src/HOL/List.thy
author blanchet
Sun Feb 16 21:33:28 2014 +0100 (2014-02-16)
changeset 55524 f41ef840f09d
parent 55473 c582a7893dcd
child 55525 70b7e91fa1f9
permissions -rw-r--r--
folded 'list_all2' with the relator generated by 'datatype_new'
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(*  Title:      HOL/List.thy
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    Author:     Tobias Nipkow
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*)
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header {* The datatype of finite lists *}
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theory List
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imports Presburger Code_Numeral Quotient Lifting_Set Lifting_Option Lifting_Product
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begin
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datatype_new 'a list (map: map rel: list_all2) =
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    =: Nil (defaults tl: "[]")  ("[]")
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  | Cons (hd: 'a) (tl: "'a list")  (infixr "#" 65)
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datatype_new_compat list
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lemma [case_names Nil Cons, cases type: list]:
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  -- {* for backward compatibility -- names of variables differ *}
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  "(y = [] \<Longrightarrow> P) \<Longrightarrow> (\<And>a list. y = a # list \<Longrightarrow> P) \<Longrightarrow> P"
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by (rule list.exhaust)
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lemma [case_names Nil Cons, induct type: list]:
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  -- {* for backward compatibility -- names of variables differ *}
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  "P [] \<Longrightarrow> (\<And>a list. P list \<Longrightarrow> P (a # list)) \<Longrightarrow> P list"
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by (rule list.induct)
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text {* Compatibility: *}
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setup {* Sign.mandatory_path "list" *}
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lemmas inducts = list.induct
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lemmas recs = list.rec
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lemmas cases = list.case
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setup {* Sign.parent_path *}
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syntax
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  -- {* list Enumeration *}
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  "_list" :: "args => 'a list"    ("[(_)]")
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translations
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  "[x, xs]" == "x#[xs]"
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  "[x]" == "x#[]"
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subsection {* Basic list processing functions *}
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primrec last :: "'a list \<Rightarrow> 'a" where
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"last (x # xs) = (if xs = [] then x else last xs)"
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primrec butlast :: "'a list \<Rightarrow> 'a list" where
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"butlast []= []" |
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"butlast (x # xs) = (if xs = [] then [] else x # butlast xs)"
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primrec set :: "'a list \<Rightarrow> 'a set" where
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"set [] = {}" |
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"set (x # xs) = insert x (set xs)"
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definition coset :: "'a list \<Rightarrow> 'a set" where
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[simp]: "coset xs = - set xs"
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primrec append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65) where
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append_Nil: "[] @ ys = ys" |
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append_Cons: "(x#xs) @ ys = x # xs @ ys"
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primrec rev :: "'a list \<Rightarrow> 'a list" where
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"rev [] = []" |
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"rev (x # xs) = rev xs @ [x]"
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primrec filter:: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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"filter P [] = []" |
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"filter P (x # xs) = (if P x then x # filter P xs else filter P xs)"
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syntax
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  -- {* Special syntax for filter *}
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  "_filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_<-_./ _])")
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translations
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  "[x<-xs . P]"== "CONST filter (%x. P) xs"
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syntax (xsymbols)
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  "_filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
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syntax (HTML output)
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  "_filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
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primrec fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where
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fold_Nil:  "fold f [] = id" |
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fold_Cons: "fold f (x # xs) = fold f xs \<circ> f x"
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primrec foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where
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foldr_Nil:  "foldr f [] = id" |
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foldr_Cons: "foldr f (x # xs) = f x \<circ> foldr f xs"
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primrec foldl :: "('b \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b" where
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foldl_Nil:  "foldl f a [] = a" |
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foldl_Cons: "foldl f a (x # xs) = foldl f (f a x) xs"
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primrec concat:: "'a list list \<Rightarrow> 'a list" where
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"concat [] = []" |
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"concat (x # xs) = x @ concat xs"
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definition (in monoid_add) listsum :: "'a list \<Rightarrow> 'a" where
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"listsum xs = foldr plus xs 0"
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primrec drop:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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drop_Nil: "drop n [] = []" |
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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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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec take:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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take_Nil:"take n [] = []" |
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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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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec nth :: "'a list => nat => 'a" (infixl "!" 100) where
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nth_Cons: "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec list_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
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"list_update [] i v = []" |
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"list_update (x # xs) i v =
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  (case i of 0 \<Rightarrow> v # xs | Suc j \<Rightarrow> x # list_update xs j v)"
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nonterminal lupdbinds and lupdbind
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syntax
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  "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
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  "" :: "lupdbind => lupdbinds"    ("_")
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  "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
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  "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
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translations
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  "_LUpdate xs (_lupdbinds b bs)" == "_LUpdate (_LUpdate xs b) bs"
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  "xs[i:=x]" == "CONST list_update xs i x"
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primrec takeWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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"takeWhile P [] = []" |
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"takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])"
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primrec dropWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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"dropWhile P [] = []" |
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"dropWhile P (x # xs) = (if P x then dropWhile P xs else x # xs)"
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primrec zip :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
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"zip xs [] = []" |
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zip_Cons: "zip xs (y # ys) =
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  (case xs of [] => [] | z # zs => (z, y) # zip zs ys)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
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primrec product :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
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"product [] _ = []" |
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"product (x#xs) ys = map (Pair x) ys @ product xs ys"
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hide_const (open) product
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primrec product_lists :: "'a list list \<Rightarrow> 'a list list" where
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"product_lists [] = [[]]" |
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"product_lists (xs # xss) = concat (map (\<lambda>x. map (Cons x) (product_lists xss)) xs)"
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primrec upt :: "nat \<Rightarrow> nat \<Rightarrow> nat list" ("(1[_..</_'])") where
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upt_0: "[i..<0] = []" |
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upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
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definition insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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"insert x xs = (if x \<in> set xs then xs else x # xs)"
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hide_const (open) insert
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hide_fact (open) insert_def
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primrec find :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a option" where
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"find _ [] = None" |
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"find P (x#xs) = (if P x then Some x else find P xs)"
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hide_const (open) find
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primrec those :: "'a option list \<Rightarrow> 'a list option"
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where
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"those [] = Some []" |
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"those (x # xs) = (case x of
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  None \<Rightarrow> None
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| Some y \<Rightarrow> map_option (Cons y) (those xs))"
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primrec remove1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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"remove1 x [] = []" |
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"remove1 x (y # xs) = (if x = y then xs else y # remove1 x xs)"
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primrec removeAll :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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"removeAll x [] = []" |
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"removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)"
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primrec distinct :: "'a list \<Rightarrow> bool" where
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"distinct [] \<longleftrightarrow> True" |
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"distinct (x # xs) \<longleftrightarrow> x \<notin> set xs \<and> distinct xs"
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primrec remdups :: "'a list \<Rightarrow> 'a list" where
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"remdups [] = []" |
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"remdups (x # xs) = (if x \<in> set xs then remdups xs else x # remdups xs)"
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fun remdups_adj :: "'a list \<Rightarrow> 'a list" where
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"remdups_adj [] = []" |
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"remdups_adj [x] = [x]" |
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"remdups_adj (x # y # xs) = (if x = y then remdups_adj (x # xs) else x # remdups_adj (y # xs))"
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primrec replicate :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
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replicate_0: "replicate 0 x = []" |
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replicate_Suc: "replicate (Suc n) x = x # replicate n x"
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text {*
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  Function @{text size} is overloaded for all datatypes. Users may
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  refer to the list version as @{text length}. *}
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abbreviation length :: "'a list \<Rightarrow> nat" where
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"length \<equiv> size"
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definition enumerate :: "nat \<Rightarrow> 'a list \<Rightarrow> (nat \<times> 'a) list" where
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enumerate_eq_zip: "enumerate n xs = zip [n..<n + length xs] xs"
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primrec rotate1 :: "'a list \<Rightarrow> 'a list" where
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"rotate1 [] = []" |
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"rotate1 (x # xs) = xs @ [x]"
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definition rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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"rotate n = rotate1 ^^ n"
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definition sublist :: "'a list => nat set => 'a list" where
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"sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"
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primrec sublists :: "'a list \<Rightarrow> 'a list list" where
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"sublists [] = [[]]" |
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"sublists (x#xs) = (let xss = sublists xs in map (Cons x) xss @ xss)"
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primrec n_lists :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list list" where
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"n_lists 0 xs = [[]]" |
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"n_lists (Suc n) xs = concat (map (\<lambda>ys. map (\<lambda>y. y # ys) xs) (n_lists n xs))"
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hide_const (open) n_lists
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fun splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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"splice [] ys = ys" |
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"splice xs [] = xs" |
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"splice (x#xs) (y#ys) = x # y # splice xs ys"
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text{*
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\begin{figure}[htbp]
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\fbox{
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\begin{tabular}{l}
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@{lemma "[a,b]@[c,d] = [a,b,c,d]" by simp}\\
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@{lemma "length [a,b,c] = 3" by simp}\\
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@{lemma "set [a,b,c] = {a,b,c}" by simp}\\
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@{lemma "map f [a,b,c] = [f a, f b, f c]" by simp}\\
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@{lemma "rev [a,b,c] = [c,b,a]" by simp}\\
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@{lemma "hd [a,b,c,d] = a" by simp}\\
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@{lemma "tl [a,b,c,d] = [b,c,d]" by simp}\\
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@{lemma "last [a,b,c,d] = d" by simp}\\
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@{lemma "butlast [a,b,c,d] = [a,b,c]" by simp}\\
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@{lemma[source] "filter (\<lambda>n::nat. n<2) [0,2,1] = [0,1]" by simp}\\
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@{lemma "concat [[a,b],[c,d,e],[],[f]] = [a,b,c,d,e,f]" by simp}\\
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@{lemma "fold f [a,b,c] x = f c (f b (f a x))" by simp}\\
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@{lemma "foldr f [a,b,c] x = f a (f b (f c x))" by simp}\\
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@{lemma "foldl f x [a,b,c] = f (f (f x a) b) c" by simp}\\
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@{lemma "zip [a,b,c] [x,y,z] = [(a,x),(b,y),(c,z)]" by simp}\\
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@{lemma "zip [a,b] [x,y,z] = [(a,x),(b,y)]" by simp}\\
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@{lemma "enumerate 3 [a,b,c] = [(3,a),(4,b),(5,c)]" by normalization}\\
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@{lemma "List.product [a,b] [c,d] = [(a, c), (a, d), (b, c), (b, d)]" by simp}\\
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@{lemma "product_lists [[a,b], [c], [d,e]] = [[a,c,d], [a,c,e], [b,c,d], [b,c,e]]" by simp}\\
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@{lemma "splice [a,b,c] [x,y,z] = [a,x,b,y,c,z]" by simp}\\
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@{lemma "splice [a,b,c,d] [x,y] = [a,x,b,y,c,d]" by simp}\\
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@{lemma "take 2 [a,b,c,d] = [a,b]" by simp}\\
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@{lemma "take 6 [a,b,c,d] = [a,b,c,d]" by simp}\\
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@{lemma "drop 2 [a,b,c,d] = [c,d]" by simp}\\
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@{lemma "drop 6 [a,b,c,d] = []" by simp}\\
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@{lemma "takeWhile (%n::nat. n<3) [1,2,3,0] = [1,2]" by simp}\\
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@{lemma "dropWhile (%n::nat. n<3) [1,2,3,0] = [3,0]" by simp}\\
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@{lemma "distinct [2,0,1::nat]" by simp}\\
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@{lemma "remdups [2,0,2,1::nat,2] = [0,1,2]" by simp}\\
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@{lemma "remdups_adj [2,2,3,1,1::nat,2,1] = [2,3,1,2,1]" by simp}\\
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@{lemma "List.insert 2 [0::nat,1,2] = [0,1,2]" by (simp add: List.insert_def)}\\
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@{lemma "List.insert 3 [0::nat,1,2] = [3,0,1,2]" by (simp add: List.insert_def)}\\
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@{lemma "List.find (%i::int. i>0) [0,0] = None" by simp}\\
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@{lemma "List.find (%i::int. i>0) [0,1,0,2] = Some 1" by simp}\\
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@{lemma "remove1 2 [2,0,2,1::nat,2] = [0,2,1,2]" by simp}\\
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@{lemma "removeAll 2 [2,0,2,1::nat,2] = [0,1]" by simp}\\
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@{lemma "nth [a,b,c,d] 2 = c" by simp}\\
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@{lemma "[a,b,c,d][2 := x] = [a,b,x,d]" by simp}\\
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@{lemma "sublist [a,b,c,d,e] {0,2,3} = [a,c,d]" by (simp add:sublist_def)}\\
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@{lemma "sublists [a,b] = [[a, b], [a], [b], []]" by simp}\\
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   291
@{lemma "List.n_lists 2 [a,b,c] = [[a, a], [b, a], [c, a], [a, b], [b, b], [c, b], [a, c], [b, c], [c, c]]" by (simp add: eval_nat_numeral)}\\
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@{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by simp}\\
blanchet@46440
   293
@{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate_def eval_nat_numeral)}\\
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   294
@{lemma "replicate 4 a = [a,a,a,a]" by (simp add:eval_nat_numeral)}\\
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   295
@{lemma "[2..<5] = [2,3,4]" by (simp add:eval_nat_numeral)}\\
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   296
@{lemma "listsum [1,2,3::nat] = 6" by (simp add: listsum_def)}
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\end{tabular}}
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\caption{Characteristic examples}
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\label{fig:Characteristic}
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\end{figure}
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   301
Figure~\ref{fig:Characteristic} shows characteristic examples
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that should give an intuitive understanding of the above functions.
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*}
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   304
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   305
text{* The following simple sort functions are intended for proofs,
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not for efficient implementations. *}
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   307
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   308
context linorder
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   309
begin
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   310
haftmann@39915
   311
inductive sorted :: "'a list \<Rightarrow> bool" where
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  Nil [iff]: "sorted []"
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| Cons: "\<forall>y\<in>set xs. x \<le> y \<Longrightarrow> sorted xs \<Longrightarrow> sorted (x # xs)"
haftmann@39915
   314
haftmann@39915
   315
lemma sorted_single [iff]:
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  "sorted [x]"
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   317
  by (rule sorted.Cons) auto
haftmann@39915
   318
haftmann@39915
   319
lemma sorted_many:
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   320
  "x \<le> y \<Longrightarrow> sorted (y # zs) \<Longrightarrow> sorted (x # y # zs)"
haftmann@39915
   321
  by (rule sorted.Cons) (cases "y # zs" rule: sorted.cases, auto)
haftmann@39915
   322
haftmann@39915
   323
lemma sorted_many_eq [simp, code]:
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   324
  "sorted (x # y # zs) \<longleftrightarrow> x \<le> y \<and> sorted (y # zs)"
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   325
  by (auto intro: sorted_many elim: sorted.cases)
haftmann@39915
   326
haftmann@39915
   327
lemma [code]:
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   328
  "sorted [] \<longleftrightarrow> True"
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   329
  "sorted [x] \<longleftrightarrow> True"
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   330
  by simp_all
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   331
hoelzl@33639
   332
primrec insort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
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   333
"insort_key f x [] = [x]" |
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   334
"insort_key f x (y#ys) =
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   335
  (if f x \<le> f y then (x#y#ys) else y#(insort_key f x ys))"
hoelzl@33639
   336
haftmann@35195
   337
definition sort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b list \<Rightarrow> 'b list" where
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   338
"sort_key f xs = foldr (insort_key f) xs []"
hoelzl@33639
   339
haftmann@40210
   340
definition insort_insert_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
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   341
"insort_insert_key f x xs =
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   342
  (if f x \<in> f ` set xs then xs else insort_key f x xs)"
haftmann@40210
   343
hoelzl@33639
   344
abbreviation "sort \<equiv> sort_key (\<lambda>x. x)"
hoelzl@33639
   345
abbreviation "insort \<equiv> insort_key (\<lambda>x. x)"
haftmann@40210
   346
abbreviation "insort_insert \<equiv> insort_insert_key (\<lambda>x. x)"
haftmann@35608
   347
wenzelm@25221
   348
end
wenzelm@25221
   349
nipkow@24616
   350
wenzelm@23388
   351
subsubsection {* List comprehension *}
nipkow@23192
   352
nipkow@24349
   353
text{* Input syntax for Haskell-like list comprehension notation.
nipkow@24349
   354
Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
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   355
the list of all pairs of distinct elements from @{text xs} and @{text ys}.
