntax (HTML output)
|
haftmann@37605
|
3008 |
"_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
|
haftmann@37605
|
3009 |
|
haftmann@37605
|
3010 |
translations -- {* Beware of argument permutation! *}
|
haftmann@37605
|
3011 |
"SUM x<-xs. b" == "CONST listsum (CONST map (%x. b) xs)"
|
haftmann@37605
|
3012 |
"\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"
|
haftmann@37605
|
3013 |
|
haftmann@40007
|
3014 |
lemma (in monoid_add) listsum_triv:
|
haftmann@40007
|
3015 |
"(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
|
haftmann@37605
|
3016 |
by (induct xs) (simp_all add: left_distrib)
|
haftmann@37605
|
3017 |
|
haftmann@40007
|
3018 |
lemma (in monoid_add) listsum_0 [simp]:
|
haftmann@40007
|
3019 |
"(\<Sum>x\<leftarrow>xs. 0) = 0"
|
haftmann@37605
|
3020 |
by (induct xs) (simp_all add: left_distrib)
|
haftmann@37605
|
3021 |
|
haftmann@37605
|
3022 |
text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
|
haftmann@40007
|
3023 |
lemma (in ab_group_add) uminus_listsum_map:
|
haftmann@40007
|
3024 |
"- listsum (map f xs) = listsum (map (uminus \<circ> f) xs)"
|
haftmann@40007
|
3025 |
by (induct xs) simp_all
|
haftmann@40007
|
3026 |
|
haftmann@40007
|
3027 |
lemma (in comm_monoid_add) listsum_addf:
|
haftmann@40007
|
3028 |
"(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
|
haftmann@40007
|
3029 |
by (induct xs) (simp_all add: algebra_simps)
|
haftmann@40007
|
3030 |
|
haftmann@40007
|
3031 |
lemma (in ab_group_add) listsum_subtractf:
|
haftmann@40007
|
3032 |
"(\<Sum>x\<leftarrow>xs. f x - g x) = listsum (map f xs) - listsum (map g xs)"
|
haftmann@40007
|
3033 |
by (induct xs) (simp_all add: algebra_simps)
|
haftmann@40007
|
3034 |
|
haftmann@40007
|
3035 |
lemma (in semiring_0) listsum_const_mult:
|
haftmann@40007
|
3036 |
"(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
|
haftmann@40007
|
3037 |
by (induct xs) (simp_all add: algebra_simps)
|
haftmann@40007
|
3038 |
|
haftmann@40007
|
3039 |
lemma (in semiring_0) listsum_mult_const:
|
haftmann@40007
|
3040 |
"(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
|
haftmann@40007
|
3041 |
by (induct xs) (simp_all add: algebra_simps)
|
haftmann@40007
|
3042 |
|
haftmann@40007
|
3043 |
lemma (in ordered_ab_group_add_abs) listsum_abs:
|
haftmann@40007
|
3044 |
"\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
|
haftmann@40007
|
3045 |
by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])
|
haftmann@37605
|
3046 |
|
haftmann@37605
|
3047 |
lemma listsum_mono:
|
haftmann@40007
|
3048 |
fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}"
|
haftmann@37605
|
3049 |
shows "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (\<Sum>x\<leftarrow>xs. f x) \<le> (\<Sum>x\<leftarrow>xs. g x)"
|
haftmann@40007
|
3050 |
by (induct xs) (simp, simp add: add_mono)
|
haftmann@40007
|
3051 |
|
haftmann@40007
|
3052 |
lemma (in monoid_add) listsum_distinct_conv_setsum_set:
|
haftmann@37605
|
3053 |
"distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)"
|
haftmann@37605
|
3054 |
by (induct xs) simp_all
|
haftmann@37605
|
3055 |
|
haftmann@40007
|
3056 |
lemma (in monoid_add) interv_listsum_conv_setsum_set_nat:
|
haftmann@37605
|
3057 |
"listsum (map f [m..<n]) = setsum f (set [m..<n])"
|
haftmann@37605
|
3058 |
by (simp add: listsum_distinct_conv_setsum_set)
|
haftmann@37605
|
3059 |
|
haftmann@40007
|
3060 |
lemma (in monoid_add) interv_listsum_conv_setsum_set_int:
|
haftmann@37605
|
3061 |
"listsum (map f [k..l]) = setsum f (set [k..l])"
|
haftmann@37605
|
3062 |
by (simp add: listsum_distinct_conv_setsum_set)
|
haftmann@37605
|
3063 |
|
haftmann@37605
|
3064 |
text {* General equivalence between @{const listsum} and @{const setsum} *}
|
haftmann@40007
|
3065 |
lemma (in monoid_add) listsum_setsum_nth:
|
haftmann@37605
|
3066 |
"listsum xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
|
haftmann@37605
|
3067 |
using interv_listsum_conv_setsum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth)
|
haftmann@37605
|
3068 |
|
haftmann@37605
|
3069 |
|
haftmann@34965
|
3070 |
subsubsection {* @{const insert} *}
|
haftmann@34965
|
3071 |
|
haftmann@34965
|
3072 |
lemma in_set_insert [simp]:
|
haftmann@34965
|
3073 |
"x \<in> set xs \<Longrightarrow> List.insert x xs = xs"
|
haftmann@34965
|
3074 |
by (simp add: List.insert_def)
|
haftmann@34965
|
3075 |
|
haftmann@34965
|
3076 |
lemma not_in_set_insert [simp]:
|
haftmann@34965
|
3077 |
"x \<notin> set xs \<Longrightarrow> List.insert x xs = x # xs"
|
haftmann@34965
|
3078 |
by (simp add: List.insert_def)
|
haftmann@34965
|
3079 |
|
haftmann@34965
|
3080 |
lemma insert_Nil [simp]:
|
haftmann@34965
|
3081 |
"List.insert x [] = [x]"
|
haftmann@34965
|
3082 |
by simp
|
haftmann@34965
|
3083 |
|
haftmann@35295
|
3084 |
lemma set_insert [simp]:
|
haftmann@34965
|
3085 |
"set (List.insert x xs) = insert x (set xs)"
|
haftmann@34965
|
3086 |
by (auto simp add: List.insert_def)
|
haftmann@34965
|
3087 |
|
haftmann@35295
|
3088 |
lemma distinct_insert [simp]:
|
haftmann@35295
|
3089 |
"distinct xs \<Longrightarrow> distinct (List.insert x xs)"
|
haftmann@35295
|
3090 |
by (simp add: List.insert_def)
|
haftmann@35295
|
3091 |
|
haftmann@36275
|
3092 |
lemma insert_remdups:
|
haftmann@36275
|
3093 |
"List.insert x (remdups xs) = remdups (List.insert x xs)"
|
haftmann@36275
|
3094 |
by (simp add: List.insert_def)
|
haftmann@36275
|
3095 |
|
haftmann@34965
|
3096 |
|
nipkow@15392
|
3097 |
subsubsection {* @{text remove1} *}
|
nipkow@15110
|
3098 |
|
nipkow@18049
|
3099 |
lemma remove1_append:
|
nipkow@18049
|
3100 |
"remove1 x (xs @ ys) =
|
nipkow@18049
|
3101 |
(if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
|
nipkow@18049
|
3102 |
by (induct xs) auto
|
nipkow@18049
|
3103 |
|
nipkow@36895
|
3104 |
lemma remove1_commute: "remove1 x (remove1 y zs) = remove1 y (remove1 x zs)"
|
nipkow@36895
|
3105 |
by (induct zs) auto
|
nipkow@36895
|
3106 |
|
nipkow@23479
|
3107 |
lemma in_set_remove1[simp]:
|
nipkow@23479
|
3108 |
"a \<noteq> b \<Longrightarrow> a : set(remove1 b xs) = (a : set xs)"
|
nipkow@23479
|
3109 |
apply (induct xs)
|
nipkow@23479
|
3110 |
apply auto
|
nipkow@23479
|
3111 |
done
|
nipkow@23479
|
3112 |
|
nipkow@15110
|
3113 |
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
|
nipkow@15110
|
3114 |
apply(induct xs)
|
nipkow@15110
|
3115 |
apply simp
|
nipkow@15110
|
3116 |
apply simp
|
nipkow@15110
|
3117 |
apply blast
|
nipkow@15110
|
3118 |
done
|
nipkow@15110
|
3119 |
|
paulson@17724
|
3120 |
lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
|
nipkow@15110
|
3121 |
apply(induct xs)
|
nipkow@15110
|
3122 |
apply simp
|
nipkow@15110
|
3123 |
apply simp
|
nipkow@15110
|
3124 |
apply blast
|
nipkow@15110
|
3125 |
done
|
nipkow@15110
|
3126 |
|
nipkow@23479
|
3127 |
lemma length_remove1:
|
huffman@30128
|
3128 |
"length(remove1 x xs) = (if x : set xs then length xs - 1 else length xs)"
|
nipkow@23479
|
3129 |
apply (induct xs)
|
nipkow@23479
|
3130 |
apply (auto dest!:length_pos_if_in_set)
|
nipkow@23479
|
3131 |
done
|
nipkow@23479
|
3132 |
|
nipkow@18049
|
3133 |
lemma remove1_filter_not[simp]:
|
nipkow@18049
|
3134 |
"\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
|
nipkow@18049
|
3135 |
by(induct xs) auto
|
nipkow@18049
|
3136 |
|
hoelzl@39307
|
3137 |
lemma filter_remove1:
|
hoelzl@39307
|
3138 |
"filter Q (remove1 x xs) = remove1 x (filter Q xs)"
|
hoelzl@39307
|
3139 |
by (induct xs) auto
|
hoelzl@39307
|
3140 |
|
nipkow@15110
|
3141 |
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
|
nipkow@15110
|
3142 |
apply(insert set_remove1_subset)
|
nipkow@15110
|
3143 |
apply fast
|
nipkow@15110
|
3144 |
done
|
nipkow@15110
|
3145 |
|
nipkow@15110
|
3146 |
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
|
nipkow@15110
|
3147 |
by (induct xs) simp_all
|
nipkow@15110
|
3148 |
|
haftmann@36275
|
3149 |
lemma remove1_remdups:
|
haftmann@36275
|
3150 |
"distinct xs \<Longrightarrow> remove1 x (remdups xs) = remdups (remove1 x xs)"
|
haftmann@36275
|
3151 |
by (induct xs) simp_all
|
haftmann@36275
|
3152 |
|
haftmann@37091
|
3153 |
lemma remove1_idem:
|
haftmann@37091
|
3154 |
assumes "x \<notin> set xs"
|
haftmann@37091
|
3155 |
shows "remove1 x xs = xs"
|
haftmann@37091
|
3156 |
using assms by (induct xs) simp_all
|
haftmann@37091
|
3157 |
|
wenzelm@13114
|
3158 |
|
nipkow@27693
|
3159 |
subsubsection {* @{text removeAll} *}
|
nipkow@27693
|
3160 |
|
haftmann@34965
|
3161 |
lemma removeAll_filter_not_eq:
|
haftmann@34965
|
3162 |
"removeAll x = filter (\<lambda>y. x \<noteq> y)"
|
haftmann@34965
|
3163 |
proof
|
haftmann@34965
|
3164 |
fix xs
|
haftmann@34965
|
3165 |
show "removeAll x xs = filter (\<lambda>y. x \<noteq> y) xs"
|
haftmann@34965
|
3166 |
by (induct xs) auto
|
haftmann@34965
|
3167 |
qed
|
haftmann@34965
|
3168 |
|
nipkow@27693
|
3169 |
lemma removeAll_append[simp]:
|
nipkow@27693
|
3170 |
"removeAll x (xs @ ys) = removeAll x xs @ removeAll x ys"
|
nipkow@27693
|
3171 |
by (induct xs) auto
|
nipkow@27693
|
3172 |
|
nipkow@27693
|
3173 |
lemma set_removeAll[simp]: "set(removeAll x xs) = set xs - {x}"
|
nipkow@27693
|
3174 |
by (induct xs) auto
|
nipkow@27693
|
3175 |
|
nipkow@27693
|
3176 |
lemma removeAll_id[simp]: "x \<notin> set xs \<Longrightarrow> removeAll x xs = xs"
|
nipkow@27693
|
3177 |
by (induct xs) auto
|
nipkow@27693
|
3178 |
|
nipkow@27693
|
3179 |
(* Needs count:: 'a \<Rightarrow> a' list \<Rightarrow> nat
|
nipkow@27693
|
3180 |
lemma length_removeAll:
|
nipkow@27693
|
3181 |
"length(removeAll x xs) = length xs - count x xs"
|
nipkow@27693
|
3182 |
*)
|
nipkow@27693
|
3183 |
|
nipkow@27693
|
3184 |
lemma removeAll_filter_not[simp]:
|
nipkow@27693
|
3185 |
"\<not> P x \<Longrightarrow> removeAll x (filter P xs) = filter P xs"
|
nipkow@27693
|
3186 |
by(induct xs) auto
|
nipkow@27693
|
3187 |
|
haftmann@34965
|
3188 |
lemma distinct_removeAll:
|
haftmann@34965
|
3189 |
"distinct xs \<Longrightarrow> distinct (removeAll x xs)"
|
haftmann@34965
|
3190 |
by (simp add: removeAll_filter_not_eq)
|
nipkow@27693
|
3191 |
|
nipkow@27693
|
3192 |
lemma distinct_remove1_removeAll:
|
nipkow@27693
|
3193 |
"distinct xs ==> remove1 x xs = removeAll x xs"
|
nipkow@27693
|
3194 |
by (induct xs) simp_all
|
nipkow@27693
|
3195 |
|
nipkow@27693
|
3196 |
lemma map_removeAll_inj_on: "inj_on f (insert x (set xs)) \<Longrightarrow>
|
nipkow@27693
|
3197 |
map f (removeAll x xs) = removeAll (f x) (map f xs)"
|
nipkow@27693
|
3198 |
by (induct xs) (simp_all add:inj_on_def)
|
nipkow@27693
|
3199 |
|
nipkow@27693
|
3200 |
lemma map_removeAll_inj: "inj f \<Longrightarrow>
|
nipkow@27693
|
3201 |
map f (removeAll x xs) = removeAll (f x) (map f xs)"
|
nipkow@27693
|
3202 |
by(metis map_removeAll_inj_on subset_inj_on subset_UNIV)
|
nipkow@27693
|
3203 |
|
nipkow@27693
|
3204 |
|
nipkow@15392
|
3205 |
subsubsection {* @{text replicate} *}
|
wenzelm@13114
|
3206 |
|
wenzelm@13142
|
3207 |
lemma length_replicate [simp]: "length (replicate n x) = n"
|
nipkow@13145
|
3208 |
by (induct n) auto
|
wenzelm@13142
|
3209 |
|
hoelzl@36610
|
3210 |
lemma Ex_list_of_length: "\<exists>xs. length xs = n"
|
hoelzl@36610
|
3211 |
by (rule exI[of _ "replicate n undefined"]) simp
|
hoelzl@36610
|
3212 |
|
wenzelm@13142
|
3213 |
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
|
nipkow@13145
|
3214 |
by (induct n) auto
|
wenzelm@13114
|
3215 |
|
hoelzl@31350
|
3216 |
lemma map_replicate_const:
|
hoelzl@31350
|
3217 |
"map (\<lambda> x. k) lst = replicate (length lst) k"
|
hoelzl@31350
|
3218 |
by (induct lst) auto
|
hoelzl@31350
|
3219 |
|
wenzelm@13114
|
3220 |
lemma replicate_app_Cons_same:
|
nipkow@13145
|
3221 |
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
|
nipkow@13145
|
3222 |
by (induct n) auto
|
wenzelm@13114
|
3223 |
|
wenzelm@13142
|
3224 |
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
|
paulson@14208
|
3225 |
apply (induct n, simp)
|
nipkow@13145
|
3226 |
apply (simp add: replicate_app_Cons_same)
|
nipkow@13145
|
3227 |
done
|
wenzelm@13114
|
3228 |
|
wenzelm@13142
|
3229 |
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
|
nipkow@13145
|
3230 |
by (induct n) auto
|
wenzelm@13114
|
3231 |
|
nipkow@16397
|
3232 |
text{* Courtesy of Matthias Daum: *}
|
nipkow@16397
|
3233 |
lemma append_replicate_commute:
|
nipkow@16397
|
3234 |
"replicate n x @ replicate k x = replicate k x @ replicate n x"
|
nipkow@16397
|
3235 |
apply (simp add: replicate_add [THEN sym])
|
nipkow@16397
|
3236 |
apply (simp add: add_commute)
|
nipkow@16397
|
3237 |
done
|
nipkow@16397
|
3238 |
|
nipkow@31080
|
3239 |
text{* Courtesy of Andreas Lochbihler: *}
|
nipkow@31080
|
3240 |
lemma filter_replicate:
|
nipkow@31080
|
3241 |
"filter P (replicate n x) = (if P x then replicate n x else [])"
|
nipkow@31080
|
3242 |
by(induct n) auto
|
nipkow@31080
|
3243 |
|
wenzelm@13142
|
3244 |
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
|
nipkow@13145
|
3245 |
by (induct n) auto
|
wenzelm@13114
|
3246 |
|
wenzelm@13142
|
3247 |
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
|
nipkow@13145
|
3248 |
by (induct n) auto
|
wenzelm@13114
|
3249 |
|
wenzelm@13142
|
3250 |
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
|
nipkow@13145
|
3251 |
by (atomize (full), induct n) auto
|
wenzelm@13114
|
3252 |
|
nipkow@24526
|
3253 |
lemma nth_replicate[simp]: "i < n ==> (replicate n x)!i = x"
|
nipkow@24526
|
3254 |
apply (induct n arbitrary: i, simp)
|
nipkow@13145
|
3255 |
apply (simp add: nth_Cons split: nat.split)
|
nipkow@13145
|
3256 |
done
|
wenzelm@13114
|
3257 |
|
nipkow@16397
|
3258 |
text{* Courtesy of Matthias Daum (2 lemmas): *}
|
nipkow@16397
|
3259 |
lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
|
nipkow@16397
|
3260 |
apply (case_tac "k \<le> i")
|
nipkow@16397
|
3261 |
apply (simp add: min_def)
|
nipkow@16397
|
3262 |
apply (drule not_leE)
|
nipkow@16397
|
3263 |
apply (simp add: min_def)
|
nipkow@16397
|
3264 |
apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
|
nipkow@16397
|
3265 |
apply simp
|
nipkow@16397
|
3266 |
apply (simp add: replicate_add [symmetric])
|
nipkow@16397
|
3267 |
done
|
nipkow@16397
|
3268 |
|
nipkow@24526
|
3269 |
lemma drop_replicate[simp]: "drop i (replicate k x) = replicate (k-i) x"
|
nipkow@24526
|
3270 |
apply (induct k arbitrary: i)
|
nipkow@16397
|
3271 |
apply simp
|
nipkow@16397
|
3272 |
apply clarsimp
|
nipkow@16397
|
3273 |
apply (case_tac i)
|
nipkow@16397
|
3274 |
apply simp
|
nipkow@16397
|
3275 |
apply clarsimp
|
nipkow@16397
|
3276 |
done
|
nipkow@16397
|
3277 |
|
nipkow@16397
|
3278 |
|
wenzelm@13142
|
3279 |
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
|
nipkow@13145
|
3280 |
by (induct n) auto
|
wenzelm@13114
|
3281 |
|
wenzelm@13142
|
3282 |
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
|
nipkow@13145
|
3283 |
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
|
wenzelm@13114
|
3284 |
|
wenzelm@13142
|
3285 |
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
|
nipkow@13145
|
3286 |
by auto
|
wenzelm@13114
|
3287 |
|
nipkow@37431
|
3288 |
lemma in_set_replicate[simp]: "(x : set (replicate n y)) = (x = y & n \<noteq> 0)"
|
nipkow@37431
|
3289 |
by (simp add: set_replicate_conv_if)
|
nipkow@37431
|
3290 |
|
nipkow@37429
|
3291 |
lemma Ball_set_replicate[simp]:
|
nipkow@37429
|
3292 |
"(ALL x : set(replicate n a). P x) = (P a | n=0)"
|
nipkow@37429
|
3293 |
by(simp add: set_replicate_conv_if)
|
nipkow@37429
|
3294 |
|
nipkow@37429
|
3295 |
lemma Bex_set_replicate[simp]:
|
nipkow@37429
|
3296 |
"(EX x : set(replicate n a). P x) = (P a & n\<noteq>0)"
|
nipkow@37429
|
3297 |
by(simp add: set_replicate_conv_if)
|
wenzelm@13114
|
3298 |
|
haftmann@24796
|
3299 |
lemma replicate_append_same:
|
haftmann@24796
|
3300 |
"replicate i x @ [x] = x # replicate i x"
|
haftmann@24796
|
3301 |
by (induct i) simp_all
|
haftmann@24796
|
3302 |
|
haftmann@24796
|
3303 |
lemma map_replicate_trivial:
|
haftmann@24796
|
3304 |
"map (\<lambda>i. x) [0..<i] = replicate i x"
|
haftmann@24796
|
3305 |
by (induct i) (simp_all add: replicate_append_same)
|
haftmann@24796
|
3306 |
|
hoelzl@31350
|
3307 |
lemma concat_replicate_trivial[simp]:
|
hoelzl@31350
|
3308 |
"concat (replicate i []) = []"
|
hoelzl@31350
|
3309 |
by (induct i) (auto simp add: map_replicate_const)
|
wenzelm@13114
|
3310 |
|
nipkow@28642
|
3311 |
lemma replicate_empty[simp]: "(replicate n x = []) \<longleftrightarrow> n=0"
|
nipkow@28642
|
3312 |
by (induct n) auto
|
nipkow@28642
|
3313 |
|
nipkow@28642
|
3314 |
lemma empty_replicate[simp]: "([] = replicate n x) \<longleftrightarrow> n=0"
|
nipkow@28642
|
3315 |
by (induct n) auto
|
nipkow@28642
|
3316 |
|
nipkow@28642
|
3317 |
lemma replicate_eq_replicate[simp]:
|
nipkow@28642
|
3318 |
"(replicate m x = replicate n y) \<longleftrightarrow> (m=n & (m\<noteq>0 \<longrightarrow> x=y))"
|
nipkow@28642
|
3319 |
apply(induct m arbitrary: n)
|
nipkow@28642
|
3320 |
apply simp
|
nipkow@28642
|
3321 |
apply(induct_tac n)
|
nipkow@28642
|
3322 |
apply auto
|
nipkow@28642
|
3323 |
done
|
nipkow@28642
|
3324 |
|
haftmann@39756
|
3325 |
lemma replicate_length_filter:
|
haftmann@39756
|
3326 |
"replicate (length (filter (\<lambda>y. x = y) xs)) x = filter (\<lambda>y. x = y) xs"
|
haftmann@39756
|
3327 |
by (induct xs) auto
|
haftmann@39756
|
3328 |
|
noschinl@43585
|
3329 |
lemma comm_append_are_replicate:
|
noschinl@43585
|
3330 |
fixes xs ys :: "'a list"
|
noschinl@43585
|
3331 |
assumes "xs \<noteq> []" "ys \<noteq> []"
|
noschinl@43585
|
3332 |
assumes "xs @ ys = ys @ xs"
|
noschinl@43585
|
3333 |
shows "\<exists>m n zs. concat (replicate m zs) = xs \<and> concat (replicate n zs) = ys"
|
noschinl@43585
|
3334 |
using assms
|
noschinl@43585
|
3335 |
proof (induct "length (xs @ ys)" arbitrary: xs ys rule: less_induct)
|
noschinl@43585
|
3336 |
case less
|
noschinl@43585
|
3337 |
|
noschinl@43585
|
3338 |
def xs' \<equiv> "if (length xs \<le> length ys) then xs else ys"
|
noschinl@43585
|
3339 |
and ys' \<equiv> "if (length xs \<le> length ys) then ys else xs"
|
noschinl@43585
|
3340 |
then have
|
noschinl@43585
|
3341 |
prems': "length xs' \<le> length ys'"
|
noschinl@43585
|
3342 |
"xs' @ ys' = ys' @ xs'"
|
noschinl@43585
|
3343 |
and "xs' \<noteq> []"
|
noschinl@43585
|
3344 |
and len: "length (xs @ ys) = length (xs' @ ys')"
|
noschinl@43585
|
3345 |
using less by (auto intro: less.hyps)
|
noschinl@43585
|
3346 |
|
noschinl@43585
|
3347 |
from prems'
|
noschinl@43585
|
3348 |
obtain ws where "ys' = xs' @ ws"
|
noschinl@43585
|
3349 |
by (auto simp: append_eq_append_conv2)
|
noschinl@43585
|
3350 |
|
noschinl@43585
|
3351 |
have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ys'"
|
noschinl@43585
|
3352 |
proof (cases "ws = []")
|
noschinl@43585
|
3353 |
case True
|
noschinl@43585
|
3354 |
then have "concat (replicate 1 xs') = xs'"
|
noschinl@43585
|
3355 |
and "concat (replicate 1 xs') = ys'"
|
noschinl@43585
|
3356 |
using `ys' = xs' @ ws` by auto
|
noschinl@43585
|
3357 |
then show ?thesis by blast
|
noschinl@43585
|
3358 |
next
|
noschinl@43585
|
3359 |
case False
|
noschinl@43585
|
3360 |
from `ys' = xs' @ ws` and `xs' @ ys' = ys' @ xs'`
|
noschinl@43585
|
3361 |
have "xs' @ ws = ws @ xs'" by simp
|
noschinl@43585
|
3362 |
then have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ws"
