src/HOL/List.thy

A few new lemmas

(*  Title:      HOL/List.thy
    ID:         $Id$
    Author:     Tobias Nipkow
*)

header {* The datatype of finite lists *}

theory List
imports PreList
begin

datatype 'a list =
    Nil    ("[]")
  | Cons 'a  "'a list"    (infixr "#" 65)

subsection{*Basic list processing functions*}

consts
  "@" :: "'a list => 'a list => 'a list"    (infixr 65)
  filter:: "('a => bool) => 'a list => 'a list"
  concat:: "'a list list => 'a list"
  foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"
  foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"
  hd:: "'a list => 'a"
  tl:: "'a list => 'a list"
  last:: "'a list => 'a"
  butlast :: "'a list => 'a list"
  set :: "'a list => 'a set"
  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool"
  map :: "('a=>'b) => ('a list => 'b list)"
  nth :: "'a list => nat => 'a"    (infixl "!" 100)
  list_update :: "'a list => nat => 'a => 'a list"
  take:: "nat => 'a list => 'a list"
  drop:: "nat => 'a list => 'a list"
  takeWhile :: "('a => bool) => 'a list => 'a list"
  dropWhile :: "('a => bool) => 'a list => 'a list"
  rev :: "'a list => 'a list"
  zip :: "'a list => 'b list => ('a * 'b) list"
  upt :: "nat => nat => nat list" ("(1[_..</_'])")
  remdups :: "'a list => 'a list"
  remove1 :: "'a => 'a list => 'a list"
  null:: "'a list => bool"
  "distinct":: "'a list => bool"
  replicate :: "nat => 'a => 'a list"
  rotate1 :: "'a list \<Rightarrow> 'a list"
  rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
  sublist :: "'a list => nat set => 'a list"
(* For efficiency *)
  mem :: "'a => 'a list => bool"    (infixl 55)
  list_inter :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
  list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
  list_all:: "('a => bool) => ('a list => bool)"
  itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
  filtermap :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list"
  map_filter :: "('a => 'b) => ('a => bool) => 'a list => 'b list"


nonterminals lupdbinds lupdbind

syntax
  -- {* list Enumeration *}
  "@list" :: "args => 'a list"    ("[(_)]")

  -- {* Special syntax for filter *}
  "@filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_:_./ _])")

  -- {* list update *}
  "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
  "" :: "lupdbind => lupdbinds"    ("_")
  "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
  "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)

  upto:: "nat => nat => nat list"    ("(1[_../_])")

translations
  "[x, xs]" == "x#[xs]"
  "[x]" == "x#[]"
  "[x:xs . P]"== "filter (%x. P) xs"

  "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"
  "xs[i:=x]" == "list_update xs i x"

  "[i..j]" == "[i..<(Suc j)]"


syntax (xsymbols)
  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
syntax (HTML output)
  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")


text {*
  Function @{text size} is overloaded for all datatypes. Users may
  refer to the list version as @{text length}. *}

syntax length :: "'a list => nat"
translations "length" => "size :: _ list => nat"

typed_print_translation {*
  let
    fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] =
          Syntax.const "length" $ t
      | size_tr' _ _ _ = raise Match;
  in [("size", size_tr')] end
*}


primrec
  "hd(x#xs) = x"

primrec
  "tl([]) = []"
  "tl(x#xs) = xs"

primrec
  "null([]) = True"
  "null(x#xs) = False"

primrec
  "last(x#xs) = (if xs=[] then x else last xs)"

primrec
  "butlast []= []"
  "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"

primrec
  "set [] = {}"
  "set (x#xs) = insert x (set xs)"

primrec
  "map f [] = []"
  "map f (x#xs) = f(x)#map f xs"

primrec
  append_Nil:"[]@ys = ys"
  append_Cons: "(x#xs)@ys = x#(xs@ys)"

primrec
  "rev([]) = []"
  "rev(x#xs) = rev(xs) @ [x]"

primrec
  "filter P [] = []"
  "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"

