src/HOL/List.thy
author wenzelm
Sun Oct 21 14:21:45 2007 +0200 (2007-10-21)
changeset 25130 d91391a8705b
parent 25112 98824cc791c0
child 25134 3d4953e88449
permissions -rw-r--r--
avoid very slow metis invocation;
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(*  Title:      HOL/List.thy
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    ID:         $Id$
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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 PreList
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uses "Tools/string_syntax.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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subsection{*Basic list processing functions*}
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consts
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  filter:: "('a => bool) => 'a list => 'a list"
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  concat:: "'a list list => 'a list"
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  foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"
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  foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"
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  hd:: "'a list => 'a"
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  tl:: "'a list => 'a list"
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  last:: "'a list => 'a"
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  butlast :: "'a list => 'a list"
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  set :: "'a list => 'a set"
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  map :: "('a=>'b) => ('a list => 'b list)"
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  listsum ::  "'a list => 'a::monoid_add"
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  nth :: "'a list => nat => 'a"    (infixl "!" 100)
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  list_update :: "'a list => nat => 'a => 'a list"
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  take:: "nat => 'a list => 'a list"
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  drop:: "nat => 'a list => 'a list"
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  takeWhile :: "('a => bool) => 'a list => 'a list"
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  dropWhile :: "('a => bool) => 'a list => 'a list"
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  rev :: "'a list => 'a list"
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  zip :: "'a list => 'b list => ('a * 'b) list"
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  upt :: "nat => nat => nat list" ("(1[_..</_'])")
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  remdups :: "'a list => 'a list"
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  remove1 :: "'a => 'a list => 'a list"
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  "distinct":: "'a list => bool"
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  replicate :: "nat => 'a => 'a list"
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  splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
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nonterminals lupdbinds lupdbind
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syntax
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  -- {* list Enumeration *}
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  "@list" :: "args => 'a list"    ("[(_)]")
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  -- {* Special syntax for filter *}
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_<-_./ _])")
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  -- {* list update *}
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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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  "[x, xs]" == "x#[xs]"
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  "[x]" == "x#[]"
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  "[x<-xs . P]"== "filter (%x. P) xs"
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  "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"
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  "xs[i:=x]" == "list_update xs i x"
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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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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 => nat" where
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  "length == size"
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primrec
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  "hd(x#xs) = x"
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primrec
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  "tl([]) = []"
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  "tl(x#xs) = xs"
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primrec
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  "last(x#xs) = (if xs=[] then x else last xs)"
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primrec
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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 [] = {}"
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  "set (x#xs) = insert x (set xs)"
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primrec
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  "map f [] = []"
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  "map f (x#xs) = f(x)#map f xs"
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fun (*authentic syntax for append -- revert to primrec
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  as soon as "authentic" primrec is available*)
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  append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65)
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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([]) = []"
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  "rev(x#xs) = rev(xs) @ [x]"
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primrec
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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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primrec
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  foldl_Nil:"foldl f a [] = a"
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  foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
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primrec
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  "foldr f [] a = a"
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  "foldr f (x#xs) a = f x (foldr f xs a)"
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primrec
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  "concat([]) = []"
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  "concat(x#xs) = x @ concat(xs)"
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primrec
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"listsum [] = 0"
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"listsum (x # xs) = x + listsum xs"
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primrec
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  drop_Nil:"drop n [] = []"
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  drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
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  -- {*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_Nil:"take n [] = []"
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  take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => 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_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => 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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  "[][i:=v] = []"
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  "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])"
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primrec
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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 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 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_0: "[i..<0] = []"
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  upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
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primrec
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  "distinct [] = True"
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  "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
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primrec
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  "remdups [] = []"
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  "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
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primrec
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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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  replicate_0: "replicate 0 x = []"
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  replicate_Suc: "replicate (Suc n) x = x # replicate n x"
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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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primrec
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  "splice [] ys = ys"
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  "splice (x#xs) ys = (if ys=[] then x#xs else x # hd ys # splice xs (tl ys))"
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    -- {*Warning: simpset does not contain the second eqn but a derived one. *}
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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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fun (in linorder) sorted :: "'a list \<Rightarrow> bool" where
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"sorted [] \<longleftrightarrow> True" |
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"sorted [x] \<longleftrightarrow> True" |
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"sorted (x#y#zs) \<longleftrightarrow> x <= y \<and> sorted (y#zs)"
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fun (in linorder) insort :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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"insort x [] = [x]" |
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"insort x (y#ys) = (if x <= y then (x#y#ys) else y#(insort x ys))"
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fun (in linorder) sort :: "'a list \<Rightarrow> 'a list" where
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"sort [] = []" |
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"sort (x#xs) = insort x (sort xs)"
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subsubsection {* List comprehension *}
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text{* Input syntax for Haskell-like list comprehension notation.
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Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
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the list of all pairs of distinct elements from @{text xs} and @{text ys}.
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The syntax is as in Haskell, except that @{text"|"} becomes a dot
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(like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
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\verb![e| x <- xs, ...]!.
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The qualifiers after the dot are
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\begin{description}
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\item[generators] @{text"p \<leftarrow> xs"},
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 where @{text p} is a pattern and @{text xs} an expression of list type, or
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\item[guards] @{text"b"}, where @{text b} is a boolean expression.
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%\item[local bindings] @ {text"let x = e"}.
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\end{description}
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Just like in Haskell, list comprehension is just a shorthand. To avoid
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misunderstandings, the translation into desugared form is not reversed
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upon output. Note that the translation of @{text"[e. x \<leftarrow> xs]"} is
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optmized to @{term"map (%x. e) xs"}.
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It is easy to write short list comprehensions which stand for complex
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expressions. During proofs, they may become unreadable (and
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mangled). In such cases it can be advisable to introduce separate
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definitions for the list comprehensions in question.  *}
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(*
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Proper theorem proving support would be nice. For example, if
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@{text"set[f x y. x \<leftarrow> xs, y \<leftarrow> ys, P x y]"}
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produced something like
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@{term"{z. EX x: set xs. EX y:set ys. P x y \<and> z = f x y}"}.
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*)
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nonterminals lc_qual lc_quals
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syntax
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"_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
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"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ <- _")
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"_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
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(*"_lc_let" :: "letbinds => lc_qual"  ("let _")*)
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"_lc_end" :: "lc_quals" ("]")
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"_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals" (", __")
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"_lc_abs" :: "'a => 'b list => 'b list"
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(* These are easier than ML code but cannot express the optimized
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   translation of [e. p<-xs]
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translations
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"[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
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"_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
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 => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
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"[e. P]" => "if P then [e] else []"
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"_listcompr e (_lc_test P) (_lc_quals Q Qs)"
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 => "if P then (_listcompr e Q Qs) else []"
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"_listcompr e (_lc_let b) (_lc_quals Q Qs)"
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 => "_Let b (_listcompr e Q Qs)"
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*)
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syntax (xsymbols)
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"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
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syntax (HTML output)
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"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
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parse_translation (advanced) {*
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let
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  val NilC = Syntax.const @{const_name Nil};
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  val ConsC = Syntax.const @{const_name Cons};
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  val mapC = Syntax.const @{const_name map};
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  val concatC = Syntax.const @{const_name concat};
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  val IfC = Syntax.const @{const_name If};
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  fun singl x = ConsC $ x $ NilC;
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   fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
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    let
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      val x = Free (Name.variant (add_term_free_names (p$e, [])) "x", dummyT);
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      val e = if opti then singl e else e;
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      val case1 = Syntax.const "_case1" $ p $ e;
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      val case2 = Syntax.const "_case1" $ Syntax.const Term.dummy_patternN
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                                        $ NilC;
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      val cs = Syntax.const "_case2" $ case1 $ case2
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      val ft = DatatypeCase.case_tr false DatatypePackage.datatype_of_constr
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                 ctxt [x, cs]
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    in lambda x ft end;
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  fun abs_tr ctxt (p as Free(s,T)) e opti =
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        let val thy = ProofContext.theory_of ctxt;
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            val s' = Sign.intern_const thy s
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        in if Sign.declared_const thy s'
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           then (pat_tr ctxt p e opti, false)
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           else (lambda p e, true)
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        end
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    | abs_tr ctxt p e opti = (pat_tr ctxt p e opti, false);
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  fun lc_tr ctxt [e, Const("_lc_test",_)$b, qs] =
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        let val res = case qs of Const("_lc_end",_) => singl e
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                      | Const("_lc_quals",_)$q$qs => lc_tr ctxt [e,q,qs];
nipkow@24476
   328
        in IfC $ b $ res $ NilC end
nipkow@24476
   329
    | lc_tr ctxt [e, Const("_lc_gen",_) $ p $ es, Const("_lc_end",_)] =
nipkow@24476
   330
        (case abs_tr ctxt p e true of
nipkow@24476
   331
           (f,true) => mapC $ f $ es
nipkow@24476
   332
         | (f, false) => concatC $ (mapC $ f $ es))
nipkow@24476
   333
    | lc_tr ctxt [e, Const("_lc_gen",_) $ p $ es, Const("_lc_quals",_)$q$qs] =
nipkow@24476
   334
        let val e' = lc_tr ctxt [e,q,qs];
nipkow@24476
   335
        in concatC $ (mapC $ (fst(abs_tr ctxt p e' false)) $ es) end
nipkow@24476
   336
nipkow@24476
   337
in [("_listcompr", lc_tr)] end
nipkow@24349
   338
*}
nipkow@23279
   339
nipkow@23240
   340
(*
nipkow@23240
   341
term "[(x,y,z). b]"
nipkow@24476
   342
term "[(x,y,z). x\<leftarrow>xs]"
nipkow@24476
   343
term "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]"
nipkow@24476
   344
term "[(x,y,z). x<a, x>b]"
nipkow@24476
   345
term "[(x,y,z). x\<leftarrow>xs, x>b]"
nipkow@24476
   346
term "[(x,y,z). x<a, x\<leftarrow>xs]"
nipkow@24349
   347
term "[(x,y). Cons True x \<leftarrow> xs]"
nipkow@24349
   348
term "[(x,y,z). Cons x [] \<leftarrow> xs]"
nipkow@23240
   349
term "[(x,y,z). x<a, x>b, x=d]"
nipkow@23240
   350
term "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
nipkow@23240
   351
term "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
nipkow@23240
   352
term "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
nipkow@23240
   353
term "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
nipkow@23240
   354
term "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
nipkow@23240
   355
term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
nipkow@23240
   356
term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
nipkow@24349
   357
term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
nipkow@23192
   358
*)
nipkow@23192
   359
haftmann@21061
   360
subsubsection {* @{const Nil} and @{const Cons} *}
haftmann@21061
   361
haftmann@21061
   362
lemma not_Cons_self [simp]:
haftmann@21061
   363
  "xs \<noteq> x # xs"
nipkow@13145
   364
by (induct xs) auto
wenzelm@13114
   365
wenzelm@13142
   366
lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
wenzelm@13114
   367
wenzelm@13142
   368
lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
nipkow@13145
   369
by (induct xs) auto
wenzelm@13114
   370
wenzelm@13142
   371
lemma length_induct:
haftmann@21061
   372
  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
nipkow@17589
   373
by (rule measure_induct [of length]) iprover
wenzelm@13114
   374
wenzelm@13114
   375
haftmann@21061
   376
subsubsection {* @{const length} *}
wenzelm@13114
   377
wenzelm@13142
   378
text {*
haftmann@21061
   379
  Needs to come before @{text "@"} because of theorem @{text
haftmann@21061
   380
  append_eq_append_conv}.
wenzelm@13142
   381
*}
wenzelm@13114
   382
wenzelm@13142
   383
lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
nipkow@13145
   384
by (induct xs) auto
wenzelm@13114
   385
wenzelm@13142
   386
lemma length_map [simp]: "length (map f xs) = length xs"
nipkow@13145
   387
by (induct xs) auto
wenzelm@13114
   388
wenzelm@13142
   389
lemma length_rev [simp]: "length (rev xs) = length xs"
nipkow@13145
   390
by (induct xs) auto
wenzelm@13114
   391
wenzelm@13142
   392
lemma length_tl [simp]: "length (tl xs) = length xs - 1"
nipkow@13145
   393
by (cases xs) auto
wenzelm@13114
   394
wenzelm@13142
   395
lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
nipkow@13145
   396
by (induct xs) auto
wenzelm@13114
   397
wenzelm@13142
   398
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
nipkow@13145
   399
by (induct xs) auto
wenzelm@13114
   400
nipkow@23479
   401
lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
nipkow@23479
   402
by auto
nipkow@23479
   403
wenzelm@13114
   404
lemma length_Suc_conv:
nipkow@13145
   405
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
nipkow@13145
   406
by (induct xs) auto
wenzelm@13142
   407
nipkow@14025
   408
lemma Suc_length_conv:
nipkow@14025
   409
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
paulson@14208
   410
apply (induct xs, simp, simp)
nipkow@14025
   411
apply blast
nipkow@14025
   412
done
nipkow@14025
   413
oheimb@14099
   414
lemma impossible_Cons [rule_format]: 
oheimb@14099
   415
  "length xs <= length ys --> xs = x # ys = False"
wenzelm@20503
   416
apply (induct xs)
wenzelm@20503
   417
apply auto
oheimb@14099
   418
done
oheimb@14099
   419
nipkow@24526
   420
lemma list_induct2[consumes 1]:
nipkow@24526
   421
  "\<lbrakk> length xs = length ys;
nipkow@14247
   422
   P [] [];
nipkow@14247
   423
   \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
nipkow@14247
   424
 \<Longrightarrow> P xs ys"
nipkow@24526
   425
apply(induct xs arbitrary: ys)
nipkow@14247
   426
 apply simp
nipkow@14247
   427
apply(case_tac ys)
nipkow@14247
   428
 apply simp
nipkow@14247
   429
apply(simp)
nipkow@14247
   430
done
wenzelm@13114
   431
krauss@22493
   432
lemma list_induct2': 
krauss@22493
   433
  "\<lbrakk> P [] [];
krauss@22493
   434
  \<And>x xs. P (x#xs) [];
krauss@22493
   435
  \<And>y ys. P [] (y#ys);
krauss@22493
   436
   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
krauss@22493
   437
 \<Longrightarrow> P xs ys"
krauss@22493
   438
by (induct xs arbitrary: ys) (case_tac x, auto)+
krauss@22493
   439
nipkow@22143
   440
lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
nipkow@24349
   441
by (rule Eq_FalseI) auto
wenzelm@24037
   442
wenzelm@24037
   443
simproc_setup list_neq ("(xs::'a list) = ys") = {*
nipkow@22143
   444
(*
nipkow@22143
   445
Reduces xs=ys to False if xs and ys cannot be of the same length.
nipkow@22143
   446
This is the case if the atomic sublists of one are a submultiset
nipkow@22143
   447
of those of the other list and there are fewer Cons's in one than the other.
