src/HOL/List.thy
author wenzelm
Fri Nov 17 02:20:03 2006 +0100 (2006-11-17)
changeset 21404 eb85850d3eb7
parent 21211 5370cfbf3070
child 21455 b6be1d1b66c5
permissions -rw-r--r--
more robust syntax for definition/abbreviation/notation;
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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 FunDef
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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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  "@" :: "'a list => 'a list => 'a list"    (infixr 65)
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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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  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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abbreviation
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  upto:: "nat => nat => nat list"  ("(1[_../_])") where
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  "[i..j] == [i..<(Suc j)]"
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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[_\<in>_ ./ _])")
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syntax (HTML output)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
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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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primrec
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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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  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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subsubsection {* @{const Nil} and @{const Cons} *}
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lemma not_Cons_self [simp]:
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  "xs \<noteq> x # xs"
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by (induct xs) auto
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lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
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lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
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by (induct xs) auto
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lemma length_induct:
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  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
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by (rule measure_induct [of length]) iprover
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subsubsection {* @{const length} *}
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text {*
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  Needs to come before @{text "@"} because of theorem @{text
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  append_eq_append_conv}.
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*}
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lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
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by (induct xs) auto
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lemma length_map [simp]: "length (map f xs) = length xs"
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by (induct xs) auto
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lemma length_rev [simp]: "length (rev xs) = length xs"
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by (induct xs) auto
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lemma length_tl [simp]: "length (tl xs) = length xs - 1"
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by (cases xs) auto
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lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
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by (induct xs) auto
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lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
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by (induct xs) auto
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lemma length_Suc_conv:
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"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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by (induct xs) auto
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lemma Suc_length_conv:
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"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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apply (induct xs, simp, simp)
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apply blast
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done
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lemma impossible_Cons [rule_format]: 
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  "length xs <= length ys --> xs = x # ys = False"
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apply (induct xs)
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apply auto
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done
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lemma list_induct2[consumes 1]: "\<And>ys.
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 \<lbrakk> length xs = length ys;
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   P [] [];
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   \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
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 \<Longrightarrow> P xs ys"
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apply(induct xs)
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 apply simp
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apply(case_tac ys)
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 apply simp
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apply(simp)
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done
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subsubsection {* @{text "@"} -- append *}
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lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
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by (induct xs) auto
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lemma append_Nil2 [simp]: "xs @ [] = xs"
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by (induct xs) auto
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lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
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by (induct xs) auto
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lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
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by (induct xs) auto
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lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
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by (induct xs) auto
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lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
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by (induct xs) auto
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lemma append_eq_append_conv [simp]:
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 "!!ys. length xs = length ys \<or> length us = length vs
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 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
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apply (induct xs)
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 apply (case_tac ys, simp, force)
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apply (case_tac ys, force, simp)
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done
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lemma append_eq_append_conv2: "!!ys zs ts.
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 (xs @ ys = zs @ ts) =
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 (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
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apply (induct xs)
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 apply fastsimp
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apply(case_tac zs)
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 apply simp
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apply fastsimp
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done
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lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
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by simp
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lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
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by simp
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lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
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by simp
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lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
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using append_same_eq [of _ _ "[]"] by auto
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lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
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using append_same_eq [of "[]"] by auto
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lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
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by (induct xs) auto
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lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
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by (induct xs) auto
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lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
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   341
by (simp add: hd_append split: list.split)
wenzelm@13114
   342
wenzelm@13142
   343
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   344
by (simp split: list.split)
wenzelm@13114
   345
wenzelm@13142
   346
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   347
by (simp add: tl_append split: list.split)
wenzelm@13114
   348
wenzelm@13114
   349
nipkow@14300
   350
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   351
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   352
by(cases ys) auto
nipkow@14300
   353
nipkow@15281
   354
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   355
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   356
by(cases ys) auto
nipkow@15281
   357
nipkow@14300
   358
wenzelm@13142
   359
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   360
wenzelm@13114
   361
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   362
by simp
wenzelm@13114
   363
wenzelm@13142
   364
lemma Cons_eq_appendI:
nipkow@13145
   365
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   366
by (drule sym) simp
wenzelm@13114
   367
wenzelm@13142
   368
lemma append_eq_appendI:
nipkow@13145
   369
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   370
by (drule sym) simp
wenzelm@13114
   371
wenzelm@13114
   372
wenzelm@13142
   373
text {*
nipkow@13145
   374
Simplification procedure for all list equalities.
nipkow@13145
   375
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   376
- both lists end in a singleton list,
nipkow@13145
   377
- or both lists end in the same list.
wenzelm@13142
   378
*}
wenzelm@13142
   379
wenzelm@13142
   380
ML_setup {*
nipkow@3507
   381
local
nipkow@3507
   382
wenzelm@13122
   383
val append_assoc = thm "append_assoc";
wenzelm@13122
   384
val append_Nil = thm "append_Nil";
wenzelm@13122
   385
val append_Cons = thm "append_Cons";
wenzelm@13122
   386
val append1_eq_conv = thm "append1_eq_conv";
wenzelm@13122
   387
val append_same_eq = thm "append_same_eq";
wenzelm@13122
   388
wenzelm@13114
   389
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462
   390
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
wenzelm@13462
   391
  | last (Const("List.op @",_) $ _ $ ys) = last ys
wenzelm@13462
   392
  | last t = t;
wenzelm@13114
   393
wenzelm@13114
   394
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462
   395
  | list1 _ = false;
wenzelm@13114
   396
wenzelm@13114
   397
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462
   398
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
wenzelm@13462
   399
  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462
   400
  | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114
   401
wenzelm@16973
   402
val rearr_ss = HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons];
wenzelm@16973
   403
wenzelm@20044
   404
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   405
  let
wenzelm@13462
   406
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   407
    fun rearr conv =
wenzelm@13462
   408
      let
wenzelm@13462
   409
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   410
        val Type(_,listT::_) = eqT
wenzelm@13462
   411
        val appT = [listT,listT] ---> listT
wenzelm@13462
   412
        val app = Const("List.op @",appT)
wenzelm@13462
   413
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   414
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@20044
   415
        val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
wenzelm@17877
   416
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
skalberg@15531
   417
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   418
wenzelm@13462
   419
  in
wenzelm@13462
   420
    if list1 lastl andalso list1 lastr then rearr append1_eq_conv
wenzelm@13462
   421
    else if lastl aconv lastr then rearr append_same_eq
skalberg@15531
   422
    else NONE
wenzelm@13462
   423
  end;
wenzelm@13462
   424
wenzelm@13114
   425
in
wenzelm@13462
   426
wenzelm@13462
   427
val list_eq_simproc =
haftmann@21061
   428
  Simplifier.simproc (the_context ()) "list_eq" ["(xs::'a list) = ys"] (K list_eq);
wenzelm@13462
   429
wenzelm@13114
   430
end;
wenzelm@13114
   431
wenzelm@13114
   432
Addsimprocs [list_eq_simproc];
wenzelm@13114
   433
*}
wenzelm@13114
   434
wenzelm@13114
   435
nipkow@15392
   436
subsubsection {* @{text map} *}
wenzelm@13114
   437
wenzelm@13142
   438
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   439
by (induct xs) simp_all
wenzelm@13114
   440
wenzelm@13142
   441
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   442
by (rule ext, induct_tac xs) auto
wenzelm@13114
   443
wenzelm@13142
   444
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   445
by (induct xs) auto
wenzelm@13114
   446
wenzelm@13142
   447
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   448
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   449
wenzelm@13142
   450
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   451
by (induct xs) auto
wenzelm@13114
   452
nipkow@13737
   453
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   454
by (induct xs) auto
nipkow@13737
   455
krauss@19770
   456
lemma map_cong [fundef_cong, recdef_cong]:
nipkow@13145
   457
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   458
-- {* a congruence rule for @{text map} *}
nipkow@13737
   459
by simp
wenzelm@13114
   460
wenzelm@13142
   461
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   462
by (cases xs) auto
wenzelm@13114
   463
wenzelm@13142
   464
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   465
by (cases xs) auto
wenzelm@13114
   466
paulson@18447
   467
lemma map_eq_Cons_conv:
nipkow@14025
   468
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   469
by (cases xs) auto
wenzelm@13114
   470
paulson@18447
   471
lemma Cons_eq_map_conv:
nipkow@14025
   472
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   473
by (cases ys) auto
nipkow@14025
   474
paulson@18447
   475
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
paulson@18447
   476
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
paulson@18447
   477
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
paulson@18447
   478
nipkow@14111
   479
lemma ex_map_conv:
nipkow@14111
   480
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
paulson@18447
   481
by(induct ys, auto simp add: Cons_eq_map_conv)
nipkow@14111
   482
nipkow@15110
   483
lemma map_eq_imp_length_eq:
nipkow@15110
   484
  "!!xs. map f xs = map f ys ==> length xs = length ys"
nipkow@15110
   485
apply (induct ys)
nipkow@15110
   486
 apply simp
nipkow@15110
   487
apply(simp (no_asm_use))
nipkow@15110
   488
apply clarify
nipkow@15110
   489
apply(simp (no_asm_use))
nipkow@15110
   490
apply fast
nipkow@15110
   491
done
nipkow@15110
   492
nipkow@15110
   493