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   356
The syntax is as in Haskell, except that @{text"|"} becomes a dot
nipkow@24349
   357
(like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
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   358
\verb![e| x <- xs, ...]!.
nipkow@24349
   359
nipkow@24349
   360
The qualifiers after the dot are
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   361
\begin{description}
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   362
\item[generators] @{text"p \<leftarrow> xs"},
nipkow@24476
   363
 where @{text p} is a pattern and @{text xs} an expression of list type, or
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   364
\item[guards] @{text"b"}, where @{text b} is a boolean expression.
nipkow@24476
   365
%\item[local bindings] @ {text"let x = e"}.
nipkow@24349
   366
\end{description}
nipkow@23240
   367
nipkow@24476
   368
Just like in Haskell, list comprehension is just a shorthand. To avoid
nipkow@24476
   369
misunderstandings, the translation into desugared form is not reversed
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   370
upon output. Note that the translation of @{text"[e. x \<leftarrow> xs]"} is
nipkow@24476
   371
optmized to @{term"map (%x. e) xs"}.
nipkow@23240
   372
nipkow@24349
   373
It is easy to write short list comprehensions which stand for complex
nipkow@24349
   374
expressions. During proofs, they may become unreadable (and
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   375
mangled). In such cases it can be advisable to introduce separate
nipkow@24349
   376
definitions for the list comprehensions in question.  *}
nipkow@24349
   377
wenzelm@46138
   378
nonterminal lc_qual and lc_quals
nipkow@23192
   379
nipkow@23192
   380
syntax
wenzelm@46138
   381
  "_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
wenzelm@46138
   382
  "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ <- _")
wenzelm@46138
   383
  "_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
wenzelm@46138
   384
  (*"_lc_let" :: "letbinds => lc_qual"  ("let _")*)
wenzelm@46138
   385
  "_lc_end" :: "lc_quals" ("]")
wenzelm@46138
   386
  "_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals"  (", __")
wenzelm@46138
   387
  "_lc_abs" :: "'a => 'b list => 'b list"
nipkow@23192
   388
nipkow@24476
   389
(* These are easier than ML code but cannot express the optimized
nipkow@24476
   390
   translation of [e. p<-xs]
nipkow@23192
   391
translations
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   392
  "[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
wenzelm@46138
   393
  "_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
wenzelm@46138
   394
   => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
wenzelm@46138
   395
  "[e. P]" => "if P then [e] else []"
wenzelm@46138
   396
  "_listcompr e (_lc_test P) (_lc_quals Q Qs)"
wenzelm@46138
   397
   => "if P then (_listcompr e Q Qs) else []"
wenzelm@46138
   398
  "_listcompr e (_lc_let b) (_lc_quals Q Qs)"
wenzelm@46138
   399
   => "_Let b (_listcompr e Q Qs)"
nipkow@24476
   400
*)
nipkow@23240
   401
nipkow@23279
   402
syntax (xsymbols)
wenzelm@46138
   403
  "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ \<leftarrow> _")
nipkow@23279
   404
syntax (HTML output)
wenzelm@46138
   405
  "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ \<leftarrow> _")
nipkow@24349
   406
wenzelm@52143
   407
parse_translation {*
wenzelm@46138
   408
  let
wenzelm@46138
   409
    val NilC = Syntax.const @{const_syntax Nil};
wenzelm@46138
   410
    val ConsC = Syntax.const @{const_syntax Cons};
wenzelm@46138
   411
    val mapC = Syntax.const @{const_syntax map};
wenzelm@46138
   412
    val concatC = Syntax.const @{const_syntax concat};
wenzelm@46138
   413
    val IfC = Syntax.const @{const_syntax If};
wenzelm@46138
   414
wenzelm@46138
   415
    fun single x = ConsC $ x $ NilC;
wenzelm@46138
   416
wenzelm@46138
   417
    fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
wenzelm@46138
   418
      let
wenzelm@46138
   419
        (* FIXME proper name context!? *)
wenzelm@46138
   420
        val x =
wenzelm@46138
   421
          Free (singleton (Name.variant_list (fold Term.add_free_names [p, e] [])) "x", dummyT);
wenzelm@46138
   422
        val e = if opti then single e else e;
wenzelm@46138
   423
        val case1 = Syntax.const @{syntax_const "_case1"} $ p $ e;
wenzelm@46138
   424
        val case2 =
wenzelm@46138
   425
          Syntax.const @{syntax_const "_case1"} $
wenzelm@46138
   426
            Syntax.const @{const_syntax dummy_pattern} $ NilC;
wenzelm@46138
   427
        val cs = Syntax.const @{syntax_const "_case2"} $ case1 $ case2;
traytel@51678
   428
      in Syntax_Trans.abs_tr [x, Case_Translation.case_tr false ctxt [x, cs]] end;
wenzelm@46138
   429
wenzelm@46138
   430
    fun abs_tr ctxt p e opti =
wenzelm@46138
   431
      (case Term_Position.strip_positions p of
wenzelm@46138
   432
        Free (s, T) =>
wenzelm@46138
   433
          let
wenzelm@46138
   434
            val thy = Proof_Context.theory_of ctxt;
wenzelm@46138
   435
            val s' = Proof_Context.intern_const ctxt s;
wenzelm@46138
   436
          in
wenzelm@46138
   437
            if Sign.declared_const thy s'
wenzelm@46138
   438
            then (pat_tr ctxt p e opti, false)
wenzelm@46138
   439
            else (Syntax_Trans.abs_tr [p, e], true)
wenzelm@46138
   440
          end
wenzelm@46138
   441
      | _ => (pat_tr ctxt p e opti, false));
wenzelm@46138
   442
wenzelm@46138
   443
    fun lc_tr ctxt [e, Const (@{syntax_const "_lc_test"}, _) $ b, qs] =
wenzelm@46138
   444
          let
wenzelm@46138
   445
            val res =
wenzelm@46138
   446
              (case qs of
wenzelm@46138
   447
                Const (@{syntax_const "_lc_end"}, _) => single e
wenzelm@46138
   448
              | Const (@{syntax_const "_lc_quals"}, _) $ q $ qs => lc_tr ctxt [e, q, qs]);
wenzelm@46138
   449
          in IfC $ b $ res $ NilC end
wenzelm@46138
   450
      | lc_tr ctxt
wenzelm@46138
   451
            [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
wenzelm@46138
   452
              Const(@{syntax_const "_lc_end"}, _)] =
wenzelm@46138
   453
          (case abs_tr ctxt p e true of
wenzelm@46138
   454
            (f, true) => mapC $ f $ es
wenzelm@46138
   455
          | (f, false) => concatC $ (mapC $ f $ es))
wenzelm@46138
   456
      | lc_tr ctxt
wenzelm@46138
   457
            [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
wenzelm@46138
   458
              Const (@{syntax_const "_lc_quals"}, _) $ q $ qs] =
wenzelm@46138
   459
          let val e' = lc_tr ctxt [e, q, qs];
wenzelm@46138
   460
          in concatC $ (mapC $ (fst (abs_tr ctxt p e' false)) $ es) end;
wenzelm@46138
   461
wenzelm@46138
   462
  in [(@{syntax_const "_listcompr"}, lc_tr)] end
nipkow@24349
   463
*}
nipkow@23279
   464
wenzelm@51272
   465
ML_val {*
wenzelm@42167
   466
  let
wenzelm@42167
   467
    val read = Syntax.read_term @{context};
wenzelm@42167
   468
    fun check s1 s2 = read s1 aconv read s2 orelse error ("Check failed: " ^ quote s1);
wenzelm@42167
   469
  in
wenzelm@42167
   470
    check "[(x,y,z). b]" "if b then [(x, y, z)] else []";
wenzelm@42167
   471
    check "[(x,y,z). x\<leftarrow>xs]" "map (\<lambda>x. (x, y, z)) xs";
wenzelm@42167
   472
    check "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]" "concat (map (\<lambda>x. map (\<lambda>y. e x y) ys) xs)";
wenzelm@42167
   473
    check "[(x,y,z). x<a, x>b]" "if x < a then if b < x then [(x, y, z)] else [] else []";
wenzelm@42167
   474
    check "[(x,y,z). x\<leftarrow>xs, x>b]" "concat (map (\<lambda>x. if b < x then [(x, y, z)] else []) xs)";
wenzelm@42167
   475
    check "[(x,y,z). x<a, x\<leftarrow>xs]" "if x < a then map (\<lambda>x. (x, y, z)) xs else []";
wenzelm@42167
   476
    check "[(x,y). Cons True x \<leftarrow> xs]"
wenzelm@42167
   477
      "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | True # x \<Rightarrow> [(x, y)] | False # x \<Rightarrow> []) xs)";
wenzelm@42167
   478
    check "[(x,y,z). Cons x [] \<leftarrow> xs]"
wenzelm@42167
   479
      "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | [x] \<Rightarrow> [(x, y, z)] | x # aa # lista \<Rightarrow> []) xs)";
wenzelm@42167
   480
    check "[(x,y,z). x<a, x>b, x=d]"
wenzelm@42167
   481
      "if x < a then if b < x then if x = d then [(x, y, z)] else [] else [] else []";
wenzelm@42167
   482
    check "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
wenzelm@42167
   483
      "if x < a then if b < x then map (\<lambda>y. (x, y, z)) ys else [] else []";
wenzelm@42167
   484
    check "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
wenzelm@42167
   485
      "if x < a then concat (map (\<lambda>x. if b < y then [(x, y, z)] else []) xs) else []";
wenzelm@42167
   486
    check "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
wenzelm@42167
   487
      "if x < a then concat (map (\<lambda>x. map (\<lambda>y. (x, y, z)) ys) xs) else []";
wenzelm@42167
   488
    check "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
wenzelm@42167
   489
      "concat (map (\<lambda>x. if b < x then if y < a then [(x, y, z)] else [] else []) xs)";
wenzelm@42167
   490
    check "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
wenzelm@42167
   491
      "concat (map (\<lambda>x. if b < x then map (\<lambda>y. (x, y, z)) ys else []) xs)";
wenzelm@42167
   492
    check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
wenzelm@42167
   493
      "concat (map (\<lambda>x. concat (map (\<lambda>y. if x < y then [(x, y, z)] else []) ys)) xs)";
wenzelm@42167
   494
    check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
wenzelm@42167
   495
      "concat (map (\<lambda>x. concat (map (\<lambda>y. map (\<lambda>z. (x, y, z)) zs) ys)) xs)"
wenzelm@42167
   496
  end;
wenzelm@42167
   497
*}
wenzelm@42167
   498
wenzelm@35115
   499
(*
nipkow@24349
   500
term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
nipkow@23192
   501
*)
nipkow@23192
   502
wenzelm@42167
   503
wenzelm@50422
   504
ML {*
wenzelm@50422
   505
(* Simproc for rewriting list comprehensions applied to List.set to set
wenzelm@50422
   506
   comprehension. *)
wenzelm@50422
   507
wenzelm@50422
   508
signature LIST_TO_SET_COMPREHENSION =
wenzelm@50422
   509
sig
wenzelm@51717
   510
  val simproc : Proof.context -> cterm -> thm option
wenzelm@50422
   511
end
wenzelm@50422
   512
wenzelm@50422
   513
structure List_to_Set_Comprehension : LIST_TO_SET_COMPREHENSION =
wenzelm@50422
   514
struct
wenzelm@50422
   515
wenzelm@50422
   516
(* conversion *)
wenzelm@50422
   517
wenzelm@50422
   518
fun all_exists_conv cv ctxt ct =
wenzelm@50422
   519
  (case Thm.term_of ct of
wenzelm@50422
   520
    Const (@{const_name HOL.Ex}, _) $ Abs _ =>
wenzelm@50422
   521
      Conv.arg_conv (Conv.abs_conv (all_exists_conv cv o #2) ctxt) ct
wenzelm@50422
   522
  | _ => cv ctxt ct)
wenzelm@50422
   523
wenzelm@50422
   524
fun all_but_last_exists_conv cv ctxt ct =
wenzelm@50422
   525
  (case Thm.term_of ct of
wenzelm@50422
   526
    Const (@{const_name HOL.Ex}, _) $ Abs (_, _, Const (@{const_name HOL.Ex}, _) $ _) =>
wenzelm@50422
   527
      Conv.arg_conv (Conv.abs_conv (all_but_last_exists_conv cv o #2) ctxt) ct
wenzelm@50422
   528
  | _ => cv ctxt ct)
wenzelm@50422
   529
wenzelm@50422
   530
fun Collect_conv cv ctxt ct =
wenzelm@50422
   531
  (case Thm.term_of ct of
wenzelm@50422
   532
    Const (@{const_name Set.Collect}, _) $ Abs _ => Conv.arg_conv (Conv.abs_conv cv ctxt) ct
wenzelm@50422
   533
  | _ => raise CTERM ("Collect_conv", [ct]))
wenzelm@50422
   534
wenzelm@50422
   535
fun rewr_conv' th = Conv.rewr_conv (mk_meta_eq th)
wenzelm@50422
   536
wenzelm@50422
   537
fun conjunct_assoc_conv ct =
wenzelm@50422
   538
  Conv.try_conv
wenzelm@51315
   539
    (rewr_conv' @{thm conj_assoc} then_conv HOLogic.conj_conv Conv.all_conv conjunct_assoc_conv) ct
wenzelm@50422
   540
wenzelm@50422
   541
fun right_hand_set_comprehension_conv conv ctxt =
wenzelm@51315
   542
  HOLogic.Trueprop_conv (HOLogic.eq_conv Conv.all_conv
wenzelm@50422
   543
    (Collect_conv (all_exists_conv conv o #2) ctxt))
wenzelm@50422
   544
wenzelm@50422
   545
wenzelm@50422
   546
(* term abstraction of list comprehension patterns *)
wenzelm@50422
   547
wenzelm@50422
   548
datatype termlets = If | Case of (typ * int)
wenzelm@50422
   549
wenzelm@51717
   550
fun simproc ctxt redex =
wenzelm@50422
   551
  let
wenzelm@50422
   552
    val set_Nil_I = @{thm trans} OF [@{thm set.simps(1)}, @{thm empty_def}]
wenzelm@50422
   553
    val set_singleton = @{lemma "set [a] = {x. x = a}" by simp}
wenzelm@50422
   554
    val inst_Collect_mem_eq = @{lemma "set A = {x. x : set A}" by simp}
wenzelm@50422
   555
    val del_refl_eq = @{lemma "(t = t & P) == P" by simp}
wenzelm@50422
   556
    fun mk_set T = Const (@{const_name List.set}, HOLogic.listT T --> HOLogic.mk_setT T)
wenzelm@50422
   557
    fun dest_set (Const (@{const_name List.set}, _) $ xs) = xs
wenzelm@50422
   558
    fun dest_singleton_list (Const (@{const_name List.Cons}, _)
wenzelm@50422
   559
          $ t $ (Const (@{const_name List.Nil}, _))) = t
wenzelm@50422
   560
      | dest_singleton_list t = raise TERM ("dest_singleton_list", [t])
wenzelm@50422
   561
    (* We check that one case returns a singleton list and all other cases
wenzelm@50422
   562
       return [], and return the index of the one singleton list case *)
wenzelm@50422
   563
    fun possible_index_of_singleton_case cases =
wenzelm@50422
   564
      let
wenzelm@50422
   565
        fun check (i, case_t) s =
wenzelm@50422
   566
          (case strip_abs_body case_t of
wenzelm@50422
   567
            (Const (@{const_name List.Nil}, _)) => s
traytel@53412
   568
          | _ => (case s of SOME NONE => SOME (SOME i) | _ => NONE))
wenzelm@50422
   569
      in
traytel@53412
   570
        fold_index check cases (SOME NONE) |> the_default NONE
wenzelm@50422
   571
      end
blanchet@54404
   572
    (* returns (case_expr type index chosen_case constr_name) option  *)
wenzelm@50422
   573
    fun dest_case case_term =
wenzelm@50422
   574
      let
wenzelm@50422
   575
        val (case_const, args) = strip_comb case_term
wenzelm@50422
   576
      in
wenzelm@50422
   577
        (case try dest_Const case_const of
wenzelm@50422
   578
          SOME (c, T) =>
blanchet@54404
   579
            (case Ctr_Sugar.ctr_sugar_of_case ctxt c of
blanchet@54404
   580
              SOME {ctrs, ...} =>
wenzelm@50422
   581
                (case possible_index_of_singleton_case (fst (split_last args)) of
wenzelm@50422
   582
                  SOME i =>
wenzelm@50422
   583
                    let
blanchet@54404
   584
                      val constr_names = map (fst o dest_Const) ctrs
wenzelm@50422
   585
                      val (Ts, _) = strip_type T
wenzelm@50422
   586
                      val T' = List.last Ts
blanchet@54404
   587
                    in SOME (List.last args, T', i, nth args i, nth constr_names i) end
wenzelm@50422
   588
                | NONE => NONE)
wenzelm@50422
   589
            | NONE => NONE)
wenzelm@50422
   590
        | NONE => NONE)
wenzelm@50422
   591
      end
wenzelm@50422
   592
    (* returns condition continuing term option *)
wenzelm@50422
   593
    fun dest_if (Const (@{const_name If}, _) $ cond $ then_t $ Const (@{const_name Nil}, _)) =
wenzelm@50422
   594
          SOME (cond, then_t)
wenzelm@50422
   595
      | dest_if _ = NONE
wenzelm@50422
   596
    fun tac _ [] = rtac set_singleton 1 ORELSE rtac inst_Collect_mem_eq 1
wenzelm@50422
   597
      | tac ctxt (If :: cont) =