nipkow@22143
   448
*)
wenzelm@24037
   449
wenzelm@24037
   450
let
nipkow@22143
   451
nipkow@22143
   452
fun len (Const("List.list.Nil",_)) acc = acc
nipkow@22143
   453
  | len (Const("List.list.Cons",_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
haftmann@23029
   454
  | len (Const("List.append",_) $ xs $ ys) acc = len xs (len ys acc)
nipkow@22143
   455
  | len (Const("List.rev",_) $ xs) acc = len xs acc
nipkow@22143
   456
  | len (Const("List.map",_) $ _ $ xs) acc = len xs acc
nipkow@22143
   457
  | len t (ts,n) = (t::ts,n);
nipkow@22143
   458
wenzelm@24037
   459
fun list_neq _ ss ct =
nipkow@22143
   460
  let
wenzelm@24037
   461
    val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
nipkow@22143
   462
    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
nipkow@22143
   463
    fun prove_neq() =
nipkow@22143
   464
      let
nipkow@22143
   465
        val Type(_,listT::_) = eqT;
haftmann@22994
   466
        val size = HOLogic.size_const listT;
nipkow@22143
   467
        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
nipkow@22143
   468
        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
nipkow@22143
   469
        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
haftmann@22633
   470
          (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
haftmann@22633
   471
      in SOME (thm RS @{thm neq_if_length_neq}) end
nipkow@22143
   472
  in
wenzelm@23214
   473
    if m < n andalso submultiset (op aconv) (ls,rs) orelse
wenzelm@23214
   474
       n < m andalso submultiset (op aconv) (rs,ls)
nipkow@22143
   475
    then prove_neq() else NONE
nipkow@22143
   476
  end;
wenzelm@24037
   477
in list_neq end;
nipkow@22143
   478
*}
nipkow@22143
   479
nipkow@22143
   480
nipkow@15392
   481
subsubsection {* @{text "@"} -- append *}
wenzelm@13114
   482
wenzelm@13142
   483
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
nipkow@13145
   484
by (induct xs) auto
wenzelm@13114
   485
wenzelm@13142
   486
lemma append_Nil2 [simp]: "xs @ [] = xs"
nipkow@13145
   487
by (induct xs) auto
nipkow@3507
   488
nipkow@24449
   489
interpretation semigroup_append: semigroup_add ["op @"]
nipkow@24449
   490
by unfold_locales simp
nipkow@24449
   491
interpretation monoid_append: monoid_add ["[]" "op @"]
nipkow@24449
   492
by unfold_locales (simp+)
nipkow@24449
   493
wenzelm@13142
   494
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
nipkow@13145
   495
by (induct xs) auto
wenzelm@13114
   496
wenzelm@13142
   497
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
nipkow@13145
   498
by (induct xs) auto
wenzelm@13114
   499
wenzelm@13142
   500
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
nipkow@13145
   501
by (induct xs) auto
wenzelm@13114
   502
wenzelm@13142
   503
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
nipkow@13145
   504
by (induct xs) auto
wenzelm@13114
   505
paulson@24286
   506
lemma append_eq_append_conv [simp,noatp]:
nipkow@24526
   507
 "length xs = length ys \<or> length us = length vs
berghofe@13883
   508
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
nipkow@24526
   509
apply (induct xs arbitrary: ys)
paulson@14208
   510
 apply (case_tac ys, simp, force)
paulson@14208
   511
apply (case_tac ys, force, simp)
nipkow@13145
   512
done
wenzelm@13142
   513
nipkow@24526
   514
lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
nipkow@24526
   515
  (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
nipkow@24526
   516
apply (induct xs arbitrary: ys zs ts)
nipkow@14495
   517
 apply fastsimp
nipkow@14495
   518
apply(case_tac zs)
nipkow@14495
   519
 apply simp
nipkow@14495
   520
apply fastsimp
nipkow@14495
   521
done
nipkow@14495
   522
wenzelm@13142
   523
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145
   524
by simp
wenzelm@13142
   525
wenzelm@13142
   526
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145
   527
by simp
wenzelm@13114
   528
wenzelm@13142
   529
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   530
by simp
wenzelm@13114
   531
wenzelm@13142
   532
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   533
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   534
wenzelm@13142
   535
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   536
using append_same_eq [of "[]"] by auto
wenzelm@13114
   537
paulson@24286
   538
lemma hd_Cons_tl [simp,noatp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   539
by (induct xs) auto
wenzelm@13114
   540
wenzelm@13142
   541
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   542
by (induct xs) auto
wenzelm@13114
   543
wenzelm@13142
   544
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   545
by (simp add: hd_append split: list.split)
wenzelm@13114
   546
wenzelm@13142
   547
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   548
by (simp split: list.split)
wenzelm@13114
   549
wenzelm@13142
   550
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   551
by (simp add: tl_append split: list.split)
wenzelm@13114
   552
wenzelm@13114
   553
nipkow@14300
   554
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   555
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   556
by(cases ys) auto
nipkow@14300
   557
nipkow@15281
   558
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   559
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   560
by(cases ys) auto
nipkow@15281
   561
nipkow@14300
   562
wenzelm@13142
   563
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   564
wenzelm@13114
   565
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   566
by simp
wenzelm@13114
   567
wenzelm@13142
   568
lemma Cons_eq_appendI:
nipkow@13145
   569
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   570
by (drule sym) simp
wenzelm@13114
   571
wenzelm@13142
   572
lemma append_eq_appendI:
nipkow@13145
   573
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   574
by (drule sym) simp
wenzelm@13114
   575
wenzelm@13114
   576
wenzelm@13142
   577
text {*
nipkow@13145
   578
Simplification procedure for all list equalities.
nipkow@13145
   579
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   580
- both lists end in a singleton list,
nipkow@13145
   581
- or both lists end in the same list.
wenzelm@13142
   582
*}
wenzelm@13142
   583
wenzelm@13142
   584
ML_setup {*
nipkow@3507
   585
local
nipkow@3507
   586
wenzelm@13114
   587
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462
   588
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
haftmann@23029
   589
  | last (Const("List.append",_) $ _ $ ys) = last ys
wenzelm@13462
   590
  | last t = t;
wenzelm@13114
   591
wenzelm@13114
   592
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462
   593
  | list1 _ = false;
wenzelm@13114
   594
wenzelm@13114
   595
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462
   596
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
haftmann@23029
   597
  | butlast ((app as Const("List.append",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462
   598
  | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114
   599
haftmann@22633
   600
val rearr_ss = HOL_basic_ss addsimps [@{thm append_assoc},
haftmann@22633
   601
  @{thm append_Nil}, @{thm append_Cons}];
wenzelm@16973
   602
wenzelm@20044
   603
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   604
  let
wenzelm@13462
   605
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   606
    fun rearr conv =
wenzelm@13462
   607
      let
wenzelm@13462
   608
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   609
        val Type(_,listT::_) = eqT
wenzelm@13462
   610
        val appT = [listT,listT] ---> listT
haftmann@23029
   611
        val app = Const("List.append",appT)
wenzelm@13462
   612
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   613
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@20044
   614
        val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
wenzelm@17877
   615
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
skalberg@15531
   616
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   617
wenzelm@13462
   618
  in
haftmann@22633
   619
    if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
haftmann@22633
   620
    else if lastl aconv lastr then rearr @{thm append_same_eq}
skalberg@15531
   621
    else NONE
wenzelm@13462
   622
  end;
wenzelm@13462
   623
wenzelm@13114
   624
in
wenzelm@13462
   625
wenzelm@13462
   626
val list_eq_simproc =
haftmann@22633
   627
  Simplifier.simproc @{theory} "list_eq" ["(xs::'a list) = ys"] (K list_eq);
wenzelm@13462
   628
wenzelm@13114
   629
end;
wenzelm@13114
   630
wenzelm@13114
   631
Addsimprocs [list_eq_simproc];
wenzelm@13114
   632
*}
wenzelm@13114
   633
wenzelm@13114
   634
nipkow@15392
   635
subsubsection {* @{text map} *}
wenzelm@13114
   636
wenzelm@13142
   637
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   638
by (induct xs) simp_all
wenzelm@13114
   639
wenzelm@13142
   640
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   641
by (rule ext, induct_tac xs) auto
wenzelm@13114
   642
wenzelm@13142
   643
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   644
by (induct xs) auto
wenzelm@13114
   645
wenzelm@13142
   646
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   647
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   648
wenzelm@13142
   649
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   650
by (induct xs) auto
wenzelm@13114
   651
nipkow@13737
   652
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   653
by (induct xs) auto
nipkow@13737
   654
krauss@19770
   655
lemma map_cong [fundef_cong, recdef_cong]:
nipkow@13145
   656
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   657
-- {* a congruence rule for @{text map} *}
nipkow@13737
   658
by simp
wenzelm@13114
   659
wenzelm@13142
   660
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   661
by (cases xs) auto
wenzelm@13114
   662
wenzelm@13142
   663
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   664
by (cases xs) auto
wenzelm@13114
   665
paulson@18447
   666
lemma map_eq_Cons_conv:
nipkow@14025
   667
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   668
by (cases xs) auto
wenzelm@13114
   669
paulson@18447
   670
lemma Cons_eq_map_conv:
nipkow@14025
   671
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   672
by (cases ys) auto
nipkow@14025
   673
paulson@18447
   674
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
paulson@18447
   675
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
paulson@18447
   676
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
paulson@18447
   677
nipkow@14111
   678
lemma ex_map_conv:
nipkow@14111
   679
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
paulson@18447
   680
by(induct ys, auto simp add: Cons_eq_map_conv)
nipkow@14111
   681
nipkow@15110
   682
lemma map_eq_imp_length_eq:
nipkow@24526
   683
  "map f xs = map f ys ==> length xs = length ys"
nipkow@24526
   684
apply (induct ys arbitrary: xs)
nipkow@15110
   685
 apply simp
paulson@24632
   686
apply (metis Suc_length_conv length_map)
nipkow@15110
   687
done
nipkow@15110
   688
nipkow@15110
   689
lemma map_inj_on:
nipkow@15110
   690
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   691
  ==> xs = ys"
nipkow@15110
   692
apply(frule map_eq_imp_length_eq)
nipkow@15110
   693
apply(rotate_tac -1)
nipkow@15110
   694
apply(induct rule:list_induct2)
nipkow@15110
   695
 apply simp
nipkow@15110
   696
apply(simp)
nipkow@15110
   697
apply (blast intro:sym)
nipkow@15110
   698
done
nipkow@15110
   699
nipkow@15110
   700
lemma inj_on_map_eq_map:
nipkow@15110
   701
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   702
by(blast dest:map_inj_on)
nipkow@15110
   703
wenzelm@13114
   704
lemma map_injective:
nipkow@24526
   705
 "map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@24526
   706
by (induct ys arbitrary: xs) (auto dest!:injD)
wenzelm@13114
   707
nipkow@14339
   708
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   709
by(blast dest:map_injective)
nipkow@14339
   710
wenzelm@13114
   711
lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@17589
   712
by (iprover dest: map_injective injD intro: inj_onI)
wenzelm@13114
   713
wenzelm@13114
   714
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   715
apply (unfold inj_on_def, clarify)
nipkow@13145
   716
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   717
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   718
apply blast
nipkow@13145
   719
done
wenzelm@13114
   720
nipkow@14339
   721
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   722
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   723
nipkow@15303
   724
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
   725
apply(rule inj_onI)
nipkow@15303
   726
apply(erule map_inj_on)
nipkow@15303
   727
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
   728
done
nipkow@15303
   729
kleing@14343
   730
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   731
by (induct xs, auto)
wenzelm@13114
   732
nipkow@14402
   733
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   734
by (induct xs) auto
nipkow@14402
   735
nipkow@15110
   736
lemma map_fst_zip[simp]:
nipkow@15110
   737
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   738
by (induct rule:list_induct2, simp_all)
nipkow@15110
   739
nipkow@15110
   740
lemma map_snd_zip[simp]:
nipkow@15110
   741
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   742
by (induct rule:list_induct2, simp_all)
nipkow@15110
   743
nipkow@15110
   744
nipkow@15392
   745
subsubsection {* @{text rev} *}
wenzelm@13114
   746
wenzelm@13142
   747
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   748
by (induct xs) auto
wenzelm@13114
   749
wenzelm@13142
   750
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   751
by (induct xs) auto
wenzelm@13114
   752
kleing@15870
   753
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870
   754
by auto
kleing@15870
   755
wenzelm@13142
   756
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   757
by (induct xs) auto
wenzelm@13114
   758
wenzelm@13142
   759
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   760
by (induct xs) auto
wenzelm@13114
   761
kleing@15870
   762
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870
   763
by (cases xs) auto
kleing@15870
   764
kleing@15870
   765
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870
   766
by (cases xs) auto
kleing@15870
   767
haftmann@21061
   768
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
haftmann@21061
   769
apply (induct xs arbitrary: ys, force)
paulson@14208
   770
apply (case_tac ys, simp, force)
nipkow@13145
   771
done
wenzelm@13114
   772
nipkow@15439
   773
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
   774
by(simp add:inj_on_def)
nipkow@15439
   775
wenzelm@13366
   776
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   777
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489
   778
apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145
   779
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   780
done
wenzelm@13114
   781
nipkow@13145
   782
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114
   783
wenzelm@13366
   784
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   785
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   786
