src/HOL/List.thy
author blanchet
Wed Feb 19 16:32:37 2014 +0100 (2014-02-19)
changeset 55584 a879f14b6f95
parent 55564 e81ee43ab290
child 55642 63beb38e9258
permissions -rw-r--r--
merged 'List.set' with BNF-generated 'set'
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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 (set: '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_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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declare list.set[simp del, code del]
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lemma set_simps[simp, code, code_post]:
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  "set [] = {}"
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  "set (x # xs) = insert x (set xs)"
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by (simp_all add: list.set)
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lemma set_rec: "set xs = rec_list {} (\<lambda>x _. insert x) xs"
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  by (induct xs) auto
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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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   292
@{lemma "nth [a,b,c,d] 2 = c" by simp}\\
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   293
@{lemma "[a,b,c,d][2 := x] = [a,b,x,d]" by simp}\\
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   294
@{lemma "sublist [a,b,c,d,e] {0,2,3} = [a,c,d]" by (simp add:sublist_def)}\\
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   295
@{lemma "sublists [a,b] = [[a, b], [a], [b], []]" by simp}\\
haftmann@49948
   296
@{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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   297
@{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by simp}\\
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@{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate_def eval_nat_numeral)}\\
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   299
@{lemma "replicate 4 a = [a,a,a,a]" by (simp add:eval_nat_numeral)}\\
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   300
@{lemma "[2..<5] = [2,3,4]" by (simp add:eval_nat_numeral)}\\
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   301
@{lemma "listsum [1,2,3::nat] = 6" by (simp add: listsum_def)}
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   302
\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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   306
Figure~\ref{fig:Characteristic} shows characteristic examples
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   307
that should give an intuitive understanding of the above functions.
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   308
*}
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   309
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   310
text{* The following simple sort functions are intended for proofs,
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   311
not for efficient implementations. *}
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   312
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   313
context linorder
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   314
begin
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   315
haftmann@39915
   316
inductive sorted :: "'a list \<Rightarrow> bool" where
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   317
  Nil [iff]: "sorted []"
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   318
| Cons: "\<forall>y\<in>set xs. x \<le> y \<Longrightarrow> sorted xs \<Longrightarrow> sorted (x # xs)"
haftmann@39915
   319
haftmann@39915
   320
lemma sorted_single [iff]:
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   321
  "sorted [x]"
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   322
  by (rule sorted.Cons) auto
haftmann@39915
   323
haftmann@39915
   324
lemma sorted_many:
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   325
  "x \<le> y \<Longrightarrow> sorted (y # zs) \<Longrightarrow> sorted (x # y # zs)"
haftmann@39915
   326
  by (rule sorted.Cons) (cases "y # zs" rule: sorted.cases, auto)
haftmann@39915
   327
haftmann@39915
   328
lemma sorted_many_eq [simp, code]:
haftmann@39915
   329
  "sorted (x # y # zs) \<longleftrightarrow> x \<le> y \<and> sorted (y # zs)"
haftmann@39915
   330
  by (auto intro: sorted_many elim: sorted.cases)
haftmann@39915
   331
haftmann@39915
   332
lemma [code]:
haftmann@39915
   333
  "sorted [] \<longleftrightarrow> True"
haftmann@39915
   334
  "sorted [x] \<longleftrightarrow> True"
haftmann@39915
   335
  by simp_all
nipkow@24697
   336
hoelzl@33639
   337
primrec insort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
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   338
"insort_key f x [] = [x]" |
nipkow@50548
   339
"insort_key f x (y#ys) =
nipkow@50548
   340
  (if f x \<le> f y then (x#y#ys) else y#(insort_key f x ys))"
hoelzl@33639
   341
haftmann@35195
   342
definition sort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b list \<Rightarrow> 'b list" where
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   343
"sort_key f xs = foldr (insort_key f) xs []"
hoelzl@33639
   344
haftmann@40210
   345
definition insort_insert_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
nipkow@50548
   346
"insort_insert_key f x xs =
nipkow@50548
   347
  (if f x \<in> f ` set xs then xs else insort_key f x xs)"
haftmann@40210
   348
hoelzl@33639
   349
abbreviation "sort \<equiv> sort_key (\<lambda>x. x)"
hoelzl@33639
   350
abbreviation "insort \<equiv> insort_key (\<lambda>x. x)"
haftmann@40210
   351
abbreviation "insort_insert \<equiv> insort_insert_key (\<lambda>x. x)"
haftmann@35608
   352
wenzelm@25221
   353
end
wenzelm@25221
   354
nipkow@24616
   355
wenzelm@23388
   356
subsubsection {* List comprehension *}
nipkow@23192
   357
nipkow@24349
   358
text{* Input syntax for Haskell-like list comprehension notation.
nipkow@24349
   359
Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
nipkow@24349
   360
the list of all pairs of distinct elements from @{text xs} and @{text ys}.
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   361
The syntax is as in Haskell, except that @{text"|"} becomes a dot
nipkow@24349
   362
(like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
nipkow@24349
   363
\verb![e| x <- xs, ...]!.
nipkow@24349
   364
nipkow@24349
   365
The qualifiers after the dot are
nipkow@24349
   366
\begin{description}
nipkow@24349
   367
\item[generators] @{text"p \<leftarrow> xs"},
nipkow@24476
   368
 where @{text p} is a pattern and @{text xs} an expression of list type, or
nipkow@24476
   369
\item[guards] @{text"b"}, where @{text b} is a boolean expression.
nipkow@24476
   370
%\item[local bindings] @ {text"let x = e"}.
nipkow@24349
   371
\end{description}
nipkow@23240
   372
nipkow@24476
   373
Just like in Haskell, list comprehension is just a shorthand. To avoid
nipkow@24476
   374
misunderstandings, the translation into desugared form is not reversed
nipkow@24476
   375
upon output. Note that the translation of @{text"[e. x \<leftarrow> xs]"} is
nipkow@24476
   376
optmized to @{term"map (%x. e) xs"}.
nipkow@23240
   377
nipkow@24349
   378
It is easy to write short list comprehensions which stand for complex
nipkow@24349
   379
expressions. During proofs, they may become unreadable (and
nipkow@24349
   380
mangled). In such cases it can be advisable to introduce separate
nipkow@24349
   381
definitions for the list comprehensions in question.  *}
nipkow@24349
   382
wenzelm@46138
   383
nonterminal lc_qual and lc_quals
nipkow@23192
   384
nipkow@23192
   385
syntax
wenzelm@46138
   386
  "_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
wenzelm@46138
   387
  "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ <- _")
wenzelm@46138
   388
  "_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
wenzelm@46138
   389
  (*"_lc_let" :: "letbinds => lc_qual"  ("let _")*)
wenzelm@46138
   390
  "_lc_end" :: "lc_quals" ("]")
wenzelm@46138
   391
  "_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals"  (", __")
wenzelm@46138
   392
  "_lc_abs" :: "'a => 'b list => 'b list"
nipkow@23192
   393
nipkow@24476
   394
(* These are easier than ML code but cannot express the optimized
nipkow@24476
   395
   translation of [e. p<-xs]
nipkow@23192
   396
translations
wenzelm@46138
   397
  "[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
wenzelm@46138
   398
  "_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
wenzelm@46138
   399
   => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
wenzelm@46138
   400
  "[e. P]" => "if P then [e] else []"
wenzelm@46138
   401
  "_listcompr e (_lc_test P) (_lc_quals Q Qs)"
wenzelm@46138
   402
   => "if P then (_listcompr e Q Qs) else []"
wenzelm@46138
   403
  "_listcompr e (_lc_let b) (_lc_quals Q Qs)"
wenzelm@46138
   404
   => "_Let b (_listcompr e Q Qs)"
nipkow@24476
   405
*)
nipkow@23240
   406
nipkow@23279
   407
syntax (xsymbols)
wenzelm@46138
   408
  "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ \<leftarrow> _")
nipkow@23279
   409
syntax (HTML output)
wenzelm@46138
   410
  "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ \<leftarrow> _")
nipkow@24349
   411
wenzelm@52143
   412
parse_translation {*
wenzelm@46138
   413
  let
wenzelm@46138
   414
    val NilC = Syntax.const @{const_syntax Nil};
wenzelm@46138
   415
    val ConsC = Syntax.const @{const_syntax Cons};
wenzelm@46138
   416
    val mapC = Syntax.const @{const_syntax map};
wenzelm@46138
   417
    val concatC = Syntax.const @{const_syntax concat};
wenzelm@46138
   418
    val IfC = Syntax.const @{const_syntax If};
wenzelm@46138
   419
wenzelm@46138
   420
    fun single x = ConsC $ x $ NilC;
wenzelm@46138
   421
wenzelm@46138
   422
    fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
wenzelm@46138
   423
      let
wenzelm@46138
   424
        (* FIXME proper name context!? *)
wenzelm@46138
   425
        val x =
wenzelm@46138
   426
          Free (singleton (Name.variant_list (fold Term.add_free_names [p, e] [])) "x", dummyT);
wenzelm@46138
   427
        val e = if opti then single e else e;
wenzelm@46138
   428
        val case1 = Syntax.const @{syntax_const "_case1"} $ p $ e;
wenzelm@46138
   429
        val case2 =
wenzelm@46138
   430
          Syntax.const @{syntax_const "_case1"} $
wenzelm@46138
   431
            Syntax.const @{const_syntax dummy_pattern} $ NilC;
wenzelm@46138
   432
        val cs = Syntax.const @{syntax_const "_case2"} $ case1 $ case2;
traytel@51678
   433
      in Syntax_Trans.abs_tr [x, Case_Translation.case_tr false ctxt [x, cs]] end;
wenzelm@46138
   434
wenzelm@46138
   435
    fun abs_tr ctxt p e opti =
wenzelm@46138
   436
      (case Term_Position.strip_positions p of
wenzelm@46138
   437
        Free (s, T) =>
wenzelm@46138
   438
          let
wenzelm@46138
   439
            val thy = Proof_Context.theory_of ctxt;
wenzelm@46138
   440
            val s' = Proof_Context.intern_const ctxt s;
wenzelm@46138
   441
          in
wenzelm@46138
   442
            if Sign.declared_const thy s'
wenzelm@46138
   443
            then (pat_tr ctxt p e opti, false)
wenzelm@46138
   444
            else (Syntax_Trans.abs_tr [p, e], true)
wenzelm@46138
   445
          end
wenzelm@46138
   446
      | _ => (pat_tr ctxt p e opti, false));
wenzelm@46138
   447
wenzelm@46138
   448
    fun lc_tr ctxt [e, Const (@{syntax_const "_lc_test"}, _) $ b, qs] =
wenzelm@46138
   449
          let
wenzelm@46138
   450
            val res =
wenzelm@46138
   451
              (case qs of
wenzelm@46138
   452
                Const (@{syntax_const "_lc_end"}, _) => single e
wenzelm@46138
   453
              | Const (@{syntax_const "_lc_quals"}, _) $ q $ qs => lc_tr ctxt [e, q, qs]);
wenzelm@46138
   454
          in IfC $ b $ res $ NilC end
wenzelm@46138
   455
      | lc_tr ctxt
wenzelm@46138
   456
            [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
wenzelm@46138
   457
              Const(@{syntax_const "_lc_end"}, _)] =
wenzelm@46138
   458
          (case abs_tr ctxt p e true of
wenzelm@46138
   459
            (f, true) => mapC $ f $ es
wenzelm@46138
   460
          | (f, false) => concatC $ (mapC $ f $ es))
wenzelm@46138
   461
      | lc_tr ctxt
wenzelm@46138
   462
            [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
wenzelm@46138
   463
              Const (@{syntax_const "_lc_quals"}, _) $ q $ qs] =
wenzelm@46138
   464
          let val e' = lc_tr ctxt [e, q, qs];
wenzelm@46138
   465
          in concatC $ (mapC $ (fst (abs_tr ctxt p e' false)) $ es) end;
wenzelm@46138
   466
wenzelm@46138
   467
  in [(@{syntax_const "_listcompr"}, lc_tr)] end
nipkow@24349
   468
*}
nipkow@23279
   469
wenzelm@51272
   470
ML_val {*
wenzelm@42167
   471
  let
wenzelm@42167
   472
    val read = Syntax.read_term @{context};
wenzelm@42167
   473
    fun check s1 s2 = read s1 aconv read s2 orelse error ("Check failed: " ^ quote s1);
wenzelm@42167
   474
  in
wenzelm@42167
   475
    check "[(x,y,z). b]" "if b then [(x, y, z)] else []";
wenzelm@42167
   476
    check "[(x,y,z). x\<leftarrow>xs]" "map (\<lambda>x. (x, y, z)) xs";
wenzelm@42167
   477
    check "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]" "concat (map (\<lambda>x. map (\<lambda>y. e x y) ys) xs)";
wenzelm@42167
   478
    check "[(x,y,z). x<a, x>b]" "if x < a then if b < x then [(x, y, z)] else [] else []";
wenzelm@42167
   479
    check "[(x,y,z). x\<leftarrow>xs, x>b]" "concat (map (\<lambda>x. if b < x then [(x, y, z)] else []) xs)";
wenzelm@42167
   480
    check "[(x,y,z). x<a, x\<leftarrow>xs]" "if x < a then map (\<lambda>x. (x, y, z)) xs else []";
wenzelm@42167
   481
    check "[(x,y). Cons True x \<leftarrow> xs]"
wenzelm@42167
   482
      "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | True # x \<Rightarrow> [(x, y)] | False # x \<Rightarrow> []) xs)";
wenzelm@42167
   483
    check "[(x,y,z). Cons x [] \<leftarrow> xs]"
wenzelm@42167
   484
      "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | [x] \<Rightarrow> [(x, y, z)] | x # aa # lista \<Rightarrow> []) xs)";
wenzelm@42167
   485
    check "[(x,y,z). x<a, x>b, x=d]"
wenzelm@42167
   486
      "if x < a then if b < x then if x = d then [(x, y, z)] else [] else [] else []";
wenzelm@42167
   487
    check "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
wenzelm@42167
   488
      "if x < a then if b < x then map (\<lambda>y. (x, y, z)) ys else [] else []";
wenzelm@42167
   489
    check "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
wenzelm@42167
   490
      "if x < a then concat (map (\<lambda>x. if b < y then [(x, y, z)] else []) xs) else []";
wenzelm@42167
   491
    check "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
wenzelm@42167
   492
      "if x < a then concat (map (\<lambda>x. map (\<lambda>y. (x, y, z)) ys) xs) else []";
wenzelm@42167
   493
    check "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
wenzelm@42167
   494
      "concat (map (\<lambda>x. if b < x then if y < a then [(x, y, z)] else [] else []) xs)";
wenzelm@42167
   495
    check "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
wenzelm@42167
   496
      "concat (map (\<lambda>x. if b < x then map (\<lambda>y. (x, y, z)) ys else []) xs)";
wenzelm@42167
   497
    check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
wenzelm@42167
   498
      "concat (map (\<lambda>x. concat (map (\<lambda>y. if x < y then [(x, y, z)] else []) ys)) xs)";
wenzelm@42167
   499
    check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
wenzelm@42167
   500
      "concat (map (\<lambda>x. concat (map (\<lambda>y. map (\<lambda>z. (x, y, z)) zs) ys)) xs)"
wenzelm@42167
   501
  end;
wenzelm@42167
   502
*}
wenzelm@42167
   503
wenzelm@35115
   504
(*
nipkow@24349
   505
term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
nipkow@23192
   506
*)
nipkow@23192
   507
wenzelm@42167
   508
wenzelm@50422
   509
ML {*
wenzelm@50422
   510