primrec
  foldl_Nil:"foldl f a [] = a"
  foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"

primrec
  "foldr f [] a = a"
  "foldr f (x#xs) a = f x (foldr f xs a)"

primrec
  "concat([]) = []"
  "concat(x#xs) = x @ concat(xs)"

primrec
  drop_Nil:"drop n [] = []"
  drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
  -- {*Warning: simpset does not contain this definition, but separate
       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}

primrec
  take_Nil:"take n [] = []"
  take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
  -- {*Warning: simpset does not contain this definition, but separate
       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}

primrec
  nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
  -- {*Warning: simpset does not contain this definition, but separate
       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}

primrec
  "[][i:=v] = []"
  "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])"

primrec
  "takeWhile P [] = []"
  "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"

primrec
  "dropWhile P [] = []"
  "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"

primrec
  "zip xs [] = []"
  zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
  -- {*Warning: simpset does not contain this definition, but separate
       theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}

primrec
  upt_0: "[i..<0] = []"
  upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"

primrec
  "distinct [] = True"
  "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"

primrec
  "remdups [] = []"
  "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"

primrec
  "remove1 x [] = []"
  "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)"

primrec
  replicate_0: "replicate 0 x = []"
  replicate_Suc: "replicate (Suc n) x = x # replicate n x"

defs
rotate1_def: "rotate1 xs == (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"
rotate_def:  "rotate n == rotate1 ^ n"

list_all2_def:
 "list_all2 P xs ys ==
  length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)"

sublist_def:
 "sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..<size xs]))"

primrec
  "x mem [] = False"
  "x mem (y#ys) = (if y=x then True else x mem ys)"

primrec
 "list_inter [] bs = []"
 "list_inter (a#as) bs =
  (if a \<in> set bs then a#(list_inter as bs) else list_inter as bs)"

primrec
  "list_all P [] = True"
  "list_all P (x#xs) = (P(x) \<and> list_all P xs)"

primrec
"list_ex P [] = False"
"list_ex P (x#xs) = (P x \<or> list_ex P xs)"

primrec
 "filtermap f [] = []"
 "filtermap f (x#xs) =
    (case f x of None \<Rightarrow> filtermap f xs
     | Some y \<Rightarrow> y # (filtermap f xs))"

primrec
  "map_filter f P [] = []"
  "map_filter f P (x#xs) = (if P x then f x # map_filter f P xs else 
               map_filter f P xs)"

primrec
"itrev [] ys = ys"
"itrev (x#xs) ys = itrev xs (x#ys)"


lemma not_Cons_self [simp]: "xs \<noteq> x # xs"
by (induct xs) auto

lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]

lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
by (induct xs) auto

lemma length_induct:
"(!!xs. \<forall>ys. length ys < length xs --> P ys ==> P xs) ==> P xs"
by (rule measure_induct [of length]) iprover


subsubsection {* @{text length} *}

text {*
Needs to come before @{text "@"} because of theorem @{text
append_eq_append_conv}.
*}

lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
by (induct xs) auto

lemma length_map [simp]: "length (map f xs) = length xs"
by (induct xs) auto

lemma length_rev [simp]: "length (rev xs) = length xs"
by (induct xs) auto

lemma length_tl [simp]: "length (tl xs) = length xs - 1"
by (cases xs) auto

lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
by (induct xs) auto

lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
by (induct xs) auto

lemma length_Suc_conv:
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
by (induct xs) auto

lemma Suc_length_conv:
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
apply (induct xs, simp, simp)
apply blast
done

lemma impossible_Cons [rule_format]: 
  "length xs <= length ys --> xs = x # ys = False"
apply (induct xs, auto)
done

lemma list_induct2[consumes 1]: "\<And>ys.
 \<lbrakk> length xs = length ys;
   P [] [];
   \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
 \<Longrightarrow> P xs ys"
apply(induct xs)
 apply simp
apply(case_tac ys)
 apply simp
apply(simp)
done

