(*  Title:      HOL/List.thy

    ID:         $Id$

    Author:     Tobias Nipkow

    Copyright   1994 TU Muenchen

*)

header {* The datatype of finite lists *}

theory List1 = PreList:

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

consts

  "@"         :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"            (infixr 65)

  filter      :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list"

  concat      :: "'a list list \<Rightarrow> 'a list"

  foldl       :: "('b \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"

  foldr       :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b"

  hd          :: "'a list \<Rightarrow> 'a"

  tl          :: "'a list \<Rightarrow> 'a list"

  last        :: "'a list \<Rightarrow> 'a"

  butlast     :: "'a list \<Rightarrow> 'a list"

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

  list_all    :: "('a \<Rightarrow> bool) \<Rightarrow> ('a list \<Rightarrow> bool)"

  list_all2   :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool"

  map         :: "('a\<Rightarrow>'b) \<Rightarrow> ('a list \<Rightarrow> 'b list)"

  mem         :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"                    (infixl 55)

  nth         :: "'a list \<Rightarrow> nat \<Rightarrow> 'a"			  (infixl "!" 100)

  list_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a list"

  take        :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"

  drop        :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"

  takeWhile   :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list"

  dropWhile   :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list"

  rev         :: "'a list \<Rightarrow> 'a list"

  zip	      :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a * 'b) list"

  upt         :: "nat \<Rightarrow> nat \<Rightarrow> nat list"                   ("(1[_../_'(])")

  remdups     :: "'a list \<Rightarrow> 'a list"

  null        :: "'a list \<Rightarrow> bool"

  "distinct"  :: "'a list \<Rightarrow> bool"

  replicate   :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list"

nonterminals

  lupdbinds  lupdbind

syntax

  (* list Enumeration *)

  "@list"     :: "args \<Rightarrow> 'a list"                          ("[(_)]")

  (* Special syntax for filter *)

  "@filter"   :: "[pttrn, 'a list, bool] \<Rightarrow> 'a list"        ("(1[_:_./ _])")

  (* list update *)

  "_lupdbind"      :: "['a, 'a] \<Rightarrow> lupdbind"            ("(2_ :=/ _)")

  ""               :: "lupdbind \<Rightarrow> lupdbinds"           ("_")

  "_lupdbinds"     :: "[lupdbind, lupdbinds] \<Rightarrow> lupdbinds" ("_,/ _")

  "_LUpdate"       :: "['a, lupdbinds] \<Rightarrow> 'a"           ("_/[(_)]" [900,0] 900)

  upto        :: "nat \<Rightarrow> nat \<Rightarrow> 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] \<Rightarrow> 'a list"        ("(1[_\<in>_ ./ _])")

consts

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

inductive "lists A"

intros

Nil:  "[]: lists A"

Cons: "\<lbrakk> a: A;  l: lists A \<rbrakk> \<Longrightarrow> a#l : lists A"

(*Function "size" is overloaded for all datatypes.  Users may refer to the

  list version as "length".*)

syntax   length :: "'a list \<Rightarrow> nat"

translations  "length"  =>  "size:: _ list \<Rightarrow> nat"

(* translating size::list -> length *)

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) & 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 \<Rightarrow> x#xs | Suc(m) \<Rightarrow> drop m xs)"

  (* Warning: simpset does not contain this definition but separate theorems 

     for n=0 / n=Suc k*)

primrec

  take_Nil:  "take n [] = []"

  take_Cons: "take n (x#xs) = (case n of 0 \<Rightarrow> [] | Suc(m) \<Rightarrow> x # take m xs)"

  (* Warning: simpset does not contain this definition but separate theorems 

     for n=0 / n=Suc k*)

primrec 

  nth_Cons:  "(x#xs)!n = (case n of 0 \<Rightarrow> x | (Suc k) \<Rightarrow> xs!k)"

  (* Warning: simpset does not contain this definition but separate theorems 

     for n=0 / n=Suc k*)

primrec

 "    [][i:=v] = []"