lemma map_inj_on:
nipkow@15110
   494
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   495
  ==> xs = ys"
nipkow@15110
   496
apply(frule map_eq_imp_length_eq)
nipkow@15110
   497
apply(rotate_tac -1)
nipkow@15110
   498
apply(induct rule:list_induct2)
nipkow@15110
   499
 apply simp
nipkow@15110
   500
apply(simp)
nipkow@15110
   501
apply (blast intro:sym)
nipkow@15110
   502
done
nipkow@15110
   503
nipkow@15110
   504
lemma inj_on_map_eq_map:
nipkow@15110
   505
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   506
by(blast dest:map_inj_on)
nipkow@15110
   507
wenzelm@13114
   508
lemma map_injective:
nipkow@14338
   509
 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@14338
   510
by (induct ys) (auto dest!:injD)
wenzelm@13114
   511
nipkow@14339
   512
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   513
by(blast dest:map_injective)
nipkow@14339
   514
wenzelm@13114
   515
lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@17589
   516
by (iprover dest: map_injective injD intro: inj_onI)
wenzelm@13114
   517
wenzelm@13114
   518
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   519
apply (unfold inj_on_def, clarify)
nipkow@13145
   520
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   521
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   522
apply blast
nipkow@13145
   523
done
wenzelm@13114
   524
nipkow@14339
   525
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   526
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   527
nipkow@15303
   528
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
   529
apply(rule inj_onI)
nipkow@15303
   530
apply(erule map_inj_on)
nipkow@15303
   531
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
   532
done
nipkow@15303
   533
kleing@14343
   534
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   535
by (induct xs, auto)
wenzelm@13114
   536
nipkow@14402
   537
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   538
by (induct xs) auto
nipkow@14402
   539
nipkow@15110
   540
lemma map_fst_zip[simp]:
nipkow@15110
   541
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   542
by (induct rule:list_induct2, simp_all)
nipkow@15110
   543
nipkow@15110
   544
lemma map_snd_zip[simp]:
nipkow@15110
   545
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   546
by (induct rule:list_induct2, simp_all)
nipkow@15110
   547
nipkow@15110
   548
nipkow@15392
   549
subsubsection {* @{text rev} *}
wenzelm@13114
   550
wenzelm@13142
   551
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   552
by (induct xs) auto
wenzelm@13114
   553
wenzelm@13142
   554
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   555
by (induct xs) auto
wenzelm@13114
   556
kleing@15870
   557
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870
   558
by auto
kleing@15870
   559
wenzelm@13142
   560
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   561
by (induct xs) auto
wenzelm@13114
   562
wenzelm@13142
   563
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   564
by (induct xs) auto
wenzelm@13114
   565
kleing@15870
   566
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870
   567
by (cases xs) auto
kleing@15870
   568
kleing@15870
   569
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870
   570
by (cases xs) auto
kleing@15870
   571
haftmann@21061
   572
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
haftmann@21061
   573
apply (induct xs arbitrary: ys, force)
paulson@14208
   574
apply (case_tac ys, simp, force)
nipkow@13145
   575
done
wenzelm@13114
   576
nipkow@15439
   577
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
   578
by(simp add:inj_on_def)
nipkow@15439
   579
wenzelm@13366
   580
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   581
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489
   582
apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145
   583
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   584
done
wenzelm@13114
   585
nipkow@13145
   586
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114
   587
wenzelm@13366
   588
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   589
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   590
by (induct xs rule: rev_induct) auto
wenzelm@13114
   591
wenzelm@13366
   592
lemmas rev_cases = rev_exhaust
wenzelm@13366
   593
nipkow@18423
   594
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
nipkow@18423
   595
by(rule rev_cases[of xs]) auto
nipkow@18423
   596
wenzelm@13114
   597
nipkow@15392
   598
subsubsection {* @{text set} *}
wenzelm@13114
   599
wenzelm@13142
   600
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   601
by (induct xs) auto
wenzelm@13114
   602
wenzelm@13142
   603
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   604
by (induct xs) auto
wenzelm@13114
   605
nipkow@17830
   606
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
nipkow@17830
   607
by(cases xs) auto
oheimb@14099
   608
wenzelm@13142
   609
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   610
by auto
wenzelm@13114
   611
oheimb@14099
   612
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   613
by auto
oheimb@14099
   614
wenzelm@13142
   615
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   616
by (induct xs) auto
wenzelm@13114
   617
nipkow@15245
   618
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   619
by(induct xs) auto
nipkow@15245
   620
wenzelm@13142
   621
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   622
by (induct xs) auto
wenzelm@13114
   623
wenzelm@13142
   624
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   625
by (induct xs) auto
wenzelm@13114
   626
wenzelm@13142
   627
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   628
by (induct xs) auto
wenzelm@13114
   629
nipkow@15425
   630
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
paulson@14208
   631
apply (induct j, simp_all)
paulson@14208
   632
apply (erule ssubst, auto)
nipkow@13145
   633
done
wenzelm@13114
   634
wenzelm@13142
   635
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
paulson@15113
   636
proof (induct xs)
paulson@15113
   637
  case Nil show ?case by simp
paulson@15113
   638
  case (Cons a xs)
paulson@15113
   639
  show ?case
paulson@15113
   640
  proof 
paulson@15113
   641
    assume "x \<in> set (a # xs)"
paulson@15113
   642
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   643
      by (simp, blast intro: Cons_eq_appendI)
paulson@15113
   644
  next
paulson@15113
   645
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   646
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
paulson@15113
   647
    show "x \<in> set (a # xs)" 
paulson@15113
   648
      by (cases ys, auto simp add: eq)
paulson@15113
   649
  qed
paulson@15113
   650
qed
wenzelm@13142
   651
nipkow@18049
   652
lemma in_set_conv_decomp_first:
nipkow@18049
   653
 "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
nipkow@18049
   654
proof (induct xs)
nipkow@18049
   655
  case Nil show ?case by simp
nipkow@18049
   656
next
nipkow@18049
   657
  case (Cons a xs)
nipkow@18049
   658
  show ?case
nipkow@18049
   659
  proof cases
nipkow@18049
   660
    assume "x = a" thus ?case using Cons by force
nipkow@18049
   661
  next
nipkow@18049
   662
    assume "x \<noteq> a"
nipkow@18049
   663
    show ?case
nipkow@18049
   664
    proof
nipkow@18049
   665
      assume "x \<in> set (a # xs)"
nipkow@18049
   666
      from prems show "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@18049
   667
	by(fastsimp intro!: Cons_eq_appendI)
nipkow@18049
   668
    next
nipkow@18049
   669
      assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@18049
   670
      then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
nipkow@18049
   671
      show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq)
nipkow@18049
   672
    qed
nipkow@18049
   673
  qed
nipkow@18049
   674
qed
nipkow@18049
   675
nipkow@18049
   676
lemmas split_list       = in_set_conv_decomp[THEN iffD1, standard]
nipkow@18049
   677
lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard]
nipkow@18049
   678
nipkow@18049
   679
paulson@13508
   680
lemma finite_list: "finite A ==> EX l. set l = A"
paulson@13508
   681
apply (erule finite_induct, auto)
paulson@13508
   682
apply (rule_tac x="x#l" in exI, auto)
paulson@13508
   683
done
paulson@13508
   684
kleing@14388
   685
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
   686
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
   687
paulson@15168
   688
nipkow@15392
   689
subsubsection {* @{text filter} *}
wenzelm@13114
   690
wenzelm@13142
   691
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   692
by (induct xs) auto
wenzelm@13114
   693
nipkow@15305
   694
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
   695
by (induct xs) simp_all
nipkow@15305
   696
wenzelm@13142
   697
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   698
by (induct xs) auto
wenzelm@13114
   699
nipkow@16998
   700
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998
   701
by (induct xs) (auto simp add: le_SucI)
nipkow@16998
   702
nipkow@18423
   703
lemma sum_length_filter_compl:
nipkow@18423
   704
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
nipkow@18423
   705
by(induct xs) simp_all
nipkow@18423
   706
wenzelm@13142
   707
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   708
by (induct xs) auto
wenzelm@13114
   709
wenzelm@13142
   710
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   711
by (induct xs) auto
wenzelm@13114
   712
nipkow@16998
   713
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@16998
   714
  by (induct xs) simp_all
nipkow@16998
   715
nipkow@16998
   716
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998
   717
apply (induct xs)
nipkow@16998
   718
 apply auto
nipkow@16998
   719
apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998
   720
apply simp
nipkow@16998
   721
done
wenzelm@13114
   722
nipkow@16965
   723
lemma filter_map:
nipkow@16965
   724
  "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965
   725
by (induct xs) simp_all
nipkow@16965
   726
nipkow@16965
   727
lemma length_filter_map[simp]:
nipkow@16965
   728
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965
   729
by (simp add:filter_map)
nipkow@16965
   730
wenzelm@13142
   731
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   732
by auto
wenzelm@13114
   733
nipkow@15246
   734
lemma length_filter_less:
nipkow@15246
   735
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
   736
proof (induct xs)
nipkow@15246
   737
  case Nil thus ?case by simp
nipkow@15246
   738
next
nipkow@15246
   739
  case (Cons x xs) thus ?case
nipkow@15246
   740
    apply (auto split:split_if_asm)
nipkow@15246
   741
    using length_filter_le[of P xs] apply arith
nipkow@15246
   742
  done
nipkow@15246
   743
qed
wenzelm@13114
   744
nipkow@15281
   745
lemma length_filter_conv_card:
nipkow@15281
   746
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
   747
proof (induct xs)
nipkow@15281
   748
  case Nil thus ?case by simp
nipkow@15281
   749
next
nipkow@15281
   750
  case (Cons x xs)
nipkow@15281
   751
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
   752
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
   753
  show ?case (is "?l = card ?S'")
nipkow@15281
   754
  proof (cases)
nipkow@15281
   755
    assume "p x"
nipkow@15281
   756
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@15281
   757
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   758
    have "length (filter p (x # xs)) = Suc(card ?S)"
nipkow@15281
   759
      using Cons by simp
nipkow@15281
   760
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
nipkow@15281
   761
      by (simp add: card_image inj_Suc)
nipkow@15281
   762
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   763
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
   764
    finally show ?thesis .
nipkow@15281
   765
  next
nipkow@15281
   766
    assume "\<not> p x"
nipkow@15281
   767
    hence eq: "?S' = Suc ` ?S"
nipkow@15281
   768
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   769
    have "length (filter p (x # xs)) = card ?S"
nipkow@15281
   770
      using Cons by simp
nipkow@15281
   771
    also have "\<dots> = card(Suc ` ?S)" using fin
nipkow@15281
   772
      by (simp add: card_image inj_Suc)
nipkow@15281
   773
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   774
      by (simp add:card_insert_if)
nipkow@15281
   775
    finally show ?thesis .
nipkow@15281
   776
  qed
nipkow@15281
   777
qed
nipkow@15281
   778
nipkow@17629
   779
lemma Cons_eq_filterD:
nipkow@17629
   780
 "x#xs = filter P ys \<Longrightarrow>
nipkow@17629
   781
  \<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
   782
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
nipkow@17629
   783
proof(induct ys)
nipkow@17629
   784
  case Nil thus ?case by simp
nipkow@17629
   785
next
nipkow@17629
   786
  case (Cons y ys)
nipkow@17629
   787
  show ?case (is "\<exists>x. ?Q x")
nipkow@17629
   788
  proof cases
nipkow@17629
   789
    assume Py: "P y"
nipkow@17629
   790
    show ?thesis
nipkow@17629
   791
    proof cases
nipkow@17629
   792
      assume xy: "x = y"
nipkow@17629
   793
      show ?thesis
nipkow@17629
   794
      proof from Py xy Cons(2) show "?Q []" by simp qed
nipkow@17629
   795
    next
nipkow@17629
   796
      assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp
nipkow@17629
   797
    qed
nipkow@17629
   798
  next
nipkow@17629
   799
    assume Py: "\<not> P y"
nipkow@17629
   800
    with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp
nipkow@17629
   801
    show ?thesis (is "? us. ?Q us")
nipkow@17629
   802
    proof show "?Q (y#us)" using 1 by simp qed
nipkow@17629
   803
  qed
nipkow@17629
   804
qed
nipkow@17629
   805
nipkow@17629
   806
lemma filter_eq_ConsD:
nipkow@17629
   807
 "filter P ys = x#xs \<Longrightarrow>
nipkow@17629
   808
  \<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
   809
by(rule Cons_eq_filterD) simp
nipkow@17629
   810
nipkow@17629
   811
lemma filter_eq_Cons_iff:
nipkow@17629
   812
 "(filter P ys = x#xs) =
nipkow@17629
   813
  (\<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
   814
by(auto dest:filter_eq_ConsD)
nipkow@17629
   815
nipkow@17629
   816
lemma Cons_eq_filter_iff:
nipkow@17629
   817
 "(x#xs = filter P ys) =
nipkow@17629
   818
  (\<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
   819
by(auto dest:Cons_eq_filterD)
nipkow@17629
   820
krauss@19770
   821
lemma filter_cong[fundef_cong, recdef_cong]:
nipkow@17501
   822
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
nipkow@17501
   823
apply simp
nipkow@17501
   824
apply(erule thin_rl)
nipkow@17501
   825
by (induct ys) simp_all
nipkow@17501
   826
nipkow@15281
   827
nipkow@15392
   828
subsubsection {* @{text concat} *}
wenzelm@13114
   829
wenzelm@13142
   830
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
   831
by (induct xs) auto
wenzelm@13114
   832
paulson@18447
   833
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   834
by (induct xss) auto
wenzelm@13114
   835
paulson@18447
   836
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   837
by (induct xss) auto
wenzelm@13114
   838
wenzelm@13142
   839
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
nipkow@13145
   840
by (induct xs) auto
wenzelm@13114
   841
wenzelm@13142
   842
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
   843
by (induct xs) auto
wenzelm@13114
   844
wenzelm@13142
   845
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
   846
by (induct xs) auto
wenzelm@13114
   847
wenzelm@13142
   848
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
   849
by (induct xs) auto
wenzelm@13114
   850
wenzelm@13114
   851
nipkow@15392
   852
subsubsection {* @{text nth} *}
wenzelm@13114
   853
wenzelm@13142
   854
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
   855
by auto
wenzelm@13114
   856
wenzelm@13142
   857
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
   858
by auto
wenzelm@13114
   859
wenzelm@13142
   860
declare nth.simps [simp del]
wenzelm@13114
   861
wenzelm@13114
   862
lemma nth_append:
nipkow@13145
   863
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
paulson@14208
   864
apply (induct "xs", simp)
paulson@14208
   865
apply (case_tac n, auto)
nipkow@13145
   866
done
wenzelm@13114
   867
nipkow@14402
   868
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
nipkow@14402
   869
by (induct "xs") auto
nipkow@14402
   870
nipkow@14402
   871
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
nipkow@14402