subsubsection {* @{text "@"} -- append *}

lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
by (induct xs) auto

lemma append_Nil2 [simp]: "xs @ [] = xs"
by (induct xs) auto

lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
by (induct xs) auto

lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
by (induct xs) auto

lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
by (induct xs) auto

lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
by (induct xs) auto

lemma append_eq_append_conv [simp]:
 "!!ys. length xs = length ys \<or> length us = length vs
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
apply (induct xs)
 apply (case_tac ys, simp, force)
apply (case_tac ys, force, simp)
done

lemma append_eq_append_conv2: "!!ys zs ts.
 (xs @ ys = zs @ ts) =
 (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
apply (induct xs)
 apply fastsimp
apply(case_tac zs)
 apply simp
apply fastsimp
done

lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
by simp

lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
by simp

lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
by simp

lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
using append_same_eq [of _ _ "[]"] by auto

lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
using append_same_eq [of "[]"] by auto

lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
by (induct xs) auto

lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
by (induct xs) auto

lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
by (simp add: hd_append split: list.split)

lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
by (simp split: list.split)

lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
by (simp add: tl_append split: list.split)


lemma Cons_eq_append_conv: "x#xs = ys@zs =
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
by(cases ys) auto

lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
by(cases ys) auto


text {* Trivial rules for solving @{text "@"}-equations automatically. *}

lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
by simp

lemma Cons_eq_appendI:
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
by (drule sym) simp

lemma append_eq_appendI:
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
by (drule sym) simp


text {*
Simplification procedure for all list equalities.
Currently only tries to rearrange @{text "@"} to see if
- both lists end in a singleton list,
- or both lists end in the same list.
*}

ML_setup {*
local

val append_assoc = thm "append_assoc";
val append_Nil = thm "append_Nil";
val append_Cons = thm "append_Cons";
val append1_eq_conv = thm "append1_eq_conv";
val append_same_eq = thm "append_same_eq";

fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
  | last (Const("List.op @",_) $ _ $ ys) = last ys
  | last t = t;

fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
  | list1 _ = false;

fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
  | butlast xs = Const("List.list.Nil",fastype_of xs);

val rearr_ss = HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons];

fun list_eq sg ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
  let
    val lastl = last lhs and lastr = last rhs;
    fun rearr conv =
      let
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
        val Type(_,listT::_) = eqT
        val appT = [listT,listT] ---> listT
        val app = Const("List.op @",appT)
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
        val thm = Goal.prove sg [] [] eq
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;

  in
    if list1 lastl andalso list1 lastr then rearr append1_eq_conv
    else if lastl aconv lastr then rearr append_same_eq
    else NONE
  end;

in

val list_eq_simproc =
  Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq;

end;

Addsimprocs [list_eq_simproc];
*}


subsubsection {* @{text map} *}

lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
by (induct xs) simp_all

lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
by (rule ext, induct_tac xs) auto

lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
by (induct xs) auto

lemma map_compose: "map (f o g) xs = map f (map g xs)"
by (induct xs) (auto simp add: o_def)

lemma rev_map: "rev (map f xs) = map f (rev xs)"
by (induct xs) auto

lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
by (induct xs) auto

lemma map_cong [recdef_cong]:
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
-- {* a congruence rule for @{text map} *}
by simp

lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
by (cases xs) auto

lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
by (cases xs) auto

lemma map_eq_Cons_conv[iff]:
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
by (cases xs) auto

lemma Cons_eq_map_conv[iff]:
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
by (cases ys) auto

lemma ex_map_conv:
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
by(induct ys, auto)

lemma map_eq_imp_length_eq:
  "!!xs. map f xs = map f ys ==> length xs = length ys"
apply (induct ys)
 apply simp
apply(simp (no_asm_use))
apply clarify
apply(simp (no_asm_use))
apply fast
done

lemma map_inj_on:
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
  ==> xs = ys"
apply(frule map_eq_imp_length_eq)
apply(rotate_tac -1)
apply(induct rule:list_induct2)
 apply simp
apply(simp)
apply (blast intro:sym)
done

lemma inj_on_map_eq_map:
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
by(blast dest:map_inj_on)

