src/HOL/List.thy
author haftmann
Fri Jan 06 10:19:49 2012 +0100 (2012-01-06)
changeset 46133 d9fe85d3d2cd
parent 46125 00cd193a48dc
child 46138 85f8d8a8c711
permissions -rw-r--r--
incorporated canonical fold combinator on lists into body of List theory; refactored passages on List.fold(l/r)
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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 Plain Presburger Code_Numeral Quotient ATP
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uses
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  ("Tools/list_code.ML")
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  ("Tools/list_to_set_comprehension.ML")
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begin
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datatype 'a list =
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    Nil    ("[]")
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  | Cons 'a  "'a list"    (infixr "#" 65)
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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
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  hd :: "'a list \<Rightarrow> 'a" where
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  "hd (x # xs) = x"
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primrec
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  tl :: "'a list \<Rightarrow> 'a list" where
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    "tl [] = []"
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  | "tl (x # xs) = xs"
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primrec
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  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
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  butlast :: "'a list \<Rightarrow> 'a list" where
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    "butlast []= []"
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  | "butlast (x # xs) = (if xs = [] then [] else x # butlast xs)"
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primrec
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  set :: "'a list \<Rightarrow> 'a set" where
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    "set [] = {}"
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  | "set (x # xs) = insert x (set xs)"
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definition
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  coset :: "'a list \<Rightarrow> 'a set" where
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  [simp]: "coset xs = - set xs"
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primrec
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  map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
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    "map f [] = []"
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  | "map f (x # xs) = f x # map f xs"
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primrec
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  append :: "'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
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  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
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  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 -- {* canonical argument order *}
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  fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where
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    "fold f [] = id"
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  | "fold f (x # xs) = fold f xs \<circ> f x"
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definition 
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  foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where
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  [code_abbrev]: "foldr f xs = fold f (rev xs)"
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definition
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  foldl :: "('b \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b" where
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  "foldl f s xs = fold (\<lambda>x s. f s x)  xs s"
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primrec
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  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)
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  listsum :: "'a list \<Rightarrow> 'a" where
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  "listsum xs = foldr plus xs 0"
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primrec
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  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
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  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
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  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
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  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 = (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
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  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
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  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
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  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) = (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 
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  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
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  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
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  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
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  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
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  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
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  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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primrec
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  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
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  length :: "'a list \<Rightarrow> nat" where
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  "length \<equiv> size"
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definition
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  rotate1 :: "'a list \<Rightarrow> 'a list" where
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  "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"
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definition
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  rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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  "rotate n = rotate1 ^^ n"
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definition
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  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
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  "list_all2 P xs ys =
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    (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"
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definition
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  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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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 add: foldr_def)}\\
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@{lemma "foldl f x [a,b,c] = f (f (f x a) b) c" by (simp add: foldl_def)}\\
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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 "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 "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 "remove1 2 [2,0,2,1::nat,2] = [0,2,1,2]" by simp}\\
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@{lemma "removeAll 2 [2,0,2,1::nat,2] = [0,1]" by simp}\\
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@{lemma "nth [a,b,c,d] 2 = c" by simp}\\
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@{lemma "[a,b,c,d][2 := x] = [a,b,x,d]" by simp}\\
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@{lemma "sublist [a,b,c,d,e] {0,2,3} = [a,c,d]" by (simp add:sublist_def)}\\
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@{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by (simp add:rotate1_def)}\\
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@{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate1_def rotate_def eval_nat_numeral)}\\
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@{lemma "replicate 4 a = [a,a,a,a]" by (simp add:eval_nat_numeral)}\\
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@{lemma "[2..<5] = [2,3,4]" by (simp add:eval_nat_numeral)}\\
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@{lemma "listsum [1,2,3::nat] = 6" by (simp add: listsum_def foldr_def)}
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\end{tabular}}
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\caption{Characteristic examples}
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\label{fig:Characteristic}
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\end{figure}
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Figure~\ref{fig:Characteristic} shows characteristic examples
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that should give an intuitive understanding of the above functions.
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*}
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text{* The following simple sort functions are intended for proofs,
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not for efficient implementations. *}
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context linorder
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begin
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inductive sorted :: "'a list \<Rightarrow> bool" where
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  Nil [iff]: "sorted []"
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| Cons: "\<forall>y\<in>set xs. x \<le> y \<Longrightarrow> sorted xs \<Longrightarrow> sorted (x # xs)"
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lemma sorted_single [iff]:
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  "sorted [x]"
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  by (rule sorted.Cons) auto
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lemma sorted_many:
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  "x \<le> y \<Longrightarrow> sorted (y # zs) \<Longrightarrow> sorted (x # y # zs)"
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  by (rule sorted.Cons) (cases "y # zs" rule: sorted.cases, auto)
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lemma sorted_many_eq [simp, code]:
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  "sorted (x # y # zs) \<longleftrightarrow> x \<le> y \<and> sorted (y # zs)"
haftmann@39915
   301
  by (auto intro: sorted_many elim: sorted.cases)
haftmann@39915
   302
haftmann@39915
   303
lemma [code]:
haftmann@39915
   304
  "sorted [] \<longleftrightarrow> True"
haftmann@39915
   305
  "sorted [x] \<longleftrightarrow> True"
haftmann@39915
   306
  by simp_all
nipkow@24697
   307
hoelzl@33639
   308
primrec insort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
haftmann@46133
   309
  "insort_key f x [] = [x]" |
haftmann@46133
   310
  "insort_key f x (y#ys) = (if f x \<le> f y then (x#y#ys) else y#(insort_key f x ys))"
hoelzl@33639
   311
haftmann@35195
   312
definition sort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b list \<Rightarrow> 'b list" where
haftmann@46133
   313
  "sort_key f xs = foldr (insort_key f) xs []"
hoelzl@33639
   314
haftmann@40210
   315
definition insort_insert_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
haftmann@40210
   316
  "insort_insert_key f x xs = (if f x \<in> f ` set xs then xs else insort_key f x xs)"
haftmann@40210
   317
hoelzl@33639
   318
abbreviation "sort \<equiv> sort_key (\<lambda>x. x)"
hoelzl@33639
   319
abbreviation "insort \<equiv> insort_key (\<lambda>x. x)"
haftmann@40210
   320
abbreviation "insort_insert \<equiv> insort_insert_key (\<lambda>x. x)"
haftmann@35608
   321
wenzelm@25221
   322
end
wenzelm@25221
   323
nipkow@24616
   324
wenzelm@23388
   325
subsubsection {* List comprehension *}
nipkow@23192
   326
nipkow@24349
   327
text{* Input syntax for Haskell-like list comprehension notation.
nipkow@24349
   328
Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
nipkow@24349
   329
the list of all pairs of distinct elements from @{text xs} and @{text ys}.
nipkow@24349
   330
The syntax is as in Haskell, except that @{text"|"} becomes a dot
nipkow@24349
   331
(like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
nipkow@24349
   332
\verb![e| x <- xs, ...]!.
nipkow@24349
   333
nipkow@24349
   334
The qualifiers after the dot are
nipkow@24349
   335
\begin{description}
nipkow@24349
   336
\item[generators] @{text"p \<leftarrow> xs"},
nipkow@24476
   337
 where @{text p} is a pattern and @{text xs} an expression of list type, or
nipkow@24476
   338
\item[guards] @{text"b"}, where @{text b} is a boolean expression.
nipkow@24476
   339
%\item[local bindings] @ {text"let x = e"}.
nipkow@24349
   340
\end{description}
nipkow@23240
   341
nipkow@24476
   342
Just like in Haskell, list comprehension is just a shorthand. To avoid
nipkow@24476
   343
misunderstandings, the translation into desugared form is not reversed
nipkow@24476
   344
upon output. Note that the translation of @{text"[e. x \<leftarrow> xs]"} is
nipkow@24476
   345
optmized to @{term"map (%x. e) xs"}.
nipkow@23240
   346
nipkow@24349
   347
It is easy to write short list comprehensions which stand for complex
nipkow@24349
   348
expressions. During proofs, they may become unreadable (and
nipkow@24349
   349
mangled). In such cases it can be advisable to introduce separate
nipkow@24349
   350
definitions for the list comprehensions in question.  *}
nipkow@24349
   351
wenzelm@42144
   352
nonterminal lc_gen and lc_qual and lc_quals
nipkow@23192
   353
nipkow@23192
   354
syntax
nipkow@23240
   355
"_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
wenzelm@42144
   356
"_lc_gen" :: "lc_gen \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ <- _")
nipkow@23240
   357
"_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
nipkow@24476
   358
(*"_lc_let" :: "letbinds => lc_qual"  ("let _")*)
nipkow@23240
   359
"_lc_end" :: "lc_quals" ("]")
nipkow@23240
   360
"_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals" (", __")
nipkow@24349
   361
"_lc_abs" :: "'a => 'b list => 'b list"
wenzelm@42144
   362
"_strip_positions" :: "'a \<Rightarrow> lc_gen"  ("_")
nipkow@23192
   363
nipkow@24476
   364
(* These are easier than ML code but cannot express the optimized
nipkow@24476
   365
   translation of [e. p<-xs]
nipkow@23192
   366
translations
nipkow@24349
   367
"[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
nipkow@23240
   368
"_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
nipkow@24349
   369
 => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
nipkow@23240
   370
"[e. P]" => "if P then [e] else []"
nipkow@23240
   371
"_listcompr e (_lc_test P) (_lc_quals Q Qs)"
nipkow@23240
   372
 => "if P then (_listcompr e Q Qs) else []"
nipkow@24349
   373
"_listcompr e (_lc_let b) (_lc_quals Q Qs)"
nipkow@24349
   374
 => "_Let b (_listcompr e Q Qs)"
nipkow@24476
   375
*)
nipkow@23240
   376
nipkow@23279
   377
syntax (xsymbols)
wenzelm@42144
   378
"_lc_gen" :: "lc_gen \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
nipkow@23279
   379
syntax (HTML output)
wenzelm@42144
   380
"_lc_gen" :: "lc_gen \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
nipkow@24349
   381
nipkow@24349
   382
parse_translation (advanced) {*
nipkow@24349
   383
let
wenzelm@35256
   384
  val NilC = Syntax.const @{const_syntax Nil};
wenzelm@35256
   385
  val ConsC = Syntax.const @{const_syntax Cons};
wenzelm@35256
   386
  val mapC = Syntax.const @{const_syntax map};
wenzelm@35256
   387
  val concatC = Syntax.const @{const_syntax concat};
wenzelm@35256
   388
  val IfC = Syntax.const @{const_syntax If};
wenzelm@35115
   389
nipkow@24476
   390
  fun singl x = ConsC $ x $ NilC;
nipkow@24476
   391
wenzelm@35115
   392
  fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
nipkow@24349
   393
    let
wenzelm@43324
   394
      (* FIXME proper name context!? *)
wenzelm@43324
   395
      val x = Free (singleton (Name.variant_list (fold Term.add_free_names [p, e] [])) "x", dummyT);
nipkow@24476
   396
      val e = if opti then singl e else e;
wenzelm@46125
   397
      val case1 = Syntax.const @{syntax_const "_case1"} $ p $ e;
wenzelm@35256
   398
      val case2 =
wenzelm@35256
   399
        Syntax.const @{syntax_const "_case1"} $
wenzelm@35256
   400
          Syntax.const @{const_syntax dummy_pattern} $ NilC;
wenzelm@35115
   401
      val cs = Syntax.const @{syntax_const "_case2"} $ case1 $ case2;
wenzelm@45891
   402
      val ft = Datatype_Case.case_tr false ctxt [x, cs];
nipkow@24349
   403
    in lambda x ft end;
nipkow@24349
   404
wenzelm@35256
   405
  fun abs_tr ctxt (p as Free (s, T)) e opti =
wenzelm@35115
   406
        let
wenzelm@42361
   407
          val thy = Proof_Context.theory_of ctxt;
wenzelm@42361
   408
          val s' = Proof_Context.intern_const ctxt s;
wenzelm@35115
   409
        in
wenzelm@35115
   410
          if Sign.declared_const thy s'
wenzelm@35115
   411
          then (pat_tr ctxt p e opti, false)
wenzelm@35115
   412
          else (lambda p e, true)
nipkow@24349
   413
        end
nipkow@24476
   414
    | abs_tr ctxt p e opti = (pat_tr ctxt p e opti, false);
nipkow@24476
   415
wenzelm@35115
   416
  fun lc_tr ctxt [e, Const (@{syntax_const "_lc_test"}, _) $ b, qs] =
wenzelm@35115
   417
        let
wenzelm@35115
   418
          val res =
wenzelm@35115
   419
            (case qs of
wenzelm@35115
   420
              Const (@{syntax_const "_lc_end"}, _) => singl e
wenzelm@35115
   421
            | Const (@{syntax_const "_lc_quals"}, _) $ q $ qs => lc_tr ctxt [e, q, qs]);
nipkow@24476
   422
        in IfC $ b $ res $ NilC end
wenzelm@35115
   423
    | lc_tr ctxt
wenzelm@35115
   424
          [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
wenzelm@35115
   425
            Const(@{syntax_const "_lc_end"}, _)] =
nipkow@24476
   426
        (case abs_tr ctxt p e true of
wenzelm@35115
   427
          (f, true) => mapC $ f $ es
wenzelm@35115
   428
        | (f, false) => concatC $ (mapC $ f $ es))
wenzelm@35115
   429
    | lc_tr ctxt
wenzelm@35115
   430
          [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
wenzelm@35115
   431
            Const (@{syntax_const "_lc_quals"}, _) $ q $ qs] =
wenzelm@35115
   432
        let val e' = lc_tr ctxt [e, q, qs];
wenzelm@35115
   433
        in concatC $ (mapC $ (fst (abs_tr ctxt p e' false)) $ es) end;
wenzelm@35115
   434
wenzelm@35115
   435
in [(@{syntax_const "_listcompr"}, lc_tr)] end
nipkow@24349
   436
*}
nipkow@23279
   437
wenzelm@42167
   438
ML {*
wenzelm@42167
   439
  let
wenzelm@42167
   440
    val read = Syntax.read_term @{context};
wenzelm@42167
   441
    fun check s1 s2 = read s1 aconv read s2 orelse error ("Check failed: " ^ quote s1);
wenzelm@42167
   442
  in
wenzelm@42167
   443
    check "[(x,y,z). b]" "if b then [(x, y, z)] else []";
wenzelm@42167
   444
    check "[(x,y,z). x\<leftarrow>xs]" "map (\<lambda>x. (x, y, z)) xs";
wenzelm@42167
   445
    check "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]" "concat (map (\<lambda>x. map (\<lambda>y. e x y) ys) xs)";
wenzelm@42167
   446
    check "[(x,y,z). x<a, x>b]" "if x < a then if b < x then [(x, y, z)] else [] else []";
wenzelm@42167
   447
    check "[(x,y,z). x\<leftarrow>xs, x>b]" "concat (map (\<lambda>x. if b < x then [(x, y, z)] else []) xs)";
wenzelm@42167
   448
    check "[(x,y,z). x<a, x\<leftarrow>xs]" "if x < a then map (\<lambda>x. (x, y, z)) xs else []";
wenzelm@42167
   449
    check "[(x,y). Cons True x \<leftarrow> xs]"
wenzelm@42167
   450
      "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | True # x \<Rightarrow> [(x, y)] | False # x \<Rightarrow> []) xs)";
wenzelm@42167
   451
    check "[(x,y,z). Cons x [] \<leftarrow> xs]"
wenzelm@42167
   452
      "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | [x] \<Rightarrow> [(x, y, z)] | x # aa # lista \<Rightarrow> []) xs)";
wenzelm@42167
   453
    check "[(x,y,z). x<a, x>b, x=d]"
wenzelm@42167
   454
      "if x < a then if b < x then if x = d then [(x, y, z)] else [] else [] else []";
wenzelm@42167
   455
    check "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
wenzelm@42167
   456
      "if x < a then if b < x then map (\<lambda>y. (x, y, z)) ys else [] else []";
wenzelm@42167
   457
    check "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
wenzelm@42167
   458
      "if x < a then concat (map (\<lambda>x. if b < y then [(x, y, z)] else []) xs) else []";
wenzelm@42167
   459
    check "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
wenzelm@42167
   460
      "if x < a then concat (map (\<lambda>x. map (\<lambda>y. (x, y, z)) ys) xs) else []";
wenzelm@42167
   461
    check "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
wenzelm@42167
   462
      "concat (map (\<lambda>x. if b < x then if y < a then [(x, y, z)] else [] else []) xs)";
wenzelm@42167
   463
    check "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
wenzelm@42167
   464
      "concat (map (\<lambda>x. if b < x then map (\<lambda>y. (x, y, z)) ys else []) xs)";
wenzelm@42167
   465
    check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
wenzelm@42167
   466
      "concat (map (\<lambda>x. concat (map (\<lambda>y. if x < y then [(x, y, z)] else []) ys)) xs)";
wenzelm@42167
   467
    check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
wenzelm@42167
   468
      "concat (map (\<lambda>x. concat (map (\<lambda>y. map (\<lambda>z. (x, y, z)) zs) ys)) xs)"
wenzelm@42167
   469
  end;
wenzelm@42167
   470
*}
wenzelm@42167
   471
wenzelm@35115
   472
(*
nipkow@24349
   473
term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
nipkow@23192
   474
*)
nipkow@23192
   475
wenzelm@42167
   476
bulwahn@41463
   477
use "Tools/list_to_set_comprehension.ML"
bulwahn@41463
   478
bulwahn@41463
   479
simproc_setup list_to_set_comprehension ("set xs") = {* K List_to_Set_Comprehension.simproc *}
bulwahn@41463
   480
haftmann@46133
   481
code_datatype set coset
haftmann@46133
   482
haftmann@46133
   483
hide_const (open) coset
wenzelm@35115
   484
haftmann@21061
   485
subsubsection {* @{const Nil} and @{const Cons} *}
haftmann@21061
   486
haftmann@21061
   487
lemma not_Cons_self [simp]:
haftmann@21061
   488
  "xs \<noteq> x # xs"
nipkow@13145
   489
by (induct xs) auto
wenzelm@13114
   490
wenzelm@41697
   491
lemma not_Cons_self2 [simp]:
wenzelm@41697
   492
  "x # xs \<noteq> xs"
wenzelm@41697
   493
by (rule not_Cons_self [symmetric])
wenzelm@13114
   494
wenzelm@13142
   495
lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
nipkow@13145
   496
by (induct xs) auto
wenzelm@13114
   497
wenzelm@13142
   498
lemma length_induct:
haftmann@21061
   499
  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
nipkow@17589
   500
by (rule measure_induct [of length]) iprover
wenzelm@13114
   501
haftmann@37289
   502
lemma list_nonempty_induct [consumes 1, case_names single cons]:
haftmann@37289
   503
  assumes "xs \<noteq> []"
haftmann@37289
   504
  assumes single: "\<And>x. P [x]"
haftmann@37289
   505
  assumes cons: "\<And>x xs. xs \<noteq> [] \<Longrightarrow> P xs \<Longrightarrow> P (x # xs)"
haftmann@37289
   506
  shows "P xs"
haftmann@37289
   507
using `xs \<noteq> []` proof (induct xs)
haftmann@37289
   508
  case Nil then show ?case by simp
haftmann@37289
   509
next
haftmann@37289
   510
  case (Cons x xs) show ?case proof (cases xs)
haftmann@37289
   511
    case Nil with single show ?thesis by simp
haftmann@37289
   512
  next
haftmann@37289
   513
    case Cons then have "xs \<noteq> []" by simp
haftmann@37289
   514
    moreover with Cons.hyps have "P xs" .