   872
by (induct "xs") auto
nipkow@14402
   873
wenzelm@13142
   874
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
paulson@14208
   875
apply (induct xs, simp)
paulson@14208
   876
apply (case_tac n, auto)
nipkow@13145
   877
done
wenzelm@13114
   878
nipkow@18423
   879
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
nipkow@18423
   880
by(cases xs) simp_all
nipkow@18423
   881
nipkow@18049
   882
nipkow@18049
   883
lemma list_eq_iff_nth_eq:
nipkow@18049
   884
 "!!ys. (xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
nipkow@18049
   885
apply(induct xs)
nipkow@18049
   886
 apply simp apply blast
nipkow@18049
   887
apply(case_tac ys)
nipkow@18049
   888
 apply simp
nipkow@18049
   889
apply(simp add:nth_Cons split:nat.split)apply blast
nipkow@18049
   890
done
nipkow@18049
   891
wenzelm@13142
   892
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
   893
apply (induct xs, simp, simp)
nipkow@13145
   894
apply safe
paulson@14208
   895
apply (rule_tac x = 0 in exI, simp)
paulson@14208
   896
 apply (rule_tac x = "Suc i" in exI, simp)
paulson@14208
   897
apply (case_tac i, simp)
nipkow@13145
   898
apply (rename_tac j)
paulson@14208
   899
apply (rule_tac x = j in exI, simp)
nipkow@13145
   900
done
wenzelm@13114
   901
nipkow@17501
   902
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
nipkow@17501
   903
by(auto simp:set_conv_nth)
nipkow@17501
   904
nipkow@13145
   905
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
   906
by (auto simp add: set_conv_nth)
wenzelm@13114
   907
wenzelm@13142
   908
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
   909
by (auto simp add: set_conv_nth)
wenzelm@13114
   910
wenzelm@13114
   911
lemma all_nth_imp_all_set:
nipkow@13145
   912
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
   913
by (auto simp add: set_conv_nth)
wenzelm@13114
   914
wenzelm@13114
   915
lemma all_set_conv_all_nth:
nipkow@13145
   916
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
   917
by (auto simp add: set_conv_nth)
wenzelm@13114
   918
wenzelm@13114
   919
nipkow@15392
   920
subsubsection {* @{text list_update} *}
wenzelm@13114
   921
wenzelm@13142
   922
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
nipkow@13145
   923
by (induct xs) (auto split: nat.split)
wenzelm@13114
   924
wenzelm@13114
   925
lemma nth_list_update:
nipkow@13145
   926
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@13145
   927
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   928
wenzelm@13142
   929
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
   930
by (simp add: nth_list_update)
wenzelm@13114
   931
wenzelm@13142
   932
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145
   933
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   934
wenzelm@13142
   935
lemma list_update_overwrite [simp]:
nipkow@13145
   936
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145
   937
by (induct xs) (auto split: nat.split)
wenzelm@13114
   938
nipkow@14402
   939
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
paulson@14208
   940
apply (induct xs, simp)
nipkow@14187
   941
apply(simp split:nat.splits)
nipkow@14187
   942
done
nipkow@14187
   943
nipkow@17501
   944
lemma list_update_beyond[simp]: "\<And>i. length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
nipkow@17501
   945
apply (induct xs)
nipkow@17501
   946
 apply simp
nipkow@17501
   947
apply (case_tac i)
nipkow@17501
   948
apply simp_all
nipkow@17501
   949
done
nipkow@17501
   950
wenzelm@13114
   951
lemma list_update_same_conv:
nipkow@13145
   952
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145
   953
by (induct xs) (auto split: nat.split)
wenzelm@13114
   954
nipkow@14187
   955
lemma list_update_append1:
nipkow@14187
   956
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
paulson@14208
   957
apply (induct xs, simp)
nipkow@14187
   958
apply(simp split:nat.split)
nipkow@14187
   959
done
nipkow@14187
   960
kleing@15868
   961
lemma list_update_append:
kleing@15868
   962
  "!!n. (xs @ ys) [n:= x] = 
kleing@15868
   963
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
kleing@15868
   964
by (induct xs) (auto split:nat.splits)
kleing@15868
   965
nipkow@14402
   966
lemma list_update_length [simp]:
nipkow@14402
   967
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
   968
by (induct xs, auto)
nipkow@14402
   969
wenzelm@13114
   970
lemma update_zip:
nipkow@13145
   971
"!!i xy xs. length xs = length ys ==>
nipkow@13145
   972
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145
   973
by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114
   974
wenzelm@13114
   975
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145
   976
by (induct xs) (auto split: nat.split)
wenzelm@13114
   977
wenzelm@13114
   978
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
   979
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
   980
kleing@15868
   981
lemma set_update_memI: "!!n. n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
kleing@15868
   982
by (induct xs) (auto split:nat.splits)
kleing@15868
   983
wenzelm@13114
   984
nipkow@15392
   985
subsubsection {* @{text last} and @{text butlast} *}
wenzelm@13114
   986
wenzelm@13142
   987
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
   988
by (induct xs) auto
wenzelm@13114
   989
wenzelm@13142
   990
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
   991
by (induct xs) auto
wenzelm@13114
   992
nipkow@14302
   993
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
   994
by(simp add:last.simps)
nipkow@14302
   995
nipkow@14302
   996
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
   997
by(simp add:last.simps)
nipkow@14302
   998
nipkow@14302
   999
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
  1000
by (induct xs) (auto)
nipkow@14302
  1001
nipkow@14302
  1002
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
  1003
by(simp add:last_append)
nipkow@14302
  1004
nipkow@14302
  1005
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
  1006
by(simp add:last_append)
nipkow@14302
  1007
nipkow@17762
  1008
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
nipkow@17762
  1009
by(rule rev_exhaust[of xs]) simp_all
nipkow@17762
  1010
nipkow@17762
  1011
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
nipkow@17762
  1012
by(cases xs) simp_all
nipkow@17762
  1013
nipkow@17765
  1014
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
nipkow@17765
  1015
by (induct as) auto
nipkow@17762
  1016
wenzelm@13142
  1017
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
  1018
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1019
wenzelm@13114
  1020
lemma butlast_append:
nipkow@13145
  1021
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145
  1022
by (induct xs) auto
wenzelm@13114
  1023
wenzelm@13142
  1024
lemma append_butlast_last_id [simp]:
nipkow@13145
  1025
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
  1026
by (induct xs) auto
wenzelm@13114
  1027
wenzelm@13142
  1028
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
  1029
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1030
wenzelm@13114
  1031
lemma in_set_butlast_appendI:
nipkow@13145
  1032
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
  1033
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
  1034
nipkow@17501
  1035
lemma last_drop[simp]: "!!n. n < length xs \<Longrightarrow> last (drop n xs) = last xs"
nipkow@17501
  1036
apply (induct xs)
nipkow@17501
  1037
 apply simp
nipkow@17501
  1038
apply (auto split:nat.split)
nipkow@17501
  1039
done
nipkow@17501
  1040
nipkow@17589
  1041
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
nipkow@17589
  1042
by(induct xs)(auto simp:neq_Nil_conv)
nipkow@17589
  1043
nipkow@15392
  1044
subsubsection {* @{text take} and @{text drop} *}
wenzelm@13114
  1045
wenzelm@13142
  1046
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
  1047
by (induct xs) auto
wenzelm@13114
  1048
wenzelm@13142
  1049
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
  1050
by (induct xs) auto
wenzelm@13114
  1051
wenzelm@13142
  1052
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
  1053
by simp
wenzelm@13114
  1054
wenzelm@13142
  1055
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
  1056
by simp
wenzelm@13114
  1057
wenzelm@13142
  1058
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
  1059
nipkow@15110
  1060
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
  1061
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
  1062
nipkow@14187
  1063
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
  1064
by(cases xs, simp_all)
nipkow@14187
  1065
nipkow@14187
  1066
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
nipkow@14187
  1067
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@14187
  1068
nipkow@14187
  1069
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
paulson@14208
  1070
apply (induct xs, simp)
nipkow@14187
  1071
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
  1072
done
nipkow@14187
  1073
nipkow@13913
  1074
lemma take_Suc_conv_app_nth:
nipkow@13913
  1075
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
paulson@14208
  1076
apply (induct xs, simp)
paulson@14208
  1077
apply (case_tac i, auto)
nipkow@13913
  1078
done
nipkow@13913
  1079
mehta@14591
  1080
lemma drop_Suc_conv_tl:
mehta@14591
  1081
  "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
mehta@14591
  1082
apply (induct xs, simp)
mehta@14591
  1083
apply (case_tac i, auto)
mehta@14591
  1084
done
mehta@14591
  1085
wenzelm@13142
  1086
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
nipkow@13145
  1087
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1088
wenzelm@13142
  1089
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
nipkow@13145
  1090
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1091
wenzelm@13142
  1092
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
nipkow@13145
  1093
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1094
wenzelm@13142
  1095
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
nipkow@13145
  1096
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1097
wenzelm@13142
  1098
lemma take_append [simp]:
nipkow@13145
  1099
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@13145
  1100
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1101
wenzelm@13142
  1102
lemma drop_append [simp]:
nipkow@13145
  1103
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@13145
  1104
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1105
wenzelm@13142
  1106
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
paulson@14208
  1107
apply (induct m, auto)
paulson@14208
  1108
apply (case_tac xs, auto)
nipkow@15236
  1109
apply (case_tac n, auto)
nipkow@13145
  1110
done
wenzelm@13114
  1111
wenzelm@13142
  1112
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
paulson@14208
  1113
apply (induct m, auto)
paulson@14208
  1114
apply (case_tac xs, auto)
nipkow@13145
  1115
done
wenzelm@13114
  1116
wenzelm@13114
  1117
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
paulson@14208
  1118
apply (induct m, auto)
paulson@14208
  1119
apply (case_tac xs, auto)
nipkow@13145
  1120
done
wenzelm@13114
  1121
nipkow@14802
  1122
lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@14802
  1123
apply(induct xs)
nipkow@14802
  1124
 apply simp
nipkow@14802
  1125
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  1126
done
nipkow@14802
  1127
wenzelm@13142
  1128
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
paulson@14208
  1129
apply (induct n, auto)
paulson@14208
  1130
apply (case_tac xs, auto)
nipkow@13145
  1131
done
wenzelm@13114
  1132
nipkow@15110
  1133
lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@15110
  1134
apply(induct xs)
nipkow@15110
  1135
 apply simp
nipkow@15110
  1136
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  1137
done
nipkow@15110
  1138
nipkow@15110
  1139
lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"
nipkow@15110
  1140
apply(induct xs)
nipkow@15110
  1141
apply simp
nipkow@15110
  1142
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1143
done
nipkow@15110
  1144
wenzelm@13114
  1145
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
paulson@14208
  1146
apply (induct n, auto)
paulson@14208
  1147
apply (case_tac xs, auto)
nipkow@13145
  1148
done
wenzelm@13114
  1149
wenzelm@13142
  1150
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
paulson@14208
  1151
apply (induct n, auto)
paulson@14208
  1152
apply (case_tac xs, auto)
nipkow@13145
  1153
done
wenzelm@13114
  1154
wenzelm@13114
  1155
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
paulson@14208
  1156
apply (induct xs, auto)
paulson@14208
  1157
apply (case_tac i, auto)
nipkow@13145
  1158
done
wenzelm@13114
  1159
wenzelm@13114
  1160
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
paulson@14208
  1161
apply (induct xs, auto)
paulson@14208
  1162
apply (case_tac i, auto)
nipkow@13145
  1163
done
wenzelm@13114
  1164
wenzelm@13142
  1165
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
paulson@14208
  1166
apply (induct xs, auto)
paulson@14208
  1167
apply (case_tac n, blast)
paulson@14208
  1168
apply (case_tac i, auto)
nipkow@13145
  1169
done
wenzelm@13114
  1170
wenzelm@13142
  1171
lemma nth_drop [simp]:
nipkow@13145
  1172
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
paulson@14208
  1173
apply (induct n, auto)
paulson@14208
  1174
apply (case_tac xs, auto)
nipkow@13145
  1175
done
nipkow@3507
  1176
nipkow@18423
  1177
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
nipkow@18423
  1178
by(simp add: hd_conv_nth)
nipkow@18423
  1179
nipkow@14025
  1180
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
nipkow@14025
  1181
by(induct xs)(auto simp:take_Cons split:nat.split)
nipkow@14025
  1182
nipkow@14025
  1183
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
nipkow@14025
  1184
by(induct xs)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1185
nipkow@14187
  1186
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1187
using set_take_subset by fast
nipkow@14187
  1188
nipkow@14187
  1189
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1190
using set_drop_subset by fast
nipkow@14187
  1191
wenzelm@13114
  1192
lemma append_eq_conv_conj:
nipkow@13145
  1193
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
paulson@14208
  1194
apply (induct xs, simp, clarsimp)
paulson@14208
  1195
apply (case_tac zs, auto)
nipkow@13145
  1196
done
wenzelm@13142
  1197
paulson@14050
  1198
lemma take_add [rule_format]: 
paulson@14050
  1199
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
paulson@14050
  1200
apply (induct xs, auto) 
paulson@14050
  1201
apply (case_tac i, simp_all) 
paulson@14050
  1202
done
paulson@14050
  1203
nipkow@14300
  1204
lemma append_eq_append_conv_if:
nipkow@14300
  1205
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1206
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1207
   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
  1208
   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@14300
  1209
apply(induct xs\<^isub>1)
nipkow@14300
  1210
 apply simp
nipkow@14300
  1211
apply(case_tac ys\<^isub>1)
nipkow@14300
  1212
apply simp_all
nipkow@14300
  1213
done
nipkow@14300
  1214
nipkow@15110
  1215
lemma take_hd_drop:
nipkow@15110
  1216
  "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
nipkow@15110
  1217
apply(induct xs)
nipkow@15110
  1218
apply simp
nipkow@15110
  1219
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1220
done
nipkow@15110
  1221
nipkow@17501
  1222
lemma id_take_nth_drop:
nipkow@17501
  1223
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
nipkow@17501
  1224
proof -
nipkow@17501
  1225
  assume si: "i < length xs"
nipkow@17501
  1226
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
nipkow@17501
  1227
  moreover
nipkow@17501
  1228
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
nipkow@17501
  1229
    apply (rule_tac take_Suc_conv_app_nth) by arith
nipkow@17501
  1230
  ultimately show ?thesis by auto
nipkow@17501
  1231
qed
nipkow@17501
  1232
  
nipkow@17501
  1233
lemma upd_conv_take_nth_drop:
nipkow@17501
  1234
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1235
proof -
nipkow@17501
  1236
  assume i: "i < length xs"
nipkow@17501
  1237
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
nipkow@17501
  1238
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
nipkow@17501
  1239
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1240
    using i by (simp add: list_update_append)
nipkow@17501
  1241
  finally show ?thesis .