lemma map_injective:
 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
by (induct ys) (auto dest!:injD)

lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
by(blast dest:map_injective)

lemma inj_mapI: "inj f ==> inj (map f)"
by (iprover dest: map_injective injD intro: inj_onI)

lemma inj_mapD: "inj (map f) ==> inj f"
apply (unfold inj_on_def, clarify)
apply (erule_tac x = "[x]" in ballE)
 apply (erule_tac x = "[y]" in ballE, simp, blast)
apply blast
done

lemma inj_map[iff]: "inj (map f) = inj f"
by (blast dest: inj_mapD intro: inj_mapI)

lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
apply(rule inj_onI)
apply(erule map_inj_on)
apply(blast intro:inj_onI dest:inj_onD)
done

lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
by (induct xs, auto)

lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
by (induct xs) auto

lemma map_fst_zip[simp]:
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
by (induct rule:list_induct2, simp_all)

lemma map_snd_zip[simp]:
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
by (induct rule:list_induct2, simp_all)


subsubsection {* @{text rev} *}

lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
by (induct xs) auto

lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
by (induct xs) auto

lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
by auto

lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
by (induct xs) auto

lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
by (induct xs) auto

lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
by (cases xs) auto

lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
by (cases xs) auto

lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"
apply (induct xs, force)
apply (case_tac ys, simp, force)
done

lemma inj_on_rev[iff]: "inj_on rev A"
by(simp add:inj_on_def)

lemma rev_induct [case_names Nil snoc]:
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
apply(simplesubst rev_rev_ident[symmetric])
apply(rule_tac list = "rev xs" in list.induct, simp_all)
done

ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"

lemma rev_exhaust [case_names Nil snoc]:
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
by (induct xs rule: rev_induct) auto

lemmas rev_cases = rev_exhaust


subsubsection {* @{text set} *}

lemma finite_set [iff]: "finite (set xs)"
by (induct xs) auto

lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
by (induct xs) auto

lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
by(cases xs) auto

lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
by auto

lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
by auto

lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
by (induct xs) auto

lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
by(induct xs) auto

lemma set_rev [simp]: "set (rev xs) = set xs"
by (induct xs) auto

lemma set_map [simp]: "set (map f xs) = f`(set xs)"
by (induct xs) auto

lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
by (induct xs) auto

lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
apply (induct j, simp_all)
apply (erule ssubst, auto)
done

lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
proof (induct xs)
  case Nil show ?case by simp
  case (Cons a xs)
  show ?case
  proof 
    assume "x \<in> set (a # xs)"
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
      by (simp, blast intro: Cons_eq_appendI)
  next
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
    show "x \<in> set (a # xs)" 
      by (cases ys, auto simp add: eq)
  qed
qed

lemma in_set_conv_decomp_first:
 "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
proof (induct xs)
  case Nil show ?case by simp
next
  case (Cons a xs)
  show ?case
  proof cases
    assume "x = a" thus ?case using Cons by force
  next
    assume "x \<noteq> a"
    show ?case
    proof
      assume "x \<in> set (a # xs)"
      from prems show "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
	by(fastsimp intro!: Cons_eq_appendI)
    next
      assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
      then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
      show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq)
    qed
  qed
qed

lemmas split_list       = in_set_conv_decomp[THEN iffD1, standard]
lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard]


lemma finite_list: "finite A ==> EX l. set l = A"
apply (erule finite_induct, auto)
apply (rule_tac x="x#l" in exI, auto)
done

lemma card_length: "card (set xs) \<le> length xs"
by (induct xs) (auto simp add: card_insert_if)


subsubsection {* @{text filter} *}

lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
by (induct xs) auto

lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
by (induct xs) simp_all

lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
by (induct xs) auto

lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
by (induct xs) (auto simp add: le_SucI)

lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
by (induct xs) auto

lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
by (induct xs) auto

lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
  by (induct xs) simp_all

lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
apply (induct xs)
 apply auto
apply(cut_tac P=P and xs=xs in length_filter_le)
apply simp
done