haftmann@37289
   515
    ultimately show ?thesis by (rule cons)
haftmann@37289
   516
  qed
haftmann@37289
   517
qed
haftmann@37289
   518
hoelzl@45714
   519
lemma inj_split_Cons: "inj_on (\<lambda>(xs, n). n#xs) X"
hoelzl@45714
   520
  by (auto intro!: inj_onI)
wenzelm@13114
   521
haftmann@21061
   522
subsubsection {* @{const length} *}
wenzelm@13114
   523
wenzelm@13142
   524
text {*
haftmann@21061
   525
  Needs to come before @{text "@"} because of theorem @{text
haftmann@21061
   526
  append_eq_append_conv}.
wenzelm@13142
   527
*}
wenzelm@13114
   528
wenzelm@13142
   529
lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
nipkow@13145
   530
by (induct xs) auto
wenzelm@13114
   531
wenzelm@13142
   532
lemma length_map [simp]: "length (map f xs) = length xs"
nipkow@13145
   533
by (induct xs) auto
wenzelm@13114
   534
wenzelm@13142
   535
lemma length_rev [simp]: "length (rev xs) = length xs"
nipkow@13145
   536
by (induct xs) auto
wenzelm@13114
   537
wenzelm@13142
   538
lemma length_tl [simp]: "length (tl xs) = length xs - 1"
nipkow@13145
   539
by (cases xs) auto
wenzelm@13114
   540
wenzelm@13142
   541
lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
nipkow@13145
   542
by (induct xs) auto
wenzelm@13114
   543
wenzelm@13142
   544
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
nipkow@13145
   545
by (induct xs) auto
wenzelm@13114
   546
nipkow@23479
   547
lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
nipkow@23479
   548
by auto
nipkow@23479
   549
wenzelm@13114
   550
lemma length_Suc_conv:
nipkow@13145
   551
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
nipkow@13145
   552
by (induct xs) auto
wenzelm@13142
   553
nipkow@14025
   554
lemma Suc_length_conv:
nipkow@14025
   555
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
paulson@14208
   556
apply (induct xs, simp, simp)
nipkow@14025
   557
apply blast
nipkow@14025
   558
done
nipkow@14025
   559
wenzelm@25221
   560
lemma impossible_Cons: "length xs <= length ys ==> xs = x # ys = False"
wenzelm@25221
   561
  by (induct xs) auto
wenzelm@25221
   562
haftmann@26442
   563
lemma list_induct2 [consumes 1, case_names Nil Cons]:
haftmann@26442
   564
  "length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
haftmann@26442
   565
   (\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
haftmann@26442
   566
   \<Longrightarrow> P xs ys"
haftmann@26442
   567
proof (induct xs arbitrary: ys)
haftmann@26442
   568
  case Nil then show ?case by simp
haftmann@26442
   569
next
haftmann@26442
   570
  case (Cons x xs ys) then show ?case by (cases ys) simp_all
haftmann@26442
   571
qed
haftmann@26442
   572
haftmann@26442
   573
lemma list_induct3 [consumes 2, case_names Nil Cons]:
haftmann@26442
   574
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
haftmann@26442
   575
   (\<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
   576
   \<Longrightarrow> P xs ys zs"
haftmann@26442
   577
proof (induct xs arbitrary: ys zs)
haftmann@26442
   578
  case Nil then show ?case by simp
haftmann@26442
   579
next
haftmann@26442
   580
  case (Cons x xs ys zs) then show ?case by (cases ys, simp_all)
haftmann@26442
   581
    (cases zs, simp_all)
haftmann@26442
   582
qed
wenzelm@13114
   583
kaliszyk@36154
   584
lemma list_induct4 [consumes 3, case_names Nil Cons]:
kaliszyk@36154
   585
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow>
kaliszyk@36154
   586
   P [] [] [] [] \<Longrightarrow> (\<And>x xs y ys z zs w ws. length xs = length ys \<Longrightarrow>
kaliszyk@36154
   587
   length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow> P xs ys zs ws \<Longrightarrow>
kaliszyk@36154
   588
   P (x#xs) (y#ys) (z#zs) (w#ws)) \<Longrightarrow> P xs ys zs ws"
kaliszyk@36154
   589
proof (induct xs arbitrary: ys zs ws)
kaliszyk@36154
   590
  case Nil then show ?case by simp
kaliszyk@36154
   591
next
kaliszyk@36154
   592
  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
   593
qed
kaliszyk@36154
   594
krauss@22493
   595
lemma list_induct2': 
krauss@22493
   596
  "\<lbrakk> P [] [];
krauss@22493
   597
  \<And>x xs. P (x#xs) [];
krauss@22493
   598
  \<And>y ys. P [] (y#ys);
krauss@22493
   599
   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
krauss@22493
   600
 \<Longrightarrow> P xs ys"
krauss@22493
   601
by (induct xs arbitrary: ys) (case_tac x, auto)+
krauss@22493
   602
nipkow@22143
   603
lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
nipkow@24349
   604
by (rule Eq_FalseI) auto
wenzelm@24037
   605
wenzelm@24037
   606
simproc_setup list_neq ("(xs::'a list) = ys") = {*
nipkow@22143
   607
(*
nipkow@22143
   608
Reduces xs=ys to False if xs and ys cannot be of the same length.
nipkow@22143
   609
This is the case if the atomic sublists of one are a submultiset
nipkow@22143
   610
of those of the other list and there are fewer Cons's in one than the other.
nipkow@22143
   611
*)
wenzelm@24037
   612
wenzelm@24037
   613
let
nipkow@22143
   614
huffman@29856
   615
fun len (Const(@{const_name Nil},_)) acc = acc
huffman@29856
   616
  | len (Const(@{const_name Cons},_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
huffman@29856
   617
  | len (Const(@{const_name append},_) $ xs $ ys) acc = len xs (len ys acc)
huffman@29856
   618
  | len (Const(@{const_name rev},_) $ xs) acc = len xs acc
huffman@29856
   619
  | len (Const(@{const_name map},_) $ _ $ xs) acc = len xs acc
nipkow@22143
   620
  | len t (ts,n) = (t::ts,n);
nipkow@22143
   621
wenzelm@24037
   622
fun list_neq _ ss ct =
nipkow@22143
   623
  let
wenzelm@24037
   624
    val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
nipkow@22143
   625
    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
nipkow@22143
   626
    fun prove_neq() =
nipkow@22143
   627
      let
nipkow@22143
   628
        val Type(_,listT::_) = eqT;
haftmann@22994
   629
        val size = HOLogic.size_const listT;
nipkow@22143
   630
        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
nipkow@22143
   631
        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
nipkow@22143
   632
        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
haftmann@22633
   633
          (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
haftmann@22633
   634
      in SOME (thm RS @{thm neq_if_length_neq}) end
nipkow@22143
   635
  in
wenzelm@23214
   636
    if m < n andalso submultiset (op aconv) (ls,rs) orelse
wenzelm@23214
   637
       n < m andalso submultiset (op aconv) (rs,ls)
nipkow@22143
   638
    then prove_neq() else NONE
nipkow@22143
   639
  end;
wenzelm@24037
   640
in list_neq end;
nipkow@22143
   641
*}
nipkow@22143
   642
nipkow@22143
   643
nipkow@15392
   644
subsubsection {* @{text "@"} -- append *}
wenzelm@13114
   645
wenzelm@13142
   646
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
nipkow@13145
   647
by (induct xs) auto
wenzelm@13114
   648
wenzelm@13142
   649
lemma append_Nil2 [simp]: "xs @ [] = xs"
nipkow@13145
   650
by (induct xs) auto
nipkow@3507
   651
wenzelm@13142
   652
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
nipkow@13145
   653
by (induct xs) auto
wenzelm@13114
   654
wenzelm@13142
   655
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
nipkow@13145
   656
by (induct xs) auto
wenzelm@13114
   657
wenzelm@13142
   658
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
nipkow@13145
   659
by (induct xs) auto
wenzelm@13114
   660
wenzelm@13142
   661
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
nipkow@13145
   662
by (induct xs) auto
wenzelm@13114
   663
blanchet@35828
   664
lemma append_eq_append_conv [simp, no_atp]:
nipkow@24526
   665
 "length xs = length ys \<or> length us = length vs
berghofe@13883
   666
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
nipkow@24526
   667
apply (induct xs arbitrary: ys)
paulson@14208
   668
 apply (case_tac ys, simp, force)
paulson@14208
   669
apply (case_tac ys, force, simp)
nipkow@13145
   670
done
wenzelm@13142
   671
nipkow@24526
   672
lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
nipkow@24526
   673
  (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
nipkow@24526
   674
apply (induct xs arbitrary: ys zs ts)
nipkow@44890
   675
 apply fastforce
nipkow@14495
   676
apply(case_tac zs)
nipkow@14495
   677
 apply simp
nipkow@44890
   678
apply fastforce
nipkow@14495
   679
done
nipkow@14495
   680
berghofe@34910
   681
lemma same_append_eq [iff, induct_simp]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145
   682
by simp
wenzelm@13142
   683
wenzelm@13142
   684
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145
   685
by simp
wenzelm@13114
   686
berghofe@34910
   687
lemma append_same_eq [iff, induct_simp]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   688
by simp
wenzelm@13114
   689
wenzelm@13142
   690
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   691
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   692
wenzelm@13142
   693
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   694
using append_same_eq [of "[]"] by auto
wenzelm@13114
   695
blanchet@35828
   696
lemma hd_Cons_tl [simp,no_atp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   697
by (induct xs) auto
wenzelm@13114
   698
wenzelm@13142
   699
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   700
by (induct xs) auto
wenzelm@13114
   701
wenzelm@13142
   702
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   703
by (simp add: hd_append split: list.split)
wenzelm@13114
   704
wenzelm@13142
   705
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   706
by (simp split: list.split)
wenzelm@13114
   707
wenzelm@13142
   708
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   709
by (simp add: tl_append split: list.split)
wenzelm@13114
   710
wenzelm@13114
   711
nipkow@14300
   712
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   713
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   714
by(cases ys) auto
nipkow@14300
   715
nipkow@15281
   716
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   717
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   718
by(cases ys) auto
nipkow@15281
   719
nipkow@14300
   720
wenzelm@13142
   721
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   722
wenzelm@13114
   723
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   724
by simp
wenzelm@13114
   725
wenzelm@13142
   726
lemma Cons_eq_appendI:
nipkow@13145
   727
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   728
by (drule sym) simp
wenzelm@13114
   729
wenzelm@13142
   730
lemma append_eq_appendI:
nipkow@13145
   731
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   732
by (drule sym) simp
wenzelm@13114
   733
wenzelm@13114
   734
wenzelm@13142
   735
text {*
nipkow@13145
   736
Simplification procedure for all list equalities.
nipkow@13145
   737
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   738
- both lists end in a singleton list,
nipkow@13145
   739
- or both lists end in the same list.