|
noschinl@43585
|
3363 |
using False and `xs' \<noteq> []` and `ys' = xs' @ ws` and len
|
noschinl@43585
|
3364 |
by (intro less.hyps) auto
|
noschinl@43585
|
3365 |
then obtain m n zs where "concat (replicate m zs) = xs'"
|
noschinl@43585
|
3366 |
and "concat (replicate n zs) = ws" by blast
|
noschinl@43585
|
3367 |
moreover
|
noschinl@43585
|
3368 |
then have "concat (replicate (m + n) zs) = ys'"
|
noschinl@43585
|
3369 |
using `ys' = xs' @ ws`
|
noschinl@43585
|
3370 |
by (simp add: replicate_add)
|
noschinl@43585
|
3371 |
ultimately
|
noschinl@43585
|
3372 |
show ?thesis by blast
|
noschinl@43585
|
3373 |
qed
|
noschinl@43585
|
3374 |
then show ?case
|
noschinl@43585
|
3375 |
using xs'_def ys'_def by metis
|
noschinl@43585
|
3376 |
qed
|
noschinl@43585
|
3377 |
|
noschinl@43585
|
3378 |
lemma comm_append_is_replicate:
|
noschinl@43585
|
3379 |
fixes xs ys :: "'a list"
|
noschinl@43585
|
3380 |
assumes "xs \<noteq> []" "ys \<noteq> []"
|
noschinl@43585
|
3381 |
assumes "xs @ ys = ys @ xs"
|
noschinl@43585
|
3382 |
shows "\<exists>n zs. n > 1 \<and> concat (replicate n zs) = xs @ ys"
|
noschinl@43585
|
3383 |
|
noschinl@43585
|
3384 |
proof -
|
noschinl@43585
|
3385 |
obtain m n zs where "concat (replicate m zs) = xs"
|
noschinl@43585
|
3386 |
and "concat (replicate n zs) = ys"
|
noschinl@43585
|
3387 |
using assms by (metis comm_append_are_replicate)
|
noschinl@43585
|
3388 |
then have "m + n > 1" and "concat (replicate (m+n) zs) = xs @ ys"
|
noschinl@43585
|
3389 |
using `xs \<noteq> []` and `ys \<noteq> []`
|
noschinl@43585
|
3390 |
by (auto simp: replicate_add)
|
noschinl@43585
|
3391 |
then show ?thesis by blast
|
noschinl@43585
|
3392 |
qed
|
noschinl@43585
|
3393 |
|
nipkow@28642
|
3394 |
|
nipkow@15392
|
3395 |
subsubsection{*@{text rotate1} and @{text rotate}*}
|
nipkow@15302
|
3396 |
|
nipkow@15302
|
3397 |
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
|
nipkow@15302
|
3398 |
by(simp add:rotate1_def)
|
nipkow@15302
|
3399 |
|
nipkow@15302
|
3400 |
lemma rotate0[simp]: "rotate 0 = id"
|
nipkow@15302
|
3401 |
by(simp add:rotate_def)
|
nipkow@15302
|
3402 |
|
nipkow@15302
|
3403 |
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
|
nipkow@15302
|
3404 |
by(simp add:rotate_def)
|
nipkow@15302
|
3405 |
|
nipkow@15302
|
3406 |
lemma rotate_add:
|
nipkow@15302
|
3407 |
"rotate (m+n) = rotate m o rotate n"
|
nipkow@15302
|
3408 |
by(simp add:rotate_def funpow_add)
|
nipkow@15302
|
3409 |
|
nipkow@15302
|
3410 |
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
|
nipkow@15302
|
3411 |
by(simp add:rotate_add)
|
nipkow@15302
|
3412 |
|
nipkow@18049
|
3413 |
lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
|
nipkow@18049
|
3414 |
by(simp add:rotate_def funpow_swap1)
|
nipkow@18049
|
3415 |
|
nipkow@15302
|
3416 |
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
|
nipkow@15302
|
3417 |
by(cases xs) simp_all
|
nipkow@15302
|
3418 |
|
nipkow@15302
|
3419 |
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
|
nipkow@15302
|
3420 |
apply(induct n)
|
nipkow@15302
|
3421 |
apply simp
|
nipkow@15302
|
3422 |
apply (simp add:rotate_def)
|
nipkow@15302
|
3423 |
done
|
nipkow@15302
|
3424 |
|
nipkow@15302
|
3425 |
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
|
nipkow@15302
|
3426 |
by(simp add:rotate1_def split:list.split)
|
nipkow@15302
|
3427 |
|
nipkow@15302
|
3428 |
lemma rotate_drop_take:
|
nipkow@15302
|
3429 |
"rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
|
nipkow@15302
|
3430 |
apply(induct n)
|
nipkow@15302
|
3431 |
apply simp
|
nipkow@15302
|
3432 |
apply(simp add:rotate_def)
|
nipkow@15302
|
3433 |
apply(cases "xs = []")
|
nipkow@15302
|
3434 |
apply (simp)
|
nipkow@15302
|
3435 |
apply(case_tac "n mod length xs = 0")
|
nipkow@15302
|
3436 |
apply(simp add:mod_Suc)
|
nipkow@15302
|
3437 |
apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
|
nipkow@15302
|
3438 |
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
|
nipkow@15302
|
3439 |
take_hd_drop linorder_not_le)
|
nipkow@15302
|
3440 |
done
|
nipkow@15302
|
3441 |
|
nipkow@15302
|
3442 |
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
|
nipkow@15302
|
3443 |
by(simp add:rotate_drop_take)
|
nipkow@15302
|
3444 |
|
nipkow@15302
|
3445 |
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
|
nipkow@15302
|
3446 |
by(simp add:rotate_drop_take)
|
nipkow@15302
|
3447 |
|
nipkow@15302
|
3448 |
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
|
nipkow@15302
|
3449 |
by(simp add:rotate1_def split:list.split)
|
nipkow@15302
|
3450 |
|
nipkow@24526
|
3451 |
lemma length_rotate[simp]: "length(rotate n xs) = length xs"
|
nipkow@24526
|
3452 |
by (induct n arbitrary: xs) (simp_all add:rotate_def)
|
nipkow@15302
|
3453 |
|
nipkow@15302
|
3454 |
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
|
nipkow@15302
|
3455 |
by(simp add:rotate1_def split:list.split) blast
|
nipkow@15302
|
3456 |
|
nipkow@15302
|
3457 |
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
|
nipkow@15302
|
3458 |
by (induct n) (simp_all add:rotate_def)
|
nipkow@15302
|
3459 |
|
nipkow@15302
|
3460 |
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
|
nipkow@15302
|
3461 |
by(simp add:rotate_drop_take take_map drop_map)
|
nipkow@15302
|
3462 |
|
nipkow@15302
|
3463 |
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
|
bulwahn@41695
|
3464 |
by (cases xs) (auto simp add:rotate1_def)
|
nipkow@15302
|
3465 |
|
nipkow@15302
|
3466 |
lemma set_rotate[simp]: "set(rotate n xs) = set xs"
|
nipkow@15302
|
3467 |
by (induct n) (simp_all add:rotate_def)
|
nipkow@15302
|
3468 |
|
nipkow@15302
|
3469 |
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
|
nipkow@15302
|
3470 |
by(simp add:rotate1_def split:list.split)
|
nipkow@15302
|
3471 |
|
nipkow@15302
|
3472 |
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
|
nipkow@15302
|
3473 |
by (induct n) (simp_all add:rotate_def)
|
nipkow@15302
|
3474 |
|
nipkow@15439
|
3475 |
lemma rotate_rev:
|
nipkow@15439
|
3476 |
"rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
|
nipkow@15439
|
3477 |
apply(simp add:rotate_drop_take rev_drop rev_take)
|
nipkow@15439
|
3478 |
apply(cases "length xs = 0")
|
nipkow@15439
|
3479 |
apply simp
|
nipkow@15439
|
3480 |
apply(cases "n mod length xs = 0")
|
nipkow@15439
|
3481 |
apply simp
|
nipkow@15439
|
3482 |
apply(simp add:rotate_drop_take rev_drop rev_take)
|
nipkow@15439
|
3483 |
done
|
nipkow@15439
|
3484 |
|
nipkow@18423
|
3485 |
lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
|
nipkow@18423
|
3486 |
apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
|
nipkow@18423
|
3487 |
apply(subgoal_tac "length xs \<noteq> 0")
|
nipkow@18423
|
3488 |
prefer 2 apply simp
|
nipkow@18423
|
3489 |
using mod_less_divisor[of "length xs" n] by arith
|
nipkow@18423
|
3490 |
|
nipkow@15302
|
3491 |
|
nipkow@15392
|
3492 |
subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
|
nipkow@15302
|
3493 |
|
nipkow@15302
|
3494 |
lemma sublist_empty [simp]: "sublist xs {} = []"
|
nipkow@15302
|
3495 |
by (auto simp add: sublist_def)
|
nipkow@15302
|
3496 |
|
nipkow@15302
|
3497 |
lemma sublist_nil [simp]: "sublist [] A = []"
|
nipkow@15302
|
3498 |
by (auto simp add: sublist_def)
|
nipkow@15302
|
3499 |
|
nipkow@15302
|
3500 |
lemma length_sublist:
|
nipkow@15302
|
3501 |
"length(sublist xs I) = card{i. i < length xs \<and> i : I}"
|
nipkow@15302
|
3502 |
by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
|
nipkow@15302
|
3503 |
|
nipkow@15302
|
3504 |
lemma sublist_shift_lemma_Suc:
|
nipkow@24526
|
3505 |
"map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
|
nipkow@24526
|
3506 |
map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
|
nipkow@24526
|
3507 |
apply(induct xs arbitrary: "is")
|
nipkow@15302
|
3508 |
apply simp
|
nipkow@15302
|
3509 |
apply (case_tac "is")
|
nipkow@15302
|
3510 |
apply simp
|
nipkow@15302
|
3511 |
apply simp
|
nipkow@15302
|
3512 |
done
|
nipkow@15302
|
3513 |
|
nipkow@15302
|
3514 |
lemma sublist_shift_lemma:
|
nipkow@23279
|
3515 |
"map fst [p<-zip xs [i..<i + length xs] . snd p : A] =
|
nipkow@23279
|
3516 |
map fst [p<-zip xs [0..<length xs] . snd p + i : A]"
|
nipkow@15302
|
3517 |
by (induct xs rule: rev_induct) (simp_all add: add_commute)
|
nipkow@15302
|
3518 |
|
nipkow@15302
|
3519 |
lemma sublist_append:
|
nipkow@15302
|
3520 |
"sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
|
nipkow@15302
|
3521 |
apply (unfold sublist_def)
|
nipkow@15302
|
3522 |
apply (induct l' rule: rev_induct, simp)
|
huffman@45792
|
3523 |
apply (simp add: upt_add_eq_append[of 0] sublist_shift_lemma)
|
nipkow@15302
|
3524 |
apply (simp add: add_commute)
|
nipkow@15302
|
3525 |
done
|
nipkow@15302
|
3526 |
|
nipkow@15302
|
3527 |
lemma sublist_Cons:
|
nipkow@15302
|
3528 |
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
|
nipkow@15302
|
3529 |
apply (induct l rule: rev_induct)
|
nipkow@15302
|
3530 |
apply (simp add: sublist_def)
|
nipkow@15302
|
3531 |
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
|
nipkow@15302
|
3532 |
done
|
nipkow@15302
|
3533 |
|
nipkow@24526
|
3534 |
lemma set_sublist: "set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
|
nipkow@24526
|
3535 |
apply(induct xs arbitrary: I)
|
nipkow@25162
|
3536 |
apply(auto simp: sublist_Cons nth_Cons split:nat.split dest!: gr0_implies_Suc)
|
nipkow@15302
|
3537 |
done
|
nipkow@15302
|
3538 |
|
nipkow@15302
|
3539 |
lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
|
nipkow@15302
|
3540 |
by(auto simp add:set_sublist)
|
nipkow@15302
|
3541 |
|
nipkow@15302
|
3542 |
lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
|
nipkow@15302
|
3543 |
by(auto simp add:set_sublist)
|
nipkow@15302
|
3544 |
|
nipkow@15302
|
3545 |
lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
|
nipkow@15302
|
3546 |
by(auto simp add:set_sublist)
|
nipkow@15302
|
3547 |
|
nipkow@15302
|
3548 |
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
|
nipkow@15302
|
3549 |
by (simp add: sublist_Cons)
|
nipkow@15302
|
3550 |
|
nipkow@15302
|
3551 |
|
nipkow@24526
|
3552 |
lemma distinct_sublistI[simp]: "distinct xs \<Longrightarrow> distinct(sublist xs I)"
|
nipkow@24526
|
3553 |
apply(induct xs arbitrary: I)
|
nipkow@15302
|
3554 |
apply simp
|
nipkow@15302
|
3555 |
apply(auto simp add:sublist_Cons)
|
nipkow@15302
|
3556 |
done
|
nipkow@15302
|
3557 |
|
nipkow@15302
|
3558 |
|
nipkow@15302
|
3559 |
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
|
nipkow@15302
|
3560 |
apply (induct l rule: rev_induct, simp)
|
nipkow@15302
|
3561 |
apply (simp split: nat_diff_split add: sublist_append)
|
nipkow@15302
|
3562 |
done
|
nipkow@15302
|
3563 |
|
nipkow@24526
|
3564 |
lemma filter_in_sublist:
|
nipkow@24526
|
3565 |
"distinct xs \<Longrightarrow> filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
|
nipkow@24526
|
3566 |
proof (induct xs arbitrary: s)
|
nipkow@17501
|
3567 |
case Nil thus ?case by simp
|
nipkow@17501
|
3568 |
next
|
nipkow@17501
|
3569 |
case (Cons a xs)
|
nipkow@17501
|
3570 |
moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
|
nipkow@17501
|
3571 |
ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
|
nipkow@17501
|
3572 |
qed
|
nipkow@17501
|
3573 |
|
nipkow@15302
|
3574 |
|
nipkow@19390
|
3575 |
subsubsection {* @{const splice} *}
|
nipkow@19390
|
3576 |
|
nipkow@40841
|
3577 |
lemma splice_Nil2 [simp, code]: "splice xs [] = xs"
|
nipkow@19390
|
3578 |
by (cases xs) simp_all
|
nipkow@19390
|
3579 |
|
nipkow@40841
|
3580 |
declare splice.simps(1,3)[code]
|
nipkow@40841
|
3581 |
declare splice.simps(2)[simp del]
|
nipkow@19390
|
3582 |
|
nipkow@24526
|
3583 |
lemma length_splice[simp]: "length(splice xs ys) = length xs + length ys"
|
nipkow@40841
|
3584 |
by (induct xs ys rule: splice.induct) auto
|
nipkow@22793
|
3585 |
|
wenzelm@35118
|
3586 |
|
wenzelm@35118
|
3587 |
subsubsection {* Transpose *}
|
hoelzl@34920
|
3588 |
|
hoelzl@34920
|
3589 |
function transpose where
|
hoelzl@34920
|
3590 |
"transpose [] = []" |
|
hoelzl@34920
|
3591 |
"transpose ([] # xss) = transpose xss" |
|
hoelzl@34920
|
3592 |
"transpose ((x#xs) # xss) =
|
hoelzl@34920
|
3593 |
(x # [h. (h#t) \<leftarrow> xss]) # transpose (xs # [t. (h#t) \<leftarrow> xss])"
|
hoelzl@34920
|
3594 |
by pat_completeness auto
|
hoelzl@34920
|
3595 |
|
hoelzl@34920
|
3596 |
lemma transpose_aux_filter_head:
|
hoelzl@34920
|
3597 |
"concat (map (list_case [] (\<lambda>h t. [h])) xss) =
|
hoelzl@34920
|
3598 |
map (\<lambda>xs. hd xs) [ys\<leftarrow>xss . ys \<noteq> []]"
|
hoelzl@34920
|
3599 |
by (induct xss) (auto split: list.split)
|
hoelzl@34920
|
3600 |
|
hoelzl@34920
|
3601 |
lemma transpose_aux_filter_tail:
|
hoelzl@34920
|
3602 |
"concat (map (list_case [] (\<lambda>h t. [t])) xss) =
|
hoelzl@34920
|
3603 |
map (\<lambda>xs. tl xs) [ys\<leftarrow>xss . ys \<noteq> []]"
|
hoelzl@34920
|
3604 |
by (induct xss) (auto split: list.split)
|
hoelzl@34920
|
3605 |
|
hoelzl@34920
|
3606 |
lemma transpose_aux_max:
|
hoelzl@34920
|
3607 |
"max (Suc (length xs)) (foldr (\<lambda>xs. max (length xs)) xss 0) =
|
hoelzl@34920
|
3608 |
Suc (max (length xs) (foldr (\<lambda>x. max (length x - Suc 0)) [ys\<leftarrow>xss . ys\<noteq>[]] 0))"
|
hoelzl@34920
|
3609 |
(is "max _ ?foldB = Suc (max _ ?foldA)")
|
hoelzl@34920
|
3610 |
proof (cases "[ys\<leftarrow>xss . ys\<noteq>[]] = []")
|
hoelzl@34920
|
3611 |
case True
|
hoelzl@34920
|
3612 |
hence "foldr (\<lambda>xs. max (length xs)) xss 0 = 0"
|
hoelzl@34920
|
3613 |
proof (induct xss)
|
hoelzl@34920
|
3614 |
case (Cons x xs)
|
hoelzl@34920
|
3615 |
moreover hence "x = []" by (cases x) auto
|
hoelzl@34920
|
3616 |
ultimately show ?case by auto
|
hoelzl@34920
|
3617 |
qed simp
|
hoelzl@34920
|
3618 |
thus ?thesis using True by simp
|
hoelzl@34920
|
3619 |
next
|
hoelzl@34920
|
3620 |
case False
|
hoelzl@34920
|
3621 |
|
hoelzl@34920
|
3622 |
have foldA: "?foldA = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0 - 1"
|
hoelzl@34920
|
3623 |
by (induct xss) auto
|
hoelzl@34920
|
3624 |
have foldB: "?foldB = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0"
|
hoelzl@34920
|
3625 |
by (induct xss) auto
|
hoelzl@34920
|
3626 |
|
hoelzl@34920
|
3627 |
have "0 < ?foldB"
|
hoelzl@34920
|
3628 |
proof -
|
hoelzl@34920
|
3629 |
from False
|
hoelzl@34920
|
3630 |
obtain z zs where zs: "[ys\<leftarrow>xss . ys \<noteq> []] = z#zs" by (auto simp: neq_Nil_conv)
|
hoelzl@34920
|
3631 |
hence "z \<in> set ([ys\<leftarrow>xss . ys \<noteq> []])" by auto
|
hoelzl@34920
|
3632 |
hence "z \<noteq> []" by auto
|
hoelzl@34920
|
3633 |
thus ?thesis
|
hoelzl@34920
|
3634 |
unfolding foldB zs
|
hoelzl@34920
|
3635 |
by (auto simp: max_def intro: less_le_trans)
|
hoelzl@34920
|
3636 |
qed
|
hoelzl@34920
|
3637 |
thus ?thesis
|
hoelzl@34920
|
3638 |
unfolding foldA foldB max_Suc_Suc[symmetric]
|
hoelzl@34920
|
3639 |
by simp
|
hoelzl@34920
|
3640 |
qed
|
hoelzl@34920
|
3641 |
|
hoelzl@34920
|
3642 |
termination transpose
|
hoelzl@34920
|
3643 |
by (relation "measure (\<lambda>xs. foldr (\<lambda>xs. max (length xs)) xs 0 + length xs)")
|
hoelzl@34920
|
3644 |
(auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max less_Suc_eq_le)
|
hoelzl@34920
|
3645 |
|
hoelzl@34920
|
3646 |
lemma transpose_empty: "(transpose xs = []) \<longleftrightarrow> (\<forall>x \<in> set xs. x = [])"
|
hoelzl@34920
|
3647 |
by (induct rule: transpose.induct) simp_all
|
hoelzl@34920
|
3648 |
|
hoelzl@34920
|
3649 |
lemma length_transpose:
|
hoelzl@34920
|
3650 |
fixes xs :: "'a list list"
|
hoelzl@34920
|
3651 |
shows "length (transpose xs) = foldr (\<lambda>xs. max (length xs)) xs 0"
|
hoelzl@34920
|
3652 |
by (induct rule: transpose.induct)
|
hoelzl@34920
|
3653 |
(auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max
|
hoelzl@34920
|
3654 |
max_Suc_Suc[symmetric] simp del: max_Suc_Suc)
|
hoelzl@34920
|
3655 |
|
hoelzl@34920
|
3656 |
lemma nth_transpose:
|
hoelzl@34920
|
3657 |
fixes xs :: "'a list list"
|
hoelzl@34920
|
3658 |
assumes "i < length (transpose xs)"
|
hoelzl@34920
|
3659 |
shows "transpose xs ! i = map (\<lambda>xs. xs ! i) [ys \<leftarrow> xs. i < length ys]"
|
hoelzl@34920
|
3660 |
using assms proof (induct arbitrary: i rule: transpose.induct)
|
hoelzl@34920
|
3661 |
case (3 x xs xss)
|
hoelzl@34920
|
3662 |
def XS == "(x # xs) # xss"
|
hoelzl@34920
|
3663 |
hence [simp]: "XS \<noteq> []" by auto
|
hoelzl@34920
|
3664 |
thus ?case
|
hoelzl@34920
|
3665 |
proof (cases i)
|
hoelzl@34920
|
3666 |
case 0
|
hoelzl@34920
|
3667 |
thus ?thesis by (simp add: transpose_aux_filter_head hd_conv_nth)
|
hoelzl@34920
|
3668 |
next
|
hoelzl@34920
|
3669 |
case (Suc j)
|
hoelzl@34920
|
3670 |
have *: "\<And>xss. xs # map tl xss = map tl ((x#xs)#xss)" by simp
|
hoelzl@34920
|
3671 |
have **: "\<And>xss. (x#xs) # filter (\<lambda>ys. ys \<noteq> []) xss = filter (\<lambda>ys. ys \<noteq> []) ((x#xs)#xss)" by simp
|
hoelzl@34920
|
3672 |
{ fix x have "Suc j < length x \<longleftrightarrow> x \<noteq> [] \<and> j < length x - Suc 0"
|
hoelzl@34920
|
3673 |
by (cases x) simp_all
|
hoelzl@34920
|
3674 |
} note *** = this
|
hoelzl@34920
|
3675 |
|
hoelzl@34920
|
3676 |
have j_less: "j < length (transpose (xs # concat (map (list_case [] (\<lambda>h t. [t])) xss)))"
|
hoelzl@34920
|
3677 |
using "3.prems" by (simp add: transpose_aux_filter_tail length_transpose Suc)
|
hoelzl@34920
|
3678 |
|
hoelzl@34920
|
3679 |
show ?thesis
|
hoelzl@34920
|
3680 |
unfolding transpose.simps `i = Suc j` nth_Cons_Suc "3.hyps"[OF j_less]
|
hoelzl@34920
|
3681 |
apply (auto simp: transpose_aux_filter_tail filter_map comp_def length_transpose * ** *** XS_def[symmetric])
|
hoelzl@34920
|
3682 |
apply (rule_tac y=x in list.exhaust)
|
hoelzl@34920
|
3683 |
by auto
|
hoelzl@34920
|
3684 |
qed
|
hoelzl@34920
|
3685 |
qed simp_all
|
hoelzl@34920
|
3686 |
|
hoelzl@34920
|
3687 |
lemma transpose_map_map:
|
hoelzl@34920
|
3688 |
"transpose (map (map f) xs) = map (map f) (transpose xs)"
|
hoelzl@34920
|
3689 |
proof (rule nth_equalityI, safe)
|
hoelzl@34920
|
3690 |
have [simp]: "length (transpose (map (map f) xs)) = length (transpose xs)"
|
hoelzl@34920
|
3691 |
by (simp add: length_transpose foldr_map comp_def)
|
hoelzl@34920
|
3692 |
show "length (transpose (map (map f) xs)) = length (map (map f) (transpose xs))" by simp
|
hoelzl@34920
|
3693 |
|
hoelzl@34920
|
3694 |
fix i assume "i < length (transpose (map (map f) xs))"
|
hoelzl@34920
|
3695 |
thus "transpose (map (map f) xs) ! i = map (map f) (transpose xs) ! i"
|
hoelzl@34920
|
3696 |
by (simp add: nth_transpose filter_map comp_def)
|
hoelzl@34920
|
3697 |
qed
|
nipkow@24616
|
3698 |
|
wenzelm@35118
|
3699 |
|
nipkow@31557
|
3700 |
subsubsection {* (In)finiteness *}
|
nipkow@28642
|
3701 |
|
nipkow@28642
|
3702 |
lemma finite_maxlen:
|
nipkow@28642
|
3703 |
"finite (M::'a list set) ==> EX n. ALL s:M. size s < n"
|
nipkow@28642
|
3704 |
proof (induct rule: finite.induct)
|
nipkow@28642
|
3705 |
case emptyI show ?case by simp
|
nipkow@28642
|
3706 |
next
|
nipkow@28642
|
3707 |
case (insertI M xs)
|
nipkow@28642
|
3708 |
then obtain n where "\<forall>s\<in>M. length s < n" by blast
|
nipkow@28642
|
3709 |
hence "ALL s:insert xs M. size s < max n (size xs) + 1" by auto
|
nipkow@28642
|
3710 |
thus ?case ..