by (induct xs rule: rev_induct) auto
wenzelm@13114
   787
wenzelm@13366
   788
lemmas rev_cases = rev_exhaust
wenzelm@13366
   789
nipkow@18423
   790
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
nipkow@18423
   791
by(rule rev_cases[of xs]) auto
nipkow@18423
   792
wenzelm@13114
   793
nipkow@15392
   794
subsubsection {* @{text set} *}
wenzelm@13114
   795
wenzelm@13142
   796
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   797
by (induct xs) auto
wenzelm@13114
   798
wenzelm@13142
   799
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   800
by (induct xs) auto
wenzelm@13114
   801
nipkow@17830
   802
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
nipkow@17830
   803
by(cases xs) auto
oheimb@14099
   804
wenzelm@13142
   805
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   806
by auto
wenzelm@13114
   807
oheimb@14099
   808
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   809
by auto
oheimb@14099
   810
wenzelm@13142
   811
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   812
by (induct xs) auto
wenzelm@13114
   813
nipkow@15245
   814
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   815
by(induct xs) auto
nipkow@15245
   816
wenzelm@13142
   817
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   818
by (induct xs) auto
wenzelm@13114
   819
wenzelm@13142
   820
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   821
by (induct xs) auto
wenzelm@13114
   822
wenzelm@13142
   823
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   824
by (induct xs) auto
wenzelm@13114
   825
nipkow@15425
   826
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
paulson@14208
   827
apply (induct j, simp_all)
paulson@14208
   828
apply (erule ssubst, auto)
nipkow@13145
   829
done
wenzelm@13114
   830
wenzelm@13142
   831
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
paulson@15113
   832
proof (induct xs)
paulson@15113
   833
  case Nil show ?case by simp
paulson@15113
   834
  case (Cons a xs)
paulson@15113
   835
  show ?case
paulson@15113
   836
  proof 
paulson@15113
   837
    assume "x \<in> set (a # xs)"
paulson@15113
   838
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   839
      by (simp, blast intro: Cons_eq_appendI)
paulson@15113
   840
  next
paulson@15113
   841
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   842
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
paulson@15113
   843
    show "x \<in> set (a # xs)" 
paulson@15113
   844
      by (cases ys, auto simp add: eq)
paulson@15113
   845
  qed
paulson@15113
   846
qed
wenzelm@13142
   847
nipkow@18049
   848
lemma in_set_conv_decomp_first:
nipkow@18049
   849
 "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
nipkow@18049
   850
proof (induct xs)
nipkow@18049
   851
  case Nil show ?case by simp
nipkow@18049
   852
next
nipkow@18049
   853
  case (Cons a xs)
nipkow@18049
   854
  show ?case
nipkow@18049
   855
  proof cases
nipkow@18049
   856
    assume "x = a" thus ?case using Cons by force
nipkow@18049
   857
  next
nipkow@18049
   858
    assume "x \<noteq> a"
nipkow@18049
   859
    show ?case
nipkow@18049
   860
    proof
nipkow@18049
   861
      assume "x \<in> set (a # xs)"
nipkow@18049
   862
      from prems show "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@18049
   863
	by(fastsimp intro!: Cons_eq_appendI)
nipkow@18049
   864
    next
nipkow@18049
   865
      assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@18049
   866
      then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
nipkow@18049
   867
      show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq)
nipkow@18049
   868
    qed
nipkow@18049
   869
  qed
nipkow@18049
   870
qed
nipkow@18049
   871
nipkow@18049
   872
lemmas split_list       = in_set_conv_decomp[THEN iffD1, standard]
nipkow@18049
   873
lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard]
nipkow@18049
   874
nipkow@18049
   875
paulson@13508
   876
lemma finite_list: "finite A ==> EX l. set l = A"
paulson@13508
   877
apply (erule finite_induct, auto)
paulson@13508
   878
apply (rule_tac x="x#l" in exI, auto)
paulson@13508
   879
done
paulson@13508
   880
kleing@14388
   881
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
   882
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
   883
paulson@15168
   884
nipkow@15392
   885
subsubsection {* @{text filter} *}
wenzelm@13114
   886
wenzelm@13142
   887
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   888
by (induct xs) auto
wenzelm@13114
   889
nipkow@15305
   890
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
   891
by (induct xs) simp_all
nipkow@15305
   892
wenzelm@13142
   893
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   894
by (induct xs) auto
wenzelm@13114
   895
nipkow@16998
   896
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998
   897
by (induct xs) (auto simp add: le_SucI)
nipkow@16998
   898
nipkow@18423
   899
lemma sum_length_filter_compl:
nipkow@18423
   900
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
nipkow@18423
   901
by(induct xs) simp_all
nipkow@18423
   902
wenzelm@13142
   903
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   904
by (induct xs) auto
wenzelm@13114
   905
wenzelm@13142
   906
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   907
by (induct xs) auto
wenzelm@13114
   908
nipkow@16998
   909
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@24349
   910
by (induct xs) simp_all
nipkow@16998
   911
nipkow@16998
   912
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998
   913
apply (induct xs)
nipkow@16998
   914
 apply auto
nipkow@16998
   915
apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998
   916
apply simp
nipkow@16998
   917
done
wenzelm@13114
   918
nipkow@16965
   919
lemma filter_map:
nipkow@16965
   920
  "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965
   921
by (induct xs) simp_all
nipkow@16965
   922
nipkow@16965
   923
lemma length_filter_map[simp]:
nipkow@16965
   924
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965
   925
by (simp add:filter_map)
nipkow@16965
   926
wenzelm@13142
   927
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   928
by auto
wenzelm@13114
   929
nipkow@15246
   930
lemma length_filter_less:
nipkow@15246
   931
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
   932
proof (induct xs)
nipkow@15246
   933
  case Nil thus ?case by simp
nipkow@15246
   934
next
nipkow@15246
   935
  case (Cons x xs) thus ?case
nipkow@15246
   936
    apply (auto split:split_if_asm)
nipkow@15246
   937
    using length_filter_le[of P xs] apply arith
nipkow@15246
   938
  done
nipkow@15246
   939
qed
wenzelm@13114
   940
nipkow@15281
   941
lemma length_filter_conv_card:
nipkow@15281
   942
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
   943
proof (induct xs)
nipkow@15281
   944
  case Nil thus ?case by simp
nipkow@15281
   945
next
nipkow@15281
   946
  case (Cons x xs)
nipkow@15281
   947
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
   948
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
   949
  show ?case (is "?l = card ?S'")
nipkow@15281
   950
  proof (cases)
nipkow@15281
   951
    assume "p x"
nipkow@15281
   952
    hence eq: "?S' = insert 0 (Suc ` ?S)"
chaieb@25112
   953
      apply (auto simp add:  nth_Cons image_def neq0_conv split:nat.split elim:lessE)
chaieb@25112
   954
      apply (rule_tac x="xa - 1" in exI, auto)
chaieb@25112
   955
      done
nipkow@15281
   956
    have "length (filter p (x # xs)) = Suc(card ?S)"
wenzelm@23388
   957
      using Cons `p x` by simp
nipkow@15281
   958
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
nipkow@15281
   959
      by (simp add: card_image inj_Suc)
nipkow@15281
   960
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   961
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
   962
    finally show ?thesis .
nipkow@15281
   963
  next
nipkow@15281
   964
    assume "\<not> p x"
nipkow@15281
   965
    hence eq: "?S' = Suc ` ?S"
chaieb@25112
   966
      apply(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
chaieb@25112
   967
      apply (rule_tac x="xa - 1" in exI, auto)
chaieb@25112
   968
      done
nipkow@15281
   969
    have "length (filter p (x # xs)) = card ?S"
wenzelm@23388
   970
      using Cons `\<not> p x` by simp
nipkow@15281
   971
    also have "\<dots> = card(Suc ` ?S)" using fin
nipkow@15281
   972
      by (simp add: card_image inj_Suc)
nipkow@15281
   973
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   974
      by (simp add:card_insert_if)
nipkow@15281
   975
    finally show ?thesis .
nipkow@15281
   976
  qed
nipkow@15281
   977
qed
nipkow@15281
   978
nipkow@17629
   979
lemma Cons_eq_filterD:
nipkow@17629
   980
 "x#xs = filter P ys \<Longrightarrow>
nipkow@17629
   981
  \<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
   982
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
nipkow@17629
   983
proof(induct ys)
nipkow@17629
   984
  case Nil thus ?case by simp
nipkow@17629
   985
next
nipkow@17629
   986
  case (Cons y ys)
nipkow@17629
   987
  show ?case (is "\<exists>x. ?Q x")
nipkow@17629
   988
  proof cases
nipkow@17629
   989
    assume Py: "P y"
nipkow@17629
   990
    show ?thesis
nipkow@17629
   991
    proof cases
nipkow@17629
   992
      assume xy: "x = y"
nipkow@17629
   993
      show ?thesis
nipkow@17629
   994
      proof from Py xy Cons(2) show "?Q []" by simp qed
nipkow@17629
   995
    next
nipkow@17629
   996
      assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp
nipkow@17629
   997
    qed
nipkow@17629
   998
  next
nipkow@17629
   999
    assume Py: "\<not> P y"
nipkow@17629
  1000
    with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp
nipkow@17629
  1001
    show ?thesis (is "? us. ?Q us")
nipkow@17629
  1002
    proof show "?Q (y#us)" using 1 by simp qed
nipkow@17629
  1003
  qed
nipkow@17629
  1004
qed
nipkow@17629
  1005
nipkow@17629
  1006
lemma filter_eq_ConsD:
nipkow@17629
  1007
 "filter P ys = x#xs \<Longrightarrow>
nipkow@17629
  1008
  \<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
  1009
by(rule Cons_eq_filterD) simp
nipkow@17629
  1010
nipkow@17629
  1011
lemma filter_eq_Cons_iff:
nipkow@17629
  1012
 "(filter P ys = x#xs) =
nipkow@17629
  1013
  (\<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
  1014
by(auto dest:filter_eq_ConsD)
nipkow@17629
  1015
nipkow@17629
  1016
lemma Cons_eq_filter_iff:
nipkow@17629
  1017
 "(x#xs = filter P ys) =
nipkow@17629
  1018
  (\<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
  1019
by(auto dest:Cons_eq_filterD)
nipkow@17629
  1020
krauss@19770
  1021
lemma filter_cong[fundef_cong, recdef_cong]:
nipkow@17501
  1022
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
nipkow@17501
  1023
apply simp
nipkow@17501
  1024
apply(erule thin_rl)
nipkow@17501
  1025
by (induct ys) simp_all
nipkow@17501
  1026
nipkow@15281
  1027
nipkow@15392
  1028
subsubsection {* @{text concat} *}
wenzelm@13114
  1029
wenzelm@13142
  1030
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
  1031
by (induct xs) auto
wenzelm@13114
  1032
paulson@18447
  1033
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
  1034
by (induct xss) auto
wenzelm@13114
  1035
paulson@18447
  1036
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
  1037
by (induct xss) auto
wenzelm@13114
  1038
nipkow@24308
  1039
lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
nipkow@13145
  1040
by (induct xs) auto
wenzelm@13114
  1041
nipkow@24476
  1042
lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs"
nipkow@24349
  1043
by (induct xs) auto
nipkow@24349
  1044
wenzelm@13142
  1045
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
  1046
by (induct xs) auto
wenzelm@13114
  1047
wenzelm@13142
  1048
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
  1049
by (induct xs) auto
wenzelm@13114
  1050
wenzelm@13142
  1051
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
  1052
by (induct xs) auto
wenzelm@13114
  1053
wenzelm@13114
  1054
nipkow@15392
  1055
subsubsection {* @{text nth} *}
wenzelm@13114
  1056
wenzelm@13142
  1057
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
  1058
by auto
wenzelm@13114
  1059
wenzelm@13142
  1060
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
  1061
by auto
wenzelm@13114
  1062
wenzelm@13142
  1063
declare nth.simps [simp del]
wenzelm@13114
  1064
wenzelm@13114
  1065
lemma nth_append:
nipkow@24526
  1066
  "(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
nipkow@24526
  1067
apply (induct xs arbitrary: n, simp)
paulson@14208
  1068
apply (case_tac n, auto)
nipkow@13145
  1069
done
wenzelm@13114
  1070
nipkow@14402
  1071
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
nipkow@14402
  1072
by (induct "xs") auto
nipkow@14402
  1073
nipkow@14402
  1074
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
nipkow@14402
  1075
by (induct "xs") auto
nipkow@14402
  1076
nipkow@24526
  1077
lemma nth_map [simp]: "n < length xs ==> (map f xs)!n = f(xs!n)"
nipkow@24526
  1078
apply (induct xs arbitrary: n, simp)
paulson@14208
  1079
apply (case_tac n, auto)
nipkow@13145
  1080
done
wenzelm@13114
  1081
nipkow@18423
  1082
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
nipkow@18423
  1083
by(cases xs) simp_all
nipkow@18423
  1084
nipkow@18049
  1085
nipkow@18049
  1086
lemma list_eq_iff_nth_eq:
nipkow@24526
  1087
 "(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
nipkow@24526
  1088
apply(induct xs arbitrary: ys)
paulson@24632
  1089
 apply force
nipkow@18049
  1090
apply(case_tac ys)
nipkow@18049
  1091
 apply simp
nipkow@18049
  1092
apply(simp add:nth_Cons split:nat.split)apply blast
nipkow@18049
  1093
done
nipkow@18049
  1094
wenzelm@13142
  1095
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
  1096
apply (induct xs, simp, simp)
nipkow@13145
  1097
apply safe
paulson@24632
  1098
apply (metis nat_case_0 nth.simps zero_less_Suc)
paulson@24632
  1099
apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
paulson@14208
  1100
apply (case_tac i, simp)
paulson@24632
  1101
apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff)
nipkow@13145
  1102
done
wenzelm@13114
  1103
nipkow@17501
  1104
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
nipkow@17501
  1105
by(auto simp:set_conv_nth)
nipkow@17501
  1106
nipkow@13145
  1107
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
  1108
by (auto simp add: set_conv_nth)
wenzelm@13114
  1109
wenzelm@13142
  1110
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
  1111
by (auto simp add: set_conv_nth)
wenzelm@13114
  1112
wenzelm@13114
  1113
lemma all_nth_imp_all_set:
nipkow@13145
  1114
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
  1115
by (auto simp add: set_conv_nth)
wenzelm@13114
  1116
wenzelm@13114
  1117
lemma all_set_conv_all_nth:
nipkow@13145
  1118
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
  1119
by (auto simp add: set_conv_nth)
wenzelm@13114
  1120
wenzelm@13114
  1121
nipkow@15392
  1122
subsubsection {* @{text list_update} *}
wenzelm@13114
  1123
nipkow@24526
  1124
lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
nipkow@24526
  1125
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1126
wenzelm@13114
  1127
lemma nth_list_update:
nipkow@24526
  1128
"i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@24526
  1129
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1130