wenzelm@13142
   740
*}
wenzelm@13142
   741
wenzelm@43594
   742
simproc_setup list_eq ("(xs::'a list) = ys")  = {*
wenzelm@13462
   743
  let
wenzelm@43594
   744
    fun last (cons as Const (@{const_name Cons}, _) $ _ $ xs) =
wenzelm@43594
   745
          (case xs of Const (@{const_name Nil}, _) => cons | _ => last xs)
wenzelm@43594
   746
      | last (Const(@{const_name append},_) $ _ $ ys) = last ys
wenzelm@43594
   747
      | last t = t;
wenzelm@43594
   748
    
wenzelm@43594
   749
    fun list1 (Const(@{const_name Cons},_) $ _ $ Const(@{const_name Nil},_)) = true
wenzelm@43594
   750
      | list1 _ = false;
wenzelm@43594
   751
    
wenzelm@43594
   752
    fun butlast ((cons as Const(@{const_name Cons},_) $ x) $ xs) =
wenzelm@43594
   753
          (case xs of Const (@{const_name Nil}, _) => xs | _ => cons $ butlast xs)
wenzelm@43594
   754
      | butlast ((app as Const (@{const_name append}, _) $ xs) $ ys) = app $ butlast ys
wenzelm@43594
   755
      | butlast xs = Const(@{const_name Nil}, fastype_of xs);
wenzelm@43594
   756
    
wenzelm@43594
   757
    val rearr_ss =
wenzelm@43594
   758
      HOL_basic_ss addsimps [@{thm append_assoc}, @{thm append_Nil}, @{thm append_Cons}];
wenzelm@43594
   759
    
wenzelm@43594
   760
    fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   761
      let
wenzelm@43594
   762
        val lastl = last lhs and lastr = last rhs;
wenzelm@43594
   763
        fun rearr conv =
wenzelm@43594
   764
          let
wenzelm@43594
   765
            val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@43594
   766
            val Type(_,listT::_) = eqT
wenzelm@43594
   767
            val appT = [listT,listT] ---> listT
wenzelm@43594
   768
            val app = Const(@{const_name append},appT)
wenzelm@43594
   769
            val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@43594
   770
            val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@43594
   771
            val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
wenzelm@43594
   772
              (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
wenzelm@43594
   773
          in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@43594
   774
      in
wenzelm@43594
   775
        if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
wenzelm@43594
   776
        else if lastl aconv lastr then rearr @{thm append_same_eq}
wenzelm@43594
   777
        else NONE
wenzelm@43594
   778
      end;
wenzelm@43594
   779
  in fn _ => fn ss => fn ct => list_eq ss (term_of ct) end;
wenzelm@13114
   780
*}
wenzelm@13114
   781
wenzelm@13114
   782
nipkow@15392
   783
subsubsection {* @{text map} *}
wenzelm@13114
   784
haftmann@40210
   785
lemma hd_map:
haftmann@40210
   786
  "xs \<noteq> [] \<Longrightarrow> hd (map f xs) = f (hd xs)"
haftmann@40210
   787
  by (cases xs) simp_all
haftmann@40210
   788
haftmann@40210
   789
lemma map_tl:
haftmann@40210
   790
  "map f (tl xs) = tl (map f xs)"
haftmann@40210
   791
  by (cases xs) simp_all
haftmann@40210
   792
wenzelm@13142
   793
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   794
by (induct xs) simp_all
wenzelm@13114
   795
wenzelm@13142
   796
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   797
by (rule ext, induct_tac xs) auto
wenzelm@13114
   798
wenzelm@13142
   799
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   800
by (induct xs) auto
wenzelm@13114
   801
hoelzl@33639
   802
lemma map_map [simp]: "map f (map g xs) = map (f \<circ> g) xs"
hoelzl@33639
   803
by (induct xs) auto
hoelzl@33639
   804
nipkow@35208
   805
lemma map_comp_map[simp]: "((map f) o (map g)) = map(f o g)"
nipkow@35208
   806
apply(rule ext)
nipkow@35208
   807
apply(simp)
nipkow@35208
   808
done
nipkow@35208
   809
wenzelm@13142
   810
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   811
by (induct xs) auto
wenzelm@13114
   812
nipkow@13737
   813
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   814
by (induct xs) auto
nipkow@13737
   815
krauss@44013
   816
lemma map_cong [fundef_cong]:
haftmann@40122
   817
  "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> map f xs = map g ys"
haftmann@40122
   818
  by simp
wenzelm@13114
   819
wenzelm@13142
   820
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   821
by (cases xs) auto
wenzelm@13114
   822
wenzelm@13142
   823
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   824
by (cases xs) auto
wenzelm@13114
   825
paulson@18447
   826
lemma map_eq_Cons_conv:
nipkow@14025
   827
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   828
by (cases xs) auto
wenzelm@13114
   829
paulson@18447
   830
lemma Cons_eq_map_conv:
nipkow@14025
   831
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   832
by (cases ys) auto
nipkow@14025
   833
paulson@18447
   834
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
paulson@18447
   835
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
paulson@18447
   836
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
paulson@18447
   837
nipkow@14111
   838
lemma ex_map_conv:
nipkow@14111
   839
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
paulson@18447
   840
by(induct ys, auto simp add: Cons_eq_map_conv)
nipkow@14111
   841
nipkow@15110
   842
lemma map_eq_imp_length_eq:
paulson@35510
   843
  assumes "map f xs = map g ys"
haftmann@26734
   844
  shows "length xs = length ys"
haftmann@26734
   845
using assms proof (induct ys arbitrary: xs)
haftmann@26734
   846
  case Nil then show ?case by simp
haftmann@26734
   847
next
haftmann@26734
   848
  case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto
paulson@35510
   849
  from Cons xs have "map f zs = map g ys" by simp
haftmann@26734
   850
  moreover with Cons have "length zs = length ys" by blast
haftmann@26734
   851
  with xs show ?case by simp
haftmann@26734
   852
qed
haftmann@26734
   853
  
nipkow@15110
   854
lemma map_inj_on:
nipkow@15110
   855
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   856
  ==> xs = ys"
nipkow@15110
   857
apply(frule map_eq_imp_length_eq)
nipkow@15110
   858
apply(rotate_tac -1)
nipkow@15110
   859
apply(induct rule:list_induct2)
nipkow@15110
   860
 apply simp
nipkow@15110
   861
apply(simp)
nipkow@15110
   862
apply (blast intro:sym)
nipkow@15110
   863
done
nipkow@15110
   864
nipkow@15110
   865
lemma inj_on_map_eq_map:
nipkow@15110
   866
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   867
by(blast dest:map_inj_on)
nipkow@15110
   868
wenzelm@13114
   869
lemma map_injective:
nipkow@24526
   870
 "map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@24526
   871
by (induct ys arbitrary: xs) (auto dest!:injD)
wenzelm@13114
   872
nipkow@14339
   873
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   874
by(blast dest:map_injective)
nipkow@14339
   875
wenzelm@13114
   876
lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@17589
   877
by (iprover dest: map_injective injD intro: inj_onI)
wenzelm@13114
   878
wenzelm@13114
   879
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   880
apply (unfold inj_on_def, clarify)
nipkow@13145
   881
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   882
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   883
apply blast
nipkow@13145
   884
done
wenzelm@13114
   885
nipkow@14339
   886
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   887
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   888
nipkow@15303
   889
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
   890
apply(rule inj_onI)
nipkow@15303
   891
apply(erule map_inj_on)
nipkow@15303
   892
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
   893
done
nipkow@15303
   894
kleing@14343
   895
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   896
by (induct xs, auto)
wenzelm@13114
   897
nipkow@14402
   898
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   899
by (induct xs) auto
nipkow@14402
   900
nipkow@15110
   901
lemma map_fst_zip[simp]:
nipkow@15110
   902
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   903
by (induct rule:list_induct2, simp_all)
nipkow@15110
   904
nipkow@15110
   905
lemma map_snd_zip[simp]:
nipkow@15110
   906
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   907
by (induct rule:list_induct2, simp_all)
nipkow@15110
   908
haftmann@41505
   909
enriched_type map: map
haftmann@41372
   910
  by (simp_all add: fun_eq_iff id_def)
haftmann@40608
   911
nipkow@15110
   912
nipkow@15392
   913
subsubsection {* @{text rev} *}
wenzelm@13114
   914
wenzelm@13142
   915
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   916
by (induct xs) auto
wenzelm@13114
   917
wenzelm@13142
   918
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   919
by (induct xs) auto
wenzelm@13114
   920
kleing@15870
   921
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870
   922
by auto
kleing@15870
   923
wenzelm@13142
   924
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   925
by (induct xs) auto
wenzelm@13114
   926
wenzelm@13142
   927
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   928
by (induct xs) auto
wenzelm@13114
   929
kleing@15870
   930
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870
   931
by (cases xs) auto
kleing@15870
   932
kleing@15870
   933
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870
   934
by (cases xs) auto
kleing@15870
   935
haftmann@21061
   936
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
haftmann@21061
   937
apply (induct xs arbitrary: ys, force)
paulson@14208
   938
apply (case_tac ys, simp, force)
nipkow@13145
   939
done
wenzelm@13114
   940
nipkow@15439
   941
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
   942
by(simp add:inj_on_def)
nipkow@15439
   943
wenzelm@13366
   944
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   945
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489
   946
apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145
   947
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   948
done
wenzelm@13114
   949
wenzelm@13366
   950
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   951
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   952
by (induct xs rule: rev_induct) auto
wenzelm@13114
   953
wenzelm@13366
   954
lemmas rev_cases = rev_exhaust
wenzelm@13366
   955
nipkow@18423
   956
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
nipkow@18423
   957
by(rule rev_cases[of xs]) auto
nipkow@18423
   958
wenzelm@13114
   959
nipkow@15392
   960
subsubsection {* @{text set} *}
wenzelm@13114
   961
wenzelm@13142
   962
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   963
by (induct xs) auto
wenzelm@13114
   964
wenzelm@13142
   965
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   966
by (induct xs) auto
wenzelm@13114
   967
nipkow@17830
   968
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
nipkow@17830
   969
by(cases xs) auto
oheimb@14099
   970
wenzelm@13142
   971
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   972
by auto
wenzelm@13114
   973
oheimb@14099
   974
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   975
by auto
oheimb@14099
   976
wenzelm@13142
   977
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   978
by (induct xs) auto
wenzelm@13114
   979
nipkow@15245
   980
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   981
by(induct xs) auto
nipkow@15245
   982
wenzelm@13142
   983
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   984
by (induct xs) auto
wenzelm@13114
   985
wenzelm@13142
   986
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   987
by (induct xs) auto
wenzelm@13114
   988
wenzelm@13142
   989
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   990
by (induct xs) auto
wenzelm@13114
   991
nipkow@32417
   992
lemma set_upt [simp]: "set[i..<j] = {i..<j}"
bulwahn@41463
   993
by (induct j) auto
wenzelm@13114
   994
wenzelm@13142
   995
wenzelm@25221
   996
lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs"
nipkow@18049
   997
proof (induct xs)
nipkow@26073
   998
  case Nil thus ?case by simp
nipkow@26073
   999
next
nipkow@26073
  1000
  case Cons thus ?case by (auto intro: Cons_eq_appendI)
nipkow@26073
  1001
qed
nipkow@26073
  1002
haftmann@26734
  1003
lemma in_set_conv_decomp: "x \<in> set xs \<longleftrightarrow> (\<exists>ys zs. xs = ys @ x # zs)"
haftmann@26734
  1004
  by (auto elim: split_list)
nipkow@26073
  1005
nipkow@26073
  1006
lemma split_list_first: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@26073
  1007
proof (induct xs)
nipkow@26073
  1008
  case Nil thus ?case by simp
nipkow@18049
  1009
next
nipkow@18049
  1010
  case (Cons a xs)
nipkow@18049
  1011
  show ?case
nipkow@18049
  1012
  proof cases
nipkow@44890
  1013
    assume "x = a" thus ?case using Cons by fastforce
nipkow@18049
  1014
  next
nipkow@44890
  1015
    assume "x \<noteq> a" thus ?case using Cons by(fastforce intro!: Cons_eq_appendI)
nipkow@26073
  1016
  qed
nipkow@26073
  1017
qed
nipkow@26073
  1018
nipkow@26073
  1019
lemma in_set_conv_decomp_first:
nipkow@26073
  1020
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
haftmann@26734
  1021
  by (auto dest!: split_list_first)
nipkow@26073
  1022
haftmann@40122
  1023
lemma split_list_last: "x \<in> set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
haftmann@40122
  1024
proof (induct xs rule: rev_induct)
nipkow@26073
  1025
  case Nil thus ?case by simp
nipkow@26073
  1026
next
nipkow@26073
  1027
  case (snoc a xs)
nipkow@26073
  1028
  show ?case
nipkow@26073
  1029
  proof cases
haftmann@40122
  1030
    assume "x = a" thus ?case using snoc by (metis List.set.simps(1) emptyE)
nipkow@26073
  1031
  next
nipkow@44890
  1032
    assume "x \<noteq> a" thus ?case using snoc by fastforce
nipkow@18049
  1033
  qed
nipkow@18049
  1034
qed
nipkow@18049
  1035
nipkow@26073
  1036
lemma in_set_conv_decomp_last:
nipkow@26073
  1037
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
haftmann@26734
  1038
  by (auto dest!: split_list_last)
nipkow@26073
  1039
nipkow@26073
  1040
lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs & P x"
nipkow@26073
  1041
proof (induct xs)
nipkow@26073
  1042
  case Nil thus ?case by simp
nipkow@26073
  1043
next
nipkow@26073
  1044
  case Cons thus ?case
nipkow@26073
  1045
    by(simp add:Bex_def)(metis append_Cons append.simps(1))
nipkow@26073
  1046
qed
nipkow@26073
  1047
nipkow@26073
  1048
lemma split_list_propE:
haftmann@26734
  1049
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
  1050
  obtains ys x zs where "xs = ys @ x # zs" and "P x"
haftmann@26734
  1051
using split_list_prop [OF assms] by blast
nipkow@26073
  1052
nipkow@26073
  1053
lemma split_list_first_prop:
nipkow@26073
  1054
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
nipkow@26073
  1055
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)"
haftmann@26734
  1056
proof (induct xs)
nipkow@26073
  1057
  case Nil thus ?case by simp
nipkow@26073
  1058
next
nipkow@26073
  1059
  case (Cons x xs)
nipkow@26073
  1060
  show ?case
nipkow@26073
  1061
  proof cases
nipkow@26073
  1062
    assume "P x"
haftmann@40122
  1063
    thus ?thesis by simp (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
nipkow@26073
  1064
  next
nipkow@26073
  1065
    assume "\<not> P x"
nipkow@26073
  1066
    hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
nipkow@26073
  1067
    thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
nipkow@26073
  1068
  qed
nipkow@26073
  1069
qed
nipkow@26073
  1070
nipkow@26073
  1071
lemma split_list_first_propE:
haftmann@26734
  1072
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
  1073
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y"
haftmann@26734
  1074
using split_list_first_prop [OF assms] by blast
nipkow@26073
  1075
nipkow@26073
  1076
lemma split_list_first_prop_iff:
nipkow@26073
  1077
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
nipkow@26073
  1078
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))"
haftmann@26734
  1079
by (rule, erule split_list_first_prop) auto
nipkow@26073
  1080
nipkow@26073
  1081
lemma split_list_last_prop:
nipkow@26073
  1082
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
nipkow@26073
  1083
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)"
nipkow@26073
  1084
proof(induct xs rule:rev_induct)
nipkow@26073
  1085
  case Nil thus ?case by simp
nipkow@26073
  1086
next
nipkow@26073
  1087
  case (snoc x xs)
nipkow@26073
  1088
  show ?case
nipkow@26073
  1089
  proof cases
nipkow@26073
  1090
    assume "P x" thus ?thesis by (metis emptyE set_empty)
nipkow@26073
  1091
  next
nipkow@26073
  1092
    assume "\<not> P x"
nipkow@26073
  1093
    hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
nipkow@44890
  1094
    thus ?thesis using `\<not> P x` snoc(1) by fastforce
nipkow@26073
  1095
  qed
nipkow@26073
  1096
qed
nipkow@26073
  1097
nipkow@26073
  1098
lemma split_list_last_propE:
haftmann@26734
  1099
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
  1100
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z"
haftmann@26734
  1101
using split_list_last_prop [OF assms] by blast
nipkow@26073
  1102
nipkow@26073
  1103
lemma split_list_last_prop_iff:
nipkow@26073
  1104
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
nipkow@26073
  1105
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
haftmann@26734
  1106
by (metis split_list_last_prop [where P=P] in_set_conv_decomp)
nipkow@26073
  1107
nipkow@26073
  1108
lemma finite_list: "finite A ==> EX xs. set xs = A"
haftmann@26734
  1109
  by (erule finite_induct)
haftmann@26734
  1110
    (auto simp add: set.simps(2) [symmetric] simp del: set.simps(2))
paulson@13508
  1111
kleing@14388
  1112
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
  1113
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
  1114
haftmann@26442
  1115
lemma set_minus_filter_out:
haftmann@26442
  1116
  "set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)"
haftmann@26442
  1117
  by (induct xs) auto
paulson@15168
  1118
wenzelm@35115
  1119
nipkow@15392
  1120
subsubsection {* @{text filter} *}
wenzelm@13114
  1121
wenzelm@13142
  1122
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
  1123
by (induct xs) auto
wenzelm@13114
  1124
nipkow@15305
  1125
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
  1126
by (induct xs) simp_all
nipkow@15305
  1127
wenzelm@13142
  1128
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
  1129
by (induct xs) auto
wenzelm@13114
  1130
nipkow@16998
  1131
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998
  1132
by (induct xs) (auto simp add: le_SucI)
nipkow@16998
  1133
nipkow@18423
  1134
lemma sum_length_filter_compl:
nipkow@18423
  1135
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
nipkow@18423
  1136
by(induct xs) simp_all
nipkow@18423
  1137
wenzelm@13142
  1138
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
  1139
by (induct xs) auto
wenzelm@13114
  1140
wenzelm@13142
  1141
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
  1142
by (induct xs) auto
wenzelm@13114
  1143
nipkow@16998
  1144
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@24349
  1145
by (induct xs) simp_all
nipkow@16998
  1146
nipkow@16998
  1147
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998
  1148
apply (induct xs)
nipkow@16998
  1149
 apply auto
nipkow@16998
  1150
apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998
  1151
apply simp
nipkow@16998
  1152
done
wenzelm@13114
  1153
nipkow@16965
  1154
lemma filter_map:
nipkow@16965
  1155
  "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965
  1156
by (induct xs) simp_all
nipkow@16965
  1157
nipkow@16965
  1158
lemma length_filter_map[simp]:
nipkow@16965
  1159
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965
  1160
by (simp add:filter_map)
nipkow@16965
  1161
wenzelm@13142
  1162
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
  1163
by auto
wenzelm@13114
  1164
nipkow@15246
  1165
lemma length_filter_less:
nipkow@15246
  1166
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
  1167
proof (induct xs)
nipkow@15246
  1168
  case Nil thus ?case by simp
nipkow@15246
  1169
next
nipkow@15246
  1170
  case (Cons x xs) thus ?case
nipkow@15246
  1171
    apply (auto split:split_if_asm)
nipkow@15246
  1172
    using length_filter_le[of P xs] apply arith
nipkow@15246
  1173
  done
nipkow@15246
  1174
qed
wenzelm@13114
  1175
nipkow@15281
  1176
lemma length_filter_conv_card:
nipkow@15281
  1177
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
  1178
proof (induct xs)
nipkow@15281
  1179
  case Nil thus ?case by simp
nipkow@15281
  1180
next
nipkow@15281
  1181
  case (Cons x xs)
nipkow@15281
  1182
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
  1183
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
  1184
  show ?case (is "?l = card ?S'")
nipkow@15281
  1185
  proof (cases)
nipkow@15281
  1186
    assume "p x"
nipkow@15281
  1187
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@25162
  1188
      by(auto simp: image_def split:nat.split dest:gr0_implies_Suc)
nipkow@15281
  1189
    have "length (filter p (x # xs)) = Suc(card ?S)"
wenzelm@23388
  1190
      using Cons `p x` by simp
nipkow@15281
  1191
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
huffman@44921
  1192
      by (simp add: card_image)
nipkow@15281
  1193
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
  1194
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
  1195
    finally show ?thesis .
