(*  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)

 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_all:: "('a => bool) => ('a list => bool)"

   list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool"

   map :: "('a=>'b) => ('a list => 'b list)"

   mem :: "'a => 'a list => bool"    (infixl 55)

   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"

 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

 "x mem [] = False"

 "x mem (y#ys) = (if y=x then True else x mem ys)"

 primrec

 "set [] = {}"

 "set (x#xs) = insert x (set xs)"

 primrec

 list_all_Nil:"list_all P [] = True"

 list_all_Cons: "list_all P (x#xs) = (P(x) \<and> list_all P 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

  list_all2_def:

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

 subsection {* Lexicographic orderings on lists *}

 consts

 lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"

 primrec

 "lexn r 0 = {}"

 "lexn r (Suc n) =

 (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int

 {(xs,ys). length xs = Suc n \<and> length ys = Suc n}"

 constdefs

 lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"

 "lex r == \<Union>n. lexn r n"

 lexico :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"

 "lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"

 sublist :: "'a list => nat set => 'a list"

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

 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]) rules

 subsection {* @{text lists}: the list-forming operator over sets *}

 consts lists :: "'a set => 'a list set"

 inductive "lists A"

  intros

   Nil [intro!]: "[]: lists A"

   Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A"

 inductive_cases listsE [elim!]: "x#l : lists A"

 lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B"

 by (unfold lists.defs) (blast intro!: lfp_mono)

 lemma lists_IntI:

   assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l

   by induct blast+

 lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B"

 proof (rule mono_Int [THEN equalityI])

   show "mono lists" by (simp add: mono_def lists_mono)

   show "lists A \<inter> lists B \<subseteq> lists (A \<inter> B)" by (blast intro: lists_IntI)

 qed

 lemma append_in_lists_conv [iff]:

      "(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"

 by (induct xs) auto

 subsection {* @{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

 subsection {* @{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

 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_tac =

   simp_tac (HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]);

 fun list_eq sg _ (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 = Tactic.prove sg [] [] eq (K (rearr_tac 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];

 *}

 subsection {* @{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 (rules 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 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)

 subsection {* @{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_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_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"

 apply (induct xs, force)

 apply (case_tac ys, simp, force)

 done

 lemma rev_induct [case_names Nil snoc]:

   "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"

 apply(subst 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

 subsection {* @{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: "l = x#xs \<Longrightarrow> x\<in>set l"

 by (case_tac l, 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_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_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"

 -- {* eliminate @{text lists} in favour of @{text set} *}

 by (induct xs) auto

 lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"

 by (rule in_lists_conv_set [THEN iffD1])

 lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"

 by (rule in_lists_conv_set [THEN iffD2])

 lemma lists_UNIV [simp]: "lists UNIV = UNIV"

 by auto

 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)

 subsection{*Sets of Lists*}

 text{*Resembles a Cartesian product: it denotes the set of lists with head

   drawn from @{term A} and tail drawn from @{term Xs}.*}

 constdefs

   set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"

    "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"

 lemma [simp]: "set_Cons A {[]} = (%x. [x])`A"

 by (auto simp add: set_Cons_def)

 text{*Yields the set of lists, all of the same length as the argument and

 with elements drawn from the corresponding element of the argument.*}

 consts  listset :: "'a set list \<Rightarrow> 'a list set"

 primrec

    "listset []    = {[]}"

    "listset(A#As) = set_Cons A (listset As)"

 subsection {* @{text mem} *}

 lemma set_mem_eq: "(x mem xs) = (x : set xs)"

 by (induct xs) auto

 subsection {* @{text list_all} *}

 lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)"

 by (induct xs) auto

 lemma list_all_append [simp]:

 "list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"

 by (induct xs) auto

 subsection {* @{text filter} *}

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

 by (induct xs) auto

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

 by (induct xs) auto

 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 length_filter [simp]: "length (filter P xs) \<le> length xs"

 by (induct xs) (auto simp add: le_SucI)