nipkow@17501
  1242
qed
nipkow@17501
  1243
wenzelm@13114
  1244
nipkow@15392
  1245
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1246
wenzelm@13142
  1247
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1248
by (induct xs) auto
wenzelm@13114
  1249
wenzelm@13142
  1250
lemma takeWhile_append1 [simp]:
nipkow@13145
  1251
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1252
by (induct xs) auto
wenzelm@13114
  1253
wenzelm@13142
  1254
lemma takeWhile_append2 [simp]:
nipkow@13145
  1255
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1256
by (induct xs) auto
wenzelm@13114
  1257
wenzelm@13142
  1258
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1259
by (induct xs) auto
wenzelm@13114
  1260
wenzelm@13142
  1261
lemma dropWhile_append1 [simp]:
nipkow@13145
  1262
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1263
by (induct xs) auto
wenzelm@13114
  1264
wenzelm@13142
  1265
lemma dropWhile_append2 [simp]:
nipkow@13145
  1266
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1267
by (induct xs) auto
wenzelm@13114
  1268
wenzelm@13142
  1269
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1270
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1271
nipkow@13913
  1272
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1273
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1274
by(induct xs, auto)
nipkow@13913
  1275
nipkow@13913
  1276
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1277
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1278
by(induct xs, auto)
nipkow@13913
  1279
nipkow@13913
  1280
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1281
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1282
by(induct xs, auto)
nipkow@13913
  1283
nipkow@17501
  1284
text{* The following two lemmmas could be generalized to an arbitrary
nipkow@17501
  1285
property. *}
nipkow@17501
  1286
nipkow@17501
  1287
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1288
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
nipkow@17501
  1289
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
nipkow@17501
  1290
nipkow@17501
  1291
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1292
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
nipkow@17501
  1293
apply(induct xs)
nipkow@17501
  1294
 apply simp
nipkow@17501
  1295
apply auto
nipkow@17501
  1296
apply(subst dropWhile_append2)
nipkow@17501
  1297
apply auto
nipkow@17501
  1298
done
nipkow@17501
  1299
nipkow@18423
  1300
lemma takeWhile_not_last:
nipkow@18423
  1301
 "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
nipkow@18423
  1302
apply(induct xs)
nipkow@18423
  1303
 apply simp
nipkow@18423
  1304
apply(case_tac xs)
nipkow@18423
  1305
apply(auto)
nipkow@18423
  1306
done
nipkow@18423
  1307
krauss@19770
  1308
lemma takeWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1309
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1310
  ==> takeWhile P l = takeWhile Q k"
wenzelm@20503
  1311
  by (induct k arbitrary: l) (simp_all)
krauss@18336
  1312
krauss@19770
  1313
lemma dropWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1314
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1315
  ==> dropWhile P l = dropWhile Q k"
wenzelm@20503
  1316
  by (induct k arbitrary: l, simp_all)
krauss@18336
  1317
wenzelm@13114
  1318
nipkow@15392
  1319
subsubsection {* @{text zip} *}
wenzelm@13114
  1320
wenzelm@13142
  1321
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1322
by (induct ys) auto
wenzelm@13114
  1323
wenzelm@13142
  1324
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1325
by simp
wenzelm@13114
  1326
wenzelm@13142
  1327
declare zip_Cons [simp del]
wenzelm@13114
  1328
nipkow@15281
  1329
lemma zip_Cons1:
nipkow@15281
  1330
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  1331
by(auto split:list.split)
nipkow@15281
  1332
wenzelm@13142
  1333
lemma length_zip [simp]:
nipkow@13145
  1334
"!!xs. length (zip xs ys) = min (length xs) (length ys)"
paulson@14208
  1335
apply (induct ys, simp)
paulson@14208
  1336
apply (case_tac xs, auto)
nipkow@13145
  1337
done
wenzelm@13114
  1338
wenzelm@13114
  1339
lemma zip_append1:
nipkow@13145
  1340
"!!xs. zip (xs @ ys) zs =
nipkow@13145
  1341
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
paulson@14208
  1342
apply (induct zs, simp)
paulson@14208
  1343
apply (case_tac xs, simp_all)
nipkow@13145
  1344
done
wenzelm@13114
  1345
wenzelm@13114
  1346
lemma zip_append2:
nipkow@13145
  1347
"!!ys. zip xs (ys @ zs) =
nipkow@13145
  1348
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
paulson@14208
  1349
apply (induct xs, simp)
paulson@14208
  1350
apply (case_tac ys, simp_all)
nipkow@13145
  1351
done
wenzelm@13114
  1352
wenzelm@13142
  1353
lemma zip_append [simp]:
wenzelm@13142
  1354
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1355
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1356
by (simp add: zip_append1)
wenzelm@13114
  1357
wenzelm@13114
  1358
lemma zip_rev:
nipkow@14247
  1359
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1360
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1361
wenzelm@13142
  1362
lemma nth_zip [simp]:
nipkow@13145
  1363
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
paulson@14208
  1364
apply (induct ys, simp)
nipkow@13145
  1365
apply (case_tac xs)
nipkow@13145
  1366
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1367
done
wenzelm@13114
  1368
wenzelm@13114
  1369
lemma set_zip:
nipkow@13145
  1370
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
  1371
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1372
wenzelm@13114
  1373
lemma zip_update:
nipkow@13145
  1374
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
  1375
by (rule sym, simp add: update_zip)
wenzelm@13114
  1376
wenzelm@13142
  1377
lemma zip_replicate [simp]:
nipkow@13145
  1378
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
paulson@14208
  1379
apply (induct i, auto)
paulson@14208
  1380
apply (case_tac j, auto)
nipkow@13145
  1381
done
wenzelm@13114
  1382
nipkow@19487
  1383
lemma take_zip:
nipkow@19487
  1384
 "!!xs ys. take n (zip xs ys) = zip (take n xs) (take n ys)"
nipkow@19487
  1385
apply (induct n)
nipkow@19487
  1386
 apply simp
nipkow@19487
  1387
apply (case_tac xs, simp)
nipkow@19487
  1388
apply (case_tac ys, simp_all)
nipkow@19487
  1389
done
nipkow@19487
  1390
nipkow@19487
  1391
lemma drop_zip:
nipkow@19487
  1392
 "!!xs ys. drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
nipkow@19487
  1393
apply (induct n)
nipkow@19487
  1394
 apply simp
nipkow@19487
  1395
apply (case_tac xs, simp)
nipkow@19487
  1396
apply (case_tac ys, simp_all)
nipkow@19487
  1397
done
nipkow@19487
  1398
wenzelm@13142
  1399
nipkow@15392
  1400
subsubsection {* @{text list_all2} *}
wenzelm@13114
  1401
kleing@14316
  1402
lemma list_all2_lengthD [intro?]: 
kleing@14316
  1403
  "list_all2 P xs ys ==> length xs = length ys"
haftmann@19607
  1404
  by (simp add: list_all2_def)
haftmann@19607
  1405
haftmann@19787
  1406
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
haftmann@19607
  1407
  by (simp add: list_all2_def)
haftmann@19607
  1408
haftmann@19787
  1409
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
haftmann@19787
  1410
  by (simp add: list_all2_def)
haftmann@19607
  1411
haftmann@19607
  1412
lemma list_all2_Cons [iff, code]:
haftmann@19607
  1413
  "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
haftmann@19607
  1414
  by (auto simp add: list_all2_def)
wenzelm@13114
  1415
wenzelm@13114
  1416
lemma list_all2_Cons1:
nipkow@13145
  1417
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1418
by (cases ys) auto
wenzelm@13114
  1419
wenzelm@13114
  1420
lemma list_all2_Cons2:
nipkow@13145
  1421
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1422
by (cases xs) auto
wenzelm@13114
  1423
wenzelm@13142
  1424
lemma list_all2_rev [iff]:
nipkow@13145
  1425
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1426
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1427
kleing@13863
  1428
lemma list_all2_rev1:
kleing@13863
  1429
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1430
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1431
wenzelm@13114
  1432
lemma list_all2_append1:
nipkow@13145
  1433
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1434
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1435
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1436
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1437
apply (rule iffI)
nipkow@13145
  1438
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1439
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  1440
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1441
apply (simp add: ball_Un)
nipkow@13145
  1442
done
wenzelm@13114
  1443
wenzelm@13114
  1444
lemma list_all2_append2:
nipkow@13145
  1445
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1446
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1447
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1448
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1449
apply (rule iffI)
nipkow@13145
  1450
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1451
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  1452
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1453
apply (simp add: ball_Un)
nipkow@13145
  1454
done
wenzelm@13114
  1455
kleing@13863
  1456
lemma list_all2_append:
nipkow@14247
  1457
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  1458
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  1459
by (induct rule:list_induct2, simp_all)
kleing@13863
  1460
kleing@13863
  1461
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1462
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
kleing@13863
  1463
  by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1464
wenzelm@13114
  1465
lemma list_all2_conv_all_nth:
nipkow@13145
  1466
"list_all2 P xs ys =
nipkow@13145
  1467
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1468
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1469
berghofe@13883
  1470
lemma list_all2_trans:
berghofe@13883
  1471
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1472
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1473
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1474
proof (induct as)
berghofe@13883
  1475
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1476
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1477
  proof (induct bs)
berghofe@13883
  1478
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1479
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1480
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1481
  qed simp
berghofe@13883
  1482
qed simp
berghofe@13883
  1483
kleing@13863
  1484
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1485
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
kleing@13863
  1486
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1487
paulson@14395
  1488
lemma list_all2I:
paulson@14395
  1489
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
paulson@14395
  1490
  by (simp add: list_all2_def)
paulson@14395
  1491
kleing@14328
  1492
lemma list_all2_nthD:
kleing@13863
  1493
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
kleing@13863
  1494
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1495
nipkow@14302
  1496
lemma list_all2_nthD2:
nipkow@14302
  1497
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@14302
  1498
  by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  1499
kleing@13863
  1500
lemma list_all2_map1: 
kleing@13863
  1501
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
kleing@13863
  1502
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1503
kleing@13863
  1504
lemma list_all2_map2: 
kleing@13863
  1505
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
kleing@13863
  1506
  by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1507
kleing@14316
  1508
lemma list_all2_refl [intro?]:
kleing@13863
  1509
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
kleing@13863
  1510
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1511
kleing@13863
  1512
lemma list_all2_update_cong:
kleing@13863
  1513
  "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
kleing@13863
  1514
  by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1515
kleing@13863
  1516
lemma list_all2_update_cong2:
kleing@13863
  1517
  "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
kleing@13863
  1518
  by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1519
nipkow@14302
  1520
lemma list_all2_takeI [simp,intro?]:
nipkow@14302
  1521
  "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@14302
  1522
  apply (induct xs)
nipkow@14302
  1523
   apply simp
nipkow@14302
  1524
  apply (clarsimp simp add: list_all2_Cons1)
nipkow@14302
  1525
  apply (case_tac n)
nipkow@14302
  1526
  apply auto
nipkow@14302
  1527
  done
nipkow@14302
  1528
nipkow@14302
  1529
lemma list_all2_dropI [simp,intro?]:
kleing@13863
  1530
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
paulson@14208
  1531
  apply (induct as, simp)
kleing@13863
  1532
  apply (clarsimp simp add: list_all2_Cons1)
paulson@14208
  1533
  apply (case_tac n, simp, simp)
kleing@13863
  1534
  done
kleing@13863
  1535
kleing@14327
  1536
lemma list_all2_mono [intro?]:
kleing@13863
  1537
  "\<And>y. list_all2 P x y \<Longrightarrow> (\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> list_all2 Q x y"
paulson@14208
  1538
  apply (induct x, simp)
paulson@14208
  1539
  apply (case_tac y, auto)
kleing@13863
  1540
  done
kleing@13863
  1541
wenzelm@13142
  1542
nipkow@15392
  1543
subsubsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  1544
wenzelm@13142
  1545
lemma foldl_append [simp]:
nipkow@13145
  1546
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145
  1547
by (induct xs) auto
wenzelm@13142
  1548
nipkow@14402
  1549
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  1550
by (induct xs) auto
nipkow@14402
  1551
krauss@19770
  1552
lemma foldl_cong [fundef_cong, recdef_cong]:
krauss@18336
  1553
  "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] 
krauss@18336
  1554
  ==> foldl f a l = foldl g b k"
wenzelm@20503
  1555
  by (induct k arbitrary: a b l) simp_all
krauss@18336
  1556
krauss@19770
  1557
lemma foldr_cong [fundef_cong, recdef_cong]:
krauss@18336
  1558
  "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] 
krauss@18336
  1559
  ==> foldr f l a = foldr g k b"
wenzelm@20503
  1560
  by (induct k arbitrary: a b l) simp_all
krauss@18336
  1561
nipkow@14402
  1562
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  1563
by (induct xs) auto
nipkow@14402
  1564
nipkow@14402
  1565
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  1566
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  1567
wenzelm@13142
  1568
text {*
nipkow@13145
  1569
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1570
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1571
*}
wenzelm@13142
  1572
wenzelm@13142
  1573
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
nipkow@13145
  1574
by (induct ns) auto
wenzelm@13142
  1575
wenzelm@13142
  1576
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1577
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1578
wenzelm@13142
  1579
lemma sum_eq_0_conv [iff]:
nipkow@13145
  1580
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145
  1581
by (induct ns) auto
wenzelm@13114
  1582
wenzelm@13114
  1583
nipkow@15392
  1584
subsubsection {* @{text upto} *}
wenzelm@13114
  1585
nipkow@17090
  1586
lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
nipkow@17090
  1587
-- {* simp does not terminate! *}
nipkow@13145
  1588
by (induct j) auto
wenzelm@13142
  1589
nipkow@15425
  1590
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
nipkow@13145
  1591
by (subst upt_rec) simp
wenzelm@13114
  1592
nipkow@15425
  1593
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
nipkow@15281
  1594
by(induct j)simp_all
nipkow@15281
  1595
nipkow@15281
  1596
lemma upt_eq_Cons_conv:
nipkow@15425
  1597
 "!!x xs. ([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
nipkow@15281
  1598
apply(induct j)
nipkow@15281
  1599
 apply simp
nipkow@15281
  1600
apply(clarsimp simp add: append_eq_Cons_conv)
nipkow@15281
  1601
apply arith
nipkow@15281
  1602
done
nipkow@15281
  1603
nipkow@15425
  1604
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
nipkow@13145
  1605