lemma filter_map:
  "filter P (map f xs) = map f (filter (P o f) xs)"
by (induct xs) simp_all

lemma length_filter_map[simp]:
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
by (simp add:filter_map)

lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
by auto

lemma length_filter_less:
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
proof (induct xs)
  case Nil thus ?case by simp
next
  case (Cons x xs) thus ?case
    apply (auto split:split_if_asm)
    using length_filter_le[of P xs] apply arith
  done
qed

lemma length_filter_conv_card:
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
proof (induct xs)
  case Nil thus ?case by simp
next
  case (Cons x xs)
  let ?S = "{i. i < length xs & p(xs!i)}"
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
  show ?case (is "?l = card ?S'")
  proof (cases)
    assume "p x"
    hence eq: "?S' = insert 0 (Suc ` ?S)"
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
    have "length (filter p (x # xs)) = Suc(card ?S)"
      using Cons by simp
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
      by (simp add: card_image inj_Suc)
    also have "\<dots> = card ?S'" using eq fin
      by (simp add:card_insert_if) (simp add:image_def)
    finally show ?thesis .
  next
    assume "\<not> p x"
    hence eq: "?S' = Suc ` ?S"
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
    have "length (filter p (x # xs)) = card ?S"
      using Cons by simp
    also have "\<dots> = card(Suc ` ?S)" using fin
      by (simp add: card_image inj_Suc)
    also have "\<dots> = card ?S'" using eq fin
      by (simp add:card_insert_if)
    finally show ?thesis .
  qed
qed

lemma Cons_eq_filterD:
 "x#xs = filter P ys \<Longrightarrow>
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
  (concl is "\<exists>us vs. ?P ys us vs")
proof(induct ys)
  case Nil thus ?case by simp
next
  case (Cons y ys)
  show ?case (is "\<exists>x. ?Q x")
  proof cases
    assume Py: "P y"
    show ?thesis
    proof cases
      assume xy: "x = y"
      show ?thesis
      proof from Py xy Cons(2) show "?Q []" by simp qed
    next
      assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp
    qed
  next
    assume Py: "\<not> P y"
    with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp
    show ?thesis (is "? us. ?Q us")
    proof show "?Q (y#us)" using 1 by simp qed
  qed
qed

lemma filter_eq_ConsD:
 "filter P ys = x#xs \<Longrightarrow>
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
by(rule Cons_eq_filterD) simp

lemma filter_eq_Cons_iff:
 "(filter P ys = x#xs) =
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
by(auto dest:filter_eq_ConsD)

lemma Cons_eq_filter_iff:
 "(x#xs = filter P ys) =
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
by(auto dest:Cons_eq_filterD)

lemma filter_cong:
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
apply simp
apply(erule thin_rl)
by (induct ys) simp_all


subsubsection {* @{text concat} *}

lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
by (induct xs) auto

lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
by (induct xss) auto

lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
by (induct xss) auto

lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
by (induct xs) auto

lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
by (induct xs) auto

lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
by (induct xs) auto

lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
by (induct xs) auto


subsubsection {* @{text nth} *}

lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
by auto

lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
by auto

declare nth.simps [simp del]

lemma nth_append:
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
apply (induct "xs", simp)
apply (case_tac n, auto)
done

lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
by (induct "xs") auto

lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
by (induct "xs") auto

lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
apply (induct xs, simp)
apply (case_tac n, auto)
done


lemma list_eq_iff_nth_eq:
 "!!ys. (xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
apply(induct xs)
 apply simp apply blast
apply(case_tac ys)
 apply simp
apply(simp add:nth_Cons split:nat.split)apply blast
done

lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
apply (induct xs, simp, simp)
apply safe
apply (rule_tac x = 0 in exI, simp)
 apply (rule_tac x = "Suc i" in exI, simp)
apply (case_tac i, simp)
apply (rename_tac j)
apply (rule_tac x = j in exI, simp)
done

lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
by(auto simp:set_conv_nth)

lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
by (auto simp add: set_conv_nth)

lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
by (auto simp add: set_conv_nth)

lemma all_nth_imp_all_set:
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
by (auto simp add: set_conv_nth)

lemma all_set_conv_all_nth:
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
by (auto simp add: set_conv_nth)


subsubsection {* @{text list_update} *}

lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
by (induct xs) (auto split: nat.split)

lemma nth_list_update:
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
by (induct xs) (auto simp add: nth_Cons split: nat.split)

lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
by (simp add: nth_list_update)

lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
by (induct xs) (auto simp add: nth_Cons split: nat.split)

lemma list_update_overwrite [simp]:
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
by (induct xs) (auto split: nat.split)

lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
apply (induct xs, simp)
apply(simp split:nat.splits)
done

lemma list_update_beyond[simp]: "\<And>i. length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
apply (induct xs)
 apply simp
apply (case_tac i)
apply simp_all
done

lemma list_update_same_conv:
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
by (induct xs) (auto split: nat.split)

lemma list_update_append1:
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
apply (induct xs, simp)
apply(simp split:nat.split)
done

lemma list_update_append:
  "!!n. (xs @ ys) [n:= x] = 
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
by (induct xs) (auto split:nat.splits)

lemma list_update_length [simp]:
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
by (induct xs, auto)

lemma update_zip:
"!!i xy xs. length xs = length ys ==>
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
by (induct ys) (auto, case_tac xs, auto split: nat.split)

lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
by (induct xs) (auto split: nat.split)

lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
by (blast dest!: set_update_subset_insert [THEN subsetD])

lemma set_update_memI: "!!n. n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
by (induct xs) (auto split:nat.splits)


subsubsection {* @{text last} and @{text butlast} *}

lemma last_snoc [simp]: "last (xs @ [x]) = x"
by (induct xs) auto

lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
by (induct xs) auto

lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
by(simp add:last.simps)

lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
by(simp add:last.simps)

lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
by (induct xs) (auto)

lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
by(simp add:last_append)

lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
by(simp add:last_append)

lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
by(rule rev_exhaust[of xs]) simp_all

lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
by(cases xs) simp_all

lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
by (induct as) auto

lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
by (induct xs rule: rev_induct) auto

lemma butlast_append:
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
by (induct xs) auto

lemma append_butlast_last_id [simp]:
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
by (induct xs) auto

lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
by (induct xs) (auto split: split_if_asm)

lemma in_set_butlast_appendI:
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
by (auto dest: in_set_butlastD simp add: butlast_append)

lemma last_drop[simp]: "!!n. n < length xs \<Longrightarrow> last (drop n xs) = last xs"
apply (induct xs)
 apply simp
apply (auto split:nat.split)
done

lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
by(induct xs)(auto simp:neq_Nil_conv)


subsubsection {* @{text take} and @{text drop} *}

lemma take_0 [simp]: "take 0 xs = []"
by (induct xs) auto

lemma drop_0 [simp]: "drop 0 xs = xs"
by (induct xs) auto

lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
by simp

lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
by simp

declare take_Cons [simp del] and drop_Cons [simp del]

lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
by(clarsimp simp add:neq_Nil_conv)

lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
by(cases xs, simp_all)

lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)

lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
apply (induct xs, simp)
apply(simp add:drop_Cons nth_Cons split:nat.splits)
done

lemma take_Suc_conv_app_nth:
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
apply (induct xs, simp)
apply (case_tac i, auto)
done

lemma drop_Suc_conv_tl:
  "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
apply (induct xs, simp)
apply (case_tac i, auto)
done

lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
by (induct n) (auto, case_tac xs, auto)

lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
by (induct n) (auto, case_tac xs, auto)

lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
by (induct n) (auto, case_tac xs, auto)

lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
by (induct n) (auto, case_tac xs, auto)

lemma take_append [simp]:
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
by (induct n) (auto, case_tac xs, auto)

lemma drop_append [simp]:
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
by (induct n) (auto, case_tac xs, auto)

lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
apply (induct m, auto)
apply (case_tac xs, auto)
apply (case_tac n, auto)
done

lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
apply (induct m, auto)
apply (case_tac xs, auto)
done

lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
apply (induct m, auto)
apply (case_tac xs, auto)
done

lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"
apply(induct xs)
 apply simp
apply(simp add: take_Cons drop_Cons split:nat.split)
done

lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
apply (induct n, auto)
apply (case_tac xs, auto)
done

lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"
apply(induct xs)
 apply simp
apply(simp add:take_Cons split:nat.split)
done

lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"
apply(induct xs)
apply simp
apply(simp add:drop_Cons split:nat.split)
done

lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
apply (induct n, auto)
apply (case_tac xs, auto)
done

lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
apply (induct n, auto)
apply (case_tac xs, auto)
done

lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
apply (induct xs, auto)
apply (case_tac i, auto)
done

lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
apply (induct xs, auto)
apply (case_tac i, auto)
done

lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
apply (induct xs, auto)
apply (case_tac n, blast)
apply (case_tac i, auto)
done

lemma nth_drop [simp]:
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
apply (induct n, auto)
apply (case_tac xs, auto)
done

lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
by(induct xs)(auto simp:take_Cons split:nat.split)

lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
by(induct xs)(auto simp:drop_Cons split:nat.split)

lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
using set_take_subset by fast

lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
using set_drop_subset by fast

lemma append_eq_conv_conj:
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
apply (induct xs, simp, clarsimp)
apply (case_tac zs, auto)
done

lemma take_add [rule_format]: 
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
apply (induct xs, auto) 
apply (case_tac i, simp_all) 
done

lemma append_eq_append_conv_if:
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
   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
   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)"
apply(induct xs\<^isub>1)
 apply simp
apply(case_tac ys\<^isub>1)
apply simp_all
done

lemma take_hd_drop:
  "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
apply(induct xs)
apply simp
apply(simp add:drop_Cons split:nat.split)
done

lemma id_take_nth_drop:
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
proof -
  assume si: "i < length xs"
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
  moreover
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
    apply (rule_tac take_Suc_conv_app_nth) by arith
  ultimately show ?thesis by auto
qed
  
lemma upd_conv_take_nth_drop:
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
proof -
  assume i: "i < length xs"
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
    using i by (simp add: list_update_append)
  finally show ?thesis .
qed


subsubsection {* @{text takeWhile} and @{text dropWhile} *}

lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
by (induct xs) auto

lemma takeWhile_append1 [simp]:
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
by (induct xs) auto

lemma takeWhile_append2 [simp]:
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
by (induct xs) auto

lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
by (induct xs) auto

lemma dropWhile_append1 [simp]:
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
by (induct xs) auto

lemma dropWhile_append2 [simp]:
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
by (induct xs) auto

lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
by (induct xs) (auto split: split_if_asm)

lemma takeWhile_eq_all_conv[simp]:
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
by(induct xs, auto)

lemma dropWhile_eq_Nil_conv[simp]:
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
by(induct xs, auto)

lemma dropWhile_eq_Cons_conv:
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
by(induct xs, auto)

text{* The following two lemmmas could be generalized to an arbitrary
property. *}

lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])

lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
apply(induct xs)
 apply simp
apply auto
apply(subst dropWhile_append2)
apply auto
done


subsubsection {* @{text zip} *}

lemma zip_Nil [simp]: "zip [] ys = []"
by (induct ys) auto

lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
by simp

declare zip_Cons [simp del]

lemma zip_Cons1:
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
by(auto split:list.split)

lemma length_zip [simp]:
"!!xs. length (zip xs ys) = min (length xs) (length ys)"
apply (induct ys, simp)
apply (case_tac xs, auto)
done

lemma zip_append1:
"!!xs. zip (xs @ ys) zs =
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
apply (induct zs, simp)
apply (case_tac xs, simp_all)
done

lemma zip_append2:
"!!ys. zip xs (ys @ zs) =
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
apply (induct xs, simp)
apply (case_tac ys, simp_all)
done