(* Simproc for rewriting list comprehensions applied to List.set to set
wenzelm@50422
   511
   comprehension. *)
wenzelm@50422
   512
wenzelm@50422
   513
signature LIST_TO_SET_COMPREHENSION =
wenzelm@50422
   514
sig
wenzelm@51717
   515
  val simproc : Proof.context -> cterm -> thm option
wenzelm@50422
   516
end
wenzelm@50422
   517
wenzelm@50422
   518
structure List_to_Set_Comprehension : LIST_TO_SET_COMPREHENSION =
wenzelm@50422
   519
struct
wenzelm@50422
   520
wenzelm@50422
   521
(* conversion *)
wenzelm@50422
   522
wenzelm@50422
   523
fun all_exists_conv cv ctxt ct =
wenzelm@50422
   524
  (case Thm.term_of ct of
wenzelm@50422
   525
    Const (@{const_name HOL.Ex}, _) $ Abs _ =>
wenzelm@50422
   526
      Conv.arg_conv (Conv.abs_conv (all_exists_conv cv o #2) ctxt) ct
wenzelm@50422
   527
  | _ => cv ctxt ct)
wenzelm@50422
   528
wenzelm@50422
   529
fun all_but_last_exists_conv cv ctxt ct =
wenzelm@50422
   530
  (case Thm.term_of ct of
wenzelm@50422
   531
    Const (@{const_name HOL.Ex}, _) $ Abs (_, _, Const (@{const_name HOL.Ex}, _) $ _) =>
wenzelm@50422
   532
      Conv.arg_conv (Conv.abs_conv (all_but_last_exists_conv cv o #2) ctxt) ct
wenzelm@50422
   533
  | _ => cv ctxt ct)
wenzelm@50422
   534
wenzelm@50422
   535
fun Collect_conv cv ctxt ct =
wenzelm@50422
   536
  (case Thm.term_of ct of
wenzelm@50422
   537
    Const (@{const_name Set.Collect}, _) $ Abs _ => Conv.arg_conv (Conv.abs_conv cv ctxt) ct
wenzelm@50422
   538
  | _ => raise CTERM ("Collect_conv", [ct]))
wenzelm@50422
   539
wenzelm@50422
   540
fun rewr_conv' th = Conv.rewr_conv (mk_meta_eq th)
wenzelm@50422
   541
wenzelm@50422
   542
fun conjunct_assoc_conv ct =
wenzelm@50422
   543
  Conv.try_conv
wenzelm@51315
   544
    (rewr_conv' @{thm conj_assoc} then_conv HOLogic.conj_conv Conv.all_conv conjunct_assoc_conv) ct
wenzelm@50422
   545
wenzelm@50422
   546
fun right_hand_set_comprehension_conv conv ctxt =
wenzelm@51315
   547
  HOLogic.Trueprop_conv (HOLogic.eq_conv Conv.all_conv
wenzelm@50422
   548
    (Collect_conv (all_exists_conv conv o #2) ctxt))
wenzelm@50422
   549
wenzelm@50422
   550
wenzelm@50422
   551
(* term abstraction of list comprehension patterns *)
wenzelm@50422
   552
wenzelm@50422
   553
datatype termlets = If | Case of (typ * int)
wenzelm@50422
   554
wenzelm@51717
   555
fun simproc ctxt redex =
wenzelm@50422
   556
  let
blanchet@55584
   557
    val set_Nil_I = @{thm trans} OF [@{thm set_simps(1)}, @{thm empty_def}]
wenzelm@50422
   558
    val set_singleton = @{lemma "set [a] = {x. x = a}" by simp}
wenzelm@50422
   559
    val inst_Collect_mem_eq = @{lemma "set A = {x. x : set A}" by simp}
wenzelm@50422
   560
    val del_refl_eq = @{lemma "(t = t & P) == P" by simp}
wenzelm@50422
   561
    fun mk_set T = Const (@{const_name List.set}, HOLogic.listT T --> HOLogic.mk_setT T)
wenzelm@50422
   562
    fun dest_set (Const (@{const_name List.set}, _) $ xs) = xs
wenzelm@50422
   563
    fun dest_singleton_list (Const (@{const_name List.Cons}, _)
wenzelm@50422
   564
          $ t $ (Const (@{const_name List.Nil}, _))) = t
wenzelm@50422
   565
      | dest_singleton_list t = raise TERM ("dest_singleton_list", [t])
wenzelm@50422
   566
    (* We check that one case returns a singleton list and all other cases
wenzelm@50422
   567
       return [], and return the index of the one singleton list case *)
wenzelm@50422
   568
    fun possible_index_of_singleton_case cases =
wenzelm@50422
   569
      let
wenzelm@50422
   570
        fun check (i, case_t) s =
wenzelm@50422
   571
          (case strip_abs_body case_t of
wenzelm@50422
   572
            (Const (@{const_name List.Nil}, _)) => s
traytel@53412
   573
          | _ => (case s of SOME NONE => SOME (SOME i) | _ => NONE))
wenzelm@50422
   574
      in
traytel@53412
   575
        fold_index check cases (SOME NONE) |> the_default NONE
wenzelm@50422
   576
      end
blanchet@54404
   577
    (* returns (case_expr type index chosen_case constr_name) option  *)
wenzelm@50422
   578
    fun dest_case case_term =
wenzelm@50422
   579
      let
wenzelm@50422
   580
        val (case_const, args) = strip_comb case_term
wenzelm@50422
   581
      in
wenzelm@50422
   582
        (case try dest_Const case_const of
wenzelm@50422
   583
          SOME (c, T) =>
blanchet@54404
   584
            (case Ctr_Sugar.ctr_sugar_of_case ctxt c of
blanchet@54404
   585
              SOME {ctrs, ...} =>
wenzelm@50422
   586
                (case possible_index_of_singleton_case (fst (split_last args)) of
wenzelm@50422
   587
                  SOME i =>
wenzelm@50422
   588
                    let
blanchet@54404
   589
                      val constr_names = map (fst o dest_Const) ctrs
wenzelm@50422
   590
                      val (Ts, _) = strip_type T
wenzelm@50422
   591
                      val T' = List.last Ts
blanchet@54404
   592
                    in SOME (List.last args, T', i, nth args i, nth constr_names i) end
wenzelm@50422
   593
                | NONE => NONE)
wenzelm@50422
   594
            | NONE => NONE)
wenzelm@50422
   595
        | NONE => NONE)
wenzelm@50422
   596
      end
wenzelm@50422
   597
    (* returns condition continuing term option *)
wenzelm@50422
   598
    fun dest_if (Const (@{const_name If}, _) $ cond $ then_t $ Const (@{const_name Nil}, _)) =
wenzelm@50422
   599
          SOME (cond, then_t)
wenzelm@50422
   600
      | dest_if _ = NONE
wenzelm@50422
   601
    fun tac _ [] = rtac set_singleton 1 ORELSE rtac inst_Collect_mem_eq 1
wenzelm@50422
   602
      | tac ctxt (If :: cont) =
wenzelm@50422
   603
          Splitter.split_tac [@{thm split_if}] 1
wenzelm@50422
   604
          THEN rtac @{thm conjI} 1
wenzelm@50422
   605
          THEN rtac @{thm impI} 1
wenzelm@50422
   606
          THEN Subgoal.FOCUS (fn {prems, context, ...} =>
wenzelm@50422
   607
            CONVERSION (right_hand_set_comprehension_conv (K
wenzelm@51315
   608
              (HOLogic.conj_conv (Conv.rewr_conv (List.last prems RS @{thm Eq_TrueI})) Conv.all_conv
wenzelm@50422
   609
               then_conv
wenzelm@50422
   610
               rewr_conv' @{lemma "(True & P) = P" by simp})) context) 1) ctxt 1
wenzelm@50422
   611
          THEN tac ctxt cont
wenzelm@50422
   612
          THEN rtac @{thm impI} 1
wenzelm@50422
   613
          THEN Subgoal.FOCUS (fn {prems, context, ...} =>
wenzelm@50422
   614
              CONVERSION (right_hand_set_comprehension_conv (K
wenzelm@51315
   615
                (HOLogic.conj_conv (Conv.rewr_conv (List.last prems RS @{thm Eq_FalseI})) Conv.all_conv
wenzelm@50422
   616
                 then_conv rewr_conv' @{lemma "(False & P) = False" by simp})) context) 1) ctxt 1
wenzelm@50422
   617
          THEN rtac set_Nil_I 1
wenzelm@50422
   618
      | tac ctxt (Case (T, i) :: cont) =
wenzelm@50422
   619
          let
blanchet@54404
   620
            val SOME {injects, distincts, case_thms, split, ...} =
blanchet@54404
   621
              Ctr_Sugar.ctr_sugar_of ctxt (fst (dest_Type T))
wenzelm@50422
   622
          in
wenzelm@50422
   623
            (* do case distinction *)
blanchet@54404
   624
            Splitter.split_tac [split] 1
wenzelm@50422
   625
            THEN EVERY (map_index (fn (i', _) =>
blanchet@54404
   626
              (if i' < length case_thms - 1 then rtac @{thm conjI} 1 else all_tac)
wenzelm@50422
   627
              THEN REPEAT_DETERM (rtac @{thm allI} 1)
wenzelm@50422
   628
              THEN rtac @{thm impI} 1
wenzelm@50422
   629
              THEN (if i' = i then
wenzelm@50422
   630
                (* continue recursively *)
wenzelm@50422
   631
                Subgoal.FOCUS (fn {prems, context, ...} =>
wenzelm@50422
   632
                  CONVERSION (Thm.eta_conversion then_conv right_hand_set_comprehension_conv (K
wenzelm@51315
   633
                      ((HOLogic.conj_conv
wenzelm@51315
   634
                        (HOLogic.eq_conv Conv.all_conv (rewr_conv' (List.last prems)) then_conv
blanchet@54404
   635
                          (Conv.try_conv (Conv.rewrs_conv (map mk_meta_eq injects))))
wenzelm@50422
   636
                        Conv.all_conv)
wenzelm@50422
   637
                        then_conv (Conv.try_conv (Conv.rewr_conv del_refl_eq))
wenzelm@50422
   638
                        then_conv conjunct_assoc_conv)) context
wenzelm@51315
   639
                    then_conv (HOLogic.Trueprop_conv (HOLogic.eq_conv Conv.all_conv (Collect_conv (fn (_, ctxt) =>
wenzelm@50422
   640
                      Conv.repeat_conv
wenzelm@50422
   641
                        (all_but_last_exists_conv
wenzelm@50422
   642
                          (K (rewr_conv'
wenzelm@50422
   643
                            @{lemma "(EX x. x = t & P x) = P t" by simp})) ctxt)) context)))) 1) ctxt 1
wenzelm@50422
   644
                THEN tac ctxt cont
wenzelm@50422
   645
              else
wenzelm@50422
   646
                Subgoal.FOCUS (fn {prems, context, ...} =>
wenzelm@50422
   647
                  CONVERSION
wenzelm@50422
   648
                    (right_hand_set_comprehension_conv (K
wenzelm@51315
   649
                      (HOLogic.conj_conv
wenzelm@51315
   650
                        ((HOLogic.eq_conv Conv.all_conv
wenzelm@50422
   651
                          (rewr_conv' (List.last prems))) then_conv
blanchet@54404
   652
                          (Conv.rewrs_conv (map (fn th => th RS @{thm Eq_FalseI}) distincts)))
wenzelm@50422
   653
                        Conv.all_conv then_conv
wenzelm@50422
   654
                        (rewr_conv' @{lemma "(False & P) = False" by simp}))) context then_conv
wenzelm@51314
   655
                      HOLogic.Trueprop_conv
wenzelm@51315
   656
                        (HOLogic.eq_conv Conv.all_conv
wenzelm@50422
   657
                          (Collect_conv (fn (_, ctxt) =>
wenzelm@50422
   658
                            Conv.repeat_conv
wenzelm@50422
   659
                              (Conv.bottom_conv
wenzelm@50422
   660
                                (K (rewr_conv'
wenzelm@50422
   661
                                  @{lemma "(EX x. P) = P" by simp})) ctxt)) context))) 1) ctxt 1
blanchet@54404
   662
                THEN rtac set_Nil_I 1)) case_thms)
wenzelm@50422
   663
          end
wenzelm@50422
   664
    fun make_inner_eqs bound_vs Tis eqs t =
wenzelm@50422
   665
      (case dest_case t of
blanchet@54404
   666
        SOME (x, T, i, cont, constr_name) =>
wenzelm@50422
   667
          let
wenzelm@52131
   668
            val (vs, body) = strip_abs (Envir.eta_long (map snd bound_vs) cont)
wenzelm@50422
   669
            val x' = incr_boundvars (length vs) x
wenzelm@50422
   670
            val eqs' = map (incr_boundvars (length vs)) eqs
wenzelm@50422
   671
            val constr_t =
wenzelm@50422
   672
              list_comb
wenzelm@50422
   673
                (Const (constr_name, map snd vs ---> T), map Bound (((length vs) - 1) downto 0))
wenzelm@50422
   674
            val constr_eq = Const (@{const_name HOL.eq}, T --> T --> @{typ bool}) $ constr_t $ x'
wenzelm@50422
   675
          in
wenzelm@50422
   676
            make_inner_eqs (rev vs @ bound_vs) (Case (T, i) :: Tis) (constr_eq :: eqs') body
wenzelm@50422
   677
          end
wenzelm@50422
   678
      | NONE =>
wenzelm@50422
   679
          (case dest_if t of
wenzelm@50422
   680
            SOME (condition, cont) => make_inner_eqs bound_vs (If :: Tis) (condition :: eqs) cont
wenzelm@50422
   681
          | NONE =>
wenzelm@50422
   682
            if eqs = [] then NONE (* no rewriting, nothing to be done *)
wenzelm@50422
   683
            else
wenzelm@50422
   684
              let
wenzelm@50422
   685
                val Type (@{type_name List.list}, [rT]) = fastype_of1 (map snd bound_vs, t)
wenzelm@50422
   686
                val pat_eq =
wenzelm@50422
   687
                  (case try dest_singleton_list t of
wenzelm@50422
   688
                    SOME t' =>
wenzelm@50422
   689
                      Const (@{const_name HOL.eq}, rT --> rT --> @{typ bool}) $
wenzelm@50422
   690
                        Bound (length bound_vs) $ t'
wenzelm@50422
   691
                  | NONE =>
wenzelm@50422
   692
                      Const (@{const_name Set.member}, rT --> HOLogic.mk_setT rT --> @{typ bool}) $
wenzelm@50422
   693
                        Bound (length bound_vs) $ (mk_set rT $ t))
wenzelm@50422
   694
                val reverse_bounds = curry subst_bounds
wenzelm@50422
   695
                  ((map Bound ((length bound_vs - 1) downto 0)) @ [Bound (length bound_vs)])
wenzelm@50422
   696
                val eqs' = map reverse_bounds eqs
wenzelm@50422
   697
                val pat_eq' = reverse_bounds pat_eq
wenzelm@50422
   698
                val inner_t =
wenzelm@50422
   699
                  fold (fn (_, T) => fn t => HOLogic.exists_const T $ absdummy T t)
wenzelm@50422
   700
                    (rev bound_vs) (fold (curry HOLogic.mk_conj) eqs' pat_eq')
wenzelm@50422
   701
                val lhs = term_of redex
wenzelm@50422
   702
                val rhs = HOLogic.mk_Collect ("x", rT, inner_t)
wenzelm@50422
   703
                val rewrite_rule_t = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
wenzelm@50422
   704
              in
wenzelm@50422
   705
                SOME
wenzelm@50422
   706
                  ((Goal.prove ctxt [] [] rewrite_rule_t
wenzelm@50422
   707
                    (fn {context, ...} => tac context (rev Tis))) RS @{thm eq_reflection})
wenzelm@50422
   708
              end))
wenzelm@50422
   709
  in
wenzelm@50422
   710
    make_inner_eqs [] [] [] (dest_set (term_of redex))
wenzelm@50422
   711
  end
wenzelm@50422
   712
wenzelm@50422
   713
end
wenzelm@50422
   714
*}
bulwahn@41463
   715
bulwahn@41463
   716
simproc_setup list_to_set_comprehension ("set xs") = {* K List_to_Set_Comprehension.simproc *}
bulwahn@41463
   717
haftmann@46133
   718
code_datatype set coset
haftmann@46133
   719
haftmann@46133
   720
hide_const (open) coset
wenzelm@35115
   721
haftmann@49948
   722
haftmann@21061
   723
subsubsection {* @{const Nil} and @{const Cons} *}
haftmann@21061
   724
haftmann@21061
   725
lemma not_Cons_self [simp]:
haftmann@21061
   726
  "xs \<noteq> x # xs"
nipkow@13145
   727
by (induct xs) auto
wenzelm@13114
   728
wenzelm@41697
   729
lemma not_Cons_self2 [simp]:
wenzelm@41697
   730
  "x # xs \<noteq> xs"
wenzelm@41697
   731
by (rule not_Cons_self [symmetric])
wenzelm@13114
   732
wenzelm@13142
   733
lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
nipkow@13145
   734
by (induct xs) auto
wenzelm@13114
   735
nipkow@53689
   736
lemma tl_Nil: "tl xs = [] \<longleftrightarrow> xs = [] \<or> (EX x. xs = [x])"
nipkow@53689
   737
by (cases xs) auto
nipkow@53689
   738
nipkow@53689
   739
lemma Nil_tl: "[] = tl xs \<longleftrightarrow> xs = [] \<or> (EX x. xs = [x])"
nipkow@53689
   740
by (cases xs) auto
nipkow@53689
   741
wenzelm@13142
   742
lemma length_induct:
haftmann@21061
   743
  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
nipkow@53689
   744
by (fact measure_induct)
wenzelm@13114
   745
haftmann@37289
   746
lemma list_nonempty_induct [consumes 1, case_names single cons]:
haftmann@37289
   747
  assumes "xs \<noteq> []"
haftmann@37289
   748
  assumes single: "\<And>x. P [x]"
haftmann@37289
   749
  assumes cons: "\<And>x xs. xs \<noteq> [] \<Longrightarrow> P xs \<Longrightarrow> P (x # xs)"
haftmann@37289
   750
  shows "P xs"
haftmann@37289
   751
using `xs \<noteq> []` proof (induct xs)
haftmann@37289
   752
  case Nil then show ?case by simp
haftmann@37289
   753
next
wenzelm@53374
   754
  case (Cons x xs)
wenzelm@53374
   755
  show ?case
wenzelm@53374
   756
  proof (cases xs)
wenzelm@53374
   757
    case Nil
wenzelm@53374
   758
    with single show ?thesis by simp
haftmann@37289
   759
  next
wenzelm@53374
   760
    case Cons
wenzelm@53374
   761
    show ?thesis
wenzelm@53374
   762
    proof (rule cons)
wenzelm@53374
   763
      from Cons show "xs \<noteq> []" by simp
wenzelm@53374
   764
      with Cons.hyps show "P xs" .