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

 by auto

 subsection {* @{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

 subsection {* @{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 set_conv_nth: "set xs = {xs!i | i. i < length xs}"

 apply (induct_tac 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 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)

 subsection {* @{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_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_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])

 subsection {* @{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 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)

 subsection {* @{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 na, 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

 subsection {* @{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)

 subsection {* @{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 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

 lemma zip_append [simp]:

  "[| length xs = length us; length ys = length vs |] ==>

 zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"

 by (simp add: zip_append1)

 lemma zip_rev:

 "length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"

 by (induct rule:list_induct2, simp_all)

 lemma nth_zip [simp]:

 "!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"

 apply (induct ys, simp)

 apply (case_tac xs)

  apply (simp_all add: nth.simps split: nat.split)

 done

 lemma set_zip:

 "set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"

 by (simp add: set_conv_nth cong: rev_conj_cong)

 lemma zip_update:

 "length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"

 by (rule sym, simp add: update_zip)

 lemma zip_replicate [simp]:

 "!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"

 apply (induct i, auto)

 apply (case_tac j, auto)

 done

 subsection {* @{text list_all2} *}

 lemma list_all2_lengthD [intro?]: 

   "list_all2 P xs ys ==> length xs = length ys"

 by (simp add: list_all2_def)

 lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])"

 by (simp add: list_all2_def)

 lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"

 by (simp add: list_all2_def)

 lemma list_all2_Cons [iff]:

 "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"

 by (auto simp add: list_all2_def)

 lemma list_all2_Cons1:

 "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"

 by (cases ys) auto

 lemma list_all2_Cons2:

 "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"

 by (cases xs) auto

 lemma list_all2_rev [iff]:

 "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"

 by (simp add: list_all2_def zip_rev cong: conj_cong)

 lemma list_all2_rev1:

 "list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"

 by (subst list_all2_rev [symmetric]) simp

 lemma list_all2_append1:

 "list_all2 P (xs @ ys) zs =

 (EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>

 list_all2 P xs us \<and> list_all2 P ys vs)"

 apply (simp add: list_all2_def zip_append1)

 apply (rule iffI)

  apply (rule_tac x = "take (length xs) zs" in exI)

  apply (rule_tac x = "drop (length xs) zs" in exI)

  apply (force split: nat_diff_split simp add: min_def, clarify)

 apply (simp add: ball_Un)

 done

 lemma list_all2_append2:

 "list_all2 P xs (ys @ zs) =

 (EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>

 list_all2 P us ys \<and> list_all2 P vs zs)"

 apply (simp add: list_all2_def zip_append2)

 apply (rule iffI)

  apply (rule_tac x = "take (length ys) xs" in exI)

  apply (rule_tac x = "drop (length ys) xs" in exI)

  apply (force split: nat_diff_split simp add: min_def, clarify)

 apply (simp add: ball_Un)

 done

 lemma list_all2_append:

   "length xs = length ys \<Longrightarrow>

   list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"

 by (induct rule:list_induct2, simp_all)

 lemma list_all2_appendI [intro?, trans]:

   "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"

   by (simp add: list_all2_append list_all2_lengthD)

 lemma list_all2_conv_all_nth:

 "list_all2 P xs ys =

 (length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"

 by (force simp add: list_all2_def set_zip)

 lemma list_all2_trans:

   assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"

   shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"

         (is "!!bs cs. PROP ?Q as bs cs")

 proof (induct as)

   fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"

   show "!!cs. PROP ?Q (x # xs) bs cs"

   proof (induct bs)

     fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"

     show "PROP ?Q (x # xs) (y # ys) cs"

       by (induct cs) (auto intro: tr I1 I2)

   qed simp

 qed simp

 lemma list_all2_all_nthI [intro?]:

   "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"

   by (simp add: list_all2_conv_all_nth)

 lemma list_all2I:

   "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"

   by (simp add: list_all2_def)

 lemma list_all2_nthD:

   "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"

   by (simp add: list_all2_conv_all_nth)

 lemma list_all2_nthD2:

   "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"

   by (frule list_all2_lengthD) (auto intro: list_all2_nthD)

 lemma list_all2_map1: 

   "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"

   by (simp add: list_all2_conv_all_nth)

 lemma list_all2_map2: 

   "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"

   by (auto simp add: list_all2_conv_all_nth)

 lemma list_all2_refl [intro?]:

   "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"

   by (simp add: list_all2_conv_all_nth)

 lemma list_all2_update_cong:

   "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"

   by (simp add: list_all2_conv_all_nth nth_list_update)

 lemma list_all2_update_cong2:

   "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"

   by (simp add: list_all2_lengthD list_all2_update_cong)

 lemma list_all2_takeI [simp,intro?]:

   "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"

   apply (induct xs)

    apply simp

   apply (clarsimp simp add: list_all2_Cons1)

   apply (case_tac n)

   apply auto

   done

 lemma list_all2_dropI [simp,intro?]:

   "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"

   apply (induct as, simp)

   apply (clarsimp simp add: list_all2_Cons1)

   apply (case_tac n, simp, simp)

   done

 lemma list_all2_mono [intro?]:

   "\<And>y. list_all2 P x y \<Longrightarrow> (\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> list_all2 Q x y"

   apply (induct x, simp)

   apply (case_tac y, auto)

   done

 subsection {* @{text foldl} and @{text foldr} *}

 lemma foldl_append [simp]:

 "!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"

 by (induct xs) auto

 lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"

 by (induct xs) auto

 lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"

 by (induct xs) auto

 lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"

 by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])

 text {*

 Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more

 difficult to use because it requires an additional transitivity step.

 *}

 lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"

 by (induct ns) auto

 lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"

 by (force intro: start_le_sum simp add: in_set_conv_decomp)

 lemma sum_eq_0_conv [iff]:

 "!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"

 by (induct ns) auto

 subsection {* @{text upto} *}

 lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])"

 -- {* Does not terminate! *}

 by (induct j) auto

 lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []"

 by (subst upt_rec) simp

 lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]"

 -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}

 by simp

 lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]"

 apply(rule trans)

 apply(subst upt_rec)

  prefer 2 apply (rule refl, simp)

 done

 lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]"

 -- {* LOOPS as a simprule, since @{text "j <= j"}. *}

 by (induct k) auto

 lemma length_upt [simp]: "length [i..j(] = j - i"

 by (induct j) (auto simp add: Suc_diff_le)

 lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k"

 apply (induct j)

 apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)

 done

 lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]"

 apply (induct m, simp)

 apply (subst upt_rec)

 apply (rule sym)

 apply (subst upt_rec)

 apply (simp del: upt.simps)

 done

 lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]"

 by (induct n) auto

 lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..n(]) ! i = f(m+i)"

 apply (induct n m rule: diff_induct)

 prefer 3 apply (subst map_Suc_upt[symmetric])

 apply (auto simp add: less_diff_conv nth_upt)

 done

 lemma nth_take_lemma:

   "!!xs ys. k <= length xs ==> k <= length ys ==>

      (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"

 apply (atomize, induct k)

 apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)

 txt {* Both lists must be non-empty *}

 apply (case_tac xs, simp)

 apply (case_tac ys, clarify)

  apply (simp (no_asm_use))

 apply clarify

 txt {* prenexing's needed, not miniscoping *}

 apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)

 apply blast

 done

 lemma nth_equalityI:

  "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"

 apply (frule nth_take_lemma [OF le_refl eq_imp_le])

 apply (simp_all add: take_all)

 done

 (* needs nth_equalityI *)

 lemma list_all2_antisym:

   "\<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> 

   \<Longrightarrow> xs = ys"

   apply (simp add: list_all2_conv_all_nth) 

   apply (rule nth_equalityI, blast, simp)

   done

 lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"