wenzelm@50422
   598
          Splitter.split_tac [@{thm split_if}] 1
wenzelm@50422
   599
          THEN rtac @{thm conjI} 1
wenzelm@50422
   600
          THEN rtac @{thm impI} 1
wenzelm@50422
   601
          THEN Subgoal.FOCUS (fn {prems, context, ...} =>
wenzelm@50422
   602
            CONVERSION (right_hand_set_comprehension_conv (K
wenzelm@51315
   603
              (HOLogic.conj_conv (Conv.rewr_conv (List.last prems RS @{thm Eq_TrueI})) Conv.all_conv
wenzelm@50422
   604
               then_conv
wenzelm@50422
   605
               rewr_conv' @{lemma "(True & P) = P" by simp})) context) 1) ctxt 1
wenzelm@50422
   606
          THEN tac ctxt cont
wenzelm@50422
   607
          THEN rtac @{thm impI} 1
wenzelm@50422
   608
          THEN Subgoal.FOCUS (fn {prems, context, ...} =>
wenzelm@50422
   609
              CONVERSION (right_hand_set_comprehension_conv (K
wenzelm@51315
   610
                (HOLogic.conj_conv (Conv.rewr_conv (List.last prems RS @{thm Eq_FalseI})) Conv.all_conv
wenzelm@50422
   611
                 then_conv rewr_conv' @{lemma "(False & P) = False" by simp})) context) 1) ctxt 1
wenzelm@50422
   612
          THEN rtac set_Nil_I 1
wenzelm@50422
   613
      | tac ctxt (Case (T, i) :: cont) =
wenzelm@50422
   614
          let
blanchet@54404
   615
            val SOME {injects, distincts, case_thms, split, ...} =
blanchet@54404
   616
              Ctr_Sugar.ctr_sugar_of ctxt (fst (dest_Type T))
wenzelm@50422
   617
          in
wenzelm@50422
   618
            (* do case distinction *)
blanchet@54404
   619
            Splitter.split_tac [split] 1
wenzelm@50422
   620
            THEN EVERY (map_index (fn (i', _) =>
blanchet@54404
   621
              (if i' < length case_thms - 1 then rtac @{thm conjI} 1 else all_tac)
wenzelm@50422
   622
              THEN REPEAT_DETERM (rtac @{thm allI} 1)
wenzelm@50422
   623
              THEN rtac @{thm impI} 1
wenzelm@50422
   624
              THEN (if i' = i then
wenzelm@50422
   625
                (* continue recursively *)
wenzelm@50422
   626
                Subgoal.FOCUS (fn {prems, context, ...} =>
wenzelm@50422
   627
                  CONVERSION (Thm.eta_conversion then_conv right_hand_set_comprehension_conv (K
wenzelm@51315
   628
                      ((HOLogic.conj_conv
wenzelm@51315
   629
                        (HOLogic.eq_conv Conv.all_conv (rewr_conv' (List.last prems)) then_conv
blanchet@54404
   630
                          (Conv.try_conv (Conv.rewrs_conv (map mk_meta_eq injects))))
wenzelm@50422
   631
                        Conv.all_conv)
wenzelm@50422
   632
                        then_conv (Conv.try_conv (Conv.rewr_conv del_refl_eq))
wenzelm@50422
   633
                        then_conv conjunct_assoc_conv)) context
wenzelm@51315
   634
                    then_conv (HOLogic.Trueprop_conv (HOLogic.eq_conv Conv.all_conv (Collect_conv (fn (_, ctxt) =>
wenzelm@50422
   635
                      Conv.repeat_conv
wenzelm@50422
   636
                        (all_but_last_exists_conv
wenzelm@50422
   637
                          (K (rewr_conv'
wenzelm@50422
   638
                            @{lemma "(EX x. x = t & P x) = P t" by simp})) ctxt)) context)))) 1) ctxt 1
wenzelm@50422
   639
                THEN tac ctxt cont
wenzelm@50422
   640
              else
wenzelm@50422
   641
                Subgoal.FOCUS (fn {prems, context, ...} =>
wenzelm@50422
   642
                  CONVERSION
wenzelm@50422
   643
                    (right_hand_set_comprehension_conv (K
wenzelm@51315
   644
                      (HOLogic.conj_conv
wenzelm@51315
   645
                        ((HOLogic.eq_conv Conv.all_conv
wenzelm@50422
   646
                          (rewr_conv' (List.last prems))) then_conv
blanchet@54404
   647
                          (Conv.rewrs_conv (map (fn th => th RS @{thm Eq_FalseI}) distincts)))
wenzelm@50422
   648
                        Conv.all_conv then_conv
wenzelm@50422
   649
                        (rewr_conv' @{lemma "(False & P) = False" by simp}))) context then_conv
wenzelm@51314
   650
                      HOLogic.Trueprop_conv
wenzelm@51315
   651
                        (HOLogic.eq_conv Conv.all_conv
wenzelm@50422
   652
                          (Collect_conv (fn (_, ctxt) =>
wenzelm@50422
   653
                            Conv.repeat_conv
wenzelm@50422
   654
                              (Conv.bottom_conv
wenzelm@50422
   655
                                (K (rewr_conv'
wenzelm@50422
   656
                                  @{lemma "(EX x. P) = P" by simp})) ctxt)) context))) 1) ctxt 1
blanchet@54404
   657
                THEN rtac set_Nil_I 1)) case_thms)
wenzelm@50422
   658
          end
wenzelm@50422
   659
    fun make_inner_eqs bound_vs Tis eqs t =
wenzelm@50422
   660
      (case dest_case t of
blanchet@54404
   661
        SOME (x, T, i, cont, constr_name) =>
wenzelm@50422
   662
          let
wenzelm@52131
   663
            val (vs, body) = strip_abs (Envir.eta_long (map snd bound_vs) cont)
wenzelm@50422
   664
            val x' = incr_boundvars (length vs) x
wenzelm@50422
   665
            val eqs' = map (incr_boundvars (length vs)) eqs
wenzelm@50422
   666
            val constr_t =
wenzelm@50422
   667
              list_comb
wenzelm@50422
   668
                (Const (constr_name, map snd vs ---> T), map Bound (((length vs) - 1) downto 0))
wenzelm@50422
   669
            val constr_eq = Const (@{const_name HOL.eq}, T --> T --> @{typ bool}) $ constr_t $ x'
wenzelm@50422
   670
          in
wenzelm@50422
   671
            make_inner_eqs (rev vs @ bound_vs) (Case (T, i) :: Tis) (constr_eq :: eqs') body
wenzelm@50422
   672
          end
wenzelm@50422
   673
      | NONE =>
wenzelm@50422
   674
          (case dest_if t of
wenzelm@50422
   675
            SOME (condition, cont) => make_inner_eqs bound_vs (If :: Tis) (condition :: eqs) cont
wenzelm@50422
   676
          | NONE =>
wenzelm@50422
   677
            if eqs = [] then NONE (* no rewriting, nothing to be done *)
wenzelm@50422
   678
            else
wenzelm@50422
   679
              let
wenzelm@50422
   680
                val Type (@{type_name List.list}, [rT]) = fastype_of1 (map snd bound_vs, t)
wenzelm@50422
   681
                val pat_eq =
wenzelm@50422
   682
                  (case try dest_singleton_list t of
wenzelm@50422
   683
                    SOME t' =>
wenzelm@50422
   684
                      Const (@{const_name HOL.eq}, rT --> rT --> @{typ bool}) $
wenzelm@50422
   685
                        Bound (length bound_vs) $ t'
wenzelm@50422
   686
                  | NONE =>
wenzelm@50422
   687
                      Const (@{const_name Set.member}, rT --> HOLogic.mk_setT rT --> @{typ bool}) $
wenzelm@50422
   688
                        Bound (length bound_vs) $ (mk_set rT $ t))
wenzelm@50422
   689
                val reverse_bounds = curry subst_bounds
wenzelm@50422
   690
                  ((map Bound ((length bound_vs - 1) downto 0)) @ [Bound (length bound_vs)])
wenzelm@50422
   691
                val eqs' = map reverse_bounds eqs
wenzelm@50422
   692
                val pat_eq' = reverse_bounds pat_eq
wenzelm@50422
   693
                val inner_t =
wenzelm@50422
   694
                  fold (fn (_, T) => fn t => HOLogic.exists_const T $ absdummy T t)
wenzelm@50422
   695
                    (rev bound_vs) (fold (curry HOLogic.mk_conj) eqs' pat_eq')
wenzelm@50422
   696
                val lhs = term_of redex
wenzelm@50422
   697
                val rhs = HOLogic.mk_Collect ("x", rT, inner_t)
wenzelm@50422
   698
                val rewrite_rule_t = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
wenzelm@50422
   699
              in
wenzelm@50422
   700
                SOME
wenzelm@50422
   701
                  ((Goal.prove ctxt [] [] rewrite_rule_t
wenzelm@50422
   702
                    (fn {context, ...} => tac context (rev Tis))) RS @{thm eq_reflection})
wenzelm@50422
   703
              end))
wenzelm@50422
   704
  in
wenzelm@50422
   705
    make_inner_eqs [] [] [] (dest_set (term_of redex))
wenzelm@50422
   706
  end
wenzelm@50422
   707
wenzelm@50422
   708
end
wenzelm@50422
   709
*}
bulwahn@41463
   710
bulwahn@41463
   711
simproc_setup list_to_set_comprehension ("set xs") = {* K List_to_Set_Comprehension.simproc *}
bulwahn@41463
   712
haftmann@46133
   713
code_datatype set coset
haftmann@46133
   714
haftmann@46133
   715
hide_const (open) coset
wenzelm@35115
   716
haftmann@49948
   717
haftmann@21061
   718
subsubsection {* @{const Nil} and @{const Cons} *}
haftmann@21061
   719
haftmann@21061
   720
lemma not_Cons_self [simp]:
haftmann@21061
   721
  "xs \<noteq> x # xs"
nipkow@13145
   722
by (induct xs) auto
wenzelm@13114
   723
wenzelm@41697
   724
lemma not_Cons_self2 [simp]:
wenzelm@41697
   725
  "x # xs \<noteq> xs"
wenzelm@41697
   726
by (rule not_Cons_self [symmetric])
wenzelm@13114
   727
wenzelm@13142
   728
lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
nipkow@13145
   729
by (induct xs) auto
wenzelm@13114
   730
nipkow@53689
   731
lemma tl_Nil: "tl xs = [] \<longleftrightarrow> xs = [] \<or> (EX x. xs = [x])"
nipkow@53689
   732
by (cases xs) auto
nipkow@53689
   733
nipkow@53689
   734
lemma Nil_tl: "[] = tl xs \<longleftrightarrow> xs = [] \<or> (EX x. xs = [x])"
nipkow@53689
   735
by (cases xs) auto
nipkow@53689
   736
wenzelm@13142
   737
lemma length_induct:
haftmann@21061
   738
  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
nipkow@53689
   739
by (fact measure_induct)
wenzelm@13114
   740
haftmann@37289
   741
lemma list_nonempty_induct [consumes 1, case_names single cons]:
haftmann@37289
   742
  assumes "xs \<noteq> []"
haftmann@37289
   743
  assumes single: "\<And>x. P [x]"
haftmann@37289
   744
  assumes cons: "\<And>x xs. xs \<noteq> [] \<Longrightarrow> P xs \<Longrightarrow> P (x # xs)"
haftmann@37289
   745
  shows "P xs"
haftmann@37289
   746
using `xs \<noteq> []` proof (induct xs)
haftmann@37289
   747
  case Nil then show ?case by simp
haftmann@37289
   748
next
wenzelm@53374
   749
  case (Cons x xs)
wenzelm@53374
   750
  show ?case
wenzelm@53374
   751
  proof (cases xs)
wenzelm@53374
   752
    case Nil
wenzelm@53374
   753
    with single show ?thesis by simp
haftmann@37289
   754
  next
wenzelm@53374
   755
    case Cons
wenzelm@53374
   756
    show ?thesis
wenzelm@53374
   757
    proof (rule cons)
wenzelm@53374
   758
      from Cons show "xs \<noteq> []" by simp
wenzelm@53374
   759
      with Cons.hyps show "P xs" .
wenzelm@53374
   760
    qed
haftmann@37289
   761
  qed
haftmann@37289
   762
qed
haftmann@37289
   763
hoelzl@45714
   764
lemma inj_split_Cons: "inj_on (\<lambda>(xs, n). n#xs) X"
hoelzl@45714
   765
  by (auto intro!: inj_onI)
wenzelm@13114
   766
haftmann@49948
   767
haftmann@21061
   768
subsubsection {* @{const length} *}
wenzelm@13114
   769
wenzelm@13142
   770
text {*
haftmann@21061
   771
  Needs to come before @{text "@"} because of theorem @{text
haftmann@21061
   772
  append_eq_append_conv}.
wenzelm@13142
   773
*}
wenzelm@13114
   774
wenzelm@13142
   775
lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
nipkow@13145
   776
by (induct xs) auto
wenzelm@13114
   777
wenzelm@13142
   778
lemma length_map [simp]: "length (map f xs) = length xs"
nipkow@13145
   779
by (induct xs) auto
wenzelm@13114
   780
wenzelm@13142
   781
lemma length_rev [simp]: "length (rev xs) = length xs"
nipkow@13145
   782
by (induct xs) auto
wenzelm@13114
   783
wenzelm@13142
   784
lemma length_tl [simp]: "length (tl xs) = length xs - 1"
nipkow@13145
   785
by (cases xs) auto
wenzelm@13114
   786
wenzelm@13142
   787
lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
nipkow@13145
   788
by (induct xs) auto
wenzelm@13114
   789
wenzelm@13142
   790
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
nipkow@13145
   791
by (induct xs) auto
wenzelm@13114
   792
nipkow@23479
   793
lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
nipkow@23479
   794
by auto
nipkow@23479
   795
wenzelm@13114
   796
lemma length_Suc_conv:
nipkow@13145
   797
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
nipkow@13145
   798
by (induct xs) auto
wenzelm@13142
   799
nipkow@14025
   800
lemma Suc_length_conv:
nipkow@14025
   801
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
paulson@14208
   802
apply (induct xs, simp, simp)
nipkow@14025
   803
apply blast
nipkow@14025
   804
done
nipkow@14025
   805
wenzelm@25221
   806
lemma impossible_Cons: "length xs <= length ys ==> xs = x # ys = False"
wenzelm@25221
   807
  by (induct xs) auto
wenzelm@25221
   808
haftmann@26442
   809
lemma list_induct2 [consumes 1, case_names Nil Cons]:
haftmann@26442
   810
  "length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
haftmann@26442
   811
   (\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
haftmann@26442
   812
   \<Longrightarrow> P xs ys"
haftmann@26442
   813
proof (induct xs arbitrary: ys)
haftmann@26442
   814
  case Nil then show ?case by simp
haftmann@26442
   815
next
haftmann@26442
   816
  case (Cons x xs ys) then show ?case by (cases ys) simp_all
haftmann@26442
   817
qed
haftmann@26442
   818
haftmann@26442
   819
lemma list_induct3 [consumes 2, case_names Nil Cons]:
haftmann@26442
   820
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
haftmann@26442
   821
   (\<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
   822
   \<Longrightarrow> P xs ys zs"
haftmann@26442
   823
proof (induct xs arbitrary: ys zs)
haftmann@26442
   824
  case Nil then show ?case by simp
haftmann@26442
   825
next
haftmann@26442
   826
  case (Cons x xs ys zs) then show ?case by (cases ys, simp_all)
haftmann@26442
   827
    (cases zs, simp_all)
haftmann@26442
   828
qed
wenzelm@13114
   829
kaliszyk@36154
   830
lemma list_induct4 [consumes 3, case_names Nil Cons]:
kaliszyk@36154
   831
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow>
kaliszyk@36154
   832
   P [] [] [] [] \<Longrightarrow> (\<And>x xs y ys z zs w ws. length xs = length ys \<Longrightarrow>
kaliszyk@36154
   833
   length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow> P xs ys zs ws \<Longrightarrow>
kaliszyk@36154
   834
   P (x#xs) (y#ys) (z#zs) (w#ws)) \<Longrightarrow> P xs ys zs ws"
kaliszyk@36154
   835
proof (induct xs arbitrary: ys zs ws)
kaliszyk@36154
   836
  case Nil then show ?case by simp
kaliszyk@36154
   837
next
kaliszyk@36154
   838
  case (Cons x xs ys zs ws) then show ?case by ((cases ys, simp_all), (cases zs,simp_all)) (cases ws, simp_all)
kaliszyk@36154
   839
qed
kaliszyk@36154
   840
krauss@22493
   841
lemma list_induct2': 
krauss@22493
   842
  "\<lbrakk> P [] [];
krauss@22493
   843
  \<And>x xs. P (x#xs) [];
krauss@22493
   844
  \<And>y ys. P [] (y#ys);
krauss@22493
   845
   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
krauss@22493
   846
 \<Longrightarrow> P xs ys"
krauss@22493
   847
by (induct xs arbitrary: ys) (case_tac x, auto)+
krauss@22493
   848
blanchet@55524
   849
lemma list_all2_iff:
blanchet@55524
   850
  "list_all2 P xs ys \<longleftrightarrow> length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)"
blanchet@55524
   851
by (induct xs ys rule: list_induct2') auto
blanchet@55524
   852
nipkow@22143
   853
lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
nipkow@24349
   854
by (rule Eq_FalseI) auto
wenzelm@24037
   855
wenzelm@24037
   856
simproc_setup list_neq ("(xs::'a list) = ys") = {*
nipkow@22143
   857
(*
nipkow@22143
   858
Reduces xs=ys to False if xs and ys cannot be of the same length.
nipkow@22143
   859
This is the case if the atomic sublists of one are a submultiset
nipkow@22143
   860
of those of the other list and there are fewer Cons's in one than the other.