nipkow@15281
  1196
  next
nipkow@15281
  1197
    assume "\<not> p x"
nipkow@15281
  1198
    hence eq: "?S' = Suc ` ?S"
nipkow@25162
  1199
      by(auto simp add: image_def split:nat.split elim:lessE)
nipkow@15281
  1200
    have "length (filter p (x # xs)) = card ?S"
wenzelm@23388
  1201
      using Cons `\<not> p x` by simp
nipkow@15281
  1202
    also have "\<dots> = card(Suc ` ?S)" using fin
huffman@44921
  1203
      by (simp add: card_image)
nipkow@15281
  1204
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
  1205
      by (simp add:card_insert_if)
nipkow@15281
  1206
    finally show ?thesis .
nipkow@15281
  1207
  qed
nipkow@15281
  1208
qed
nipkow@15281
  1209
nipkow@17629
  1210
lemma Cons_eq_filterD:
nipkow@17629
  1211
 "x#xs = filter P ys \<Longrightarrow>
nipkow@17629
  1212
  \<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
  1213
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
nipkow@17629
  1214
proof(induct ys)
nipkow@17629
  1215
  case Nil thus ?case by simp
nipkow@17629
  1216
next
nipkow@17629
  1217
  case (Cons y ys)
nipkow@17629
  1218
  show ?case (is "\<exists>x. ?Q x")
nipkow@17629
  1219
  proof cases
nipkow@17629
  1220
    assume Py: "P y"
nipkow@17629
  1221
    show ?thesis
nipkow@17629
  1222
    proof cases
wenzelm@25221
  1223
      assume "x = y"
wenzelm@25221
  1224
      with Py Cons.prems have "?Q []" by simp
wenzelm@25221
  1225
      then show ?thesis ..
nipkow@17629
  1226
    next
wenzelm@25221
  1227
      assume "x \<noteq> y"
wenzelm@25221
  1228
      with Py Cons.prems show ?thesis by simp
nipkow@17629
  1229
    qed
nipkow@17629
  1230
  next
wenzelm@25221
  1231
    assume "\<not> P y"
nipkow@44890
  1232
    with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastforce
wenzelm@25221
  1233
    then have "?Q (y#us)" by simp
wenzelm@25221
  1234
    then show ?thesis ..
nipkow@17629
  1235
  qed
nipkow@17629
  1236
qed
nipkow@17629
  1237
nipkow@17629
  1238
lemma filter_eq_ConsD:
nipkow@17629
  1239
 "filter P ys = x#xs \<Longrightarrow>
nipkow@17629
  1240
  \<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
  1241
by(rule Cons_eq_filterD) simp
nipkow@17629
  1242
nipkow@17629
  1243
lemma filter_eq_Cons_iff:
nipkow@17629
  1244
 "(filter P ys = x#xs) =
nipkow@17629
  1245
  (\<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
  1246
by(auto dest:filter_eq_ConsD)
nipkow@17629
  1247
nipkow@17629
  1248
lemma Cons_eq_filter_iff:
nipkow@17629
  1249
 "(x#xs = filter P ys) =
nipkow@17629
  1250
  (\<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
  1251
by(auto dest:Cons_eq_filterD)
nipkow@17629
  1252
krauss@44013
  1253
lemma filter_cong[fundef_cong]:
nipkow@17501
  1254
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
nipkow@17501
  1255
apply simp
nipkow@17501
  1256
apply(erule thin_rl)
nipkow@17501
  1257
by (induct ys) simp_all
nipkow@17501
  1258
nipkow@15281
  1259
haftmann@26442
  1260
subsubsection {* List partitioning *}
haftmann@26442
  1261
haftmann@26442
  1262
primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where
haftmann@26442
  1263
  "partition P [] = ([], [])"
haftmann@26442
  1264
  | "partition P (x # xs) = 
haftmann@26442
  1265
      (let (yes, no) = partition P xs
haftmann@26442
  1266
      in if P x then (x # yes, no) else (yes, x # no))"
haftmann@26442
  1267
haftmann@26442
  1268
lemma partition_filter1:
haftmann@26442
  1269
    "fst (partition P xs) = filter P xs"
haftmann@26442
  1270
by (induct xs) (auto simp add: Let_def split_def)
haftmann@26442
  1271
haftmann@26442
  1272
lemma partition_filter2:
haftmann@26442
  1273
    "snd (partition P xs) = filter (Not o P) xs"
haftmann@26442
  1274
by (induct xs) (auto simp add: Let_def split_def)
haftmann@26442
  1275
haftmann@26442
  1276
lemma partition_P:
haftmann@26442
  1277
  assumes "partition P xs = (yes, no)"
haftmann@26442
  1278
  shows "(\<forall>p \<in> set yes.  P p) \<and> (\<forall>p  \<in> set no. \<not> P p)"
haftmann@26442
  1279
proof -
haftmann@26442
  1280
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
haftmann@26442
  1281
    by simp_all
haftmann@26442
  1282
  then show ?thesis by (simp_all add: partition_filter1 partition_filter2)
haftmann@26442
  1283
qed
haftmann@26442
  1284
haftmann@26442
  1285
lemma partition_set:
haftmann@26442
  1286
  assumes "partition P xs = (yes, no)"
haftmann@26442
  1287
  shows "set yes \<union> set no = set xs"
haftmann@26442
  1288
proof -
haftmann@26442
  1289
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
haftmann@26442
  1290
    by simp_all
haftmann@26442
  1291
  then show ?thesis by (auto simp add: partition_filter1 partition_filter2) 
haftmann@26442
  1292
qed
haftmann@26442
  1293
hoelzl@33639
  1294
lemma partition_filter_conv[simp]:
hoelzl@33639
  1295
  "partition f xs = (filter f xs,filter (Not o f) xs)"
hoelzl@33639
  1296
unfolding partition_filter2[symmetric]
hoelzl@33639
  1297
unfolding partition_filter1[symmetric] by simp
hoelzl@33639
  1298
hoelzl@33639
  1299
declare partition.simps[simp del]
haftmann@26442
  1300
wenzelm@35115
  1301
nipkow@15392
  1302
subsubsection {* @{text concat} *}
wenzelm@13114
  1303
wenzelm@13142
  1304
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
  1305
by (induct xs) auto
wenzelm@13114
  1306
paulson@18447
  1307
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
  1308
by (induct xss) auto
wenzelm@13114
  1309
paulson@18447
  1310
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
  1311
by (induct xss) auto
wenzelm@13114
  1312
nipkow@24308
  1313
lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
nipkow@13145
  1314
by (induct xs) auto
wenzelm@13114
  1315
nipkow@24476
  1316
lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs"
nipkow@24349
  1317
by (induct xs) auto
nipkow@24349
  1318
wenzelm@13142
  1319
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
  1320
by (induct xs) auto
wenzelm@13114
  1321
wenzelm@13142
  1322
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
  1323
by (induct xs) auto
wenzelm@13114
  1324
wenzelm@13142
  1325
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
  1326
by (induct xs) auto
wenzelm@13114
  1327
bulwahn@40365
  1328
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
  1329
proof (induct xs arbitrary: ys)
bulwahn@40365
  1330
  case (Cons x xs ys)
bulwahn@40365
  1331
  thus ?case by (cases ys) auto
bulwahn@40365
  1332
qed (auto)
bulwahn@40365
  1333
bulwahn@40365
  1334
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
  1335
by (simp add: concat_eq_concat_iff)
bulwahn@40365
  1336
wenzelm@13114
  1337
nipkow@15392
  1338
subsubsection {* @{text nth} *}
wenzelm@13114
  1339
haftmann@29827
  1340
lemma nth_Cons_0 [simp, code]: "(x # xs)!0 = x"
nipkow@13145
  1341
by auto
wenzelm@13114
  1342
haftmann@29827
  1343
lemma nth_Cons_Suc [simp, code]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
  1344
by auto
wenzelm@13114
  1345
wenzelm@13142
  1346
declare nth.simps [simp del]
wenzelm@13114
  1347
nipkow@41842
  1348
lemma nth_Cons_pos[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
nipkow@41842
  1349
by(auto simp: Nat.gr0_conv_Suc)
nipkow@41842
  1350
wenzelm@13114
  1351
lemma nth_append:
nipkow@24526
  1352
  "(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
nipkow@24526
  1353
apply (induct xs arbitrary: n, simp)
paulson@14208
  1354
apply (case_tac n, auto)
nipkow@13145
  1355
done
wenzelm@13114
  1356
nipkow@14402
  1357
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
wenzelm@25221
  1358
by (induct xs) auto
nipkow@14402
  1359
nipkow@14402
  1360
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
wenzelm@25221
  1361
by (induct xs) auto
nipkow@14402
  1362
nipkow@24526
  1363
lemma nth_map [simp]: "n < length xs ==> (map f xs)!n = f(xs!n)"
nipkow@24526
  1364
apply (induct xs arbitrary: n, simp)
paulson@14208
  1365
apply (case_tac n, auto)
nipkow@13145
  1366
done
wenzelm@13114
  1367
noschinl@45841
  1368
lemma nth_tl:
noschinl@45841
  1369
  assumes "n < length (tl x)" shows "tl x ! n = x ! Suc n"
noschinl@45841
  1370
using assms by (induct x) auto
noschinl@45841
  1371
nipkow@18423
  1372
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
nipkow@18423
  1373
by(cases xs) simp_all
nipkow@18423
  1374
nipkow@18049
  1375
nipkow@18049
  1376
lemma list_eq_iff_nth_eq:
nipkow@24526
  1377
 "(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
nipkow@24526
  1378
apply(induct xs arbitrary: ys)
paulson@24632
  1379
 apply force
nipkow@18049
  1380
apply(case_tac ys)
nipkow@18049
  1381
 apply simp
nipkow@18049
  1382
apply(simp add:nth_Cons split:nat.split)apply blast
nipkow@18049
  1383
done
nipkow@18049
  1384
wenzelm@13142
  1385
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
  1386
apply (induct xs, simp, simp)
nipkow@13145
  1387
apply safe
paulson@24632
  1388
apply (metis nat_case_0 nth.simps zero_less_Suc)
paulson@24632
  1389
apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
paulson@14208
  1390
apply (case_tac i, simp)
paulson@24632
  1391
apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff)
nipkow@13145
  1392
done
wenzelm@13114
  1393
nipkow@17501
  1394
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
nipkow@17501
  1395
by(auto simp:set_conv_nth)
nipkow@17501
  1396
nipkow@13145
  1397
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
  1398
by (auto simp add: set_conv_nth)
wenzelm@13114
  1399
wenzelm@13142
  1400
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
  1401
by (auto simp add: set_conv_nth)
wenzelm@13114
  1402
wenzelm@13114
  1403
lemma all_nth_imp_all_set:
nipkow@13145
  1404
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
  1405
by (auto simp add: set_conv_nth)
wenzelm@13114
  1406
wenzelm@13114
  1407
lemma all_set_conv_all_nth:
nipkow@13145
  1408
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
  1409
by (auto simp add: set_conv_nth)
wenzelm@13114
  1410
kleing@25296
  1411
lemma rev_nth:
kleing@25296
  1412
  "n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)"
kleing@25296
  1413
proof (induct xs arbitrary: n)
kleing@25296
  1414
  case Nil thus ?case by simp
kleing@25296
  1415
next
kleing@25296
  1416
  case (Cons x xs)
kleing@25296
  1417
  hence n: "n < Suc (length xs)" by simp
kleing@25296
  1418
  moreover
kleing@25296
  1419
  { assume "n < length xs"
kleing@25296
  1420
    with n obtain n' where "length xs - n = Suc n'"
kleing@25296
  1421
      by (cases "length xs - n", auto)
kleing@25296
  1422
    moreover
kleing@25296
  1423
    then have "length xs - Suc n = n'" by simp
kleing@25296
  1424
    ultimately
kleing@25296
  1425
    have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp
kleing@25296
  1426
  }
kleing@25296
  1427
  ultimately
kleing@25296
  1428
  show ?case by (clarsimp simp add: Cons nth_append)
kleing@25296
  1429
qed
wenzelm@13114
  1430
nipkow@31159
  1431
lemma Skolem_list_nth:
nipkow@31159
  1432
  "(ALL i<k. EX x. P i x) = (EX xs. size xs = k & (ALL i<k. P i (xs!i)))"
nipkow@31159
  1433
  (is "_ = (EX xs. ?P k xs)")
nipkow@31159
  1434
proof(induct k)
nipkow@31159
  1435
  case 0 show ?case by simp
nipkow@31159
  1436
next
nipkow@31159
  1437
  case (Suc k)
nipkow@31159
  1438
  show ?case (is "?L = ?R" is "_ = (EX xs. ?P' xs)")
nipkow@31159
  1439
  proof
nipkow@31159
  1440
    assume "?R" thus "?L" using Suc by auto
nipkow@31159
  1441
  next
nipkow@31159
  1442
    assume "?L"
nipkow@31159
  1443
    with Suc obtain x xs where "?P k xs & P k x" by (metis less_Suc_eq)
nipkow@31159
  1444
    hence "?P'(xs@[x])" by(simp add:nth_append less_Suc_eq)
nipkow@31159
  1445
    thus "?R" ..