 -- {* The famous take-lemma. *}

 apply (drule_tac x = "max (length xs) (length ys)" in spec)

 apply (simp add: le_max_iff_disj take_all)

 done

 subsection {* @{text "distinct"} and @{text remdups} *}

 lemma distinct_append [simp]:

 "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"

 by (induct xs) auto

 lemma set_remdups [simp]: "set (remdups xs) = set xs"

 by (induct xs) (auto simp add: insert_absorb)

 lemma distinct_remdups [iff]: "distinct (remdups xs)"

 by (induct xs) auto

 lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"

   by (induct_tac x, auto) 

 lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"

   by (induct_tac x, auto)

 lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"

 by (induct xs) auto

 text {*

 It is best to avoid this indexed version of distinct, but sometimes

 it is useful. *}

 lemma distinct_conv_nth:

 "distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)"

 apply (induct_tac xs, simp, simp)

 apply (rule iffI, clarsimp)

  apply (case_tac i)

 apply (case_tac j, simp)

 apply (simp add: set_conv_nth)

  apply (case_tac j)

 apply (clarsimp simp add: set_conv_nth, simp)

 apply (rule conjI)

  apply (clarsimp simp add: set_conv_nth)

  apply (erule_tac x = 0 in allE)

  apply (erule_tac x = "Suc i" in allE, simp, clarsimp)

 apply (erule_tac x = "Suc i" in allE)

 apply (erule_tac x = "Suc j" in allE, simp)

 done

 lemma distinct_card: "distinct xs ==> card (set xs) = size xs"

   by (induct xs) auto

 lemma card_distinct: "card (set xs) = size xs ==> distinct xs"

 proof (induct xs)

   case Nil thus ?case by simp

 next

   case (Cons x xs)

   show ?case

   proof (cases "x \<in> set xs")

     case False with Cons show ?thesis by simp

   next

     case True with Cons.prems

     have "card (set xs) = Suc (length xs)" 

       by (simp add: card_insert_if split: split_if_asm)

     moreover have "card (set xs) \<le> length xs" by (rule card_length)

     ultimately have False by simp

     thus ?thesis ..

   qed

 qed

 lemma inj_on_setI: "distinct(map f xs) ==> inj_on f (set xs)"

 apply(induct xs)

  apply simp

 apply fastsimp

 done

 lemma inj_on_set_conv:

  "distinct xs \<Longrightarrow> inj_on f (set xs) = distinct(map f xs)"

 apply(induct xs)

  apply simp

 apply fastsimp

 done

 subsection {* @{text remove1} *}

 lemma set_remove1_subset: "set(remove1 x xs) <= set xs"

 apply(induct xs)

  apply simp

 apply simp

 apply blast

 done

 lemma [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"

 apply(induct xs)

  apply simp

 apply simp

 apply blast

 done

 lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"

 apply(insert set_remove1_subset)

 apply fast

 done

 lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"

 by (induct xs) simp_all

 subsection {* @{text replicate} *}

 lemma length_replicate [simp]: "length (replicate n x) = n"

 by (induct n) auto

 lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"

 by (induct n) auto

 lemma replicate_app_Cons_same:

 "(replicate n x) @ (x # xs) = x # replicate n x @ xs"

 by (induct n) auto

 lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"

 apply (induct n, simp)

 apply (simp add: replicate_app_Cons_same)

 done

 lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"

 by (induct n) auto

 lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"

 by (induct n) auto

 lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"

 by (induct n) auto

 lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"

 by (atomize (full), induct n) auto

 lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"

 apply (induct n, simp)

 apply (simp add: nth_Cons split: nat.split)

 done

 lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"

 by (induct n) auto

 lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"

 by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)

 lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"

 by auto

 lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"

 by (simp add: set_replicate_conv_if split: split_if_asm)

 subsection {* Lexicographic orderings on lists *}

 lemma wf_lexn: "wf r ==> wf (lexn r n)"

 apply (induct_tac n, simp, simp)

 apply(rule wf_subset)

  prefer 2 apply (rule Int_lower1)

 apply(rule wf_prod_fun_image)

  prefer 2 apply (rule inj_onI, auto)

 done

 lemma lexn_length:

      "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"

 by (induct n) auto

 lemma wf_lex [intro!]: "wf r ==> wf (lex r)"

 apply (unfold lex_def)

 apply (rule wf_UN)

 apply (blast intro: wf_lexn, clarify)

 apply (rename_tac m n)

 apply (subgoal_tac "m \<noteq> n")

  prefer 2 apply blast

 apply (blast dest: lexn_length not_sym)

 done

 lemma lexn_conv:

 "lexn r n =

 {(xs,ys). length xs = n \<and> length ys = n \<and>

 (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"

 apply (induct_tac n, simp, blast)

 apply (simp add: image_Collect lex_prod_def, safe, blast)

  apply (rule_tac x = "ab # xys" in exI, simp)

 apply (case_tac xys, simp_all, blast)

 done

 lemma lex_conv:

 "lex r =

 {(xs,ys). length xs = length ys \<and>

 (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"

 by (force simp add: lex_def lexn_conv)

 lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)"

 by (unfold lexico_def) blast

 lemma lexico_conv:

 "lexico r = {(xs,ys). length xs < length ys |

 length xs = length ys \<and> (xs, ys) : lex r}"

 by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def)

 lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"

 by (simp add: lex_conv)

 lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"

 by (simp add:lex_conv)

 lemma Cons_in_lex [iff]:

 "((x # xs, y # ys) : lex r) =

 ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"

 apply (simp add: lex_conv)

 apply (rule iffI)

  prefer 2 apply (blast intro: Cons_eq_appendI, clarify)

 apply (case_tac xys, simp, simp)

 apply blast

 done

 subsection {* @{text sublist} --- a generalization of @{text nth} to sets *}

 lemma sublist_empty [simp]: "sublist xs {} = []"

 by (auto simp add: sublist_def)

 lemma sublist_nil [simp]: "sublist [] A = []"

 by (auto simp add: sublist_def)

 lemma sublist_shift_lemma:

      "map fst [p:zip xs [i..i + length xs(] . snd p : A] =

       map fst [p:zip xs [0..length xs(] . snd p + i : A]"

 by (induct xs rule: rev_induct) (simp_all add: add_commute)

 lemma sublist_append:

      "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"

 apply (unfold sublist_def)

 apply (induct l' rule: rev_induct, simp)

 apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)

 apply (simp add: add_commute)

 done

 lemma sublist_Cons:

 "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"

 apply (induct l rule: rev_induct)

  apply (simp add: sublist_def)

 apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)

 done

 lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"

 by (simp add: sublist_Cons)

 lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"

 apply (induct l rule: rev_induct, simp)

 apply (simp split: nat_diff_split add: sublist_append)

 done

 lemma take_Cons':

      "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"

 by (cases n) simp_all

 lemma drop_Cons':

      "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"

 by (cases n) simp_all

 lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"

 by (cases n) simp_all

 lemmas [simp] = take_Cons'[of "number_of v",standard]

                 drop_Cons'[of "number_of v",standard]

                 nth_Cons'[of _ _ "number_of v",standard]

 lemma "card (set xs) = size xs \<Longrightarrow> distinct xs"

 proof (induct xs)

   case Nil thus ?case by simp

 next

   case (Cons x xs)

   show ?case

   proof (cases "x \<in> set xs")

     case False with Cons show ?thesis by simp

   next

     case True with Cons.prems

     have "card (set xs) = Suc (length xs)" 

       by (simp add: card_insert_if split: split_if_asm)

     moreover have "card (set xs) \<le> length xs" by (rule card_length)

     ultimately have False by simp

     thus ?thesis ..