 "(x#xs)[i:=v] = (case i of 0     \<Rightarrow> v # xs 

			  | Suc j \<Rightarrow> 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 [] \<Rightarrow> [] | z#zs \<Rightarrow> (z,y)#zip zs ys)"

  (* Warning: simpset does not contain this definition but separate theorems 

     for xs=[] / 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 & 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

 list_all2_def:

 "list_all2 P xs ys == length xs = length ys & (!(x,y):set(zip xs ys). P x y)"

(** Lexicographic orderings on lists **)

consts

 lexn :: "('a * 'a)set \<Rightarrow> nat \<Rightarrow> ('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 & length ys = Suc n}"

constdefs

  lex :: "('a * 'a)set \<Rightarrow> ('a list * 'a list)set"

    "lex r == UN n. lexn r n"

  lexico :: "('a * 'a)set \<Rightarrow> ('a list * 'a list)set"

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

  sublist :: "['a list, nat set] \<Rightarrow> 'a list"

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

lemma not_Cons_self[simp]: "\<And>x. xs ~= x#xs"

by(induct_tac "xs", auto)

lemmas not_Cons_self2[simp] = not_Cons_self[THEN not_sym]

lemma neq_Nil_conv: "(xs ~= []) = (? y ys. xs = y#ys)";

by(induct_tac "xs", auto)

(* Induction over the length of a list: *)

(* "(!!xs. (!ys. length ys < length xs --> P ys) ==> P xs) ==> P(xs)" *)

lemmas length_induct = measure_induct[of length]

(** "lists": the list-forming operator over sets **)

lemma lists_mono: "A<=B ==> lists A <= lists B"

apply(unfold lists.defs)

apply(blast intro!:lfp_mono)

done

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

declare lists.intros[intro!]

lemma lists_IntI[rule_format]:

 "l: lists A ==> l: lists B --> l: lists (A Int B)";

apply(erule lists.induct)

apply blast+

done

lemma lists_Int_eq[simp]: "lists (A Int B) = lists A Int 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 & ys : lists A)"

by(induct_tac "xs", auto)

(** length **)

(* needs to come before "@" because of thm append_eq_append_conv *)

section "length"

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

by(induct_tac "xs", auto)

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

by(induct_tac "xs", auto)

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

by(induct_tac "xs", auto)

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

by(case_tac "xs", auto)

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

by(induct_tac "xs", auto)

lemma length_greater_0_conv[iff]: "(0 < length xs) = (xs ~= [])"

by(induct_tac xs, auto)

lemma length_Suc_conv:

 "(length xs = Suc n) = (? y ys. xs = y#ys & length ys = n)"

by(induct_tac "xs", auto)

(** @ - append **)

section "@ - append"

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

by(induct_tac "xs", auto)

lemma append_Nil2[simp]: "xs @ [] = xs"

by(induct_tac "xs", auto)

lemma append_is_Nil_conv[iff]: "(xs@ys = []) = (xs=[] & ys=[])"

by(induct_tac "xs", auto)

lemma Nil_is_append_conv[iff]: "([] = xs@ys) = (xs=[] & ys=[])"

by(induct_tac "xs", auto)

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

by(induct_tac "xs", auto)

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

by(induct_tac "xs", auto)

lemma append_eq_append_conv[rule_format,simp]:

 "!ys. length xs = length ys | length us = length vs

       --> (xs@us = ys@vs) = (xs=ys & us=vs)"

apply(induct_tac "xs")

 apply(rule allI)

 apply(case_tac "ys")

  apply simp

 apply force

apply(rule allI)

apply(case_tac "ys")

 apply force

apply simp

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 & 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=[])"

by(insert append_same_eq[of _ _ "[]"], auto)

lemma self_append_conv2[iff]: "(ys = xs @ ys) = (xs=[])"

by(auto simp add: append_same_eq[of "[]", simplified])

lemma hd_Cons_tl[rule_format,simp]: "xs ~= [] --> hd xs # tl xs = xs"

by(induct_tac "xs", auto)

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

by(induct_tac "xs", auto)

lemma hd_append2[simp]: "xs ~= [] ==> 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 ~= [] ==> tl(xs @ ys) = (tl xs) @ ys"

by(simp add: tl_append split: list.split)

(* trivial rules for solving @-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)

(***

Simplification procedure for all list equalities.