|
nipkow@28642
|
3711 |
qed
|
nipkow@28642
|
3712 |
|
nipkow@31557
|
3713 |
lemma finite_lists_length_eq:
|
nipkow@31557
|
3714 |
assumes "finite A"
|
nipkow@31557
|
3715 |
shows "finite {xs. set xs \<subseteq> A \<and> length xs = n}" (is "finite (?S n)")
|
nipkow@31557
|
3716 |
proof(induct n)
|
nipkow@31557
|
3717 |
case 0 show ?case by simp
|
nipkow@31557
|
3718 |
next
|
nipkow@31557
|
3719 |
case (Suc n)
|
nipkow@31557
|
3720 |
have "?S (Suc n) = (\<Union>x\<in>A. (\<lambda>xs. x#xs) ` ?S n)"
|
nipkow@31557
|
3721 |
by (auto simp:length_Suc_conv)
|
nipkow@31557
|
3722 |
then show ?case using `finite A`
|
nipkow@41030
|
3723 |
by (auto intro: Suc) (* FIXME metis? *)
|
nipkow@31557
|
3724 |
qed
|
nipkow@31557
|
3725 |
|
nipkow@31557
|
3726 |
lemma finite_lists_length_le:
|
nipkow@31557
|
3727 |
assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs \<le> n}"
|
nipkow@31557
|
3728 |
(is "finite ?S")
|
nipkow@31557
|
3729 |
proof-
|
nipkow@31557
|
3730 |
have "?S = (\<Union>n\<in>{0..n}. {xs. set xs \<subseteq> A \<and> length xs = n})" by auto
|
nipkow@31557
|
3731 |
thus ?thesis by (auto intro: finite_lists_length_eq[OF `finite A`])
|
nipkow@31557
|
3732 |
qed
|
nipkow@31557
|
3733 |
|
nipkow@28642
|
3734 |
lemma infinite_UNIV_listI: "~ finite(UNIV::'a list set)"
|
nipkow@28642
|
3735 |
apply(rule notI)
|
nipkow@28642
|
3736 |
apply(drule finite_maxlen)
|
nipkow@28642
|
3737 |
apply (metis UNIV_I length_replicate less_not_refl)
|
nipkow@28642
|
3738 |
done
|
nipkow@28642
|
3739 |
|
nipkow@28642
|
3740 |
|
wenzelm@35118
|
3741 |
subsection {* Sorting *}
|
nipkow@24616
|
3742 |
|
nipkow@24617
|
3743 |
text{* Currently it is not shown that @{const sort} returns a
|
nipkow@24617
|
3744 |
permutation of its input because the nicest proof is via multisets,
|
nipkow@24617
|
3745 |
which are not yet available. Alternatively one could define a function
|
nipkow@24617
|
3746 |
that counts the number of occurrences of an element in a list and use
|
nipkow@24617
|
3747 |
that instead of multisets to state the correctness property. *}
|
nipkow@24617
|
3748 |
|
nipkow@24616
|
3749 |
context linorder
|
nipkow@24616
|
3750 |
begin
|
nipkow@24616
|
3751 |
|
haftmann@40451
|
3752 |
lemma length_insort [simp]:
|
haftmann@40451
|
3753 |
"length (insort_key f x xs) = Suc (length xs)"
|
haftmann@40451
|
3754 |
by (induct xs) simp_all
|
haftmann@40451
|
3755 |
|
haftmann@40451
|
3756 |
lemma insort_key_left_comm:
|
haftmann@40451
|
3757 |
assumes "f x \<noteq> f y"
|
haftmann@40451
|
3758 |
shows "insort_key f y (insort_key f x xs) = insort_key f x (insort_key f y xs)"
|
haftmann@40451
|
3759 |
by (induct xs) (auto simp add: assms dest: antisym)
|
hoelzl@33639
|
3760 |
|
haftmann@35195
|
3761 |
lemma insort_left_comm:
|
haftmann@35195
|
3762 |
"insort x (insort y xs) = insort y (insort x xs)"
|
haftmann@40451
|
3763 |
by (cases "x = y") (auto intro: insort_key_left_comm)
|
haftmann@35195
|
3764 |
|
haftmann@43740
|
3765 |
lemma comp_fun_commute_insort:
|
haftmann@43740
|
3766 |
"comp_fun_commute insort"
|
haftmann@35195
|
3767 |
proof
|
haftmann@43670
|
3768 |
qed (simp add: insort_left_comm fun_eq_iff)
|
haftmann@35195
|
3769 |
|
haftmann@35195
|
3770 |
lemma sort_key_simps [simp]:
|
haftmann@35195
|
3771 |
"sort_key f [] = []"
|
haftmann@35195
|
3772 |
"sort_key f (x#xs) = insort_key f x (sort_key f xs)"
|
haftmann@35195
|
3773 |
by (simp_all add: sort_key_def)
|
haftmann@35195
|
3774 |
|
haftmann@35195
|
3775 |
lemma sort_foldl_insort:
|
haftmann@35195
|
3776 |
"sort xs = foldl (\<lambda>ys x. insort x ys) [] xs"
|
haftmann@35195
|
3777 |
by (simp add: sort_key_def foldr_foldl foldl_rev insort_left_comm)
|
haftmann@35195
|
3778 |
|
hoelzl@33639
|
3779 |
lemma length_sort[simp]: "length (sort_key f xs) = length xs"
|
hoelzl@33639
|
3780 |
by (induct xs, auto)
|
hoelzl@33639
|
3781 |
|
haftmann@25062
|
3782 |
lemma sorted_Cons: "sorted (x#xs) = (sorted xs & (ALL y:set xs. x <= y))"
|
nipkow@24616
|
3783 |
apply(induct xs arbitrary: x) apply simp
|
nipkow@24616
|
3784 |
by simp (blast intro: order_trans)
|
nipkow@24616
|
3785 |
|
haftmann@40451
|
3786 |
lemma sorted_tl:
|
haftmann@40451
|
3787 |
"sorted xs \<Longrightarrow> sorted (tl xs)"
|
haftmann@40451
|
3788 |
by (cases xs) (simp_all add: sorted_Cons)
|
haftmann@40451
|
3789 |
|
nipkow@24616
|
3790 |
lemma sorted_append:
|
haftmann@25062
|
3791 |
"sorted (xs@ys) = (sorted xs & sorted ys & (\<forall>x \<in> set xs. \<forall>y \<in> set ys. x\<le>y))"
|
nipkow@24616
|
3792 |
by (induct xs) (auto simp add:sorted_Cons)
|
nipkow@24616
|
3793 |
|
nipkow@31201
|
3794 |
lemma sorted_nth_mono:
|
hoelzl@33639
|
3795 |
"sorted xs \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!i \<le> xs!j"
|
nipkow@31201
|
3796 |
by (induct xs arbitrary: i j) (auto simp:nth_Cons' sorted_Cons)
|
nipkow@31201
|
3797 |
|
hoelzl@33639
|
3798 |
lemma sorted_rev_nth_mono:
|
hoelzl@33639
|
3799 |
"sorted (rev xs) \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!j \<le> xs!i"
|
hoelzl@33639
|
3800 |
using sorted_nth_mono[ of "rev xs" "length xs - j - 1" "length xs - i - 1"]
|
hoelzl@33639
|
3801 |
rev_nth[of "length xs - i - 1" "xs"] rev_nth[of "length xs - j - 1" "xs"]
|
hoelzl@33639
|
3802 |
by auto
|
hoelzl@33639
|
3803 |
|
hoelzl@33639
|
3804 |
lemma sorted_nth_monoI:
|
hoelzl@33639
|
3805 |
"(\<And> i j. \<lbrakk> i \<le> j ; j < length xs \<rbrakk> \<Longrightarrow> xs ! i \<le> xs ! j) \<Longrightarrow> sorted xs"
|
hoelzl@33639
|
3806 |
proof (induct xs)
|
hoelzl@33639
|
3807 |
case (Cons x xs)
|
hoelzl@33639
|
3808 |
have "sorted xs"
|
hoelzl@33639
|
3809 |
proof (rule Cons.hyps)
|
hoelzl@33639
|
3810 |
fix i j assume "i \<le> j" and "j < length xs"
|
hoelzl@33639
|
3811 |
with Cons.prems[of "Suc i" "Suc j"]
|
hoelzl@33639
|
3812 |
show "xs ! i \<le> xs ! j" by auto
|
hoelzl@33639
|
3813 |
qed
|
hoelzl@33639
|
3814 |
moreover
|
hoelzl@33639
|
3815 |
{
|
hoelzl@33639
|
3816 |
fix y assume "y \<in> set xs"
|
hoelzl@33639
|
3817 |
then obtain j where "j < length xs" and "xs ! j = y"
|
hoelzl@33639
|
3818 |
unfolding in_set_conv_nth by blast
|
hoelzl@33639
|
3819 |
with Cons.prems[of 0 "Suc j"]
|
hoelzl@33639
|
3820 |
have "x \<le> y"
|
hoelzl@33639
|
3821 |
by auto
|
hoelzl@33639
|
3822 |
}
|
hoelzl@33639
|
3823 |
ultimately
|
hoelzl@33639
|
3824 |
show ?case
|
hoelzl@33639
|
3825 |
unfolding sorted_Cons by auto
|
hoelzl@33639
|
3826 |
qed simp
|
hoelzl@33639
|
3827 |
|
hoelzl@33639
|
3828 |
lemma sorted_equals_nth_mono:
|
hoelzl@33639
|
3829 |
"sorted xs = (\<forall>j < length xs. \<forall>i \<le> j. xs ! i \<le> xs ! j)"
|
hoelzl@33639
|
3830 |
by (auto intro: sorted_nth_monoI sorted_nth_mono)
|
hoelzl@33639
|
3831 |
|
hoelzl@33639
|
3832 |
lemma set_insort: "set(insort_key f x xs) = insert x (set xs)"
|
nipkow@24616
|
3833 |
by (induct xs) auto
|
nipkow@24616
|
3834 |
|
hoelzl@33639
|
3835 |
lemma set_sort[simp]: "set(sort_key f xs) = set xs"
|
nipkow@24616
|
3836 |
by (induct xs) (simp_all add:set_insort)
|
nipkow@24616
|
3837 |
|
hoelzl@33639
|
3838 |
lemma distinct_insort: "distinct (insort_key f x xs) = (x \<notin> set xs \<and> distinct xs)"
|
nipkow@24616
|
3839 |
by(induct xs)(auto simp:set_insort)
|
nipkow@24616
|
3840 |
|
hoelzl@33639
|
3841 |
lemma distinct_sort[simp]: "distinct (sort_key f xs) = distinct xs"
|
huffman@45792
|
3842 |
by (induct xs) (simp_all add: distinct_insort)
|
nipkow@24616
|
3843 |
|
hoelzl@33639
|
3844 |
lemma sorted_insort_key: "sorted (map f (insort_key f x xs)) = sorted (map f xs)"
|
haftmann@40451
|
3845 |
by (induct xs) (auto simp:sorted_Cons set_insort)
|
hoelzl@33639
|
3846 |
|
nipkow@24616
|
3847 |
lemma sorted_insort: "sorted (insort x xs) = sorted xs"
|
haftmann@40451
|
3848 |
using sorted_insort_key [where f="\<lambda>x. x"] by simp
|
haftmann@40451
|
3849 |
|
haftmann@40451
|
3850 |
theorem sorted_sort_key [simp]: "sorted (map f (sort_key f xs))"
|
haftmann@40451
|
3851 |
by (induct xs) (auto simp:sorted_insort_key)
|
haftmann@40451
|
3852 |
|
haftmann@40451
|
3853 |
theorem sorted_sort [simp]: "sorted (sort xs)"
|
haftmann@40451
|
3854 |
using sorted_sort_key [where f="\<lambda>x. x"] by simp
|
hoelzl@33639
|
3855 |
|
haftmann@36846
|
3856 |
lemma sorted_butlast:
|
haftmann@36846
|
3857 |
assumes "xs \<noteq> []" and "sorted xs"
|
haftmann@36846
|
3858 |
shows "sorted (butlast xs)"
|
haftmann@36846
|
3859 |
proof -
|
haftmann@36846
|
3860 |
from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
|
haftmann@36846
|
3861 |
with `sorted xs` show ?thesis by (simp add: sorted_append)
|
haftmann@36846
|
3862 |
qed
|
haftmann@36846
|
3863 |
|
haftmann@36846
|
3864 |
lemma insort_not_Nil [simp]:
|
haftmann@36846
|
3865 |
"insort_key f a xs \<noteq> []"
|
haftmann@36846
|
3866 |
by (induct xs) simp_all
|
haftmann@36846
|
3867 |
|
hoelzl@33639
|
3868 |
lemma insort_is_Cons: "\<forall>x\<in>set xs. f a \<le> f x \<Longrightarrow> insort_key f a xs = a # xs"
|
bulwahn@26143
|
3869 |
by (cases xs) auto
|
bulwahn@26143
|
3870 |
|
bulwahn@45787
|
3871 |
lemma sorted_sort_id: "sorted xs \<Longrightarrow> sort xs = xs"
|
bulwahn@45787
|
3872 |
by (induct xs) (auto simp add: sorted_Cons insort_is_Cons)
|
bulwahn@45787
|
3873 |
|
haftmann@39756
|
3874 |
lemma sorted_map_remove1:
|
haftmann@39756
|
3875 |
"sorted (map f xs) \<Longrightarrow> sorted (map f (remove1 x xs))"
|
haftmann@39756
|
3876 |
by (induct xs) (auto simp add: sorted_Cons)
|
haftmann@39756
|
3877 |
|
bulwahn@26143
|
3878 |
lemma sorted_remove1: "sorted xs \<Longrightarrow> sorted (remove1 a xs)"
|
haftmann@39756
|
3879 |
using sorted_map_remove1 [of "\<lambda>x. x"] by simp
|
haftmann@39756
|
3880 |
|
haftmann@39756
|
3881 |
lemma insort_key_remove1:
|
haftmann@39756
|
3882 |
assumes "a \<in> set xs" and "sorted (map f xs)" and "hd (filter (\<lambda>x. f a = f x) xs) = a"
|
haftmann@39756
|
3883 |
shows "insort_key f a (remove1 a xs) = xs"
|
haftmann@39756
|
3884 |
using assms proof (induct xs)
|
hoelzl@33639
|
3885 |
case (Cons x xs)
|
haftmann@39756
|
3886 |
then show ?case
|
hoelzl@33639
|
3887 |
proof (cases "x = a")
|
hoelzl@33639
|
3888 |
case False
|
haftmann@39756
|
3889 |
then have "f x \<noteq> f a" using Cons.prems by auto
|
haftmann@39756
|
3890 |
then have "f x < f a" using Cons.prems by (auto simp: sorted_Cons)
|
haftmann@39756
|
3891 |
with `f x \<noteq> f a` show ?thesis using Cons by (auto simp: sorted_Cons insort_is_Cons)
|
hoelzl@33639
|
3892 |
qed (auto simp: sorted_Cons insort_is_Cons)
|
hoelzl@33639
|
3893 |
qed simp
|
bulwahn@26143
|
3894 |
|
haftmann@39756
|
3895 |
lemma insort_remove1:
|
haftmann@39756
|
3896 |
assumes "a \<in> set xs" and "sorted xs"
|
haftmann@39756
|
3897 |
shows "insort a (remove1 a xs) = xs"
|
haftmann@39756
|
3898 |
proof (rule insort_key_remove1)
|
haftmann@39756
|
3899 |
from `a \<in> set xs` show "a \<in> set xs" .