wenzelm@13142
  1131
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
  1132
by (simp add: nth_list_update)
wenzelm@13114
  1133
nipkow@24526
  1134
lemma nth_list_update_neq [simp]: "i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@24526
  1135
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1136
wenzelm@13142
  1137
lemma list_update_overwrite [simp]:
nipkow@24526
  1138
"i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@24526
  1139
by (induct xs arbitrary: i) (auto split: nat.split)
nipkow@24526
  1140
nipkow@24526
  1141
lemma list_update_id[simp]: "xs[i := xs!i] = xs"
nipkow@24526
  1142
by (induct xs arbitrary: i) (simp_all split:nat.splits)
nipkow@24526
  1143
nipkow@24526
  1144
lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
nipkow@24526
  1145
apply (induct xs arbitrary: i)
nipkow@17501
  1146
 apply simp
nipkow@17501
  1147
apply (case_tac i)
nipkow@17501
  1148
apply simp_all
nipkow@17501
  1149
done
nipkow@17501
  1150
wenzelm@13114
  1151
lemma list_update_same_conv:
nipkow@24526
  1152
"i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@24526
  1153
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1154
nipkow@14187
  1155
lemma list_update_append1:
nipkow@24526
  1156
 "i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
nipkow@24526
  1157
apply (induct xs arbitrary: i, simp)
nipkow@14187
  1158
apply(simp split:nat.split)
nipkow@14187
  1159
done
nipkow@14187
  1160
kleing@15868
  1161
lemma list_update_append:
nipkow@24526
  1162
  "(xs @ ys) [n:= x] = 
kleing@15868
  1163
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
nipkow@24526
  1164
by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868
  1165
nipkow@14402
  1166
lemma list_update_length [simp]:
nipkow@14402
  1167
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
  1168
by (induct xs, auto)
nipkow@14402
  1169
wenzelm@13114
  1170
lemma update_zip:
nipkow@24526
  1171
  "length xs = length ys ==>
nipkow@24526
  1172
  (zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@24526
  1173
by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split)
nipkow@24526
  1174
nipkow@24526
  1175
lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)"
nipkow@24526
  1176
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1177
wenzelm@13114
  1178
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
  1179
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
  1180
nipkow@24526
  1181
lemma set_update_memI: "n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
nipkow@24526
  1182
by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868
  1183
haftmann@24796
  1184
lemma list_update_overwrite:
haftmann@24796
  1185
  "xs [i := x, i := y] = xs [i := y]"
haftmann@24796
  1186
apply (induct xs arbitrary: i)
haftmann@24796
  1187
apply simp
haftmann@24796
  1188
apply (case_tac i)
haftmann@24796
  1189
apply simp_all
haftmann@24796
  1190
done
haftmann@24796
  1191
haftmann@24796
  1192
lemma list_update_swap:
haftmann@24796
  1193
  "i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]"
haftmann@24796
  1194
apply (induct xs arbitrary: i i')
haftmann@24796
  1195
apply simp
haftmann@24796
  1196
apply (case_tac i, case_tac i')
haftmann@24796
  1197
apply auto
haftmann@24796
  1198
apply (case_tac i')
haftmann@24796
  1199
apply auto
haftmann@24796
  1200
done
haftmann@24796
  1201
wenzelm@13114
  1202
nipkow@15392
  1203
subsubsection {* @{text last} and @{text butlast} *}
wenzelm@13114
  1204
wenzelm@13142
  1205
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
  1206
by (induct xs) auto
wenzelm@13114
  1207
wenzelm@13142
  1208
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
  1209
by (induct xs) auto
wenzelm@13114
  1210
nipkow@14302
  1211
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
  1212
by(simp add:last.simps)
nipkow@14302
  1213
nipkow@14302
  1214
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
  1215
by(simp add:last.simps)
nipkow@14302
  1216
nipkow@14302
  1217
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
  1218
by (induct xs) (auto)
nipkow@14302
  1219
nipkow@14302
  1220
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
  1221
by(simp add:last_append)
nipkow@14302
  1222
nipkow@14302
  1223
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
  1224
by(simp add:last_append)
nipkow@14302
  1225
nipkow@17762
  1226
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
nipkow@17762
  1227
by(rule rev_exhaust[of xs]) simp_all
nipkow@17762
  1228
nipkow@17762
  1229
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
nipkow@17762
  1230
by(cases xs) simp_all
nipkow@17762
  1231
nipkow@17765
  1232
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
nipkow@17765
  1233
by (induct as) auto
nipkow@17762
  1234
wenzelm@13142
  1235
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
  1236
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1237
wenzelm@13114
  1238
lemma butlast_append:
nipkow@24526
  1239
  "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@24526
  1240
by (induct xs arbitrary: ys) auto
wenzelm@13114
  1241
wenzelm@13142
  1242
lemma append_butlast_last_id [simp]:
nipkow@13145
  1243
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
  1244
by (induct xs) auto
wenzelm@13114
  1245
wenzelm@13142
  1246
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
  1247
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1248
wenzelm@13114
  1249
lemma in_set_butlast_appendI:
nipkow@13145
  1250
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
  1251
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
  1252
nipkow@24526
  1253
lemma last_drop[simp]: "n < length xs \<Longrightarrow> last (drop n xs) = last xs"
nipkow@24526
  1254
apply (induct xs arbitrary: n)
nipkow@17501
  1255
 apply simp
nipkow@17501
  1256
apply (auto split:nat.split)
nipkow@17501
  1257
done
nipkow@17501
  1258
nipkow@17589
  1259
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
nipkow@17589
  1260
by(induct xs)(auto simp:neq_Nil_conv)
nipkow@17589
  1261
haftmann@24796
  1262
nipkow@15392
  1263
subsubsection {* @{text take} and @{text drop} *}
wenzelm@13114
  1264
wenzelm@13142
  1265
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
  1266
by (induct xs) auto
wenzelm@13114
  1267
wenzelm@13142
  1268
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
  1269
by (induct xs) auto
wenzelm@13114
  1270
wenzelm@13142
  1271
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
  1272
by simp
wenzelm@13114
  1273
wenzelm@13142
  1274
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
  1275
by simp
wenzelm@13114
  1276
wenzelm@13142
  1277
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
  1278
nipkow@15110
  1279
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
  1280
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
  1281
nipkow@14187
  1282
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
  1283
by(cases xs, simp_all)
nipkow@14187
  1284
nipkow@24526
  1285
lemma drop_tl: "drop n (tl xs) = tl(drop n xs)"
nipkow@24526
  1286
by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@24526
  1287
nipkow@24526
  1288
lemma nth_via_drop: "drop n xs = y#ys \<Longrightarrow> xs!n = y"
nipkow@24526
  1289
apply (induct xs arbitrary: n, simp)
nipkow@14187
  1290
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
  1291
done
nipkow@14187
  1292
nipkow@13913
  1293
lemma take_Suc_conv_app_nth:
nipkow@24526
  1294
  "i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
nipkow@24526
  1295
apply (induct xs arbitrary: i, simp)
paulson@14208
  1296
apply (case_tac i, auto)
nipkow@13913
  1297
done
nipkow@13913
  1298
mehta@14591
  1299
lemma drop_Suc_conv_tl:
nipkow@24526
  1300
  "i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
nipkow@24526
  1301
apply (induct xs arbitrary: i, simp)
mehta@14591
  1302
apply (case_tac i, auto)
mehta@14591
  1303
done
mehta@14591
  1304
nipkow@24526
  1305
lemma length_take [simp]: "length (take n xs) = min (length xs) n"
nipkow@24526
  1306
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1307
nipkow@24526
  1308
lemma length_drop [simp]: "length (drop n xs) = (length xs - n)"
nipkow@24526
  1309
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1310
nipkow@24526
  1311
lemma take_all [simp]: "length xs <= n ==> take n xs = xs"
nipkow@24526
  1312
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1313
nipkow@24526
  1314
lemma drop_all [simp]: "length xs <= n ==> drop n xs = []"
nipkow@24526
  1315
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114
  1316
wenzelm@13142
  1317
lemma take_append [simp]:
nipkow@24526
  1318
  "take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@24526
  1319
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114
  1320
wenzelm@13142
  1321
lemma drop_append [simp]:
nipkow@24526
  1322
  "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@24526
  1323
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1324
nipkow@24526
  1325
lemma take_take [simp]: "take n (take m xs) = take (min n m) xs"
nipkow@24526
  1326
apply (induct m arbitrary: xs n, auto)
paulson@14208
  1327
apply (case_tac xs, auto)
nipkow@15236
  1328
apply (case_tac n, auto)
nipkow@13145
  1329
done
wenzelm@13114
  1330
nipkow@24526
  1331
lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs"
nipkow@24526
  1332
apply (induct m arbitrary: xs, auto)
paulson@14208
  1333
apply (case_tac xs, auto)
nipkow@13145
  1334
done
wenzelm@13114
  1335
nipkow@24526
  1336
lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)"
nipkow@24526
  1337
apply (induct m arbitrary: xs n, auto)
paulson@14208
  1338
apply (case_tac xs, auto)
nipkow@13145
  1339
done
wenzelm@13114
  1340
nipkow@24526
  1341
lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@24526
  1342
apply(induct xs arbitrary: m n)
nipkow@14802
  1343
 apply simp
nipkow@14802
  1344
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  1345
done
nipkow@14802
  1346
nipkow@24526
  1347
lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs"
nipkow@24526
  1348
apply (induct n arbitrary: xs, auto)
paulson@14208
  1349
apply (case_tac xs, auto)
nipkow@13145
  1350
done
wenzelm@13114
  1351
nipkow@24526
  1352
lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@24526
  1353
apply(induct xs arbitrary: n)
nipkow@15110
  1354
 apply simp
nipkow@15110
  1355
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  1356
done
nipkow@15110
  1357
nipkow@24526
  1358
lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)"
nipkow@24526
  1359
apply(induct xs arbitrary: n)
nipkow@15110
  1360
apply simp
nipkow@15110
  1361
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1362
done
nipkow@15110
  1363
nipkow@24526
  1364
lemma take_map: "take n (map f xs) = map f (take n xs)"
nipkow@24526
  1365
apply (induct n arbitrary: xs, auto)
paulson@14208
  1366
apply (case_tac xs, auto)
nipkow@13145
  1367
done
wenzelm@13114
  1368
nipkow@24526
  1369
lemma drop_map: "drop n (map f xs) = map f (drop n xs)"
nipkow@24526
  1370
apply (induct n arbitrary: xs, auto)
paulson@14208
  1371
apply (case_tac xs, auto)
nipkow@13145
  1372
done
wenzelm@13114
  1373
nipkow@24526
  1374
lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)"
nipkow@24526
  1375
apply (induct xs arbitrary: i, auto)
paulson@14208
  1376
apply (case_tac i, auto)
nipkow@13145
  1377
done
wenzelm@13114
  1378
nipkow@24526
  1379
lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)"
nipkow@24526
  1380
apply (induct xs arbitrary: i, auto)
paulson@14208
  1381
apply (case_tac i, auto)
nipkow@13145
  1382
done
wenzelm@13114
  1383
nipkow@24526
  1384
lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i"
nipkow@24526
  1385
apply (induct xs arbitrary: i n, auto)
paulson@14208
  1386
apply (case_tac n, blast)
paulson@14208
  1387
apply (case_tac i, auto)
nipkow@13145
  1388
done
wenzelm@13114
  1389
wenzelm@13142
  1390
lemma nth_drop [simp]:
nipkow@24526
  1391
  "n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
nipkow@24526
  1392
apply (induct n arbitrary: xs i, auto)
paulson@14208
  1393
apply (case_tac xs, auto)
nipkow@13145
  1394
done
nipkow@3507
  1395
nipkow@18423
  1396
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
nipkow@18423
  1397
by(simp add: hd_conv_nth)
nipkow@18423
  1398
nipkow@24526
  1399
lemma set_take_subset: "set(take n xs) \<subseteq> set xs"
nipkow@24526
  1400
by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split)
nipkow@24526
  1401
nipkow@24526
  1402
lemma set_drop_subset: "set(drop n xs) \<subseteq> set xs"
nipkow@24526
  1403
by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1404
nipkow@14187
  1405
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1406
using set_take_subset by fast
nipkow@14187
  1407
nipkow@14187
  1408
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1409
using set_drop_subset by fast
nipkow@14187
  1410
wenzelm@13114
  1411
lemma append_eq_conv_conj:
nipkow@24526
  1412
  "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
nipkow@24526
  1413
apply (induct xs arbitrary: zs, simp, clarsimp)
paulson@14208
  1414
apply (case_tac zs, auto)
nipkow@13145
  1415
done
wenzelm@13142
  1416
nipkow@24526
  1417
lemma take_add: 
nipkow@24526
  1418
  "i+j \<le> length(xs) \<Longrightarrow> take (i+j) xs = take i xs @ take j (drop i xs)"
nipkow@24526
  1419
apply (induct xs arbitrary: i, auto) 
nipkow@24526
  1420
apply (case_tac i, simp_all)
paulson@14050
  1421
done
paulson@14050
  1422
nipkow@14300
  1423
lemma append_eq_append_conv_if:
nipkow@24526
  1424
 "(xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1425
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1426
   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
  1427
   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
  1428
apply(induct xs\<^isub>1 arbitrary: ys\<^isub>1)
nipkow@14300
  1429
 apply simp
nipkow@14300
  1430
apply(case_tac ys\<^isub>1)
nipkow@14300
  1431
apply simp_all
nipkow@14300
  1432
done
nipkow@14300
  1433
nipkow@15110
  1434
lemma take_hd_drop:
nipkow@24526
  1435
  "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
nipkow@24526
  1436
apply(induct xs arbitrary: n)
nipkow@15110
  1437
apply simp
nipkow@15110
  1438
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1439
done
nipkow@15110
  1440
nipkow@17501
  1441
lemma id_take_nth_drop:
nipkow@17501
  1442
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
nipkow@17501
  1443
proof -
nipkow@17501
  1444
  assume si: "i < length xs"
nipkow@17501
  1445
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
nipkow@17501
  1446
  moreover
nipkow@17501
  1447
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
nipkow@17501
  1448
    apply (rule_tac take_Suc_conv_app_nth) by arith
nipkow@17501
  1449
  ultimately show ?thesis by auto
nipkow@17501
  1450
qed
nipkow@17501
  1451
  
nipkow@17501
  1452
lemma upd_conv_take_nth_drop:
nipkow@17501
  1453
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1454
proof -
nipkow@17501
  1455
  assume i: "i < length xs"
nipkow@17501
  1456
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
nipkow@17501
  1457
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
nipkow@17501
  1458
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1459
    using i by (simp add: list_update_append)
nipkow@17501
  1460
  finally show ?thesis .
