src/HOL/List.thy

tuned document;

(*  Title:      HOL/List.thy
    ID:         $Id$
    Author:     Tobias Nipkow
    Copyright   1994 TU Muenchen
*)

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"
  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"
  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>_ ./ _])")


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
  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: "A \<subseteq> B ==> lists A \<subseteq> lists B"
  by (unfold lists.defs) (blast intro!: lfp_mono)

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 \<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


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 [rule_format, simp]:
 "\<forall>ys. length xs = length ys \<or> length us = length vs
       --> (xs@us = ys@vs) = (xs=ys \<and> 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 \<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)


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";

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];
*}


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_cong:
  "xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
  -- {* a congruence rule for @{text map} *}
  by (clarify, induct ys) auto

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:
  "(map f xs = y # ys) = (\<exists>x xs'. xs = x # xs' \<and> f x = y \<and> map f xs' = ys)"
  by (cases xs) auto

lemma map_injective:
    "!!xs. map f xs = map f ys ==> (\<forall>x y. f x = f y --> x = y) ==> xs = ys"
  by (induct ys) (auto simp add: map_eq_Cons)

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


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)
   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

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

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


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 set_subset_Cons: "set xs \<subseteq> set (x # 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)
   apply simp_all
  apply(erule ssubst)
  apply auto
  apply arith
  done

lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
  apply (induct 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

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])


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")
   apply simp
  apply (case_tac n)
   apply auto
  done

lemma nth_map [simp]: "!!n. n < length xs ==> (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: "[| 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_same_conv:
    "!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
  by (induct xs) (auto split: nat.split)

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 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 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 \<and> ys = drop (length xs) zs)"
  apply(induct xs)
   apply simp
  apply clarsimp
  apply (case_tac zs)
  apply auto
  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)


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)
   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. [| i < length xs; i < length ys  |] ==> (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


subsection {* @{text 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 \<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_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)
  apply 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)
  apply clarify
  apply (simp add: ball_Un)
  done

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[rule_format]:
  "\<forall>a b c. P1 a b --> P2 b c --> P3 a c ==>
    \<forall>bs cs. list_all2 P1 as bs --> list_all2 P2 bs cs --> list_all2 P3 as cs"
  apply(induct_tac as)
   apply simp
  apply(rule allI)
  apply(induct_tac bs)
   apply simp
  apply(rule allI)
  apply(induct_tac cs)
   apply auto
  done


subsection {* @{text foldl} *}

lemma foldl_append [simp]:
  "!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
  by (induct xs) auto

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)
  apply 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)
   apply 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 [rule_format]:
  "ALL xs ys. k <= length xs --> k <= length ys
    --> (ALL i. i < k --> xs!i = ys!i)
    --> take k xs = take k ys"
  apply (induct k)
  apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib)
  apply clarify
  txt {* Both lists must be non-empty *}
  apply (case_tac xs)
   apply simp
  apply (case_tac ys)
   apply 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

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 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)
   apply simp
  apply simp
  apply (rule iffI)
   apply clarsimp
   apply (case_tac i)
    apply (case_tac j)
     apply simp
    apply (simp add: set_conv_nth)
   apply (case_tac j)
    apply (clarsimp simp add: set_conv_nth)
   apply 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)
   apply simp
  apply clarsimp
  apply (erule_tac x = "Suc i" in allE)
  apply (erule_tac x = "Suc j" in allE)
  apply simp
  done


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)
   apply 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)
   apply 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 {* Lexcicographic orderings on lists *}

lemma wf_lexn: "wf r ==> wf (lexn r n)"
  apply (induct_tac n)
   apply simp
  apply simp
  apply(rule wf_subset)
   prefer 2 apply (rule Int_lower1)
  apply(rule wf_prod_fun_image)
   prefer 2 apply (rule injI)
  apply 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)
  apply 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)
   apply simp
   apply blast
  apply (simp add: image_Collect lex_prod_def)
  apply auto
    apply blast
   apply (rename_tac a xys x xs' y ys')
   apply (rule_tac x = "a # xys" in exI)
   apply simp
  apply (case_tac xys)
   apply simp_all
  apply 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)
  apply clarify
  apply (case_tac xys)
   apply simp
  apply 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)
   apply 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)
   apply 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 [of "number_of v", standard, simp] =
  take_Cons' drop_Cons' nth_Cons'

end