|
haftmann@39756
|
3900 |
from `sorted xs` show "sorted (map (\<lambda>x. x) xs)" by simp
|
haftmann@39756
|
3901 |
from `a \<in> set xs` have "a \<in> set (filter (op = a) xs)" by auto
|
haftmann@39756
|
3902 |
then have "set (filter (op = a) xs) \<noteq> {}" by auto
|
haftmann@39756
|
3903 |
then have "filter (op = a) xs \<noteq> []" by (auto simp only: set_empty)
|
haftmann@39756
|
3904 |
then have "length (filter (op = a) xs) > 0" by simp
|
haftmann@39756
|
3905 |
then obtain n where n: "Suc n = length (filter (op = a) xs)"
|
haftmann@39756
|
3906 |
by (cases "length (filter (op = a) xs)") simp_all
|
haftmann@39756
|
3907 |
moreover have "replicate (Suc n) a = a # replicate n a"
|
haftmann@39756
|
3908 |
by simp
|
haftmann@39756
|
3909 |
ultimately show "hd (filter (op = a) xs) = a" by (simp add: replicate_length_filter)
|
haftmann@39756
|
3910 |
qed
|
bulwahn@26143
|
3911 |
|
bulwahn@26143
|
3912 |
lemma sorted_remdups[simp]:
|
bulwahn@26143
|
3913 |
"sorted l \<Longrightarrow> sorted (remdups l)"
|
bulwahn@26143
|
3914 |
by (induct l) (auto simp: sorted_Cons)
|
bulwahn@26143
|
3915 |
|
nipkow@24645
|
3916 |
lemma sorted_distinct_set_unique:
|
nipkow@24645
|
3917 |
assumes "sorted xs" "distinct xs" "sorted ys" "distinct ys" "set xs = set ys"
|
nipkow@24645
|
3918 |
shows "xs = ys"
|
nipkow@24645
|
3919 |
proof -
|
haftmann@26734
|
3920 |
from assms have 1: "length xs = length ys" by (auto dest!: distinct_card)
|
nipkow@24645
|
3921 |
from assms show ?thesis
|
nipkow@24645
|
3922 |
proof(induct rule:list_induct2[OF 1])
|
nipkow@24645
|
3923 |
case 1 show ?case by simp
|
nipkow@24645
|
3924 |
next
|
nipkow@24645
|
3925 |
case 2 thus ?case by (simp add:sorted_Cons)
|
nipkow@24645
|
3926 |
(metis Diff_insert_absorb antisym insertE insert_iff)
|
nipkow@24645
|
3927 |
qed
|
nipkow@24645
|
3928 |
qed
|
nipkow@24645
|
3929 |
|
haftmann@35603
|
3930 |
lemma map_sorted_distinct_set_unique:
|
haftmann@35603
|
3931 |
assumes "inj_on f (set xs \<union> set ys)"
|
haftmann@35603
|
3932 |
assumes "sorted (map f xs)" "distinct (map f xs)"
|
haftmann@35603
|
3933 |
"sorted (map f ys)" "distinct (map f ys)"
|
haftmann@35603
|
3934 |
assumes "set xs = set ys"
|
haftmann@35603
|
3935 |
shows "xs = ys"
|
haftmann@35603
|
3936 |
proof -
|
haftmann@35603
|
3937 |
from assms have "map f xs = map f ys"
|
haftmann@35603
|
3938 |
by (simp add: sorted_distinct_set_unique)
|
haftmann@35603
|
3939 |
moreover with `inj_on f (set xs \<union> set ys)` show "xs = ys"
|
haftmann@35603
|
3940 |
by (blast intro: map_inj_on)
|
haftmann@35603
|
3941 |
qed
|
haftmann@35603
|
3942 |
|
nipkow@24645
|
3943 |
lemma finite_sorted_distinct_unique:
|
nipkow@24645
|
3944 |
shows "finite A \<Longrightarrow> EX! xs. set xs = A & sorted xs & distinct xs"
|
nipkow@24645
|
3945 |
apply(drule finite_distinct_list)
|
nipkow@24645
|
3946 |
apply clarify
|
nipkow@24645
|
3947 |
apply(rule_tac a="sort xs" in ex1I)
|
nipkow@24645
|
3948 |
apply (auto simp: sorted_distinct_set_unique)
|
nipkow@24645
|
3949 |
done
|
nipkow@24645
|
3950 |
|
haftmann@40096
|
3951 |
lemma
|
haftmann@40096
|
3952 |
assumes "sorted xs"
|
haftmann@40096
|
3953 |
shows sorted_take: "sorted (take n xs)"
|
haftmann@40096
|
3954 |
and sorted_drop: "sorted (drop n xs)"
|
haftmann@40096
|
3955 |
proof -
|
haftmann@40096
|
3956 |
from assms have "sorted (take n xs @ drop n xs)" by simp
|
haftmann@40096
|
3957 |
then show "sorted (take n xs)" and "sorted (drop n xs)"
|
haftmann@40096
|
3958 |
unfolding sorted_append by simp_all
|
haftmann@29626
|
3959 |
qed
|
haftmann@29626
|
3960 |
|
hoelzl@33639
|
3961 |
lemma sorted_dropWhile: "sorted xs \<Longrightarrow> sorted (dropWhile P xs)"
|
haftmann@40096
|
3962 |
by (auto dest: sorted_drop simp add: dropWhile_eq_drop)
|
hoelzl@33639
|
3963 |
|
hoelzl@33639
|
3964 |
lemma sorted_takeWhile: "sorted xs \<Longrightarrow> sorted (takeWhile P xs)"
|
haftmann@40096
|
3965 |
by (subst takeWhile_eq_take) (auto dest: sorted_take)
|
haftmann@29626
|
3966 |
|
hoelzl@34920
|
3967 |
lemma sorted_filter:
|
hoelzl@34920
|
3968 |
"sorted (map f xs) \<Longrightarrow> sorted (map f (filter P xs))"
|
hoelzl@34920
|
3969 |
by (induct xs) (simp_all add: sorted_Cons)
|
hoelzl@34920
|
3970 |
|
hoelzl@34920
|
3971 |
lemma foldr_max_sorted:
|
hoelzl@34920
|
3972 |
assumes "sorted (rev xs)"
|
hoelzl@34920
|
3973 |
shows "foldr max xs y = (if xs = [] then y else max (xs ! 0) y)"
|
hoelzl@34920
|
3974 |
using assms proof (induct xs)
|
hoelzl@34920
|
3975 |
case (Cons x xs)
|
hoelzl@34920
|
3976 |
moreover hence "sorted (rev xs)" using sorted_append by auto
|
hoelzl@34920
|
3977 |
ultimately show ?case
|
hoelzl@34920
|
3978 |
by (cases xs, auto simp add: sorted_append max_def)
|
hoelzl@34920
|
3979 |
qed simp
|
hoelzl@34920
|
3980 |
|
hoelzl@34920
|
3981 |
lemma filter_equals_takeWhile_sorted_rev:
|
hoelzl@34920
|
3982 |
assumes sorted: "sorted (rev (map f xs))"
|
hoelzl@34920
|
3983 |
shows "[x \<leftarrow> xs. t < f x] = takeWhile (\<lambda> x. t < f x) xs"
|
hoelzl@34920
|
3984 |
(is "filter ?P xs = ?tW")
|
hoelzl@34920
|
3985 |
proof (rule takeWhile_eq_filter[symmetric])
|
hoelzl@34920
|
3986 |
let "?dW" = "dropWhile ?P xs"
|
hoelzl@34920
|
3987 |
fix x assume "x \<in> set ?dW"
|
hoelzl@34920
|
3988 |
then obtain i where i: "i < length ?dW" and nth_i: "x = ?dW ! i"
|
hoelzl@34920
|
3989 |
unfolding in_set_conv_nth by auto
|
hoelzl@34920
|
3990 |
hence "length ?tW + i < length (?tW @ ?dW)"
|
hoelzl@34920
|
3991 |
unfolding length_append by simp
|
hoelzl@34920
|
3992 |
hence i': "length (map f ?tW) + i < length (map f xs)" by simp
|
hoelzl@34920
|
3993 |
have "(map f ?tW @ map f ?dW) ! (length (map f ?tW) + i) \<le>
|
hoelzl@34920
|
3994 |
(map f ?tW @ map f ?dW) ! (length (map f ?tW) + 0)"
|
hoelzl@34920
|
3995 |
using sorted_rev_nth_mono[OF sorted _ i', of "length ?tW"]
|
hoelzl@34920
|
3996 |
unfolding map_append[symmetric] by simp
|
hoelzl@34920
|
3997 |
hence "f x \<le> f (?dW ! 0)"
|
hoelzl@34920
|
3998 |
unfolding nth_append_length_plus nth_i
|
hoelzl@34920
|
3999 |
using i preorder_class.le_less_trans[OF le0 i] by simp
|
hoelzl@34920
|
4000 |
also have "... \<le> t"
|
hoelzl@34920
|
4001 |
using hd_dropWhile[of "?P" xs] le0[THEN preorder_class.le_less_trans, OF i]
|
hoelzl@34920
|
4002 |
using hd_conv_nth[of "?dW"] by simp
|
hoelzl@34920
|
4003 |
finally show "\<not> t < f x" by simp
|
hoelzl@34920
|
4004 |
qed
|
hoelzl@34920
|
4005 |
|
haftmann@40451
|
4006 |
lemma insort_insert_key_triv:
|
haftmann@40451
|
4007 |
"f x \<in> f ` set xs \<Longrightarrow> insort_insert_key f x xs = xs"
|
haftmann@40451
|
4008 |
by (simp add: insort_insert_key_def)
|
haftmann@40451
|
4009 |
|
haftmann@40451
|
4010 |
lemma insort_insert_triv:
|
haftmann@40451
|
4011 |
"x \<in> set xs \<Longrightarrow> insort_insert x xs = xs"
|
haftmann@40451
|
4012 |
using insort_insert_key_triv [of "\<lambda>x. x"] by simp
|
haftmann@40451
|
4013 |
|
haftmann@40451
|
4014 |
lemma insort_insert_insort_key:
|
haftmann@40451
|
4015 |
"f x \<notin> f ` set xs \<Longrightarrow> insort_insert_key f x xs = insort_key f x xs"
|
haftmann@40451
|
4016 |
by (simp add: insort_insert_key_def)
|
haftmann@40451
|
4017 |
|
haftmann@40451
|
4018 |
lemma insort_insert_insort:
|
haftmann@40451
|
4019 |
"x \<notin> set xs \<Longrightarrow> insort_insert x xs = insort x xs"
|
haftmann@40451
|
4020 |
using insort_insert_insort_key [of "\<lambda>x. x"] by simp
|
haftmann@40451
|
4021 |
|
haftmann@35608
|
4022 |
lemma set_insort_insert:
|
haftmann@35608
|
4023 |
"set (insort_insert x xs) = insert x (set xs)"
|
haftmann@40451
|
4024 |
by (auto simp add: insort_insert_key_def set_insort)
|
haftmann@35608
|
4025 |
|
haftmann@35608
|
4026 |
lemma distinct_insort_insert:
|
haftmann@35608
|
4027 |
assumes "distinct xs"
|
haftmann@40451
|
4028 |
shows "distinct (insort_insert_key f x xs)"
|
haftmann@40451
|
4029 |
using assms by (induct xs) (auto simp add: insort_insert_key_def set_insort)
|
haftmann@40451
|
4030 |
|
haftmann@40451
|
4031 |
lemma sorted_insort_insert_key:
|
haftmann@40451
|
4032 |
assumes "sorted (map f xs)"
|
haftmann@40451
|
4033 |
shows "sorted (map f (insort_insert_key f x xs))"
|
haftmann@40451
|
4034 |
using assms by (simp add: insort_insert_key_def sorted_insort_key)
|
haftmann@35608
|
4035 |
|
haftmann@35608
|
4036 |
lemma sorted_insort_insert:
|
haftmann@35608
|
4037 |
assumes "sorted xs"
|
haftmann@35608
|
4038 |
shows "sorted (insort_insert x xs)"
|
haftmann@40451
|
4039 |
using assms sorted_insort_insert_key [of "\<lambda>x. x"] by simp
|
haftmann@40451
|
4040 |
|
haftmann@40451
|
4041 |
lemma filter_insort_triv:
|
haftmann@37091
|
4042 |
"\<not> P x \<Longrightarrow> filter P (insort_key f x xs) = filter P xs"
|
haftmann@37091
|
4043 |
by (induct xs) simp_all
|
haftmann@37091
|
4044 |
|
haftmann@40451
|
4045 |
lemma filter_insort:
|
haftmann@37091
|
4046 |
"sorted (map f xs) \<Longrightarrow> P x \<Longrightarrow> filter P (insort_key f x xs) = insort_key f x (filter P xs)"
|
haftmann@37091
|
4047 |
using assms by (induct xs)
|
haftmann@37091
|
4048 |
(auto simp add: sorted_Cons, subst insort_is_Cons, auto)
|
haftmann@37091
|
4049 |
|
haftmann@40451
|
4050 |
lemma filter_sort:
|
haftmann@37091
|
4051 |
"filter P (sort_key f xs) = sort_key f (filter P xs)"
|
haftmann@40451
|
4052 |
by (induct xs) (simp_all add: filter_insort_triv filter_insort)
|
haftmann@37091
|
4053 |
|
haftmann@40547
|
4054 |
lemma sorted_map_same:
|
haftmann@40547
|
4055 |
"sorted (map f [x\<leftarrow>xs. f x = g xs])"
|
haftmann@40547
|
4056 |
proof (induct xs arbitrary: g)
|
haftmann@37091
|
4057 |
case Nil then show ?case by simp
|
haftmann@37091
|
4058 |
next
|
haftmann@37091
|
4059 |
case (Cons x xs)
|
haftmann@40547
|
4060 |
then have "sorted (map f [y\<leftarrow>xs . f y = (\<lambda>xs. f x) xs])" .
|
haftmann@40547
|
4061 |
moreover from Cons have "sorted (map f [y\<leftarrow>xs . f y = (g \<circ> Cons x) xs])" .
|
haftmann@37091
|
4062 |
ultimately show ?case by (simp_all add: sorted_Cons)
|
haftmann@37091
|
4063 |
qed
|
haftmann@37091
|
4064 |
|
haftmann@40547
|
4065 |
lemma sorted_same:
|
haftmann@40547
|
4066 |
"sorted [x\<leftarrow>xs. x = g xs]"
|
haftmann@40547
|
4067 |
using sorted_map_same [of "\<lambda>x. x"] by simp
|
haftmann@40547
|
4068 |
|
haftmann@37091
|
4069 |
lemma remove1_insort [simp]:
|
haftmann@37091
|
4070 |
"remove1 x (insort x xs) = xs"
|
haftmann@37091
|
4071 |
by (induct xs) simp_all
|
haftmann@37091
|
4072 |
|
nipkow@24616
|
4073 |
end
|
nipkow@24616
|
4074 |
|
nipkow@25277
|
4075 |
lemma sorted_upt[simp]: "sorted[i..<j]"
|
nipkow@25277
|
4076 |
by (induct j) (simp_all add:sorted_append)
|
nipkow@25277
|
4077 |
|
nipkow@32415
|
4078 |
lemma sorted_upto[simp]: "sorted[i..j]"
|
nipkow@32415
|
4079 |
apply(induct i j rule:upto.induct)
|
nipkow@32415
|
4080 |
apply(subst upto.simps)
|
nipkow@32415
|
4081 |
apply(simp add:sorted_Cons)
|
nipkow@32415
|
4082 |
done
|
nipkow@32415
|
4083 |
|
wenzelm@35118
|
4084 |
|
wenzelm@35118
|
4085 |
subsubsection {* @{const transpose} on sorted lists *}
|
hoelzl@34920
|
4086 |
|
hoelzl@34920
|
4087 |
lemma sorted_transpose[simp]:
|
hoelzl@34920
|
4088 |
shows "sorted (rev (map length (transpose xs)))"
|
hoelzl@34920
|
4089 |
by (auto simp: sorted_equals_nth_mono rev_nth nth_transpose
|
hoelzl@34920
|
4090 |
length_filter_conv_card intro: card_mono)
|
hoelzl@34920
|
4091 |
|
hoelzl@34920
|
4092 |
lemma transpose_max_length:
|
hoelzl@34920
|
4093 |
"foldr (\<lambda>xs. max (length xs)) (transpose xs) 0 = length [x \<leftarrow> xs. x \<noteq> []]"
|
hoelzl@34920
|
4094 |
(is "?L = ?R")
|
hoelzl@34920
|
4095 |
proof (cases "transpose xs = []")
|
hoelzl@34920
|
4096 |
case False
|
hoelzl@34920
|
4097 |
have "?L = foldr max (map length (transpose xs)) 0"
|
hoelzl@34920
|
4098 |
by (simp add: foldr_map comp_def)
|
hoelzl@34920
|
4099 |
also have "... = length (transpose xs ! 0)"
|
hoelzl@34920
|
4100 |
using False sorted_transpose by (simp add: foldr_max_sorted)
|
hoelzl@34920
|
4101 |
finally show ?thesis
|
hoelzl@34920
|
4102 |
using False by (simp add: nth_transpose)
|
hoelzl@34920
|
4103 |
next
|
hoelzl@34920
|
4104 |
case True
|
hoelzl@34920
|
4105 |
hence "[x \<leftarrow> xs. x \<noteq> []] = []"
|
hoelzl@34920
|
4106 |
by (auto intro!: filter_False simp: transpose_empty)
|
hoelzl@34920
|
4107 |
thus ?thesis by (simp add: transpose_empty True)
|
hoelzl@34920
|
4108 |
qed
|
hoelzl@34920
|
4109 |
|
hoelzl@34920
|
4110 |
lemma length_transpose_sorted:
|
hoelzl@34920
|
4111 |
fixes xs :: "'a list list"
|
hoelzl@34920
|
4112 |
assumes sorted: "sorted (rev (map length xs))"
|
hoelzl@34920
|
4113 |
shows "length (transpose xs) = (if xs = [] then 0 else length (xs ! 0))"
|
hoelzl@34920
|
4114 |
proof (cases "xs = []")
|
hoelzl@34920
|
4115 |
case False
|
hoelzl@34920
|
4116 |
thus ?thesis
|
hoelzl@34920
|
4117 |
using foldr_max_sorted[OF sorted] False
|
hoelzl@34920
|
4118 |
unfolding length_transpose foldr_map comp_def
|
hoelzl@34920
|
4119 |
by simp
|
hoelzl@34920
|
4120 |
qed simp
|
hoelzl@34920
|
4121 |
|
hoelzl@34920
|
4122 |
lemma nth_nth_transpose_sorted[simp]:
|
hoelzl@34920
|
4123 |
fixes xs :: "'a list list"
|
hoelzl@34920
|
4124 |
assumes sorted: "sorted (rev (map length xs))"
|
hoelzl@34920
|
4125 |
and i: "i < length (transpose xs)"
|
hoelzl@34920
|
4126 |
and j: "j < length [ys \<leftarrow> xs. i < length ys]"
|
hoelzl@34920
|
4127 |
shows "transpose xs ! i ! j = xs ! j ! i"
|
hoelzl@34920
|
4128 |
using j filter_equals_takeWhile_sorted_rev[OF sorted, of i]
|
hoelzl@34920
|
4129 |
nth_transpose[OF i] nth_map[OF j]
|
hoelzl@34920
|
4130 |
by (simp add: takeWhile_nth)
|
hoelzl@34920
|
4131 |
|
hoelzl@34920
|
4132 |
lemma transpose_column_length:
|
hoelzl@34920
|
4133 |
fixes xs :: "'a list list"
|
hoelzl@34920
|
4134 |
assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
|
hoelzl@34920
|
4135 |
shows "length (filter (\<lambda>ys. i < length ys) (transpose xs)) = length (xs ! i)"
|
hoelzl@34920
|
4136 |
proof -
|
hoelzl@34920
|
4137 |
have "xs \<noteq> []" using `i < length xs` by auto
|
hoelzl@34920
|
4138 |
note filter_equals_takeWhile_sorted_rev[OF sorted, simp]
|
hoelzl@34920
|
4139 |
{ fix j assume "j \<le> i"
|
hoelzl@34920
|
4140 |
note sorted_rev_nth_mono[OF sorted, of j i, simplified, OF this `i < length xs`]
|
hoelzl@34920
|
4141 |
} note sortedE = this[consumes 1]
|
hoelzl@34920
|
4142 |
|
hoelzl@34920
|
4143 |
have "{j. j < length (transpose xs) \<and> i < length (transpose xs ! j)}
|
hoelzl@34920
|
4144 |
= {..< length (xs ! i)}"
|
hoelzl@34920
|
4145 |
proof safe
|
hoelzl@34920
|
4146 |
fix j
|
hoelzl@34920
|
4147 |
assume "j < length (transpose xs)" and "i < length (transpose xs ! j)"
|
hoelzl@34920
|
4148 |
with this(2) nth_transpose[OF this(1)]
|
hoelzl@34920
|
4149 |
have "i < length (takeWhile (\<lambda>ys. j < length ys) xs)" by simp
|
hoelzl@34920
|
4150 |
from nth_mem[OF this] takeWhile_nth[OF this]
|
hoelzl@34920
|
4151 |
show "j < length (xs ! i)" by (auto dest: set_takeWhileD)
|
hoelzl@34920
|
4152 |
next
|
hoelzl@34920
|
4153 |
fix j assume "j < length (xs ! i)"
|
hoelzl@34920
|
4154 |
thus "j < length (transpose xs)"
|
hoelzl@34920
|
4155 |
using foldr_max_sorted[OF sorted] `xs \<noteq> []` sortedE[OF le0]
|
hoelzl@34920
|
4156 |
by (auto simp: length_transpose comp_def foldr_map)
|
hoelzl@34920
|
4157 |
|
hoelzl@34920
|
4158 |
have "Suc i \<le> length (takeWhile (\<lambda>ys. j < length ys) xs)"
|
hoelzl@34920
|
4159 |
using `i < length xs` `j < length (xs ! i)` less_Suc_eq_le
|
hoelzl@34920
|
4160 |
by (auto intro!: length_takeWhile_less_P_nth dest!: sortedE)
|
hoelzl@34920
|
4161 |
with nth_transpose[OF `j < length (transpose xs)`]
|
hoelzl@34920
|
4162 |
show "i < length (transpose xs ! j)" by simp
|
hoelzl@34920
|
4163 |
qed
|
hoelzl@34920
|
4164 |
thus ?thesis by (simp add: length_filter_conv_card)
|
hoelzl@34920
|
4165 |
qed
|
hoelzl@34920
|
4166 |
|
hoelzl@34920
|
4167 |
lemma transpose_column:
|
hoelzl@34920
|
4168 |
fixes xs :: "'a list list"
|
hoelzl@34920
|
4169 |
assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
|
hoelzl@34920
|
4170 |
shows "map (\<lambda>ys. ys ! i) (filter (\<lambda>ys. i < length ys) (transpose xs))
|
hoelzl@34920
|
4171 |
= xs ! i" (is "?R = _")
|
hoelzl@34920
|
4172 |
proof (rule nth_equalityI, safe)
|
hoelzl@34920
|
4173 |
show length: "length ?R = length (xs ! i)"
|
hoelzl@34920
|
4174 |
using transpose_column_length[OF assms] by simp
|
hoelzl@34920
|
4175 |
|
hoelzl@34920
|
4176 |
fix j assume j: "j < length ?R"
|
hoelzl@34920
|
4177 |
note * = less_le_trans[OF this, unfolded length_map, OF length_filter_le]
|
hoelzl@34920
|
4178 |
from j have j_less: "j < length (xs ! i)" using length by simp
|
hoelzl@34920
|
4179 |
have i_less_tW: "Suc i \<le> length (takeWhile (\<lambda>ys. Suc j \<le> length ys) xs)"
|
hoelzl@34920
|
4180 |
proof (rule length_takeWhile_less_P_nth)
|
hoelzl@34920
|
4181 |
show "Suc i \<le> length xs" using `i < length xs` by simp
|
hoelzl@34920
|
4182 |
fix k assume "k < Suc i"
|
hoelzl@34920
|
4183 |
hence "k \<le> i" by auto
|
hoelzl@34920
|
4184 |
with sorted_rev_nth_mono[OF sorted this] `i < length xs`