nipkow@17501
  1461
qed
nipkow@17501
  1462
haftmann@24796
  1463
lemma nth_drop':
haftmann@24796
  1464
  "i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs"
haftmann@24796
  1465
apply (induct i arbitrary: xs)
haftmann@24796
  1466
apply (simp add: neq_Nil_conv)
haftmann@24796
  1467
apply (erule exE)+
haftmann@24796
  1468
apply simp
haftmann@24796
  1469
apply (case_tac xs)
haftmann@24796
  1470
apply simp_all
haftmann@24796
  1471
done
haftmann@24796
  1472
wenzelm@13114
  1473
nipkow@15392
  1474
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1475
wenzelm@13142
  1476
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1477
by (induct xs) auto
wenzelm@13114
  1478
wenzelm@13142
  1479
lemma takeWhile_append1 [simp]:
nipkow@13145
  1480
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1481
by (induct xs) auto
wenzelm@13114
  1482
wenzelm@13142
  1483
lemma takeWhile_append2 [simp]:
nipkow@13145
  1484
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1485
by (induct xs) auto
wenzelm@13114
  1486
wenzelm@13142
  1487
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1488
by (induct xs) auto
wenzelm@13114
  1489
wenzelm@13142
  1490
lemma dropWhile_append1 [simp]:
nipkow@13145
  1491
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1492
by (induct xs) auto
wenzelm@13114
  1493
wenzelm@13142
  1494
lemma dropWhile_append2 [simp]:
nipkow@13145
  1495
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1496
by (induct xs) auto
wenzelm@13114
  1497
krauss@23971
  1498
lemma set_takeWhileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1499
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1500
nipkow@13913
  1501
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1502
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1503
by(induct xs, auto)
nipkow@13913
  1504
nipkow@13913
  1505
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1506
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1507
by(induct xs, auto)
nipkow@13913
  1508
nipkow@13913
  1509
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1510
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1511
by(induct xs, auto)
nipkow@13913
  1512
nipkow@17501
  1513
text{* The following two lemmmas could be generalized to an arbitrary
nipkow@17501
  1514
property. *}
nipkow@17501
  1515
nipkow@17501
  1516
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1517
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
nipkow@17501
  1518
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
nipkow@17501
  1519
nipkow@17501
  1520
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1521
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
nipkow@17501
  1522
apply(induct xs)
nipkow@17501
  1523
 apply simp
nipkow@17501
  1524
apply auto
nipkow@17501
  1525
apply(subst dropWhile_append2)
nipkow@17501
  1526
apply auto
nipkow@17501
  1527
done
nipkow@17501
  1528
nipkow@18423
  1529
lemma takeWhile_not_last:
nipkow@18423
  1530
 "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
nipkow@18423
  1531
apply(induct xs)
nipkow@18423
  1532
 apply simp
nipkow@18423
  1533
apply(case_tac xs)
nipkow@18423
  1534
apply(auto)
nipkow@18423
  1535
done
nipkow@18423
  1536
krauss@19770
  1537
lemma takeWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1538
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1539
  ==> takeWhile P l = takeWhile Q k"
nipkow@24349
  1540
by (induct k arbitrary: l) (simp_all)
krauss@18336
  1541
krauss@19770
  1542
lemma dropWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1543
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1544
  ==> dropWhile P l = dropWhile Q k"
nipkow@24349
  1545
by (induct k arbitrary: l, simp_all)
krauss@18336
  1546
wenzelm@13114
  1547
nipkow@15392
  1548
subsubsection {* @{text zip} *}
wenzelm@13114
  1549
wenzelm@13142
  1550
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1551
by (induct ys) auto
wenzelm@13114
  1552
wenzelm@13142
  1553
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1554
by simp
wenzelm@13114
  1555
wenzelm@13142
  1556
declare zip_Cons [simp del]
wenzelm@13114
  1557
nipkow@15281
  1558
lemma zip_Cons1:
nipkow@15281
  1559
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  1560
by(auto split:list.split)
nipkow@15281
  1561
wenzelm@13142
  1562
lemma length_zip [simp]:
krauss@22493
  1563
"length (zip xs ys) = min (length xs) (length ys)"
krauss@22493
  1564
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1565
wenzelm@13114
  1566
lemma zip_append1:
krauss@22493
  1567
"zip (xs @ ys) zs =
nipkow@13145
  1568
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
krauss@22493
  1569
by (induct xs zs rule:list_induct2') auto
wenzelm@13114
  1570
wenzelm@13114
  1571
lemma zip_append2:
krauss@22493
  1572
"zip xs (ys @ zs) =
nipkow@13145
  1573
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
krauss@22493
  1574
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1575
wenzelm@13142
  1576
lemma zip_append [simp]:
wenzelm@13142
  1577
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1578
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1579
by (simp add: zip_append1)
wenzelm@13114
  1580
wenzelm@13114
  1581
lemma zip_rev:
nipkow@14247
  1582
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1583
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1584
nipkow@23096
  1585
lemma map_zip_map:
nipkow@23096
  1586
 "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
nipkow@23096
  1587
apply(induct xs arbitrary:ys) apply simp
nipkow@23096
  1588
apply(case_tac ys)
nipkow@23096
  1589
apply simp_all
nipkow@23096
  1590
done
nipkow@23096
  1591
nipkow@23096
  1592
lemma map_zip_map2:
nipkow@23096
  1593
 "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
nipkow@23096
  1594
apply(induct xs arbitrary:ys) apply simp
nipkow@23096
  1595
apply(case_tac ys)
nipkow@23096
  1596
apply simp_all
nipkow@23096
  1597
done
nipkow@23096
  1598
wenzelm@13142
  1599
lemma nth_zip [simp]:
nipkow@24526
  1600
"[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
nipkow@24526
  1601
apply (induct ys arbitrary: i xs, simp)
nipkow@13145
  1602
apply (case_tac xs)
nipkow@13145
  1603
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1604
done
wenzelm@13114
  1605
wenzelm@13114
  1606
lemma set_zip:
nipkow@13145
  1607
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
  1608
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1609
wenzelm@13114
  1610
lemma zip_update:
nipkow@13145
  1611
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
  1612
by (rule sym, simp add: update_zip)
wenzelm@13114
  1613
wenzelm@13142
  1614
lemma zip_replicate [simp]:
nipkow@24526
  1615
  "zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
nipkow@24526
  1616
apply (induct i arbitrary: j, auto)
paulson@14208
  1617
apply (case_tac j, auto)
nipkow@13145
  1618
done
wenzelm@13114
  1619
nipkow@19487
  1620
lemma take_zip:
nipkow@24526
  1621
  "take n (zip xs ys) = zip (take n xs) (take n ys)"
nipkow@24526
  1622
apply (induct n arbitrary: xs ys)
nipkow@19487
  1623
 apply simp
nipkow@19487
  1624
apply (case_tac xs, simp)
nipkow@19487
  1625
apply (case_tac ys, simp_all)
nipkow@19487
  1626
done
nipkow@19487
  1627
nipkow@19487
  1628
lemma drop_zip:
nipkow@24526
  1629
  "drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
nipkow@24526
  1630
apply (induct n arbitrary: xs ys)
nipkow@19487
  1631
 apply simp
nipkow@19487
  1632
apply (case_tac xs, simp)
nipkow@19487
  1633
apply (case_tac ys, simp_all)
nipkow@19487
  1634
done
nipkow@19487
  1635
krauss@22493
  1636
lemma set_zip_leftD:
krauss@22493
  1637
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
krauss@22493
  1638
by (induct xs ys rule:list_induct2') auto
krauss@22493
  1639
krauss@22493
  1640
lemma set_zip_rightD:
krauss@22493
  1641
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
krauss@22493
  1642
by (induct xs ys rule:list_induct2') auto
wenzelm@13142
  1643
nipkow@23983
  1644
lemma in_set_zipE:
nipkow@23983
  1645
  "(x,y) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
nipkow@23983
  1646
by(blast dest: set_zip_leftD set_zip_rightD)
nipkow@23983
  1647
nipkow@15392
  1648
subsubsection {* @{text list_all2} *}
wenzelm@13114
  1649
kleing@14316
  1650
lemma list_all2_lengthD [intro?]: 
kleing@14316
  1651
  "list_all2 P xs ys ==> length xs = length ys"
nipkow@24349
  1652
by (simp add: list_all2_def)
haftmann@19607
  1653
haftmann@19787
  1654
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
nipkow@24349
  1655
by (simp add: list_all2_def)
haftmann@19607
  1656
haftmann@19787
  1657
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
nipkow@24349
  1658
by (simp add: list_all2_def)
haftmann@19607
  1659
haftmann@19607
  1660
lemma list_all2_Cons [iff, code]:
haftmann@19607
  1661
  "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@24349
  1662
by (auto simp add: list_all2_def)
wenzelm@13114
  1663
wenzelm@13114
  1664
lemma list_all2_Cons1:
nipkow@13145
  1665
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1666
by (cases ys) auto
wenzelm@13114
  1667
wenzelm@13114
  1668
lemma list_all2_Cons2:
nipkow@13145
  1669
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1670
by (cases xs) auto
wenzelm@13114
  1671
wenzelm@13142
  1672
lemma list_all2_rev [iff]:
nipkow@13145
  1673
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1674
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1675
kleing@13863
  1676
lemma list_all2_rev1:
kleing@13863
  1677
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1678
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1679
wenzelm@13114
  1680
lemma list_all2_append1:
nipkow@13145
  1681
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1682
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1683
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1684
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1685
apply (rule iffI)
nipkow@13145
  1686
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1687
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  1688
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1689
apply (simp add: ball_Un)
nipkow@13145
  1690
done
wenzelm@13114
  1691
wenzelm@13114
  1692
lemma list_all2_append2:
nipkow@13145
  1693
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1694
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1695
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1696
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1697
apply (rule iffI)
nipkow@13145
  1698
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1699
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  1700
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1701
apply (simp add: ball_Un)
nipkow@13145
  1702
done
wenzelm@13114
  1703
kleing@13863
  1704
lemma list_all2_append:
nipkow@14247
  1705
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  1706
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  1707
by (induct rule:list_induct2, simp_all)
kleing@13863
  1708
kleing@13863
  1709
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1710
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
nipkow@24349
  1711
by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1712
wenzelm@13114
  1713
lemma list_all2_conv_all_nth:
nipkow@13145
  1714
"list_all2 P xs ys =
nipkow@13145
  1715
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1716
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1717
berghofe@13883
  1718
lemma list_all2_trans:
berghofe@13883
  1719
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1720
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1721
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1722
proof (induct as)
berghofe@13883
  1723
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1724
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1725
  proof (induct bs)
berghofe@13883
  1726
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1727
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1728
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1729
  qed simp
berghofe@13883
  1730
qed simp
berghofe@13883
  1731
kleing@13863
  1732
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1733
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
nipkow@24349
  1734
by (simp add: list_all2_conv_all_nth)
kleing@13863
  1735
paulson@14395
  1736
lemma list_all2I:
paulson@14395
  1737
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
nipkow@24349
  1738
by (simp add: list_all2_def)
paulson@14395
  1739
kleing@14328
  1740
lemma list_all2_nthD:
kleing@13863
  1741
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@24349
  1742
by (simp add: list_all2_conv_all_nth)
kleing@13863
  1743
nipkow@14302
  1744
lemma list_all2_nthD2:
nipkow@14302
  1745
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@24349
  1746
by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  1747
kleing@13863
  1748
lemma list_all2_map1: 
kleing@13863
  1749
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
nipkow@24349
  1750
by (simp add: list_all2_conv_all_nth)
kleing@13863
  1751
kleing@13863
  1752
lemma list_all2_map2: 
kleing@13863
  1753
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
nipkow@24349
  1754
by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1755
kleing@14316
  1756
lemma list_all2_refl [intro?]:
kleing@13863
  1757
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
nipkow@24349
  1758
by (simp add: list_all2_conv_all_nth)
kleing@13863
  1759
kleing@13863
  1760
lemma list_all2_update_cong:
kleing@13863
  1761
  "\<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
  1762
by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1763
kleing@13863
  1764
lemma list_all2_update_cong2:
kleing@13863
  1765
  "\<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
  1766
by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1767
nipkow@14302
  1768
lemma list_all2_takeI [simp,intro?]:
nipkow@24526
  1769
  "list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@24526
  1770
apply (induct xs arbitrary: n ys)
nipkow@24526
  1771
 apply simp
nipkow@24526
  1772
apply (clarsimp simp add: list_all2_Cons1)
nipkow@24526
  1773
apply (case_tac n)
nipkow@24526
  1774
apply auto
nipkow@24526
  1775
done
nipkow@14302
  1776
nipkow@14302
  1777
lemma list_all2_dropI [simp,intro?]:
nipkow@24526
  1778
  "list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
nipkow@24526
  1779
apply (induct as arbitrary: n bs, simp)
nipkow@24526
  1780
apply (clarsimp simp add: list_all2_Cons1)
nipkow@24526
  1781
apply (case_tac n, simp, simp)
nipkow@24526
  1782
done
kleing@13863
  1783
kleing@14327
  1784
lemma list_all2_mono [intro?]:
nipkow@24526
  1785
  "list_all2 P xs ys \<Longrightarrow> (\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys) \<Longrightarrow> list_all2 Q xs ys"
nipkow@24526
  1786
apply (induct xs arbitrary: ys, simp)
nipkow@24526
  1787
apply (case_tac ys, auto)
nipkow@24526
  1788
done
kleing@13863
  1789
haftmann@22551
  1790
lemma list_all2_eq:
haftmann@22551
  1791
  "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
nipkow@24349
  1792
by (induct xs ys rule: list_induct2') auto
haftmann@22551
  1793
wenzelm@13142
  1794
nipkow@15392
  1795
subsubsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  1796
wenzelm@13142
  1797
lemma foldl_append [simp]:
nipkow@24526
  1798
  "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@24526
  1799
by (induct xs arbitrary: a) auto
wenzelm@13142
  1800
nipkow@14402
  1801
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  1802
by (induct xs) auto
nipkow@14402
  1803
nipkow@23096
  1804
lemma foldr_map: "foldr g (map f xs) a = foldr (g o f) xs a"
nipkow@23096
  1805
by(induct xs) simp_all
nipkow@23096
  1806
nipkow@24449
  1807
text{* For efficient code generation: avoid intermediate list. *}
nipkow@24449
  1808
lemma foldl_map[code unfold]:
nipkow@24449
  1809
  "foldl g a (map f xs) = foldl (%a x. g a (f x)) a xs"
nipkow@23096
  1810
by(induct xs arbitrary:a) simp_all
nipkow@23096
  1811
krauss@19770
  1812
lemma foldl_cong [fundef_cong, recdef_cong]:
krauss@18336
  1813
  "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] 
krauss@18336
  1814
  ==> foldl f a l = foldl g b k"
nipkow@24349
  1815
by (induct k arbitrary: a b l) simp_all
krauss@18336
  1816
krauss@19770
  1817
lemma foldr_cong [fundef_cong, recdef_cong]:
krauss@18336
  1818
  "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] 
krauss@18336
  1819
  ==> foldr f l a = foldr g k b"
nipkow@24349
  1820
by (induct k arbitrary: a b l) simp_all
krauss@18336
  1821
nipkow@24449
  1822
lemma (in semigroup_add) foldl_assoc:
haftmann@25062
  1823
shows "foldl op+ (x+y) zs = x + (foldl op+ y zs)"
nipkow@24449
  1824
by (induct zs arbitrary: y) (simp_all add:add_assoc)
nipkow@24449
  1825
nipkow@24449
  1826
lemma (in monoid_add) foldl_absorb0:
haftmann@25062
  1827
shows "x + (foldl op+ 0 zs) = foldl op+ x zs"
nipkow@24449
  1828