nipkow@22143
   861
*)
wenzelm@24037
   862
wenzelm@24037
   863
let
nipkow@22143
   864
huffman@29856
   865
fun len (Const(@{const_name Nil},_)) acc = acc
huffman@29856
   866
  | len (Const(@{const_name Cons},_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
huffman@29856
   867
  | len (Const(@{const_name append},_) $ xs $ ys) acc = len xs (len ys acc)
huffman@29856
   868
  | len (Const(@{const_name rev},_) $ xs) acc = len xs acc
huffman@29856
   869
  | len (Const(@{const_name map},_) $ _ $ xs) acc = len xs acc
nipkow@22143
   870
  | len t (ts,n) = (t::ts,n);
nipkow@22143
   871
wenzelm@51717
   872
val ss = simpset_of @{context};
wenzelm@51717
   873
wenzelm@51717
   874
fun list_neq ctxt ct =
nipkow@22143
   875
  let
wenzelm@24037
   876
    val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
nipkow@22143
   877
    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
nipkow@22143
   878
    fun prove_neq() =
nipkow@22143
   879
      let
nipkow@22143
   880
        val Type(_,listT::_) = eqT;
haftmann@22994
   881
        val size = HOLogic.size_const listT;
nipkow@22143
   882
        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
nipkow@22143
   883
        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
wenzelm@51717
   884
        val thm = Goal.prove ctxt [] [] neq_len
wenzelm@51717
   885
          (K (simp_tac (put_simpset ss ctxt) 1));
haftmann@22633
   886
      in SOME (thm RS @{thm neq_if_length_neq}) end
nipkow@22143
   887
  in
wenzelm@23214
   888
    if m < n andalso submultiset (op aconv) (ls,rs) orelse
wenzelm@23214
   889
       n < m andalso submultiset (op aconv) (rs,ls)
nipkow@22143
   890
    then prove_neq() else NONE
nipkow@22143
   891
  end;
wenzelm@51717
   892
in K list_neq end;
nipkow@22143
   893
*}
nipkow@22143
   894
nipkow@22143
   895
nipkow@15392
   896
subsubsection {* @{text "@"} -- append *}
wenzelm@13114
   897
wenzelm@13142
   898
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
nipkow@13145
   899
by (induct xs) auto
wenzelm@13114
   900
wenzelm@13142
   901
lemma append_Nil2 [simp]: "xs @ [] = xs"
nipkow@13145
   902
by (induct xs) auto
nipkow@3507
   903
wenzelm@13142
   904
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
nipkow@13145
   905
by (induct xs) auto
wenzelm@13114
   906
wenzelm@13142
   907
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
nipkow@13145
   908
by (induct xs) auto
wenzelm@13114
   909
wenzelm@13142
   910
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
nipkow@13145
   911
by (induct xs) auto
wenzelm@13114
   912
wenzelm@13142
   913
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
nipkow@13145
   914
by (induct xs) auto
wenzelm@13114
   915
blanchet@54147
   916
lemma append_eq_append_conv [simp]:
nipkow@24526
   917
 "length xs = length ys \<or> length us = length vs
berghofe@13883
   918
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
nipkow@24526
   919
apply (induct xs arbitrary: ys)
paulson@14208
   920
 apply (case_tac ys, simp, force)
paulson@14208
   921
apply (case_tac ys, force, simp)
nipkow@13145
   922
done
wenzelm@13142
   923
nipkow@24526
   924
lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
nipkow@24526
   925
  (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
nipkow@24526
   926
apply (induct xs arbitrary: ys zs ts)
nipkow@44890
   927
 apply fastforce
nipkow@14495
   928
apply(case_tac zs)
nipkow@14495
   929
 apply simp
nipkow@44890
   930
apply fastforce
nipkow@14495
   931
done
nipkow@14495
   932
berghofe@34910
   933
lemma same_append_eq [iff, induct_simp]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145
   934
by simp
wenzelm@13142
   935
wenzelm@13142
   936
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145
   937
by simp
wenzelm@13114
   938
berghofe@34910
   939
lemma append_same_eq [iff, induct_simp]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   940
by simp
wenzelm@13114
   941
wenzelm@13142
   942
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   943
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   944
wenzelm@13142
   945
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   946
using append_same_eq [of "[]"] by auto
wenzelm@13114
   947
blanchet@54147
   948
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   949
by (induct xs) auto
wenzelm@13114
   950
wenzelm@13142
   951
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   952
by (induct xs) auto
wenzelm@13114
   953
wenzelm@13142
   954
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   955
by (simp add: hd_append split: list.split)
wenzelm@13114
   956
wenzelm@13142
   957
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   958
by (simp split: list.split)
wenzelm@13114
   959
wenzelm@13142
   960
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   961
by (simp add: tl_append split: list.split)
wenzelm@13114
   962
wenzelm@13114
   963
nipkow@14300
   964
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   965
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   966
by(cases ys) auto
nipkow@14300
   967
nipkow@15281
   968
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   969
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   970
by(cases ys) auto
nipkow@15281
   971
nipkow@14300
   972
wenzelm@13142
   973
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   974
wenzelm@13114
   975
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   976
by simp
wenzelm@13114
   977
wenzelm@13142
   978
lemma Cons_eq_appendI:
nipkow@13145
   979
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   980
by (drule sym) simp
wenzelm@13114
   981
wenzelm@13142
   982
lemma append_eq_appendI:
nipkow@13145
   983
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   984
by (drule sym) simp
wenzelm@13114
   985
wenzelm@13114
   986
wenzelm@13142
   987
text {*
nipkow@13145
   988
Simplification procedure for all list equalities.
nipkow@13145
   989
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   990
- both lists end in a singleton list,
nipkow@13145
   991
- or both lists end in the same list.
wenzelm@13142
   992
*}
wenzelm@13142
   993
wenzelm@43594
   994
simproc_setup list_eq ("(xs::'a list) = ys")  = {*
wenzelm@13462
   995
  let
wenzelm@43594
   996
    fun last (cons as Const (@{const_name Cons}, _) $ _ $ xs) =
wenzelm@43594
   997
          (case xs of Const (@{const_name Nil}, _) => cons | _ => last xs)
wenzelm@43594
   998
      | last (Const(@{const_name append},_) $ _ $ ys) = last ys
wenzelm@43594
   999
      | last t = t;
wenzelm@43594
  1000
    
wenzelm@43594
  1001
    fun list1 (Const(@{const_name Cons},_) $ _ $ Const(@{const_name Nil},_)) = true
wenzelm@43594
  1002
      | list1 _ = false;
wenzelm@43594
  1003
    
wenzelm@43594
  1004
    fun butlast ((cons as Const(@{const_name Cons},_) $ x) $ xs) =
wenzelm@43594
  1005
          (case xs of Const (@{const_name Nil}, _) => xs | _ => cons $ butlast xs)
wenzelm@43594
  1006
      | butlast ((app as Const (@{const_name append}, _) $ xs) $ ys) = app $ butlast ys
wenzelm@43594
  1007
      | butlast xs = Const(@{const_name Nil}, fastype_of xs);
wenzelm@43594
  1008
    
wenzelm@43594
  1009
    val rearr_ss =
wenzelm@51717
  1010
      simpset_of (put_simpset HOL_basic_ss @{context}
wenzelm@51717
  1011
        addsimps [@{thm append_assoc}, @{thm append_Nil}, @{thm append_Cons}]);
wenzelm@43594
  1012
    
wenzelm@51717
  1013
    fun list_eq ctxt (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
  1014
      let
wenzelm@43594
  1015
        val lastl = last lhs and lastr = last rhs;
wenzelm@43594
  1016
        fun rearr conv =
wenzelm@43594
  1017
          let
wenzelm@43594
  1018
            val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@43594
  1019
            val Type(_,listT::_) = eqT
wenzelm@43594
  1020
            val appT = [listT,listT] ---> listT
wenzelm@43594
  1021
            val app = Const(@{const_name append},appT)
wenzelm@43594
  1022
            val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@43594
  1023
            val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@51717
  1024
            val thm = Goal.prove ctxt [] [] eq
wenzelm@51717
  1025
              (K (simp_tac (put_simpset rearr_ss ctxt) 1));
wenzelm@43594
  1026
          in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@43594
  1027
      in
wenzelm@43594
  1028
        if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
wenzelm@43594
  1029
        else if lastl aconv lastr then rearr @{thm append_same_eq}
wenzelm@43594
  1030
        else NONE
wenzelm@43594
  1031
      end;
wenzelm@51717
  1032
  in fn _ => fn ctxt => fn ct => list_eq ctxt (term_of ct) end;
wenzelm@13114
  1033
*}
wenzelm@13114
  1034
wenzelm@13114
  1035
haftmann@49948
  1036
subsubsection {* @{const map} *}
wenzelm@13114
  1037
haftmann@40210
  1038
lemma hd_map:
haftmann@40210
  1039
  "xs \<noteq> [] \<Longrightarrow> hd (map f xs) = f (hd xs)"
haftmann@40210
  1040
  by (cases xs) simp_all
haftmann@40210
  1041
haftmann@40210
  1042
lemma map_tl:
haftmann@40210
  1043
  "map f (tl xs) = tl (map f xs)"
haftmann@40210
  1044
  by (cases xs) simp_all
haftmann@40210
  1045
wenzelm@13142
  1046
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
  1047
by (induct xs) simp_all
wenzelm@13114
  1048
wenzelm@13142
  1049
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
  1050
by (rule ext, induct_tac xs) auto
wenzelm@13114
  1051
wenzelm@13142
  1052
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
  1053
by (induct xs) auto
wenzelm@13114
  1054
hoelzl@33639
  1055
lemma map_map [simp]: "map f (map g xs) = map (f \<circ> g) xs"
hoelzl@33639
  1056
by (induct xs) auto
hoelzl@33639
  1057
nipkow@35208
  1058
lemma map_comp_map[simp]: "((map f) o (map g)) = map(f o g)"
nipkow@35208
  1059
apply(rule ext)
nipkow@35208
  1060
apply(simp)
nipkow@35208
  1061
done
nipkow@35208
  1062
wenzelm@13142
  1063
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
  1064
by (induct xs) auto
wenzelm@13114
  1065
nipkow@13737
  1066
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
  1067
by (induct xs) auto
nipkow@13737
  1068
krauss@44013
  1069
lemma map_cong [fundef_cong]:
haftmann@40122
  1070
  "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> map f xs = map g ys"
haftmann@40122
  1071
  by simp
wenzelm@13114
  1072
wenzelm@13142
  1073
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
  1074
by (cases xs) auto
wenzelm@13114
  1075
wenzelm@13142
  1076
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
  1077
by (cases xs) auto
wenzelm@13114
  1078
paulson@18447
  1079
lemma map_eq_Cons_conv:
nipkow@14025
  1080
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
  1081
by (cases xs) auto
wenzelm@13114
  1082
paulson@18447
  1083
lemma Cons_eq_map_conv:
nipkow@14025
  1084
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
  1085
by (cases ys) auto
nipkow@14025
  1086
paulson@18447
  1087
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
paulson@18447
  1088
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
paulson@18447
  1089
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
paulson@18447
  1090
nipkow@14111
  1091
lemma ex_map_conv:
nipkow@14111
  1092
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
paulson@18447
  1093
by(induct ys, auto simp add: Cons_eq_map_conv)
nipkow@14111
  1094
nipkow@15110
  1095
lemma map_eq_imp_length_eq:
paulson@35510
  1096
  assumes "map f xs = map g ys"
haftmann@26734
  1097
  shows "length xs = length ys"
wenzelm@53374
  1098
  using assms
wenzelm@53374
  1099
proof (induct ys arbitrary: xs)
haftmann@26734
  1100
  case Nil then show ?case by simp
haftmann@26734
  1101
next
haftmann@26734
  1102
  case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto
paulson@35510
  1103
  from Cons xs have "map f zs = map g ys" by simp
wenzelm@53374
  1104
  with Cons have "length zs = length ys" by blast
haftmann@26734
  1105
  with xs show ?case by simp
haftmann@26734
  1106
qed
haftmann@26734
  1107
  
nipkow@15110
  1108
lemma map_inj_on:
nipkow@15110
  1109
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
  1110
  ==> xs = ys"
nipkow@15110
  1111
apply(frule map_eq_imp_length_eq)
nipkow@15110
  1112
apply(rotate_tac -1)
nipkow@15110
  1113
apply(induct rule:list_induct2)
nipkow@15110
  1114
 apply simp
nipkow@15110
  1115
apply(simp)
nipkow@15110
  1116
apply (blast intro:sym)
nipkow@15110
  1117
done
nipkow@15110
  1118
nipkow@15110
  1119
lemma inj_on_map_eq_map:
nipkow@15110
  1120
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
  1121
by(blast dest:map_inj_on)
nipkow@15110
  1122
wenzelm@13114
  1123
lemma map_injective:
nipkow@24526
  1124
 "map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@24526
  1125
by (induct ys arbitrary: xs) (auto dest!:injD)
wenzelm@13114
  1126
nipkow@14339
  1127
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
  1128
by(blast dest:map_injective)
nipkow@14339
  1129
wenzelm@13114
  1130
lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@17589
  1131
by (iprover dest: map_injective injD intro: inj_onI)
wenzelm@13114
  1132
wenzelm@13114
  1133
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
  1134
apply (unfold inj_on_def, clarify)
nipkow@13145
  1135
apply (erule_tac x = "[x]" in ballE)
paulson@14208
  1136
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
  1137
apply blast
nipkow@13145
  1138
done
wenzelm@13114
  1139
nipkow@14339
  1140
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
  1141
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
  1142
nipkow@15303
  1143
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
  1144
apply(rule inj_onI)
nipkow@15303
  1145
apply(erule map_inj_on)
nipkow@15303
  1146
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
  1147
done
nipkow@15303
  1148
kleing@14343
  1149
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
  1150
by (induct xs, auto)
wenzelm@13114
  1151
nipkow@14402
  1152
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
  1153
by (induct xs) auto
nipkow@14402
  1154
nipkow@15110
  1155
lemma map_fst_zip[simp]:
nipkow@15110
  1156
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
  1157
by (induct rule:list_induct2, simp_all)
nipkow@15110
  1158
nipkow@15110
  1159
lemma map_snd_zip[simp]:
nipkow@15110
  1160
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
  1161
by (induct rule:list_induct2, simp_all)
nipkow@15110
  1162
blanchet@55467
  1163
functor map: map
nipkow@47122
  1164
by (simp_all add: id_def)
nipkow@47122
  1165
haftmann@49948
  1166
declare map.id [simp]
haftmann@49948
  1167
haftmann@49948
  1168
haftmann@49948
  1169
subsubsection {* @{const rev} *}
wenzelm@13114
  1170
wenzelm@13142
  1171
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
  1172
by (induct xs) auto
wenzelm@13114
  1173
wenzelm@13142
  1174
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
  1175
by (induct xs) auto
wenzelm@13114
  1176
kleing@15870
  1177
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870
  1178
by auto
kleing@15870
  1179
wenzelm@13142
  1180
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
  1181
by (induct xs) auto
wenzelm@13114
  1182
wenzelm@13142
  1183
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
  1184
by (induct xs) auto
wenzelm@13114
  1185
kleing@15870
  1186
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870
  1187
by (cases xs) auto
kleing@15870
  1188
kleing@15870
  1189
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870
  1190
by (cases xs) auto
kleing@15870
  1191
blanchet@54147
  1192
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
haftmann@21061
  1193
apply (induct xs arbitrary: ys, force)
paulson@14208
  1194
apply (case_tac ys, simp, force)
nipkow@13145
  1195
done
wenzelm@13114
  1196
nipkow@15439
  1197
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
  1198
by(simp add:inj_on_def)
nipkow@15439
  1199
wenzelm@13366
  1200
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
  1201
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489
  1202
apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145
  1203
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
  1204
done
wenzelm@13114
  1205
wenzelm@13366
  1206
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
  1207
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
  1208
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1209
wenzelm@13366
  1210
lemmas rev_cases = rev_exhaust
wenzelm@13366
  1211
nipkow@18423
  1212
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
nipkow@18423
  1213
by(rule rev_cases[of xs]) auto
nipkow@18423
  1214
wenzelm@13114
  1215
haftmann@49948
  1216
subsubsection {* @{const set} *}
wenzelm@13114
  1217
nipkow@46698
  1218
declare set.simps [code_post]  --"pretty output"
nipkow@46698
  1219
wenzelm@13142
  1220
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