nipkow@31159
  1446
  qed
nipkow@31159
  1447
qed
nipkow@31159
  1448
nipkow@31159
  1449
nipkow@15392
  1450
subsubsection {* @{text list_update} *}
wenzelm@13114
  1451
nipkow@24526
  1452
lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
nipkow@24526
  1453
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1454
wenzelm@13114
  1455
lemma nth_list_update:
nipkow@24526
  1456
"i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@24526
  1457
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1458
wenzelm@13142
  1459
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
  1460
by (simp add: nth_list_update)
wenzelm@13114
  1461
nipkow@24526
  1462
lemma nth_list_update_neq [simp]: "i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@24526
  1463
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1464
nipkow@24526
  1465
lemma list_update_id[simp]: "xs[i := xs!i] = xs"
nipkow@24526
  1466
by (induct xs arbitrary: i) (simp_all split:nat.splits)
nipkow@24526
  1467
nipkow@24526
  1468
lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
nipkow@24526
  1469
apply (induct xs arbitrary: i)
nipkow@17501
  1470
 apply simp
nipkow@17501
  1471
apply (case_tac i)
nipkow@17501
  1472
apply simp_all
nipkow@17501
  1473
done
nipkow@17501
  1474
nipkow@31077
  1475
lemma list_update_nonempty[simp]: "xs[k:=x] = [] \<longleftrightarrow> xs=[]"
nipkow@31077
  1476
by(metis length_0_conv length_list_update)
nipkow@31077
  1477
wenzelm@13114
  1478
lemma list_update_same_conv:
nipkow@24526
  1479
"i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@24526
  1480
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1481
nipkow@14187
  1482
lemma list_update_append1:
nipkow@24526
  1483
 "i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
nipkow@24526
  1484
apply (induct xs arbitrary: i, simp)
nipkow@14187
  1485
apply(simp split:nat.split)
nipkow@14187
  1486
done
nipkow@14187
  1487
kleing@15868
  1488
lemma list_update_append:
nipkow@24526
  1489
  "(xs @ ys) [n:= x] = 
kleing@15868
  1490
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
nipkow@24526
  1491
by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868
  1492
nipkow@14402
  1493
lemma list_update_length [simp]:
nipkow@14402
  1494
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
  1495
by (induct xs, auto)
nipkow@14402
  1496
nipkow@31264
  1497
lemma map_update: "map f (xs[k:= y]) = (map f xs)[k := f y]"
nipkow@31264
  1498
by(induct xs arbitrary: k)(auto split:nat.splits)
nipkow@31264
  1499
nipkow@31264
  1500
lemma rev_update:
nipkow@31264
  1501
  "k < length xs \<Longrightarrow> rev (xs[k:= y]) = (rev xs)[length xs - k - 1 := y]"
nipkow@31264
  1502
by (induct xs arbitrary: k) (auto simp: list_update_append split:nat.splits)
nipkow@31264
  1503
wenzelm@13114
  1504
lemma update_zip:
nipkow@31080
  1505
  "(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@24526
  1506
by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split)
nipkow@24526
  1507
nipkow@24526
  1508
lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)"
nipkow@24526
  1509
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1510
wenzelm@13114
  1511
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
  1512
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
  1513
nipkow@24526
  1514
lemma set_update_memI: "n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
nipkow@24526
  1515
by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868
  1516
nipkow@31077
  1517
lemma list_update_overwrite[simp]:
haftmann@24796
  1518
  "xs [i := x, i := y] = xs [i := y]"
nipkow@31077
  1519
apply (induct xs arbitrary: i) apply simp
nipkow@31077
  1520
apply (case_tac i, simp_all)
haftmann@24796
  1521
done
haftmann@24796
  1522
haftmann@24796
  1523
lemma list_update_swap:
haftmann@24796
  1524
  "i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]"
haftmann@24796
  1525
apply (induct xs arbitrary: i i')
haftmann@24796
  1526
apply simp
haftmann@24796
  1527
apply (case_tac i, case_tac i')
haftmann@24796
  1528
apply auto
haftmann@24796
  1529
apply (case_tac i')
haftmann@24796
  1530
apply auto
haftmann@24796
  1531
done
haftmann@24796
  1532
haftmann@29827
  1533
lemma list_update_code [code]:
haftmann@29827
  1534
  "[][i := y] = []"
haftmann@29827
  1535
  "(x # xs)[0 := y] = y # xs"
haftmann@29827
  1536
  "(x # xs)[Suc i := y] = x # xs[i := y]"
haftmann@29827
  1537
  by simp_all
haftmann@29827
  1538
wenzelm@13114
  1539
nipkow@15392
  1540
subsubsection {* @{text last} and @{text butlast} *}
wenzelm@13114
  1541
wenzelm@13142
  1542
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
  1543
by (induct xs) auto
wenzelm@13114
  1544
wenzelm@13142
  1545
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
  1546
by (induct xs) auto
wenzelm@13114
  1547
nipkow@14302
  1548
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
huffman@44921
  1549
  by simp
nipkow@14302
  1550
nipkow@14302
  1551
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
huffman@44921
  1552
  by simp
nipkow@14302
  1553
nipkow@14302
  1554
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
  1555
by (induct xs) (auto)
nipkow@14302
  1556
nipkow@14302
  1557
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
  1558
by(simp add:last_append)
nipkow@14302
  1559
nipkow@14302
  1560
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
  1561
by(simp add:last_append)
nipkow@14302
  1562
noschinl@45841
  1563
lemma last_tl: "xs = [] \<or> tl xs \<noteq> [] \<Longrightarrow>last (tl xs) = last xs"
noschinl@45841
  1564
by (induct xs) simp_all
noschinl@45841
  1565
noschinl@45841
  1566
lemma butlast_tl: "butlast (tl xs) = tl (butlast xs)"
noschinl@45841
  1567
by (induct xs) simp_all
noschinl@45841
  1568
nipkow@17762
  1569
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
nipkow@17762
  1570
by(rule rev_exhaust[of xs]) simp_all
nipkow@17762
  1571
nipkow@17762
  1572
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
nipkow@17762
  1573
by(cases xs) simp_all
nipkow@17762
  1574
nipkow@17765
  1575
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
nipkow@17765
  1576
by (induct as) auto
nipkow@17762
  1577
wenzelm@13142
  1578
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
  1579
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1580
wenzelm@13114
  1581
lemma butlast_append:
nipkow@24526
  1582
  "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@24526
  1583
by (induct xs arbitrary: ys) auto
wenzelm@13114
  1584
wenzelm@13142
  1585
lemma append_butlast_last_id [simp]:
nipkow@13145
  1586
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
  1587
by (induct xs) auto
wenzelm@13114
  1588
wenzelm@13142
  1589
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
  1590
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1591
wenzelm@13114
  1592
lemma in_set_butlast_appendI:
nipkow@13145
  1593
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
  1594
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
  1595
nipkow@24526
  1596
lemma last_drop[simp]: "n < length xs \<Longrightarrow> last (drop n xs) = last xs"
nipkow@24526
  1597
apply (induct xs arbitrary: n)
nipkow@17501
  1598
 apply simp
nipkow@17501
  1599
apply (auto split:nat.split)
nipkow@17501
  1600
done
nipkow@17501
  1601
noschinl@45841
  1602
lemma nth_butlast:
noschinl@45841
  1603
  assumes "n < length (butlast xs)" shows "butlast xs ! n = xs ! n"
noschinl@45841
  1604
proof (cases xs)
noschinl@45841
  1605
  case (Cons y ys)
noschinl@45841
  1606
  moreover from assms have "butlast xs ! n = (butlast xs @ [last xs]) ! n"
noschinl@45841
  1607
    by (simp add: nth_append)
noschinl@45841
  1608
  ultimately show ?thesis using append_butlast_last_id by simp
noschinl@45841
  1609
qed simp
noschinl@45841
  1610
huffman@30128
  1611
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
nipkow@17589
  1612
by(induct xs)(auto simp:neq_Nil_conv)
nipkow@17589
  1613
huffman@30128
  1614
lemma butlast_conv_take: "butlast xs = take (length xs - 1) xs"
huffman@26584
  1615
by (induct xs, simp, case_tac xs, simp_all)
huffman@26584
  1616
nipkow@31077
  1617
lemma last_list_update:
nipkow@31077
  1618
  "xs \<noteq> [] \<Longrightarrow> last(xs[k:=x]) = (if k = size xs - 1 then x else last xs)"
nipkow@31077
  1619
by (auto simp: last_conv_nth)
nipkow@31077
  1620
nipkow@31077
  1621
lemma butlast_list_update:
nipkow@31077
  1622
  "butlast(xs[k:=x]) =
nipkow@31077
  1623
 (if k = size xs - 1 then butlast xs else (butlast xs)[k:=x])"
nipkow@31077
  1624
apply(cases xs rule:rev_cases)
nipkow@31077
  1625
apply simp
nipkow@31077
  1626
apply(simp add:list_update_append split:nat.splits)
nipkow@31077
  1627
done
nipkow@31077
  1628
haftmann@36851
  1629
lemma last_map:
haftmann@36851
  1630
  "xs \<noteq> [] \<Longrightarrow> last (map f xs) = f (last xs)"
haftmann@36851
  1631
  by (cases xs rule: rev_cases) simp_all
haftmann@36851
  1632
haftmann@36851
  1633
lemma map_butlast:
haftmann@36851
  1634
  "map f (butlast xs) = butlast (map f xs)"
haftmann@36851
  1635
  by (induct xs) simp_all
haftmann@36851
  1636
nipkow@40230
  1637
lemma snoc_eq_iff_butlast:
nipkow@40230
  1638
  "xs @ [x] = ys \<longleftrightarrow> (ys \<noteq> [] & butlast ys = xs & last ys = x)"
nipkow@40230
  1639
by (metis append_butlast_last_id append_is_Nil_conv butlast_snoc last_snoc not_Cons_self)
nipkow@40230
  1640
haftmann@24796
  1641
nipkow@15392
  1642
subsubsection {* @{text take} and @{text drop} *}
wenzelm@13114
  1643
wenzelm@13142
  1644
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
  1645
by (induct xs) auto
wenzelm@13114
  1646
wenzelm@13142
  1647
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
  1648
by (induct xs) auto
wenzelm@13114
  1649
wenzelm@13142
  1650
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
  1651
by simp
wenzelm@13114
  1652
wenzelm@13142
  1653
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
  1654
by simp
wenzelm@13114
  1655
wenzelm@13142
  1656
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
  1657
huffman@30128
  1658
lemma take_1_Cons [simp]: "take 1 (x # xs) = [x]"
huffman@30128
  1659
  unfolding One_nat_def by simp
huffman@30128
  1660
huffman@30128
  1661
lemma drop_1_Cons [simp]: "drop 1 (x # xs) = xs"
huffman@30128
  1662
  unfolding One_nat_def by simp
huffman@30128
  1663
nipkow@15110
  1664
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
  1665
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
  1666
nipkow@14187
  1667
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
  1668
by(cases xs, simp_all)
nipkow@14187
  1669
huffman@26584
  1670
lemma take_tl: "take n (tl xs) = tl (take (Suc n) xs)"
huffman@26584
  1671
by (induct xs arbitrary: n) simp_all
huffman@26584
  1672
nipkow@24526
  1673
lemma drop_tl: "drop n (tl xs) = tl(drop n xs)"
nipkow@24526
  1674
by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@24526
  1675
huffman@26584
  1676
lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)"
huffman@26584
  1677
by (cases n, simp, cases xs, auto)
huffman@26584
  1678
huffman@26584
  1679
lemma tl_drop: "tl (drop n xs) = drop n (tl xs)"
huffman@26584
  1680
by (simp only: drop_tl)
huffman@26584
  1681
nipkow@24526
  1682
lemma nth_via_drop: "drop n xs = y#ys \<Longrightarrow> xs!n = y"
nipkow@24526
  1683
apply (induct xs arbitrary: n, simp)
nipkow@14187
  1684
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
  1685
done
nipkow@14187
  1686
nipkow@13913
  1687
lemma take_Suc_conv_app_nth:
nipkow@24526
  1688
  "i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
nipkow@24526
  1689
apply (induct xs arbitrary: i, simp)
paulson@14208
  1690
apply (case_tac i, auto)
nipkow@13913
  1691
done
nipkow@13913
  1692
mehta@14591
  1693
lemma drop_Suc_conv_tl:
nipkow@24526
  1694
  "i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
nipkow@24526
  1695
apply (induct xs arbitrary: i, simp)
mehta@14591
  1696
apply (case_tac i, auto)
mehta@14591
  1697
done
mehta@14591
  1698
nipkow@24526
  1699
lemma length_take [simp]: "length (take n xs) = min (length xs) n"
nipkow@24526
  1700
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1701
nipkow@24526
  1702
lemma length_drop [simp]: "length (drop n xs) = (length xs - n)"
nipkow@24526
  1703
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1704
nipkow@24526
  1705
lemma take_all [simp]: "length xs <= n ==> take n xs = xs"
nipkow@24526
  1706
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1707
nipkow@24526
  1708
lemma drop_all [simp]: "length xs <= n ==> drop n xs = []"
nipkow@24526
  1709
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114
  1710
wenzelm@13142
  1711
lemma take_append [simp]:
nipkow@24526
  1712
  "take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@24526
  1713
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114
  1714
wenzelm@13142
  1715
lemma drop_append [simp]:
nipkow@24526
  1716
  "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@24526
  1717
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1718
nipkow@24526
  1719
lemma take_take [simp]: "take n (take m xs) = take (min n m) xs"
nipkow@24526
  1720
apply (induct m arbitrary: xs n, auto)
paulson@14208
  1721
apply (case_tac xs, auto)
nipkow@15236
  1722
apply (case_tac n, auto)