wenzelm@53374
   765
    qed
haftmann@37289
   766
  qed
haftmann@37289
   767
qed
haftmann@37289
   768
hoelzl@45714
   769
lemma inj_split_Cons: "inj_on (\<lambda>(xs, n). n#xs) X"
hoelzl@45714
   770
  by (auto intro!: inj_onI)
wenzelm@13114
   771
haftmann@49948
   772
haftmann@21061
   773
subsubsection {* @{const length} *}
wenzelm@13114
   774
wenzelm@13142
   775
text {*
haftmann@21061
   776
  Needs to come before @{text "@"} because of theorem @{text
haftmann@21061
   777
  append_eq_append_conv}.
wenzelm@13142
   778
*}
wenzelm@13114
   779
wenzelm@13142
   780
lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
nipkow@13145
   781
by (induct xs) auto
wenzelm@13114
   782
wenzelm@13142
   783
lemma length_map [simp]: "length (map f xs) = length xs"
nipkow@13145
   784
by (induct xs) auto
wenzelm@13114
   785
wenzelm@13142
   786
lemma length_rev [simp]: "length (rev xs) = length xs"
nipkow@13145
   787
by (induct xs) auto
wenzelm@13114
   788
wenzelm@13142
   789
lemma length_tl [simp]: "length (tl xs) = length xs - 1"
nipkow@13145
   790
by (cases xs) auto
wenzelm@13114
   791
wenzelm@13142
   792
lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
nipkow@13145
   793
by (induct xs) auto
wenzelm@13114
   794
wenzelm@13142
   795
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
nipkow@13145
   796
by (induct xs) auto
wenzelm@13114
   797
nipkow@23479
   798
lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
nipkow@23479
   799
by auto
nipkow@23479
   800
wenzelm@13114
   801
lemma length_Suc_conv:
nipkow@13145
   802
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
nipkow@13145
   803
by (induct xs) auto
wenzelm@13142
   804
nipkow@14025
   805
lemma Suc_length_conv:
nipkow@14025
   806
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
paulson@14208
   807
apply (induct xs, simp, simp)
nipkow@14025
   808
apply blast
nipkow@14025
   809
done
nipkow@14025
   810
wenzelm@25221
   811
lemma impossible_Cons: "length xs <= length ys ==> xs = x # ys = False"
wenzelm@25221
   812
  by (induct xs) auto
wenzelm@25221
   813
haftmann@26442
   814
lemma list_induct2 [consumes 1, case_names Nil Cons]:
haftmann@26442
   815
  "length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
haftmann@26442
   816
   (\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
haftmann@26442
   817
   \<Longrightarrow> P xs ys"
haftmann@26442
   818
proof (induct xs arbitrary: ys)
haftmann@26442
   819
  case Nil then show ?case by simp
haftmann@26442
   820
next
haftmann@26442
   821
  case (Cons x xs ys) then show ?case by (cases ys) simp_all
haftmann@26442
   822
qed
haftmann@26442
   823
haftmann@26442
   824
lemma list_induct3 [consumes 2, case_names Nil Cons]:
haftmann@26442
   825
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
haftmann@26442
   826
   (\<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
   827
   \<Longrightarrow> P xs ys zs"
haftmann@26442
   828
proof (induct xs arbitrary: ys zs)
haftmann@26442
   829
  case Nil then show ?case by simp
haftmann@26442
   830
next
haftmann@26442
   831
  case (Cons x xs ys zs) then show ?case by (cases ys, simp_all)
haftmann@26442
   832
    (cases zs, simp_all)
haftmann@26442
   833
qed
wenzelm@13114
   834
kaliszyk@36154
   835
lemma list_induct4 [consumes 3, case_names Nil Cons]:
kaliszyk@36154
   836
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow>
kaliszyk@36154
   837
   P [] [] [] [] \<Longrightarrow> (\<And>x xs y ys z zs w ws. length xs = length ys \<Longrightarrow>
kaliszyk@36154
   838
   length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow> P xs ys zs ws \<Longrightarrow>
kaliszyk@36154
   839
   P (x#xs) (y#ys) (z#zs) (w#ws)) \<Longrightarrow> P xs ys zs ws"
kaliszyk@36154
   840
proof (induct xs arbitrary: ys zs ws)
kaliszyk@36154
   841
  case Nil then show ?case by simp
kaliszyk@36154
   842
next
kaliszyk@36154
   843
  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
   844
qed
kaliszyk@36154
   845
krauss@22493
   846
lemma list_induct2': 
krauss@22493
   847
  "\<lbrakk> P [] [];
krauss@22493
   848
  \<And>x xs. P (x#xs) [];
krauss@22493
   849
  \<And>y ys. P [] (y#ys);
krauss@22493
   850
   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
krauss@22493
   851
 \<Longrightarrow> P xs ys"
krauss@22493
   852
by (induct xs arbitrary: ys) (case_tac x, auto)+
krauss@22493
   853
blanchet@55524
   854
lemma list_all2_iff:
blanchet@55524
   855
  "list_all2 P xs ys \<longleftrightarrow> length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)"
blanchet@55524
   856
by (induct xs ys rule: list_induct2') auto
blanchet@55524
   857
nipkow@22143
   858
lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
nipkow@24349
   859
by (rule Eq_FalseI) auto
wenzelm@24037
   860
wenzelm@24037
   861
simproc_setup list_neq ("(xs::'a list) = ys") = {*
nipkow@22143
   862
(*
nipkow@22143
   863
Reduces xs=ys to False if xs and ys cannot be of the same length.
nipkow@22143
   864
This is the case if the atomic sublists of one are a submultiset
nipkow@22143
   865
of those of the other list and there are fewer Cons's in one than the other.
nipkow@22143
   866
*)
wenzelm@24037
   867
wenzelm@24037
   868
let
nipkow@22143
   869
huffman@29856
   870
fun len (Const(@{const_name Nil},_)) acc = acc
huffman@29856
   871
  | len (Const(@{const_name Cons},_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
huffman@29856
   872
  | len (Const(@{const_name append},_) $ xs $ ys) acc = len xs (len ys acc)
huffman@29856
   873
  | len (Const(@{const_name rev},_) $ xs) acc = len xs acc
huffman@29856
   874
  | len (Const(@{const_name map},_) $ _ $ xs) acc = len xs acc
nipkow@22143
   875
  | len t (ts,n) = (t::ts,n);
nipkow@22143
   876
wenzelm@51717
   877
val ss = simpset_of @{context};
wenzelm@51717
   878
wenzelm@51717
   879
fun list_neq ctxt ct =
nipkow@22143
   880
  let
wenzelm@24037
   881
    val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
nipkow@22143
   882
    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
nipkow@22143
   883
    fun prove_neq() =
nipkow@22143
   884
      let
nipkow@22143
   885
        val Type(_,listT::_) = eqT;
haftmann@22994
   886
        val size = HOLogic.size_const listT;
nipkow@22143
   887
        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
nipkow@22143
   888
        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
wenzelm@51717
   889
        val thm = Goal.prove ctxt [] [] neq_len
wenzelm@51717
   890
          (K (simp_tac (put_simpset ss ctxt) 1));
haftmann@22633
   891
      in SOME (thm RS @{thm neq_if_length_neq}) end
nipkow@22143
   892
  in
wenzelm@23214
   893
    if m < n andalso submultiset (op aconv) (ls,rs) orelse
wenzelm@23214
   894
       n < m andalso submultiset (op aconv) (rs,ls)
nipkow@22143
   895
    then prove_neq() else NONE
nipkow@22143
   896
  end;
wenzelm@51717
   897
in K list_neq end;
nipkow@22143
   898
*}
nipkow@22143
   899
nipkow@22143
   900
nipkow@15392
   901
subsubsection {* @{text "@"} -- append *}
wenzelm@13114
   902
wenzelm@13142
   903
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
nipkow@13145
   904
by (induct xs) auto
wenzelm@13114
   905
wenzelm@13142
   906
lemma append_Nil2 [simp]: "xs @ [] = xs"
nipkow@13145
   907
by (induct xs) auto
nipkow@3507
   908
wenzelm@13142
   909
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
nipkow@13145
   910
by (induct xs) auto
wenzelm@13114
   911
wenzelm@13142
   912
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
nipkow@13145
   913
by (induct xs) auto
wenzelm@13114
   914
wenzelm@13142
   915
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
nipkow@13145
   916
by (induct xs) auto
wenzelm@13114
   917
wenzelm@13142
   918
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
nipkow@13145
   919
by (induct xs) auto
wenzelm@13114
   920
blanchet@54147
   921
lemma append_eq_append_conv [simp]:
nipkow@24526
   922
 "length xs = length ys \<or> length us = length vs
berghofe@13883
   923
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
nipkow@24526
   924
apply (induct xs arbitrary: ys)
paulson@14208
   925
 apply (case_tac ys, simp, force)
paulson@14208
   926
apply (case_tac ys, force, simp)
nipkow@13145
   927
done
wenzelm@13142
   928
nipkow@24526
   929
lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
nipkow@24526
   930
  (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
nipkow@24526
   931
apply (induct xs arbitrary: ys zs ts)
nipkow@44890
   932
 apply fastforce
nipkow@14495
   933
apply(case_tac zs)
nipkow@14495
   934
 apply simp
nipkow@44890
   935
apply fastforce
nipkow@14495
   936
done
nipkow@14495
   937
berghofe@34910
   938
lemma same_append_eq [iff, induct_simp]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145
   939
by simp
wenzelm@13142
   940
wenzelm@13142
   941
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145
   942
by simp
wenzelm@13114
   943
berghofe@34910
   944
lemma append_same_eq [iff, induct_simp]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   945
by simp
wenzelm@13114
   946
wenzelm@13142
   947
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   948
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   949
wenzelm@13142
   950
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   951
using append_same_eq [of "[]"] by auto
wenzelm@13114
   952
blanchet@54147
   953
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   954
by (induct xs) auto
wenzelm@13114
   955
wenzelm@13142
   956
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   957
by (induct xs) auto
wenzelm@13114
   958
wenzelm@13142
   959
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   960
by (simp add: hd_append split: list.split)
wenzelm@13114
   961
wenzelm@13142
   962
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   963
by (simp split: list.split)
wenzelm@13114
   964
wenzelm@13142
   965
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   966
by (simp add: tl_append split: list.split)
wenzelm@13114
   967
wenzelm@13114
   968
nipkow@14300
   969
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   970
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   971
by(cases ys) auto
nipkow@14300
   972
nipkow@15281
   973
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   974
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   975
by(cases ys) auto
nipkow@15281
   976
nipkow@14300
   977
wenzelm@13142
   978
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   979
wenzelm@13114
   980
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   981
by simp
wenzelm@13114
   982
wenzelm@13142
   983
lemma Cons_eq_appendI:
nipkow@13145
   984
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   985
by (drule sym) simp
wenzelm@13114
   986
wenzelm@13142
   987
lemma append_eq_appendI:
nipkow@13145
   988
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   989
by (drule sym) simp
wenzelm@13114
   990
wenzelm@13114
   991
wenzelm@13142
   992
text {*
nipkow@13145
   993
Simplification procedure for all list equalities.
nipkow@13145
   994
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   995
- both lists end in a singleton list,
nipkow@13145
   996
- or both lists end in the same list.