by (induct zs) (simp_all add:foldl_assoc)
nipkow@24449
  1829
nipkow@24449
  1830
nipkow@23096
  1831
text{* The ``First Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096
  1832
nipkow@23096
  1833
lemma foldl_foldr1_lemma:
nipkow@23096
  1834
 "foldl op + a xs = a + foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096
  1835
by (induct xs arbitrary: a) (auto simp:add_assoc)
nipkow@23096
  1836
nipkow@23096
  1837
corollary foldl_foldr1:
nipkow@23096
  1838
 "foldl op + 0 xs = foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096
  1839
by (simp add:foldl_foldr1_lemma)
nipkow@23096
  1840
nipkow@23096
  1841
nipkow@23096
  1842
text{* The ``Third Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096
  1843
nipkow@14402
  1844
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  1845
by (induct xs) auto
nipkow@14402
  1846
nipkow@14402
  1847
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  1848
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  1849
haftmann@25062
  1850
lemma (in ab_semigroup_add) foldr_conv_foldl: "foldr op + xs a = foldl op + a xs"
chaieb@24471
  1851
  by (induct xs, auto simp add: foldl_assoc add_commute)
chaieb@24471
  1852
wenzelm@13142
  1853
text {*
nipkow@13145
  1854
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1855
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1856
*}
wenzelm@13142
  1857
nipkow@24526
  1858
lemma start_le_sum: "(m::nat) <= n ==> m <= foldl (op +) n ns"
nipkow@24526
  1859
by (induct ns arbitrary: n) auto
nipkow@24526
  1860
nipkow@24526
  1861
lemma elem_le_sum: "(n::nat) : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1862
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1863
wenzelm@13142
  1864
lemma sum_eq_0_conv [iff]:
nipkow@24526
  1865
  "(foldl (op +) (m::nat) ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@24526
  1866
by (induct ns arbitrary: m) auto
wenzelm@13114
  1867
chaieb@24471
  1868
lemma foldr_invariant: 
chaieb@24471
  1869
  "\<lbrakk>Q x ; \<forall> x\<in> set xs. P x; \<forall> x y. P x \<and> Q y \<longrightarrow> Q (f x y) \<rbrakk> \<Longrightarrow> Q (foldr f xs x)"
chaieb@24471
  1870
  by (induct xs, simp_all)
chaieb@24471
  1871
chaieb@24471
  1872
lemma foldl_invariant: 
chaieb@24471
  1873
  "\<lbrakk>Q x ; \<forall> x\<in> set xs. P x; \<forall> x y. P x \<and> Q y \<longrightarrow> Q (f y x) \<rbrakk> \<Longrightarrow> Q (foldl f x xs)"
chaieb@24471
  1874
  by (induct xs arbitrary: x, simp_all)
chaieb@24471
  1875
nipkow@24449
  1876
text{* @{const foldl} and @{text concat} *}
nipkow@24449
  1877
nipkow@24449
  1878
lemma concat_conv_foldl: "concat xss = foldl op@ [] xss"
nipkow@24449
  1879
by (induct xss) (simp_all add:monoid_append.foldl_absorb0)
nipkow@24449
  1880
nipkow@24449
  1881
lemma foldl_conv_concat:
nipkow@24449
  1882
  "foldl (op @) xs xxs = xs @ (concat xxs)"
nipkow@24449
  1883
by(simp add:concat_conv_foldl monoid_append.foldl_absorb0)
nipkow@24449
  1884
nipkow@23096
  1885
subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
nipkow@23096
  1886
nipkow@24449
  1887
lemma listsum_append[simp]: "listsum (xs @ ys) = listsum xs + listsum ys"
nipkow@24449
  1888
by (induct xs) (simp_all add:add_assoc)
nipkow@24449
  1889
nipkow@24449
  1890
lemma listsum_rev[simp]:
nipkow@24449
  1891
fixes xs :: "'a::comm_monoid_add list"
nipkow@24449
  1892
shows "listsum (rev xs) = listsum xs"
nipkow@24449
  1893
by (induct xs) (simp_all add:add_ac)
nipkow@24449
  1894
nipkow@23096
  1895
lemma listsum_foldr:
nipkow@23096
  1896
 "listsum xs = foldr (op +) xs 0"
nipkow@23096
  1897
by(induct xs) auto
nipkow@23096
  1898
nipkow@24449
  1899
text{* For efficient code generation ---
nipkow@24449
  1900
       @{const listsum} is not tail recursive but @{const foldl} is. *}
nipkow@24449
  1901
lemma listsum[code unfold]: "listsum xs = foldl (op +) 0 xs"
nipkow@23096
  1902
by(simp add:listsum_foldr foldl_foldr1)
nipkow@23096
  1903
nipkow@24449
  1904
nipkow@23096
  1905
text{* Some syntactic sugar for summing a function over a list: *}
nipkow@23096
  1906
nipkow@23096
  1907
syntax
nipkow@23096
  1908
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
nipkow@23096
  1909
syntax (xsymbols)
nipkow@23096
  1910
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096
  1911
syntax (HTML output)
nipkow@23096
  1912
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096
  1913
nipkow@23096
  1914
translations -- {* Beware of argument permutation! *}
nipkow@23096
  1915
  "SUM x<-xs. b" == "CONST listsum (map (%x. b) xs)"
nipkow@23096
  1916
  "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (map (%x. b) xs)"
nipkow@23096
  1917
nipkow@23096
  1918
lemma listsum_0 [simp]: "(\<Sum>x\<leftarrow>xs. 0) = 0"
nipkow@23096
  1919
by (induct xs) simp_all
nipkow@23096
  1920
nipkow@23096
  1921
text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
nipkow@23096
  1922
lemma uminus_listsum_map:
nipkow@23096
  1923
 "- listsum (map f xs) = (listsum (map (uminus o f) xs) :: 'a::ab_group_add)"
nipkow@23096
  1924
by(induct xs) simp_all
nipkow@23096
  1925
wenzelm@13114
  1926
nipkow@24645
  1927
subsubsection {* @{text upt} *}
wenzelm@13114
  1928
nipkow@17090
  1929
lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
nipkow@17090
  1930
-- {* simp does not terminate! *}
nipkow@13145
  1931
by (induct j) auto
wenzelm@13142
  1932
nipkow@15425
  1933
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
nipkow@13145
  1934
by (subst upt_rec) simp
wenzelm@13114
  1935
nipkow@15425
  1936
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
nipkow@15281
  1937
by(induct j)simp_all
nipkow@15281
  1938
nipkow@15281
  1939
lemma upt_eq_Cons_conv:
nipkow@24526
  1940
 "([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
nipkow@24526
  1941
apply(induct j arbitrary: x xs)
nipkow@15281
  1942
 apply simp
nipkow@15281
  1943
apply(clarsimp simp add: append_eq_Cons_conv)
nipkow@15281
  1944
apply arith
nipkow@15281
  1945
done
nipkow@15281
  1946
nipkow@15425
  1947
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
nipkow@13145
  1948
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1949
by simp
wenzelm@13114
  1950
nipkow@15425
  1951
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
paulson@24632
  1952
by (metis upt_rec)
wenzelm@13114
  1953
nipkow@15425
  1954
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
nipkow@13145
  1955
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1956
by (induct k) auto
wenzelm@13114
  1957
nipkow@15425
  1958
lemma length_upt [simp]: "length [i..<j] = j - i"
nipkow@13145
  1959
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1960
nipkow@15425
  1961
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
nipkow@13145
  1962
apply (induct j)
nipkow@13145
  1963
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1964
done
wenzelm@13114
  1965
nipkow@17906
  1966
nipkow@17906
  1967
lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
nipkow@17906
  1968
by(simp add:upt_conv_Cons)
nipkow@17906
  1969
nipkow@17906
  1970
lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
nipkow@17906
  1971
apply(cases j)
nipkow@17906
  1972
 apply simp
nipkow@17906
  1973
by(simp add:upt_Suc_append)
nipkow@17906
  1974
nipkow@24526
  1975
lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]"
nipkow@24526
  1976
apply (induct m arbitrary: i, simp)
nipkow@13145
  1977
apply (subst upt_rec)
nipkow@13145
  1978
apply (rule sym)
nipkow@13145
  1979
apply (subst upt_rec)
nipkow@13145
  1980
apply (simp del: upt.simps)
nipkow@13145
  1981
done
nipkow@3507
  1982
nipkow@17501
  1983
lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
nipkow@17501
  1984
apply(induct j)
nipkow@17501
  1985
apply auto
nipkow@17501
  1986
done
nipkow@17501
  1987
nipkow@24645
  1988
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]"
nipkow@13145
  1989
by (induct n) auto
wenzelm@13114
  1990
nipkow@24526
  1991
lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
nipkow@24526
  1992
apply (induct n m  arbitrary: i rule: diff_induct)
nipkow@13145
  1993
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1994
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1995
done
wenzelm@13114
  1996
berghofe@13883
  1997
lemma nth_take_lemma:
nipkow@24526
  1998
  "k <= length xs ==> k <= length ys ==>
berghofe@13883
  1999
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
nipkow@24526
  2000
apply (atomize, induct k arbitrary: xs ys)
paulson@14208
  2001
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  2002
txt {* Both lists must be non-empty *}
paulson@14208
  2003
apply (case_tac xs, simp)
paulson@14208
  2004
apply (case_tac ys, clarify)
nipkow@13145
  2005
 apply (simp (no_asm_use))
nipkow@13145
  2006
apply clarify
nipkow@13145
  2007
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  2008
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  2009
apply blast
nipkow@13145
  2010
done
wenzelm@13114
  2011
wenzelm@13114
  2012
lemma nth_equalityI:
wenzelm@13114
  2013
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  2014
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  2015
apply (simp_all add: take_all)
nipkow@13145
  2016
done
wenzelm@13142
  2017
haftmann@24796
  2018
lemma map_nth:
haftmann@24796
  2019
  "map (\<lambda>i. xs ! i) [0..<length xs] = xs"
haftmann@24796
  2020
  by (rule nth_equalityI, auto)
haftmann@24796
  2021
kleing@13863
  2022
(* needs nth_equalityI *)
kleing@13863
  2023
lemma list_all2_antisym:
kleing@13863
  2024
  "\<lbrakk> (\<And>x y. \<lbrakk>P x y; Q y x\<rbrakk> \<Longrightarrow> x = y); list_all2 P xs ys; list_all2 Q ys xs \<rbrakk> 
kleing@13863
  2025
  \<Longrightarrow> xs = ys"
kleing@13863
  2026
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  2027
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  2028
  done
kleing@13863
  2029
wenzelm@13142
  2030
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  2031
-- {* The famous take-lemma. *}
nipkow@13145
  2032
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  2033
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  2034
done
wenzelm@13142
  2035
wenzelm@13142
  2036
nipkow@15302
  2037
lemma take_Cons':
nipkow@15302
  2038
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@15302
  2039
by (cases n) simp_all
nipkow@15302
  2040
nipkow@15302
  2041
lemma drop_Cons':
nipkow@15302
  2042
     "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@15302
  2043
by (cases n) simp_all
nipkow@15302
  2044
nipkow@15302
  2045
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@15302
  2046
by (cases n) simp_all
nipkow@15302
  2047
paulson@18622
  2048
lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard]
paulson@18622
  2049
lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard]
paulson@18622
  2050
lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard]
paulson@18622
  2051
paulson@18622
  2052
declare take_Cons_number_of [simp] 
paulson@18622
  2053
        drop_Cons_number_of [simp] 
paulson@18622
  2054
        nth_Cons_number_of [simp] 
nipkow@15302
  2055
nipkow@15302
  2056
nipkow@15392
  2057
subsubsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  2058
wenzelm@13142
  2059
lemma distinct_append [simp]:
nipkow@13145
  2060
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  2061
by (induct xs) auto
wenzelm@13142
  2062
nipkow@15305
  2063
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
nipkow@15305
  2064
by(induct xs) auto
nipkow@15305
  2065
wenzelm@13142
  2066
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  2067
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  2068
wenzelm@13142
  2069
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  2070
by (induct xs) auto
wenzelm@13142
  2071
nipkow@24566
  2072
lemma finite_distinct_list: "finite A \<Longrightarrow> EX xs. set xs = A & distinct xs"
paulson@24632
  2073
by (metis distinct_remdups finite_list set_remdups)
nipkow@24566
  2074
paulson@15072
  2075
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
nipkow@24349
  2076
by (induct x, auto) 
paulson@15072
  2077
paulson@15072
  2078
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
nipkow@24349
  2079
by (induct x, auto)
paulson@15072
  2080
nipkow@15245
  2081
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245
  2082
by (induct xs) auto
nipkow@15245
  2083
nipkow@15245
  2084
lemma length_remdups_eq[iff]:
nipkow@15245
  2085
  "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245
  2086
apply(induct xs)
nipkow@15245
  2087
 apply auto
nipkow@15245
  2088
apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245
  2089
 apply arith
nipkow@15245
  2090
apply(rule length_remdups_leq)
nipkow@15245
  2091
done
nipkow@15245
  2092
nipkow@18490
  2093
nipkow@18490
  2094
lemma distinct_map:
nipkow@18490
  2095
  "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
nipkow@18490
  2096
by (induct xs) auto
nipkow@18490
  2097
nipkow@18490
  2098
wenzelm@13142
  2099
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  2100
by (induct xs) auto
wenzelm@13114
  2101
nipkow@17501
  2102
lemma distinct_upt[simp]: "distinct[i..<j]"
nipkow@17501
  2103
by (induct j) auto
nipkow@17501
  2104
nipkow@24526
  2105
lemma distinct_take[simp]: "distinct xs \<Longrightarrow> distinct (take i xs)"
nipkow@24526
  2106
apply(induct xs arbitrary: i)
nipkow@17501
  2107
 apply simp
nipkow@17501
  2108
apply (case_tac i)
nipkow@17501
  2109
 apply simp_all
nipkow@17501
  2110
apply(blast dest:in_set_takeD)
nipkow@17501
  2111
done
nipkow@17501
  2112
nipkow@24526
  2113
lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)"
nipkow@24526
  2114
apply(induct xs arbitrary: i)
nipkow@17501
  2115
 apply simp
nipkow@17501
  2116
apply (case_tac i)
nipkow@17501
  2117
 apply simp_all
nipkow@17501
  2118
done
nipkow@17501
  2119
nipkow@17501
  2120
lemma distinct_list_update:
nipkow@17501
  2121
assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
nipkow@17501
  2122
shows "distinct (xs[i:=a])"
nipkow@17501
  2123
proof (cases "i < length xs")
nipkow@17501
  2124
  case True
nipkow@17501
  2125
  with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
nipkow@17501
  2126
    apply (drule_tac id_take_nth_drop) by simp
nipkow@17501
  2127
  with d True show ?thesis
nipkow@17501
  2128
    apply (simp add: upd_conv_take_nth_drop)
nipkow@17501
  2129
    apply (drule subst [OF id_take_nth_drop]) apply assumption
nipkow@17501
  2130
    apply simp apply (cases "a = xs!i") apply simp by blast
nipkow@17501
  2131
next
nipkow@17501
  2132
  case False with d show ?thesis by auto
nipkow@17501
  2133
qed
nipkow@17501
  2134
nipkow@17501
  2135
nipkow@17501
  2136
text {* It is best to avoid this indexed version of distinct, but
nipkow@17501
  2137
sometimes it is useful. *}
nipkow@17501
  2138
wenzelm@13142
  2139
lemma distinct_conv_nth:
nipkow@17501
  2140
"distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@15251
  2141
apply (induct xs, simp, simp)
paulson@14208
  2142
apply (rule iffI, clarsimp)
nipkow@13145
  2143
 apply (case_tac i)
paulson@14208
  2144
apply (case_tac j, simp)
nipkow@13145
  2145
apply (simp add: set_conv_nth)
nipkow@13145
  2146
 apply (case_tac j)
paulson@24648
  2147
apply (clarsimp simp add: set_conv_nth, simp) 
nipkow@13145
  2148
apply (rule conjI)
paulson@24648
  2149
(*TOO SLOW
paulson@24632
  2150
apply (metis Zero_neq_Suc gr0_conv_Suc in_set_conv_nth lessI less_trans_Suc nth_Cons' nth_Cons_Suc)
paulson@24648
  2151
*)
paulson@24648
  2152
 apply (clarsimp simp add: set_conv_nth)
paulson@24648
  2153
 apply (erule_tac x = 0 in allE, simp)
paulson@24648
  2154
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
wenzelm@25130
  2155
(*TOO SLOW
paulson@24632
  2156
apply (metis Suc_Suc_eq lessI less_trans_Suc nth_Cons_Suc)
wenzelm@25130
  2157
*)
wenzelm@25130
  2158
apply (erule_tac x = "Suc i" in allE, simp)
wenzelm@25130
  2159
apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145
  2160
done
wenzelm@13114
  2161
nipkow@18490
  2162
lemma nth_eq_iff_index_eq:
nipkow@18490
  2163
 "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
nipkow@18490
  2164
by(auto simp: distinct_conv_nth)
nipkow@18490
  2165
nipkow@15110
  2166
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
nipkow@24349
  2167
by (induct xs) auto
kleing@14388
  2168
nipkow@15110
  2169
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388
  2170
proof (induct xs)
kleing@14388
  2171
  case Nil thus ?case by simp
kleing@14388
  2172
next
kleing@14388
  2173
  case (Cons x xs)
kleing@14388
  2174
  show ?case
kleing@14388
  2175
  proof (cases "x \<in> set xs")
kleing@14388
  2176
    case False with Cons show ?thesis by simp
kleing@14388
  2177
  next
kleing@14388
  2178
    case True with Cons.prems
kleing@14388
  2179
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  2180
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  2181
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  2182
    ultimately have False by simp
kleing@14388
  2183
    thus ?thesis ..