  1221
by (induct xs) auto
wenzelm@13114
  1222
wenzelm@13142
  1223
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
  1224
by (induct xs) auto
wenzelm@13114
  1225
nipkow@17830
  1226
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
nipkow@17830
  1227
by(cases xs) auto
oheimb@14099
  1228
wenzelm@13142
  1229
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
  1230
by auto
wenzelm@13114
  1231
oheimb@14099
  1232
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
  1233
by auto
oheimb@14099
  1234
wenzelm@13142
  1235
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
  1236
by (induct xs) auto
wenzelm@13114
  1237
nipkow@15245
  1238
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
  1239
by(induct xs) auto
nipkow@15245
  1240
wenzelm@13142
  1241
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
  1242
by (induct xs) auto
wenzelm@13114
  1243
wenzelm@13142
  1244
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
  1245
by (induct xs) auto
wenzelm@13114
  1246
wenzelm@13142
  1247
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
  1248
by (induct xs) auto
wenzelm@13114
  1249
nipkow@32417
  1250
lemma set_upt [simp]: "set[i..<j] = {i..<j}"
bulwahn@41463
  1251
by (induct j) auto
wenzelm@13114
  1252
wenzelm@13142
  1253
wenzelm@25221
  1254
lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs"
nipkow@18049
  1255
proof (induct xs)
nipkow@26073
  1256
  case Nil thus ?case by simp
nipkow@26073
  1257
next
nipkow@26073
  1258
  case Cons thus ?case by (auto intro: Cons_eq_appendI)
nipkow@26073
  1259
qed
nipkow@26073
  1260
haftmann@26734
  1261
lemma in_set_conv_decomp: "x \<in> set xs \<longleftrightarrow> (\<exists>ys zs. xs = ys @ x # zs)"
haftmann@26734
  1262
  by (auto elim: split_list)
nipkow@26073
  1263
nipkow@26073
  1264
lemma split_list_first: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@26073
  1265
proof (induct xs)
nipkow@26073
  1266
  case Nil thus ?case by simp
nipkow@18049
  1267
next
nipkow@18049
  1268
  case (Cons a xs)
nipkow@18049
  1269
  show ?case
nipkow@18049
  1270
  proof cases
nipkow@44890
  1271
    assume "x = a" thus ?case using Cons by fastforce
nipkow@18049
  1272
  next
nipkow@44890
  1273
    assume "x \<noteq> a" thus ?case using Cons by(fastforce intro!: Cons_eq_appendI)
nipkow@26073
  1274
  qed
nipkow@26073
  1275
qed
nipkow@26073
  1276
nipkow@26073
  1277
lemma in_set_conv_decomp_first:
nipkow@26073
  1278
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
haftmann@26734
  1279
  by (auto dest!: split_list_first)
nipkow@26073
  1280
haftmann@40122
  1281
lemma split_list_last: "x \<in> set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
haftmann@40122
  1282
proof (induct xs rule: rev_induct)
nipkow@26073
  1283
  case Nil thus ?case by simp
nipkow@26073
  1284
next
nipkow@26073
  1285
  case (snoc a xs)
nipkow@26073
  1286
  show ?case
nipkow@26073
  1287
  proof cases
haftmann@40122
  1288
    assume "x = a" thus ?case using snoc by (metis List.set.simps(1) emptyE)
nipkow@26073
  1289
  next
nipkow@44890
  1290
    assume "x \<noteq> a" thus ?case using snoc by fastforce
nipkow@18049
  1291
  qed
nipkow@18049
  1292
qed
nipkow@18049
  1293
nipkow@26073
  1294
lemma in_set_conv_decomp_last:
nipkow@26073
  1295
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
haftmann@26734
  1296
  by (auto dest!: split_list_last)
nipkow@26073
  1297
nipkow@26073
  1298
lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs & P x"
nipkow@26073
  1299
proof (induct xs)
nipkow@26073
  1300
  case Nil thus ?case by simp
nipkow@26073
  1301
next
nipkow@26073
  1302
  case Cons thus ?case
nipkow@26073
  1303
    by(simp add:Bex_def)(metis append_Cons append.simps(1))
nipkow@26073
  1304
qed
nipkow@26073
  1305
nipkow@26073
  1306
lemma split_list_propE:
haftmann@26734
  1307
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
  1308
  obtains ys x zs where "xs = ys @ x # zs" and "P x"
haftmann@26734
  1309
using split_list_prop [OF assms] by blast
nipkow@26073
  1310
nipkow@26073
  1311
lemma split_list_first_prop:
nipkow@26073
  1312
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
nipkow@26073
  1313
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)"
haftmann@26734
  1314
proof (induct xs)
nipkow@26073
  1315
  case Nil thus ?case by simp
nipkow@26073
  1316
next
nipkow@26073
  1317
  case (Cons x xs)
nipkow@26073
  1318
  show ?case
nipkow@26073
  1319
  proof cases
nipkow@26073
  1320
    assume "P x"
haftmann@40122
  1321
    thus ?thesis by simp (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
nipkow@26073
  1322
  next
nipkow@26073
  1323
    assume "\<not> P x"
nipkow@26073
  1324
    hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
nipkow@26073
  1325
    thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
nipkow@26073
  1326
  qed
nipkow@26073
  1327
qed
nipkow@26073
  1328
nipkow@26073
  1329
lemma split_list_first_propE:
haftmann@26734
  1330
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
  1331
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y"
haftmann@26734
  1332
using split_list_first_prop [OF assms] by blast
nipkow@26073
  1333
nipkow@26073
  1334
lemma split_list_first_prop_iff:
nipkow@26073
  1335
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
nipkow@26073
  1336
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))"
haftmann@26734
  1337
by (rule, erule split_list_first_prop) auto
nipkow@26073
  1338
nipkow@26073
  1339
lemma split_list_last_prop:
nipkow@26073
  1340
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
nipkow@26073
  1341
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)"
nipkow@26073
  1342
proof(induct xs rule:rev_induct)
nipkow@26073
  1343
  case Nil thus ?case by simp
nipkow@26073
  1344
next
nipkow@26073
  1345
  case (snoc x xs)
nipkow@26073
  1346
  show ?case
nipkow@26073
  1347
  proof cases
nipkow@26073
  1348
    assume "P x" thus ?thesis by (metis emptyE set_empty)
nipkow@26073
  1349
  next
nipkow@26073
  1350
    assume "\<not> P x"
nipkow@26073
  1351
    hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
nipkow@44890
  1352
    thus ?thesis using `\<not> P x` snoc(1) by fastforce
nipkow@26073
  1353
  qed
nipkow@26073
  1354
qed
nipkow@26073
  1355
nipkow@26073
  1356
lemma split_list_last_propE:
haftmann@26734
  1357
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
  1358
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z"
haftmann@26734
  1359
using split_list_last_prop [OF assms] by blast
nipkow@26073
  1360
nipkow@26073
  1361
lemma split_list_last_prop_iff:
nipkow@26073
  1362
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
nipkow@26073
  1363
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
haftmann@26734
  1364
by (metis split_list_last_prop [where P=P] in_set_conv_decomp)
nipkow@26073
  1365
nipkow@26073
  1366
lemma finite_list: "finite A ==> EX xs. set xs = A"
haftmann@26734
  1367
  by (erule finite_induct)
haftmann@26734
  1368
    (auto simp add: set.simps(2) [symmetric] simp del: set.simps(2))
paulson@13508
  1369
kleing@14388
  1370
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
  1371
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
  1372
haftmann@26442
  1373
lemma set_minus_filter_out:
haftmann@26442
  1374
  "set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)"
haftmann@26442
  1375
  by (induct xs) auto
paulson@15168
  1376
wenzelm@35115
  1377
haftmann@49948
  1378
subsubsection {* @{const filter} *}
wenzelm@13114
  1379
wenzelm@13142
  1380
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
  1381
by (induct xs) auto
wenzelm@13114
  1382
nipkow@15305
  1383
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
  1384
by (induct xs) simp_all
nipkow@15305
  1385
wenzelm@13142
  1386
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
  1387
by (induct xs) auto
wenzelm@13114
  1388
nipkow@16998
  1389
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998
  1390
by (induct xs) (auto simp add: le_SucI)
nipkow@16998
  1391
nipkow@18423
  1392
lemma sum_length_filter_compl:
nipkow@18423
  1393
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
nipkow@18423
  1394
by(induct xs) simp_all
nipkow@18423
  1395
wenzelm@13142
  1396
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
  1397
by (induct xs) auto
wenzelm@13114
  1398
wenzelm@13142
  1399
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
  1400
by (induct xs) auto
wenzelm@13114
  1401
nipkow@16998
  1402
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@24349
  1403
by (induct xs) simp_all
nipkow@16998
  1404
nipkow@16998
  1405
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998
  1406
apply (induct xs)
nipkow@16998
  1407
 apply auto
nipkow@16998
  1408
apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998
  1409
apply simp
nipkow@16998
  1410
done
wenzelm@13114
  1411
nipkow@16965
  1412
lemma filter_map:
nipkow@16965
  1413
  "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965
  1414
by (induct xs) simp_all
nipkow@16965
  1415
nipkow@16965
  1416
lemma length_filter_map[simp]:
nipkow@16965
  1417
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965
  1418
by (simp add:filter_map)
nipkow@16965
  1419
wenzelm@13142
  1420
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
  1421
by auto
wenzelm@13114
  1422
nipkow@15246
  1423
lemma length_filter_less:
nipkow@15246
  1424
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
  1425
proof (induct xs)
nipkow@15246
  1426
  case Nil thus ?case by simp
nipkow@15246
  1427
next
nipkow@15246
  1428
  case (Cons x xs) thus ?case
nipkow@15246
  1429
    apply (auto split:split_if_asm)
nipkow@15246
  1430
    using length_filter_le[of P xs] apply arith
nipkow@15246
  1431
  done
nipkow@15246
  1432
qed
wenzelm@13114
  1433
nipkow@15281
  1434
lemma length_filter_conv_card:
nipkow@15281
  1435
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
  1436
proof (induct xs)
nipkow@15281
  1437
  case Nil thus ?case by simp
nipkow@15281
  1438
next
nipkow@15281
  1439
  case (Cons x xs)
nipkow@15281
  1440
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
  1441
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
  1442
  show ?case (is "?l = card ?S'")
nipkow@15281
  1443
  proof (cases)
nipkow@15281
  1444
    assume "p x"
nipkow@15281
  1445
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@25162
  1446
      by(auto simp: image_def split:nat.split dest:gr0_implies_Suc)
nipkow@15281
  1447
    have "length (filter p (x # xs)) = Suc(card ?S)"
wenzelm@23388
  1448
      using Cons `p x` by simp
nipkow@15281
  1449
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
huffman@44921
  1450
      by (simp add: card_image)
nipkow@15281
  1451
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
  1452
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
  1453
    finally show ?thesis .
nipkow@15281
  1454
  next
nipkow@15281
  1455
    assume "\<not> p x"
nipkow@15281
  1456
    hence eq: "?S' = Suc ` ?S"
nipkow@25162
  1457
      by(auto simp add: image_def split:nat.split elim:lessE)
nipkow@15281
  1458
    have "length (filter p (x # xs)) = card ?S"
wenzelm@23388
  1459
      using Cons `\<not> p x` by simp
nipkow@15281
  1460
    also have "\<dots> = card(Suc ` ?S)" using fin
huffman@44921
  1461
      by (simp add: card_image)
nipkow@15281
  1462
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
  1463
      by (simp add:card_insert_if)
nipkow@15281
  1464
    finally show ?thesis .
nipkow@15281
  1465
  qed
nipkow@15281
  1466
qed
nipkow@15281
  1467
nipkow@17629
  1468
lemma Cons_eq_filterD:
nipkow@17629
  1469
 "x#xs = filter P ys \<Longrightarrow>
nipkow@17629
  1470
  \<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
  1471
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
nipkow@17629
  1472
proof(induct ys)
nipkow@17629
  1473
  case Nil thus ?case by simp
nipkow@17629
  1474
next
nipkow@17629
  1475
  case (Cons y ys)
nipkow@17629
  1476
  show ?case (is "\<exists>x. ?Q x")
nipkow@17629
  1477
  proof cases
nipkow@17629
  1478
    assume Py: "P y"
nipkow@17629
  1479
    show ?thesis
nipkow@17629
  1480
    proof cases
wenzelm@25221
  1481
      assume "x = y"
wenzelm@25221
  1482
      with Py Cons.prems have "?Q []" by simp
wenzelm@25221
  1483
      then show ?thesis ..
nipkow@17629
  1484
    next
wenzelm@25221
  1485
      assume "x \<noteq> y"
wenzelm@25221
  1486
      with Py Cons.prems show ?thesis by simp
nipkow@17629
  1487
    qed
nipkow@17629
  1488
  next
wenzelm@25221
  1489
    assume "\<not> P y"
nipkow@44890
  1490
    with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastforce
wenzelm@25221
  1491
    then have "?Q (y#us)" by simp
wenzelm@25221
  1492
    then show ?thesis ..
nipkow@17629
  1493
  qed
nipkow@17629
  1494
qed
nipkow@17629
  1495
nipkow@17629
  1496
lemma filter_eq_ConsD:
nipkow@17629
  1497
 "filter P ys = x#xs \<Longrightarrow>
nipkow@17629
  1498
  \<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
  1499
by(rule Cons_eq_filterD) simp
nipkow@17629
  1500
nipkow@17629
  1501
lemma filter_eq_Cons_iff:
nipkow@17629
  1502
 "(filter P ys = x#xs) =
nipkow@17629
  1503
  (\<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
  1504
by(auto dest:filter_eq_ConsD)
nipkow@17629
  1505
nipkow@17629
  1506
lemma Cons_eq_filter_iff:
nipkow@17629
  1507
 "(x#xs = filter P ys) =
nipkow@17629
  1508
  (\<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
  1509
by(auto dest:Cons_eq_filterD)
nipkow@17629
  1510
krauss@44013
  1511
lemma filter_cong[fundef_cong]:
nipkow@17501
  1512
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
nipkow@17501
  1513
apply simp
nipkow@17501
  1514
apply(erule thin_rl)
nipkow@17501
  1515
by (induct ys) simp_all
nipkow@17501
  1516
nipkow@15281
  1517
haftmann@26442
  1518
subsubsection {* List partitioning *}
haftmann@26442
  1519
haftmann@26442
  1520
primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where
nipkow@50548
  1521
"partition P [] = ([], [])" |
nipkow@50548
  1522
"partition P (x # xs) = 
nipkow@50548
  1523
  (let (yes, no) = partition P xs
nipkow@50548
  1524
   in if P x then (x # yes, no) else (yes, x # no))"
haftmann@26442
  1525
haftmann@26442
  1526
lemma partition_filter1:
haftmann@26442
  1527
    "fst (partition P xs) = filter P xs"
haftmann@26442
  1528
by (induct xs) (auto simp add: Let_def split_def)
haftmann@26442
  1529
haftmann@26442
  1530
lemma partition_filter2:
haftmann@26442
  1531
    "snd (partition P xs) = filter (Not o P) xs"
haftmann@26442
  1532
by (induct xs) (auto simp add: Let_def split_def)
haftmann@26442
  1533
haftmann@26442
  1534
lemma partition_P:
haftmann@26442
  1535
  assumes "partition P xs = (yes, no)"
haftmann@26442
  1536
  shows "(\<forall>p \<in> set yes.  P p) \<and> (\<forall>p  \<in> set no. \<not> P p)"
haftmann@26442
  1537
proof -
haftmann@26442
  1538
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
haftmann@26442
  1539
    by simp_all
haftmann@26442
  1540
  then show ?thesis by (simp_all add: partition_filter1 partition_filter2)
haftmann@26442
  1541
qed
haftmann@26442
  1542
haftmann@26442
  1543
lemma partition_set:
haftmann@26442
  1544
  assumes "partition P xs = (yes, no)"
haftmann@26442
  1545
  shows "set yes \<union> set no = set xs"
haftmann@26442
  1546
proof -
haftmann@26442
  1547
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
haftmann@26442
  1548
    by simp_all
haftmann@26442
  1549
  then show ?thesis by (auto simp add: partition_filter1 partition_filter2) 
haftmann@26442
  1550
qed
haftmann@26442
  1551
hoelzl@33639
  1552
lemma partition_filter_conv[simp]:
hoelzl@33639
  1553
  "partition f xs = (filter f xs,filter (Not o f) xs)"
hoelzl@33639
  1554
unfolding partition_filter2[symmetric]
hoelzl@33639
  1555
unfolding partition_filter1[symmetric] by simp
hoelzl@33639
  1556
hoelzl@33639
  1557
declare partition.simps[simp del]
haftmann@26442
  1558
wenzelm@35115
  1559
haftmann@49948
  1560
subsubsection {* @{const concat} *}
wenzelm@13114
  1561
wenzelm@13142
  1562
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
  1563
by (induct xs) auto
wenzelm@13114
  1564
paulson@18447
  1565
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
  1566
by (induct xss) auto
wenzelm@13114
  1567
paulson@18447
  1568
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
  1569
by (induct xss) auto
wenzelm@13114
  1570
nipkow@24308
  1571
lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
nipkow@13145
  1572
by (induct xs) auto
wenzelm@13114
  1573
nipkow@24476
  1574
lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs"
nipkow@24349
  1575
by (induct xs) auto
nipkow@24349
  1576
wenzelm@13142
  1577
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
  1578
by (induct xs) auto
wenzelm@13114
  1579
wenzelm@13142
  1580
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
  1581
by (induct xs) auto
wenzelm@13114
  1582