wenzelm@13142
   997
*}
wenzelm@13142
   998
wenzelm@43594
   999
simproc_setup list_eq ("(xs::'a list) = ys")  = {*
wenzelm@13462
  1000
  let
wenzelm@43594
  1001
    fun last (cons as Const (@{const_name Cons}, _) $ _ $ xs) =
wenzelm@43594
  1002
          (case xs of Const (@{const_name Nil}, _) => cons | _ => last xs)
wenzelm@43594
  1003
      | last (Const(@{const_name append},_) $ _ $ ys) = last ys
wenzelm@43594
  1004
      | last t = t;
wenzelm@43594
  1005
    
wenzelm@43594
  1006
    fun list1 (Const(@{const_name Cons},_) $ _ $ Const(@{const_name Nil},_)) = true
wenzelm@43594
  1007
      | list1 _ = false;
wenzelm@43594
  1008
    
wenzelm@43594
  1009
    fun butlast ((cons as Const(@{const_name Cons},_) $ x) $ xs) =
wenzelm@43594
  1010
          (case xs of Const (@{const_name Nil}, _) => xs | _ => cons $ butlast xs)
wenzelm@43594
  1011
      | butlast ((app as Const (@{const_name append}, _) $ xs) $ ys) = app $ butlast ys
wenzelm@43594
  1012
      | butlast xs = Const(@{const_name Nil}, fastype_of xs);
wenzelm@43594
  1013
    
wenzelm@43594
  1014
    val rearr_ss =
wenzelm@51717
  1015
      simpset_of (put_simpset HOL_basic_ss @{context}
wenzelm@51717
  1016
        addsimps [@{thm append_assoc}, @{thm append_Nil}, @{thm append_Cons}]);
wenzelm@43594
  1017
    
wenzelm@51717
  1018
    fun list_eq ctxt (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
  1019
      let
wenzelm@43594
  1020
        val lastl = last lhs and lastr = last rhs;
wenzelm@43594
  1021
        fun rearr conv =
wenzelm@43594
  1022
          let
wenzelm@43594
  1023
            val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@43594
  1024
            val Type(_,listT::_) = eqT
wenzelm@43594
  1025
            val appT = [listT,listT] ---> listT
wenzelm@43594
  1026
            val app = Const(@{const_name append},appT)
wenzelm@43594
  1027
            val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@43594
  1028
            val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@51717
  1029
            val thm = Goal.prove ctxt [] [] eq
wenzelm@51717
  1030
              (K (simp_tac (put_simpset rearr_ss ctxt) 1));
wenzelm@43594
  1031
          in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@43594
  1032
      in
wenzelm@43594
  1033
        if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
wenzelm@43594
  1034
        else if lastl aconv lastr then rearr @{thm append_same_eq}
wenzelm@43594
  1035
        else NONE
wenzelm@43594
  1036
      end;
wenzelm@51717
  1037
  in fn _ => fn ctxt => fn ct => list_eq ctxt (term_of ct) end;
wenzelm@13114
  1038
*}
wenzelm@13114
  1039
wenzelm@13114
  1040
haftmann@49948
  1041
subsubsection {* @{const map} *}
wenzelm@13114
  1042
haftmann@40210
  1043
lemma hd_map:
haftmann@40210
  1044
  "xs \<noteq> [] \<Longrightarrow> hd (map f xs) = f (hd xs)"
haftmann@40210
  1045
  by (cases xs) simp_all
haftmann@40210
  1046
haftmann@40210
  1047
lemma map_tl:
haftmann@40210
  1048
  "map f (tl xs) = tl (map f xs)"
haftmann@40210
  1049
  by (cases xs) simp_all
haftmann@40210
  1050
wenzelm@13142
  1051
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
  1052
by (induct xs) simp_all
wenzelm@13114
  1053
wenzelm@13142
  1054
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
  1055
by (rule ext, induct_tac xs) auto
wenzelm@13114
  1056
wenzelm@13142
  1057
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
  1058
by (induct xs) auto
wenzelm@13114
  1059
hoelzl@33639
  1060
lemma map_map [simp]: "map f (map g xs) = map (f \<circ> g) xs"
hoelzl@33639
  1061
by (induct xs) auto
hoelzl@33639
  1062
nipkow@35208
  1063
lemma map_comp_map[simp]: "((map f) o (map g)) = map(f o g)"
nipkow@35208
  1064
apply(rule ext)
nipkow@35208
  1065
apply(simp)
nipkow@35208
  1066
done
nipkow@35208
  1067
wenzelm@13142
  1068
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
  1069
by (induct xs) auto
wenzelm@13114
  1070
nipkow@13737
  1071
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
  1072
by (induct xs) auto
nipkow@13737
  1073
krauss@44013
  1074
lemma map_cong [fundef_cong]:
haftmann@40122
  1075
  "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> map f xs = map g ys"
haftmann@40122
  1076
  by simp
wenzelm@13114
  1077
wenzelm@13142
  1078
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
  1079
by (cases xs) auto
wenzelm@13114
  1080
wenzelm@13142
  1081
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
  1082
by (cases xs) auto
wenzelm@13114
  1083
paulson@18447
  1084
lemma map_eq_Cons_conv:
nipkow@14025
  1085
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
  1086
by (cases xs) auto
wenzelm@13114
  1087
paulson@18447
  1088
lemma Cons_eq_map_conv:
nipkow@14025
  1089
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
  1090
by (cases ys) auto
nipkow@14025
  1091
paulson@18447
  1092
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
paulson@18447
  1093
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
paulson@18447
  1094
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
paulson@18447
  1095
nipkow@14111
  1096
lemma ex_map_conv:
nipkow@14111
  1097
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
paulson@18447
  1098
by(induct ys, auto simp add: Cons_eq_map_conv)
nipkow@14111
  1099
nipkow@15110
  1100
lemma map_eq_imp_length_eq:
paulson@35510
  1101
  assumes "map f xs = map g ys"
haftmann@26734
  1102
  shows "length xs = length ys"
wenzelm@53374
  1103
  using assms
wenzelm@53374
  1104
proof (induct ys arbitrary: xs)
haftmann@26734
  1105
  case Nil then show ?case by simp
haftmann@26734
  1106
next
haftmann@26734
  1107
  case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto
paulson@35510
  1108
  from Cons xs have "map f zs = map g ys" by simp
wenzelm@53374
  1109
  with Cons have "length zs = length ys" by blast
haftmann@26734
  1110
  with xs show ?case by simp
haftmann@26734
  1111
qed
haftmann@26734
  1112
  
nipkow@15110
  1113
lemma map_inj_on:
nipkow@15110
  1114
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
  1115
  ==> xs = ys"
nipkow@15110
  1116
apply(frule map_eq_imp_length_eq)
nipkow@15110
  1117
apply(rotate_tac -1)
nipkow@15110
  1118
apply(induct rule:list_induct2)
nipkow@15110
  1119
 apply simp
nipkow@15110
  1120
apply(simp)
nipkow@15110
  1121
apply (blast intro:sym)
nipkow@15110
  1122
done
nipkow@15110
  1123
nipkow@15110
  1124
lemma inj_on_map_eq_map:
nipkow@15110
  1125
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
  1126
by(blast dest:map_inj_on)
nipkow@15110
  1127
wenzelm@13114
  1128
lemma map_injective:
nipkow@24526
  1129
 "map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@24526
  1130
by (induct ys arbitrary: xs) (auto dest!:injD)
wenzelm@13114
  1131
nipkow@14339
  1132
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
  1133
by(blast dest:map_injective)
nipkow@14339
  1134
wenzelm@13114
  1135
lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@17589
  1136
by (iprover dest: map_injective injD intro: inj_onI)
wenzelm@13114
  1137
wenzelm@13114
  1138
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
  1139
apply (unfold inj_on_def, clarify)
nipkow@13145
  1140
apply (erule_tac x = "[x]" in ballE)
paulson@14208
  1141
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
  1142
apply blast
nipkow@13145
  1143
done
wenzelm@13114
  1144
nipkow@14339
  1145
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
  1146
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
  1147
nipkow@15303
  1148
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
  1149
apply(rule inj_onI)
nipkow@15303
  1150
apply(erule map_inj_on)
nipkow@15303
  1151
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
  1152
done
nipkow@15303
  1153
kleing@14343
  1154
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
  1155
by (induct xs, auto)
wenzelm@13114
  1156
nipkow@14402
  1157
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
  1158
by (induct xs) auto
nipkow@14402
  1159
nipkow@15110
  1160
lemma map_fst_zip[simp]:
nipkow@15110
  1161
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
  1162
by (induct rule:list_induct2, simp_all)
nipkow@15110
  1163
nipkow@15110
  1164
lemma map_snd_zip[simp]:
nipkow@15110
  1165
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
  1166
by (induct rule:list_induct2, simp_all)
nipkow@15110
  1167
blanchet@55467
  1168
functor map: map
nipkow@47122
  1169
by (simp_all add: id_def)
nipkow@47122
  1170
haftmann@49948
  1171
declare map.id [simp]
haftmann@49948
  1172
haftmann@49948
  1173
haftmann@49948
  1174
subsubsection {* @{const rev} *}
wenzelm@13114
  1175
wenzelm@13142
  1176
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
  1177
by (induct xs) auto
wenzelm@13114
  1178
wenzelm@13142
  1179
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
  1180
by (induct xs) auto
wenzelm@13114
  1181
kleing@15870
  1182
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870
  1183
by auto
kleing@15870
  1184
wenzelm@13142
  1185
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
  1186
by (induct xs) auto
wenzelm@13114
  1187
wenzelm@13142
  1188
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
  1189
by (induct xs) auto
wenzelm@13114
  1190
kleing@15870
  1191
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870
  1192
by (cases xs) auto
kleing@15870
  1193
kleing@15870
  1194
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870
  1195
by (cases xs) auto
kleing@15870
  1196
blanchet@54147
  1197
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
haftmann@21061
  1198
apply (induct xs arbitrary: ys, force)
paulson@14208
  1199
apply (case_tac ys, simp, force)
nipkow@13145
  1200
done
wenzelm@13114
  1201
nipkow@15439
  1202
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
  1203
by(simp add:inj_on_def)
nipkow@15439
  1204
wenzelm@13366
  1205
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
  1206
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489
  1207
apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145
  1208
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
  1209
done
wenzelm@13114
  1210
wenzelm@13366
  1211
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
  1212
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
  1213
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1214
wenzelm@13366
  1215
lemmas rev_cases = rev_exhaust
wenzelm@13366
  1216
nipkow@18423
  1217
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
nipkow@18423
  1218
by(rule rev_cases[of xs]) auto
nipkow@18423
  1219
wenzelm@13114
  1220
haftmann@49948
  1221
subsubsection {* @{const set} *}
wenzelm@13114
  1222
wenzelm@13142
  1223
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
  1224
by (induct xs) auto
wenzelm@13114
  1225
wenzelm@13142
  1226
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
  1227
by (induct xs) auto
wenzelm@13114
  1228
nipkow@17830
  1229
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
nipkow@17830
  1230
by(cases xs) auto
oheimb@14099
  1231
wenzelm@13142
  1232
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
  1233
by auto
wenzelm@13114
  1234
oheimb@14099
  1235
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
  1236
by auto
oheimb@14099
  1237
wenzelm@13142
  1238
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
  1239
by (induct xs) auto
wenzelm@13114
  1240
nipkow@15245
  1241
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
  1242
by(induct xs) auto
nipkow@15245
  1243
wenzelm@13142
  1244
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
  1245
by (induct xs) auto
wenzelm@13114
  1246
wenzelm@13142
  1247
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
  1248
by (induct xs) auto
wenzelm@13114
  1249
wenzelm@13142
  1250
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
  1251
by (induct xs) auto
wenzelm@13114
  1252
nipkow@32417
  1253
lemma set_upt [simp]: "set[i..<j] = {i..<j}"
bulwahn@41463
  1254
by (induct j) auto
wenzelm@13114
  1255
wenzelm@13142
  1256
wenzelm@25221
  1257
lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs"
nipkow@18049
  1258
proof (induct xs)
nipkow@26073
  1259
  case Nil thus ?case by simp
nipkow@26073
  1260
next
nipkow@26073
  1261
  case Cons thus ?case by (auto intro: Cons_eq_appendI)
nipkow@26073
  1262
qed
nipkow@26073
  1263
haftmann@26734
  1264
lemma in_set_conv_decomp: "x \<in> set xs \<longleftrightarrow> (\<exists>ys zs. xs = ys @ x # zs)"
haftmann@26734
  1265
  by (auto elim: split_list)
nipkow@26073
  1266
nipkow@26073
  1267
lemma split_list_first: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@26073
  1268
proof (induct xs)
nipkow@26073
  1269
  case Nil thus ?case by simp
nipkow@18049
  1270
next
nipkow@18049
  1271
  case (Cons a xs)
nipkow@18049
  1272
  show ?case
nipkow@18049
  1273
  proof cases
nipkow@44890
  1274
    assume "x = a" thus ?case using Cons by fastforce
nipkow@18049
  1275
  next
nipkow@44890
  1276
    assume "x \<noteq> a" thus ?case using Cons by(fastforce intro!: Cons_eq_appendI)
nipkow@26073
  1277
  qed
nipkow@26073
  1278
qed
nipkow@26073
  1279
nipkow@26073
  1280
lemma in_set_conv_decomp_first:
nipkow@26073
  1281
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
haftmann@26734
  1282
  by (auto dest!: split_list_first)
nipkow@26073
  1283
haftmann@40122
  1284
lemma split_list_last: "x \<in> set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
haftmann@40122
  1285
proof (induct xs rule: rev_induct)
nipkow@26073
  1286
  case Nil thus ?case by simp
nipkow@26073
  1287
next
nipkow@26073
  1288
  case (snoc a xs)
nipkow@26073
  1289
  show ?case
nipkow@26073
  1290
  proof cases
blanchet@55584
  1291
    assume "x = a" thus ?case using snoc by (metis set_simps(1) emptyE)
nipkow@26073
  1292
  next
nipkow@44890
  1293
    assume "x \<noteq> a" thus ?case using snoc by fastforce
nipkow@18049
  1294
  qed
nipkow@18049
  1295
qed
nipkow@18049
  1296
nipkow@26073
  1297
lemma in_set_conv_decomp_last:
nipkow@26073
  1298
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
haftmann@26734
  1299
  by (auto dest!: split_list_last)
nipkow@26073
  1300
nipkow@26073
  1301
lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs & P x"
nipkow@26073
  1302
proof (induct xs)
nipkow@26073
  1303
  case Nil thus ?case by simp
nipkow@26073
  1304
next
nipkow@26073
  1305
  case Cons thus ?case
nipkow@26073
  1306
    by(simp add:Bex_def)(metis append_Cons append.simps(1))
nipkow@26073
  1307
qed
nipkow@26073
  1308
nipkow@26073
  1309
lemma split_list_propE:
haftmann@26734
  1310
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
  1311
  obtains ys x zs where "xs = ys @ x # zs" and "P x"
haftmann@26734
  1312
using split_list_prop [OF assms] by blast
nipkow@26073
  1313
nipkow@26073
  1314
lemma split_list_first_prop:
nipkow@26073
  1315
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
nipkow@26073
  1316
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)"
haftmann@26734
  1317
proof (induct xs)
nipkow@26073
  1318
  case Nil thus ?case by simp
nipkow@26073
  1319
next
nipkow@26073
  1320
  case (Cons x xs)
nipkow@26073
  1321
  show ?case
nipkow@26073
  1322
  proof cases
nipkow@26073
  1323
    assume "P x"
haftmann@40122
  1324
    thus ?thesis by simp (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
nipkow@26073
  1325
  next
nipkow@26073
  1326
    assume "\<not> P x"
nipkow@26073
  1327
    hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
nipkow@26073
  1328
    thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
nipkow@26073
  1329
  qed
nipkow@26073
  1330
qed
nipkow@26073
  1331
nipkow@26073
  1332
lemma split_list_first_propE:
haftmann@26734
  1333
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
  1334
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y"
haftmann@26734
  1335
using split_list_first_prop [OF assms] by blast
nipkow@26073
  1336
nipkow@26073
  1337
lemma split_list_first_prop_iff:
nipkow@26073
  1338
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
nipkow@26073
  1339
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))"
haftmann@26734
  1340
by (rule, erule split_list_first_prop) auto
nipkow@26073
  1341
nipkow@26073
  1342
lemma split_list_last_prop:
nipkow@26073
  1343
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
nipkow@26073
  1344
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)"
nipkow@26073
  1345
proof(induct xs rule:rev_induct)
nipkow@26073
  1346
  case Nil thus ?case by simp
nipkow@26073
  1347
next
nipkow@26073
  1348
  case (snoc x xs)
nipkow@26073
  1349
  show ?case
nipkow@26073
  1350
  proof cases
nipkow@26073
  1351
    assume "P x" thus ?thesis by (metis emptyE set_empty)
nipkow@26073
  1352
  next
nipkow@26073
  1353
    assume "\<not> P x"
nipkow@26073
  1354
    hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
nipkow@44890
  1355
    thus ?thesis using `\<not> P x` snoc(1) by fastforce
nipkow@26073
  1356
  qed
nipkow@26073
  1357
qed
nipkow@26073
  1358
nipkow@26073
  1359
lemma split_list_last_propE:
haftmann@26734
  1360
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
  1361
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z"
haftmann@26734
  1362
using split_list_last_prop [OF assms] by blast
nipkow@26073
  1363
nipkow@26073
  1364
lemma split_list_last_prop_iff:
nipkow@26073
  1365
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
nipkow@26073
  1366
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
haftmann@26734
  1367
by (metis split_list_last_prop [where P=P] in_set_conv_decomp)
nipkow@26073
  1368
nipkow@26073
  1369
lemma finite_list: "finite A ==> EX xs. set xs = A"
blanchet@55584
  1370
  by (erule finite_induct) (auto simp add: set_simps(2) [symmetric] simp del: set_simps(2))
paulson@13508
  1371
kleing@14388
  1372
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
  1373
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
  1374
haftmann@26442
  1375
lemma set_minus_filter_out:
haftmann@26442
  1376
  "set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)"
haftmann@26442
  1377
  by (induct xs) auto
paulson@15168
  1378
wenzelm@35115
  1379
haftmann@49948
  1380
subsubsection {* @{const filter} *}
wenzelm@13114
  1381
wenzelm@13142
  1382
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
  1383
by (induct xs) auto
wenzelm@13114
  1384
nipkow@15305
  1385
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
  1386
by (induct xs) simp_all
nipkow@15305
  1387
wenzelm@13142
  1388
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
  1389
by (induct xs) auto
wenzelm@13114
  1390
nipkow@16998
  1391
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998
  1392
by (induct xs) (auto simp add: le_SucI)
nipkow@16998
  1393
nipkow@18423
  1394
lemma sum_length_filter_compl:
nipkow@18423
  1395
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
nipkow@18423
  1396
by(induct xs) simp_all
nipkow@18423
  1397
wenzelm@13142
  1398
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
  1399
by (induct xs) auto
wenzelm@13114
  1400
wenzelm@13142
  1401
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
  1402
by (induct xs) auto
wenzelm@13114
  1403
nipkow@16998
  1404
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@24349
  1405
by (induct xs) simp_all
nipkow@16998
  1406
nipkow@16998
  1407
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998
  1408
apply (induct xs)
nipkow@16998
  1409
 apply auto
nipkow@16998
  1410
apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998
  1411
apply simp
nipkow@16998
  1412
done
wenzelm@13114
  1413
nipkow@16965
  1414
lemma filter_map:
nipkow@16965
  1415
  "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965
  1416
by (induct xs) simp_all
nipkow@16965
  1417
nipkow@16965
  1418
lemma length_filter_map[simp]:
nipkow@16965
  1419
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965
  1420
by (simp add:filter_map)
nipkow@16965
  1421
wenzelm@13142
  1422
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
  1423
by auto
wenzelm@13114
  1424
nipkow@15246
  1425
lemma length_filter_less:
nipkow@15246
  1426
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
  1427
proof (induct xs)
nipkow@15246
  1428
  case Nil thus ?case by simp
nipkow@15246
  1429
next
nipkow@15246
  1430
  case (Cons x xs) thus ?case
nipkow@15246
  1431
    apply (auto split:split_if_asm)
nipkow@15246
  1432
    using length_filter_le[of P xs] apply arith
nipkow@15246
  1433
  done
nipkow@15246
  1434
qed
wenzelm@13114
  1435
nipkow@15281
  1436
lemma length_filter_conv_card:
nipkow@15281
  1437
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
  1438
proof (induct xs)
nipkow@15281
  1439
  case Nil thus ?case by simp
nipkow@15281
  1440
next
nipkow@15281
  1441
  case (Cons x xs)
nipkow@15281
  1442
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
  1443
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
  1444
  show ?case (is "?l = card ?S'")
nipkow@15281
  1445
  proof (cases)
nipkow@15281
  1446
    assume "p x"
nipkow@15281
  1447
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@25162
  1448
      by(auto simp: image_def split:nat.split dest:gr0_implies_Suc)
nipkow@15281
  1449
    have "length (filter p (x # xs)) = Suc(card ?S)"
wenzelm@23388
  1450
      using Cons `p x` by simp
nipkow@15281
  1451
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
huffman@44921
  1452
      by (simp add: card_image)
nipkow@15281
  1453
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
  1454
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
  1455
    finally show ?thesis .