kleing@14388
  2184
  qed
kleing@14388
  2185
qed
kleing@14388
  2186
nipkow@18490
  2187
nipkow@18490
  2188
lemma length_remdups_concat:
nipkow@18490
  2189
 "length(remdups(concat xss)) = card(\<Union>xs \<in> set xss. set xs)"
nipkow@24308
  2190
by(simp add: set_concat distinct_card[symmetric])
nipkow@17906
  2191
nipkow@17906
  2192
nipkow@15392
  2193
subsubsection {* @{text remove1} *}
nipkow@15110
  2194
nipkow@18049
  2195
lemma remove1_append:
nipkow@18049
  2196
  "remove1 x (xs @ ys) =
nipkow@18049
  2197
  (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
nipkow@18049
  2198
by (induct xs) auto
nipkow@18049
  2199
nipkow@23479
  2200
lemma in_set_remove1[simp]:
nipkow@23479
  2201
  "a \<noteq> b \<Longrightarrow> a : set(remove1 b xs) = (a : set xs)"
nipkow@23479
  2202
apply (induct xs)
nipkow@23479
  2203
apply auto
nipkow@23479
  2204
done
nipkow@23479
  2205
nipkow@15110
  2206
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110
  2207
apply(induct xs)
nipkow@15110
  2208
 apply simp
nipkow@15110
  2209
apply simp
nipkow@15110
  2210
apply blast
nipkow@15110
  2211
done
nipkow@15110
  2212
paulson@17724
  2213
lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110
  2214
apply(induct xs)
nipkow@15110
  2215
 apply simp
nipkow@15110
  2216
apply simp
nipkow@15110
  2217
apply blast
nipkow@15110
  2218
done
nipkow@15110
  2219
nipkow@23479
  2220
lemma length_remove1:
nipkow@23479
  2221
  "length(remove1 x xs) = (if x : set xs then length xs - 1 else length xs)"
nipkow@23479
  2222
apply (induct xs)
nipkow@23479
  2223
 apply (auto dest!:length_pos_if_in_set)
nipkow@23479
  2224
done
nipkow@23479
  2225
nipkow@18049
  2226
lemma remove1_filter_not[simp]:
nipkow@18049
  2227
  "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
nipkow@18049
  2228
by(induct xs) auto
nipkow@18049
  2229
nipkow@15110
  2230
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110
  2231
apply(insert set_remove1_subset)
nipkow@15110
  2232
apply fast
nipkow@15110
  2233
done
nipkow@15110
  2234
nipkow@15110
  2235
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110
  2236
by (induct xs) simp_all
nipkow@15110
  2237
wenzelm@13114
  2238
nipkow@15392
  2239
subsubsection {* @{text replicate} *}
wenzelm@13114
  2240
wenzelm@13142
  2241
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  2242
by (induct n) auto
nipkow@13124
  2243
wenzelm@13142
  2244
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  2245
by (induct n) auto
wenzelm@13114
  2246
wenzelm@13114
  2247
lemma replicate_app_Cons_same:
nipkow@13145
  2248
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  2249
by (induct n) auto
wenzelm@13114
  2250
wenzelm@13142
  2251
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208
  2252
apply (induct n, simp)
nipkow@13145
  2253
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  2254
done
wenzelm@13114
  2255
wenzelm@13142
  2256
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  2257
by (induct n) auto
wenzelm@13114
  2258
nipkow@16397
  2259
text{* Courtesy of Matthias Daum: *}
nipkow@16397
  2260
lemma append_replicate_commute:
nipkow@16397
  2261
  "replicate n x @ replicate k x = replicate k x @ replicate n x"
nipkow@16397
  2262
apply (simp add: replicate_add [THEN sym])
nipkow@16397
  2263
apply (simp add: add_commute)
nipkow@16397
  2264
done
nipkow@16397
  2265
wenzelm@13142
  2266
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  2267
by (induct n) auto
wenzelm@13114
  2268
wenzelm@13142
  2269
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  2270
by (induct n) auto
wenzelm@13114
  2271
wenzelm@13142
  2272
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  2273
by (atomize (full), induct n) auto
wenzelm@13114
  2274
nipkow@24526
  2275
lemma nth_replicate[simp]: "i < n ==> (replicate n x)!i = x"
nipkow@24526
  2276
apply (induct n arbitrary: i, simp)
nipkow@13145
  2277
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  2278
done
wenzelm@13114
  2279
nipkow@16397
  2280
text{* Courtesy of Matthias Daum (2 lemmas): *}
nipkow@16397
  2281
lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
nipkow@16397
  2282
apply (case_tac "k \<le> i")
nipkow@16397
  2283
 apply  (simp add: min_def)
nipkow@16397
  2284
apply (drule not_leE)
nipkow@16397
  2285
apply (simp add: min_def)
nipkow@16397
  2286
apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
nipkow@16397
  2287
 apply  simp
nipkow@16397
  2288
apply (simp add: replicate_add [symmetric])
nipkow@16397
  2289
done
nipkow@16397
  2290
nipkow@24526
  2291
lemma drop_replicate[simp]: "drop i (replicate k x) = replicate (k-i) x"
nipkow@24526
  2292
apply (induct k arbitrary: i)
nipkow@16397
  2293
 apply simp
nipkow@16397
  2294
apply clarsimp
nipkow@16397
  2295
apply (case_tac i)
nipkow@16397
  2296
 apply simp
nipkow@16397
  2297
apply clarsimp
nipkow@16397
  2298
done
nipkow@16397
  2299
nipkow@16397
  2300
wenzelm@13142
  2301
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  2302
by (induct n) auto
wenzelm@13114
  2303
wenzelm@13142
  2304
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  2305
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  2306
wenzelm@13142
  2307
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  2308
by auto
wenzelm@13114
  2309
wenzelm@13142
  2310
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  2311
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  2312
haftmann@24796
  2313
lemma replicate_append_same:
haftmann@24796
  2314
  "replicate i x @ [x] = x # replicate i x"
haftmann@24796
  2315
  by (induct i) simp_all
haftmann@24796
  2316
haftmann@24796
  2317
lemma map_replicate_trivial:
haftmann@24796
  2318
  "map (\<lambda>i. x) [0..<i] = replicate i x"
haftmann@24796
  2319
  by (induct i) (simp_all add: replicate_append_same)
haftmann@24796
  2320
wenzelm@13114
  2321
nipkow@15392
  2322
subsubsection{*@{text rotate1} and @{text rotate}*}
nipkow@15302
  2323
nipkow@15302
  2324
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
nipkow@15302
  2325
by(simp add:rotate1_def)
nipkow@15302
  2326
nipkow@15302
  2327
lemma rotate0[simp]: "rotate 0 = id"
nipkow@15302
  2328
by(simp add:rotate_def)
nipkow@15302
  2329
nipkow@15302
  2330
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
nipkow@15302
  2331
by(simp add:rotate_def)
nipkow@15302
  2332
nipkow@15302
  2333
lemma rotate_add:
nipkow@15302
  2334
  "rotate (m+n) = rotate m o rotate n"
nipkow@15302
  2335
by(simp add:rotate_def funpow_add)
nipkow@15302
  2336
nipkow@15302
  2337
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
nipkow@15302
  2338
by(simp add:rotate_add)
nipkow@15302
  2339
nipkow@18049
  2340
lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
nipkow@18049
  2341
by(simp add:rotate_def funpow_swap1)
nipkow@18049
  2342
nipkow@15302
  2343
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
nipkow@15302
  2344
by(cases xs) simp_all
nipkow@15302
  2345
nipkow@15302
  2346
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2347
apply(induct n)
nipkow@15302
  2348
 apply simp
nipkow@15302
  2349
apply (simp add:rotate_def)
nipkow@13145
  2350
done
wenzelm@13114
  2351
nipkow@15302
  2352
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
nipkow@15302
  2353
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2354
nipkow@15302
  2355
lemma rotate_drop_take:
nipkow@15302
  2356
  "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
nipkow@15302
  2357
apply(induct n)
nipkow@15302
  2358
 apply simp
nipkow@15302
  2359
apply(simp add:rotate_def)
nipkow@15302
  2360
apply(cases "xs = []")
nipkow@15302
  2361
 apply (simp)
nipkow@15302
  2362
apply(case_tac "n mod length xs = 0")
nipkow@15302
  2363
 apply(simp add:mod_Suc)
nipkow@15302
  2364
 apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
nipkow@15302
  2365
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
nipkow@15302
  2366
                take_hd_drop linorder_not_le)
nipkow@13145
  2367
done
wenzelm@13114
  2368
nipkow@15302
  2369
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
nipkow@15302
  2370
by(simp add:rotate_drop_take)
nipkow@15302
  2371
nipkow@15302
  2372
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2373
by(simp add:rotate_drop_take)
nipkow@15302
  2374
nipkow@15302
  2375
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
nipkow@15302
  2376
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2377
nipkow@24526
  2378
lemma length_rotate[simp]: "length(rotate n xs) = length xs"
nipkow@24526
  2379
by (induct n arbitrary: xs) (simp_all add:rotate_def)
nipkow@15302
  2380
nipkow@15302
  2381
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
nipkow@15302
  2382
by(simp add:rotate1_def split:list.split) blast
nipkow@15302
  2383
nipkow@15302
  2384
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
nipkow@15302
  2385
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2386
nipkow@15302
  2387
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
nipkow@15302
  2388
by(simp add:rotate_drop_take take_map drop_map)
nipkow@15302
  2389
nipkow@15302
  2390
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
nipkow@15302
  2391
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2392
nipkow@15302
  2393
lemma set_rotate[simp]: "set(rotate n xs) = set xs"
nipkow@15302
  2394
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2395
nipkow@15302
  2396
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
nipkow@15302
  2397
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2398
nipkow@15302
  2399
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
nipkow@15302
  2400
by (induct n) (simp_all add:rotate_def)
wenzelm@13114
  2401
nipkow@15439
  2402
lemma rotate_rev:
nipkow@15439
  2403
  "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
nipkow@15439
  2404
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2405
apply(cases "length xs = 0")
nipkow@15439
  2406
 apply simp
nipkow@15439
  2407
apply(cases "n mod length xs = 0")
nipkow@15439
  2408
 apply simp
nipkow@15439
  2409
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2410
done
nipkow@15439
  2411
nipkow@18423
  2412
lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
nipkow@18423
  2413
apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
nipkow@18423
  2414
apply(subgoal_tac "length xs \<noteq> 0")
nipkow@18423
  2415
 prefer 2 apply simp
nipkow@18423
  2416
using mod_less_divisor[of "length xs" n] by arith
nipkow@18423
  2417
wenzelm@13114
  2418
nipkow@15392
  2419
subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  2420
wenzelm@13142
  2421
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  2422
by (auto simp add: sublist_def)
wenzelm@13114
  2423
wenzelm@13142
  2424
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  2425
by (auto simp add: sublist_def)
wenzelm@13114
  2426
nipkow@15281
  2427
lemma length_sublist:
nipkow@15281
  2428
  "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
nipkow@15281
  2429
by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
nipkow@15281
  2430
nipkow@15281
  2431
lemma sublist_shift_lemma_Suc:
nipkow@24526
  2432
  "map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
nipkow@24526
  2433
   map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
nipkow@24526
  2434
apply(induct xs arbitrary: "is")
nipkow@15281
  2435
 apply simp
nipkow@15281
  2436
apply (case_tac "is")
nipkow@15281
  2437
 apply simp
nipkow@15281
  2438
apply simp
nipkow@15281
  2439
done
nipkow@15281
  2440
wenzelm@13114
  2441
lemma sublist_shift_lemma:
nipkow@23279
  2442
     "map fst [p<-zip xs [i..<i + length xs] . snd p : A] =
nipkow@23279
  2443
      map fst [p<-zip xs [0..<length xs] . snd p + i : A]"
nipkow@13145
  2444
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  2445
wenzelm@13114
  2446
lemma sublist_append:
paulson@15168
  2447
     "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  2448
apply (unfold sublist_def)
paulson@14208
  2449
apply (induct l' rule: rev_induct, simp)
nipkow@13145
  2450
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  2451
apply (simp add: add_commute)
nipkow@13145
  2452
done
wenzelm@13114
  2453
wenzelm@13114
  2454
lemma sublist_Cons:
nipkow@13145
  2455
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  2456
apply (induct l rule: rev_induct)
nipkow@13145
  2457
 apply (simp add: sublist_def)
nipkow@13145
  2458
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  2459
done
wenzelm@13114
  2460
nipkow@24526
  2461
lemma set_sublist: "set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
nipkow@24526
  2462
apply(induct xs arbitrary: I)
nipkow@15281
  2463
 apply simp
chaieb@25112
  2464
apply(auto simp add: neq0_conv sublist_Cons nth_Cons split:nat.split elim: lessE)
nipkow@15281
  2465
 apply(erule lessE)
nipkow@15281
  2466
  apply auto
nipkow@15281
  2467
apply(erule lessE)
nipkow@15281
  2468
apply auto
nipkow@15281
  2469
done
nipkow@15281
  2470
nipkow@15281
  2471
lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
nipkow@15281
  2472
by(auto simp add:set_sublist)
nipkow@15281
  2473
nipkow@15281
  2474
lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
nipkow@15281
  2475
by(auto simp add:set_sublist)
nipkow@15281
  2476
nipkow@15281
  2477
lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
nipkow@15281
  2478
by(auto simp add:set_sublist)
nipkow@15281
  2479
wenzelm@13142
  2480
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  2481
by (simp add: sublist_Cons)
wenzelm@13114
  2482
nipkow@15281
  2483
nipkow@24526
  2484
lemma distinct_sublistI[simp]: "distinct xs \<Longrightarrow> distinct(sublist xs I)"
nipkow@24526
  2485
apply(induct xs arbitrary: I)