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1606
by simp
wenzelm@13114
  1607
nipkow@15425
  1608
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
nipkow@13145
  1609
apply(rule trans)
nipkow@13145
  1610
apply(subst upt_rec)
paulson@14208
  1611
 prefer 2 apply (rule refl, simp)
nipkow@13145
  1612
done
wenzelm@13114
  1613
nipkow@15425
  1614
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
nipkow@13145
  1615
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1616
by (induct k) auto
wenzelm@13114
  1617
nipkow@15425
  1618
lemma length_upt [simp]: "length [i..<j] = j - i"
nipkow@13145
  1619
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1620
nipkow@15425
  1621
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
nipkow@13145
  1622
apply (induct j)
nipkow@13145
  1623
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1624
done
wenzelm@13114
  1625
nipkow@17906
  1626
nipkow@17906
  1627
lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
nipkow@17906
  1628
by(simp add:upt_conv_Cons)
nipkow@17906
  1629
nipkow@17906
  1630
lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
nipkow@17906
  1631
apply(cases j)
nipkow@17906
  1632
 apply simp
nipkow@17906
  1633
by(simp add:upt_Suc_append)
nipkow@17906
  1634
nipkow@15425
  1635
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..<n] = [i..<i+m]"
paulson@14208
  1636
apply (induct m, simp)
nipkow@13145
  1637
apply (subst upt_rec)
nipkow@13145
  1638
apply (rule sym)
nipkow@13145
  1639
apply (subst upt_rec)
nipkow@13145
  1640
apply (simp del: upt.simps)
nipkow@13145
  1641
done
nipkow@3507
  1642
nipkow@17501
  1643
lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
nipkow@17501
  1644
apply(induct j)
nipkow@17501
  1645
apply auto
nipkow@17501
  1646
done
nipkow@17501
  1647
nipkow@15425
  1648
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..n]"
nipkow@13145
  1649
by (induct n) auto
wenzelm@13114
  1650
nipkow@15425
  1651
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
nipkow@13145
  1652
apply (induct n m rule: diff_induct)
nipkow@13145
  1653
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1654
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1655
done
wenzelm@13114
  1656
berghofe@13883
  1657
lemma nth_take_lemma:
berghofe@13883
  1658
  "!!xs ys. k <= length xs ==> k <= length ys ==>
berghofe@13883
  1659
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
berghofe@13883
  1660
apply (atomize, induct k)
paulson@14208
  1661
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  1662
txt {* Both lists must be non-empty *}
paulson@14208
  1663
apply (case_tac xs, simp)
paulson@14208
  1664
apply (case_tac ys, clarify)
nipkow@13145
  1665
 apply (simp (no_asm_use))
nipkow@13145
  1666
apply clarify
nipkow@13145
  1667
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  1668
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  1669
apply blast
nipkow@13145
  1670
done
wenzelm@13114
  1671
wenzelm@13114
  1672
lemma nth_equalityI:
wenzelm@13114
  1673
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  1674
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  1675
apply (simp_all add: take_all)
nipkow@13145
  1676
done
wenzelm@13142
  1677
kleing@13863
  1678
(* needs nth_equalityI *)
kleing@13863
  1679
lemma list_all2_antisym:
kleing@13863
  1680
  "\<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
  1681
  \<Longrightarrow> xs = ys"
kleing@13863
  1682
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  1683
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  1684
  done
kleing@13863
  1685
wenzelm@13142
  1686
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  1687
-- {* The famous take-lemma. *}
nipkow@13145
  1688
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  1689
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  1690
done
wenzelm@13142
  1691
wenzelm@13142
  1692
nipkow@15302
  1693
lemma take_Cons':
nipkow@15302
  1694
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@15302
  1695
by (cases n) simp_all
nipkow@15302
  1696
nipkow@15302
  1697
lemma drop_Cons':
nipkow@15302
  1698
     "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@15302
  1699
by (cases n) simp_all
nipkow@15302
  1700
nipkow@15302
  1701
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@15302
  1702
by (cases n) simp_all
nipkow@15302
  1703
paulson@18622
  1704
lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard]
paulson@18622
  1705
lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard]
paulson@18622
  1706
lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard]
paulson@18622
  1707
paulson@18622
  1708
declare take_Cons_number_of [simp] 
paulson@18622
  1709
        drop_Cons_number_of [simp] 
paulson@18622
  1710
        nth_Cons_number_of [simp] 
nipkow@15302
  1711
nipkow@15302
  1712
nipkow@15392
  1713
subsubsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  1714
wenzelm@13142
  1715
lemma distinct_append [simp]:
nipkow@13145
  1716
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  1717
by (induct xs) auto
wenzelm@13142
  1718
nipkow@15305
  1719
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
nipkow@15305
  1720
by(induct xs) auto
nipkow@15305
  1721
wenzelm@13142
  1722
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  1723
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  1724
wenzelm@13142
  1725
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  1726
by (induct xs) auto
wenzelm@13142
  1727
paulson@15072
  1728
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
paulson@15251
  1729
  by (induct x, auto) 
paulson@15072
  1730
paulson@15072
  1731
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
paulson@15251
  1732
  by (induct x, auto)
paulson@15072
  1733
nipkow@15245
  1734
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245
  1735
by (induct xs) auto
nipkow@15245
  1736
nipkow@15245
  1737
lemma length_remdups_eq[iff]:
nipkow@15245
  1738
  "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245
  1739
apply(induct xs)
nipkow@15245
  1740
 apply auto
nipkow@15245
  1741
apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245
  1742
 apply arith
nipkow@15245
  1743
apply(rule length_remdups_leq)
nipkow@15245
  1744
done
nipkow@15245
  1745
nipkow@18490
  1746
nipkow@18490
  1747
lemma distinct_map:
nipkow@18490
  1748
  "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
nipkow@18490
  1749
by (induct xs) auto
nipkow@18490
  1750
nipkow@18490
  1751
wenzelm@13142
  1752
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  1753
by (induct xs) auto
wenzelm@13114
  1754
nipkow@17501
  1755
lemma distinct_upt[simp]: "distinct[i..<j]"
nipkow@17501
  1756
by (induct j) auto
nipkow@17501
  1757
nipkow@17501
  1758
lemma distinct_take[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (take i xs)"
nipkow@17501
  1759
apply(induct xs)
nipkow@17501
  1760
 apply simp
nipkow@17501
  1761
apply (case_tac i)
nipkow@17501
  1762
 apply simp_all
nipkow@17501
  1763
apply(blast dest:in_set_takeD)
nipkow@17501
  1764
done
nipkow@17501
  1765
nipkow@17501
  1766
lemma distinct_drop[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (drop i xs)"
nipkow@17501
  1767
apply(induct xs)
nipkow@17501
  1768
 apply simp
nipkow@17501
  1769
apply (case_tac i)
nipkow@17501
  1770
 apply simp_all
nipkow@17501
  1771
done
nipkow@17501
  1772
nipkow@17501
  1773
lemma distinct_list_update:
nipkow@17501
  1774
assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
nipkow@17501
  1775
shows "distinct (xs[i:=a])"
nipkow@17501
  1776
proof (cases "i < length xs")
nipkow@17501
  1777
  case True
nipkow@17501
  1778
  with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
nipkow@17501
  1779
    apply (drule_tac id_take_nth_drop) by simp
nipkow@17501
  1780
  with d True show ?thesis
nipkow@17501
  1781
    apply (simp add: upd_conv_take_nth_drop)
nipkow@17501
  1782
    apply (drule subst [OF id_take_nth_drop]) apply assumption
nipkow@17501
  1783
    apply simp apply (cases "a = xs!i") apply simp by blast
nipkow@17501
  1784
next
nipkow@17501
  1785
  case False with d show ?thesis by auto
nipkow@17501
  1786
qed
nipkow@17501
  1787
nipkow@17501
  1788
nipkow@17501
  1789
text {* It is best to avoid this indexed version of distinct, but
nipkow@17501
  1790
sometimes it is useful. *}
nipkow@17501
  1791
wenzelm@13142
  1792
lemma distinct_conv_nth:
nipkow@17501
  1793
"distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@15251
  1794
apply (induct xs, simp, simp)
paulson@14208
  1795
apply (rule iffI, clarsimp)
nipkow@13145
  1796
 apply (case_tac i)
paulson@14208
  1797
apply (case_tac j, simp)
nipkow@13145
  1798
apply (simp add: set_conv_nth)
nipkow@13145
  1799
 apply (case_tac j)
paulson@14208
  1800
apply (clarsimp simp add: set_conv_nth, simp)
nipkow@13145
  1801
apply (rule conjI)
nipkow@13145
  1802
 apply (clarsimp simp add: set_conv_nth)
nipkow@17501
  1803
 apply (erule_tac x = 0 in allE, simp)
paulson@14208
  1804
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
nipkow@17501
  1805
apply (erule_tac x = "Suc i" in allE, simp)
paulson@14208
  1806
apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145
  1807
done
wenzelm@13114
  1808
nipkow@18490
  1809
lemma nth_eq_iff_index_eq:
nipkow@18490
  1810
 "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
nipkow@18490
  1811
by(auto simp: distinct_conv_nth)
nipkow@18490
  1812
nipkow@15110
  1813
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
kleing@14388
  1814
  by (induct xs) auto
kleing@14388
  1815
nipkow@15110
  1816
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388
  1817
proof (induct xs)
kleing@14388
  1818
  case Nil thus ?case by simp
kleing@14388
  1819
next
kleing@14388
  1820
  case (Cons x xs)
kleing@14388
  1821
  show ?case
kleing@14388
  1822
  proof (cases "x \<in> set xs")
kleing@14388
  1823
    case False with Cons show ?thesis by simp
kleing@14388
  1824
  next
kleing@14388
  1825
    case True with Cons.prems
kleing@14388
  1826
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  1827
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  1828
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  1829
    ultimately have False by simp
kleing@14388
  1830
    thus ?thesis ..
kleing@14388
  1831
  qed
kleing@14388
  1832
qed
kleing@14388
  1833
nipkow@18490
  1834
nipkow@18490
  1835
lemma length_remdups_concat:
nipkow@18490
  1836
 "length(remdups(concat xss)) = card(\<Union>xs \<in> set xss. set xs)"
nipkow@18490
  1837
by(simp add: distinct_card[symmetric])
nipkow@17906
  1838
nipkow@17906
  1839
nipkow@15392
  1840
subsubsection {* @{text remove1} *}
nipkow@15110
  1841
nipkow@18049
  1842
lemma remove1_append:
nipkow@18049
  1843
  "remove1 x (xs @ ys) =
nipkow@18049
  1844
  (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
nipkow@18049
  1845
by (induct xs) auto
nipkow@18049
  1846
nipkow@15110
  1847
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110
  1848
apply(induct xs)
nipkow@15110
  1849
 apply simp
nipkow@15110
  1850
apply simp
nipkow@15110
  1851
apply blast
nipkow@15110
  1852
done
nipkow@15110
  1853
paulson@17724
  1854
lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110
  1855
apply(induct xs)
nipkow@15110
  1856
 apply simp
nipkow@15110
  1857
apply simp
nipkow@15110
  1858
apply blast
nipkow@15110
  1859
done
nipkow@15110
  1860
nipkow@18049
  1861
lemma remove1_filter_not[simp]:
nipkow@18049
  1862
  "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
nipkow@18049
  1863
by(induct xs) auto
nipkow@18049
  1864
nipkow@15110
  1865
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110
  1866
apply(insert set_remove1_subset)
nipkow@15110
  1867
apply fast
nipkow@15110
  1868
done
nipkow@15110
  1869
nipkow@15110
  1870
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110
  1871
by (induct xs) simp_all
nipkow@15110
  1872
wenzelm@13114
  1873
nipkow@15392
  1874
subsubsection {* @{text replicate} *}
wenzelm@13114
  1875
wenzelm@13142
  1876
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  1877
by (induct n) auto
nipkow@13124
  1878
wenzelm@13142
  1879
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  1880
by (induct n) auto
wenzelm@13114
  1881
wenzelm@13114
  1882
lemma replicate_app_Cons_same:
nipkow@13145
  1883
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  1884
by (induct n) auto
wenzelm@13114
  1885
wenzelm@13142
  1886
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208
  1887
apply (induct n, simp)
nipkow@13145
  1888
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  1889
done
wenzelm@13114
  1890
wenzelm@13142
  1891
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  1892
by (induct n) auto
wenzelm@13114
  1893
nipkow@16397
  1894
text{* Courtesy of Matthias Daum: *}
nipkow@16397
  1895
lemma append_replicate_commute:
nipkow@16397
  1896
  "replicate n x @ replicate k x = replicate k x @ replicate n x"
nipkow@16397
  1897
apply (simp add: replicate_add [THEN sym])
nipkow@16397
  1898
apply (simp add: add_commute)
nipkow@16397
  1899
done
nipkow@16397
  1900
wenzelm@13142
  1901
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  1902
by (induct n) auto
wenzelm@13114
  1903
wenzelm@13142
  1904
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  1905
by (induct n) auto
wenzelm@13114
  1906
wenzelm@13142
  1907
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  1908
by (atomize (full), induct n) auto
wenzelm@13114
  1909
wenzelm@13142
  1910
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
paulson@14208
  1911
apply (induct n, simp)
nipkow@13145
  1912
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  1913
done
wenzelm@13114
  1914
nipkow@16397
  1915
text{* Courtesy of Matthias Daum (2 lemmas): *}
nipkow@16397
  1916
lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
nipkow@16397
  1917
apply (case_tac "k \<le> i")
nipkow@16397
  1918
 apply  (simp add: min_def)
nipkow@16397
  1919
apply (drule not_leE)
nipkow@16397
  1920
apply (simp add: min_def)
nipkow@16397
  1921
apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
nipkow@16397
  1922
 apply  simp
nipkow@16397
  1923
apply (simp add: replicate_add [symmetric])
nipkow@16397
  1924
done
nipkow@16397
  1925
nipkow@16397
  1926
lemma drop_replicate[simp]: "!!i. drop i (replicate k x) = replicate (k-i) x"
nipkow@16397
  1927
apply (induct k)
nipkow@16397
  1928
 apply simp
nipkow@16397
  1929
apply clarsimp
nipkow@16397
  1930
apply (case_tac i)
nipkow@16397
  1931
 apply simp
nipkow@16397
  1932
apply clarsimp
nipkow@16397
  1933
done
nipkow@16397
  1934
nipkow@16397
  1935
wenzelm@13142
  1936
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  1937
by (induct n) auto
wenzelm@13114
  1938
wenzelm@13142
  1939
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  1940
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  1941
wenzelm@13142
  1942
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  1943
by auto
wenzelm@13114
  1944
wenzelm@13142
  1945
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  1946
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  1947
wenzelm@13114
  1948
nipkow@15392
  1949
subsubsection{*@{text rotate1} and @{text rotate}*}
nipkow@15302
  1950
nipkow@15302
  1951
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
nipkow@15302
  1952
by(simp add:rotate1_def)
nipkow@15302
  1953
nipkow@15302
  1954
lemma rotate0[simp]: "rotate 0 = id"
nipkow@15302
  1955
by(simp add:rotate_def)
nipkow@15302
  1956
nipkow@15302
  1957
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
nipkow@15302
  1958
by(simp add:rotate_def)
nipkow@15302
  1959
nipkow@15302
  1960
lemma rotate_add:
nipkow@15302
  1961
  "rotate (m+n) = rotate m o rotate n"
nipkow@15302
  1962
by(simp add:rotate_def funpow_add)
nipkow@15302
  1963
nipkow@15302
  1964
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
nipkow@15302
  1965
by(simp add:rotate_add)
nipkow@15302
  1966
nipkow@18049
  1967
lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
nipkow@18049
  1968
by(simp add:rotate_def funpow_swap1)
nipkow@18049
  1969
nipkow@15302
  1970
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
nipkow@15302
  1971
by(cases xs) simp_all
nipkow@15302
  1972
nipkow@15302
  1973
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  1974
apply(induct n)
nipkow@15302
  1975
 apply simp
nipkow@15302
  1976
apply (simp add:rotate_def)
nipkow@13145
  1977