nipkow@15281
  1456
  next
nipkow@15281
  1457
    assume "\<not> p x"
nipkow@15281
  1458
    hence eq: "?S' = Suc ` ?S"
nipkow@25162
  1459
      by(auto simp add: image_def split:nat.split elim:lessE)
nipkow@15281
  1460
    have "length (filter p (x # xs)) = card ?S"
wenzelm@23388
  1461
      using Cons `\<not> p x` by simp
nipkow@15281
  1462
    also have "\<dots> = card(Suc ` ?S)" using fin
huffman@44921
  1463
      by (simp add: card_image)
nipkow@15281
  1464
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
  1465
      by (simp add:card_insert_if)
nipkow@15281
  1466
    finally show ?thesis .
nipkow@15281
  1467
  qed
nipkow@15281
  1468
qed
nipkow@15281
  1469
nipkow@17629
  1470
lemma Cons_eq_filterD:
nipkow@17629
  1471
 "x#xs = filter P ys \<Longrightarrow>
nipkow@17629
  1472
  \<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
  1473
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
nipkow@17629
  1474
proof(induct ys)
nipkow@17629
  1475
  case Nil thus ?case by simp
nipkow@17629
  1476
next
nipkow@17629
  1477
  case (Cons y ys)
nipkow@17629
  1478
  show ?case (is "\<exists>x. ?Q x")
nipkow@17629
  1479
  proof cases
nipkow@17629
  1480
    assume Py: "P y"
nipkow@17629
  1481
    show ?thesis
nipkow@17629
  1482
    proof cases
wenzelm@25221
  1483
      assume "x = y"
wenzelm@25221
  1484
      with Py Cons.prems have "?Q []" by simp
wenzelm@25221
  1485
      then show ?thesis ..
nipkow@17629
  1486
    next
wenzelm@25221
  1487
      assume "x \<noteq> y"
wenzelm@25221
  1488
      with Py Cons.prems show ?thesis by simp
nipkow@17629
  1489
    qed
nipkow@17629
  1490
  next
wenzelm@25221
  1491
    assume "\<not> P y"
nipkow@44890
  1492
    with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastforce
wenzelm@25221
  1493
    then have "?Q (y#us)" by simp
wenzelm@25221
  1494
    then show ?thesis ..
nipkow@17629
  1495
  qed
nipkow@17629
  1496
qed
nipkow@17629
  1497
nipkow@17629
  1498
lemma filter_eq_ConsD:
nipkow@17629
  1499
 "filter P ys = x#xs \<Longrightarrow>
nipkow@17629
  1500
  \<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
  1501
by(rule Cons_eq_filterD) simp
nipkow@17629
  1502
nipkow@17629
  1503
lemma filter_eq_Cons_iff:
nipkow@17629
  1504
 "(filter P ys = x#xs) =
nipkow@17629
  1505
  (\<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
  1506
by(auto dest:filter_eq_ConsD)
nipkow@17629
  1507
nipkow@17629
  1508
lemma Cons_eq_filter_iff:
nipkow@17629
  1509
 "(x#xs = filter P ys) =
nipkow@17629
  1510
  (\<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
  1511
by(auto dest:Cons_eq_filterD)
nipkow@17629
  1512
krauss@44013
  1513
lemma filter_cong[fundef_cong]:
nipkow@17501
  1514
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
nipkow@17501
  1515
apply simp
nipkow@17501
  1516
apply(erule thin_rl)
nipkow@17501
  1517
by (induct ys) simp_all
nipkow@17501
  1518
nipkow@15281
  1519
haftmann@26442
  1520
subsubsection {* List partitioning *}
haftmann@26442
  1521
haftmann@26442
  1522
primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where
nipkow@50548
  1523
"partition P [] = ([], [])" |
nipkow@50548
  1524
"partition P (x # xs) = 
nipkow@50548
  1525
  (let (yes, no) = partition P xs
nipkow@50548
  1526
   in if P x then (x # yes, no) else (yes, x # no))"
haftmann@26442
  1527
haftmann@26442
  1528
lemma partition_filter1:
haftmann@26442
  1529
    "fst (partition P xs) = filter P xs"
haftmann@26442
  1530
by (induct xs) (auto simp add: Let_def split_def)
haftmann@26442
  1531
haftmann@26442
  1532
lemma partition_filter2:
haftmann@26442
  1533
    "snd (partition P xs) = filter (Not o P) xs"
haftmann@26442
  1534
by (induct xs) (auto simp add: Let_def split_def)
haftmann@26442
  1535
haftmann@26442
  1536
lemma partition_P:
haftmann@26442
  1537
  assumes "partition P xs = (yes, no)"
haftmann@26442
  1538
  shows "(\<forall>p \<in> set yes.  P p) \<and> (\<forall>p  \<in> set no. \<not> P p)"
haftmann@26442
  1539
proof -
haftmann@26442
  1540
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
haftmann@26442
  1541
    by simp_all
haftmann@26442
  1542
  then show ?thesis by (simp_all add: partition_filter1 partition_filter2)
haftmann@26442
  1543
qed
haftmann@26442
  1544
haftmann@26442
  1545
lemma partition_set:
haftmann@26442
  1546
  assumes "partition P xs = (yes, no)"
haftmann@26442
  1547
  shows "set yes \<union> set no = set xs"
haftmann@26442
  1548
proof -
haftmann@26442
  1549
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
haftmann@26442
  1550
    by simp_all
haftmann@26442
  1551
  then show ?thesis by (auto simp add: partition_filter1 partition_filter2) 
haftmann@26442
  1552
qed
haftmann@26442
  1553
hoelzl@33639
  1554
lemma partition_filter_conv[simp]:
hoelzl@33639
  1555
  "partition f xs = (filter f xs,filter (Not o f) xs)"
hoelzl@33639
  1556
unfolding partition_filter2[symmetric]
hoelzl@33639
  1557
unfolding partition_filter1[symmetric] by simp
hoelzl@33639
  1558
hoelzl@33639
  1559
declare partition.simps[simp del]
haftmann@26442
  1560
wenzelm@35115
  1561
haftmann@49948
  1562
subsubsection {* @{const concat} *}
wenzelm@13114
  1563
wenzelm@13142
  1564
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
  1565
by (induct xs) auto
wenzelm@13114
  1566
paulson@18447
  1567
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
  1568
by (induct xss) auto
wenzelm@13114
  1569
paulson@18447
  1570
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
  1571
by (induct xss) auto
wenzelm@13114
  1572
nipkow@24308
  1573
lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
nipkow@13145
  1574
by (induct xs) auto
wenzelm@13114
  1575
nipkow@24476
  1576
lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs"
nipkow@24349
  1577
by (induct xs) auto
nipkow@24349
  1578
wenzelm@13142
  1579
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
  1580
by (induct xs) auto
wenzelm@13114
  1581
wenzelm@13142
  1582
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
  1583
by (induct xs) auto
wenzelm@13114
  1584
wenzelm@13142
  1585
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
  1586
by (induct xs) auto
wenzelm@13114
  1587
bulwahn@40365
  1588
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
  1589
proof (induct xs arbitrary: ys)
bulwahn@40365
  1590
  case (Cons x xs ys)
bulwahn@40365
  1591
  thus ?case by (cases ys) auto
bulwahn@40365
  1592
qed (auto)
bulwahn@40365
  1593
bulwahn@40365
  1594
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
  1595
by (simp add: concat_eq_concat_iff)
bulwahn@40365
  1596
wenzelm@13114
  1597
haftmann@49948
  1598
subsubsection {* @{const nth} *}
wenzelm@13114
  1599
haftmann@29827
  1600
lemma nth_Cons_0 [simp, code]: "(x # xs)!0 = x"
nipkow@13145
  1601
by auto
wenzelm@13114
  1602
haftmann@29827
  1603
lemma nth_Cons_Suc [simp, code]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
  1604
by auto
wenzelm@13114
  1605
wenzelm@13142
  1606
declare nth.simps [simp del]
wenzelm@13114
  1607
nipkow@41842
  1608
lemma nth_Cons_pos[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
nipkow@41842
  1609
by(auto simp: Nat.gr0_conv_Suc)
nipkow@41842
  1610
wenzelm@13114
  1611
lemma nth_append:
nipkow@24526
  1612
  "(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
nipkow@24526
  1613
apply (induct xs arbitrary: n, simp)
paulson@14208
  1614
apply (case_tac n, auto)
nipkow@13145
  1615
done
wenzelm@13114
  1616
nipkow@14402
  1617
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
wenzelm@25221
  1618
by (induct xs) auto
nipkow@14402
  1619
nipkow@14402
  1620
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
wenzelm@25221
  1621
by (induct xs) auto
nipkow@14402
  1622
nipkow@24526
  1623
lemma nth_map [simp]: "n < length xs ==> (map f xs)!n = f(xs!n)"
nipkow@24526
  1624
apply (induct xs arbitrary: n, simp)
paulson@14208
  1625
apply (case_tac n, auto)
nipkow@13145
  1626
done
wenzelm@13114
  1627
noschinl@45841
  1628
lemma nth_tl:
noschinl@45841
  1629
  assumes "n < length (tl x)" shows "tl x ! n = x ! Suc n"
noschinl@45841
  1630
using assms by (induct x) auto
noschinl@45841
  1631
nipkow@18423
  1632
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
nipkow@18423
  1633
by(cases xs) simp_all
nipkow@18423
  1634
nipkow@18049
  1635
nipkow@18049
  1636
lemma list_eq_iff_nth_eq:
nipkow@24526
  1637
 "(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
nipkow@24526
  1638
apply(induct xs arbitrary: ys)
paulson@24632
  1639
 apply force
nipkow@18049
  1640
apply(case_tac ys)
nipkow@18049
  1641
 apply simp
nipkow@18049
  1642
apply(simp add:nth_Cons split:nat.split)apply blast
nipkow@18049
  1643
done
nipkow@18049
  1644
wenzelm@13142
  1645
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
  1646
apply (induct xs, simp, simp)
nipkow@13145
  1647
apply safe
blanchet@55417
  1648
apply (metis nat.cases(1) nth.simps zero_less_Suc)
paulson@24632
  1649
apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
paulson@14208
  1650
apply (case_tac i, simp)
blanchet@55417
  1651
apply (metis diff_Suc_Suc nat.cases(2) nth.simps zero_less_diff)
nipkow@13145
  1652
done
wenzelm@13114
  1653
nipkow@17501
  1654
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
nipkow@17501
  1655
by(auto simp:set_conv_nth)
nipkow@17501
  1656
haftmann@51160
  1657
lemma nth_equal_first_eq:
haftmann@51160
  1658
  assumes "x \<notin> set xs"
haftmann@51160
  1659
  assumes "n \<le> length xs"
haftmann@51160
  1660
  shows "(x # xs) ! n = x \<longleftrightarrow> n = 0" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@51160
  1661
proof
haftmann@51160
  1662
  assume ?lhs
haftmann@51160
  1663
  show ?rhs
haftmann@51160
  1664
  proof (rule ccontr)
haftmann@51160
  1665
    assume "n \<noteq> 0"
haftmann@51160
  1666
    then have "n > 0" by simp
haftmann@51160
  1667
    with `?lhs` have "xs ! (n - 1) = x" by simp
haftmann@51160
  1668
    moreover from `n > 0` `n \<le> length xs` have "n - 1 < length xs" by simp
haftmann@51160
  1669
    ultimately have "\<exists>i<length xs. xs ! i = x" by auto
haftmann@51160
  1670
    with `x \<notin> set xs` in_set_conv_nth [of x xs] show False by simp
haftmann@51160
  1671
  qed
haftmann@51160
  1672
next
haftmann@51160
  1673
  assume ?rhs then show ?lhs by simp
haftmann@51160
  1674
qed
haftmann@51160
  1675
haftmann@51160
  1676
lemma nth_non_equal_first_eq:
haftmann@51160
  1677
  assumes "x \<noteq> y"
haftmann@51160
  1678
  shows "(x # xs) ! n = y \<longleftrightarrow> xs ! (n - 1) = y \<and> n > 0" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@51160
  1679
proof
haftmann@51160
  1680
  assume "?lhs" with assms have "n > 0" by (cases n) simp_all
haftmann@51160
  1681
  with `?lhs` show ?rhs by simp
haftmann@51160
  1682
next
haftmann@51160
  1683
  assume "?rhs" then show "?lhs" by simp
haftmann@51160
  1684
qed
haftmann@51160
  1685
nipkow@13145
  1686
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
  1687
by (auto simp add: set_conv_nth)
wenzelm@13114
  1688
wenzelm@13142
  1689
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
  1690
by (auto simp add: set_conv_nth)
wenzelm@13114
  1691
wenzelm@13114
  1692
lemma all_nth_imp_all_set:
nipkow@13145
  1693
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
  1694
by (auto simp add: set_conv_nth)
wenzelm@13114
  1695
wenzelm@13114
  1696
lemma all_set_conv_all_nth:
nipkow@13145
  1697
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
  1698
by (auto simp add: set_conv_nth)
wenzelm@13114
  1699
kleing@25296
  1700
lemma rev_nth:
kleing@25296
  1701
  "n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)"
kleing@25296
  1702
proof (induct xs arbitrary: n)
kleing@25296
  1703
  case Nil thus ?case by simp
kleing@25296
  1704
next
kleing@25296
  1705
  case (Cons x xs)
kleing@25296
  1706
  hence n: "n < Suc (length xs)" by simp
kleing@25296
  1707
  moreover
kleing@25296
  1708
  { assume "n < length xs"
wenzelm@53374
  1709
    with n obtain n' where n': "length xs - n = Suc n'"
kleing@25296
  1710
      by (cases "length xs - n", auto)
kleing@25296
  1711
    moreover
wenzelm@53374
  1712
    from n' have "length xs - Suc n = n'" by simp
kleing@25296
  1713
    ultimately
kleing@25296
  1714
    have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp
kleing@25296
  1715
  }
kleing@25296
  1716
  ultimately
kleing@25296
  1717
  show ?case by (clarsimp simp add: Cons nth_append)
kleing@25296
  1718
qed
wenzelm@13114
  1719
nipkow@31159
  1720
lemma Skolem_list_nth:
nipkow@31159
  1721
  "(ALL i<k. EX x. P i x) = (EX xs. size xs = k & (ALL i<k. P i (xs!i)))"
nipkow@31159
  1722
  (is "_ = (EX xs. ?P k xs)")
nipkow@31159
  1723
proof(induct k)
nipkow@31159
  1724
  case 0 show ?case by simp
nipkow@31159
  1725
next
nipkow@31159
  1726
  case (Suc k)
nipkow@31159
  1727
  show ?case (is "?L = ?R" is "_ = (EX xs. ?P' xs)")
nipkow@31159
  1728
  proof
nipkow@31159
  1729
    assume "?R" thus "?L" using Suc by auto
nipkow@31159
  1730
  next
nipkow@31159
  1731
    assume "?L"
nipkow@31159
  1732
    with Suc obtain x xs where "?P k xs & P k x" by (metis less_Suc_eq)
nipkow@31159
  1733
    hence "?P'(xs@[x])" by(simp add:nth_append less_Suc_eq)
nipkow@31159
  1734
    thus "?R" ..