   qed

 qed

 subsection {* Characters and strings *}

 datatype nibble =

     Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7

   | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF

 datatype char = Char nibble nibble

   -- "Note: canonical order of character encoding coincides with standard term ordering"

 types string = "char list"

 syntax

   "_Char" :: "xstr => char"    ("CHR _")

   "_String" :: "xstr => string"    ("_")

 parse_ast_translation {*

   let

     val constants = Syntax.Appl o map Syntax.Constant;

     fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10));

     fun mk_char c =

       if Symbol.is_ascii c andalso Symbol.is_printable c then

         constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)]

       else error ("Printable ASCII character expected: " ^ quote c);

     fun mk_string [] = Syntax.Constant "Nil"

       | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs];

     fun char_ast_tr [Syntax.Variable xstr] =

         (case Syntax.explode_xstr xstr of

           [c] => mk_char c

         | _ => error ("Single character expected: " ^ xstr))

       | char_ast_tr asts = raise AST ("char_ast_tr", asts);

     fun string_ast_tr [Syntax.Variable xstr] =

         (case Syntax.explode_xstr xstr of

           [] => constants [Syntax.constrainC, "Nil", "string"]

         | cs => mk_string cs)

       | string_ast_tr asts = raise AST ("string_tr", asts);

   in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end;

 *}

 ML {*

 fun int_of_nibble h =

   if "0" <= h andalso h <= "9" then ord h - ord "0"

   else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10

   else raise Match;

 fun nibble_of_int i =

   if i <= 9 then chr (ord "0" + i) else chr (ord "A" + i - 10);

 *}

 print_ast_translation {*

   let

     fun dest_nib (Syntax.Constant c) =

         (case explode c of

           ["N", "i", "b", "b", "l", "e", h] => int_of_nibble h

         | _ => raise Match)

       | dest_nib _ = raise Match;

     fun dest_chr c1 c2 =

       let val c = chr (dest_nib c1 * 16 + dest_nib c2)

       in if Symbol.is_printable c then c else raise Match end;

     fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2

       | dest_char _ = raise Match;

     fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)];

     fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]]

       | char_ast_tr' _ = raise Match;

     fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String",

             xstr (map dest_char (Syntax.unfold_ast "_args" args))]

       | list_ast_tr' ts = raise Match;

   in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end;

 *}

 subsection {* Code generator setup *}

 ML {*

 local

 fun list_codegen thy gr dep b t =

   let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy dep false)

     (gr, HOLogic.dest_list t)

   in Some (gr', Pretty.list "[" "]" ps) end handle TERM _ => None;

 fun dest_nibble (Const (s, _)) = int_of_nibble (unprefix "List.nibble.Nibble" s)

   | dest_nibble _ = raise Match;

 fun char_codegen thy gr dep b (Const ("List.char.Char", _) $ c1 $ c2) =

     (let val c = chr (dest_nibble c1 * 16 + dest_nibble c2)

      in if Symbol.is_printable c then Some (gr, Pretty.quote (Pretty.str c))

        else None

      end handle LIST _ => None | Match => None)

   | char_codegen thy gr dep b _ = None;

 in

 val list_codegen_setup =

   [Codegen.add_codegen "list_codegen" list_codegen,

    Codegen.add_codegen "char_codegen" char_codegen];

 end;

 val term_of_list = HOLogic.mk_list;

 fun gen_list' aG i j = frequency

   [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] ()

 and gen_list aG i = gen_list' aG i i;

 val nibbleT = Type ("List.nibble", []);

 fun term_of_char c =

   Const ("List.char.Char", nibbleT --> nibbleT --> Type ("List.char", [])) $

     Const ("List.nibble.Nibble" ^ nibble_of_int (ord c div 16), nibbleT) $

     Const ("List.nibble.Nibble" ^ nibble_of_int (ord c mod 16), nibbleT);

 fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z")));

 *}

 types_code

   "list" ("_ list")

   "char" ("string")

 consts_code "Cons" ("(_ ::/ _)")

 setup list_codegen_setup

 end