|
hoelzl@34920
|
4185 |
have "length (xs ! i) \<le> length (xs ! k)" by simp
|
hoelzl@34920
|
4186 |
thus "Suc j \<le> length (xs ! k)" using j_less by simp
|
hoelzl@34920
|
4187 |
qed
|
hoelzl@34920
|
4188 |
have i_less_filter: "i < length [ys\<leftarrow>xs . j < length ys]"
|
hoelzl@34920
|
4189 |
unfolding filter_equals_takeWhile_sorted_rev[OF sorted, of j]
|
hoelzl@34920
|
4190 |
using i_less_tW by (simp_all add: Suc_le_eq)
|
hoelzl@34920
|
4191 |
from j show "?R ! j = xs ! i ! j"
|
hoelzl@34920
|
4192 |
unfolding filter_equals_takeWhile_sorted_rev[OF sorted_transpose, of i]
|
hoelzl@34920
|
4193 |
by (simp add: takeWhile_nth nth_nth_transpose_sorted[OF sorted * i_less_filter])
|
hoelzl@34920
|
4194 |
qed
|
hoelzl@34920
|
4195 |
|
hoelzl@34920
|
4196 |
lemma transpose_transpose:
|
hoelzl@34920
|
4197 |
fixes xs :: "'a list list"
|
hoelzl@34920
|
4198 |
assumes sorted: "sorted (rev (map length xs))"
|
hoelzl@34920
|
4199 |
shows "transpose (transpose xs) = takeWhile (\<lambda>x. x \<noteq> []) xs" (is "?L = ?R")
|
hoelzl@34920
|
4200 |
proof -
|
hoelzl@34920
|
4201 |
have len: "length ?L = length ?R"
|
hoelzl@34920
|
4202 |
unfolding length_transpose transpose_max_length
|
hoelzl@34920
|
4203 |
using filter_equals_takeWhile_sorted_rev[OF sorted, of 0]
|
hoelzl@34920
|
4204 |
by simp
|
hoelzl@34920
|
4205 |
|
hoelzl@34920
|
4206 |
{ fix i assume "i < length ?R"
|
hoelzl@34920
|
4207 |
with less_le_trans[OF _ length_takeWhile_le[of _ xs]]
|
hoelzl@34920
|
4208 |
have "i < length xs" by simp
|
hoelzl@34920
|
4209 |
} note * = this
|
hoelzl@34920
|
4210 |
show ?thesis
|
hoelzl@34920
|
4211 |
by (rule nth_equalityI)
|
hoelzl@34920
|
4212 |
(simp_all add: len nth_transpose transpose_column[OF sorted] * takeWhile_nth)
|
hoelzl@34920
|
4213 |
qed
|
haftmann@34064
|
4214 |
|
hoelzl@34921
|
4215 |
theorem transpose_rectangle:
|
hoelzl@34921
|
4216 |
assumes "xs = [] \<Longrightarrow> n = 0"
|
hoelzl@34921
|
4217 |
assumes rect: "\<And> i. i < length xs \<Longrightarrow> length (xs ! i) = n"
|
hoelzl@34921
|
4218 |
shows "transpose xs = map (\<lambda> i. map (\<lambda> j. xs ! j ! i) [0..<length xs]) [0..<n]"
|
hoelzl@34921
|
4219 |
(is "?trans = ?map")
|
hoelzl@34921
|
4220 |
proof (rule nth_equalityI)
|
hoelzl@34921
|
4221 |
have "sorted (rev (map length xs))"
|
hoelzl@34921
|
4222 |
by (auto simp: rev_nth rect intro!: sorted_nth_monoI)
|
hoelzl@34921
|
4223 |
from foldr_max_sorted[OF this] assms
|
hoelzl@34921
|
4224 |
show len: "length ?trans = length ?map"
|
hoelzl@34921
|
4225 |
by (simp_all add: length_transpose foldr_map comp_def)
|
hoelzl@34921
|
4226 |
moreover
|
hoelzl@34921
|
4227 |
{ fix i assume "i < n" hence "[ys\<leftarrow>xs . i < length ys] = xs"
|
hoelzl@34921
|
4228 |
using rect by (auto simp: in_set_conv_nth intro!: filter_True) }
|
hoelzl@34921
|
4229 |
ultimately show "\<forall>i < length ?trans. ?trans ! i = ?map ! i"
|
hoelzl@34921
|
4230 |
by (auto simp: nth_transpose intro: nth_equalityI)
|
hoelzl@34921
|
4231 |
qed
|
nipkow@24616
|
4232 |
|
wenzelm@35118
|
4233 |
|
nipkow@25069
|
4234 |
subsubsection {* @{text sorted_list_of_set} *}
|
nipkow@25069
|
4235 |
|
nipkow@25069
|
4236 |
text{* This function maps (finite) linearly ordered sets to sorted
|
nipkow@25069
|
4237 |
lists. Warning: in most cases it is not a good idea to convert from
|
nipkow@25069
|
4238 |
sets to lists but one should convert in the other direction (via
|
nipkow@25069
|
4239 |
@{const set}). *}
|
nipkow@25069
|
4240 |
|
nipkow@25069
|
4241 |
context linorder
|
nipkow@25069
|
4242 |
begin
|
nipkow@25069
|
4243 |
|
haftmann@35195
|
4244 |
definition sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where
|
haftmann@35195
|
4245 |
"sorted_list_of_set = Finite_Set.fold insort []"
|
haftmann@35195
|
4246 |
|
haftmann@35195
|
4247 |
lemma sorted_list_of_set_empty [simp]:
|
haftmann@35195
|
4248 |
"sorted_list_of_set {} = []"
|
haftmann@35195
|
4249 |
by (simp add: sorted_list_of_set_def)
|
haftmann@35195
|
4250 |
|
haftmann@35195
|
4251 |
lemma sorted_list_of_set_insert [simp]:
|
haftmann@35195
|
4252 |
assumes "finite A"
|
haftmann@35195
|
4253 |
shows "sorted_list_of_set (insert x A) = insort x (sorted_list_of_set (A - {x}))"
|
haftmann@35195
|
4254 |
proof -
|
haftmann@43740
|
4255 |
interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
|
haftmann@35195
|
4256 |
with assms show ?thesis by (simp add: sorted_list_of_set_def fold_insert_remove)
|
haftmann@35195
|
4257 |
qed
|
haftmann@35195
|
4258 |
|
haftmann@35195
|
4259 |
lemma sorted_list_of_set [simp]:
|
haftmann@35195
|
4260 |
"finite A \<Longrightarrow> set (sorted_list_of_set A) = A \<and> sorted (sorted_list_of_set A)
|
haftmann@35195
|
4261 |
\<and> distinct (sorted_list_of_set A)"
|
haftmann@35195
|
4262 |
by (induct A rule: finite_induct) (simp_all add: set_insort sorted_insort distinct_insort)
|
haftmann@35195
|
4263 |
|
haftmann@35195
|
4264 |
lemma sorted_list_of_set_sort_remdups:
|
haftmann@35195
|
4265 |
"sorted_list_of_set (set xs) = sort (remdups xs)"
|
haftmann@35195
|
4266 |
proof -
|
haftmann@43740
|
4267 |
interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
|
haftmann@35195
|
4268 |
show ?thesis by (simp add: sort_foldl_insort sorted_list_of_set_def fold_set_remdups)
|
haftmann@35195
|
4269 |
qed
|
nipkow@25069
|
4270 |
|
haftmann@37091
|
4271 |
lemma sorted_list_of_set_remove:
|
haftmann@37091
|
4272 |
assumes "finite A"
|
haftmann@37091
|
4273 |
shows "sorted_list_of_set (A - {x}) = remove1 x (sorted_list_of_set A)"
|
haftmann@37091
|
4274 |
proof (cases "x \<in> A")
|
haftmann@37091
|
4275 |
case False with assms have "x \<notin> set (sorted_list_of_set A)" by simp
|
haftmann@37091
|
4276 |
with False show ?thesis by (simp add: remove1_idem)
|
haftmann@37091
|
4277 |
next
|
haftmann@37091
|
4278 |
case True then obtain B where A: "A = insert x B" by (rule Set.set_insert)
|
haftmann@37091
|
4279 |
with assms show ?thesis by simp
|
haftmann@37091
|
4280 |
qed
|
haftmann@37091
|
4281 |
|
nipkow@25069
|
4282 |
end
|
nipkow@25069
|
4283 |
|
haftmann@37091
|
4284 |
lemma sorted_list_of_set_range [simp]:
|
haftmann@37091
|
4285 |
"sorted_list_of_set {m..<n} = [m..<n]"
|
haftmann@37091
|
4286 |
by (rule sorted_distinct_set_unique) simp_all
|
haftmann@37091
|
4287 |
|
haftmann@37091
|
4288 |
|
nipkow@15392
|
4289 |
subsubsection {* @{text lists}: the list-forming operator over sets *}
|
nipkow@15302
|
4290 |
|
berghofe@23740
|
4291 |
inductive_set
|
berghofe@23740
|
4292 |
lists :: "'a set => 'a list set"
|
berghofe@23740
|
4293 |
for A :: "'a set"
|
berghofe@22262
|
4294 |
where
|
nipkow@39859
|
4295 |
Nil [intro!, simp]: "[]: lists A"
|
nipkow@39859
|
4296 |
| Cons [intro!, simp, no_atp]: "[| a: A; l: lists A|] ==> a#l : lists A"
|
blanchet@35828
|
4297 |
|
blanchet@35828
|
4298 |
inductive_cases listsE [elim!,no_atp]: "x#l : lists A"
|
blanchet@35828
|
4299 |
inductive_cases listspE [elim!,no_atp]: "listsp A (x # l)"
|
berghofe@23740
|
4300 |
|
berghofe@23740
|
4301 |
lemma listsp_mono [mono]: "A \<le> B ==> listsp A \<le> listsp B"
|
haftmann@34064
|
4302 |
by (rule predicate1I, erule listsp.induct, (blast dest: predicate1D)+)
|
berghofe@26795
|
4303 |
|
berghofe@26795
|
4304 |
lemmas lists_mono = listsp_mono [to_set pred_subset_eq]
|
berghofe@22262
|
4305 |
|
haftmann@22422
|
4306 |
lemma listsp_infI:
|
haftmann@22422
|
4307 |
assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
|
nipkow@24349
|
4308 |
by induct blast+
|
nipkow@15302
|
4309 |
|
haftmann@22422
|
4310 |
lemmas lists_IntI = listsp_infI [to_set]
|
haftmann@22422
|
4311 |
|
haftmann@22422
|
4312 |
lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
|
haftmann@22422
|
4313 |
proof (rule mono_inf [where f=listsp, THEN order_antisym])
|
berghofe@22262
|
4314 |
show "mono listsp" by (simp add: mono_def listsp_mono)
|
berghofe@26795
|
4315 |
show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro!: listsp_infI predicate1I)
|
nipkow@15302
|
4316 |
qed
|
nipkow@15302
|
4317 |
|
haftmann@41323
|
4318 |
lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_def inf_bool_def]
|
haftmann@22422
|
4319 |
|
berghofe@26795
|
4320 |
lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set pred_equals_eq]
|
berghofe@22262
|
4321 |
|
nipkow@39859
|
4322 |
lemma Cons_in_lists_iff[simp]: "x#xs : lists A \<longleftrightarrow> x:A \<and> xs : lists A"
|
nipkow@39859
|
4323 |
by auto
|
nipkow@39859
|
4324 |
|
berghofe@22262
|
4325 |
lemma append_in_listsp_conv [iff]:
|
berghofe@22262
|
4326 |
"(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
|
nipkow@15302
|
4327 |
by (induct xs) auto
|
nipkow@15302
|
4328 |
|
berghofe@22262
|
4329 |
lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]
|
berghofe@22262
|
4330 |
|
berghofe@22262
|
4331 |
lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
|
berghofe@22262
|
4332 |
-- {* eliminate @{text listsp} in favour of @{text set} *}
|
nipkow@15302
|
4333 |
by (induct xs) auto
|
nipkow@15302
|
4334 |
|
berghofe@22262
|
4335 |
lemmas in_lists_conv_set = in_listsp_conv_set [to_set]
|
berghofe@22262
|
4336 |
|
blanchet@35828
|
4337 |
lemma in_listspD [dest!,no_atp]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
|
berghofe@22262
|
4338 |
by (rule in_listsp_conv_set [THEN iffD1])
|
berghofe@22262
|
4339 |
|
blanchet@35828
|
4340 |
lemmas in_listsD [dest!,no_atp] = in_listspD [to_set]
|
blanchet@35828
|
4341 |
|
blanchet@35828
|
4342 |
lemma in_listspI [intro!,no_atp]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
|
berghofe@22262
|
4343 |
by (rule in_listsp_conv_set [THEN iffD2])
|
berghofe@22262
|
4344 |
|
blanchet@35828
|
4345 |
lemmas in_listsI [intro!,no_atp] = in_listspI [to_set]
|
nipkow@15302
|
4346 |
|
nipkow@39821
|
4347 |
lemma lists_eq_set: "lists A = {xs. set xs <= A}"
|
nipkow@39821
|
4348 |
by auto
|
nipkow@39821
|
4349 |
|
nipkow@39859
|
4350 |
lemma lists_empty [simp]: "lists {} = {[]}"
|
nipkow@39859
|
4351 |
by auto
|
nipkow@39859
|
4352 |
|
nipkow@15302
|
4353 |
lemma lists_UNIV [simp]: "lists UNIV = UNIV"
|
nipkow@15302
|
4354 |
by auto
|
nipkow@15302
|
4355 |
|
nipkow@17086
|
4356 |
|
wenzelm@35118
|
4357 |
subsubsection {* Inductive definition for membership *}
|
nipkow@17086
|
4358 |
|
berghofe@23740
|
4359 |
inductive ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
|
berghofe@22262
|
4360 |
where
|
berghofe@22262
|
4361 |
elem: "ListMem x (x # xs)"
|
berghofe@22262
|
4362 |
| insert: "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"
|
berghofe@22262
|
4363 |
|
berghofe@22262
|
4364 |
lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
|
nipkow@17086
|
4365 |
apply (rule iffI)
|
nipkow@17086
|
4366 |
apply (induct set: ListMem)
|
nipkow@17086
|
4367 |
apply auto
|
nipkow@17086
|
4368 |
apply (induct xs)
|
nipkow@17086
|
4369 |
apply (auto intro: ListMem.intros)
|
nipkow@17086
|
4370 |
done
|
nipkow@17086
|
4371 |
|
nipkow@17086
|
4372 |
|
wenzelm@35118
|
4373 |
subsubsection {* Lists as Cartesian products *}
|
nipkow@15302
|
4374 |
|
nipkow@15302
|
4375 |
text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
|
nipkow@15302
|
4376 |
@{term A} and tail drawn from @{term Xs}.*}
|
nipkow@15302
|
4377 |
|
haftmann@34928
|
4378 |
definition
|
haftmann@34928
|
4379 |
set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set" where
|
haftmann@37767
|
4380 |
"set_Cons A XS = {z. \<exists>x xs. z = x # xs \<and> x \<in> A \<and> xs \<in> XS}"
|
nipkow@15302
|
4381 |
|
paulson@17724
|
4382 |
lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
|
nipkow@15302
|
4383 |
by (auto simp add: set_Cons_def)
|
nipkow@15302
|
4384 |
|
nipkow@15302
|
4385 |
text{*Yields the set of lists, all of the same length as the argument and
|
nipkow@15302
|
4386 |
with elements drawn from the corresponding element of the argument.*}
|
nipkow@15302
|
4387 |
|
nipkow@15302
|
4388 |
primrec
|
haftmann@34928
|
4389 |
listset :: "'a set list \<Rightarrow> 'a list set" where
|
haftmann@34928
|
4390 |
"listset [] = {[]}"
|
haftmann@34928
|
4391 |
| "listset (A # As) = set_Cons A (listset As)"
|
nipkow@15302
|
4392 |
|
nipkow@15302
|
4393 |
|
wenzelm@35118
|
4394 |
subsection {* Relations on Lists *}
|
paulson@15656
|
4395 |
|
paulson@15656
|
4396 |
subsubsection {* Length Lexicographic Ordering *}
|
paulson@15656
|
4397 |
|
paulson@15656
|
4398 |
text{*These orderings preserve well-foundedness: shorter lists
|
paulson@15656
|
4399 |
precede longer lists. These ordering are not used in dictionaries.*}
|
haftmann@34928
|
4400 |
|
haftmann@34928
|
4401 |
primrec -- {*The lexicographic ordering for lists of the specified length*}
|
haftmann@34928
|
4402 |
lexn :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> ('a list \<times> 'a list) set" where
|
haftmann@37767
|
4403 |
"lexn r 0 = {}"
|
haftmann@40856
|
4404 |
| "lexn r (Suc n) = (map_pair (%(x, xs). x#xs) (%(x, xs). x#xs) ` (r <*lex*> lexn r n)) Int
|
haftmann@34928
|
4405 |
{(xs, ys). length xs = Suc n \<and> length ys = Suc n}"
|
haftmann@34928
|
4406 |
|
haftmann@34928
|
4407 |
definition
|
haftmann@34928
|
4408 |
lex :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
|
haftmann@37767
|
4409 |
"lex r = (\<Union>n. lexn r n)" -- {*Holds only between lists of the same length*}
|
haftmann@34928
|
4410 |
|
haftmann@34928
|
4411 |
definition
|
haftmann@34928
|
4412 |
lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" where
|
haftmann@37767
|
4413 |
"lenlex r = inv_image (less_than <*lex*> lex r) (\<lambda>xs. (length xs, xs))"
|
haftmann@34928
|
4414 |
-- {*Compares lists by their length and then lexicographically*}
|
nipkow@15302
|
4415 |
|
wenzelm@13142
|
4416 |
lemma wf_lexn: "wf r ==> wf (lexn r n)"
|
paulson@15251
|
4417 |
apply (induct n, simp, simp)
|
nipkow@13145
|
4418 |
apply(rule wf_subset)
|
nipkow@13145
|
4419 |
prefer 2 apply (rule Int_lower1)
|
haftmann@40856
|
4420 |
apply(rule wf_map_pair_image)
|
paulson@14208
|
4421 |
prefer 2 apply (rule inj_onI, auto)
|
nipkow@13145
|
4422 |
done
|
wenzelm@13114
|
4423 |
|
wenzelm@13114
|
4424 |
lemma lexn_length:
|
nipkow@24526
|
4425 |
"(xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
|
nipkow@24526
|
4426 |
by (induct n arbitrary: xs ys) auto
|
wenzelm@13114
|
4427 |
|
wenzelm@13142
|
4428 |
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
|
nipkow@13145
|
4429 |
apply (unfold lex_def)
|
nipkow@13145
|
4430 |
apply (rule wf_UN)
|
paulson@14208
|
4431 |
apply (blast intro: wf_lexn, clarify)
|
nipkow@13145
|
4432 |
apply (rename_tac m n)
|
nipkow@13145
|
4433 |
apply (subgoal_tac "m \<noteq> n")
|
nipkow@13145
|
4434 |
prefer 2 apply blast
|
nipkow@13145
|
4435 |
apply (blast dest: lexn_length not_sym)
|
nipkow@13145
|
4436 |
done
|
wenzelm@13114
|
4437 |
|
wenzelm@13114
|
4438 |
lemma lexn_conv:
|
paulson@15656
|
4439 |
"lexn r n =
|
paulson@15656
|
4440 |
{(xs,ys). length xs = n \<and> length ys = n \<and>
|
paulson@15656
|
4441 |
(\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
|
nipkow@18423
|
4442 |
apply (induct n, simp)
|
paulson@14208
|
4443 |
apply (simp add: image_Collect lex_prod_def, safe, blast)
|
paulson@14208
|
4444 |
apply (rule_tac x = "ab # xys" in exI, simp)
|
paulson@14208
|
4445 |
apply (case_tac xys, simp_all, blast)
|
nipkow@13145
|
4446 |
done
|
wenzelm@13114
|
4447 |
|
wenzelm@13114
|
4448 |
lemma lex_conv:
|
paulson@15656
|
4449 |
"lex r =
|
paulson@15656
|
4450 |
{(xs,ys). length xs = length ys \<and>
|
paulson@15656
|
4451 |
(\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
|
nipkow@13145
|
4452 |
by (force simp add: lex_def lexn_conv)
|
wenzelm@13114
|
4453 |
|
nipkow@15693
|
4454 |
lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
|
nipkow@15693
|
4455 |
by (unfold lenlex_def) blast
|
nipkow@15693
|
4456 |
|
nipkow@15693
|
4457 |
lemma lenlex_conv:
|
nipkow@15693
|
4458 |
"lenlex r = {(xs,ys). length xs < length ys |
|
paulson@15656
|
4459 |
length xs = length ys \<and> (xs, ys) : lex r}"
|
nipkow@30198
|
4460 |
by (simp add: lenlex_def Id_on_def lex_prod_def inv_image_def)
|
wenzelm@13114
|
4461 |
|
wenzelm@13142
|
4462 |
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
|
nipkow@13145
|
4463 |
by (simp add: lex_conv)
|
wenzelm@13114
|
4464 |
|
wenzelm@13142
|
4465 |
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
|
nipkow@13145
|
4466 |
by (simp add:lex_conv)
|
wenzelm@13114
|
4467 |
|
paulson@18447
|
4468 |
lemma Cons_in_lex [simp]:
|
paulson@15656
|
4469 |
"((x # xs, y # ys) : lex r) =
|
paulson@15656
|
4470 |
((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
|
nipkow@13145
|
4471 |
apply (simp add: lex_conv)
|
nipkow@13145
|
4472 |
apply (rule iffI)
|
paulson@14208
|
4473 |
prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
|
paulson@14208
|
4474 |
apply (case_tac xys, simp, simp)
|
nipkow@13145
|
4475 |
apply blast
|
nipkow@13145
|
4476 |
done
|
wenzelm@13114
|
4477 |
|
wenzelm@13114
|
4478 |
|
paulson@15656
|
4479 |
subsubsection {* Lexicographic Ordering *}
|
paulson@15656
|
4480 |
|
paulson@15656
|
4481 |
text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
|
paulson@15656
|
4482 |
This ordering does \emph{not} preserve well-foundedness.