wenzelm@13142
  1583
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
  1584
by (induct xs) auto
wenzelm@13114
  1585
bulwahn@40365
  1586
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@40365
  1587
proof (induct xs arbitrary: ys)
bulwahn@40365
  1588
  case (Cons x xs ys)
bulwahn@40365
  1589
  thus ?case by (cases ys) auto
bulwahn@40365
  1590
qed (auto)
bulwahn@40365
  1591
bulwahn@40365
  1592
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@40365
  1593
by (simp add: concat_eq_concat_iff)
bulwahn@40365
  1594
wenzelm@13114
  1595
haftmann@49948
  1596
subsubsection {* @{const nth} *}
wenzelm@13114
  1597
haftmann@29827
  1598
lemma nth_Cons_0 [simp, code]: "(x # xs)!0 = x"
nipkow@13145
  1599
by auto
wenzelm@13114
  1600
haftmann@29827
  1601
lemma nth_Cons_Suc [simp, code]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
  1602
by auto
wenzelm@13114
  1603
wenzelm@13142
  1604
declare nth.simps [simp del]
wenzelm@13114
  1605
nipkow@41842
  1606
lemma nth_Cons_pos[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
nipkow@41842
  1607
by(auto simp: Nat.gr0_conv_Suc)
nipkow@41842
  1608
wenzelm@13114
  1609
lemma nth_append:
nipkow@24526
  1610
  "(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
nipkow@24526
  1611
apply (induct xs arbitrary: n, simp)
paulson@14208
  1612
apply (case_tac n, auto)
nipkow@13145
  1613
done
wenzelm@13114
  1614
nipkow@14402
  1615
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
wenzelm@25221
  1616
by (induct xs) auto
nipkow@14402
  1617
nipkow@14402
  1618
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
wenzelm@25221
  1619
by (induct xs) auto
nipkow@14402
  1620
nipkow@24526
  1621
lemma nth_map [simp]: "n < length xs ==> (map f xs)!n = f(xs!n)"
nipkow@24526
  1622
apply (induct xs arbitrary: n, simp)
paulson@14208
  1623
apply (case_tac n, auto)
nipkow@13145
  1624
done
wenzelm@13114
  1625
noschinl@45841
  1626
lemma nth_tl:
noschinl@45841
  1627
  assumes "n < length (tl x)" shows "tl x ! n = x ! Suc n"
noschinl@45841
  1628
using assms by (induct x) auto
noschinl@45841
  1629
nipkow@18423
  1630
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
nipkow@18423
  1631
by(cases xs) simp_all
nipkow@18423
  1632
nipkow@18049
  1633
nipkow@18049
  1634
lemma list_eq_iff_nth_eq:
nipkow@24526
  1635
 "(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
nipkow@24526
  1636
apply(induct xs arbitrary: ys)
paulson@24632
  1637
 apply force
nipkow@18049
  1638
apply(case_tac ys)
nipkow@18049
  1639
 apply simp
nipkow@18049
  1640
apply(simp add:nth_Cons split:nat.split)apply blast
nipkow@18049
  1641
done
nipkow@18049
  1642
wenzelm@13142
  1643
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
  1644
apply (induct xs, simp, simp)
nipkow@13145
  1645
apply safe
blanchet@55417
  1646
apply (metis nat.cases(1) nth.simps zero_less_Suc)
paulson@24632
  1647
apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
paulson@14208
  1648
apply (case_tac i, simp)
blanchet@55417
  1649
apply (metis diff_Suc_Suc nat.cases(2) nth.simps zero_less_diff)
nipkow@13145
  1650
done
wenzelm@13114
  1651
nipkow@17501
  1652
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
nipkow@17501
  1653
by(auto simp:set_conv_nth)
nipkow@17501
  1654
haftmann@51160
  1655
lemma nth_equal_first_eq:
haftmann@51160
  1656
  assumes "x \<notin> set xs"
haftmann@51160
  1657
  assumes "n \<le> length xs"
haftmann@51160
  1658
  shows "(x # xs) ! n = x \<longleftrightarrow> n = 0" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@51160
  1659
proof
haftmann@51160
  1660
  assume ?lhs
haftmann@51160
  1661
  show ?rhs
haftmann@51160
  1662
  proof (rule ccontr)
haftmann@51160
  1663
    assume "n \<noteq> 0"
haftmann@51160
  1664
    then have "n > 0" by simp
haftmann@51160
  1665
    with `?lhs` have "xs ! (n - 1) = x" by simp
haftmann@51160
  1666
    moreover from `n > 0` `n \<le> length xs` have "n - 1 < length xs" by simp
haftmann@51160
  1667
    ultimately have "\<exists>i<length xs. xs ! i = x" by auto
haftmann@51160
  1668
    with `x \<notin> set xs` in_set_conv_nth [of x xs] show False by simp
haftmann@51160
  1669
  qed
haftmann@51160
  1670
next
haftmann@51160
  1671
  assume ?rhs then show ?lhs by simp
haftmann@51160
  1672
qed
haftmann@51160
  1673
haftmann@51160
  1674
lemma nth_non_equal_first_eq:
haftmann@51160
  1675
  assumes "x \<noteq> y"
haftmann@51160
  1676
  shows "(x # xs) ! n = y \<longleftrightarrow> xs ! (n - 1) = y \<and> n > 0" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@51160
  1677
proof
haftmann@51160
  1678
  assume "?lhs" with assms have "n > 0" by (cases n) simp_all
haftmann@51160
  1679
  with `?lhs` show ?rhs by simp
haftmann@51160
  1680
next
haftmann@51160
  1681
  assume "?rhs" then show "?lhs" by simp
haftmann@51160
  1682
qed
haftmann@51160
  1683
nipkow@13145
  1684
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
  1685
by (auto simp add: set_conv_nth)
wenzelm@13114
  1686
wenzelm@13142
  1687
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
  1688
by (auto simp add: set_conv_nth)
wenzelm@13114
  1689
wenzelm@13114
  1690
lemma all_nth_imp_all_set:
nipkow@13145
  1691
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
  1692
by (auto simp add: set_conv_nth)
wenzelm@13114
  1693
wenzelm@13114
  1694
lemma all_set_conv_all_nth:
nipkow@13145
  1695
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
  1696
by (auto simp add: set_conv_nth)
wenzelm@13114
  1697
kleing@25296
  1698
lemma rev_nth:
kleing@25296
  1699
  "n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)"
kleing@25296
  1700
proof (induct xs arbitrary: n)
kleing@25296
  1701
  case Nil thus ?case by simp
kleing@25296
  1702
next
kleing@25296
  1703
  case (Cons x xs)
kleing@25296
  1704
  hence n: "n < Suc (length xs)" by simp
kleing@25296
  1705
  moreover
kleing@25296
  1706
  { assume "n < length xs"
wenzelm@53374
  1707
    with n obtain n' where n': "length xs - n = Suc n'"
kleing@25296
  1708
      by (cases "length xs - n", auto)
kleing@25296
  1709
    moreover
wenzelm@53374
  1710
    from n' have "length xs - Suc n = n'" by simp
kleing@25296
  1711
    ultimately
kleing@25296
  1712
    have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp
kleing@25296
  1713
  }
kleing@25296
  1714
  ultimately
kleing@25296
  1715
  show ?case by (clarsimp simp add: Cons nth_append)
kleing@25296
  1716
qed
wenzelm@13114
  1717
nipkow@31159
  1718
lemma Skolem_list_nth:
nipkow@31159
  1719
  "(ALL i<k. EX x. P i x) = (EX xs. size xs = k & (ALL i<k. P i (xs!i)))"
nipkow@31159
  1720
  (is "_ = (EX xs. ?P k xs)")
nipkow@31159
  1721
proof(induct k)
nipkow@31159
  1722
  case 0 show ?case by simp
nipkow@31159
  1723
next
nipkow@31159
  1724
  case (Suc k)
nipkow@31159
  1725
  show ?case (is "?L = ?R" is "_ = (EX xs. ?P' xs)")
nipkow@31159
  1726
  proof
nipkow@31159
  1727
    assume "?R" thus "?L" using Suc by auto
nipkow@31159
  1728
  next
nipkow@31159
  1729
    assume "?L"
nipkow@31159
  1730
    with Suc obtain x xs where "?P k xs & P k x" by (metis less_Suc_eq)
nipkow@31159
  1731
    hence "?P'(xs@[x])" by(simp add:nth_append less_Suc_eq)
nipkow@31159
  1732
    thus "?R" ..
nipkow@31159
  1733
  qed
nipkow@31159
  1734
qed
nipkow@31159
  1735
nipkow@31159
  1736
haftmann@49948
  1737
subsubsection {* @{const list_update} *}
wenzelm@13114
  1738
nipkow@24526
  1739
lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
nipkow@24526
  1740
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1741
wenzelm@13114
  1742
lemma nth_list_update:
nipkow@24526
  1743
"i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@24526
  1744
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1745
wenzelm@13142
  1746
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
  1747
by (simp add: nth_list_update)
wenzelm@13114
  1748
nipkow@24526
  1749
lemma nth_list_update_neq [simp]: "i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@24526
  1750
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1751
nipkow@24526
  1752
lemma list_update_id[simp]: "xs[i := xs!i] = xs"
nipkow@24526
  1753
by (induct xs arbitrary: i) (simp_all split:nat.splits)
nipkow@24526
  1754
nipkow@24526
  1755
lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
nipkow@24526
  1756
apply (induct xs arbitrary: i)
nipkow@17501
  1757
 apply simp
nipkow@17501
  1758
apply (case_tac i)
nipkow@17501
  1759
apply simp_all
nipkow@17501
  1760
done
nipkow@17501
  1761
nipkow@31077
  1762
lemma list_update_nonempty[simp]: "xs[k:=x] = [] \<longleftrightarrow> xs=[]"
nipkow@31077
  1763
by(metis length_0_conv length_list_update)
nipkow@31077
  1764
wenzelm@13114
  1765
lemma list_update_same_conv:
nipkow@24526
  1766
"i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@24526
  1767
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1768
nipkow@14187
  1769
lemma list_update_append1:
nipkow@24526
  1770
 "i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
nipkow@24526
  1771
apply (induct xs arbitrary: i, simp)
nipkow@14187
  1772
apply(simp split:nat.split)
nipkow@14187
  1773
done
nipkow@14187
  1774
kleing@15868
  1775
lemma list_update_append:
nipkow@24526
  1776
  "(xs @ ys) [n:= x] = 
kleing@15868
  1777
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
nipkow@24526
  1778
by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868
  1779
nipkow@14402
  1780
lemma list_update_length [simp]:
nipkow@14402
  1781
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
  1782
by (induct xs, auto)
nipkow@14402
  1783
nipkow@31264
  1784
lemma map_update: "map f (xs[k:= y]) = (map f xs)[k := f y]"
nipkow@31264
  1785
by(induct xs arbitrary: k)(auto split:nat.splits)
nipkow@31264
  1786
nipkow@31264
  1787
lemma rev_update:
nipkow@31264
  1788
  "k < length xs \<Longrightarrow> rev (xs[k:= y]) = (rev xs)[length xs - k - 1 := y]"
nipkow@31264
  1789
by (induct xs arbitrary: k) (auto simp: list_update_append split:nat.splits)
nipkow@31264
  1790
wenzelm@13114
  1791
lemma update_zip:
nipkow@31080
  1792
  "(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@24526
  1793
by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split)
nipkow@24526
  1794
nipkow@24526
  1795
lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)"
nipkow@24526
  1796
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1797
wenzelm@13114
  1798
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
  1799
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
  1800
nipkow@24526
  1801
lemma set_update_memI: "n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
nipkow@24526
  1802
by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868
  1803
nipkow@31077
  1804
lemma list_update_overwrite[simp]:
haftmann@24796
  1805
  "xs [i := x, i := y] = xs [i := y]"
nipkow@31077
  1806
apply (induct xs arbitrary: i) apply simp
nipkow@31077
  1807
apply (case_tac i, simp_all)
haftmann@24796
  1808
done
haftmann@24796
  1809
haftmann@24796
  1810
lemma list_update_swap:
haftmann@24796
  1811
  "i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]"
haftmann@24796
  1812
apply (induct xs arbitrary: i i')
haftmann@24796
  1813
apply simp
haftmann@24796
  1814
apply (case_tac i, case_tac i')
haftmann@24796
  1815
apply auto
haftmann@24796
  1816
apply (case_tac i')
haftmann@24796
  1817
apply auto
haftmann@24796
  1818
done
haftmann@24796
  1819
haftmann@29827
  1820
lemma list_update_code [code]:
haftmann@29827
  1821
  "[][i := y] = []"
haftmann@29827
  1822
  "(x # xs)[0 := y] = y # xs"
haftmann@29827
  1823
  "(x # xs)[Suc i := y] = x # xs[i := y]"
haftmann@29827
  1824
  by simp_all
haftmann@29827
  1825
wenzelm@13114
  1826
haftmann@49948
  1827
subsubsection {* @{const last} and @{const butlast} *}
wenzelm@13114
  1828
wenzelm@13142
  1829
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
  1830
by (induct xs) auto
wenzelm@13114
  1831
wenzelm@13142
  1832
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
  1833
by (induct xs) auto
wenzelm@13114
  1834
nipkow@14302
  1835
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
huffman@44921
  1836
  by simp
nipkow@14302
  1837
nipkow@14302
  1838
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
huffman@44921
  1839
  by simp
nipkow@14302
  1840
nipkow@14302
  1841
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
  1842
by (induct xs) (auto)
nipkow@14302
  1843
nipkow@14302
  1844
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
  1845
by(simp add:last_append)
nipkow@14302
  1846
nipkow@14302
  1847
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
  1848
by(simp add:last_append)
nipkow@14302
  1849
noschinl@45841
  1850
lemma last_tl: "xs = [] \<or> tl xs \<noteq> [] \<Longrightarrow>last (tl xs) = last xs"
noschinl@45841
  1851
by (induct xs) simp_all
noschinl@45841
  1852
noschinl@45841
  1853
lemma butlast_tl: "butlast (tl xs) = tl (butlast xs)"
noschinl@45841
  1854
by (induct xs) simp_all
noschinl@45841
  1855
nipkow@17762
  1856
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
nipkow@17762
  1857
by(rule rev_exhaust[of xs]) simp_all
nipkow@17762
  1858
nipkow@17762
  1859
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
nipkow@17762
  1860
by(cases xs) simp_all
nipkow@17762
  1861
nipkow@17765
  1862
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
nipkow@17765
  1863
by (induct as) auto
nipkow@17762
  1864
wenzelm@13142
  1865
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
  1866
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1867
wenzelm@13114
  1868
lemma butlast_append:
nipkow@24526
  1869
  "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@24526
  1870
by (induct xs arbitrary: ys) auto
wenzelm@13114
  1871
wenzelm@13142
  1872
lemma append_butlast_last_id [simp]:
nipkow@13145
  1873
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
  1874
by (induct xs) auto
wenzelm@13114
  1875
wenzelm@13142
  1876
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
  1877
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1878
wenzelm@13114
  1879
lemma in_set_butlast_appendI:
nipkow@13145
  1880
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
  1881
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
  1882
nipkow@24526
  1883
lemma last_drop[simp]: "n < length xs \<Longrightarrow> last (drop n xs) = last xs"
nipkow@24526
  1884
apply (induct xs arbitrary: n)
nipkow@17501
  1885
 apply simp
nipkow@17501
  1886
apply (auto split:nat.split)
nipkow@17501
  1887
done
nipkow@17501
  1888
noschinl@45841
  1889
lemma nth_butlast:
noschinl@45841
  1890
  assumes "n < length (butlast xs)" shows "butlast xs ! n = xs ! n"
noschinl@45841
  1891
proof (cases xs)
noschinl@45841
  1892
  case (Cons y ys)
noschinl@45841
  1893
  moreover from assms have "butlast xs ! n = (butlast xs @ [last xs]) ! n"
noschinl@45841
  1894
    by (simp add: nth_append)
noschinl@45841
  1895
  ultimately show ?thesis using append_butlast_last_id by simp
noschinl@45841
  1896
qed simp
noschinl@45841
  1897
huffman@30128
  1898
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
nipkow@17589
  1899
by(induct xs)(auto simp:neq_Nil_conv)
nipkow@17589
  1900
huffman@30128
  1901
lemma butlast_conv_take: "butlast xs = take (length xs - 1) xs"
huffman@26584
  1902
by (induct xs, simp, case_tac xs, simp_all)
huffman@26584
  1903
nipkow@31077
  1904
lemma last_list_update:
nipkow@31077
  1905
  "xs \<noteq> [] \<Longrightarrow> last(xs[k:=x]) = (if k = size xs - 1 then x else last xs)"
nipkow@31077
  1906
by (auto simp: last_conv_nth)
nipkow@31077
  1907
nipkow@31077
  1908
lemma butlast_list_update:
nipkow@31077
  1909
  "butlast(xs[k:=x]) =
nipkow@31077
  1910
 (if k = size xs - 1 then butlast xs else (butlast xs)[k:=x])"
nipkow@31077
  1911
apply(cases xs rule:rev_cases)
nipkow@31077
  1912
apply simp
nipkow@31077
  1913
apply(simp add:list_update_append split:nat.splits)
nipkow@31077
  1914
done
nipkow@31077
  1915
haftmann@36851
  1916
lemma last_map:
haftmann@36851
  1917
  "xs \<noteq> [] \<Longrightarrow> last (map f xs) = f (last xs)"
haftmann@36851
  1918
  by (cases xs rule: rev_cases) simp_all
haftmann@36851
  1919
haftmann@36851
  1920
lemma map_butlast:
haftmann@36851
  1921
  "map f (butlast xs) = butlast (map f xs)"
haftmann@36851
  1922
  by (induct xs) simp_all
haftmann@36851
  1923
nipkow@40230
  1924
lemma snoc_eq_iff_butlast:
nipkow@40230
  1925
  "xs @ [x] = ys \<longleftrightarrow> (ys \<noteq> [] & butlast ys = xs & last ys = x)"
nipkow@40230
  1926
by (metis append_butlast_last_id append_is_Nil_conv butlast_snoc last_snoc not_Cons_self)
nipkow@40230
  1927
haftmann@24796
  1928
haftmann@49948
  1929
subsubsection {* @{const take} and @{const drop} *}
wenzelm@13114
  1930
wenzelm@13142
  1931
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
  1932
by (induct xs) auto
wenzelm@13114
  1933
wenzelm@13142
  1934
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
  1935
by (induct xs) auto
wenzelm@13114