nipkow@13145
  1723
done
wenzelm@13114
  1724
nipkow@24526
  1725
lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs"
nipkow@24526
  1726
apply (induct m arbitrary: xs, auto)
paulson@14208
  1727
apply (case_tac xs, auto)
nipkow@13145
  1728
done
wenzelm@13114
  1729
nipkow@24526
  1730
lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)"
nipkow@24526
  1731
apply (induct m arbitrary: xs n, auto)
paulson@14208
  1732
apply (case_tac xs, auto)
nipkow@13145
  1733
done
wenzelm@13114
  1734
nipkow@24526
  1735
lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@24526
  1736
apply(induct xs arbitrary: m n)
nipkow@14802
  1737
 apply simp
nipkow@14802
  1738
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  1739
done
nipkow@14802
  1740
nipkow@24526
  1741
lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs"
nipkow@24526
  1742
apply (induct n arbitrary: xs, auto)
paulson@14208
  1743
apply (case_tac xs, auto)
nipkow@13145
  1744
done
wenzelm@13114
  1745
nipkow@24526
  1746
lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@24526
  1747
apply(induct xs arbitrary: n)
nipkow@15110
  1748
 apply simp
nipkow@15110
  1749
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  1750
done
nipkow@15110
  1751
nipkow@24526
  1752
lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)"
nipkow@24526
  1753
apply(induct xs arbitrary: n)
nipkow@15110
  1754
apply simp
nipkow@15110
  1755
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1756
done
nipkow@15110
  1757
nipkow@24526
  1758
lemma take_map: "take n (map f xs) = map f (take n xs)"
nipkow@24526
  1759
apply (induct n arbitrary: xs, auto)
paulson@14208
  1760
apply (case_tac xs, auto)
nipkow@13145
  1761
done
wenzelm@13114
  1762
nipkow@24526
  1763
lemma drop_map: "drop n (map f xs) = map f (drop n xs)"
nipkow@24526
  1764
apply (induct n arbitrary: xs, auto)
paulson@14208
  1765
apply (case_tac xs, auto)
nipkow@13145
  1766
done
wenzelm@13114
  1767
nipkow@24526
  1768
lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)"
nipkow@24526
  1769
apply (induct xs arbitrary: i, auto)
paulson@14208
  1770
apply (case_tac i, auto)
nipkow@13145
  1771
done
wenzelm@13114
  1772
nipkow@24526
  1773
lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)"
nipkow@24526
  1774
apply (induct xs arbitrary: i, auto)
paulson@14208
  1775
apply (case_tac i, auto)
nipkow@13145
  1776
done
wenzelm@13114
  1777
nipkow@24526
  1778
lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i"
nipkow@24526
  1779
apply (induct xs arbitrary: i n, auto)
paulson@14208
  1780
apply (case_tac n, blast)
paulson@14208
  1781
apply (case_tac i, auto)
nipkow@13145
  1782
done
wenzelm@13114
  1783
wenzelm@13142
  1784
lemma nth_drop [simp]:
nipkow@24526
  1785
  "n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
nipkow@24526
  1786
apply (induct n arbitrary: xs i, auto)
paulson@14208
  1787
apply (case_tac xs, auto)
nipkow@13145
  1788
done
nipkow@3507
  1789
huffman@26584
  1790
lemma butlast_take:
huffman@30128
  1791
  "n <= length xs ==> butlast (take n xs) = take (n - 1) xs"
huffman@26584
  1792
by (simp add: butlast_conv_take min_max.inf_absorb1 min_max.inf_absorb2)
huffman@26584
  1793
huffman@26584
  1794
lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)"
huffman@30128
  1795
by (simp add: butlast_conv_take drop_take add_ac)
huffman@26584
  1796
huffman@26584
  1797
lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs"
huffman@26584
  1798
by (simp add: butlast_conv_take min_max.inf_absorb1)
huffman@26584
  1799
huffman@26584
  1800
lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)"
huffman@30128
  1801
by (simp add: butlast_conv_take drop_take add_ac)
huffman@26584
  1802
nipkow@18423
  1803
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
nipkow@18423
  1804
by(simp add: hd_conv_nth)
nipkow@18423
  1805
nipkow@35248
  1806
lemma set_take_subset_set_take:
nipkow@35248
  1807
  "m <= n \<Longrightarrow> set(take m xs) <= set(take n xs)"
bulwahn@41463
  1808
apply (induct xs arbitrary: m n)
bulwahn@41463
  1809
apply simp
bulwahn@41463
  1810
apply (case_tac n)
bulwahn@41463
  1811
apply (auto simp: take_Cons)
bulwahn@41463
  1812
done
nipkow@35248
  1813
nipkow@24526
  1814
lemma set_take_subset: "set(take n xs) \<subseteq> set xs"
nipkow@24526
  1815
by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split)
nipkow@24526
  1816
nipkow@24526
  1817
lemma set_drop_subset: "set(drop n xs) \<subseteq> set xs"
nipkow@24526
  1818
by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1819
nipkow@35248
  1820
lemma set_drop_subset_set_drop:
nipkow@35248
  1821
  "m >= n \<Longrightarrow> set(drop m xs) <= set(drop n xs)"
nipkow@35248
  1822
apply(induct xs arbitrary: m n)
nipkow@35248
  1823
apply(auto simp:drop_Cons split:nat.split)
nipkow@35248
  1824
apply (metis set_drop_subset subset_iff)
nipkow@35248
  1825
done
nipkow@35248
  1826
nipkow@14187
  1827
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1828
using set_take_subset by fast
nipkow@14187
  1829
nipkow@14187
  1830
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1831
using set_drop_subset by fast
nipkow@14187
  1832
wenzelm@13114
  1833
lemma append_eq_conv_conj:
nipkow@24526
  1834
  "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
nipkow@24526
  1835
apply (induct xs arbitrary: zs, simp, clarsimp)
paulson@14208
  1836
apply (case_tac zs, auto)
nipkow@13145
  1837
done
wenzelm@13142
  1838
nipkow@24526
  1839
lemma take_add: 
noschinl@42713
  1840
  "take (i+j) xs = take i xs @ take j (drop i xs)"
nipkow@24526
  1841
apply (induct xs arbitrary: i, auto) 
nipkow@24526
  1842
apply (case_tac i, simp_all)
paulson@14050
  1843
done
paulson@14050
  1844
nipkow@14300
  1845
lemma append_eq_append_conv_if:
nipkow@24526
  1846
 "(xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1847
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1848
   then xs\<^isub>1 = take (size xs\<^isub>1) ys\<^isub>1 \<and> xs\<^isub>2 = drop (size xs\<^isub>1) ys\<^isub>1 @ ys\<^isub>2
nipkow@14300
  1849
   else take (size ys\<^isub>1) xs\<^isub>1 = ys\<^isub>1 \<and> drop (size ys\<^isub>1) xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>2)"
nipkow@24526
  1850
apply(induct xs\<^isub>1 arbitrary: ys\<^isub>1)
nipkow@14300
  1851
 apply simp
nipkow@14300
  1852
apply(case_tac ys\<^isub>1)
nipkow@14300
  1853
apply simp_all
nipkow@14300
  1854
done
nipkow@14300
  1855
nipkow@15110
  1856
lemma take_hd_drop:
huffman@30079
  1857
  "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (Suc n) xs"
nipkow@24526
  1858
apply(induct xs arbitrary: n)
nipkow@15110
  1859
apply simp
nipkow@15110
  1860
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1861
done
nipkow@15110
  1862
nipkow@17501
  1863
lemma id_take_nth_drop:
nipkow@17501
  1864
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
nipkow@17501
  1865
proof -
nipkow@17501
  1866
  assume si: "i < length xs"
nipkow@17501
  1867
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
nipkow@17501
  1868
  moreover
nipkow@17501
  1869
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
nipkow@17501
  1870
    apply (rule_tac take_Suc_conv_app_nth) by arith
nipkow@17501
  1871
  ultimately show ?thesis by auto
nipkow@17501
  1872
qed
nipkow@17501
  1873
  
nipkow@17501
  1874
lemma upd_conv_take_nth_drop:
nipkow@17501
  1875
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1876
proof -
nipkow@17501
  1877
  assume i: "i < length xs"
nipkow@17501
  1878
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
nipkow@17501
  1879
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
nipkow@17501
  1880
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1881
    using i by (simp add: list_update_append)
nipkow@17501
  1882
  finally show ?thesis .
nipkow@17501
  1883
qed
nipkow@17501
  1884
haftmann@24796
  1885
lemma nth_drop':
haftmann@24796
  1886
  "i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs"
haftmann@24796
  1887
apply (induct i arbitrary: xs)
haftmann@24796
  1888
apply (simp add: neq_Nil_conv)
haftmann@24796
  1889
apply (erule exE)+
haftmann@24796
  1890
apply simp
haftmann@24796
  1891
apply (case_tac xs)
haftmann@24796
  1892
apply simp_all
haftmann@24796
  1893
done
haftmann@24796
  1894
wenzelm@13114
  1895
nipkow@15392
  1896
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1897
hoelzl@33639
  1898
lemma length_takeWhile_le: "length (takeWhile P xs) \<le> length xs"
hoelzl@33639
  1899
  by (induct xs) auto
hoelzl@33639
  1900
wenzelm@13142
  1901
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1902
by (induct xs) auto
wenzelm@13114
  1903
wenzelm@13142
  1904
lemma takeWhile_append1 [simp]:
nipkow@13145
  1905
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1906
by (induct xs) auto
wenzelm@13114
  1907
wenzelm@13142
  1908
lemma takeWhile_append2 [simp]:
nipkow@13145
  1909
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1910
by (induct xs) auto
wenzelm@13114
  1911
wenzelm@13142
  1912
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1913
by (induct xs) auto
wenzelm@13114
  1914
hoelzl@33639
  1915
lemma takeWhile_nth: "j < length (takeWhile P xs) \<Longrightarrow> takeWhile P xs ! j = xs ! j"
hoelzl@33639
  1916
apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
hoelzl@33639
  1917
hoelzl@33639
  1918
lemma dropWhile_nth: "j < length (dropWhile P xs) \<Longrightarrow> dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))"
hoelzl@33639
  1919
apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
hoelzl@33639
  1920
hoelzl@33639
  1921
lemma length_dropWhile_le: "length (dropWhile P xs) \<le> length xs"
hoelzl@33639
  1922
by (induct xs) auto
hoelzl@33639
  1923
wenzelm@13142
  1924
lemma dropWhile_append1 [simp]:
nipkow@13145
  1925
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1926
by (induct xs) auto
wenzelm@13114
  1927
wenzelm@13142
  1928
lemma dropWhile_append2 [simp]:
nipkow@13145
  1929
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1930
by (induct xs) auto
wenzelm@13114
  1931
noschinl@45841
  1932
lemma dropWhile_append3:
noschinl@45841
  1933
  "\<not> P y \<Longrightarrow>dropWhile P (xs @ y # ys) = dropWhile P xs @ y # ys"
noschinl@45841
  1934
by (induct xs) auto
noschinl@45841
  1935
noschinl@45841
  1936
lemma dropWhile_last:
noschinl@45841
  1937
  "x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> last (dropWhile P xs) = last xs"
noschinl@45841
  1938
by (auto simp add: dropWhile_append3 in_set_conv_decomp)
noschinl@45841
  1939
noschinl@45841
  1940
lemma set_dropWhileD: "x \<in> set (dropWhile P xs) \<Longrightarrow> x \<in> set xs"
noschinl@45841
  1941
by (induct xs) (auto split: split_if_asm)
noschinl@45841
  1942
krauss@23971
  1943
lemma set_takeWhileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1944
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1945
nipkow@13913
  1946
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1947
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1948
by(induct xs, auto)
nipkow@13913
  1949
nipkow@13913
  1950
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1951
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1952
by(induct xs, auto)
nipkow@13913
  1953
nipkow@13913
  1954
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1955
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1956
by(induct xs, auto)
nipkow@13913
  1957
nipkow@31077
  1958
lemma distinct_takeWhile[simp]: "distinct xs ==> distinct (takeWhile P xs)"
nipkow@31077
  1959
by (induct xs) (auto dest: set_takeWhileD)
nipkow@31077
  1960
nipkow@31077
  1961
lemma distinct_dropWhile[simp]: "distinct xs ==> distinct (dropWhile P xs)"
nipkow@31077
  1962
by (induct xs) auto
nipkow@31077
  1963
hoelzl@33639
  1964
lemma takeWhile_map: "takeWhile P (map f xs) = map f (takeWhile (P \<circ> f) xs)"
hoelzl@33639
  1965
by (induct xs) auto
hoelzl@33639
  1966
hoelzl@33639
  1967
lemma dropWhile_map: "dropWhile P (map f xs) = map f (dropWhile (P \<circ> f) xs)"
hoelzl@33639
  1968
by (induct xs) auto
hoelzl@33639
  1969
hoelzl@33639
  1970
lemma takeWhile_eq_take: "takeWhile P xs = take (length (takeWhile P xs)) xs"
hoelzl@33639
  1971
by (induct xs) auto
hoelzl@33639
  1972
hoelzl@33639
  1973
lemma dropWhile_eq_drop: "dropWhile P xs = drop (length (takeWhile P xs)) xs"
hoelzl@33639
  1974
by (induct xs) auto
hoelzl@33639
  1975
hoelzl@33639
  1976
lemma hd_dropWhile:
hoelzl@33639
  1977
  "dropWhile P xs \<noteq> [] \<Longrightarrow> \<not> P (hd (dropWhile P xs))"
hoelzl@33639
  1978
using assms by (induct xs) auto
hoelzl@33639
  1979
hoelzl@33639
  1980
lemma takeWhile_eq_filter:
hoelzl@33639
  1981
  assumes "\<And> x. x \<in> set (dropWhile P xs) \<Longrightarrow> \<not> P x"
hoelzl@33639
  1982
  shows "takeWhile P xs = filter P xs"
hoelzl@33639
  1983
proof -
hoelzl@33639
  1984
  have A: "filter P xs = filter P (takeWhile P xs @ dropWhile P xs)"
hoelzl@33639
  1985
    by simp
hoelzl@33639
  1986
  have B: "filter P (dropWhile P xs) = []"
hoelzl@33639
  1987
    unfolding filter_empty_conv using assms by blast
hoelzl@33639
  1988
  have "filter P xs = takeWhile P xs"
hoelzl@33639
  1989
    unfolding A filter_append B
hoelzl@33639
  1990
    by (auto simp add: filter_id_conv dest: set_takeWhileD)
hoelzl@33639
  1991
  thus ?thesis ..