|
nipkow@17090
|
4483 |
Author: N. Voelker, March 2005. *}
|
paulson@15656
|
4484 |
|
haftmann@34928
|
4485 |
definition
|
haftmann@34928
|
4486 |
lexord :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
|
haftmann@37767
|
4487 |
"lexord r = {(x,y ). \<exists> a v. y = x @ a # v \<or>
|
paulson@15656
|
4488 |
(\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
|
paulson@15656
|
4489 |
|
paulson@15656
|
4490 |
lemma lexord_Nil_left[simp]: "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
|
nipkow@24349
|
4491 |
by (unfold lexord_def, induct_tac y, auto)
|
paulson@15656
|
4492 |
|
paulson@15656
|
4493 |
lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
|
nipkow@24349
|
4494 |
by (unfold lexord_def, induct_tac x, auto)
|
paulson@15656
|
4495 |
|
paulson@15656
|
4496 |
lemma lexord_cons_cons[simp]:
|
paulson@15656
|
4497 |
"((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
|
paulson@15656
|
4498 |
apply (unfold lexord_def, safe, simp_all)
|
paulson@15656
|
4499 |
apply (case_tac u, simp, simp)
|
paulson@15656
|
4500 |
apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
|
paulson@15656
|
4501 |
apply (erule_tac x="b # u" in allE)
|
paulson@15656
|
4502 |
by force
|
paulson@15656
|
4503 |
|
paulson@15656
|
4504 |
lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
|
paulson@15656
|
4505 |
|
paulson@15656
|
4506 |
lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
|
nipkow@24349
|
4507 |
by (induct_tac x, auto)
|
paulson@15656
|
4508 |
|
paulson@15656
|
4509 |
lemma lexord_append_left_rightI:
|
paulson@15656
|
4510 |
"(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
|
nipkow@24349
|
4511 |
by (induct_tac u, auto)
|
paulson@15656
|
4512 |
|
paulson@15656
|
4513 |
lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
|
nipkow@24349
|
4514 |
by (induct x, auto)
|
paulson@15656
|
4515 |
|
paulson@15656
|
4516 |
lemma lexord_append_leftD:
|
paulson@15656
|
4517 |
"\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
|
nipkow@24349
|
4518 |
by (erule rev_mp, induct_tac x, auto)
|
paulson@15656
|
4519 |
|
paulson@15656
|
4520 |
lemma lexord_take_index_conv:
|
paulson@15656
|
4521 |
"((x,y) : lexord r) =
|
paulson@15656
|
4522 |
((length x < length y \<and> take (length x) y = x) \<or>
|
paulson@15656
|
4523 |
(\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
|
paulson@15656
|
4524 |
apply (unfold lexord_def Let_def, clarsimp)
|
paulson@15656
|
4525 |
apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
|
paulson@15656
|
4526 |
apply auto
|
paulson@15656
|
4527 |
apply (rule_tac x="hd (drop (length x) y)" in exI)
|
paulson@15656
|
4528 |
apply (rule_tac x="tl (drop (length x) y)" in exI)
|
paulson@15656
|
4529 |
apply (erule subst, simp add: min_def)
|
paulson@15656
|
4530 |
apply (rule_tac x ="length u" in exI, simp)
|
paulson@15656
|
4531 |
apply (rule_tac x ="take i x" in exI)
|
paulson@15656
|
4532 |
apply (rule_tac x ="x ! i" in exI)
|
paulson@15656
|
4533 |
apply (rule_tac x ="y ! i" in exI, safe)
|
paulson@15656
|
4534 |
apply (rule_tac x="drop (Suc i) x" in exI)
|
paulson@15656
|
4535 |
apply (drule sym, simp add: drop_Suc_conv_tl)
|
paulson@15656
|
4536 |
apply (rule_tac x="drop (Suc i) y" in exI)
|
paulson@15656
|
4537 |
by (simp add: drop_Suc_conv_tl)
|
paulson@15656
|
4538 |
|
paulson@15656
|
4539 |
-- {* lexord is extension of partial ordering List.lex *}
|
nipkow@42857
|
4540 |
lemma lexord_lex: "(x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
|
paulson@15656
|
4541 |
apply (rule_tac x = y in spec)
|
paulson@15656
|
4542 |
apply (induct_tac x, clarsimp)
|
paulson@15656
|
4543 |
by (clarify, case_tac x, simp, force)
|
paulson@15656
|
4544 |
|
nipkow@42857
|
4545 |
lemma lexord_irreflexive: "ALL x. (x,x) \<notin> r \<Longrightarrow> (xs,xs) \<notin> lexord r"
|
nipkow@42857
|
4546 |
by (induct xs) auto
|
nipkow@42857
|
4547 |
|
nipkow@42857
|
4548 |
text{* By Ren\'e Thiemann: *}
|
nipkow@42857
|
4549 |
lemma lexord_partial_trans:
|
nipkow@42857
|
4550 |
"(\<And>x y z. x \<in> set xs \<Longrightarrow> (x,y) \<in> r \<Longrightarrow> (y,z) \<in> r \<Longrightarrow> (x,z) \<in> r)
|
nipkow@42857
|
4551 |
\<Longrightarrow> (xs,ys) \<in> lexord r \<Longrightarrow> (ys,zs) \<in> lexord r \<Longrightarrow> (xs,zs) \<in> lexord r"
|
nipkow@42857
|
4552 |
proof (induct xs arbitrary: ys zs)
|
nipkow@42857
|
4553 |
case Nil
|
nipkow@42857
|
4554 |
from Nil(3) show ?case unfolding lexord_def by (cases zs, auto)
|
nipkow@42857
|
4555 |
next
|
nipkow@42857
|
4556 |
case (Cons x xs yys zzs)
|
nipkow@42857
|
4557 |
from Cons(3) obtain y ys where yys: "yys = y # ys" unfolding lexord_def
|
nipkow@42857
|
4558 |
by (cases yys, auto)
|
nipkow@42857
|
4559 |
note Cons = Cons[unfolded yys]
|
nipkow@42857
|
4560 |
from Cons(3) have one: "(x,y) \<in> r \<or> x = y \<and> (xs,ys) \<in> lexord r" by auto
|
nipkow@42857
|
4561 |
from Cons(4) obtain z zs where zzs: "zzs = z # zs" unfolding lexord_def
|
nipkow@42857
|
4562 |
by (cases zzs, auto)
|
nipkow@42857
|
4563 |
note Cons = Cons[unfolded zzs]
|
nipkow@42857
|
4564 |
from Cons(4) have two: "(y,z) \<in> r \<or> y = z \<and> (ys,zs) \<in> lexord r" by auto
|
nipkow@42857
|
4565 |
{
|
nipkow@42857
|
4566 |
assume "(xs,ys) \<in> lexord r" and "(ys,zs) \<in> lexord r"
|
nipkow@42857
|
4567 |
from Cons(1)[OF _ this] Cons(2)
|
nipkow@42857
|
4568 |
have "(xs,zs) \<in> lexord r" by auto
|
nipkow@42857
|
4569 |
} note ind1 = this
|
nipkow@42857
|
4570 |
{
|
nipkow@42857
|
4571 |
assume "(x,y) \<in> r" and "(y,z) \<in> r"
|
nipkow@42857
|
4572 |
from Cons(2)[OF _ this] have "(x,z) \<in> r" by auto
|
nipkow@42857
|
4573 |
} note ind2 = this
|
nipkow@42857
|
4574 |
from one two ind1 ind2
|
nipkow@42857
|
4575 |
have "(x,z) \<in> r \<or> x = z \<and> (xs,zs) \<in> lexord r" by blast
|
nipkow@42857
|
4576 |
thus ?case unfolding zzs by auto
|
nipkow@42857
|
4577 |
qed
|
paulson@15656
|
4578 |
|
paulson@15656
|
4579 |
lemma lexord_trans:
|
paulson@15656
|
4580 |
"\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
|
nipkow@42857
|
4581 |
by(auto simp: trans_def intro:lexord_partial_trans)
|
paulson@15656
|
4582 |
|
paulson@15656
|
4583 |
lemma lexord_transI: "trans r \<Longrightarrow> trans (lexord r)"
|
nipkow@24349
|
4584 |
by (rule transI, drule lexord_trans, blast)
|
paulson@15656
|
4585 |
|
paulson@15656
|
4586 |
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
|
4587 |
apply (rule_tac x = y in spec)
|
paulson@15656
|
4588 |
apply (induct_tac x, rule allI)
|
paulson@15656
|
4589 |
apply (case_tac x, simp, simp)
|
paulson@15656
|
4590 |
apply (rule allI, case_tac x, simp, simp)
|
paulson@15656
|
4591 |
by blast
|
paulson@15656
|
4592 |
|
paulson@15656
|
4593 |
|
nipkow@40476
|
4594 |
subsubsection {* Lexicographic combination of measure functions *}
|
krauss@21103
|
4595 |
|
krauss@21103
|
4596 |
text {* These are useful for termination proofs *}
|
krauss@21103
|
4597 |
|
krauss@21103
|
4598 |
definition
|
krauss@21103
|
4599 |
"measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"
|
krauss@21103
|
4600 |
|
krauss@44884
|
4601 |
lemma wf_measures[simp]: "wf (measures fs)"
|
nipkow@24349
|
4602 |
unfolding measures_def
|
nipkow@24349
|
4603 |
by blast
|
krauss@21103
|
4604 |
|
krauss@21103
|
4605 |
lemma in_measures[simp]:
|
krauss@21103
|
4606 |
"(x, y) \<in> measures [] = False"
|
krauss@21103
|
4607 |
"(x, y) \<in> measures (f # fs)
|
krauss@21103
|
4608 |
= (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"
|
nipkow@24349
|
4609 |
unfolding measures_def
|
nipkow@24349
|
4610 |
by auto
|
krauss@21103
|
4611 |
|
krauss@21103
|
4612 |
lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
|
nipkow@24349
|
4613 |
by simp
|
krauss@21103
|
4614 |
|
krauss@21103
|
4615 |
lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
|
nipkow@24349
|
4616 |
by auto
|
krauss@21103
|
4617 |
|
krauss@21103
|
4618 |
|
nipkow@40476
|
4619 |
subsubsection {* Lifting Relations to Lists: one element *}
|
nipkow@40476
|
4620 |
|
nipkow@40476
|
4621 |
definition listrel1 :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
|
nipkow@40476
|
4622 |
"listrel1 r = {(xs,ys).
|
nipkow@40476
|
4623 |
\<exists>us z z' vs. xs = us @ z # vs \<and> (z,z') \<in> r \<and> ys = us @ z' # vs}"
|
nipkow@40476
|
4624 |
|
nipkow@40476
|
4625 |
lemma listrel1I:
|
nipkow@40476
|
4626 |
"\<lbrakk> (x, y) \<in> r; xs = us @ x # vs; ys = us @ y # vs \<rbrakk> \<Longrightarrow>
|
nipkow@40476
|
4627 |
(xs, ys) \<in> listrel1 r"
|
nipkow@40476
|
4628 |
unfolding listrel1_def by auto
|
nipkow@40476
|
4629 |
|
nipkow@40476
|
4630 |
lemma listrel1E:
|
nipkow@40476
|
4631 |
"\<lbrakk> (xs, ys) \<in> listrel1 r;
|
nipkow@40476
|
4632 |
!!x y us vs. \<lbrakk> (x, y) \<in> r; xs = us @ x # vs; ys = us @ y # vs \<rbrakk> \<Longrightarrow> P
|
nipkow@40476
|
4633 |
\<rbrakk> \<Longrightarrow> P"
|
nipkow@40476
|
4634 |
unfolding listrel1_def by auto
|
nipkow@40476
|
4635 |
|
nipkow@40476
|
4636 |
lemma not_Nil_listrel1 [iff]: "([], xs) \<notin> listrel1 r"
|
nipkow@40476
|
4637 |
unfolding listrel1_def by blast
|
nipkow@40476
|
4638 |
|
nipkow@40476
|
4639 |
lemma not_listrel1_Nil [iff]: "(xs, []) \<notin> listrel1 r"
|
nipkow@40476
|
4640 |
unfolding listrel1_def by blast
|
nipkow@40476
|
4641 |
|
nipkow@40476
|
4642 |
lemma Cons_listrel1_Cons [iff]:
|
nipkow@40476
|
4643 |
"(x # xs, y # ys) \<in> listrel1 r \<longleftrightarrow>
|
nipkow@40476
|
4644 |
(x,y) \<in> r \<and> xs = ys \<or> x = y \<and> (xs, ys) \<in> listrel1 r"
|
nipkow@40476
|
4645 |
by (simp add: listrel1_def Cons_eq_append_conv) (blast)
|
nipkow@40476
|
4646 |
|
nipkow@40476
|
4647 |
lemma listrel1I1: "(x,y) \<in> r \<Longrightarrow> (x # xs, y # xs) \<in> listrel1 r"
|
nipkow@40476
|
4648 |
by (metis Cons_listrel1_Cons)
|
nipkow@40476
|
4649 |
|
nipkow@40476
|
4650 |
lemma listrel1I2: "(xs, ys) \<in> listrel1 r \<Longrightarrow> (x # xs, x # ys) \<in> listrel1 r"
|
nipkow@40476
|
4651 |
by (metis Cons_listrel1_Cons)
|
nipkow@40476
|
4652 |
|
nipkow@40476
|
4653 |
lemma append_listrel1I:
|
nipkow@40476
|
4654 |
"(xs, ys) \<in> listrel1 r \<and> us = vs \<or> xs = ys \<and> (us, vs) \<in> listrel1 r
|
nipkow@40476
|
4655 |
\<Longrightarrow> (xs @ us, ys @ vs) \<in> listrel1 r"
|
nipkow@40476
|
4656 |
unfolding listrel1_def
|
nipkow@40476
|
4657 |
by auto (blast intro: append_eq_appendI)+
|
nipkow@40476
|
4658 |
|
nipkow@40476
|
4659 |
lemma Cons_listrel1E1[elim!]:
|
nipkow@40476
|
4660 |
assumes "(x # xs, ys) \<in> listrel1 r"
|
nipkow@40476
|
4661 |
and "\<And>y. ys = y # xs \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
|
nipkow@40476
|
4662 |
and "\<And>zs. ys = x # zs \<Longrightarrow> (xs, zs) \<in> listrel1 r \<Longrightarrow> R"
|
nipkow@40476
|
4663 |
shows R
|
nipkow@40476
|
4664 |
using assms by (cases ys) blast+
|
nipkow@40476
|
4665 |
|
nipkow@40476
|
4666 |
lemma Cons_listrel1E2[elim!]:
|
nipkow@40476
|
4667 |
assumes "(xs, y # ys) \<in> listrel1 r"
|
nipkow@40476
|
4668 |
and "\<And>x. xs = x # ys \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
|
nipkow@40476
|
4669 |
and "\<And>zs. xs = y # zs \<Longrightarrow> (zs, ys) \<in> listrel1 r \<Longrightarrow> R"
|
nipkow@40476
|
4670 |
shows R
|
nipkow@40476
|
4671 |
using assms by (cases xs) blast+
|
nipkow@40476
|
4672 |
|
nipkow@40476
|
4673 |
lemma snoc_listrel1_snoc_iff:
|
nipkow@40476
|
4674 |
"(xs @ [x], ys @ [y]) \<in> listrel1 r
|
nipkow@40476
|
4675 |
\<longleftrightarrow> (xs, ys) \<in> listrel1 r \<and> x = y \<or> xs = ys \<and> (x,y) \<in> r" (is "?L \<longleftrightarrow> ?R")
|
nipkow@40476
|
4676 |
proof
|
nipkow@40476
|
4677 |
assume ?L thus ?R
|
nipkow@45761
|
4678 |
by (fastforce simp: listrel1_def snoc_eq_iff_butlast butlast_append)
|
nipkow@40476
|
4679 |
next
|
nipkow@40476
|
4680 |
assume ?R then show ?L unfolding listrel1_def by force
|
nipkow@40476
|
4681 |
qed
|
nipkow@40476
|
4682 |
|
nipkow@40476
|
4683 |
lemma listrel1_eq_len: "(xs,ys) \<in> listrel1 r \<Longrightarrow> length xs = length ys"
|
nipkow@40476
|
4684 |
unfolding listrel1_def by auto
|
nipkow@40476
|
4685 |
|
nipkow@40476
|
4686 |
lemma listrel1_mono:
|
nipkow@40476
|
4687 |
"r \<subseteq> s \<Longrightarrow> listrel1 r \<subseteq> listrel1 s"
|
nipkow@40476
|
4688 |
unfolding listrel1_def by blast
|
nipkow@40476
|
4689 |
|
nipkow@40476
|
4690 |
|
nipkow@40476
|
4691 |
lemma listrel1_converse: "listrel1 (r^-1) = (listrel1 r)^-1"
|
nipkow@40476
|
4692 |
unfolding listrel1_def by blast
|
nipkow@40476
|
4693 |
|
nipkow@40476
|
4694 |
lemma in_listrel1_converse:
|
nipkow@40476
|
4695 |
"(x,y) : listrel1 (r^-1) \<longleftrightarrow> (x,y) : (listrel1 r)^-1"
|
nipkow@40476
|
4696 |
unfolding listrel1_def by blast
|
nipkow@40476
|
4697 |
|
nipkow@40476
|
4698 |
lemma listrel1_iff_update:
|
nipkow@40476
|
4699 |
"(xs,ys) \<in> (listrel1 r)
|
nipkow@40476
|
4700 |
\<longleftrightarrow> (\<exists>y n. (xs ! n, y) \<in> r \<and> n < length xs \<and> ys = xs[n:=y])" (is "?L \<longleftrightarrow> ?R")
|
nipkow@40476
|
4701 |
proof
|
nipkow@40476
|
4702 |
assume "?L"
|
nipkow@40476
|
4703 |
then obtain x y u v where "xs = u @ x # v" "ys = u @ y # v" "(x,y) \<in> r"
|
nipkow@40476
|
4704 |
unfolding listrel1_def by auto
|
nipkow@40476
|
4705 |
then have "ys = xs[length u := y]" and "length u < length xs"
|
nipkow@40476
|
4706 |
and "(xs ! length u, y) \<in> r" by auto
|
nipkow@40476
|
4707 |
then show "?R" by auto
|
nipkow@40476
|
4708 |
next
|
nipkow@40476
|
4709 |
assume "?R"
|
nipkow@40476
|
4710 |
then obtain x y n where "(xs!n, y) \<in> r" "n < size xs" "ys = xs[n:=y]" "x = xs!n"
|
nipkow@40476
|
4711 |
by auto
|
nipkow@40476
|
4712 |
then obtain u v where "xs = u @ x # v" and "ys = u @ y # v" and "(x, y) \<in> r"
|
nipkow@40476
|
4713 |
by (auto intro: upd_conv_take_nth_drop id_take_nth_drop)
|
nipkow@40476
|
4714 |
then show "?L" by (auto simp: listrel1_def)
|
nipkow@40476
|
4715 |
qed
|
nipkow@40476
|
4716 |
|
nipkow@40476
|
4717 |
|
nipkow@45365
|
4718 |
text{* Accessible part and wellfoundedness: *}
|
nipkow@40476
|
4719 |
|
nipkow@40476
|
4720 |
lemma Cons_acc_listrel1I [intro!]:
|
nipkow@40476
|
4721 |
"x \<in> acc r \<Longrightarrow> xs \<in> acc (listrel1 r) \<Longrightarrow> (x # xs) \<in> acc (listrel1 r)"
|
nipkow@40476
|
4722 |
apply (induct arbitrary: xs set: acc)
|
nipkow@40476
|
4723 |
apply (erule thin_rl)
|
nipkow@40476
|
4724 |
apply (erule acc_induct)
|
nipkow@40476
|
4725 |
apply (rule accI)
|
nipkow@40476
|
4726 |
apply (blast)
|
nipkow@40476
|
4727 |
done
|
nipkow@40476
|
4728 |
|
nipkow@40476
|
4729 |
lemma lists_accD: "xs \<in> lists (acc r) \<Longrightarrow> xs \<in> acc (listrel1 r)"
|
nipkow@40476
|
4730 |
apply (induct set: lists)
|
nipkow@40476
|
4731 |
apply (rule accI)
|
nipkow@40476
|
4732 |
apply simp
|
nipkow@40476
|
4733 |
apply (rule accI)
|
nipkow@40476
|
4734 |
apply (fast dest: acc_downward)
|
nipkow@40476
|
4735 |
done
|
nipkow@40476
|
4736 |
|
nipkow@40476
|
4737 |
lemma lists_accI: "xs \<in> acc (listrel1 r) \<Longrightarrow> xs \<in> lists (acc r)"
|
nipkow@40476
|
4738 |
apply (induct set: acc)
|
nipkow@40476
|
4739 |
apply clarify
|
nipkow@40476
|
4740 |
apply (rule accI)
|
nipkow@45761
|
4741 |
apply (fastforce dest!: in_set_conv_decomp[THEN iffD1] simp: listrel1_def)
