(*  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]) rules

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 = Tactic.prove sg [] [] eq

          (K (simp_tac (Simplifier.inherit_bounds 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 (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 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: "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_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 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 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 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 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

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