nipkow@31159
  1735
  qed
nipkow@31159
  1736
qed
nipkow@31159
  1737
nipkow@31159
  1738
haftmann@49948
  1739
subsubsection {* @{const list_update} *}
wenzelm@13114
  1740
nipkow@24526
  1741
lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
nipkow@24526
  1742
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1743
wenzelm@13114
  1744
lemma nth_list_update:
nipkow@24526
  1745
"i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@24526
  1746
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1747
wenzelm@13142
  1748
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
  1749
by (simp add: nth_list_update)
wenzelm@13114
  1750
nipkow@24526
  1751
lemma nth_list_update_neq [simp]: "i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@24526
  1752
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1753
nipkow@24526
  1754
lemma list_update_id[simp]: "xs[i := xs!i] = xs"
nipkow@24526
  1755
by (induct xs arbitrary: i) (simp_all split:nat.splits)
nipkow@24526
  1756
nipkow@24526
  1757
lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
nipkow@24526
  1758
apply (induct xs arbitrary: i)
nipkow@17501
  1759
 apply simp
nipkow@17501
  1760
apply (case_tac i)
nipkow@17501
  1761
apply simp_all
nipkow@17501
  1762
done
nipkow@17501
  1763
nipkow@31077
  1764
lemma list_update_nonempty[simp]: "xs[k:=x] = [] \<longleftrightarrow> xs=[]"
nipkow@31077
  1765
by(metis length_0_conv length_list_update)
nipkow@31077
  1766
wenzelm@13114
  1767
lemma list_update_same_conv:
nipkow@24526
  1768
"i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@24526
  1769
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1770
nipkow@14187
  1771
lemma list_update_append1:
nipkow@24526
  1772
 "i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
nipkow@24526
  1773
apply (induct xs arbitrary: i, simp)
nipkow@14187
  1774
apply(simp split:nat.split)
nipkow@14187
  1775
done
nipkow@14187
  1776
kleing@15868
  1777
lemma list_update_append:
nipkow@24526
  1778
  "(xs @ ys) [n:= x] = 
kleing@15868
  1779
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
nipkow@24526
  1780
by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868
  1781
nipkow@14402
  1782
lemma list_update_length [simp]:
nipkow@14402
  1783
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
  1784
by (induct xs, auto)
nipkow@14402
  1785
nipkow@31264
  1786
lemma map_update: "map f (xs[k:= y]) = (map f xs)[k := f y]"
nipkow@31264
  1787
by(induct xs arbitrary: k)(auto split:nat.splits)
nipkow@31264
  1788
nipkow@31264
  1789
lemma rev_update:
nipkow@31264
  1790
  "k < length xs \<Longrightarrow> rev (xs[k:= y]) = (rev xs)[length xs - k - 1 := y]"
nipkow@31264
  1791
by (induct xs arbitrary: k) (auto simp: list_update_append split:nat.splits)
nipkow@31264
  1792
wenzelm@13114
  1793
lemma update_zip:
nipkow@31080
  1794
  "(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@24526
  1795
by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split)
nipkow@24526
  1796
nipkow@24526
  1797
lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)"
nipkow@24526
  1798
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1799
wenzelm@13114
  1800
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
  1801
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
  1802
nipkow@24526
  1803
lemma set_update_memI: "n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
nipkow@24526
  1804
by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868
  1805
nipkow@31077
  1806
lemma list_update_overwrite[simp]:
haftmann@24796
  1807
  "xs [i := x, i := y] = xs [i := y]"
nipkow@31077
  1808
apply (induct xs arbitrary: i) apply simp
nipkow@31077
  1809
apply (case_tac i, simp_all)
haftmann@24796
  1810
done
haftmann@24796
  1811
haftmann@24796
  1812
lemma list_update_swap:
haftmann@24796
  1813
  "i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]"
haftmann@24796
  1814
apply (induct xs arbitrary: i i')
haftmann@24796
  1815
apply simp
haftmann@24796
  1816
apply (case_tac i, case_tac i')
haftmann@24796
  1817
apply auto
haftmann@24796
  1818
apply (case_tac i')
haftmann@24796
  1819
apply auto
haftmann@24796
  1820
done
haftmann@24796
  1821
haftmann@29827
  1822
lemma list_update_code [code]:
haftmann@29827
  1823
  "[][i := y] = []"
haftmann@29827
  1824
  "(x # xs)[0 := y] = y # xs"
haftmann@29827
  1825
  "(x # xs)[Suc i := y] = x # xs[i := y]"
haftmann@29827
  1826
  by simp_all
haftmann@29827
  1827
wenzelm@13114
  1828
haftmann@49948
  1829
subsubsection {* @{const last} and @{const butlast} *}
wenzelm@13114
  1830
wenzelm@13142
  1831
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
  1832
by (induct xs) auto
wenzelm@13114
  1833
wenzelm@13142
  1834
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
  1835
by (induct xs) auto
wenzelm@13114
  1836
nipkow@14302
  1837
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
huffman@44921
  1838
  by simp
nipkow@14302
  1839
nipkow@14302
  1840
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
huffman@44921
  1841
  by simp
nipkow@14302
  1842
nipkow@14302
  1843
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
  1844
by (induct xs) (auto)
nipkow@14302
  1845
nipkow@14302
  1846
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
  1847
by(simp add:last_append)
nipkow@14302
  1848
nipkow@14302
  1849
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
  1850
by(simp add:last_append)
nipkow@14302
  1851
noschinl@45841
  1852
lemma last_tl: "xs = [] \<or> tl xs \<noteq> [] \<Longrightarrow>last (tl xs) = last xs"
noschinl@45841
  1853
by (induct xs) simp_all
noschinl@45841
  1854
noschinl@45841
  1855
lemma butlast_tl: "butlast (tl xs) = tl (butlast xs)"
noschinl@45841
  1856
by (induct xs) simp_all
noschinl@45841
  1857
nipkow@17762
  1858
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
nipkow@17762
  1859
by(rule rev_exhaust[of xs]) simp_all
nipkow@17762
  1860
nipkow@17762
  1861
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
nipkow@17762
  1862
by(cases xs) simp_all
nipkow@17762
  1863
nipkow@17765
  1864
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
nipkow@17765
  1865
by (induct as) auto
nipkow@17762
  1866
wenzelm@13142
  1867
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
  1868
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1869
wenzelm@13114
  1870
lemma butlast_append:
nipkow@24526
  1871
  "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@24526
  1872
by (induct xs arbitrary: ys) auto
wenzelm@13114
  1873
wenzelm@13142
  1874
lemma append_butlast_last_id [simp]:
nipkow@13145
  1875
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
  1876
by (induct xs) auto
wenzelm@13114
  1877
wenzelm@13142
  1878
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
  1879
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1880
wenzelm@13114
  1881
lemma in_set_butlast_appendI:
nipkow@13145
  1882
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
  1883
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
  1884
nipkow@24526
  1885
lemma last_drop[simp]: "n < length xs \<Longrightarrow> last (drop n xs) = last xs"
nipkow@24526
  1886
apply (induct xs arbitrary: n)
nipkow@17501
  1887
 apply simp
nipkow@17501
  1888
apply (auto split:nat.split)
nipkow@17501
  1889
done
nipkow@17501
  1890
noschinl@45841
  1891
lemma nth_butlast:
noschinl@45841
  1892
  assumes "n < length (butlast xs)" shows "butlast xs ! n = xs ! n"
noschinl@45841
  1893
proof (cases xs)
noschinl@45841
  1894
  case (Cons y ys)
noschinl@45841
  1895
  moreover from assms have "butlast xs ! n = (butlast xs @ [last xs]) ! n"
noschinl@45841
  1896
    by (simp add: nth_append)
noschinl@45841
  1897
  ultimately show ?thesis using append_butlast_last_id by simp
noschinl@45841
  1898
qed simp
noschinl@45841
  1899
huffman@30128
  1900
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
nipkow@17589
  1901
by(induct xs)(auto simp:neq_Nil_conv)
nipkow@17589
  1902
huffman@30128
  1903
lemma butlast_conv_take: "butlast xs = take (length xs - 1) xs"
huffman@26584
  1904
by (induct xs, simp, case_tac xs, simp_all)
huffman@26584
  1905
nipkow@31077
  1906
lemma last_list_update:
nipkow@31077
  1907
  "xs \<noteq> [] \<Longrightarrow> last(xs[k:=x]) = (if k = size xs - 1 then x else last xs)"
nipkow@31077
  1908
by (auto simp: last_conv_nth)
nipkow@31077
  1909
nipkow@31077
  1910
lemma butlast_list_update:
nipkow@31077
  1911
  "butlast(xs[k:=x]) =
nipkow@31077
  1912
 (if k = size xs - 1 then butlast xs else (butlast xs)[k:=x])"
nipkow@31077
  1913
apply(cases xs rule:rev_cases)
nipkow@31077
  1914
apply simp
nipkow@31077
  1915
apply(simp add:list_update_append split:nat.splits)
nipkow@31077
  1916
done
nipkow@31077
  1917
haftmann@36851
  1918
lemma last_map:
haftmann@36851
  1919
  "xs \<noteq> [] \<Longrightarrow> last (map f xs) = f (last xs)"
haftmann@36851
  1920
  by (cases xs rule: rev_cases) simp_all
haftmann@36851
  1921
haftmann@36851
  1922
lemma map_butlast:
haftmann@36851
  1923
  "map f (butlast xs) = butlast (map f xs)"
haftmann@36851
  1924
  by (induct xs) simp_all
haftmann@36851
  1925
nipkow@40230
  1926
lemma snoc_eq_iff_butlast:
nipkow@40230
  1927
  "xs @ [x] = ys \<longleftrightarrow> (ys \<noteq> [] & butlast ys = xs & last ys = x)"
nipkow@40230
  1928
by (metis append_butlast_last_id append_is_Nil_conv butlast_snoc last_snoc not_Cons_self)
nipkow@40230
  1929
haftmann@24796
  1930
haftmann@49948
  1931
subsubsection {* @{const take} and @{const drop} *}
wenzelm@13114
  1932
wenzelm@13142
  1933
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
  1934
by (induct xs) auto
wenzelm@13114
  1935
wenzelm@13142
  1936
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
  1937
by (induct xs) auto
wenzelm@13114
  1938
wenzelm@13142
  1939
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
  1940
by simp
wenzelm@13114
  1941
wenzelm@13142
  1942
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
  1943
by simp
wenzelm@13114
  1944
wenzelm@13142
  1945
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
  1946
huffman@30128
  1947
lemma take_1_Cons [simp]: "take 1 (x # xs) = [x]"
huffman@30128
  1948
  unfolding One_nat_def by simp
huffman@30128
  1949
huffman@30128
  1950
lemma drop_1_Cons [simp]: "drop 1 (x # xs) = xs"
huffman@30128
  1951
  unfolding One_nat_def by simp
huffman@30128
  1952
nipkow@15110
  1953
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
  1954
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
  1955
nipkow@14187
  1956
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
  1957
by(cases xs, simp_all)
nipkow@14187
  1958
huffman@26584
  1959
lemma take_tl: "take n (tl xs) = tl (take (Suc n) xs)"
huffman@26584
  1960
by (induct xs arbitrary: n) simp_all
huffman@26584
  1961
nipkow@24526
  1962
lemma drop_tl: "drop n (tl xs) = tl(drop n xs)"
nipkow@24526
  1963
by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@24526
  1964
huffman@26584
  1965
lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)"
huffman@26584
  1966
by (cases n, simp, cases xs, auto)
huffman@26584
  1967
huffman@26584
  1968
lemma tl_drop: "tl (drop n xs) = drop n (tl xs)"
huffman@26584
  1969
by (simp only: drop_tl)
huffman@26584
  1970
nipkow@24526
  1971
lemma nth_via_drop: "drop n xs = y#ys \<Longrightarrow> xs!n = y"
nipkow@24526
  1972
apply (induct xs arbitrary: n, simp)
nipkow@14187
  1973
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
  1974
done
nipkow@14187
  1975
nipkow@13913
  1976
lemma take_Suc_conv_app_nth:
nipkow@24526
  1977
  "i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
nipkow@24526
  1978
apply (induct xs arbitrary: i, simp)
paulson@14208
  1979
apply (case_tac i, auto)
nipkow@13913
  1980
done
nipkow@13913
  1981
mehta@14591
  1982
lemma drop_Suc_conv_tl:
nipkow@24526
  1983
  "i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
nipkow@24526
  1984
apply (induct xs arbitrary: i, simp)
mehta@14591
  1985
apply (case_tac i, auto)
mehta@14591
  1986
done
mehta@14591
  1987
nipkow@24526
  1988
lemma length_take [simp]: "length (take n xs) = min (length xs) n"
nipkow@24526
  1989
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1990
nipkow@24526
  1991
lemma length_drop [simp]: "length (drop n xs) = (length xs - n)"
nipkow@24526
  1992
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1993
nipkow@24526
  1994
lemma take_all [simp]: "length xs <= n ==> take n xs = xs"
nipkow@24526
  1995
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1996
nipkow@24526
  1997
lemma drop_all [simp]: "length xs <= n ==> drop n xs = []"
nipkow@24526
  1998
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114
  1999
wenzelm@13142
  2000
lemma take_append [simp]:
nipkow@24526
  2001
  "take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@24526
  2002
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114
  2003
wenzelm@13142
  2004
lemma drop_append [simp]:
nipkow@24526
  2005
  "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@24526
  2006
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  2007
nipkow@24526
  2008
lemma take_take [simp]: "take n (take m xs) = take (min n m) xs"
nipkow@24526
  2009
apply (induct m arbitrary: xs n, auto)
paulson@14208
  2010
apply (case_tac xs, auto)