|
nipkow@40476
|
4742 |
done
|
nipkow@40476
|
4743 |
|
nipkow@45365
|
4744 |
lemma wf_listrel1_iff[simp]: "wf(listrel1 r) = wf r"
|
nipkow@45365
|
4745 |
by(metis wf_acc_iff in_lists_conv_set lists_accI lists_accD Cons_in_lists_iff)
|
nipkow@45365
|
4746 |
|
nipkow@40476
|
4747 |
|
nipkow@40476
|
4748 |
subsubsection {* Lifting Relations to Lists: all elements *}
|
nipkow@15302
|
4749 |
|
berghofe@23740
|
4750 |
inductive_set
|
berghofe@23740
|
4751 |
listrel :: "('a * 'a)set => ('a list * 'a list)set"
|
berghofe@23740
|
4752 |
for r :: "('a * 'a)set"
|
berghofe@22262
|
4753 |
where
|
berghofe@23740
|
4754 |
Nil: "([],[]) \<in> listrel r"
|
berghofe@23740
|
4755 |
| Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
|
berghofe@23740
|
4756 |
|
berghofe@23740
|
4757 |
inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
|
berghofe@23740
|
4758 |
inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
|
berghofe@23740
|
4759 |
inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
|
berghofe@23740
|
4760 |
inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
|
nipkow@15302
|
4761 |
|
nipkow@15302
|
4762 |
|
nipkow@40476
|
4763 |
lemma listrel_eq_len: "(xs, ys) \<in> listrel r \<Longrightarrow> length xs = length ys"
|
nipkow@40476
|
4764 |
by(induct rule: listrel.induct) auto
|
nipkow@40476
|
4765 |
|
nipkow@40476
|
4766 |
lemma listrel_iff_zip: "(xs,ys) : listrel r \<longleftrightarrow>
|
nipkow@40476
|
4767 |
length xs = length ys & (\<forall>(x,y) \<in> set(zip xs ys). (x,y) \<in> r)" (is "?L \<longleftrightarrow> ?R")
|
nipkow@40476
|
4768 |
proof
|
nipkow@40476
|
4769 |
assume ?L thus ?R by induct (auto intro: listrel_eq_len)
|
nipkow@40476
|
4770 |
next
|
nipkow@40476
|
4771 |
assume ?R thus ?L
|
nipkow@40476
|
4772 |
apply (clarify)
|
nipkow@40476
|
4773 |
by (induct rule: list_induct2) (auto intro: listrel.intros)
|
nipkow@40476
|
4774 |
qed
|
nipkow@40476
|
4775 |
|
nipkow@40476
|
4776 |
lemma listrel_iff_nth: "(xs,ys) : listrel r \<longleftrightarrow>
|
nipkow@40476
|
4777 |
length xs = length ys & (\<forall>n < length xs. (xs!n, ys!n) \<in> r)" (is "?L \<longleftrightarrow> ?R")
|
nipkow@40476
|
4778 |
by (auto simp add: all_set_conv_all_nth listrel_iff_zip)
|
nipkow@40476
|
4779 |
|
nipkow@40476
|
4780 |
|
nipkow@15302
|
4781 |
lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
|
nipkow@15302
|
4782 |
apply clarify
|
berghofe@23740
|
4783 |
apply (erule listrel.induct)
|
berghofe@23740
|
4784 |
apply (blast intro: listrel.intros)+
|
nipkow@15281
|
4785 |
done
|
nipkow@15281
|
4786 |
|
nipkow@15302
|
4787 |
lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
|
nipkow@15302
|
4788 |
apply clarify
|
berghofe@23740
|
4789 |
apply (erule listrel.induct, auto)
|
nipkow@13145
|
4790 |
done
|
wenzelm@13114
|
4791 |
|
nipkow@30198
|
4792 |
lemma listrel_refl_on: "refl_on A r \<Longrightarrow> refl_on (lists A) (listrel r)"
|
nipkow@30198
|
4793 |
apply (simp add: refl_on_def listrel_subset Ball_def)
|
nipkow@15302
|
4794 |
apply (rule allI)
|
nipkow@15302
|
4795 |
apply (induct_tac x)
|
berghofe@23740
|
4796 |
apply (auto intro: listrel.intros)
|
nipkow@13145
|
4797 |
done
|
wenzelm@13114
|
4798 |
|
nipkow@15302
|
4799 |
lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)"
|
nipkow@15302
|
4800 |
apply (auto simp add: sym_def)
|
berghofe@23740
|
4801 |
apply (erule listrel.induct)
|
berghofe@23740
|
4802 |
apply (blast intro: listrel.intros)+
|
nipkow@15281
|
4803 |
done
|
nipkow@15281
|
4804 |
|
nipkow@15302
|
4805 |
lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)"
|
nipkow@15302
|
4806 |
apply (simp add: trans_def)
|
nipkow@15302
|
4807 |
apply (intro allI)
|
nipkow@15302
|
4808 |
apply (rule impI)
|
berghofe@23740
|
4809 |
apply (erule listrel.induct)
|
berghofe@23740
|
4810 |
apply (blast intro: listrel.intros)+
|
nipkow@15281
|
4811 |
done
|
nipkow@15281
|
4812 |
|
nipkow@15302
|
4813 |
theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
|
nipkow@30198
|
4814 |
by (simp add: equiv_def listrel_refl_on listrel_sym listrel_trans)
|
nipkow@15302
|
4815 |
|
nipkow@40476
|
4816 |
lemma listrel_rtrancl_refl[iff]: "(xs,xs) : listrel(r^*)"
|
nipkow@40476
|
4817 |
using listrel_refl_on[of UNIV, OF refl_rtrancl]
|
nipkow@40476
|
4818 |
by(auto simp: refl_on_def)
|
nipkow@40476
|
4819 |
|
nipkow@40476
|
4820 |
lemma listrel_rtrancl_trans:
|
nipkow@40476
|
4821 |
"\<lbrakk> (xs,ys) : listrel(r^*); (ys,zs) : listrel(r^*) \<rbrakk>
|
nipkow@40476
|
4822 |
\<Longrightarrow> (xs,zs) : listrel(r^*)"
|
nipkow@40476
|
4823 |
by (metis listrel_trans trans_def trans_rtrancl)
|
nipkow@40476
|
4824 |
|
nipkow@40476
|
4825 |
|
nipkow@15302
|
4826 |
lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
|
berghofe@23740
|
4827 |
by (blast intro: listrel.intros)
|
nipkow@15302
|
4828 |
|
nipkow@15302
|
4829 |
lemma listrel_Cons:
|
haftmann@33301
|
4830 |
"listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})"
|
haftmann@33301
|
4831 |
by (auto simp add: set_Cons_def intro: listrel.intros)
|
nipkow@15302
|
4832 |
|
nipkow@40476
|
4833 |
text {* Relating @{term listrel1}, @{term listrel} and closures: *}
|
nipkow@40476
|
4834 |
|
nipkow@40476
|
4835 |
lemma listrel1_rtrancl_subset_rtrancl_listrel1:
|
nipkow@40476
|
4836 |
"listrel1 (r^*) \<subseteq> (listrel1 r)^*"
|
nipkow@40476
|
4837 |
proof (rule subrelI)
|
nipkow@40476
|
4838 |
fix xs ys assume 1: "(xs,ys) \<in> listrel1 (r^*)"
|
nipkow@40476
|
4839 |
{ fix x y us vs
|
nipkow@40476
|
4840 |
have "(x,y) : r^* \<Longrightarrow> (us @ x # vs, us @ y # vs) : (listrel1 r)^*"
|
nipkow@40476
|
4841 |
proof(induct rule: rtrancl.induct)
|
nipkow@40476
|
4842 |
case rtrancl_refl show ?case by simp
|
nipkow@40476
|
4843 |
next
|
nipkow@40476
|
4844 |
case rtrancl_into_rtrancl thus ?case
|
nipkow@40476
|
4845 |
by (metis listrel1I rtrancl.rtrancl_into_rtrancl)
|
nipkow@40476
|
4846 |
qed }
|
nipkow@40476
|
4847 |
thus "(xs,ys) \<in> (listrel1 r)^*" using 1 by(blast elim: listrel1E)
|
nipkow@40476
|
4848 |
qed
|
nipkow@40476
|
4849 |
|
nipkow@40476
|
4850 |
lemma rtrancl_listrel1_eq_len: "(x,y) \<in> (listrel1 r)^* \<Longrightarrow> length x = length y"
|
nipkow@40476
|
4851 |
by (induct rule: rtrancl.induct) (auto intro: listrel1_eq_len)
|
nipkow@40476
|
4852 |
|
nipkow@40476
|
4853 |
lemma rtrancl_listrel1_ConsI1:
|
nipkow@40476
|
4854 |
"(xs,ys) : (listrel1 r)^* \<Longrightarrow> (x#xs,x#ys) : (listrel1 r)^*"
|
nipkow@40476
|
4855 |
apply(induct rule: rtrancl.induct)
|
nipkow@40476
|
4856 |
apply simp
|
nipkow@40476
|
4857 |
by (metis listrel1I2 rtrancl.rtrancl_into_rtrancl)
|
nipkow@40476
|
4858 |
|
nipkow@40476
|
4859 |
lemma rtrancl_listrel1_ConsI2:
|
nipkow@40476
|
4860 |
"(x,y) \<in> r^* \<Longrightarrow> (xs, ys) \<in> (listrel1 r)^*
|
nipkow@40476
|
4861 |
\<Longrightarrow> (x # xs, y # ys) \<in> (listrel1 r)^*"
|
nipkow@40476
|
4862 |
by (blast intro: rtrancl_trans rtrancl_listrel1_ConsI1
|
nipkow@40476
|
4863 |
subsetD[OF listrel1_rtrancl_subset_rtrancl_listrel1 listrel1I1])
|
nipkow@40476
|
4864 |
|
nipkow@40476
|
4865 |
lemma listrel1_subset_listrel:
|
nipkow@40476
|
4866 |
"r \<subseteq> r' \<Longrightarrow> refl r' \<Longrightarrow> listrel1 r \<subseteq> listrel(r')"
|
nipkow@40476
|
4867 |
by(auto elim!: listrel1E simp add: listrel_iff_zip set_zip refl_on_def)
|
nipkow@40476
|
4868 |
|
nipkow@40476
|
4869 |
lemma listrel_reflcl_if_listrel1:
|
nipkow@40476
|
4870 |
"(xs,ys) : listrel1 r \<Longrightarrow> (xs,ys) : listrel(r^*)"
|
nipkow@40476
|
4871 |
by(erule listrel1E)(auto simp add: listrel_iff_zip set_zip)
|
nipkow@40476
|
4872 |
|
nipkow@40476
|
4873 |
lemma listrel_rtrancl_eq_rtrancl_listrel1: "listrel (r^*) = (listrel1 r)^*"
|
nipkow@40476
|
4874 |
proof
|
nipkow@40476
|
4875 |
{ fix x y assume "(x,y) \<in> listrel (r^*)"
|
nipkow@40476
|
4876 |
then have "(x,y) \<in> (listrel1 r)^*"
|
nipkow@40476
|
4877 |
by induct (auto intro: rtrancl_listrel1_ConsI2) }
|
nipkow@40476
|
4878 |
then show "listrel (r^*) \<subseteq> (listrel1 r)^*"
|
nipkow@40476
|
4879 |
by (rule subrelI)
|
nipkow@40476
|
4880 |
next
|
nipkow@40476
|
4881 |
show "listrel (r^*) \<supseteq> (listrel1 r)^*"
|
nipkow@40476
|
4882 |
proof(rule subrelI)
|
nipkow@40476
|
4883 |
fix xs ys assume "(xs,ys) \<in> (listrel1 r)^*"
|
nipkow@40476
|
4884 |
then show "(xs,ys) \<in> listrel (r^*)"
|
nipkow@40476
|
4885 |
proof induct
|
nipkow@40476
|
4886 |
case base show ?case by(auto simp add: listrel_iff_zip set_zip)
|
nipkow@40476
|
4887 |
next
|
nipkow@40476
|
4888 |
case (step ys zs)
|
nipkow@40476
|
4889 |
thus ?case by (metis listrel_reflcl_if_listrel1 listrel_rtrancl_trans)
|
nipkow@40476
|
4890 |
qed
|
nipkow@40476
|
4891 |
qed
|
nipkow@40476
|
4892 |
qed
|
nipkow@40476
|
4893 |
|
nipkow@40476
|
4894 |
lemma rtrancl_listrel1_if_listrel:
|
nipkow@40476
|
4895 |
"(xs,ys) : listrel r \<Longrightarrow> (xs,ys) : (listrel1 r)^*"
|
nipkow@40476
|
4896 |
by(metis listrel_rtrancl_eq_rtrancl_listrel1 subsetD[OF listrel_mono] r_into_rtrancl subsetI)
|
nipkow@40476
|
4897 |
|
nipkow@40476
|
4898 |
lemma listrel_subset_rtrancl_listrel1: "listrel r \<subseteq> (listrel1 r)^*"
|
nipkow@40476
|
4899 |
by(fast intro:rtrancl_listrel1_if_listrel)
|
nipkow@40476
|
4900 |
|
nipkow@15302
|
4901 |
|
krauss@26749
|
4902 |
subsection {* Size function *}
|
krauss@26749
|
4903 |
|
krauss@26875
|
4904 |
lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (list_size f)"
|
krauss@26875
|
4905 |
by (rule is_measure_trivial)
|
krauss@26875
|
4906 |
|
krauss@26875
|
4907 |
lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (option_size f)"
|
krauss@26875
|
4908 |
by (rule is_measure_trivial)
|
krauss@26875
|
4909 |
|
krauss@26875
|
4910 |
lemma list_size_estimation[termination_simp]:
|
krauss@26875
|
4911 |
"x \<in> set xs \<Longrightarrow> y < f x \<Longrightarrow> y < list_size f xs"
|
krauss@26749
|
4912 |
by (induct xs) auto
|
krauss@26749
|
4913 |
|
krauss@26875
|
4914 |
lemma list_size_estimation'[termination_simp]:
|
krauss@26875
|
4915 |
"x \<in> set xs \<Longrightarrow> y \<le> f x \<Longrightarrow> y \<le> list_size f xs"
|
krauss@26875
|
4916 |
by (induct xs) auto
|
krauss@26875
|
4917 |
|
krauss@26875
|
4918 |
lemma list_size_map[simp]: "list_size f (map g xs) = list_size (f o g) xs"
|
krauss@26875
|
4919 |
by (induct xs) auto
|
krauss@26875
|
4920 |
|
bulwahn@45477
|
4921 |
lemma list_size_append[simp]: "list_size f (xs @ ys) = list_size f xs + list_size f ys"
|
bulwahn@45477
|
4922 |
by (induct xs, auto)
|
bulwahn@45477
|
4923 |
|
krauss@26875
|
4924 |
lemma list_size_pointwise[termination_simp]:
|
bulwahn@45476
|
4925 |
"(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> list_size f xs \<le> list_size g xs"
|
krauss@26875
|
4926 |
by (induct xs) force+
|
krauss@26749
|
4927 |
|
haftmann@31048
|
4928 |
|
haftmann@33301
|
4929 |
subsection {* Transfer *}
|
haftmann@33301
|
4930 |
|
haftmann@33301
|
4931 |
definition
|
haftmann@33301
|
4932 |
embed_list :: "nat list \<Rightarrow> int list"
|
haftmann@33301
|
4933 |
where
|
haftmann@33301
|
4934 |
"embed_list l = map int l"
|
haftmann@33301
|
4935 |
|
haftmann@33301
|
4936 |
definition
|
haftmann@33301
|
4937 |
nat_list :: "int list \<Rightarrow> bool"
|
haftmann@33301
|
4938 |
where
|
haftmann@33301
|
4939 |
"nat_list l = nat_set (set l)"
|
haftmann@33301
|
4940 |
|
haftmann@33301
|
4941 |
definition
|
haftmann@33301
|
4942 |
return_list :: "int list \<Rightarrow> nat list"
|
haftmann@33301
|
4943 |
where
|
haftmann@33301
|
4944 |
"return_list l = map nat l"
|
haftmann@33301
|
4945 |
|
haftmann@33301
|
4946 |
lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow>
|
haftmann@33301
|
4947 |
embed_list (return_list l) = l"
|
haftmann@33301
|
4948 |
unfolding embed_list_def return_list_def nat_list_def nat_set_def
|
haftmann@33301
|
4949 |
apply (induct l)
|
haftmann@33301
|
4950 |
apply auto
|
haftmann@33301
|
4951 |
done
|
haftmann@33301
|
4952 |
|
haftmann@33301
|
4953 |
lemma transfer_nat_int_list_functions:
|
haftmann@33301
|
4954 |
"l @ m = return_list (embed_list l @ embed_list m)"
|
haftmann@33301
|
4955 |
"[] = return_list []"
|
haftmann@33301
|
4956 |
unfolding return_list_def embed_list_def
|
haftmann@33301
|
4957 |
apply auto
|
haftmann@33301
|
4958 |
apply (induct l, auto)
|
haftmann@33301
|
4959 |
apply (induct m, auto)
|
haftmann@33301
|
4960 |
done
|
haftmann@33301
|
4961 |
|
haftmann@33301
|
4962 |
(*
|
haftmann@33301
|
4963 |
lemma transfer_nat_int_fold1: "fold f l x =
|
haftmann@33301
|
4964 |
fold (%x. f (nat x)) (embed_list l) x";
|
haftmann@33301
|
4965 |
*)
|
haftmann@33301
|
4966 |
|
haftmann@33301
|
4967 |
|
haftmann@37605
|
4968 |
subsection {* Code generation *}
|
haftmann@37605
|
4969 |
|
haftmann@37605
|
4970 |
subsubsection {* Counterparts for set-related operations *}
|
haftmann@37605
|
4971 |
|
haftmann@37605
|
4972 |
definition member :: "'a list \<Rightarrow> 'a \<Rightarrow> bool" where
|
haftmann@37605
|
4973 |
[code_post]: "member xs x \<longleftrightarrow> x \<in> set xs"
|
haftmann@37605
|
4974 |
|
haftmann@37605
|
4975 |
text {*
|
haftmann@37605
|
4976 |
Only use @{text member} for generating executable code. Otherwise use
|
haftmann@37605
|
4977 |
@{prop "x \<in> set xs"} instead --- it is much easier to reason about.
|
haftmann@37605
|
4978 |
*}
|
haftmann@37605
|
4979 |
|
haftmann@37605
|
4980 |
lemma member_set:
|
haftmann@37605
|
4981 |
"member = set"
|
nipkow@39535
|
4982 |
by (simp add: fun_eq_iff member_def mem_def)
|
haftmann@37605
|
4983 |
|
haftmann@37605
|
4984 |
lemma member_rec [code]:
|
haftmann@37605
|
4985 |
"member (x # xs) y \<longleftrightarrow> x = y \<or> member xs y"
|
haftmann@37605
|
4986 |
"member [] y \<longleftrightarrow> False"
|
haftmann@37605
|
4987 |
by (auto simp add: member_def)
|
haftmann@37605
|
4988 |
|
haftmann@37605
|
4989 |
lemma in_set_member [code_unfold]:
|
haftmann@37605
|
4990 |
"x \<in> set xs \<longleftrightarrow> member xs x"
|
haftmann@37605
|
4991 |
by (simp add: member_def)
|
haftmann@37605
|
4992 |
|
hoelzl@45799
|
4993 |
declare INF_def [code_unfold]
|
hoelzl@45799
|
4994 |
declare SUP_def [code_unfold]
|
haftmann@37605
|
4995 |
|
haftmann@37605
|
4996 |
declare set_map [symmetric, code_unfold]
|
haftmann@37605
|
4997 |
|
haftmann@37605
|
4998 |
definition list_all :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
|
haftmann@37605
|
4999 |
list_all_iff [code_post]: "list_all P xs \<longleftrightarrow> (\<forall>x \<in> set xs. P x)"
|
haftmann@37605
|
5000 |
|
haftmann@37605
|
5001 |
definition list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
|
haftmann@37605
|
5002 |
list_ex_iff [code_post]: "list_ex P xs \<longleftrightarrow> (\<exists>x \<in> set xs. P x)"
|
haftmann@37605
|
5003 |
|
bulwahn@40900
|
5004 |
definition list_ex1
|
bulwahn@40900
|
5005 |
where
|
bulwahn@40900
|
5006 |
list_ex1_iff: "list_ex1 P xs \<longleftrightarrow> (\<exists>! x. x \<in> set xs \<and> P x)"
|
bulwahn@40900
|
5007 |
|
haftmann@37605
|
5008 |
text {*
|
haftmann@37605
|
5009 |
Usually you should prefer @{text "\<forall>x\<in>set xs"} and @{text "\<exists>x\<in>set xs"}
|
haftmann@37605
|
5010 |
over @{const list_all} and @{const list_ex} in specifications.