hoelzl@33639
  1992
qed
hoelzl@33639
  1993
hoelzl@33639
  1994
lemma takeWhile_eq_take_P_nth:
hoelzl@33639
  1995
  "\<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
  1996
  takeWhile P xs = take n xs"
hoelzl@33639
  1997
proof (induct xs arbitrary: n)
hoelzl@33639
  1998
  case (Cons x xs)
hoelzl@33639
  1999
  thus ?case
hoelzl@33639
  2000
  proof (cases n)
hoelzl@33639
  2001
    case (Suc n') note this[simp]
hoelzl@33639
  2002
    have "P x" using Cons.prems(1)[of 0] by simp
hoelzl@33639
  2003
    moreover have "takeWhile P xs = take n' xs"
hoelzl@33639
  2004
    proof (rule Cons.hyps)
hoelzl@33639
  2005
      case goal1 thus "P (xs ! i)" using Cons.prems(1)[of "Suc i"] by simp
hoelzl@33639
  2006
    next case goal2 thus ?case using Cons by auto
hoelzl@33639
  2007
    qed
hoelzl@33639
  2008
    ultimately show ?thesis by simp
hoelzl@33639
  2009
   qed simp
hoelzl@33639
  2010
qed simp
hoelzl@33639
  2011
hoelzl@33639
  2012
lemma nth_length_takeWhile:
hoelzl@33639
  2013
  "length (takeWhile P xs) < length xs \<Longrightarrow> \<not> P (xs ! length (takeWhile P xs))"
hoelzl@33639
  2014
by (induct xs) auto
hoelzl@33639
  2015
hoelzl@33639
  2016
lemma length_takeWhile_less_P_nth:
hoelzl@33639
  2017
  assumes all: "\<And> i. i < j \<Longrightarrow> P (xs ! i)" and "j \<le> length xs"
hoelzl@33639
  2018
  shows "j \<le> length (takeWhile P xs)"
hoelzl@33639
  2019
proof (rule classical)
hoelzl@33639
  2020
  assume "\<not> ?thesis"
hoelzl@33639
  2021
  hence "length (takeWhile P xs) < length xs" using assms by simp
hoelzl@33639
  2022
  thus ?thesis using all `\<not> ?thesis` nth_length_takeWhile[of P xs] by auto
hoelzl@33639
  2023
qed
nipkow@31077
  2024
nipkow@17501
  2025
text{* The following two lemmmas could be generalized to an arbitrary
nipkow@17501
  2026
property. *}
nipkow@17501
  2027
nipkow@17501
  2028
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  2029
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
nipkow@17501
  2030
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
nipkow@17501
  2031
nipkow@17501
  2032
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  2033
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
nipkow@17501
  2034
apply(induct xs)
nipkow@17501
  2035
 apply simp
nipkow@17501
  2036
apply auto
nipkow@17501
  2037
apply(subst dropWhile_append2)
nipkow@17501
  2038
apply auto
nipkow@17501
  2039
done
nipkow@17501
  2040
nipkow@18423
  2041
lemma takeWhile_not_last:
nipkow@18423
  2042
 "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
nipkow@18423
  2043
apply(induct xs)
nipkow@18423
  2044
 apply simp
nipkow@18423
  2045
apply(case_tac xs)
nipkow@18423
  2046
apply(auto)
nipkow@18423
  2047
done
nipkow@18423
  2048
krauss@44013
  2049
lemma takeWhile_cong [fundef_cong]:
krauss@18336
  2050
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  2051
  ==> takeWhile P l = takeWhile Q k"
nipkow@24349
  2052
by (induct k arbitrary: l) (simp_all)
krauss@18336
  2053
krauss@44013
  2054
lemma dropWhile_cong [fundef_cong]:
krauss@18336
  2055
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  2056
  ==> dropWhile P l = dropWhile Q k"
nipkow@24349
  2057
by (induct k arbitrary: l, simp_all)
krauss@18336
  2058
wenzelm@13114
  2059
nipkow@15392
  2060
subsubsection {* @{text zip} *}
wenzelm@13114
  2061
wenzelm@13142
  2062
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  2063
by (induct ys) auto
wenzelm@13114
  2064
wenzelm@13142
  2065
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  2066
by simp
wenzelm@13114
  2067
wenzelm@13142
  2068
declare zip_Cons [simp del]
wenzelm@13114
  2069
haftmann@36198
  2070
lemma [code]:
haftmann@36198
  2071
  "zip [] ys = []"
haftmann@36198
  2072
  "zip xs [] = []"
haftmann@36198
  2073
  "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
haftmann@36198
  2074
  by (fact zip_Nil zip.simps(1) zip_Cons_Cons)+
haftmann@36198
  2075
nipkow@15281
  2076
lemma zip_Cons1:
nipkow@15281
  2077
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  2078
by(auto split:list.split)
nipkow@15281
  2079
wenzelm@13142
  2080
lemma length_zip [simp]:
krauss@22493
  2081
"length (zip xs ys) = min (length xs) (length ys)"
krauss@22493
  2082
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  2083
haftmann@34978
  2084
lemma zip_obtain_same_length:
haftmann@34978
  2085
  assumes "\<And>zs ws n. length zs = length ws \<Longrightarrow> n = min (length xs) (length ys)
haftmann@34978
  2086
    \<Longrightarrow> zs = take n xs \<Longrightarrow> ws = take n ys \<Longrightarrow> P (zip zs ws)"
haftmann@34978
  2087
  shows "P (zip xs ys)"
haftmann@34978
  2088
proof -
haftmann@34978
  2089
  let ?n = "min (length xs) (length ys)"
haftmann@34978
  2090
  have "P (zip (take ?n xs) (take ?n ys))"
haftmann@34978
  2091
    by (rule assms) simp_all
haftmann@34978
  2092
  moreover have "zip xs ys = zip (take ?n xs) (take ?n ys)"
haftmann@34978
  2093
  proof (induct xs arbitrary: ys)
haftmann@34978
  2094
    case Nil then show ?case by simp
haftmann@34978
  2095
  next
haftmann@34978
  2096
    case (Cons x xs) then show ?case by (cases ys) simp_all
haftmann@34978
  2097
  qed
haftmann@34978
  2098
  ultimately show ?thesis by simp
haftmann@34978
  2099
qed
haftmann@34978
  2100
wenzelm@13114
  2101
lemma zip_append1:
krauss@22493
  2102
"zip (xs @ ys) zs =
nipkow@13145
  2103
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
krauss@22493
  2104
by (induct xs zs rule:list_induct2') auto
wenzelm@13114
  2105
wenzelm@13114
  2106
lemma zip_append2:
krauss@22493
  2107
"zip xs (ys @ zs) =
nipkow@13145
  2108
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
krauss@22493
  2109
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  2110
wenzelm@13142
  2111
lemma zip_append [simp]:
wenzelm@13142
  2112
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  2113
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  2114
by (simp add: zip_append1)
wenzelm@13114
  2115
wenzelm@13114
  2116
lemma zip_rev:
nipkow@14247
  2117
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  2118
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  2119
hoelzl@33639
  2120
lemma zip_map_map:
hoelzl@33639
  2121
  "zip (map f xs) (map g ys) = map (\<lambda> (x, y). (f x, g y)) (zip xs ys)"
hoelzl@33639
  2122
proof (induct xs arbitrary: ys)
hoelzl@33639
  2123
  case (Cons x xs) note Cons_x_xs = Cons.hyps
hoelzl@33639
  2124
  show ?case
hoelzl@33639
  2125
  proof (cases ys)
hoelzl@33639
  2126
    case (Cons y ys')
hoelzl@33639
  2127
    show ?thesis unfolding Cons using Cons_x_xs by simp
hoelzl@33639
  2128
  qed simp
hoelzl@33639
  2129
qed simp
hoelzl@33639
  2130
hoelzl@33639
  2131
lemma zip_map1:
hoelzl@33639
  2132
  "zip (map f xs) ys = map (\<lambda>(x, y). (f x, y)) (zip xs ys)"
hoelzl@33639
  2133
using zip_map_map[of f xs "\<lambda>x. x" ys] by simp
hoelzl@33639
  2134
hoelzl@33639
  2135
lemma zip_map2:
hoelzl@33639
  2136
  "zip xs (map f ys) = map (\<lambda>(x, y). (x, f y)) (zip xs ys)"
hoelzl@33639
  2137
using zip_map_map[of "\<lambda>x. x" xs f ys] by simp
hoelzl@33639
  2138
nipkow@23096
  2139
lemma map_zip_map:
hoelzl@33639
  2140
  "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
hoelzl@33639
  2141
unfolding zip_map1 by auto
nipkow@23096
  2142
nipkow@23096
  2143
lemma map_zip_map2:
hoelzl@33639
  2144
  "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
hoelzl@33639
  2145
unfolding zip_map2 by auto
nipkow@23096
  2146
nipkow@31080
  2147
text{* Courtesy of Andreas Lochbihler: *}
nipkow@31080
  2148
lemma zip_same_conv_map: "zip xs xs = map (\<lambda>x. (x, x)) xs"
nipkow@31080
  2149
by(induct xs) auto
nipkow@31080
  2150
wenzelm@13142
  2151
lemma nth_zip [simp]:
nipkow@24526
  2152
"[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
nipkow@24526
  2153
apply (induct ys arbitrary: i xs, simp)
nipkow@13145
  2154
apply (case_tac xs)
nipkow@13145
  2155
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  2156
done
wenzelm@13114
  2157
wenzelm@13114
  2158
lemma set_zip:
nipkow@13145
  2159
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@31080
  2160
by(simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  2161
hoelzl@33639
  2162
lemma zip_same: "((a,b) \<in> set (zip xs xs)) = (a \<in> set xs \<and> a = b)"
hoelzl@33639
  2163
by(induct xs) auto
hoelzl@33639
  2164
wenzelm@13114
  2165
lemma zip_update:
nipkow@31080
  2166
  "zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@31080
  2167
by(rule sym, simp add: update_zip)
wenzelm@13114
  2168
wenzelm@13142
  2169
lemma zip_replicate [simp]:
nipkow@24526
  2170
  "zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
nipkow@24526
  2171
apply (induct i arbitrary: j, auto)
paulson@14208
  2172
apply (case_tac j, auto)
nipkow@13145
  2173
done
wenzelm@13114
  2174
nipkow@19487
  2175
lemma take_zip:
nipkow@24526
  2176
  "take n (zip xs ys) = zip (take n xs) (take n ys)"
nipkow@24526
  2177
apply (induct n arbitrary: xs ys)
nipkow@19487
  2178
 apply simp
nipkow@19487
  2179
apply (case_tac xs, simp)
nipkow@19487
  2180
apply (case_tac ys, simp_all)
nipkow@19487
  2181
done
nipkow@19487
  2182
nipkow@19487
  2183
lemma drop_zip:
nipkow@24526
  2184
  "drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
nipkow@24526
  2185
apply (induct n arbitrary: xs ys)
nipkow@19487
  2186
 apply simp
nipkow@19487
  2187
apply (case_tac xs, simp)
nipkow@19487
  2188
apply (case_tac ys, simp_all)
nipkow@19487
  2189
done
nipkow@19487
  2190
hoelzl@33639
  2191
lemma zip_takeWhile_fst: "zip (takeWhile P xs) ys = takeWhile (P \<circ> fst) (zip xs ys)"
hoelzl@33639
  2192
proof (induct xs arbitrary: ys)
hoelzl@33639
  2193
  case (Cons x xs) thus ?case by (cases ys) auto
hoelzl@33639
  2194
qed simp
hoelzl@33639
  2195
hoelzl@33639
  2196
lemma zip_takeWhile_snd: "zip xs (takeWhile P ys) = takeWhile (P \<circ> snd) (zip xs ys)"
hoelzl@33639
  2197
proof (induct xs arbitrary: ys)
hoelzl@33639
  2198
  case (Cons x xs) thus ?case by (cases ys) auto
hoelzl@33639
  2199
qed simp
hoelzl@33639
  2200
krauss@22493
  2201
lemma set_zip_leftD:
krauss@22493
  2202
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
krauss@22493
  2203
by (induct xs ys rule:list_induct2') auto
krauss@22493
  2204
krauss@22493
  2205
lemma set_zip_rightD:
krauss@22493
  2206
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
krauss@22493
  2207
by (induct xs ys rule:list_induct2') auto
wenzelm@13142
  2208
nipkow@23983
  2209
lemma in_set_zipE:
nipkow@23983
  2210
  "(x,y) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
nipkow@23983
  2211
by(blast dest: set_zip_leftD set_zip_rightD)
nipkow@23983
  2212
haftmann@29829
  2213
lemma zip_map_fst_snd:
haftmann@29829
  2214
  "zip (map fst zs) (map snd zs) = zs"
haftmann@29829
  2215
  by (induct zs) simp_all
haftmann@29829
  2216
haftmann@29829
  2217
lemma zip_eq_conv:
haftmann@29829
  2218
  "length xs = length ys \<Longrightarrow> zip xs ys = zs \<longleftrightarrow> map fst zs = xs \<and> map snd zs = ys"
haftmann@29829
  2219
  by (auto simp add: zip_map_fst_snd)
haftmann@29829
  2220
wenzelm@35115
  2221
nipkow@15392
  2222
subsubsection {* @{text list_all2} *}
wenzelm@13114
  2223
kleing@14316
  2224
lemma list_all2_lengthD [intro?]: 
kleing@14316
  2225
  "list_all2 P xs ys ==> length xs = length ys"
nipkow@24349
  2226
by (simp add: list_all2_def)
haftmann@19607
  2227
haftmann@19787
  2228
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
nipkow@24349
  2229
by (simp add: list_all2_def)
haftmann@19607
  2230
haftmann@19787
  2231
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
nipkow@24349
  2232
by (simp add: list_all2_def)
haftmann@19607
  2233
haftmann@19607
  2234
lemma list_all2_Cons [iff, code]:
haftmann@19607
  2235
  "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@24349
  2236
by (auto simp add: list_all2_def)
wenzelm@13114
  2237
wenzelm@13114
  2238
lemma list_all2_Cons1:
nipkow@13145
  2239
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  2240
by (cases ys) auto
wenzelm@13114
  2241
wenzelm@13114
  2242
lemma list_all2_Cons2:
nipkow@13145
  2243
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  2244
by (cases xs) auto
wenzelm@13114
  2245
huffman@45794
  2246
lemma list_all2_induct
huffman@45794
  2247
  [consumes 1, case_names Nil Cons, induct set: list_all2]:
huffman@45794
  2248
  assumes P: "list_all2 P xs ys"
huffman@45794
  2249
  assumes Nil: "R [] []"
huffman@45794
  2250
  assumes Cons: "\<And>x xs y ys. \<lbrakk>P x y; R xs ys\<rbrakk> \<Longrightarrow> R (x # xs) (y # ys)"
huffman@45794
  2251
  shows "R xs ys"
huffman@45794
  2252
using P
huffman@45794
  2253
by (induct xs arbitrary: ys) (auto simp add: list_all2_Cons1 Nil Cons)
huffman@45794
  2254
wenzelm@13142
  2255
lemma list_all2_rev [iff]:
nipkow@13145
  2256
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  2257
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  2258
kleing@13863
  2259
lemma list_all2_rev1:
kleing@13863
  2260
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  2261
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  2262
wenzelm@13114
  2263
lemma list_all2_append1:
nipkow@13145
  2264
"list_all2 P (xs @ ys) zs =
nipkow@13145
  2265
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  2266
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  2267
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  2268
apply (rule iffI)
nipkow@13145
  2269
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  2270
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  2271
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  2272
apply (simp add: ball_Un)
nipkow@13145
  2273
done
wenzelm@13114
  2274
wenzelm@13114
  2275
lemma list_all2_append2:
nipkow@13145
  2276
"list_all2 P xs (ys @ zs) =
nipkow@13145
  2277
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  2278
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  2279
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  2280
apply (rule iffI)
nipkow@13145
  2281
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  2282
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  2283
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  2284
apply (simp add: ball_Un)
nipkow@13145
  2285
done
wenzelm@13114
  2286
kleing@13863
  2287
lemma list_all2_append:
nipkow@14247
  2288
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  2289
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  2290
by (induct rule:list_induct2, simp_all)
kleing@13863
  2291
kleing@13863
  2292
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  2293
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
nipkow@24349
  2294
by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  2295
wenzelm@13114
  2296
lemma list_all2_conv_all_nth:
nipkow@13145
  2297
"list_all2 P xs ys =
nipkow@13145
  2298
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  2299
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  2300
berghofe@13883
  2301
lemma list_all2_trans:
berghofe@13883
  2302
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  2303
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  2304
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  2305
proof (induct as)
berghofe@13883
  2306
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  2307
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  2308
  proof (induct bs)
berghofe@13883
  2309
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  2310
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  2311
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  2312
  qed simp
berghofe@13883
  2313
qed simp
berghofe@13883
  2314
kleing@13863
  2315
lemma list_all2_all_nthI [intro?]:
kleing@13863
  2316
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
nipkow@24349
  2317
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2318
paulson@14395
  2319
lemma list_all2I:
paulson@14395
  2320
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
nipkow@24349
  2321
by (simp add: list_all2_def)
paulson@14395
  2322
kleing@14328
  2323
lemma list_all2_nthD:
kleing@13863
  2324
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@24349
  2325
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2326
nipkow@14302
  2327
lemma list_all2_nthD2:
nipkow@14302
  2328
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@24349