nipkow@15236
  2011
apply (case_tac n, auto)
nipkow@13145
  2012
done
wenzelm@13114
  2013
nipkow@24526
  2014
lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs"
nipkow@24526
  2015
apply (induct m arbitrary: xs, auto)
paulson@14208
  2016
apply (case_tac xs, auto)
nipkow@13145
  2017
done
wenzelm@13114
  2018
nipkow@24526
  2019
lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)"
nipkow@24526
  2020
apply (induct m arbitrary: xs n, auto)
paulson@14208
  2021
apply (case_tac xs, auto)
nipkow@13145
  2022
done
wenzelm@13114
  2023
nipkow@24526
  2024
lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@24526
  2025
apply(induct xs arbitrary: m n)
nipkow@14802
  2026
 apply simp
nipkow@14802
  2027
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  2028
done
nipkow@14802
  2029
nipkow@24526
  2030
lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs"
nipkow@24526
  2031
apply (induct n arbitrary: xs, auto)
paulson@14208
  2032
apply (case_tac xs, auto)
nipkow@13145
  2033
done
wenzelm@13114
  2034
nipkow@24526
  2035
lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@24526
  2036
apply(induct xs arbitrary: n)
nipkow@15110
  2037
 apply simp
nipkow@15110
  2038
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  2039
done
nipkow@15110
  2040
nipkow@24526
  2041
lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)"
nipkow@24526
  2042
apply(induct xs arbitrary: n)
nipkow@15110
  2043
apply simp
nipkow@15110
  2044
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  2045
done
nipkow@15110
  2046
nipkow@24526
  2047
lemma take_map: "take n (map f xs) = map f (take n xs)"
nipkow@24526
  2048
apply (induct n arbitrary: xs, auto)
paulson@14208
  2049
apply (case_tac xs, auto)
nipkow@13145
  2050
done
wenzelm@13114
  2051
nipkow@24526
  2052
lemma drop_map: "drop n (map f xs) = map f (drop n xs)"
nipkow@24526
  2053
apply (induct n arbitrary: xs, auto)
paulson@14208
  2054
apply (case_tac xs, auto)
nipkow@13145
  2055
done
wenzelm@13114
  2056
nipkow@24526
  2057
lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)"
nipkow@24526
  2058
apply (induct xs arbitrary: i, auto)
paulson@14208
  2059
apply (case_tac i, auto)
nipkow@13145
  2060
done
wenzelm@13114
  2061
nipkow@24526
  2062
lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)"
nipkow@24526
  2063
apply (induct xs arbitrary: i, auto)
paulson@14208
  2064
apply (case_tac i, auto)
nipkow@13145
  2065
done
wenzelm@13114
  2066
nipkow@24526
  2067
lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i"
nipkow@24526
  2068
apply (induct xs arbitrary: i n, auto)
paulson@14208
  2069
apply (case_tac n, blast)
paulson@14208
  2070
apply (case_tac i, auto)
nipkow@13145
  2071
done
wenzelm@13114
  2072
wenzelm@13142
  2073
lemma nth_drop [simp]:
nipkow@24526
  2074
  "n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
nipkow@24526
  2075
apply (induct n arbitrary: xs i, auto)
paulson@14208
  2076
apply (case_tac xs, auto)
nipkow@13145
  2077
done
nipkow@3507
  2078
huffman@26584
  2079
lemma butlast_take:
huffman@30128
  2080
  "n <= length xs ==> butlast (take n xs) = take (n - 1) xs"
haftmann@54863
  2081
by (simp add: butlast_conv_take min.absorb1 min.absorb2)
huffman@26584
  2082
huffman@26584
  2083
lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)"
huffman@30128
  2084
by (simp add: butlast_conv_take drop_take add_ac)
huffman@26584
  2085
huffman@26584
  2086
lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs"
haftmann@54863
  2087
by (simp add: butlast_conv_take min.absorb1)
huffman@26584
  2088
huffman@26584
  2089
lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)"
huffman@30128
  2090
by (simp add: butlast_conv_take drop_take add_ac)
huffman@26584
  2091
bulwahn@46500
  2092
lemma hd_drop_conv_nth: "n < length xs \<Longrightarrow> hd(drop n xs) = xs!n"
nipkow@18423
  2093
by(simp add: hd_conv_nth)
nipkow@18423
  2094
nipkow@35248
  2095
lemma set_take_subset_set_take:
nipkow@35248
  2096
  "m <= n \<Longrightarrow> set(take m xs) <= set(take n xs)"
bulwahn@41463
  2097
apply (induct xs arbitrary: m n)
bulwahn@41463
  2098
apply simp
bulwahn@41463
  2099
apply (case_tac n)
bulwahn@41463
  2100
apply (auto simp: take_Cons)
bulwahn@41463
  2101
done
nipkow@35248
  2102
nipkow@24526
  2103
lemma set_take_subset: "set(take n xs) \<subseteq> set xs"
nipkow@24526
  2104
by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split)
nipkow@24526
  2105
nipkow@24526
  2106
lemma set_drop_subset: "set(drop n xs) \<subseteq> set xs"
nipkow@24526
  2107
by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  2108
nipkow@35248
  2109
lemma set_drop_subset_set_drop:
nipkow@35248
  2110
  "m >= n \<Longrightarrow> set(drop m xs) <= set(drop n xs)"
nipkow@35248
  2111
apply(induct xs arbitrary: m n)
nipkow@35248
  2112
apply(auto simp:drop_Cons split:nat.split)
nipkow@35248
  2113
apply (metis set_drop_subset subset_iff)
nipkow@35248
  2114
done
nipkow@35248
  2115
nipkow@14187
  2116
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  2117
using set_take_subset by fast
nipkow@14187
  2118
nipkow@14187
  2119
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  2120
using set_drop_subset by fast
nipkow@14187
  2121
wenzelm@13114
  2122
lemma append_eq_conv_conj:
nipkow@24526
  2123
  "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
nipkow@24526
  2124
apply (induct xs arbitrary: zs, simp, clarsimp)
paulson@14208
  2125
apply (case_tac zs, auto)
nipkow@13145
  2126
done
wenzelm@13142
  2127
nipkow@24526
  2128
lemma take_add: 
noschinl@42713
  2129
  "take (i+j) xs = take i xs @ take j (drop i xs)"
nipkow@24526
  2130
apply (induct xs arbitrary: i, auto) 
nipkow@24526
  2131
apply (case_tac i, simp_all)
paulson@14050
  2132
done
paulson@14050
  2133
nipkow@14300
  2134
lemma append_eq_append_conv_if:
wenzelm@53015
  2135
 "(xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>1 @ ys\<^sub>2) =
wenzelm@53015
  2136
  (if size xs\<^sub>1 \<le> size ys\<^sub>1
wenzelm@53015
  2137
   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
  2138
   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
  2139
apply(induct xs\<^sub>1 arbitrary: ys\<^sub>1)
nipkow@14300
  2140
 apply simp
wenzelm@53015
  2141
apply(case_tac ys\<^sub>1)
nipkow@14300
  2142
apply simp_all
nipkow@14300
  2143
done
nipkow@14300
  2144
nipkow@15110
  2145
lemma take_hd_drop:
huffman@30079
  2146
  "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (Suc n) xs"
nipkow@24526
  2147
apply(induct xs arbitrary: n)
nipkow@15110
  2148
apply simp
nipkow@15110
  2149
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  2150
done
nipkow@15110
  2151
nipkow@17501
  2152
lemma id_take_nth_drop:
nipkow@17501
  2153
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
nipkow@17501
  2154
proof -
nipkow@17501
  2155
  assume si: "i < length xs"
nipkow@17501
  2156
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
nipkow@17501
  2157
  moreover
nipkow@17501
  2158
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
nipkow@17501
  2159
    apply (rule_tac take_Suc_conv_app_nth) by arith
nipkow@17501
  2160
  ultimately show ?thesis by auto
nipkow@17501
  2161
qed
nipkow@17501
  2162
  
nipkow@17501
  2163
lemma upd_conv_take_nth_drop:
nipkow@17501
  2164
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  2165
proof -
nipkow@17501
  2166
  assume i: "i < length xs"
nipkow@17501
  2167
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
nipkow@17501
  2168
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
nipkow@17501
  2169
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  2170
    using i by (simp add: list_update_append)
nipkow@17501
  2171
  finally show ?thesis .
nipkow@17501
  2172
qed
nipkow@17501
  2173
haftmann@24796
  2174
lemma nth_drop':
haftmann@24796
  2175
  "i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs"
haftmann@24796
  2176
apply (induct i arbitrary: xs)
haftmann@24796
  2177
apply (simp add: neq_Nil_conv)
haftmann@24796
  2178
apply (erule exE)+
haftmann@24796
  2179
apply simp
haftmann@24796
  2180
apply (case_tac xs)
haftmann@24796
  2181
apply simp_all
haftmann@24796
  2182
done
haftmann@24796
  2183
wenzelm@13114
  2184
haftmann@49948
  2185
subsubsection {* @{const takeWhile} and @{const dropWhile} *}
wenzelm@13114
  2186
hoelzl@33639
  2187
lemma length_takeWhile_le: "length (takeWhile P xs) \<le> length xs"
hoelzl@33639
  2188
  by (induct xs) auto
hoelzl@33639
  2189
wenzelm@13142
  2190
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  2191
by (induct xs) auto
wenzelm@13114
  2192
wenzelm@13142
  2193
lemma takeWhile_append1 [simp]:
nipkow@13145
  2194
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  2195
by (induct xs) auto
wenzelm@13114
  2196
wenzelm@13142
  2197
lemma takeWhile_append2 [simp]:
nipkow@13145
  2198
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  2199
by (induct xs) auto
wenzelm@13114
  2200
wenzelm@13142
  2201
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  2202
by (induct xs) auto
wenzelm@13114
  2203
hoelzl@33639
  2204
lemma takeWhile_nth: "j < length (takeWhile P xs) \<Longrightarrow> takeWhile P xs ! j = xs ! j"
hoelzl@33639
  2205
apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
hoelzl@33639
  2206
hoelzl@33639
  2207
lemma dropWhile_nth: "j < length (dropWhile P xs) \<Longrightarrow> dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))"
hoelzl@33639
  2208
apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
hoelzl@33639
  2209
hoelzl@33639
  2210
lemma length_dropWhile_le: "length (dropWhile P xs) \<le> length xs"
hoelzl@33639
  2211
by (induct xs) auto
hoelzl@33639
  2212
wenzelm@13142
  2213
lemma dropWhile_append1 [simp]:
nipkow@13145
  2214
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  2215
by (induct xs) auto
wenzelm@13114
  2216
wenzelm@13142
  2217
lemma dropWhile_append2 [simp]:
nipkow@13145
  2218
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  2219
by (induct xs) auto
wenzelm@13114
  2220
noschinl@45841
  2221
lemma dropWhile_append3:
noschinl@45841
  2222
  "\<not> P y \<Longrightarrow>dropWhile P (xs @ y # ys) = dropWhile P xs @ y # ys"
noschinl@45841
  2223
by (induct xs) auto
noschinl@45841
  2224
noschinl@45841
  2225
lemma dropWhile_last:
noschinl@45841
  2226
  "x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> last (dropWhile P xs) = last xs"
noschinl@45841
  2227
by (auto simp add: dropWhile_append3 in_set_conv_decomp)
noschinl@45841
  2228
noschinl@45841
  2229
lemma set_dropWhileD: "x \<in> set (dropWhile P xs) \<Longrightarrow> x \<in> set xs"
noschinl@45841
  2230
by (induct xs) (auto split: split_if_asm)
noschinl@45841
  2231
krauss@23971
  2232
lemma set_takeWhileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  2233
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  2234
nipkow@13913
  2235
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  2236
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  2237
by(induct xs, auto)
nipkow@13913
  2238
nipkow@13913
  2239
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  2240
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  2241
by(induct xs, auto)
nipkow@13913
  2242
nipkow@13913
  2243
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  2244
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  2245
by(induct xs, auto)
nipkow@13913
  2246
nipkow@31077
  2247
lemma distinct_takeWhile[simp]: "distinct xs ==> distinct (takeWhile P xs)"
nipkow@31077
  2248
by (induct xs) (auto dest: set_takeWhileD)
nipkow@31077
  2249
nipkow@31077
  2250
lemma distinct_dropWhile[simp]: "distinct xs ==> distinct (dropWhile P xs)"
nipkow@31077
  2251
by (induct xs) auto
nipkow@31077
  2252
hoelzl@33639
  2253
lemma takeWhile_map: "takeWhile P (map f xs) = map f (takeWhile (P \<circ> f) xs)"
hoelzl@33639
  2254
by (induct xs) auto
hoelzl@33639
  2255
hoelzl@33639
  2256
lemma dropWhile_map: "dropWhile P (map f xs) = map f (dropWhile (P \<circ> f) xs)"
hoelzl@33639
  2257
by (induct xs) auto
hoelzl@33639
  2258
hoelzl@33639
  2259
lemma takeWhile_eq_take: "takeWhile P xs = take (length (takeWhile P xs)) xs"
hoelzl@33639
  2260
by (induct xs) auto
hoelzl@33639
  2261
hoelzl@33639
  2262
lemma dropWhile_eq_drop: "dropWhile P xs = drop (length (takeWhile P xs)) xs"
hoelzl@33639
  2263
by (induct xs) auto
hoelzl@33639
  2264
hoelzl@33639
  2265
lemma hd_dropWhile:
hoelzl@33639
  2266
  "dropWhile P xs \<noteq> [] \<Longrightarrow> \<not> P (hd (dropWhile P xs))"
hoelzl@33639
  2267
using assms by (induct xs) auto
hoelzl@33639
  2268
hoelzl@33639
  2269
lemma takeWhile_eq_filter:
hoelzl@33639
  2270
  assumes "\<And> x. x \<in> set (dropWhile P xs) \<Longrightarrow> \<not> P x"
hoelzl@33639
  2271
  shows "takeWhile P xs = filter P xs"
hoelzl@33639
  2272
proof -
hoelzl@33639
  2273
  have A: "filter P xs = filter P (takeWhile P xs @ dropWhile P xs)"
hoelzl@33639
  2274
    by simp
hoelzl@33639
  2275
  have B: "filter P (dropWhile P xs) = []"
hoelzl@33639
  2276
    unfolding filter_empty_conv using assms by blast
hoelzl@33639
  2277
  have "filter P xs = takeWhile P xs"
hoelzl@33639
  2278
    unfolding A filter_append B
hoelzl@33639
  2279
    by (auto simp add: filter_id_conv dest: set_takeWhileD)
hoelzl@33639
  2280
  thus ?thesis ..