done
wenzelm@13114
  1978
nipkow@15302
  1979
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
nipkow@15302
  1980
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1981
nipkow@15302
  1982
lemma rotate_drop_take:
nipkow@15302
  1983
  "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
nipkow@15302
  1984
apply(induct n)
nipkow@15302
  1985
 apply simp
nipkow@15302
  1986
apply(simp add:rotate_def)
nipkow@15302
  1987
apply(cases "xs = []")
nipkow@15302
  1988
 apply (simp)
nipkow@15302
  1989
apply(case_tac "n mod length xs = 0")
nipkow@15302
  1990
 apply(simp add:mod_Suc)
nipkow@15302
  1991
 apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
nipkow@15302
  1992
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
nipkow@15302
  1993
                take_hd_drop linorder_not_le)
nipkow@13145
  1994
done
wenzelm@13114
  1995
nipkow@15302
  1996
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
nipkow@15302
  1997
by(simp add:rotate_drop_take)
nipkow@15302
  1998
nipkow@15302
  1999
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2000
by(simp add:rotate_drop_take)
nipkow@15302
  2001
nipkow@15302
  2002
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
nipkow@15302
  2003
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2004
nipkow@15302
  2005
lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs"
nipkow@15302
  2006
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2007
nipkow@15302
  2008
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
nipkow@15302
  2009
by(simp add:rotate1_def split:list.split) blast
nipkow@15302
  2010
nipkow@15302
  2011
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
nipkow@15302
  2012
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2013
nipkow@15302
  2014
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
nipkow@15302
  2015
by(simp add:rotate_drop_take take_map drop_map)
nipkow@15302
  2016
nipkow@15302
  2017
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
nipkow@15302
  2018
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2019
nipkow@15302
  2020
lemma set_rotate[simp]: "set(rotate n xs) = set xs"
nipkow@15302
  2021
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2022
nipkow@15302
  2023
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
nipkow@15302
  2024
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2025
nipkow@15302
  2026
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
nipkow@15302
  2027
by (induct n) (simp_all add:rotate_def)
wenzelm@13114
  2028
nipkow@15439
  2029
lemma rotate_rev:
nipkow@15439
  2030
  "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
nipkow@15439
  2031
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2032
apply(cases "length xs = 0")
nipkow@15439
  2033
 apply simp
nipkow@15439
  2034
apply(cases "n mod length xs = 0")
nipkow@15439
  2035
 apply simp
nipkow@15439
  2036
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2037
done
nipkow@15439
  2038
nipkow@18423
  2039
lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
nipkow@18423
  2040
apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
nipkow@18423
  2041
apply(subgoal_tac "length xs \<noteq> 0")
nipkow@18423
  2042
 prefer 2 apply simp
nipkow@18423
  2043
using mod_less_divisor[of "length xs" n] by arith
nipkow@18423
  2044
wenzelm@13114
  2045
nipkow@15392
  2046
subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  2047
wenzelm@13142
  2048
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  2049
by (auto simp add: sublist_def)
wenzelm@13114
  2050
wenzelm@13142
  2051
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  2052
by (auto simp add: sublist_def)
wenzelm@13114
  2053
nipkow@15281
  2054
lemma length_sublist:
nipkow@15281
  2055
  "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
nipkow@15281
  2056
by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
nipkow@15281
  2057
nipkow@15281
  2058
lemma sublist_shift_lemma_Suc:
nipkow@15281
  2059
  "!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
nipkow@15281
  2060
         map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
nipkow@15281
  2061
apply(induct xs)
nipkow@15281
  2062
 apply simp
nipkow@15281
  2063
apply (case_tac "is")
nipkow@15281
  2064
 apply simp
nipkow@15281
  2065
apply simp
nipkow@15281
  2066
done
nipkow@15281
  2067
wenzelm@13114
  2068
lemma sublist_shift_lemma:
nipkow@15425
  2069
     "map fst [p:zip xs [i..<i + length xs] . snd p : A] =
nipkow@15425
  2070
      map fst [p:zip xs [0..<length xs] . snd p + i : A]"
nipkow@13145
  2071
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  2072
wenzelm@13114
  2073
lemma sublist_append:
paulson@15168
  2074
     "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  2075
apply (unfold sublist_def)
paulson@14208
  2076
apply (induct l' rule: rev_induct, simp)
nipkow@13145
  2077
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  2078
apply (simp add: add_commute)
nipkow@13145
  2079
done
wenzelm@13114
  2080
wenzelm@13114
  2081
lemma sublist_Cons:
nipkow@13145
  2082
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  2083
apply (induct l rule: rev_induct)
nipkow@13145
  2084
 apply (simp add: sublist_def)
nipkow@13145
  2085
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  2086
done
wenzelm@13114
  2087
nipkow@15281
  2088
lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
nipkow@15281
  2089
apply(induct xs)
nipkow@15281
  2090
 apply simp
nipkow@15281
  2091
apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE)
nipkow@15281
  2092
 apply(erule lessE)
nipkow@15281
  2093
  apply auto
nipkow@15281
  2094
apply(erule lessE)
nipkow@15281
  2095
apply auto
nipkow@15281
  2096
done
nipkow@15281
  2097
nipkow@15281
  2098
lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
nipkow@15281
  2099
by(auto simp add:set_sublist)
nipkow@15281
  2100
nipkow@15281
  2101
lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
nipkow@15281
  2102
by(auto simp add:set_sublist)
nipkow@15281
  2103
nipkow@15281
  2104
lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
nipkow@15281
  2105
by(auto simp add:set_sublist)
nipkow@15281
  2106
wenzelm@13142
  2107
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  2108
by (simp add: sublist_Cons)
wenzelm@13114
  2109
nipkow@15281
  2110
nipkow@15281
  2111
lemma distinct_sublistI[simp]: "!!I. distinct xs \<Longrightarrow> distinct(sublist xs I)"
nipkow@15281
  2112
apply(induct xs)
nipkow@15281
  2113
 apply simp
nipkow@15281
  2114
apply(auto simp add:sublist_Cons)
nipkow@15281
  2115
done
nipkow@15281
  2116
nipkow@15281
  2117
nipkow@15045
  2118
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
paulson@14208
  2119
apply (induct l rule: rev_induct, simp)
nipkow@13145
  2120
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  2121
done
wenzelm@13114
  2122
nipkow@17501
  2123
lemma filter_in_sublist: "\<And>s. distinct xs \<Longrightarrow>
nipkow@17501
  2124
  filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
nipkow@17501
  2125
proof (induct xs)
nipkow@17501
  2126
  case Nil thus ?case by simp
nipkow@17501
  2127
next
nipkow@17501
  2128
  case (Cons a xs)
nipkow@17501
  2129
  moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
nipkow@17501
  2130
  ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
nipkow@17501
  2131
qed
nipkow@17501
  2132
wenzelm@13114
  2133
nipkow@19390
  2134
subsubsection {* @{const splice} *}
nipkow@19390
  2135
haftmann@19607
  2136
lemma splice_Nil2 [simp, code]:
nipkow@19390
  2137
 "splice xs [] = xs"
nipkow@19390
  2138
by (cases xs) simp_all
nipkow@19390
  2139
haftmann@19607
  2140
lemma splice_Cons_Cons [simp, code]:
nipkow@19390
  2141
 "splice (x#xs) (y#ys) = x # y # splice xs ys"
nipkow@19390
  2142
by simp
nipkow@19390
  2143
haftmann@19607
  2144
declare splice.simps(2) [simp del, code del]
nipkow@19390
  2145
nipkow@15392
  2146
subsubsection{*Sets of Lists*}
nipkow@15392
  2147
nipkow@15392
  2148
subsubsection {* @{text lists}: the list-forming operator over sets *}
nipkow@15302
  2149
nipkow@15302
  2150
consts lists :: "'a set => 'a list set"
nipkow@15302
  2151
inductive "lists A"
nipkow@15302
  2152
 intros
nipkow@15302
  2153
  Nil [intro!]: "[]: lists A"
nipkow@15302
  2154
  Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A"
nipkow@15302
  2155
nipkow@15302
  2156
inductive_cases listsE [elim!]: "x#l : lists A"
nipkow@15302
  2157
nipkow@15302
  2158
lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B"
nipkow@15302
  2159
by (unfold lists.defs) (blast intro!: lfp_mono)
nipkow@15302
  2160
nipkow@15302
  2161
lemma lists_IntI:
nipkow@15302
  2162
  assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l
nipkow@15302
  2163
  by induct blast+
nipkow@15302
  2164
nipkow@15302
  2165
lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B"
nipkow@15302
  2166
proof (rule mono_Int [THEN equalityI])
nipkow@15302
  2167
  show "mono lists" by (simp add: mono_def lists_mono)
nipkow@15302
  2168
  show "lists A \<inter> lists B \<subseteq> lists (A \<inter> B)" by (blast intro: lists_IntI)
kleing@14388
  2169
qed
kleing@14388
  2170
nipkow@15302
  2171
lemma append_in_lists_conv [iff]:
nipkow@15302
  2172
     "(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"
nipkow@15302
  2173
by (induct xs) auto
nipkow@15302
  2174
nipkow@15302
  2175
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"
nipkow@15302
  2176
-- {* eliminate @{text lists} in favour of @{text set} *}
nipkow@15302
  2177
by (induct xs) auto
nipkow@15302
  2178
nipkow@15302
  2179
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"
nipkow@15302
  2180
by (rule in_lists_conv_set [THEN iffD1])
nipkow@15302
  2181
nipkow@15302
  2182
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"
nipkow@15302
  2183
by (rule in_lists_conv_set [THEN iffD2])
nipkow@15302
  2184
nipkow@15302
  2185
lemma lists_UNIV [simp]: "lists UNIV = UNIV"
nipkow@15302
  2186
by auto
nipkow@15302
  2187
nipkow@17086
  2188
nipkow@17086
  2189
nipkow@17086
  2190
subsubsection{* Inductive definition for membership *}
nipkow@17086
  2191
nipkow@17086
  2192
consts ListMem :: "('a \<times> 'a list)set"
nipkow@17086
  2193
inductive ListMem
nipkow@17086
  2194
intros
nipkow@17086
  2195
 elem:  "(x,x#xs) \<in> ListMem"
nipkow@17086
  2196
 insert:  "(x,xs) \<in> ListMem \<Longrightarrow> (x,y#xs) \<in> ListMem"
nipkow@17086
  2197
nipkow@17086
  2198
lemma ListMem_iff: "((x,xs) \<in> ListMem) = (x \<in> set xs)"
nipkow@17086
  2199
apply (rule iffI)
nipkow@17086
  2200
 apply (induct set: ListMem)
nipkow@17086
  2201
  apply auto
nipkow@17086
  2202
apply (induct xs)
nipkow@17086
  2203
 apply (auto intro: ListMem.intros)
nipkow@17086
  2204
done
nipkow@17086
  2205
nipkow@17086
  2206
nipkow@17086
  2207
nipkow@15392
  2208
subsubsection{*Lists as Cartesian products*}
nipkow@15302
  2209
nipkow@15302
  2210
text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
nipkow@15302
  2211
@{term A} and tail drawn from @{term Xs}.*}
nipkow@15302
  2212
nipkow@15302
  2213
constdefs
nipkow@15302
  2214
  set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
nipkow@15302
  2215
  "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"
nipkow@15302
  2216
paulson@17724
  2217
lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
nipkow@15302
  2218
by (auto simp add: set_Cons_def)
nipkow@15302
  2219
nipkow@15302
  2220
text{*Yields the set of lists, all of the same length as the argument and
nipkow@15302
  2221
with elements drawn from the corresponding element of the argument.*}
nipkow@15302
  2222
nipkow@15302
  2223
consts  listset :: "'a set list \<Rightarrow> 'a list set"
nipkow@15302
  2224
primrec
nipkow@15302
  2225
   "listset []    = {[]}"
nipkow@15302
  2226
   "listset(A#As) = set_Cons A (listset As)"
nipkow@15302
  2227
nipkow@15302
  2228
paulson@15656
  2229
subsection{*Relations on Lists*}
paulson@15656
  2230
paulson@15656
  2231
subsubsection {* Length Lexicographic Ordering *}
paulson@15656
  2232
paulson@15656
  2233
text{*These orderings preserve well-foundedness: shorter lists 
paulson@15656
  2234
  precede longer lists. These ordering are not used in dictionaries.*}
paulson@15656
  2235
paulson@15656
  2236
consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
paulson@15656
  2237
        --{*The lexicographic ordering for lists of the specified length*}
nipkow@15302
  2238
primrec
paulson@15656
  2239
  "lexn r 0 = {}"
paulson@15656
  2240
  "lexn r (Suc n) =
paulson@15656
  2241
    (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
paulson@15656
  2242
    {(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
nipkow@15302
  2243
nipkow@15302
  2244
constdefs
paulson@15656
  2245
  lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
paulson@15656
  2246
    "lex r == \<Union>n. lexn r n"
paulson@15656
  2247
        --{*Holds only between lists of the same length*}
paulson@15656
  2248
nipkow@15693
  2249
  lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@15693
  2250
    "lenlex r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
paulson@15656
  2251
        --{*Compares lists by their length and then lexicographically*}
nipkow@15302
  2252
nipkow@15302
  2253
nipkow@15302
  2254
lemma wf_lexn: "wf r ==> wf (lexn r n)"
nipkow@15302
  2255
apply (induct n, simp, simp)
nipkow@15302
  2256
apply(rule wf_subset)
nipkow@15302
  2257
 prefer 2 apply (rule Int_lower1)
nipkow@15302
  2258
apply(rule wf_prod_fun_image)
nipkow@15302
  2259
 prefer 2 apply (rule inj_onI, auto)
nipkow@15302
  2260
done
nipkow@15302
  2261
nipkow@15302
  2262
lemma lexn_length:
nipkow@15302
  2263
     "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@15302
  2264
by (induct n) auto
nipkow@15302
  2265
nipkow@15302
  2266
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@15302
  2267
apply (unfold lex_def)
nipkow@15302
  2268
apply (rule wf_UN)
nipkow@15302
  2269
apply (blast intro: wf_lexn, clarify)
nipkow@15302
  2270
apply (rename_tac m n)
nipkow@15302
  2271
apply (subgoal_tac "m \<noteq> n")
nipkow@15302
  2272
 prefer 2 apply blast
nipkow@15302
  2273
apply (blast dest: lexn_length not_sym)
nipkow@15302
  2274
done
nipkow@15302
  2275
nipkow@15302
  2276
lemma lexn_conv:
paulson@15656
  2277
  "lexn r n =
paulson@15656
  2278
    {(xs,ys). length xs = n \<and> length ys = n \<and>
paulson@15656
  2279
    (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
nipkow@18423
  2280
apply (induct n, simp)
nipkow@15302
  2281
apply (simp add: image_Collect lex_prod_def, safe, blast)
nipkow@15302
  2282
 apply (rule_tac x = "ab # xys" in exI, simp)
nipkow@15302
  2283
apply (case_tac xys, simp_all, blast)
nipkow@15302
  2284
done
nipkow@15302
  2285
nipkow@15302
  2286
lemma lex_conv:
paulson@15656
  2287
  "lex r =
paulson@15656
  2288
    {(xs,ys). length xs = length ys \<and>
paulson@15656
  2289
    (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@15302
  2290
by (force simp add: lex_def lexn_conv)
nipkow@15302
  2291
nipkow@15693
  2292
lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
nipkow@15693
  2293
by (unfold lenlex_def) blast
nipkow@15693
  2294
nipkow@15693
  2295
lemma lenlex_conv:
nipkow@15693
  2296
    "lenlex r = {(xs,ys). length xs < length ys |
paulson@15656
  2297
                 length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@19623
  2298
by (simp add: lenlex_def diag_def lex_prod_def inv_image_def)
nipkow@15302
  2299
nipkow@15302
  2300
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@15302
  2301
by (simp add: lex_conv)
nipkow@15302
  2302
nipkow@15302
  2303
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@15302
  2304
by (simp add:lex_conv)
nipkow@15302
  2305
paulson@18447
  2306
lemma Cons_in_lex [simp]:
paulson@15656
  2307
    "((x # xs, y # ys) : lex r) =
paulson@15656
  2308
      ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@15302
  2309
apply (simp add: lex_conv)
nipkow@15302
  2310
apply (rule iffI)
nipkow@15302
  2311
 prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
nipkow@15302
  2312
apply (case_tac xys, simp, simp)
nipkow@15302
  2313
apply blast
nipkow@15302
  2314
done
nipkow@15302
  2315
nipkow@15302
  2316
paulson@15656
  2317
subsubsection {* Lexicographic Ordering *}
paulson@15656
  2318
paulson@15656
  2319
text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
paulson@15656
  2320
    This ordering does \emph{not} preserve well-foundedness.