|
haftmann@37605
|
5011 |
*}
|
haftmann@37605
|
5012 |
|
haftmann@37605
|
5013 |
lemma list_all_simps [simp, code]:
|
haftmann@37605
|
5014 |
"list_all P (x # xs) \<longleftrightarrow> P x \<and> list_all P xs"
|
haftmann@37605
|
5015 |
"list_all P [] \<longleftrightarrow> True"
|
haftmann@37605
|
5016 |
by (simp_all add: list_all_iff)
|
haftmann@37605
|
5017 |
|
haftmann@37605
|
5018 |
lemma list_ex_simps [simp, code]:
|
haftmann@37605
|
5019 |
"list_ex P (x # xs) \<longleftrightarrow> P x \<or> list_ex P xs"
|
haftmann@37605
|
5020 |
"list_ex P [] \<longleftrightarrow> False"
|
haftmann@37605
|
5021 |
by (simp_all add: list_ex_iff)
|
haftmann@37605
|
5022 |
|
bulwahn@40900
|
5023 |
lemma list_ex1_simps [simp, code]:
|
bulwahn@40900
|
5024 |
"list_ex1 P [] = False"
|
bulwahn@40900
|
5025 |
"list_ex1 P (x # xs) = (if P x then list_all (\<lambda>y. \<not> P y \<or> x = y) xs else list_ex1 P xs)"
|
bulwahn@40900
|
5026 |
unfolding list_ex1_iff list_all_iff by auto
|
bulwahn@40900
|
5027 |
|
haftmann@37605
|
5028 |
lemma Ball_set_list_all [code_unfold]:
|
haftmann@37605
|
5029 |
"Ball (set xs) P \<longleftrightarrow> list_all P xs"
|
haftmann@37605
|
5030 |
by (simp add: list_all_iff)
|
haftmann@37605
|
5031 |
|
haftmann@37605
|
5032 |
lemma Bex_set_list_ex [code_unfold]:
|
haftmann@37605
|
5033 |
"Bex (set xs) P \<longleftrightarrow> list_ex P xs"
|
haftmann@37605
|
5034 |
by (simp add: list_ex_iff)
|
haftmann@37605
|
5035 |
|
haftmann@37605
|
5036 |
lemma list_all_append [simp]:
|
haftmann@37605
|
5037 |
"list_all P (xs @ ys) \<longleftrightarrow> list_all P xs \<and> list_all P ys"
|
haftmann@37605
|
5038 |
by (auto simp add: list_all_iff)
|
haftmann@37605
|
5039 |
|
haftmann@37605
|
5040 |
lemma list_ex_append [simp]:
|
haftmann@37605
|
5041 |
"list_ex P (xs @ ys) \<longleftrightarrow> list_ex P xs \<or> list_ex P ys"
|
haftmann@37605
|
5042 |
by (auto simp add: list_ex_iff)
|
haftmann@37605
|
5043 |
|
haftmann@37605
|
5044 |
lemma list_all_rev [simp]:
|
haftmann@37605
|
5045 |
"list_all P (rev xs) \<longleftrightarrow> list_all P xs"
|
haftmann@37605
|
5046 |
by (simp add: list_all_iff)
|
haftmann@37605
|
5047 |
|
haftmann@37605
|
5048 |
lemma list_ex_rev [simp]:
|
haftmann@37605
|
5049 |
"list_ex P (rev xs) \<longleftrightarrow> list_ex P xs"
|
haftmann@37605
|
5050 |
by (simp add: list_ex_iff)
|
haftmann@37605
|
5051 |
|
haftmann@37605
|
5052 |
lemma list_all_length:
|
haftmann@37605
|
5053 |
"list_all P xs \<longleftrightarrow> (\<forall>n < length xs. P (xs ! n))"
|
haftmann@37605
|
5054 |
by (auto simp add: list_all_iff set_conv_nth)
|
haftmann@37605
|
5055 |
|
haftmann@37605
|
5056 |
lemma list_ex_length:
|
haftmann@37605
|
5057 |
"list_ex P xs \<longleftrightarrow> (\<exists>n < length xs. P (xs ! n))"
|
haftmann@37605
|
5058 |
by (auto simp add: list_ex_iff set_conv_nth)
|
haftmann@37605
|
5059 |
|
haftmann@37605
|
5060 |
lemma list_all_cong [fundef_cong]:
|
haftmann@37605
|
5061 |
"xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_all f xs = list_all g ys"
|
haftmann@37605
|
5062 |
by (simp add: list_all_iff)
|
haftmann@37605
|
5063 |
|
haftmann@37605
|
5064 |
lemma list_any_cong [fundef_cong]:
|
haftmann@37605
|
5065 |
"xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_ex f xs = list_ex g ys"
|
haftmann@37605
|
5066 |
by (simp add: list_ex_iff)
|
haftmann@37605
|
5067 |
|
haftmann@37605
|
5068 |
text {* Bounded quantification and summation over nats. *}
|
haftmann@37605
|
5069 |
|
haftmann@37605
|
5070 |
lemma atMost_upto [code_unfold]:
|
haftmann@37605
|
5071 |
"{..n} = set [0..<Suc n]"
|
haftmann@37605
|
5072 |
by auto
|
haftmann@37605
|
5073 |
|
haftmann@37605
|
5074 |
lemma atLeast_upt [code_unfold]:
|
haftmann@37605
|
5075 |
"{..<n} = set [0..<n]"
|
haftmann@37605
|
5076 |
by auto
|
haftmann@37605
|
5077 |
|
haftmann@37605
|
5078 |
lemma greaterThanLessThan_upt [code_unfold]:
|
haftmann@37605
|
5079 |
"{n<..<m} = set [Suc n..<m]"
|
haftmann@37605
|
5080 |
by auto
|
haftmann@37605
|
5081 |
|
haftmann@37605
|
5082 |
lemmas atLeastLessThan_upt [code_unfold] = set_upt [symmetric]
|
haftmann@37605
|
5083 |
|
haftmann@37605
|
5084 |
lemma greaterThanAtMost_upt [code_unfold]:
|
haftmann@37605
|
5085 |
"{n<..m} = set [Suc n..<Suc m]"
|
haftmann@37605
|
5086 |
by auto
|
haftmann@37605
|
5087 |
|
haftmann@37605
|
5088 |
lemma atLeastAtMost_upt [code_unfold]:
|
haftmann@37605
|
5089 |
"{n..m} = set [n..<Suc m]"
|
haftmann@37605
|
5090 |
by auto
|
haftmann@37605
|
5091 |
|
haftmann@37605
|
5092 |
lemma all_nat_less_eq [code_unfold]:
|
haftmann@37605
|
5093 |
"(\<forall>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..<n}. P m)"
|
haftmann@37605
|
5094 |
by auto
|
haftmann@37605
|
5095 |
|
haftmann@37605
|
5096 |
lemma ex_nat_less_eq [code_unfold]:
|
haftmann@37605
|
5097 |
"(\<exists>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..<n}. P m)"
|
haftmann@37605
|
5098 |
by auto
|
haftmann@37605
|
5099 |
|
haftmann@37605
|
5100 |
lemma all_nat_less [code_unfold]:
|
haftmann@37605
|
5101 |
"(\<forall>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..n}. P m)"
|
haftmann@37605
|
5102 |
by auto
|
haftmann@37605
|
5103 |
|
haftmann@37605
|
5104 |
lemma ex_nat_less [code_unfold]:
|
haftmann@37605
|
5105 |
"(\<exists>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..n}. P m)"
|
haftmann@37605
|
5106 |
by auto
|
haftmann@37605
|
5107 |
|
haftmann@37605
|
5108 |
lemma setsum_set_upt_conv_listsum_nat [code_unfold]:
|
haftmann@37605
|
5109 |
"setsum f (set [m..<n]) = listsum (map f [m..<n])"
|
haftmann@37605
|
5110 |
by (simp add: interv_listsum_conv_setsum_set_nat)
|
haftmann@37605
|
5111 |
|
haftmann@37605
|
5112 |
text {* Summation over ints. *}
|
haftmann@37605
|
5113 |
|
haftmann@37605
|
5114 |
lemma greaterThanLessThan_upto [code_unfold]:
|
haftmann@37605
|
5115 |
"{i<..<j::int} = set [i+1..j - 1]"
|
haftmann@37605
|
5116 |
by auto
|
haftmann@37605
|
5117 |
|
haftmann@37605
|
5118 |
lemma atLeastLessThan_upto [code_unfold]:
|
haftmann@37605
|
5119 |
"{i..<j::int} = set [i..j - 1]"
|
haftmann@37605
|
5120 |
by auto
|
haftmann@37605
|
5121 |
|
haftmann@37605
|
5122 |
lemma greaterThanAtMost_upto [code_unfold]:
|
haftmann@37605
|
5123 |
"{i<..j::int} = set [i+1..j]"
|
haftmann@37605
|
5124 |
by auto
|
haftmann@37605
|
5125 |
|
haftmann@37605
|
5126 |
lemmas atLeastAtMost_upto [code_unfold] = set_upto [symmetric]
|
haftmann@37605
|
5127 |
|
haftmann@37605
|
5128 |
lemma setsum_set_upto_conv_listsum_int [code_unfold]:
|
haftmann@37605
|
5129 |
"setsum f (set [i..j::int]) = listsum (map f [i..j])"
|
haftmann@37605
|
5130 |
by (simp add: interv_listsum_conv_setsum_set_int)
|
haftmann@37605
|
5131 |
|
haftmann@37605
|
5132 |
|
haftmann@37605
|
5133 |
subsubsection {* Optimizing by rewriting *}
|
haftmann@37605
|
5134 |
|
haftmann@37605
|
5135 |
definition null :: "'a list \<Rightarrow> bool" where
|
haftmann@37605
|
5136 |
[code_post]: "null xs \<longleftrightarrow> xs = []"
|
haftmann@37605
|
5137 |
|
haftmann@37605
|
5138 |
text {*
|
haftmann@37605
|
5139 |
Efficient emptyness check is implemented by @{const null}.
|
haftmann@37605
|
5140 |
*}
|
haftmann@37605
|
5141 |
|
haftmann@37605
|
5142 |
lemma null_rec [code]:
|
haftmann@37605
|
5143 |
"null (x # xs) \<longleftrightarrow> False"
|
haftmann@37605
|
5144 |
"null [] \<longleftrightarrow> True"
|
haftmann@37605
|
5145 |
by (simp_all add: null_def)
|
haftmann@37605
|
5146 |
|
haftmann@37605
|
5147 |
lemma eq_Nil_null [code_unfold]:
|
haftmann@37605
|
5148 |
"xs = [] \<longleftrightarrow> null xs"
|
haftmann@37605
|
5149 |
by (simp add: null_def)
|
haftmann@37605
|
5150 |
|
haftmann@37605
|
5151 |
lemma equal_Nil_null [code_unfold]:
|
haftmann@39086
|
5152 |
"HOL.equal xs [] \<longleftrightarrow> null xs"
|
haftmann@39086
|
5153 |
by (simp add: equal eq_Nil_null)
|
haftmann@37605
|
5154 |
|
haftmann@37605
|
5155 |
definition maps :: "('a \<Rightarrow> 'b list) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
|
haftmann@37605
|
5156 |
[code_post]: "maps f xs = concat (map f xs)"
|
haftmann@37605
|
5157 |
|
haftmann@37605
|
5158 |
definition map_filter :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
|
haftmann@37605
|
5159 |
[code_post]: "map_filter f xs = map (the \<circ> f) (filter (\<lambda>x. f x \<noteq> None) xs)"
|
haftmann@37605
|
5160 |
|
haftmann@37605
|
5161 |
text {*
|
haftmann@37605
|
5162 |
Operations @{const maps} and @{const map_filter} avoid
|
haftmann@37605
|
5163 |
intermediate lists on execution -- do not use for proving.
|
haftmann@37605
|
5164 |
*}
|
haftmann@37605
|
5165 |
|
haftmann@37605
|
5166 |
lemma maps_simps [code]:
|
haftmann@37605
|
5167 |
"maps f (x # xs) = f x @ maps f xs"
|
haftmann@37605
|
5168 |
"maps f [] = []"
|
haftmann@37605
|
5169 |
by (simp_all add: maps_def)
|
haftmann@37605
|
5170 |
|
haftmann@37605
|
5171 |
lemma map_filter_simps [code]:
|
haftmann@37605
|
5172 |
"map_filter f (x # xs) = (case f x of None \<Rightarrow> map_filter f xs | Some y \<Rightarrow> y # map_filter f xs)"
|
haftmann@37605
|
5173 |
"map_filter f [] = []"
|
haftmann@37605
|
5174 |
by (simp_all add: map_filter_def split: option.split)
|
haftmann@37605
|
5175 |
|
haftmann@37605
|
5176 |
lemma concat_map_maps [code_unfold]:
|
haftmann@37605
|
5177 |
"concat (map f xs) = maps f xs"
|
haftmann@37605
|
5178 |
by (simp add: maps_def)
|
haftmann@37605
|
5179 |
|
haftmann@37605
|
5180 |
lemma map_filter_map_filter [code_unfold]:
|
haftmann@37605
|
5181 |
"map f (filter P xs) = map_filter (\<lambda>x. if P x then Some (f x) else None) xs"
|
haftmann@37605
|
5182 |
by (simp add: map_filter_def)
|
haftmann@37605
|
5183 |
|
haftmann@37605
|
5184 |
text {* Optimized code for @{text"\<forall>i\<in>{a..b::int}"} and @{text"\<forall>n:{a..<b::nat}"}
|
haftmann@37605
|
5185 |
and similiarly for @{text"\<exists>"}. *}
|
haftmann@37605
|
5186 |
|
haftmann@37605
|
5187 |
definition all_interval_nat :: "(nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
|
haftmann@37605
|
5188 |
"all_interval_nat P i j \<longleftrightarrow> (\<forall>n \<in> {i..<j}. P n)"
|
haftmann@37605
|
5189 |
|
haftmann@37605
|
5190 |
lemma [code]:
|
haftmann@37605
|
5191 |
"all_interval_nat P i j \<longleftrightarrow> i \<ge> j \<or> P i \<and> all_interval_nat P (Suc i) j"
|
haftmann@37605
|
5192 |
proof -
|
haftmann@37605
|
5193 |
have *: "\<And>n. P i \<Longrightarrow> \<forall>n\<in>{Suc i..<j}. P n \<Longrightarrow> i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n"
|
haftmann@37605
|
5194 |
proof -
|
haftmann@37605
|
5195 |
fix n
|
haftmann@37605
|
5196 |
assume "P i" "\<forall>n\<in>{Suc i..<j}. P n" "i \<le> n" "n < j"
|
haftmann@37605
|
5197 |
then show "P n" by (cases "n = i") simp_all
|
haftmann@37605
|
5198 |
qed
|
haftmann@37605
|
5199 |
show ?thesis by (auto simp add: all_interval_nat_def intro: *)
|
haftmann@37605
|
5200 |
qed
|
haftmann@37605
|
5201 |
|
haftmann@37605
|
5202 |
lemma list_all_iff_all_interval_nat [code_unfold]:
|
haftmann@37605
|
5203 |
"list_all P [i..<j] \<longleftrightarrow> all_interval_nat P i j"
|
haftmann@37605
|
5204 |
by (simp add: list_all_iff all_interval_nat_def)
|
haftmann@37605
|
5205 |
|
haftmann@37605
|
5206 |
lemma list_ex_iff_not_all_inverval_nat [code_unfold]:
|
haftmann@37605
|
5207 |
"list_ex P [i..<j] \<longleftrightarrow> \<not> (all_interval_nat (Not \<circ> P) i j)"
|
haftmann@37605
|
5208 |
by (simp add: list_ex_iff all_interval_nat_def)
|
haftmann@37605
|
5209 |
|
haftmann@37605
|
5210 |
definition all_interval_int :: "(int \<Rightarrow> bool) \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool" where
|
haftmann@37605
|
5211 |
"all_interval_int P i j \<longleftrightarrow> (\<forall>k \<in> {i..j}. P k)"
|
haftmann@37605
|
5212 |
|
haftmann@37605
|
5213 |
lemma [code]:
|
haftmann@37605
|
5214 |
"all_interval_int P i j \<longleftrightarrow> i > j \<or> P i \<and> all_interval_int P (i + 1) j"
|
haftmann@37605
|
5215 |
proof -
|
haftmann@37605
|
5216 |
have *: "\<And>k. P i \<Longrightarrow> \<forall>k\<in>{i+1..j}. P k \<Longrightarrow> i \<le> k \<Longrightarrow> k \<le> j \<Longrightarrow> P k"
|
haftmann@37605
|
5217 |
proof -
|
haftmann@37605
|
5218 |
fix k
|
haftmann@37605
|
5219 |
assume "P i" "\<forall>k\<in>{i+1..j}. P k" "i \<le> k" "k \<le> j"
|
haftmann@37605
|
5220 |
then show "P k" by (cases "k = i") simp_all
|
haftmann@37605
|
5221 |
qed
|
haftmann@37605
|
5222 |
show ?thesis by (auto simp add: all_interval_int_def intro: *)
|
haftmann@37605
|
5223 |
qed
|
haftmann@37605
|
5224 |
|
haftmann@37605
|
5225 |
lemma list_all_iff_all_interval_int [code_unfold]:
|
haftmann@37605
|
5226 |
"list_all P [i..j] \<longleftrightarrow> all_interval_int P i j"
|
haftmann@37605
|
5227 |
by (simp add: list_all_iff all_interval_int_def)
|
haftmann@37605
|
5228 |
|
haftmann@37605
|
5229 |
lemma list_ex_iff_not_all_inverval_int [code_unfold]:
|
haftmann@37605
|
5230 |
"list_ex P [i..j] \<longleftrightarrow> \<not> (all_interval_int (Not \<circ> P) i j)"
|
haftmann@37605
|
5231 |
by (simp add: list_ex_iff all_interval_int_def)
|
haftmann@37605
|
5232 |
|
haftmann@37605
|
5233 |
hide_const (open) member null maps map_filter all_interval_nat all_interval_int
|
haftmann@37605
|
5234 |
|
haftmann@37605
|
5235 |
|
haftmann@37605
|
5236 |
subsubsection {* Pretty lists *}
|
berghofe@15064
|
5237 |
|
haftmann@31055
|
5238 |
use "Tools/list_code.ML"
|
haftmann@31055
|
5239 |
|
haftmann@31048
|
5240 |
code_type list
|
haftmann@31048
|
5241 |
(SML "_ list")
|
haftmann@31048
|
5242 |
(OCaml "_ list")
|
haftmann@34886
|
5243 |
(Haskell "![(_)]")
|
haftmann@34886
|
5244 |
(Scala "List[(_)]")
|
haftmann@31048
|
5245 |
|
haftmann@31048
|
5246 |
code_const Nil
|
haftmann@31048
|
5247 |
(SML "[]")
|
haftmann@31048
|
5248 |
(OCaml "[]")
|
haftmann@31048
|
5249 |
(Haskell "[]")
|
haftmann@37853
|
5250 |
(Scala "!Nil")
|
haftmann@31048
|
5251 |
|
haftmann@39086
|
5252 |
code_instance list :: equal
|
haftmann@31048
|
5253 |
(Haskell -)
|
haftmann@31048
|
5254 |
|
haftmann@39086
|
5255 |
code_const "HOL.equal \<Colon> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
|
haftmann@39499
|
5256 |
(Haskell infix 4 "==")
|
haftmann@31048
|
5257 |
|
haftmann@31048
|
5258 |
code_reserved SML
|
haftmann@31048
|
5259 |
list
|
haftmann@31048
|
5260 |
|
haftmann@31048
|
5261 |
code_reserved OCaml
|
haftmann@31048
|
5262 |
list
|
haftmann@31048
|
5263 |
|
berghofe@16770
|
5264 |
types_code
|
berghofe@16770
|
5265 |
"list" ("_ list")
|
berghofe@16770
|
5266 |
attach (term_of) {*
|
wenzelm@21760
|
5267 |
fun term_of_list f T = HOLogic.mk_list T o map f;
|
berghofe@16770
|
5268 |
*}
|
berghofe@16770
|
5269 |
attach (test) {*
|
berghofe@25885
|
5270 |
fun gen_list' aG aT i j = frequency
|
berghofe@25885
|
5271 |
[(i, fn () =>
|
berghofe@25885
|
5272 |
let
|
berghofe@25885
|
5273 |
val (x, t) = aG j;
|
berghofe@25885
|
5274 |
val (xs, ts) = gen_list' aG aT (i-1) j
|
berghofe@25885
|
5275 |
in (x :: xs, fn () => HOLogic.cons_const aT $ t () $ ts ()) end),
|
berghofe@25885
|
5276 |
(1, fn () => ([], fn () => HOLogic.nil_const aT))] ()
|
berghofe@25885
|
5277 |
and gen_list aG aT i = gen_list' aG aT i i;
|
berghofe@16770
|
5278 |
*}
|
haftmann@31048
|
5279 |
|
haftmann@31048
|
5280 |
consts_code Cons ("(_ ::/ _)")
|
haftmann@20588
|
5281 |
|
haftmann@20453
|
5282 |
setup {*
|
haftmann@20453
|
5283 |
let
|
wenzelm@43292
|
5284 |
fun list_codegen thy mode defs dep thyname b t gr =
|
haftmann@31055
|
5285 |
let
|
haftmann@31055
|
5286 |
val ts = HOLogic.dest_list t;
|
wenzelm@43292
|
5287 |
val (_, gr') = Codegen.invoke_tycodegen thy mode defs dep thyname false
|
haftmann@31055
|
5288 |
(fastype_of t) gr;
|
haftmann@31055
|
5289 |
val (ps, gr'') = fold_map
|
wenzelm@43292
|
5290 |
(Codegen.invoke_codegen thy mode defs dep thyname false) ts gr'
|
haftmann@31055
|
5291 |
in SOME (Pretty.list "[" "]" ps, gr'') end handle TERM _ => NONE;
|
haftmann@31055
|
5292 |
in
|
haftmann@34886
|
5293 |
fold (List_Code.add_literal_list) ["SML", "OCaml", "Haskell", "Scala"]
|
haftmann@31055
|
5294 |
#> Codegen.add_codegen "list_codegen" list_codegen
|
haftmann@31055
|
5295 |
end
|
haftmann@20453
|
5296 |
*}
|
berghofe@15064
|
5297 |
|
haftmann@21061
|
5298 |
|
haftmann@37399
|
5299 |
subsubsection {* Use convenient predefined operations *}
|
haftmann@37399
|
5300 |
|
haftmann@37399
|
5301 |
code_const "op @"
|
haftmann@37399
|
5302 |
(SML infixr 7 "@")
|
haftmann@37399
|
5303 |
(OCaml infixr 6 "@")
|
haftmann@37399
|
5304 |
(Haskell infixr 5 "++")
|
haftmann@37399
|
5305 |
(Scala infixl 7 "++")
|
haftmann@37399
|
5306 |
|
haftmann@37399
|
5307 |
code_const map
|
haftmann@37399
|
5308 |
(Haskell "map")
|
haftmann@37399
|
5309 |
|
haftmann@37399
|
5310 |
code_const filter
|
haftmann@37399
|
5311 |
(Haskell "filter")
|
haftmann@37399
|
5312 |
|
haftmann@37399
|
5313 |
code_const concat
|
haftmann@37399
|
5314 |
(Haskell "concat")
|
haftmann@37399
|
5315 |
|
haftmann@37605
|
5316 |
code_const List.maps
|
haftmann@37605
|
5317 |
(Haskell "concatMap")
|
haftmann@37605
|
5318 |
|
haftmann@37399
|
5319 |
code_const rev
|
haftmann@37426
|
5320 |
(Haskell "reverse")
|
haftmann@37399
|
5321 |
|
haftmann@37399
|
5322 |
code_const zip
|
haftmann@37399
|
5323 |
(Haskell "zip")
|
haftmann@37399
|
5324 |
|
haftmann@37605
|
5325 |
code_const List.null
|
haftmann@37605
|
5326 |
(Haskell "null")
|
haftmann@37605
|
5327 |
|
haftmann@37399
|
5328 |
code_const takeWhile
|
haftmann@37399
|
5329 |
(Haskell "takeWhile")
|
haftmann@37399
|
5330 |
|
haftmann@37399
|
5331 |
code_const dropWhile
|
haftmann@37399
|
5332 |
(Haskell "dropWhile")
|
haftmann@37399
|
5333 |
|
haftmann@37399
|
5334 |
code_const hd
|
haftmann@37399
|
5335 |
(Haskell "head")
|
haftmann@37399
|
5336 |
|
haftmann@37399
|
5337 |
code_const last
|
haftmann@37399
|
5338 |
(Haskell "last")
|
haftmann@37399
|
5339 |
|
haftmann@37605
|
5340 |
code_const list_all
|
haftmann@37605
|
5341 |
(Haskell "all")
|
haftmann@37605
|
5342 |
|
haftmann@37605
|
5343 |
code_const list_ex
|
haftmann@37605
|
5344 |
(Haskell "any")
|
haftmann@37605
|
5345 |
|
wenzelm@23388
|
5346 |
end
|