  2329
by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  2330
kleing@13863
  2331
lemma list_all2_map1: 
kleing@13863
  2332
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
nipkow@24349
  2333
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2334
kleing@13863
  2335
lemma list_all2_map2: 
kleing@13863
  2336
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
nipkow@24349
  2337
by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  2338
kleing@14316
  2339
lemma list_all2_refl [intro?]:
kleing@13863
  2340
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
nipkow@24349
  2341
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2342
kleing@13863
  2343
lemma list_all2_update_cong:
kleing@13863
  2344
  "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
nipkow@24349
  2345
by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  2346
kleing@13863
  2347
lemma list_all2_update_cong2:
kleing@13863
  2348
  "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
nipkow@24349
  2349
by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  2350
nipkow@14302
  2351
lemma list_all2_takeI [simp,intro?]:
nipkow@24526
  2352
  "list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@24526
  2353
apply (induct xs arbitrary: n ys)
nipkow@24526
  2354
 apply simp
nipkow@24526
  2355
apply (clarsimp simp add: list_all2_Cons1)
nipkow@24526
  2356
apply (case_tac n)
nipkow@24526
  2357
apply auto
nipkow@24526
  2358
done
nipkow@14302
  2359
nipkow@14302
  2360
lemma list_all2_dropI [simp,intro?]:
nipkow@24526
  2361
  "list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
nipkow@24526
  2362
apply (induct as arbitrary: n bs, simp)
nipkow@24526
  2363
apply (clarsimp simp add: list_all2_Cons1)
nipkow@24526
  2364
apply (case_tac n, simp, simp)
nipkow@24526
  2365
done
kleing@13863
  2366
kleing@14327
  2367
lemma list_all2_mono [intro?]:
nipkow@24526
  2368
  "list_all2 P xs ys \<Longrightarrow> (\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys) \<Longrightarrow> list_all2 Q xs ys"
nipkow@24526
  2369
apply (induct xs arbitrary: ys, simp)
nipkow@24526
  2370
apply (case_tac ys, auto)
nipkow@24526
  2371
done
kleing@13863
  2372
haftmann@22551
  2373
lemma list_all2_eq:
haftmann@22551
  2374
  "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
nipkow@24349
  2375
by (induct xs ys rule: list_induct2') auto
haftmann@22551
  2376
nipkow@40230
  2377
lemma list_eq_iff_zip_eq:
nipkow@40230
  2378
  "xs = ys \<longleftrightarrow> length xs = length ys \<and> (\<forall>(x,y) \<in> set (zip xs ys). x = y)"
nipkow@40230
  2379
by(auto simp add: set_zip list_all2_eq list_all2_conv_all_nth cong: conj_cong)
nipkow@40230
  2380
wenzelm@13142
  2381
haftmann@46133
  2382
subsubsection {* @{const fold} with canonical argument order *}
haftmann@46133
  2383
haftmann@46133
  2384
lemma fold_remove1_split:
haftmann@46133
  2385
  assumes f: "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
haftmann@46133
  2386
    and x: "x \<in> set xs"
haftmann@46133
  2387
  shows "fold f xs = fold f (remove1 x xs) \<circ> f x"
haftmann@46133
  2388
  using assms by (induct xs) (auto simp add: o_assoc [symmetric])
haftmann@46133
  2389
haftmann@46133
  2390
lemma fold_cong [fundef_cong]:
haftmann@46133
  2391
  "a = b \<Longrightarrow> xs = ys \<Longrightarrow> (\<And>x. x \<in> set xs \<Longrightarrow> f x = g x)
haftmann@46133
  2392
    \<Longrightarrow> fold f xs a = fold g ys b"
haftmann@46133
  2393
  by (induct ys arbitrary: a b xs) simp_all
haftmann@46133
  2394
haftmann@46133
  2395
lemma fold_id:
haftmann@46133
  2396
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x = id"
haftmann@46133
  2397
  shows "fold f xs = id"
haftmann@46133
  2398
  using assms by (induct xs) simp_all
haftmann@46133
  2399
haftmann@46133
  2400
lemma fold_commute:
haftmann@46133
  2401
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
haftmann@46133
  2402
  shows "h \<circ> fold g xs = fold f xs \<circ> h"
haftmann@46133
  2403
  using assms by (induct xs) (simp_all add: fun_eq_iff)
haftmann@46133
  2404
haftmann@46133
  2405
lemma fold_commute_apply:
haftmann@46133
  2406
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
haftmann@46133
  2407
  shows "h (fold g xs s) = fold f xs (h s)"
haftmann@46133
  2408
proof -
haftmann@46133
  2409
  from assms have "h \<circ> fold g xs = fold f xs \<circ> h" by (rule fold_commute)
haftmann@46133
  2410
  then show ?thesis by (simp add: fun_eq_iff)
haftmann@37605
  2411
qed
haftmann@37605
  2412
haftmann@46133
  2413
lemma fold_invariant: 
haftmann@46133
  2414
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> Q x" and "P s"
haftmann@46133
  2415
    and "\<And>x s. Q x \<Longrightarrow> P s \<Longrightarrow> P (f x s)"
haftmann@46133
  2416
  shows "P (fold f xs s)"
haftmann@34978
  2417
  using assms by (induct xs arbitrary: s) simp_all
haftmann@34978
  2418
haftmann@46133
  2419
lemma fold_append [simp]:
haftmann@46133
  2420
  "fold f (xs @ ys) = fold f ys \<circ> fold f xs"
haftmann@46133
  2421
  by (induct xs) simp_all
haftmann@46133
  2422
haftmann@46133
  2423
lemma fold_map [code_unfold]:
haftmann@46133
  2424
  "fold g (map f xs) = fold (g o f) xs"
haftmann@46133
  2425
  by (induct xs) simp_all
haftmann@46133
  2426
haftmann@46133
  2427
lemma fold_rev:
haftmann@46133
  2428
  assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
haftmann@46133
  2429
  shows "fold f (rev xs) = fold f xs"
haftmann@46133
  2430
using assms by (induct xs) (simp_all add: fold_commute_apply fun_eq_iff)
haftmann@46133
  2431
haftmann@46133
  2432
lemma fold_Cons_rev:
haftmann@46133
  2433
  "fold Cons xs = append (rev xs)"
haftmann@46133
  2434
  by (induct xs) simp_all
haftmann@46133
  2435
haftmann@46133
  2436
lemma rev_conv_fold [code]:
haftmann@46133
  2437
  "rev xs = fold Cons xs []"
haftmann@46133
  2438
  by (simp add: fold_Cons_rev)
haftmann@46133
  2439
haftmann@46133
  2440
lemma fold_append_concat_rev:
haftmann@46133
  2441
  "fold append xss = append (concat (rev xss))"
haftmann@46133
  2442
  by (induct xss) simp_all
haftmann@46133
  2443
haftmann@46133
  2444
text {* @{const Finite_Set.fold} and @{const fold} *}
haftmann@46133
  2445
haftmann@46133
  2446
lemma (in comp_fun_commute) fold_set_fold_remdups:
haftmann@46133
  2447
  "Finite_Set.fold f y (set xs) = fold f (remdups xs) y"
haftmann@35195
  2448
  by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm insert_absorb)
haftmann@35195
  2449
haftmann@46133
  2450
lemma (in comp_fun_idem) fold_set_fold:
haftmann@46133
  2451
  "Finite_Set.fold f y (set xs) = fold f xs y"
haftmann@31455
  2452
  by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm)
haftmann@31455
  2453
haftmann@46133
  2454
lemma (in ab_semigroup_idem_mult) fold1_set_fold:
haftmann@32681
  2455
  assumes "xs \<noteq> []"
haftmann@46133
  2456
  shows "Finite_Set.fold1 times (set xs) = fold times (tl xs) (hd xs)"
haftmann@32681
  2457
proof -
haftmann@42871
  2458
  interpret comp_fun_idem times by (fact comp_fun_idem)
haftmann@32681
  2459
  from assms obtain y ys where xs: "xs = y # ys"
haftmann@32681
  2460
    by (cases xs) auto
haftmann@32681
  2461
  show ?thesis
haftmann@32681
  2462
  proof (cases "set ys = {}")
haftmann@32681
  2463
    case True with xs show ?thesis by simp
haftmann@32681
  2464
  next
haftmann@32681
  2465
    case False
haftmann@46034
  2466
    then have "fold1 times (insert y (set ys)) = Finite_Set.fold times y (set ys)"
haftmann@32681
  2467
      by (simp only: finite_set fold1_eq_fold_idem)
haftmann@46133
  2468
    with xs show ?thesis by (simp add: fold_set_fold mult_commute)
haftmann@32681
  2469
  qed
haftmann@32681
  2470
qed
haftmann@32681
  2471
haftmann@46133
  2472
lemma (in lattice) Inf_fin_set_fold:
haftmann@46133
  2473
  "Inf_fin (set (x # xs)) = fold inf xs x"
haftmann@46133
  2474
proof -
haftmann@46133
  2475
  interpret ab_semigroup_idem_mult "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@46133
  2476
    by (fact ab_semigroup_idem_mult_inf)
haftmann@46133
  2477
  show ?thesis
haftmann@46133
  2478
    by (simp add: Inf_fin_def fold1_set_fold del: set.simps)
haftmann@46133
  2479
qed
haftmann@46133
  2480
haftmann@46133
  2481
lemma (in lattice) Sup_fin_set_fold:
haftmann@46133
  2482
  "Sup_fin (set (x # xs)) = fold sup xs x"
haftmann@46133
  2483
proof -
haftmann@46133
  2484
  interpret ab_semigroup_idem_mult "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@46133
  2485
    by (fact ab_semigroup_idem_mult_sup)
haftmann@46133
  2486
  show ?thesis
haftmann@46133
  2487
    by (simp add: Sup_fin_def fold1_set_fold del: set.simps)
haftmann@46133
  2488
qed
haftmann@46133
  2489
haftmann@46133
  2490
lemma (in linorder) Min_fin_set_fold:
haftmann@46133
  2491
  "Min (set (x # xs)) = fold min xs x"
haftmann@46133
  2492
proof -
haftmann@46133
  2493
  interpret ab_semigroup_idem_mult "min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@46133
  2494
    by (fact ab_semigroup_idem_mult_min)
haftmann@46133
  2495
  show ?thesis
haftmann@46133
  2496
    by (simp add: Min_def fold1_set_fold del: set.simps)
haftmann@46133
  2497
qed
haftmann@46133
  2498
haftmann@46133
  2499
lemma (in linorder) Max_fin_set_fold:
haftmann@46133
  2500
  "Max (set (x # xs)) = fold max xs x"
haftmann@46133
  2501
proof -
haftmann@46133
  2502
  interpret ab_semigroup_idem_mult "max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@46133
  2503
    by (fact ab_semigroup_idem_mult_max)
haftmann@46133
  2504
  show ?thesis
haftmann@46133
  2505
    by (simp add: Max_def fold1_set_fold del: set.simps)
haftmann@46133
  2506
qed
haftmann@46133
  2507
haftmann@46133
  2508
lemma (in complete_lattice) Inf_set_fold:
haftmann@46133
  2509
  "Inf (set xs) = fold inf xs top"
haftmann@46133
  2510
proof -
haftmann@46133
  2511
  interpret comp_fun_idem "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@46133
  2512
    by (fact comp_fun_idem_inf)
haftmann@46133
  2513
  show ?thesis by (simp add: Inf_fold_inf fold_set_fold inf_commute)
haftmann@46133
  2514
qed
haftmann@46133
  2515
haftmann@46133
  2516
lemma (in complete_lattice) Sup_set_fold:
haftmann@46133
  2517
  "Sup (set xs) = fold sup xs bot"
haftmann@46133
  2518
proof -
haftmann@46133
  2519
  interpret comp_fun_idem "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@46133
  2520
    by (fact comp_fun_idem_sup)
haftmann@46133
  2521
  show ?thesis by (simp add: Sup_fold_sup fold_set_fold sup_commute)
haftmann@46133
  2522
qed
haftmann@46133
  2523
haftmann@46133
  2524
lemma (in complete_lattice) INF_set_fold:
haftmann@46133
  2525
  "INFI (set xs) f = fold (inf \<circ> f) xs top"
haftmann@46133
  2526
  unfolding INF_def set_map [symmetric] Inf_set_fold fold_map ..
haftmann@46133
  2527
haftmann@46133
  2528
lemma (in complete_lattice) SUP_set_fold:
haftmann@46133
  2529
  "SUPR (set xs) f = fold (sup \<circ> f) xs bot"
haftmann@46133
  2530
  unfolding SUP_def set_map [symmetric] Sup_set_fold fold_map ..
haftmann@46133
  2531
haftmann@46133
  2532
haftmann@46133
  2533
subsubsection {* Fold variants: @{const foldr} and @{const foldl} *}
haftmann@46133
  2534
haftmann@46133
  2535
text {* Correspondence *}
haftmann@46133
  2536
haftmann@46133
  2537
lemma foldr_foldl: -- {* The ``Third Duality Theorem'' in Bird \& Wadler: *}
haftmann@46133
  2538
  "foldr f xs a = foldl (\<lambda>x y. f y x) a (rev xs)"
haftmann@46133
  2539
  by (simp add: foldr_def foldl_def)
haftmann@46133
  2540
haftmann@46133
  2541
lemma foldl_foldr:
haftmann@46133
  2542
  "foldl f a xs = foldr (\<lambda>x y. f y x) (rev xs) a"
haftmann@46133
  2543
  by (simp add: foldr_def foldl_def)
haftmann@46133
  2544
haftmann@46133
  2545
lemma foldr_fold:
haftmann@46133
  2546
  assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
haftmann@46133
  2547
  shows "foldr f xs = fold f xs"
haftmann@46133
  2548
  using assms unfolding foldr_def by (rule fold_rev)
haftmann@46133
  2549
haftmann@46133
  2550
lemma
haftmann@46133
  2551
  foldr_Nil [code, simp]: "foldr f [] = id"
haftmann@46133
  2552
  and foldr_Cons [code, simp]: "foldr f (x # xs) = f x \<circ> foldr f xs"
haftmann@46133
  2553
  by (simp_all add: foldr_def)
haftmann@46133
  2554
haftmann@46133
  2555
lemma
haftmann@46133
  2556
  foldl_Nil [simp]: "foldl f a [] = a"
haftmann@46133
  2557
  and foldl_Cons [simp]: "foldl f a (x # xs) = foldl f (f a x) xs"
haftmann@46133
  2558
  by (simp_all add: foldl_def)
haftmann@46133
  2559
haftmann@46133
  2560
lemma foldr_cong [fundef_cong]:
haftmann@46133
  2561
  "a = b \<Longrightarrow> l = k \<Longrightarrow> (\<And>a x. x \<in> set l \<Longrightarrow> f x a = g x a) \<Longrightarrow> foldr f l a = foldr g k b"
haftmann@46133
  2562
  by (auto simp add: foldr_def intro!: fold_cong)
haftmann@46133
  2563
haftmann@46133
  2564
lemma foldl_cong [fundef_cong]:
haftmann@46133
  2565
  "a = b \<Longrightarrow> l = k \<Longrightarrow> (\<And>a x. x \<in> set l \<Longrightarrow> f a x = g a x) \<Longrightarrow> foldl f a l = foldl g b k"
haftmann@46133
  2566
  by (auto simp add: foldl_def intro!: fold_cong)
haftmann@46133
  2567
haftmann@46133
  2568
lemma foldr_append [simp]:
haftmann@46133
  2569
  "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
haftmann@46133
  2570
  by (simp add: foldr_def)
haftmann@46133
  2571
haftmann@46133
  2572
lemma foldl_append [simp]:
haftmann@46133
  2573
  "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
haftmann@46133
  2574
  by (simp add: foldl_def)
haftmann@46133
  2575
haftmann@46133
  2576
lemma foldr_map [code_unfold]:
haftmann@46133
  2577
  "foldr g (map f xs) a = foldr (g o f) xs a"
haftmann@46133
  2578
  by (simp add: foldr_def fold_map rev_map)
haftmann@46133
  2579
haftmann@46133
  2580
lemma foldl_map [code_unfold]:
haftmann@46133
  2581
  "foldl g a (map f xs) = foldl (\<lambda>a x. g a (f x)) a xs"
haftmann@46133
  2582
  by (simp add: foldl_def fold_map comp_def)
haftmann@46133
  2583
haftmann@46133
  2584
text {* Executing operations in terms of @{const foldr} -- tail-recursive! *}
haftmann@46133
  2585
haftmann@46133
  2586
lemma concat_conv_foldr [code]:
haftmann@46133
  2587
  "concat xss = foldr append xss []"
haftmann@46133
  2588
  by (simp add: fold_append_concat_rev foldr_def)
haftmann@46133
  2589
haftmann@46133
  2590
lemma (in lattice) Inf_fin_set_foldr [code]:
haftmann@46133
  2591
  "Inf_fin (set (x # xs)) = foldr inf xs x"
haftmann@46133
  2592
  by (simp add: Inf_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
haftmann@46133
  2593
haftmann@46133
  2594
lemma (in lattice) Sup_fin_set_foldr [code]:
haftmann@46133
  2595
  "Sup_fin (set (x # xs)) = foldr sup xs x"
haftmann@46133
  2596
  by (simp add: Sup_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
haftmann@46133
  2597
haftmann@46133
  2598
lemma (in linorder) Min_fin_set_foldr [code]:
haftmann@46133
  2599
  "Min (set (x # xs)) = foldr min xs x"
haftmann@46133
  2600
  by (simp add: Min_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
haftmann@46133
  2601
haftmann@46133
  2602
lemma (in linorder) Max_fin_set_foldr [code]:
haftmann@46133
  2603
  "Max (set (x # xs)) = foldr max xs x"
haftmann@46133
  2604
  by (simp add: Max_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
haftmann@46133
  2605
haftmann@46133
  2606
lemma (in complete_lattice) Inf_set_foldr:
haftmann@46133
  2607
  "Inf (set xs) = foldr inf xs top"
haftmann@46133
  2608
  by (simp add: Inf_set_fold ac_simps foldr_fold fun_eq_iff)
haftmann@46133
  2609
haftmann@46133
  2610
lemma (in complete_lattice) Sup_set_foldr:
haftmann@46133
  2611
  "Sup (set xs) = foldr sup xs bot"
haftmann@46133
  2612
  by (simp add: Sup_set_fold ac_simps foldr_fold fun_eq_iff)
haftmann@46133
  2613
haftmann@46133
  2614
lemma (in complete_lattice) INF_set_foldr [code]:
haftmann@46133
  2615
  "INFI (set xs) f = foldr (inf \<circ> f) xs top"
haftmann@46133
  2616
  by (simp only: INF_def Inf_set_foldr foldr_map set_map [symmetric])
haftmann@46133
  2617
haftmann@46133
  2618
lemma (in complete_lattice) SUP_set_foldr [code]:
haftmann@46133
  2619
  "SUPR (set xs) f = foldr (sup \<circ> f) xs bot"
haftmann@46133
  2620
  by (simp only: SUP_def Sup_set_foldr foldr_map set_map [symmetric])
haftmann@46133
  2621
wenzelm@35115
  2622
nipkow@24645
  2623
subsubsection {* @{text upt} *}
wenzelm@13114
  2624