  1936
wenzelm@13142
  1937
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
  1938
by simp
wenzelm@13114
  1939
wenzelm@13142
  1940
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
  1941
by simp
wenzelm@13114
  1942
wenzelm@13142
  1943
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
  1944
huffman@30128
  1945
lemma take_1_Cons [simp]: "take 1 (x # xs) = [x]"
huffman@30128
  1946
  unfolding One_nat_def by simp
huffman@30128
  1947
huffman@30128
  1948
lemma drop_1_Cons [simp]: "drop 1 (x # xs) = xs"
huffman@30128
  1949
  unfolding One_nat_def by simp
huffman@30128
  1950
nipkow@15110
  1951
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
  1952
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
  1953
nipkow@14187
  1954
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
  1955
by(cases xs, simp_all)
nipkow@14187
  1956
huffman@26584
  1957
lemma take_tl: "take n (tl xs) = tl (take (Suc n) xs)"
huffman@26584
  1958
by (induct xs arbitrary: n) simp_all
huffman@26584
  1959
nipkow@24526
  1960
lemma drop_tl: "drop n (tl xs) = tl(drop n xs)"
nipkow@24526
  1961
by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@24526
  1962
huffman@26584
  1963
lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)"
huffman@26584
  1964
by (cases n, simp, cases xs, auto)
huffman@26584
  1965
huffman@26584
  1966
lemma tl_drop: "tl (drop n xs) = drop n (tl xs)"
huffman@26584
  1967
by (simp only: drop_tl)
huffman@26584
  1968
nipkow@24526
  1969
lemma nth_via_drop: "drop n xs = y#ys \<Longrightarrow> xs!n = y"
nipkow@24526
  1970
apply (induct xs arbitrary: n, simp)
nipkow@14187
  1971
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
  1972
done
nipkow@14187
  1973
nipkow@13913
  1974
lemma take_Suc_conv_app_nth:
nipkow@24526
  1975
  "i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
nipkow@24526
  1976
apply (induct xs arbitrary: i, simp)
paulson@14208
  1977
apply (case_tac i, auto)
nipkow@13913
  1978
done
nipkow@13913
  1979
mehta@14591
  1980
lemma drop_Suc_conv_tl:
nipkow@24526
  1981
  "i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
nipkow@24526
  1982
apply (induct xs arbitrary: i, simp)
mehta@14591
  1983
apply (case_tac i, auto)
mehta@14591
  1984
done
mehta@14591
  1985
nipkow@24526
  1986
lemma length_take [simp]: "length (take n xs) = min (length xs) n"
nipkow@24526
  1987
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1988
nipkow@24526
  1989
lemma length_drop [simp]: "length (drop n xs) = (length xs - n)"
nipkow@24526
  1990
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1991
nipkow@24526
  1992
lemma take_all [simp]: "length xs <= n ==> take n xs = xs"
nipkow@24526
  1993
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1994
nipkow@24526
  1995
lemma drop_all [simp]: "length xs <= n ==> drop n xs = []"
nipkow@24526
  1996
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114
  1997
wenzelm@13142
  1998
lemma take_append [simp]:
nipkow@24526
  1999
  "take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@24526
  2000
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114
  2001
wenzelm@13142
  2002
lemma drop_append [simp]:
nipkow@24526
  2003
  "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@24526
  2004
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  2005
nipkow@24526
  2006
lemma take_take [simp]: "take n (take m xs) = take (min n m) xs"
nipkow@24526
  2007
apply (induct m arbitrary: xs n, auto)
paulson@14208
  2008
apply (case_tac xs, auto)
nipkow@15236
  2009
apply (case_tac n, auto)
nipkow@13145
  2010
done
wenzelm@13114
  2011
nipkow@24526
  2012
lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs"
nipkow@24526
  2013
apply (induct m arbitrary: xs, auto)
paulson@14208
  2014
apply (case_tac xs, auto)
nipkow@13145
  2015
done
wenzelm@13114
  2016
nipkow@24526
  2017
lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)"
nipkow@24526
  2018
apply (induct m arbitrary: xs n, auto)
paulson@14208
  2019
apply (case_tac xs, auto)
nipkow@13145
  2020
done
wenzelm@13114
  2021
nipkow@24526
  2022
lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@24526
  2023
apply(induct xs arbitrary: m n)
nipkow@14802
  2024
 apply simp
nipkow@14802
  2025
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  2026
done
nipkow@14802
  2027
nipkow@24526
  2028
lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs"
nipkow@24526
  2029
apply (induct n arbitrary: xs, auto)
paulson@14208
  2030
apply (case_tac xs, auto)
nipkow@13145
  2031
done
wenzelm@13114
  2032
nipkow@24526
  2033
lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@24526
  2034
apply(induct xs arbitrary: n)
nipkow@15110
  2035
 apply simp
nipkow@15110
  2036
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  2037
done
nipkow@15110
  2038
nipkow@24526
  2039
lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)"
nipkow@24526
  2040
apply(induct xs arbitrary: n)
nipkow@15110
  2041
apply simp
nipkow@15110
  2042
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  2043
done
nipkow@15110
  2044
nipkow@24526
  2045
lemma take_map: "take n (map f xs) = map f (take n xs)"
nipkow@24526
  2046
apply (induct n arbitrary: xs, auto)
paulson@14208
  2047
apply (case_tac xs, auto)
nipkow@13145
  2048
done
wenzelm@13114
  2049
nipkow@24526
  2050
lemma drop_map: "drop n (map f xs) = map f (drop n xs)"
nipkow@24526
  2051
apply (induct n arbitrary: xs, auto)
paulson@14208
  2052
apply (case_tac xs, auto)
nipkow@13145
  2053
done
wenzelm@13114
  2054
nipkow@24526
  2055
lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)"
nipkow@24526
  2056
apply (induct xs arbitrary: i, auto)
paulson@14208
  2057
apply (case_tac i, auto)
nipkow@13145
  2058
done
wenzelm@13114
  2059
nipkow@24526
  2060
lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)"
nipkow@24526
  2061
apply (induct xs arbitrary: i, auto)
paulson@14208
  2062
apply (case_tac i, auto)
nipkow@13145
  2063
done
wenzelm@13114
  2064
nipkow@24526
  2065
lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i"
nipkow@24526
  2066
apply (induct xs arbitrary: i n, auto)
paulson@14208
  2067
apply (case_tac n, blast)
paulson@14208
  2068
apply (case_tac i, auto)
nipkow@13145
  2069
done
wenzelm@13114
  2070
wenzelm@13142
  2071
lemma nth_drop [simp]:
nipkow@24526
  2072
  "n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
nipkow@24526
  2073
apply (induct n arbitrary: xs i, auto)
paulson@14208
  2074
apply (case_tac xs, auto)
nipkow@13145
  2075
done
nipkow@3507
  2076
huffman@26584
  2077
lemma butlast_take:
huffman@30128
  2078
  "n <= length xs ==> butlast (take n xs) = take (n - 1) xs"
haftmann@54863
  2079
by (simp add: butlast_conv_take min.absorb1 min.absorb2)
huffman@26584
  2080
huffman@26584
  2081
lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)"
huffman@30128
  2082
by (simp add: butlast_conv_take drop_take add_ac)
huffman@26584
  2083
huffman@26584
  2084
lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs"
haftmann@54863
  2085
by (simp add: butlast_conv_take min.absorb1)
huffman@26584
  2086
huffman@26584
  2087
lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)"
huffman@30128
  2088
by (simp add: butlast_conv_take drop_take add_ac)
huffman@26584
  2089
bulwahn@46500
  2090
lemma hd_drop_conv_nth: "n < length xs \<Longrightarrow> hd(drop n xs) = xs!n"
nipkow@18423
  2091
by(simp add: hd_conv_nth)
nipkow@18423
  2092
nipkow@35248
  2093
lemma set_take_subset_set_take:
nipkow@35248
  2094
  "m <= n \<Longrightarrow> set(take m xs) <= set(take n xs)"
bulwahn@41463
  2095
apply (induct xs arbitrary: m n)
bulwahn@41463
  2096
apply simp
bulwahn@41463
  2097
apply (case_tac n)
bulwahn@41463
  2098
apply (auto simp: take_Cons)
bulwahn@41463
  2099
done
nipkow@35248
  2100
nipkow@24526
  2101
lemma set_take_subset: "set(take n xs) \<subseteq> set xs"
nipkow@24526
  2102
by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split)
nipkow@24526
  2103
nipkow@24526
  2104
lemma set_drop_subset: "set(drop n xs) \<subseteq> set xs"
nipkow@24526
  2105
by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  2106
nipkow@35248
  2107
lemma set_drop_subset_set_drop:
nipkow@35248
  2108
  "m >= n \<Longrightarrow> set(drop m xs) <= set(drop n xs)"
nipkow@35248
  2109
apply(induct xs arbitrary: m n)
nipkow@35248
  2110
apply(auto simp:drop_Cons split:nat.split)
nipkow@35248
  2111
apply (metis set_drop_subset subset_iff)
nipkow@35248
  2112
done
nipkow@35248
  2113
nipkow@14187
  2114
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  2115
using set_take_subset by fast
nipkow@14187
  2116
nipkow@14187
  2117
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  2118
using set_drop_subset by fast
nipkow@14187
  2119
wenzelm@13114
  2120
lemma append_eq_conv_conj:
nipkow@24526
  2121
  "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
nipkow@24526
  2122
apply (induct xs arbitrary: zs, simp, clarsimp)
paulson@14208
  2123
apply (case_tac zs, auto)
nipkow@13145
  2124
done
wenzelm@13142
  2125
nipkow@24526
  2126
lemma take_add: 
noschinl@42713
  2127
  "take (i+j) xs = take i xs @ take j (drop i xs)"
nipkow@24526
  2128
apply (induct xs arbitrary: i, auto) 
nipkow@24526
  2129
apply (case_tac i, simp_all)
paulson@14050
  2130
done
paulson@14050
  2131
nipkow@14300
  2132
lemma append_eq_append_conv_if:
wenzelm@53015
  2133
 "(xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>1 @ ys\<^sub>2) =
wenzelm@53015
  2134
  (if size xs\<^sub>1 \<le> size ys\<^sub>1
wenzelm@53015
  2135
   then xs\<^sub>1 = take (size xs\<^sub>1) ys\<^sub>1 \<and> xs\<^sub>2 = drop (size xs\<^sub>1) ys\<^sub>1 @ ys\<^sub>2
wenzelm@53015
  2136
   else take (size ys\<^sub>1) xs\<^sub>1 = ys\<^sub>1 \<and> drop (size ys\<^sub>1) xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>2)"
wenzelm@53015
  2137
apply(induct xs\<^sub>1 arbitrary: ys\<^sub>1)
nipkow@14300
  2138
 apply simp
wenzelm@53015
  2139
apply(case_tac ys\<^sub>1)
nipkow@14300
  2140
apply simp_all
nipkow@14300
  2141
done
nipkow@14300
  2142
nipkow@15110
  2143
lemma take_hd_drop:
huffman@30079
  2144
  "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (Suc n) xs"
nipkow@24526
  2145
apply(induct xs arbitrary: n)
nipkow@15110
  2146
apply simp
nipkow@15110
  2147
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  2148
done
nipkow@15110
  2149
nipkow@17501
  2150
lemma id_take_nth_drop:
nipkow@17501
  2151
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
nipkow@17501
  2152
proof -
nipkow@17501
  2153
  assume si: "i < length xs"
nipkow@17501
  2154
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
nipkow@17501
  2155
  moreover
nipkow@17501
  2156
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
nipkow@17501
  2157
    apply (rule_tac take_Suc_conv_app_nth) by arith
nipkow@17501
  2158
  ultimately show ?thesis by auto
nipkow@17501
  2159
qed
nipkow@17501
  2160
  
nipkow@17501
  2161
lemma upd_conv_take_nth_drop:
nipkow@17501
  2162
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  2163
proof -
nipkow@17501
  2164
  assume i: "i < length xs"
nipkow@17501
  2165
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
nipkow@17501
  2166
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
nipkow@17501
  2167
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  2168
    using i by (simp add: list_update_append)
nipkow@17501
  2169
  finally show ?thesis .
nipkow@17501
  2170
qed
nipkow@17501
  2171
haftmann@24796
  2172
lemma nth_drop':
haftmann@24796
  2173
  "i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs"
haftmann@24796
  2174
apply (induct i arbitrary: xs)
haftmann@24796
  2175
apply (simp add: neq_Nil_conv)
haftmann@24796
  2176
apply (erule exE)+
haftmann@24796
  2177
apply simp
haftmann@24796
  2178
apply (case_tac xs)
haftmann@24796
  2179
apply simp_all
haftmann@24796
  2180
done
haftmann@24796
  2181
wenzelm@13114
  2182
haftmann@49948
  2183
subsubsection {* @{const takeWhile} and @{const dropWhile} *}
wenzelm@13114
  2184
hoelzl@33639
  2185
lemma length_takeWhile_le: "length (takeWhile P xs) \<le> length xs"
hoelzl@33639
  2186
  by (induct xs) auto
hoelzl@33639
  2187
wenzelm@13142
  2188
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  2189
by (induct xs) auto
wenzelm@13114
  2190
wenzelm@13142
  2191
lemma takeWhile_append1 [simp]:
nipkow@13145
  2192
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  2193
by (induct xs) auto
wenzelm@13114
  2194
wenzelm@13142
  2195
lemma takeWhile_append2 [simp]:
nipkow@13145
  2196
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  2197
by (induct xs) auto
wenzelm@13114
  2198
wenzelm@13142
  2199
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  2200
by (induct xs) auto
wenzelm@13114
  2201
hoelzl@33639
  2202
lemma takeWhile_nth: "j < length (takeWhile P xs) \<Longrightarrow> takeWhile P xs ! j = xs ! j"
hoelzl@33639
  2203
apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
hoelzl@33639
  2204
hoelzl@33639
  2205
lemma dropWhile_nth: "j < length (dropWhile P xs) \<Longrightarrow> dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))"
hoelzl@33639
  2206
apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
hoelzl@33639
  2207
hoelzl@33639
  2208
lemma length_dropWhile_le: "length (dropWhile P xs) \<le> length xs"
hoelzl@33639
  2209
by (induct xs) auto
hoelzl@33639
  2210
wenzelm@13142
  2211
lemma dropWhile_append1 [simp]:
nipkow@13145
  2212
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  2213
by (induct xs) auto
wenzelm@13114
  2214
wenzelm@13142
  2215
lemma dropWhile_append2 [simp]:
nipkow@13145
  2216
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  2217
by (induct xs) auto
wenzelm@13114
  2218
noschinl@45841
  2219
lemma dropWhile_append3:
noschinl@45841
  2220
  "\<not> P y \<Longrightarrow>dropWhile P (xs @ y # ys) = dropWhile P xs @ y # ys"
noschinl@45841
  2221
by (induct xs) auto
noschinl@45841
  2222
noschinl@45841
  2223
lemma dropWhile_last:
noschinl@45841
  2224
  "x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> last (dropWhile P xs) = last xs"
noschinl@45841
  2225
by (auto simp add: dropWhile_append3 in_set_conv_decomp)
noschinl@45841
  2226
noschinl@45841
  2227
lemma set_dropWhileD: "x \<in> set (dropWhile P xs) \<Longrightarrow> x \<in> set xs"
noschinl@45841
  2228
by (induct xs) (auto split: split_if_asm)
noschinl@45841
  2229
krauss@23971
  2230
lemma set_takeWhileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  2231
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  2232
nipkow@13913
  2233
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  2234
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  2235
by(induct xs, auto)
nipkow@13913
  2236
nipkow@13913
  2237
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  2238
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  2239
by(induct xs, auto)
nipkow@13913
  2240
nipkow@13913
  2241
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  2242
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  2243
by(induct xs, auto)
nipkow@13913
  2244
nipkow@31077
  2245
lemma distinct_takeWhile[simp]: "distinct xs ==> distinct (takeWhile P xs)"
nipkow@31077
  2246
by (induct xs) (auto dest: set_takeWhileD)
nipkow@31077
  2247
nipkow@31077
  2248
lemma distinct_dropWhile[simp]: "distinct xs ==> distinct (dropWhile P xs)"
nipkow@31077
  2249
by (induct xs) auto
nipkow@31077
  2250
hoelzl@33639
  2251
lemma takeWhile_map: "takeWhile P (map f xs) = map f (takeWhile (P \<circ> f) xs)"
hoelzl@33639
  2252
by (induct xs) auto
hoelzl@33639
  2253
hoelzl@33639
  2254
lemma dropWhile_map: "dropWhile P (map f xs) = map f (dropWhile (P \<circ> f) xs)"
hoelzl@33639
  2255
by (induct xs) auto
hoelzl@33639
  2256
hoelzl@33639
  2257
lemma takeWhile_eq_take: "takeWhile P xs = take (length (takeWhile P xs)) xs"
hoelzl@33639
  2258
by (induct xs) auto
hoelzl@33639
  2259
hoelzl@33639
  2260
lemma dropWhile_eq_drop: "dropWhile P xs = drop (length (takeWhile P xs)) xs"
hoelzl@33639
  2261
by (induct xs) auto
hoelzl@33639
  2262
hoelzl@33639
  2263
lemma hd_dropWhile:
hoelzl@33639
  2264
  "dropWhile P xs \<noteq> [] \<Longrightarrow> \<not> P (hd (dropWhile P xs))"
hoelzl@33639
  2265
using assms by (induct xs) auto
hoelzl@33639
  2266
hoelzl@33639
  2267
lemma takeWhile_eq_filter:
hoelzl@33639
  2268
  assumes "\<And> x. x \<in> set (dropWhile P xs) \<Longrightarrow> \<not> P x"
hoelzl@33639
  2269
  shows "takeWhile P xs = filter P xs"
hoelzl@33639
  2270
proof -
hoelzl@33639
  2271
  have A: "filter P xs = filter P (takeWhile P xs @ dropWhile P xs)"
hoelzl@33639
  2272
    by simp
hoelzl@33639
  2273
  have B: "filter P (dropWhile P xs) = []"
hoelzl@33639
  2274
    unfolding filter_empty_conv using assms by blast
hoelzl@33639
  2275
  have "filter P xs = takeWhile P xs"
hoelzl@33639
  2276
    unfolding A filter_append B
hoelzl@33639
  2277
    by (auto simp add: filter_id_conv dest: set_takeWhileD)
hoelzl@33639
  2278
  thus ?thesis ..