nipkow@17090
  2321
     Author: N. Voelker, March 2005. *} 
paulson@15656
  2322
paulson@15656
  2323
constdefs 
paulson@15656
  2324
  lexord :: "('a * 'a)set \<Rightarrow> ('a list * 'a list) set" 
paulson@15656
  2325
  "lexord  r == {(x,y). \<exists> a v. y = x @ a # v \<or> 
paulson@15656
  2326
            (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
paulson@15656
  2327
paulson@15656
  2328
lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
paulson@15656
  2329
  by (unfold lexord_def, induct_tac y, auto) 
paulson@15656
  2330
paulson@15656
  2331
lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
paulson@15656
  2332
  by (unfold lexord_def, induct_tac x, auto)
paulson@15656
  2333
paulson@15656
  2334
lemma lexord_cons_cons[simp]:
paulson@15656
  2335
     "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
paulson@15656
  2336
  apply (unfold lexord_def, safe, simp_all)
paulson@15656
  2337
  apply (case_tac u, simp, simp)
paulson@15656
  2338
  apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
paulson@15656
  2339
  apply (erule_tac x="b # u" in allE)
paulson@15656
  2340
  by force
paulson@15656
  2341
paulson@15656
  2342
lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
paulson@15656
  2343
paulson@15656
  2344
lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
paulson@15656
  2345
  by (induct_tac x, auto)  
paulson@15656
  2346
paulson@15656
  2347
lemma lexord_append_left_rightI:
paulson@15656
  2348
     "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
paulson@15656
  2349
  by (induct_tac u, auto)
paulson@15656
  2350
paulson@15656
  2351
lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
paulson@15656
  2352
  by (induct x, auto)
paulson@15656
  2353
paulson@15656
  2354
lemma lexord_append_leftD:
paulson@15656
  2355
     "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
paulson@15656
  2356
  by (erule rev_mp, induct_tac x, auto)
paulson@15656
  2357
paulson@15656
  2358
lemma lexord_take_index_conv: 
paulson@15656
  2359
   "((x,y) : lexord r) = 
paulson@15656
  2360
    ((length x < length y \<and> take (length x) y = x) \<or> 
paulson@15656
  2361
     (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
paulson@15656
  2362
  apply (unfold lexord_def Let_def, clarsimp) 
paulson@15656
  2363
  apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
paulson@15656
  2364
  apply auto 
paulson@15656
  2365
  apply (rule_tac x="hd (drop (length x) y)" in exI)
paulson@15656
  2366
  apply (rule_tac x="tl (drop (length x) y)" in exI)
paulson@15656
  2367
  apply (erule subst, simp add: min_def) 
paulson@15656
  2368
  apply (rule_tac x ="length u" in exI, simp) 
paulson@15656
  2369
  apply (rule_tac x ="take i x" in exI) 
paulson@15656
  2370
  apply (rule_tac x ="x ! i" in exI) 
paulson@15656
  2371
  apply (rule_tac x ="y ! i" in exI, safe) 
paulson@15656
  2372
  apply (rule_tac x="drop (Suc i) x" in exI)
paulson@15656
  2373
  apply (drule sym, simp add: drop_Suc_conv_tl) 
paulson@15656
  2374
  apply (rule_tac x="drop (Suc i) y" in exI)
paulson@15656
  2375
  by (simp add: drop_Suc_conv_tl) 
paulson@15656
  2376
paulson@15656
  2377
-- {* lexord is extension of partial ordering List.lex *} 
paulson@15656
  2378
lemma lexord_lex: " (x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
paulson@15656
  2379
  apply (rule_tac x = y in spec) 
paulson@15656
  2380
  apply (induct_tac x, clarsimp) 
paulson@15656
  2381
  by (clarify, case_tac x, simp, force)
paulson@15656
  2382
paulson@15656
  2383
lemma lexord_irreflexive: "(! x. (x,x) \<notin> r) \<Longrightarrow> (y,y) \<notin> lexord r"
paulson@15656
  2384
  by (induct y, auto)
paulson@15656
  2385
paulson@15656
  2386
lemma lexord_trans: 
paulson@15656
  2387
    "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
paulson@15656
  2388
   apply (erule rev_mp)+
paulson@15656
  2389
   apply (rule_tac x = x in spec) 
paulson@15656
  2390
  apply (rule_tac x = z in spec) 
paulson@15656
  2391
  apply ( induct_tac y, simp, clarify)
paulson@15656
  2392
  apply (case_tac xa, erule ssubst) 
paulson@15656
  2393
  apply (erule allE, erule allE) -- {* avoid simp recursion *} 
paulson@15656
  2394
  apply (case_tac x, simp, simp) 
paulson@15656
  2395
  apply (case_tac x, erule allE, erule allE, simp) 
paulson@15656
  2396
  apply (erule_tac x = listb in allE) 
paulson@15656
  2397
  apply (erule_tac x = lista in allE, simp)
paulson@15656
  2398
  apply (unfold trans_def)
paulson@15656
  2399
  by blast
paulson@15656
  2400
paulson@15656
  2401
lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
paulson@15656
  2402
  by (rule transI, drule lexord_trans, blast) 
paulson@15656
  2403
paulson@15656
  2404
lemma lexord_linear: "(! a b. (a,b)\<in> r | a = b | (b,a) \<in> r) \<Longrightarrow> (x,y) : lexord r | x = y | (y,x) : lexord r"
paulson@15656
  2405
  apply (rule_tac x = y in spec) 
paulson@15656
  2406
  apply (induct_tac x, rule allI) 
paulson@15656
  2407
  apply (case_tac x, simp, simp) 
paulson@15656
  2408
  apply (rule allI, case_tac x, simp, simp) 
paulson@15656
  2409
  by blast
paulson@15656
  2410
paulson@15656
  2411
krauss@21103
  2412
subsection {* Lexicographic combination of measure functions *}
krauss@21103
  2413
krauss@21103
  2414
text {* These are useful for termination proofs *}
krauss@21103
  2415
krauss@21103
  2416
definition
krauss@21103
  2417
  "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"
krauss@21103
  2418
krauss@21106
  2419
lemma wf_measures[recdef_wf, simp]: "wf (measures fs)"
krauss@21103
  2420
  unfolding measures_def
krauss@21103
  2421
  by blast
krauss@21103
  2422
krauss@21103
  2423
lemma in_measures[simp]: 
krauss@21103
  2424
  "(x, y) \<in> measures [] = False"
krauss@21103
  2425
  "(x, y) \<in> measures (f # fs)
krauss@21103
  2426
         = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"  
krauss@21103
  2427
  unfolding measures_def
krauss@21103
  2428
  by auto
krauss@21103
  2429
krauss@21103
  2430
lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
krauss@21103
  2431
  by simp
krauss@21103
  2432
krauss@21103
  2433
lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
krauss@21103
  2434
  by auto
krauss@21103
  2435
krauss@21211
  2436
(* install the lexicographic_order method and the "fun" command *)
bulwahn@21131
  2437
use "Tools/function_package/lexicographic_order.ML"
krauss@21211
  2438
use "Tools/function_package/fundef_datatype.ML"
krauss@21211
  2439
setup LexicographicOrder.setup
krauss@21211
  2440
setup FundefDatatype.setup
krauss@21211
  2441
krauss@21103
  2442
nipkow@15392
  2443
subsubsection{*Lifting a Relation on List Elements to the Lists*}
nipkow@15302
  2444
nipkow@15302
  2445
consts  listrel :: "('a * 'a)set => ('a list * 'a list)set"
nipkow@15302
  2446
nipkow@15302
  2447
inductive "listrel(r)"
nipkow@15302
  2448
 intros
nipkow@15302
  2449
   Nil:  "([],[]) \<in> listrel r"
nipkow@15302
  2450
   Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
nipkow@15302
  2451
nipkow@15302
  2452
inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
nipkow@15302
  2453
inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
nipkow@15302
  2454
inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
nipkow@15302
  2455
inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
nipkow@15302
  2456
nipkow@15302
  2457
nipkow@15302
  2458
lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
nipkow@15302
  2459
apply clarify  
nipkow@15302
  2460
apply (erule listrel.induct)
nipkow@15302
  2461
apply (blast intro: listrel.intros)+
nipkow@15302
  2462
done
nipkow@15302
  2463
nipkow@15302
  2464
lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
nipkow@15302
  2465
apply clarify 
nipkow@15302
  2466
apply (erule listrel.induct, auto) 
nipkow@15302
  2467
done
nipkow@15302
  2468
nipkow@15302
  2469
lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" 
nipkow@15302
  2470
apply (simp add: refl_def listrel_subset Ball_def)
nipkow@15302
  2471
apply (rule allI) 
nipkow@15302
  2472
apply (induct_tac x) 
nipkow@15302
  2473
apply (auto intro: listrel.intros)
nipkow@15302
  2474
done
nipkow@15302
  2475
nipkow@15302
  2476
lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
nipkow@15302
  2477
apply (auto simp add: sym_def)
nipkow@15302
  2478
apply (erule listrel.induct) 
nipkow@15302
  2479
apply (blast intro: listrel.intros)+
nipkow@15302
  2480
done
nipkow@15302
  2481
nipkow@15302
  2482
lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
nipkow@15302
  2483
apply (simp add: trans_def)
nipkow@15302
  2484
apply (intro allI) 
nipkow@15302
  2485
apply (rule impI) 
nipkow@15302
  2486
apply (erule listrel.induct) 
nipkow@15302
  2487
apply (blast intro: listrel.intros)+
nipkow@15302
  2488
done
nipkow@15302
  2489
nipkow@15302
  2490
theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
nipkow@15302
  2491
by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) 
nipkow@15302
  2492
nipkow@15302
  2493
lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
nipkow@15302
  2494
by (blast intro: listrel.intros)
nipkow@15302
  2495
nipkow@15302
  2496
lemma listrel_Cons:
nipkow@15302
  2497
     "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})";
nipkow@15302
  2498
by (auto simp add: set_Cons_def intro: listrel.intros) 
nipkow@15302
  2499
nipkow@15302
  2500
nipkow@15392
  2501
subsection{*Miscellany*}
nipkow@15392
  2502
nipkow@15392
  2503
subsubsection {* Characters and strings *}
wenzelm@13366
  2504
wenzelm@13366
  2505
datatype nibble =
wenzelm@13366
  2506
    Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
wenzelm@13366
  2507
  | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
wenzelm@13366
  2508
wenzelm@13366
  2509
datatype char = Char nibble nibble
wenzelm@13366
  2510
  -- "Note: canonical order of character encoding coincides with standard term ordering"
wenzelm@13366
  2511
wenzelm@13366
  2512
types string = "char list"
wenzelm@13366
  2513
wenzelm@13366
  2514
syntax
wenzelm@13366
  2515
  "_Char" :: "xstr => char"    ("CHR _")
wenzelm@13366
  2516
  "_String" :: "xstr => string"    ("_")
wenzelm@13366
  2517
wenzelm@13366
  2518
parse_ast_translation {*
wenzelm@13366
  2519
  let
wenzelm@13366
  2520
    val constants = Syntax.Appl o map Syntax.Constant;
wenzelm@13366
  2521
wenzelm@13366
  2522
    fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10));
wenzelm@13366
  2523
    fun mk_char c =
wenzelm@13366
  2524
      if Symbol.is_ascii c andalso Symbol.is_printable c then
wenzelm@13366
  2525
        constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)]
wenzelm@13366
  2526
      else error ("Printable ASCII character expected: " ^ quote c);
wenzelm@13366
  2527
wenzelm@13366
  2528
    fun mk_string [] = Syntax.Constant "Nil"
wenzelm@13366
  2529
      | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs];
wenzelm@13366
  2530
wenzelm@13366
  2531
    fun char_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  2532
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  2533