Currently only tries to rearrange @ to see if

- both lists end in a singleton list,

- or both lists end in the same list.

***)

ML_setup{*

local

val list_eq_pattern =

  Thm.read_cterm (Theory.sign_of (the_context ())) ("(xs::'a list) = ys",HOLogic.boolT)

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 ct = cterm_of sg (HOLogic.mk_Trueprop(HOLogic.mk_eq(F,F2)))

          val thm = prove_goalw_cterm [] ct (K [rearr_tac 1])

            handle ERROR =>

            error("The error(s) above occurred while trying to prove " ^

                  string_of_cterm ct)

      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 = mk_simproc "list_eq" [list_eq_pattern] list_eq

end;

Addsimprocs [list_eq_simproc];

*}

(** map **)

section "map"

lemma map_ext: "(\<And>x. x : set xs \<longrightarrow> f x = g x) \<Longrightarrow> map f xs = map g xs"

by (induct xs, simp_all)

lemma map_ident[simp]: "map (%x. x) = (%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_tac "xs", auto)

lemma map_compose(*[simp]*): "map (f o g) xs = map f (map g xs)"

by(unfold o_def, induct_tac "xs", auto)

lemma rev_map: "rev(map f xs) = map f (rev xs)"

by(induct_tac xs, auto)

(* a congruence rule for map: *)

lemma map_cong:

 "xs=ys ==> (!!x. x : set ys \<Longrightarrow> f x = g x) \<Longrightarrow> map f xs = map g ys"

by (clarify, induct ys, auto)

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

by(case_tac xs, auto)

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

by(case_tac xs, auto)

lemma map_eq_Cons:

 "(map f xs = y#ys) = (? x xs'. xs = x#xs' & f x = y & map f xs' = ys)"

by(case_tac xs, auto)

lemma map_injective:

 "\<And>xs. map f xs = map f ys \<Longrightarrow> (!x y. f x = f y --> x=y) \<Longrightarrow> xs=ys"

by(induct "ys", simp, fastsimp simp add:map_eq_Cons)

lemma inj_mapI: "inj f ==> inj (map f)"

by(blast dest:map_injective injD intro:injI)

lemma inj_mapD: "inj (map f) ==> inj f"

apply(unfold inj_on_def)

apply clarify

apply(erule_tac x = "[x]" in ballE)

 apply(erule_tac x = "[y]" in ballE)

  apply simp

 apply blast

apply blast

done

lemma inj_map: "inj (map f) = inj f"

by(blast dest:inj_mapD intro:inj_mapI)

(** rev **)

section "rev"

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

by(induct_tac xs, auto)

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

by(induct_tac xs, auto)

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

by(induct_tac xs, auto)

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

by(induct_tac xs, auto)

lemma rev_is_rev_conv[iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"

apply(induct "xs" )

 apply force

apply(case_tac ys)

 apply simp

apply force

done

lemma rev_induct: "[| 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

(* Instead of (rev_induct_tac xs) use (induct_tac xs rule: rev_induct) *)

lemma rev_exhaust: "(xs = [] \<Longrightarrow> P) \<Longrightarrow>  (!!ys y. xs = ys@[y] \<Longrightarrow> P) \<Longrightarrow> P"

by(induct xs rule: rev_induct, auto)

(** set **)

section "set"

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

by(induct_tac xs, auto)

lemma set_append[simp]: "set (xs@ys) = (set xs Un set ys)"

by(induct_tac xs, auto)

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

by auto

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

by(induct_tac xs, auto)

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

by(induct_tac xs, auto)

