(*  Title:      HOL/List.thy

    ID:         $Id$

    Author:     Tobias Nipkow

    License:    GPL (GNU GENERAL PUBLIC LICENSE)

*)

header {* The datatype of finite lists *}

theory List = PreList:

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"

  fold_rel :: "('a * 'c * 'a) set => ('a * 'c list * 'a) set"

  hd:: "'a list => 'a"

  tl:: "'a list => 'a list"

  last:: "'a list => 'a"

  butlast :: "'a list => 'a list"

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

  o2l :: "'a option => 'a list"

  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"

  null:: "'a list => bool"

  "distinct":: "'a list => bool"

  replicate :: "nat => 'a => 'a list"

  postfix :: "'a list => 'a list => bool"

syntax (xsymbols)

  postfix :: "'a list => 'a list => bool"             ("(_/ \<sqsupseteq> _)" [51, 51] 50)

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>_ ./ _])")

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

 "o2l  None    = []"

 "o2l (Some x) = [x]"

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

replicate_0: "replicate 0 x = []"

replicate_Suc: "replicate (Suc n) x = x # replicate n x"

defs

 postfix_def: "postfix xs ys == \<exists>zs. xs = zs @ ys"

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"

apply (rule mono_Int [THEN equalityI])

apply (simp add: mono_def lists_mono)

apply (blast intro!: lists_IntI)

done

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_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

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)"

apply (induct xs, simp, simp)

apply (rule iffI)

 apply (blast intro: eq_Nil_appendI Cons_eq_appendI)

apply (erule exE)+

apply (case_tac ys, auto)

done

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 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 {* @{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 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 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 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_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

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)