          [c] => mk_char c
wenzelm@13366
  2534
        | _ => error ("Single character expected: " ^ xstr))
wenzelm@13366
  2535
      | char_ast_tr asts = raise AST ("char_ast_tr", asts);
wenzelm@13366
  2536
wenzelm@13366
  2537
    fun string_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  2538
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  2539
          [] => constants [Syntax.constrainC, "Nil", "string"]
wenzelm@13366
  2540
        | cs => mk_string cs)
wenzelm@13366
  2541
      | string_ast_tr asts = raise AST ("string_tr", asts);
wenzelm@13366
  2542
  in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end;
wenzelm@13366
  2543
*}
wenzelm@13366
  2544
berghofe@15064
  2545
ML {*
haftmann@20184
  2546
structure HOList =
haftmann@20184
  2547
struct
haftmann@20184
  2548
haftmann@20184
  2549
local
haftmann@20184
  2550
  val thy = the_context ();
haftmann@20184
  2551
in
haftmann@20184
  2552
  val typ_string = Type (Sign.intern_type thy "string", []);
haftmann@20184
  2553
  fun typ_list ty = Type (Sign.intern_type thy "list", [ty]);
haftmann@20184
  2554
  fun term_list ty f [] = Const (Sign.intern_const thy "Nil", typ_list ty)
haftmann@20184
  2555
    | term_list ty f (x::xs) = Const (Sign.intern_const thy "Cons",
haftmann@20184
  2556
        ty --> typ_list ty --> typ_list ty) $ f x $ term_list ty f xs;
haftmann@20184
  2557
end;
haftmann@20184
  2558
berghofe@15064
  2559
fun int_of_nibble h =
berghofe@15064
  2560
  if "0" <= h andalso h <= "9" then ord h - ord "0"
berghofe@15064
  2561
  else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10
berghofe@15064
  2562
  else raise Match;
berghofe@15064
  2563
berghofe@15064
  2564
fun nibble_of_int i =
berghofe@15064
  2565
  if i <= 9 then chr (ord "0" + i) else chr (ord "A" + i - 10);
haftmann@20181
  2566
haftmann@20453
  2567
fun dest_char (Const ("List.char.Char", _) $ c1 $ c2) =
haftmann@20453
  2568
      let
haftmann@20453
  2569
        fun dest_nibble (Const (s, _)) = (int_of_nibble o unprefix "List.nibble.Nibble") s
haftmann@20453
  2570
          | dest_nibble _ = raise Match;
haftmann@20453
  2571
      in
haftmann@20453
  2572
        (SOME (dest_nibble c1 * 16 + dest_nibble c2)
haftmann@20453
  2573
        handle Fail _ => NONE | Match => NONE)
haftmann@20453
  2574
      end
haftmann@20453
  2575
  | dest_char _ = NONE;
haftmann@20453
  2576
haftmann@20453
  2577
val print_list = Pretty.enum "," "[" "]";
haftmann@20453
  2578
haftmann@20453
  2579
fun print_char c =
haftmann@20453
  2580
  let
haftmann@20453
  2581
    val i = ord c
haftmann@20453
  2582
  in if i < 32
haftmann@20453
  2583
    then prefix "\\" (string_of_int i)
haftmann@20453
  2584
    else c
haftmann@20453
  2585
  end;
haftmann@20453
  2586
haftmann@20453
  2587
val print_string = quote;
haftmann@20453
  2588
haftmann@20184
  2589
fun term_string s =
haftmann@20181
  2590
  let
haftmann@20181
  2591
    val ty_n = Type ("List.nibble", []);
haftmann@20181
  2592
    val ty_c = Type ("List.char", []);
haftmann@20181
  2593
    val ty_l = Type ("List.list", [ty_c]);
haftmann@20181
  2594
    fun mk_nib n = Const ("List.nibble.Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10)), ty_n);
haftmann@20181
  2595
    fun mk_char c =
haftmann@20181
  2596
      if Symbol.is_ascii c andalso Symbol.is_printable c then
haftmann@20181
  2597
        Const ("List.char.Char", ty_n --> ty_n --> ty_c) $ mk_nib (ord c div 16) $ mk_nib (ord c mod 16)
haftmann@20181
  2598
      else error ("Printable ASCII character expected: " ^ quote c);
haftmann@20181
  2599
    fun mk_string c t = Const ("List.list.Cons", ty_c --> ty_l --> ty_l)
haftmann@20181
  2600
      $ mk_char c $ t;
haftmann@20181
  2601
  in fold_rev mk_string (explode s) (Const ("List.list.Nil", ty_l)) end;
haftmann@20184
  2602
haftmann@20184
  2603
end;
berghofe@15064
  2604
*}
berghofe@15064
  2605
wenzelm@13366
  2606
print_ast_translation {*
wenzelm@13366
  2607
  let
wenzelm@13366
  2608
    fun dest_nib (Syntax.Constant c) =
wenzelm@13366
  2609
        (case explode c of
haftmann@20184
  2610
          ["N", "i", "b", "b", "l", "e", h] => HOList.int_of_nibble h
wenzelm@13366
  2611
        | _ => raise Match)
wenzelm@13366
  2612
      | dest_nib _ = raise Match;
wenzelm@13366
  2613
wenzelm@13366
  2614
    fun dest_chr c1 c2 =
wenzelm@13366
  2615
      let val c = chr (dest_nib c1 * 16 + dest_nib c2)
wenzelm@13366
  2616
      in if Symbol.is_printable c then c else raise Match end;
wenzelm@13366
  2617
wenzelm@13366
  2618
    fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2
wenzelm@13366
  2619
      | dest_char _ = raise Match;
wenzelm@13366
  2620
wenzelm@13366
  2621
    fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)];
wenzelm@13366
  2622
wenzelm@13366
  2623
    fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]]
wenzelm@13366
  2624
      | char_ast_tr' _ = raise Match;
wenzelm@13366
  2625
wenzelm@13366
  2626
    fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String",
wenzelm@13366
  2627
            xstr (map dest_char (Syntax.unfold_ast "_args" args))]
wenzelm@13366
  2628
      | list_ast_tr' ts = raise Match;
wenzelm@13366
  2629
  in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end;
wenzelm@13366
  2630
*}
wenzelm@13366
  2631
haftmann@20453
  2632
haftmann@21061
  2633
subsection {* Code generator *}
haftmann@21061
  2634
haftmann@21061
  2635
subsubsection {* Setup *}
berghofe@15064
  2636
berghofe@16770
  2637
types_code
berghofe@16770
  2638
  "list" ("_ list")
berghofe@16770
  2639
attach (term_of) {*
berghofe@15064
  2640
val term_of_list = HOLogic.mk_list;
berghofe@16770
  2641
*}
berghofe@16770
  2642
attach (test) {*
berghofe@15064
  2643
fun gen_list' aG i j = frequency
berghofe@15064
  2644
  [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] ()
berghofe@15064
  2645
and gen_list aG i = gen_list' aG i i;
berghofe@16770
  2646
*}
berghofe@16770
  2647
  "char" ("string")
berghofe@16770
  2648
attach (term_of) {*
berghofe@15064
  2649
val nibbleT = Type ("List.nibble", []);
berghofe@15064
  2650
berghofe@15064
  2651
fun term_of_char c =
berghofe@15064
  2652
  Const ("List.char.Char", nibbleT --> nibbleT --> Type ("List.char", [])) $
haftmann@20184
  2653
    Const ("List.nibble.Nibble" ^ HOList.nibble_of_int (ord c div 16), nibbleT) $
haftmann@20184
  2654
    Const ("List.nibble.Nibble" ^ HOList.nibble_of_int (ord c mod 16), nibbleT);
berghofe@16770
  2655
*}
berghofe@16770
  2656
attach (test) {*
berghofe@15064
  2657
fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z")));
berghofe@15064
  2658
*}
berghofe@15064
  2659
berghofe@15064
  2660
consts_code "Cons" ("(_ ::/ _)")
berghofe@15064
  2661
haftmann@20453
  2662
code_type list
haftmann@20453
  2663
  (SML "_ list")
haftmann@21113
  2664
  (Haskell "![_]")
haftmann@20453
  2665
haftmann@20453
  2666
code_const Nil
haftmann@21113
  2667
  (SML "[]")
haftmann@21113
  2668
  (Haskell "[]")
haftmann@20453
  2669
haftmann@20453
  2670
code_type char
haftmann@21113
  2671
  (SML "char")
haftmann@21113
  2672
  (Haskell "Char")
haftmann@20453
  2673
haftmann@20453
  2674
code_const Char
haftmann@21126
  2675
  (SML "!((_),/ (_))")
haftmann@21126
  2676
  (Haskell "!((_),/ (_))")
haftmann@20453
  2677
haftmann@20588
  2678
code_instance list :: eq and char :: eq
haftmann@20588
  2679
  (Haskell - and -)
haftmann@20588
  2680
haftmann@21046
  2681
code_const "Code_Generator.eq \<Colon> 'a\<Colon>eq list \<Rightarrow> 'a list \<Rightarrow> bool"
haftmann@20588
  2682
  (Haskell infixl 4 "==")
haftmann@20588
  2683
haftmann@21046
  2684
code_const "Code_Generator.eq \<Colon> char \<Rightarrow> char \<Rightarrow> bool"
haftmann@20588
  2685
  (Haskell infixl 4 "==")
haftmann@20588
  2686
haftmann@21079
  2687
code_reserved SML
haftmann@21079
  2688
  list char
haftmann@21079
  2689
haftmann@21079
  2690
code_reserved Haskell
haftmann@21079
  2691
  Char
haftmann@21079
  2692
haftmann@20453
  2693
setup {*
haftmann@20453
  2694
let
haftmann@20453
  2695
haftmann@20453
  2696
fun list_codegen thy defs gr dep thyname b t =
haftmann@20453
  2697
  let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy defs dep thyname false)
haftmann@20453
  2698
    (gr, HOLogic.dest_list t)
haftmann@20453
  2699
  in SOME (gr', Pretty.list "[" "]" ps) end handle TERM _ => NONE;
haftmann@20453
  2700
haftmann@20453
  2701
fun char_codegen thy defs gr dep thyname b t =
haftmann@20453
  2702
  case (Option.map chr o HOList.dest_char) t 
haftmann@20453
  2703
   of SOME c =>
haftmann@20453
  2704
        if Symbol.is_printable c
haftmann@20453
  2705
        then SOME (gr, (Pretty.quote o Pretty.str) c)
haftmann@20453
  2706
        else NONE
haftmann@20453
  2707
    | NONE => NONE;
haftmann@20453
  2708
haftmann@20453
  2709
in
haftmann@20453
  2710
haftmann@20453
  2711
  Codegen.add_codegen "list_codegen" list_codegen
haftmann@20453
  2712
  #> Codegen.add_codegen "char_codegen" char_codegen
haftmann@20699
  2713
  #> CodegenSerializer.add_pretty_list "SML" "List.list.Nil" "List.list.Cons"
haftmann@20453
  2714
       HOList.print_list NONE (7, "::")
haftmann@20699
  2715
  #> CodegenSerializer.add_pretty_list "Haskell" "List.list.Nil" "List.list.Cons"
haftmann@20453
  2716
       HOList.print_list (SOME (HOList.print_char, HOList.print_string)) (5, ":")
haftmann@20453
  2717
  #> CodegenPackage.add_appconst
haftmann@20453
  2718
       ("List.char.Char", CodegenPackage.appgen_char HOList.dest_char)
haftmann@20453
  2719
haftmann@20453
  2720
end;
haftmann@20453
  2721
*}
berghofe@15064
  2722
haftmann@21061
  2723
haftmann@21061
  2724
subsubsection {* Generation of efficient code *}
haftmann@21061
  2725
haftmann@21061
  2726
consts
haftmann@21079
  2727
  memberl :: "'a \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "mem" 55)
haftmann@21061
  2728
  null:: "'a list \<Rightarrow> bool"
haftmann@21061
  2729
  list_inter :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
haftmann@21061
  2730
  list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
haftmann@21061
  2731
  list_all :: "('a \<Rightarrow> bool) \<Rightarrow> ('a list \<Rightarrow> bool)"
haftmann@21061
  2732
  itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
haftmann@21061
  2733
  filtermap :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list"
haftmann@21061
  2734
  map_filter :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list"
haftmann@21061
  2735
haftmann@21061
  2736
primrec
haftmann@21061
  2737
  "x mem [] = False"
haftmann@21079
  2738
  "x mem (y#ys) = (x = y \<or> x mem ys)"
haftmann@21061
  2739
haftmann@21061
  2740
primrec
haftmann@21061
  2741
  "null [] = True"
haftmann@21061
  2742
  "null (x#xs) = False"
haftmann@21061
  2743
haftmann@21061
  2744
primrec
haftmann@21061
  2745
  "list_inter [] bs = []"
haftmann@21061
  2746
  "list_inter (a#as) bs =
haftmann@21061
  2747
     (if a \<in> set bs then a # list_inter as bs else list_inter as bs)"
haftmann@21061