hoelzl@33639
  2281
qed
hoelzl@33639
  2282
hoelzl@33639
  2283
lemma takeWhile_eq_take_P_nth:
hoelzl@33639
  2284
  "\<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
  2285
  takeWhile P xs = take n xs"
hoelzl@33639
  2286
proof (induct xs arbitrary: n)
hoelzl@33639
  2287
  case (Cons x xs)
hoelzl@33639
  2288
  thus ?case
hoelzl@33639
  2289
  proof (cases n)
hoelzl@33639
  2290
    case (Suc n') note this[simp]
hoelzl@33639
  2291
    have "P x" using Cons.prems(1)[of 0] by simp
hoelzl@33639
  2292
    moreover have "takeWhile P xs = take n' xs"
hoelzl@33639
  2293
    proof (rule Cons.hyps)
hoelzl@33639
  2294
      case goal1 thus "P (xs ! i)" using Cons.prems(1)[of "Suc i"] by simp
hoelzl@33639
  2295
    next case goal2 thus ?case using Cons by auto
hoelzl@33639
  2296
    qed
hoelzl@33639
  2297
    ultimately show ?thesis by simp
hoelzl@33639
  2298
   qed simp
hoelzl@33639
  2299
qed simp
hoelzl@33639
  2300
hoelzl@33639
  2301
lemma nth_length_takeWhile:
hoelzl@33639
  2302
  "length (takeWhile P xs) < length xs \<Longrightarrow> \<not> P (xs ! length (takeWhile P xs))"
hoelzl@33639
  2303
by (induct xs) auto
hoelzl@33639
  2304
hoelzl@33639
  2305
lemma length_takeWhile_less_P_nth:
hoelzl@33639
  2306
  assumes all: "\<And> i. i < j \<Longrightarrow> P (xs ! i)" and "j \<le> length xs"
hoelzl@33639
  2307
  shows "j \<le> length (takeWhile P xs)"
hoelzl@33639
  2308
proof (rule classical)
hoelzl@33639
  2309
  assume "\<not> ?thesis"
hoelzl@33639
  2310
  hence "length (takeWhile P xs) < length xs" using assms by simp
hoelzl@33639
  2311
  thus ?thesis using all `\<not> ?thesis` nth_length_takeWhile[of P xs] by auto
hoelzl@33639
  2312
qed
nipkow@31077
  2313
nipkow@17501
  2314
text{* The following two lemmmas could be generalized to an arbitrary
nipkow@17501
  2315
property. *}
nipkow@17501
  2316
nipkow@17501
  2317
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  2318
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
nipkow@17501
  2319
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
nipkow@17501
  2320
nipkow@17501
  2321
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  2322
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
nipkow@17501
  2323
apply(induct xs)
nipkow@17501
  2324
 apply simp
nipkow@17501
  2325
apply auto
nipkow@17501
  2326
apply(subst dropWhile_append2)
nipkow@17501
  2327
apply auto
nipkow@17501
  2328
done
nipkow@17501
  2329
nipkow@18423
  2330
lemma takeWhile_not_last:
bulwahn@46500
  2331
 "distinct xs \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
nipkow@18423
  2332
apply(induct xs)
nipkow@18423
  2333
 apply simp
nipkow@18423
  2334
apply(case_tac xs)
nipkow@18423
  2335
apply(auto)
nipkow@18423
  2336
done
nipkow@18423
  2337
krauss@44013
  2338
lemma takeWhile_cong [fundef_cong]:
krauss@18336
  2339
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  2340
  ==> takeWhile P l = takeWhile Q k"
nipkow@24349
  2341
by (induct k arbitrary: l) (simp_all)
krauss@18336
  2342
krauss@44013
  2343
lemma dropWhile_cong [fundef_cong]:
krauss@18336
  2344
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  2345
  ==> dropWhile P l = dropWhile Q k"
nipkow@24349
  2346
by (induct k arbitrary: l, simp_all)
krauss@18336
  2347
haftmann@52380
  2348
lemma takeWhile_idem [simp]:
haftmann@52380
  2349
  "takeWhile P (takeWhile P xs) = takeWhile P xs"
haftmann@52380
  2350
  by (induct xs) auto
haftmann@52380
  2351
haftmann@52380
  2352
lemma dropWhile_idem [simp]:
haftmann@52380
  2353
  "dropWhile P (dropWhile P xs) = dropWhile P xs"
haftmann@52380
  2354
  by (induct xs) auto
haftmann@52380
  2355
wenzelm@13114
  2356
haftmann@49948
  2357
subsubsection {* @{const zip} *}
wenzelm@13114
  2358
wenzelm@13142
  2359
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  2360
by (induct ys) auto
wenzelm@13114
  2361
wenzelm@13142
  2362
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  2363
by simp
wenzelm@13114
  2364
wenzelm@13142
  2365
declare zip_Cons [simp del]
wenzelm@13114
  2366
haftmann@36198
  2367
lemma [code]:
haftmann@36198
  2368
  "zip [] ys = []"
haftmann@36198
  2369
  "zip xs [] = []"
haftmann@36198
  2370
  "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
haftmann@36198
  2371
  by (fact zip_Nil zip.simps(1) zip_Cons_Cons)+
haftmann@36198
  2372
nipkow@15281
  2373
lemma zip_Cons1:
nipkow@15281
  2374
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  2375
by(auto split:list.split)
nipkow@15281
  2376
wenzelm@13142
  2377
lemma length_zip [simp]:
krauss@22493
  2378
"length (zip xs ys) = min (length xs) (length ys)"
krauss@22493
  2379
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  2380
haftmann@34978
  2381
lemma zip_obtain_same_length:
haftmann@34978
  2382
  assumes "\<And>zs ws n. length zs = length ws \<Longrightarrow> n = min (length xs) (length ys)
haftmann@34978
  2383
    \<Longrightarrow> zs = take n xs \<Longrightarrow> ws = take n ys \<Longrightarrow> P (zip zs ws)"
haftmann@34978
  2384
  shows "P (zip xs ys)"
haftmann@34978
  2385
proof -
haftmann@34978
  2386
  let ?n = "min (length xs) (length ys)"
haftmann@34978
  2387
  have "P (zip (take ?n xs) (take ?n ys))"
haftmann@34978
  2388
    by (rule assms) simp_all
haftmann@34978
  2389
  moreover have "zip xs ys = zip (take ?n xs) (take ?n ys)"
haftmann@34978
  2390
  proof (induct xs arbitrary: ys)
haftmann@34978
  2391
    case Nil then show ?case by simp
haftmann@34978
  2392
  next
haftmann@34978
  2393
    case (Cons x xs) then show ?case by (cases ys) simp_all
haftmann@34978
  2394
  qed
haftmann@34978
  2395
  ultimately show ?thesis by simp
haftmann@34978
  2396
qed
haftmann@34978
  2397
wenzelm@13114
  2398
lemma zip_append1:
krauss@22493
  2399
"zip (xs @ ys) zs =
nipkow@13145
  2400
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
krauss@22493
  2401
by (induct xs zs rule:list_induct2') auto
wenzelm@13114
  2402
wenzelm@13114
  2403
lemma zip_append2:
krauss@22493
  2404
"zip xs (ys @ zs) =
nipkow@13145
  2405
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
krauss@22493
  2406
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  2407
wenzelm@13142
  2408
lemma zip_append [simp]:
bulwahn@46500
  2409
 "[| length xs = length us |] ==>
nipkow@13145
  2410
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  2411
by (simp add: zip_append1)
wenzelm@13114
  2412
wenzelm@13114
  2413
lemma zip_rev:
nipkow@14247
  2414
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  2415
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  2416
hoelzl@33639
  2417
lemma zip_map_map:
hoelzl@33639
  2418
  "zip (map f xs) (map g ys) = map (\<lambda> (x, y). (f x, g y)) (zip xs ys)"
hoelzl@33639
  2419
proof (induct xs arbitrary: ys)
hoelzl@33639
  2420
  case (Cons x xs) note Cons_x_xs = Cons.hyps
hoelzl@33639
  2421
  show ?case
hoelzl@33639
  2422
  proof (cases ys)
hoelzl@33639
  2423
    case (Cons y ys')
hoelzl@33639
  2424
    show ?thesis unfolding Cons using Cons_x_xs by simp
hoelzl@33639
  2425
  qed simp
hoelzl@33639
  2426
qed simp
hoelzl@33639
  2427
hoelzl@33639
  2428
lemma zip_map1:
hoelzl@33639
  2429
  "zip (map f xs) ys = map (\<lambda>(x, y). (f x, y)) (zip xs ys)"
hoelzl@33639
  2430
using zip_map_map[of f xs "\<lambda>x. x" ys] by simp
hoelzl@33639
  2431
hoelzl@33639
  2432
lemma zip_map2:
hoelzl@33639
  2433
  "zip xs (map f ys) = map (\<lambda>(x, y). (x, f y)) (zip xs ys)"
hoelzl@33639
  2434
using zip_map_map[of "\<lambda>x. x" xs f ys] by simp
hoelzl@33639
  2435
nipkow@23096
  2436
lemma map_zip_map:
hoelzl@33639
  2437
  "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
hoelzl@33639
  2438
unfolding zip_map1 by auto
nipkow@23096
  2439
nipkow@23096
  2440
lemma map_zip_map2:
hoelzl@33639
  2441
  "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
hoelzl@33639
  2442
unfolding zip_map2 by auto
nipkow@23096
  2443
nipkow@31080
  2444
text{* Courtesy of Andreas Lochbihler: *}
nipkow@31080
  2445
lemma zip_same_conv_map: "zip xs xs = map (\<lambda>x. (x, x)) xs"
nipkow@31080
  2446
by(induct xs) auto
nipkow@31080
  2447
wenzelm@13142
  2448
lemma nth_zip [simp]:
nipkow@24526
  2449
"[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
nipkow@24526
  2450
apply (induct ys arbitrary: i xs, simp)
nipkow@13145
  2451
apply (case_tac xs)
nipkow@13145
  2452
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  2453
done
wenzelm@13114
  2454
wenzelm@13114
  2455
lemma set_zip:
nipkow@13145
  2456
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@31080
  2457
by(simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  2458
hoelzl@33639
  2459
lemma zip_same: "((a,b) \<in> set (zip xs xs)) = (a \<in> set xs \<and> a = b)"
hoelzl@33639
  2460
by(induct xs) auto
hoelzl@33639
  2461
wenzelm@13114
  2462
lemma zip_update:
nipkow@31080
  2463
  "zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@31080
  2464
by(rule sym, simp add: update_zip)
wenzelm@13114
  2465
wenzelm@13142
  2466
lemma zip_replicate [simp]:
nipkow@24526
  2467
  "zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
nipkow@24526
  2468
apply (induct i arbitrary: j, auto)
paulson@14208
  2469
apply (case_tac j, auto)
nipkow@13145
  2470
done
wenzelm@13114
  2471
nipkow@19487
  2472
lemma take_zip:
nipkow@24526
  2473
  "take n (zip xs ys) = zip (take n xs) (take n ys)"
nipkow@24526
  2474
apply (induct n arbitrary: xs ys)
nipkow@19487
  2475
 apply simp
nipkow@19487
  2476
apply (case_tac xs, simp)
nipkow@19487
  2477
apply (case_tac ys, simp_all)
nipkow@19487
  2478
done
nipkow@19487
  2479
nipkow@19487
  2480
lemma drop_zip:
nipkow@24526
  2481
  "drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
nipkow@24526
  2482
apply (induct n arbitrary: xs ys)
nipkow@19487
  2483
 apply simp
nipkow@19487
  2484
apply (case_tac xs, simp)
nipkow@19487
  2485
apply (case_tac ys, simp_all)
nipkow@19487
  2486
done
nipkow@19487
  2487
hoelzl@33639
  2488
lemma zip_takeWhile_fst: "zip (takeWhile P xs) ys = takeWhile (P \<circ> fst) (zip xs ys)"
hoelzl@33639
  2489
proof (induct xs arbitrary: ys)
hoelzl@33639
  2490
  case (Cons x xs) thus ?case by (cases ys) auto
hoelzl@33639
  2491
qed simp
hoelzl@33639
  2492
hoelzl@33639
  2493
lemma zip_takeWhile_snd: "zip xs (takeWhile P ys) = takeWhile (P \<circ> snd) (zip xs ys)"
hoelzl@33639
  2494
proof (induct xs arbitrary: ys)
hoelzl@33639
  2495
  case (Cons x xs) thus ?case by (cases ys) auto
hoelzl@33639
  2496
qed simp
hoelzl@33639
  2497
krauss@22493
  2498
lemma set_zip_leftD:
krauss@22493
  2499
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
krauss@22493
  2500
by (induct xs ys rule:list_induct2') auto
krauss@22493
  2501
krauss@22493
  2502
lemma set_zip_rightD:
krauss@22493
  2503
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
krauss@22493
  2504
by (induct xs ys rule:list_induct2') auto
wenzelm@13142
  2505
nipkow@23983
  2506
lemma in_set_zipE:
nipkow@23983
  2507
  "(x,y) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
nipkow@23983
  2508
by(blast dest: set_zip_leftD set_zip_rightD)
nipkow@23983
  2509
haftmann@29829
  2510
lemma zip_map_fst_snd:
haftmann@29829
  2511
  "zip (map fst zs) (map snd zs) = zs"
haftmann@29829
  2512
  by (induct zs) simp_all
haftmann@29829
  2513
haftmann@29829
  2514
lemma zip_eq_conv:
haftmann@29829
  2515
  "length xs = length ys \<Longrightarrow> zip xs ys = zs \<longleftrightarrow> map fst zs = xs \<and> map snd zs = ys"
haftmann@29829
  2516
  by (auto simp add: zip_map_fst_snd)
haftmann@29829
  2517
haftmann@51173
  2518
lemma in_set_zip:
haftmann@51173
  2519
  "p \<in> set (zip xs ys) \<longleftrightarrow> (\<exists>n. xs ! n = fst p \<and> ys ! n = snd p
haftmann@51173
  2520
    \<and> n < length xs \<and> n < length ys)"
haftmann@51173
  2521
  by (cases p) (auto simp add: set_zip)
haftmann@51173
  2522
haftmann@51173
  2523
lemma pair_list_eqI:
haftmann@51173
  2524
  assumes "map fst xs = map fst ys" and "map snd xs = map snd ys"
haftmann@51173
  2525
  shows "xs = ys"
haftmann@51173
  2526
proof -
haftmann@51173
  2527
  from assms(1) have "length xs = length ys" by (rule map_eq_imp_length_eq)
haftmann@51173
  2528
  from this assms show ?thesis
haftmann@51173
  2529
    by (induct xs ys rule: list_induct2) (simp_all add: prod_eqI)
haftmann@51173
  2530
qed
haftmann@51173
  2531
wenzelm@35115
  2532
haftmann@49948
  2533
subsubsection {* @{const list_all2} *}
wenzelm@13114
  2534
kleing@14316
  2535
lemma list_all2_lengthD [intro?]: 
kleing@14316
  2536
  "list_all2 P xs ys ==> length xs = length ys"
blanchet@55524
  2537
by (simp add: list_all2_iff)
haftmann@19607
  2538
haftmann@19787
  2539
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
blanchet@55524
  2540
by (simp add: list_all2_iff)
haftmann@19607
  2541
haftmann@19787
  2542
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
blanchet@55524
  2543
by (simp add: list_all2_iff)
haftmann@19607
  2544
haftmann@19607
  2545
lemma list_all2_Cons [iff, code]:
haftmann@19607
  2546
  "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
blanchet@55524
  2547
by (auto simp add: list_all2_iff)
wenzelm@13114
  2548
wenzelm@13114
  2549
lemma list_all2_Cons1:
nipkow@13145
  2550
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  2551
by (cases ys) auto
wenzelm@13114
  2552
wenzelm@13114
  2553
lemma list_all2_Cons2:
nipkow@13145
  2554
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  2555
by (cases xs) auto
wenzelm@13114
  2556
huffman@45794
  2557
lemma list_all2_induct
huffman@45794
  2558
  [consumes 1, case_names Nil Cons, induct set: list_all2]:
huffman@45794
  2559
  assumes P: "list_all2 P xs ys"
huffman@45794
  2560
  assumes Nil: "R [] []"
huffman@47640
  2561
  assumes Cons: "\<And>x xs y ys.
huffman@47640
  2562
    \<lbrakk>P x y; list_all2 P xs ys; R xs ys\<rbrakk> \<Longrightarrow> R (x # xs) (y # ys)"
huffman@45794
  2563
  shows "R xs ys"
huffman@45794
  2564
using P
huffman@45794
  2565
by (induct xs arbitrary: ys) (auto simp add: list_all2_Cons1 Nil Cons)
huffman@45794
  2566
wenzelm@13142
  2567
lemma list_all2_rev [iff]:
nipkow@13145
  2568
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
blanchet@55524
  2569
by (simp add: list_all2_iff zip_rev cong: conj_cong)
wenzelm@13114
  2570
kleing@13863
  2571
lemma list_all2_rev1:
kleing@13863
  2572
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  2573
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  2574
wenzelm@13114
  2575
lemma list_all2_append1:
nipkow@13145
  2576
"list_all2 P (xs @ ys) zs =