hoelzl@33639
  2279
qed
hoelzl@33639
  2280
hoelzl@33639
  2281
lemma takeWhile_eq_take_P_nth:
hoelzl@33639
  2282
  "\<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
  2283
  takeWhile P xs = take n xs"
hoelzl@33639
  2284
proof (induct xs arbitrary: n)
hoelzl@33639
  2285
  case (Cons x xs)
hoelzl@33639
  2286
  thus ?case
hoelzl@33639
  2287
  proof (cases n)
hoelzl@33639
  2288
    case (Suc n') note this[simp]
hoelzl@33639
  2289
    have "P x" using Cons.prems(1)[of 0] by simp
hoelzl@33639
  2290
    moreover have "takeWhile P xs = take n' xs"
hoelzl@33639
  2291
    proof (rule Cons.hyps)
hoelzl@33639
  2292
      case goal1 thus "P (xs ! i)" using Cons.prems(1)[of "Suc i"] by simp
hoelzl@33639
  2293
    next case goal2 thus ?case using Cons by auto
hoelzl@33639
  2294
    qed
hoelzl@33639
  2295
    ultimately show ?thesis by simp
hoelzl@33639
  2296
   qed simp
hoelzl@33639
  2297
qed simp
hoelzl@33639
  2298
hoelzl@33639
  2299
lemma nth_length_takeWhile:
hoelzl@33639
  2300
  "length (takeWhile P xs) < length xs \<Longrightarrow> \<not> P (xs ! length (takeWhile P xs))"
hoelzl@33639
  2301
by (induct xs) auto
hoelzl@33639
  2302
hoelzl@33639
  2303
lemma length_takeWhile_less_P_nth:
hoelzl@33639
  2304
  assumes all: "\<And> i. i < j \<Longrightarrow> P (xs ! i)" and "j \<le> length xs"
hoelzl@33639
  2305
  shows "j \<le> length (takeWhile P xs)"
hoelzl@33639
  2306
proof (rule classical)
hoelzl@33639
  2307
  assume "\<not> ?thesis"
hoelzl@33639
  2308
  hence "length (takeWhile P xs) < length xs" using assms by simp
hoelzl@33639
  2309
  thus ?thesis using all `\<not> ?thesis` nth_length_takeWhile[of P xs] by auto
hoelzl@33639
  2310
qed
nipkow@31077
  2311
nipkow@17501
  2312
text{* The following two lemmmas could be generalized to an arbitrary
nipkow@17501
  2313
property. *}
nipkow@17501
  2314
nipkow@17501
  2315
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  2316
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
nipkow@17501
  2317
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
nipkow@17501
  2318
nipkow@17501
  2319
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  2320
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
nipkow@17501
  2321
apply(induct xs)
nipkow@17501
  2322
 apply simp
nipkow@17501
  2323
apply auto
nipkow@17501
  2324
apply(subst dropWhile_append2)
nipkow@17501
  2325
apply auto
nipkow@17501
  2326
done
nipkow@17501
  2327
nipkow@18423
  2328
lemma takeWhile_not_last:
bulwahn@46500
  2329
 "distinct xs \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
nipkow@18423
  2330
apply(induct xs)
nipkow@18423
  2331
 apply simp
nipkow@18423
  2332
apply(case_tac xs)
nipkow@18423
  2333
apply(auto)
nipkow@18423
  2334
done
nipkow@18423
  2335
krauss@44013
  2336
lemma takeWhile_cong [fundef_cong]:
krauss@18336
  2337
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  2338
  ==> takeWhile P l = takeWhile Q k"
nipkow@24349
  2339
by (induct k arbitrary: l) (simp_all)
krauss@18336
  2340
krauss@44013
  2341
lemma dropWhile_cong [fundef_cong]:
krauss@18336
  2342
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  2343
  ==> dropWhile P l = dropWhile Q k"
nipkow@24349
  2344
by (induct k arbitrary: l, simp_all)
krauss@18336
  2345
haftmann@52380
  2346
lemma takeWhile_idem [simp]:
haftmann@52380
  2347
  "takeWhile P (takeWhile P xs) = takeWhile P xs"
haftmann@52380
  2348
  by (induct xs) auto
haftmann@52380
  2349
haftmann@52380
  2350
lemma dropWhile_idem [simp]:
haftmann@52380
  2351
  "dropWhile P (dropWhile P xs) = dropWhile P xs"
haftmann@52380
  2352
  by (induct xs) auto
haftmann@52380
  2353
wenzelm@13114
  2354
haftmann@49948
  2355
subsubsection {* @{const zip} *}
wenzelm@13114
  2356
wenzelm@13142
  2357
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  2358
by (induct ys) auto
wenzelm@13114
  2359
wenzelm@13142
  2360
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  2361
by simp
wenzelm@13114
  2362
wenzelm@13142
  2363
declare zip_Cons [simp del]
wenzelm@13114
  2364
haftmann@36198
  2365
lemma [code]:
haftmann@36198
  2366
  "zip [] ys = []"
haftmann@36198
  2367
  "zip xs [] = []"
haftmann@36198
  2368
  "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
haftmann@36198
  2369
  by (fact zip_Nil zip.simps(1) zip_Cons_Cons)+
haftmann@36198
  2370
nipkow@15281
  2371
lemma zip_Cons1:
nipkow@15281
  2372
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  2373
by(auto split:list.split)
nipkow@15281
  2374
wenzelm@13142
  2375
lemma length_zip [simp]:
krauss@22493
  2376
"length (zip xs ys) = min (length xs) (length ys)"
krauss@22493
  2377
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  2378
haftmann@34978
  2379
lemma zip_obtain_same_length:
haftmann@34978
  2380
  assumes "\<And>zs ws n. length zs = length ws \<Longrightarrow> n = min (length xs) (length ys)
haftmann@34978
  2381
    \<Longrightarrow> zs = take n xs \<Longrightarrow> ws = take n ys \<Longrightarrow> P (zip zs ws)"
haftmann@34978
  2382
  shows "P (zip xs ys)"
haftmann@34978
  2383
proof -
haftmann@34978
  2384
  let ?n = "min (length xs) (length ys)"
haftmann@34978
  2385
  have "P (zip (take ?n xs) (take ?n ys))"
haftmann@34978
  2386
    by (rule assms) simp_all
haftmann@34978
  2387
  moreover have "zip xs ys = zip (take ?n xs) (take ?n ys)"
haftmann@34978
  2388
  proof (induct xs arbitrary: ys)
haftmann@34978
  2389
    case Nil then show ?case by simp
haftmann@34978
  2390
  next
haftmann@34978
  2391
    case (Cons x xs) then show ?case by (cases ys) simp_all
haftmann@34978
  2392
  qed
haftmann@34978
  2393
  ultimately show ?thesis by simp
haftmann@34978
  2394
qed
haftmann@34978
  2395
wenzelm@13114
  2396
lemma zip_append1:
krauss@22493
  2397
"zip (xs @ ys) zs =
nipkow@13145
  2398
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
krauss@22493
  2399
by (induct xs zs rule:list_induct2') auto
wenzelm@13114
  2400
wenzelm@13114
  2401
lemma zip_append2:
krauss@22493
  2402
"zip xs (ys @ zs) =
nipkow@13145
  2403
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
krauss@22493
  2404
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  2405
wenzelm@13142
  2406
lemma zip_append [simp]:
bulwahn@46500
  2407
 "[| length xs = length us |] ==>
nipkow@13145
  2408
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  2409
by (simp add: zip_append1)
wenzelm@13114
  2410
wenzelm@13114
  2411
lemma zip_rev:
nipkow@14247
  2412
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  2413
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  2414
hoelzl@33639
  2415
lemma zip_map_map:
hoelzl@33639
  2416
  "zip (map f xs) (map g ys) = map (\<lambda> (x, y). (f x, g y)) (zip xs ys)"
hoelzl@33639
  2417
proof (induct xs arbitrary: ys)
hoelzl@33639
  2418
  case (Cons x xs) note Cons_x_xs = Cons.hyps
hoelzl@33639
  2419
  show ?case
hoelzl@33639
  2420
  proof (cases ys)
hoelzl@33639
  2421
    case (Cons y ys')
hoelzl@33639
  2422
    show ?thesis unfolding Cons using Cons_x_xs by simp
hoelzl@33639
  2423
  qed simp
hoelzl@33639
  2424
qed simp
hoelzl@33639
  2425
hoelzl@33639
  2426
lemma zip_map1:
hoelzl@33639
  2427
  "zip (map f xs) ys = map (\<lambda>(x, y). (f x, y)) (zip xs ys)"
hoelzl@33639
  2428
using zip_map_map[of f xs "\<lambda>x. x" ys] by simp
hoelzl@33639
  2429
hoelzl@33639
  2430
lemma zip_map2:
hoelzl@33639
  2431
  "zip xs (map f ys) = map (\<lambda>(x, y). (x, f y)) (zip xs ys)"
hoelzl@33639
  2432
using zip_map_map[of "\<lambda>x. x" xs f ys] by simp
hoelzl@33639
  2433
nipkow@23096
  2434
lemma map_zip_map:
hoelzl@33639
  2435
  "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
hoelzl@33639
  2436
unfolding zip_map1 by auto
nipkow@23096
  2437
nipkow@23096
  2438
lemma map_zip_map2:
hoelzl@33639
  2439
  "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
hoelzl@33639
  2440
unfolding zip_map2 by auto
nipkow@23096
  2441
nipkow@31080
  2442
text{* Courtesy of Andreas Lochbihler: *}
nipkow@31080
  2443
lemma zip_same_conv_map: "zip xs xs = map (\<lambda>x. (x, x)) xs"
nipkow@31080
  2444
by(induct xs) auto
nipkow@31080
  2445
wenzelm@13142
  2446
lemma nth_zip [simp]:
nipkow@24526
  2447
"[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
nipkow@24526
  2448
apply (induct ys arbitrary: i xs, simp)
nipkow@13145
  2449
apply (case_tac xs)
nipkow@13145
  2450
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  2451
done
wenzelm@13114
  2452
wenzelm@13114
  2453
lemma set_zip:
nipkow@13145
  2454
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@31080
  2455
by(simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  2456
hoelzl@33639
  2457
lemma zip_same: "((a,b) \<in> set (zip xs xs)) = (a \<in> set xs \<and> a = b)"
hoelzl@33639
  2458
by(induct xs) auto
hoelzl@33639
  2459
wenzelm@13114
  2460
lemma zip_update:
nipkow@31080
  2461
  "zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@31080
  2462
by(rule sym, simp add: update_zip)
wenzelm@13114
  2463
wenzelm@13142
  2464
lemma zip_replicate [simp]:
nipkow@24526
  2465
  "zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
nipkow@24526
  2466
apply (induct i arbitrary: j, auto)
paulson@14208
  2467
apply (case_tac j, auto)
nipkow@13145
  2468
done
wenzelm@13114
  2469
nipkow@19487
  2470
lemma take_zip:
nipkow@24526
  2471
  "take n (zip xs ys) = zip (take n xs) (take n ys)"
nipkow@24526
  2472
apply (induct n arbitrary: xs ys)
nipkow@19487
  2473
 apply simp
nipkow@19487
  2474
apply (case_tac xs, simp)
nipkow@19487
  2475
apply (case_tac ys, simp_all)
nipkow@19487
  2476
done
nipkow@19487
  2477
nipkow@19487
  2478
lemma drop_zip:
nipkow@24526
  2479
  "drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
nipkow@24526
  2480
apply (induct n arbitrary: xs ys)
nipkow@19487
  2481
 apply simp
nipkow@19487
  2482
apply (case_tac xs, simp)
nipkow@19487
  2483
apply (case_tac ys, simp_all)
nipkow@19487
  2484
done
nipkow@19487
  2485
hoelzl@33639
  2486
lemma zip_takeWhile_fst: "zip (takeWhile P xs) ys = takeWhile (P \<circ> fst) (zip xs ys)"
hoelzl@33639
  2487
proof (induct xs arbitrary: ys)
hoelzl@33639
  2488
  case (Cons x xs) thus ?case by (cases ys) auto
hoelzl@33639
  2489
qed simp
hoelzl@33639
  2490
hoelzl@33639
  2491
lemma zip_takeWhile_snd: "zip xs (takeWhile P ys) = takeWhile (P \<circ> snd) (zip xs ys)"
hoelzl@33639
  2492
proof (induct xs arbitrary: ys)
hoelzl@33639
  2493
  case (Cons x xs) thus ?case by (cases ys) auto
hoelzl@33639
  2494
qed simp
hoelzl@33639
  2495
krauss@22493
  2496
lemma set_zip_leftD:
krauss@22493
  2497
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
krauss@22493
  2498
by (induct xs ys rule:list_induct2') auto
krauss@22493
  2499
krauss@22493
  2500
lemma set_zip_rightD:
krauss@22493
  2501
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
krauss@22493
  2502
by (induct xs ys rule:list_induct2') auto
wenzelm@13142
  2503
nipkow@23983
  2504
lemma in_set_zipE:
nipkow@23983
  2505
  "(x,y) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
nipkow@23983
  2506
by(blast dest: set_zip_leftD set_zip_rightD)
nipkow@23983
  2507
haftmann@29829
  2508
lemma zip_map_fst_snd:
haftmann@29829
  2509
  "zip (map fst zs) (map snd zs) = zs"
haftmann@29829
  2510
  by (induct zs) simp_all
haftmann@29829
  2511
haftmann@29829
  2512
lemma zip_eq_conv:
haftmann@29829
  2513
  "length xs = length ys \<Longrightarrow> zip xs ys = zs \<longleftrightarrow> map fst zs = xs \<and> map snd zs = ys"
haftmann@29829
  2514
  by (auto simp add: zip_map_fst_snd)
haftmann@29829
  2515
haftmann@51173
  2516
lemma in_set_zip:
haftmann@51173
  2517
  "p \<in> set (zip xs ys) \<longleftrightarrow> (\<exists>n. xs ! n = fst p \<and> ys ! n = snd p
haftmann@51173
  2518
    \<and> n < length xs \<and> n < length ys)"
haftmann@51173
  2519
  by (cases p) (auto simp add: set_zip)
haftmann@51173
  2520
haftmann@51173
  2521
lemma pair_list_eqI:
haftmann@51173
  2522
  assumes "map fst xs = map fst ys" and "map snd xs = map snd ys"
haftmann@51173
  2523
  shows "xs = ys"
haftmann@51173
  2524
proof -
haftmann@51173
  2525
  from assms(1) have "length xs = length ys" by (rule map_eq_imp_length_eq)
haftmann@51173
  2526
  from this assms show ?thesis
haftmann@51173
  2527
    by (induct xs ys rule: list_induct2) (simp_all add: prod_eqI)
haftmann@51173
  2528
qed
haftmann@51173
  2529
wenzelm@35115
  2530
haftmann@49948
  2531
subsubsection {* @{const list_all2} *}
wenzelm@13114
  2532
kleing@14316
  2533
lemma list_all2_lengthD [intro?]: 
kleing@14316
  2534
  "list_all2 P xs ys ==> length xs = length ys"
blanchet@55524
  2535
by (simp add: list_all2_iff)
haftmann@19607
  2536
haftmann@19787
  2537
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
blanchet@55524
  2538
by (simp add: list_all2_iff)
haftmann@19607
  2539
haftmann@19787
  2540
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
blanchet@55524
  2541
by (simp add: list_all2_iff)
haftmann@19607
  2542
haftmann@19607
  2543
lemma list_all2_Cons [iff, code]:
haftmann@19607
  2544
  "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
blanchet@55524
  2545
by (auto simp add: list_all2_iff)
wenzelm@13114
  2546
wenzelm@13114
  2547
lemma list_all2_Cons1:
nipkow@13145
  2548
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  2549
by (cases ys) auto
wenzelm@13114
  2550
wenzelm@13114
  2551
lemma list_all2_Cons2:
nipkow@13145
  2552
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  2553
by (cases xs) auto
wenzelm@13114
  2554
huffman@45794
  2555
lemma list_all2_induct
huffman@45794
  2556
  [consumes 1, case_names Nil Cons, induct set: list_all2]:
huffman@45794
  2557
  assumes P: "list_all2 P xs ys"
huffman@45794
  2558
  assumes Nil: "R [] []"
huffman@47640
  2559
  assumes Cons: "\<And>x xs y ys.
huffman@47640
  2560
    \<lbrakk>P x y; list_all2 P xs ys; R xs ys\<rbrakk> \<Longrightarrow> R (x # xs) (y # ys)"
huffman@45794
  2561
  shows "R xs ys"
huffman@45794
  2562
using P
huffman@45794
  2563
by (induct xs arbitrary: ys) (auto simp add: list_all2_Cons1 Nil Cons)
huffman@45794
  2564
wenzelm@13142
  2565
lemma list_all2_rev [iff]:
nipkow@13145
  2566
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
blanchet@55524
  2567
by (simp add: list_all2_iff zip_rev cong: conj_cong)
wenzelm@13114
  2568
kleing@13863
  2569
lemma list_all2_rev1:
kleing@13863
  2570
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  2571
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  2572
wenzelm@13114
  2573
lemma list_all2_append1:
nipkow@13145
  2574
"list_all2 P (xs @ ys) zs =
nipkow@13145
  2575
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>