nipkow@15281
  2486
 apply simp
nipkow@15281
  2487
apply(auto simp add:sublist_Cons)
nipkow@15281
  2488
done
nipkow@15281
  2489
nipkow@15281
  2490
nipkow@15045
  2491
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
paulson@14208
  2492
apply (induct l rule: rev_induct, simp)
nipkow@13145
  2493
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  2494
done
wenzelm@13114
  2495
nipkow@24526
  2496
lemma filter_in_sublist:
nipkow@24526
  2497
 "distinct xs \<Longrightarrow> filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
nipkow@24526
  2498
proof (induct xs arbitrary: s)
nipkow@17501
  2499
  case Nil thus ?case by simp
nipkow@17501
  2500
next
nipkow@17501
  2501
  case (Cons a xs)
nipkow@17501
  2502
  moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
nipkow@17501
  2503
  ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
nipkow@17501
  2504
qed
nipkow@17501
  2505
wenzelm@13114
  2506
nipkow@19390
  2507
subsubsection {* @{const splice} *}
nipkow@19390
  2508
haftmann@19607
  2509
lemma splice_Nil2 [simp, code]:
nipkow@19390
  2510
 "splice xs [] = xs"
nipkow@19390
  2511
by (cases xs) simp_all
nipkow@19390
  2512
haftmann@19607
  2513
lemma splice_Cons_Cons [simp, code]:
nipkow@19390
  2514
 "splice (x#xs) (y#ys) = x # y # splice xs ys"
nipkow@19390
  2515
by simp
nipkow@19390
  2516
haftmann@19607
  2517
declare splice.simps(2) [simp del, code del]
nipkow@19390
  2518
nipkow@24526
  2519
lemma length_splice[simp]: "length(splice xs ys) = length xs + length ys"
nipkow@24526
  2520
apply(induct xs arbitrary: ys) apply simp
nipkow@22793
  2521
apply(case_tac ys)
nipkow@22793
  2522
 apply auto
nipkow@22793
  2523
done
nipkow@22793
  2524
nipkow@24616
  2525
nipkow@24616
  2526
subsection {*Sorting*}
nipkow@24616
  2527
nipkow@24617
  2528
text{* Currently it is not shown that @{const sort} returns a
nipkow@24617
  2529
permutation of its input because the nicest proof is via multisets,
nipkow@24617
  2530
which are not yet available. Alternatively one could define a function
nipkow@24617
  2531
that counts the number of occurrences of an element in a list and use
nipkow@24617
  2532
that instead of multisets to state the correctness property. *}
nipkow@24617
  2533
nipkow@24616
  2534
context linorder
nipkow@24616
  2535
begin
nipkow@24616
  2536
haftmann@25062
  2537
lemma sorted_Cons: "sorted (x#xs) = (sorted xs & (ALL y:set xs. x <= y))"
nipkow@24616
  2538
apply(induct xs arbitrary: x) apply simp
nipkow@24616
  2539
by simp (blast intro: order_trans)
nipkow@24616
  2540
nipkow@24616
  2541
lemma sorted_append:
haftmann@25062
  2542
  "sorted (xs@ys) = (sorted xs & sorted ys & (\<forall>x \<in> set xs. \<forall>y \<in> set ys. x\<le>y))"
nipkow@24616
  2543
by (induct xs) (auto simp add:sorted_Cons)
nipkow@24616
  2544
nipkow@24616
  2545
lemma set_insort: "set(insort x xs) = insert x (set xs)"
nipkow@24616
  2546
by (induct xs) auto
nipkow@24616
  2547
nipkow@24617
  2548
lemma set_sort[simp]: "set(sort xs) = set xs"
nipkow@24616
  2549
by (induct xs) (simp_all add:set_insort)
nipkow@24616
  2550
nipkow@24616
  2551
lemma distinct_insort: "distinct (insort x xs) = (x \<notin> set xs \<and> distinct xs)"
nipkow@24616
  2552
by(induct xs)(auto simp:set_insort)
nipkow@24616
  2553
nipkow@24617
  2554
lemma distinct_sort[simp]: "distinct (sort xs) = distinct xs"
nipkow@24616
  2555
by(induct xs)(simp_all add:distinct_insort set_sort)
nipkow@24616
  2556
nipkow@24616
  2557
lemma sorted_insort: "sorted (insort x xs) = sorted xs"
nipkow@24616
  2558
apply (induct xs)
nipkow@24650
  2559
 apply(auto simp:sorted_Cons set_insort)
nipkow@24616
  2560
done
nipkow@24616
  2561
nipkow@24616
  2562
theorem sorted_sort[simp]: "sorted (sort xs)"
nipkow@24616
  2563
by (induct xs) (auto simp:sorted_insort)
nipkow@24616
  2564
nipkow@24645
  2565
lemma sorted_distinct_set_unique:
nipkow@24645
  2566
assumes "sorted xs" "distinct xs" "sorted ys" "distinct ys" "set xs = set ys"
nipkow@24645
  2567
shows "xs = ys"
nipkow@24645
  2568
proof -
nipkow@24645
  2569
  from assms have 1: "length xs = length ys" by (metis distinct_card)
nipkow@24645
  2570
  from assms show ?thesis
nipkow@24645
  2571
  proof(induct rule:list_induct2[OF 1])
nipkow@24645
  2572
    case 1 show ?case by simp
nipkow@24645
  2573
  next
nipkow@24645
  2574
    case 2 thus ?case by (simp add:sorted_Cons)
nipkow@24645
  2575
       (metis Diff_insert_absorb antisym insertE insert_iff)
nipkow@24645
  2576
  qed
nipkow@24645
  2577
qed
nipkow@24645
  2578
nipkow@24645
  2579
lemma finite_sorted_distinct_unique:
nipkow@24645
  2580
shows "finite A \<Longrightarrow> EX! xs. set xs = A & sorted xs & distinct xs"
nipkow@24645
  2581
apply(drule finite_distinct_list)
nipkow@24645
  2582
apply clarify
nipkow@24645
  2583
apply(rule_tac a="sort xs" in ex1I)
nipkow@24645
  2584
apply (auto simp: sorted_distinct_set_unique)
nipkow@24645
  2585
done
nipkow@24645
  2586
nipkow@24616
  2587
end
nipkow@24616
  2588
nipkow@24616
  2589
nipkow@25069
  2590
subsubsection {* @{text sorted_list_of_set} *}
nipkow@25069
  2591
nipkow@25069
  2592
text{* This function maps (finite) linearly ordered sets to sorted
nipkow@25069
  2593
lists. Warning: in most cases it is not a good idea to convert from
nipkow@25069
  2594
sets to lists but one should convert in the other direction (via
nipkow@25069
  2595
@{const set}). *}
nipkow@25069
  2596
nipkow@25069
  2597
nipkow@25069
  2598
context linorder
nipkow@25069
  2599
begin
nipkow@25069
  2600
nipkow@25069
  2601
definition
nipkow@25069
  2602
 sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where
nipkow@25069
  2603
"sorted_list_of_set A == THE xs. set xs = A & sorted xs & distinct xs"
nipkow@25069
  2604
nipkow@25069
  2605
lemma sorted_list_of_set[simp]: "finite A \<Longrightarrow>
nipkow@25069
  2606
  set(sorted_list_of_set A) = A &
nipkow@25069
  2607
  sorted(sorted_list_of_set A) & distinct(sorted_list_of_set A)"
nipkow@25069
  2608
apply(simp add:sorted_list_of_set_def)
nipkow@25069
  2609
apply(rule the1I2)
nipkow@25069
  2610
 apply(simp_all add: finite_sorted_distinct_unique)
nipkow@25069
  2611
done
nipkow@25069
  2612
nipkow@25069
  2613
lemma sorted_list_of_empty[simp]: "sorted_list_of_set {} = []"
nipkow@25069
  2614
unfolding sorted_list_of_set_def
nipkow@25069
  2615
apply(subst the_equality[of _ "[]"])
nipkow@25069
  2616
apply simp_all
nipkow@25069
  2617
done
nipkow@25069
  2618
nipkow@25069
  2619
end
nipkow@25069
  2620
nipkow@25069
  2621
nipkow@24645
  2622
subsubsection {* @{text upto}: the generic interval-list *}
nipkow@24645
  2623
nipkow@24697
  2624
class finite_intvl_succ = linorder +
nipkow@24697
  2625
fixes successor :: "'a \<Rightarrow> 'a"
nipkow@25069
  2626
assumes finite_intvl: "finite{a..b}"
haftmann@25062
  2627
and successor_incr: "a < successor a"
haftmann@25062
  2628
and ord_discrete: "\<not>(\<exists>x. a < x & x < successor a)"
nipkow@24697
  2629
nipkow@24697
  2630
context finite_intvl_succ
nipkow@24697
  2631
begin
nipkow@24697
  2632
nipkow@24697
  2633
definition
haftmann@25062
  2634
 upto :: "'a \<Rightarrow> 'a \<Rightarrow> 'a list" ("(1[_../_])") where
nipkow@25069
  2635
"upto i j == sorted_list_of_set {i..j}"
nipkow@25069
  2636
nipkow@25069
  2637
lemma upto[simp]: "set[a..b] = {a..b} & sorted[a..b] & distinct[a..b]"
nipkow@25069
  2638
by(simp add:upto_def finite_intvl)
nipkow@24697
  2639
haftmann@25062
  2640
lemma insert_intvl: "i \<le> j \<Longrightarrow> insert i {successor i..j} = {i..j}"
nipkow@24697
  2641
apply(insert successor_incr[of i])
nipkow@24697
  2642
apply(auto simp: atLeastAtMost_def atLeast_def atMost_def)
nipkow@24697
  2643
apply (metis ord_discrete less_le not_le)
nipkow@24645
  2644
done
nipkow@24645
  2645
nipkow@25069
  2646
lemma sorted_list_of_set_rec: "i \<le> j \<Longrightarrow>
nipkow@25069
  2647
  sorted_list_of_set {i..j} = i # sorted_list_of_set {successor i..j}"
nipkow@25069
  2648
apply(simp add:sorted_list_of_set_def upto_def)
nipkow@25069
  2649
apply (rule the1_equality[OF finite_sorted_distinct_unique])
nipkow@25069
  2650
 apply (simp add:finite_intvl)
nipkow@25069
  2651
apply(rule the1I2[OF finite_sorted_distinct_unique])
nipkow@25069
  2652
 apply (simp add:finite_intvl)
nipkow@25069
  2653
apply (simp add: sorted_Cons insert_intvl Ball_def)
nipkow@25069
  2654
apply (metis successor_incr leD less_imp_le order_trans)
nipkow@25069
  2655
done
nipkow@25069
  2656
haftmann@25062
  2657
lemma upto_rec[code]: "[i..j] = (if i \<le> j then i # [successor i..j] else [])"
nipkow@25069
  2658
by(simp add: upto_def sorted_list_of_set_rec)
nipkow@24697
  2659
nipkow@24697
  2660
end
nipkow@24697
  2661
nipkow@24697
  2662
text{* The integers are an instance of the above class: *}
nipkow@24697
  2663
nipkow@24697
  2664
instance int:: finite_intvl_succ
nipkow@24697
  2665
  successor_int_def: "successor == (%i. i+1)"
haftmann@25082
  2666
by intro_classes (simp_all add: successor_int_def)
nipkow@24645
  2667
nipkow@24697
  2668
text{* Now @{term"[i..j::int]"} is defined for integers. *}
nipkow@24697
  2669
nipkow@24698
  2670
hide (open) const successor
nipkow@24698
  2671
nipkow@24645
  2672
nipkow@15392
  2673
subsubsection {* @{text lists}: the list-forming operator over sets *}
nipkow@15302
  2674
berghofe@23740
  2675
inductive_set
berghofe@22262
  2676
  lists :: "'a set => 'a list set"
berghofe@23740
  2677
  for A :: "'a set"
berghofe@23740
  2678
where
berghofe@23740
  2679
    Nil [intro!]: "[]: lists A"
paulson@24286
  2680
  | Cons [intro!,noatp]: "[| a: A;l: lists A|] ==> a#l : lists A"
paulson@24286
  2681
paulson@24286
  2682
inductive_cases listsE [elim!,noatp]: "x#l : lists A"
paulson@24286
  2683
inductive_cases listspE [elim!,noatp]: "listsp A (x # l)"
berghofe@23740
  2684
berghofe@23740
  2685
lemma listsp_mono [mono]: "A \<le> B ==> listsp A \<le> listsp B"
nipkow@24349
  2686
by (clarify, erule listsp.induct, blast+)
berghofe@22262
  2687
berghofe@23740
  2688
lemmas lists_mono = listsp_mono [to_set]
berghofe@22262
  2689
haftmann@22422
  2690
lemma listsp_infI:
haftmann@22422
  2691
  assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
nipkow@24349
  2692
by induct blast+
nipkow@15302
  2693
haftmann@22422
  2694
lemmas lists_IntI = listsp_infI [to_set]
haftmann@22422
  2695
haftmann@22422
  2696
lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
haftmann@22422
  2697
proof (rule mono_inf [where f=listsp, THEN order_antisym])
berghofe@22262
  2698
  show "mono listsp" by (simp add: mono_def listsp_mono)
haftmann@22422
  2699
  show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro: listsp_infI)
kleing@14388
  2700
qed
kleing@14388
  2701
haftmann@22422
  2702
lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_eq inf_bool_eq]
haftmann@22422
  2703
haftmann@22422
  2704
lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set]
berghofe@22262
  2705
berghofe@22262
  2706
lemma append_in_listsp_conv [iff]:
berghofe@22262
  2707
     "(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
nipkow@15302
  2708
by (induct xs) auto
nipkow@15302
  2709
berghofe@22262
  2710
lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]
berghofe@22262
  2711
berghofe@22262
  2712
lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
berghofe@22262
  2713
-- {* eliminate @{text listsp} in favour of @{text set} *}
nipkow@15302
  2714
by (induct xs) auto
nipkow@15302
  2715
berghofe@22262
  2716
lemmas in_lists_conv_set = in_listsp_conv_set [to_set]
berghofe@22262
  2717
paulson@24286
  2718
lemma in_listspD [dest!,noatp]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
berghofe@22262
  2719
by (rule in_listsp_conv_set [THEN iffD1])
berghofe@22262
  2720
paulson@24286
  2721
lemmas in_listsD [dest!,noatp] = in_listspD [to_set]
paulson@24286
  2722
paulson@24286
  2723
lemma in_listspI [intro!,noatp]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
berghofe@22262
  2724
by (rule in_listsp_conv_set [THEN iffD2])
berghofe@22262
  2725
paulson@24286
  2726
lemmas in_listsI [intro!,noatp] = in_listspI [to_set]
nipkow@15302
  2727
nipkow@15302
  2728
lemma lists_UNIV [simp]: "lists UNIV = UNIV"
nipkow@15302
  2729
by auto
nipkow@15302
  2730
nipkow@17086
  2731
nipkow@17086
  2732
nipkow@17086
  2733
subsubsection{* Inductive definition for membership *}
nipkow@17086
  2734
berghofe@23740
  2735
inductive ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
berghofe@22262
  2736
where
berghofe@22262
  2737
    elem:  "ListMem x (x # xs)"
berghofe@22262
  2738
  | insert:  "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"
berghofe@22262
  2739
berghofe@22262
  2740
lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
nipkow@17086
  2741
apply (rule iffI)
nipkow@17086
  2742
 apply (induct set: ListMem)
nipkow@17086
  2743
  apply auto
nipkow@17086
  2744
apply (induct xs)
nipkow@17086
  2745
 apply (auto intro: ListMem.intros)
nipkow@17086
  2746
done
nipkow@17086
  2747
nipkow@17086
  2748
nipkow@17086
  2749
nipkow@15392
  2750
subsubsection{*Lists as Cartesian products*}
nipkow@15302
  2751
nipkow@15302
  2752
text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
nipkow@15302
  2753
@{term A} and tail drawn from @{term Xs}.*}
nipkow@15302
  2754
nipkow@15302
  2755
constdefs
nipkow@15302
  2756
  set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
nipkow@15302
  2757
  "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"
nipkow@15302
  2758
paulson@17724
  2759
lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
nipkow@15302
  2760
by (auto simp add: set_Cons_def)
nipkow@15302
  2761
nipkow@15302
  2762
text{*Yields the set of lists, all of the same length as the argument and