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

by(induct_tac xs, auto)

lemma set_filter[simp]: "set(filter P xs) = {x. x : set xs & P x}"

by(induct_tac xs, auto)

lemma set_upt[simp]: "set[i..j(] = {k. i <= k & k < j}"

apply(induct_tac j)

 apply simp_all

apply(erule ssubst)

apply auto

apply arith

done

lemma in_set_conv_decomp: "(x : set xs) = (? ys zs. xs = ys@x#zs)"

apply(induct_tac "xs")

 apply simp

apply simp

apply(rule iffI)

 apply(blast intro: eq_Nil_appendI Cons_eq_appendI)

apply(erule exE)+

apply(case_tac "ys")

apply auto

done

(* eliminate `lists' in favour of `set' *)

lemma in_lists_conv_set: "(xs : lists A) = (!x : set xs. x : A)"

by(induct_tac xs, auto)

lemmas in_listsD[dest!] = in_lists_conv_set[THEN iffD1]

lemmas in_listsI[intro!] = in_lists_conv_set[THEN iffD2]

(** mem **)

section "mem"

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

by(induct_tac xs, auto)

(** list_all **)

section "list_all"

lemma list_all_conv: "list_all P xs = (!x:set xs. P x)"

by(induct_tac xs, auto)

lemma list_all_append[simp]:

 "list_all P (xs@ys) = (list_all P xs & list_all P ys)"

by(induct_tac xs, auto)

(** filter **)

section "filter"

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

by(induct_tac xs, auto)

lemma filter_filter[simp]: "filter P (filter Q xs) = filter (%x. Q x & P x) xs"

by(induct_tac xs, auto)

lemma filter_True[simp]: "!x : set xs. P x \<Longrightarrow> filter P xs = xs"

by(induct xs, auto)

lemma filter_False[simp]: "!x : set xs. ~P x \<Longrightarrow> filter P xs = []"

by(induct xs, auto)

lemma length_filter[simp]: "length (filter P xs) <= length xs"

by(induct xs, auto simp add: le_SucI)

lemma filter_is_subset[simp]: "set (filter P xs) <= set xs"

by auto

section "concat"

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

by(induct xs, auto)

lemma concat_eq_Nil_conv[iff]: "(concat xss = []) = (!xs:set xss. xs=[])"

by(induct xss, auto)

lemma Nil_eq_concat_conv[iff]: "([] = concat xss) = (!xs: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)

(** nth **)

section "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")

 apply simp

apply(case_tac "n" )

 apply auto

done

lemma nth_map[simp]: "!!n. n < length xs \<Longrightarrow> (map f xs)!n = f(xs!n)"

apply(induct "xs" )

 apply simp

apply(case_tac "n")

 apply auto

done

lemma set_conv_nth: "set xs = {xs!i |i. i < length xs}"

apply(induct_tac "xs")

 apply simp

apply simp

apply safe

  apply(rule_tac x = 0 in exI)

  apply simp

 apply(rule_tac x = "Suc i" in exI)

 apply simp

apply(case_tac "i")

 apply simp

apply(rename_tac "j")

apply(rule_tac x = "j" in exI)

apply simp

done

lemma list_ball_nth: "\<lbrakk> n < length xs; !x : set xs. P x \<rbrakk> \<Longrightarrow> P(xs!n)"

by(simp add:set_conv_nth, blast)

lemma nth_mem[simp]: "n < length xs ==> xs!n : set xs"

by(simp add:set_conv_nth, blast)

lemma all_nth_imp_all_set:

 "\<lbrakk> !i < length xs. P(xs!i); x : set xs \<rbrakk> \<Longrightarrow> P x"

by(simp add:set_conv_nth, blast)

lemma all_set_conv_all_nth:

 "(!x : set xs. P x) = (!i. i<length xs --> P (xs ! i))"

by(simp add:set_conv_nth, blast)

(** list update **)

section "list update"

lemma length_list_update[simp]: "!!i. length(xs[i:=x]) = length xs"

by(induct xs, simp, simp split:nat.split)

lemma nth_list_update:

 "!!i j. i < length xs  \<Longrightarrow> (xs[i:=x])!j = (if i=j then x else xs!j)"

by(induct xs, simp, 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 ~= j \<Longrightarrow> xs[i:=x]!j = xs!j"

by(induct xs, simp, 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, simp, simp split:nat.split)

lemma list_update_same_conv:

 "!!i. i < length xs \<Longrightarrow> (xs[i := x] = xs) = (xs!i = x)"

by(induct xs, simp, simp split:nat.split, blast)

lemma update_zip:

"!!i xy xs. length xs = length ys \<Longrightarrow>

    (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, simp, simp split:nat.split, fast)

lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"

by(fast dest!:set_update_subset_insert[THEN subsetD])

(** last & butlast **)

section "last / 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 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 ~= [] --> 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)

(** take  & drop **)

section "take & 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] drop_Cons[simp del]

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)

 apply auto

apply(case_tac xs)

 apply auto

apply(case_tac na)

 apply auto

done

lemma drop_drop[simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs" 

apply(induct m)

 apply auto

apply(case_tac xs)

 apply auto

done

lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"

apply(induct m)

 apply auto

apply(case_tac xs)

 apply auto

done

lemma append_take_drop_id[simp]: "!!xs. take n xs @ drop n xs = xs"

apply(induct n)

 apply auto

apply(case_tac xs)

 apply auto

done

lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"

apply(induct n)

 apply auto

apply(case_tac xs)

 apply auto

done

lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)" 

apply(induct n)

 apply auto

apply(case_tac xs)

 apply auto

done

lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"

apply(induct xs)

 apply auto

apply(case_tac i)

 apply auto

done

lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"

apply(induct xs)

 apply auto

apply(case_tac i)

 apply auto

done

lemma nth_take[simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"

apply(induct xs)

 apply auto

apply(case_tac n)

 apply(blast )

apply(case_tac i)

 apply auto

done

lemma nth_drop[simp]: "!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n+i)"

apply(induct n)

 apply auto

apply(case_tac xs)

 apply auto

done

lemma append_eq_conv_conj:

 "!!zs. (xs@ys = zs) = (xs = take (length xs) zs & ys = drop (length xs) zs)"

apply(induct xs)

 apply simp

apply clarsimp

apply(case_tac zs)

apply auto

done

(** takeWhile & dropWhile **)

section "takeWhile & dropWhile"

lemma takeWhile_dropWhile_id[simp]: "takeWhile P xs @ dropWhile P xs = xs"

by(induct xs, auto)

lemma  takeWhile_append1[simp]:

 "\<lbrakk> x:set xs; ~P(x) \<rbrakk> \<Longrightarrow> takeWhile P (xs @ ys) = takeWhile P xs"

by(induct xs, auto)

lemma takeWhile_append2[simp]:

 "(!!x. x : set xs \<Longrightarrow> P(x)) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"

by(induct xs, auto)

lemma takeWhile_tail: "~P(x) ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"

by(induct xs, auto)

lemma dropWhile_append1[simp]:

 "\<lbrakk> x : set xs; ~P(x) \<rbrakk> \<Longrightarrow> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"

by(induct xs, auto)

lemma dropWhile_append2[simp]:

 "(!!x. x:set xs \<Longrightarrow> 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 & P x"

by(induct xs, auto split:split_if_asm)

(** zip **)

section "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)

 apply simp

apply(case_tac xs)

 apply 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)

 apply simp

apply(case_tac xs)

 apply 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)

 apply simp

apply(case_tac ys)

 apply 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:

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

apply(induct ys)

 apply simp

apply(case_tac xs)

 apply simp_all

done

lemma nth_zip[simp]:

"!!i xs. \<lbrakk> i < length xs; i < length ys \<rbrakk> \<Longrightarrow> (zip xs ys)!i = (xs!i, ys!i)"

apply(induct ys)

 apply 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)

 apply auto

apply(case_tac j)

 apply auto

done

(** list_all2 **)

section "list_all2"

lemma list_all2_lengthD: "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 & list_all2 P xs ys)"

by(auto simp add:list_all2_def)

lemma list_all2_Cons1:

 "list_all2 P (x#xs) ys = (? z zs. ys = z#zs & P x z & list_all2 P xs zs)"

by(case_tac ys, auto)

lemma list_all2_Cons2:

 "list_all2 P xs (y#ys) = (? z zs. xs = z#zs & P z y & list_all2 P zs ys)"

by(case_tac 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)