src/HOL/List.thy

added allpairs

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

header {* The datatype of finite lists *}

theory List
imports PreList Recdef
uses "Tools/string_syntax.ML"
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"
  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"
  "distinct":: "'a list => bool"
  replicate :: "nat => 'a => 'a list"
  splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
  allpairs :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a * 'b)list"

abbreviation
  upto:: "nat => nat => nat list"  ("(1[_../_])") where
  "[i..j] == [i..<(Suc j)]"


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)

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"


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}. *}

abbreviation
  length :: "'a list => nat" where
  "length == size"

primrec
  "hd(x#xs) = x"

primrec
  "tl([]) = []"
  "tl(x#xs) = xs"

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"

definition
  rotate1 :: "'a list \<Rightarrow> 'a list" where
  "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"

definition
  rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
  "rotate n = rotate1 ^ n"

definition
  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
  "list_all2 P xs ys =
    (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"

definition
  sublist :: "'a list => nat set => 'a list" where
  "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"

primrec
  "splice [] ys = ys"
  "splice (x#xs) ys = (if ys=[] then x#xs else x # hd ys # splice xs (tl ys))"
    -- {*Warning: simpset does not contain the second eqn but a derived one. *}

primrec
"allpairs [] ys = []"
"allpairs (x # xs) ys = map (Pair x) ys @ allpairs xs ys"

subsubsection {* @{const Nil} and @{const Cons} *}

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:
  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
by (rule measure_induct [of length]) iprover


subsubsection {* @{const 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_allpairs[simp]:
 "length(allpairs xs ys) = length xs * length ys"
by(induct 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)
apply 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

lemma list_induct2': 
  "\<lbrakk> P [] [];
  \<And>x xs. P (x#xs) [];
  \<And>y ys. P [] (y#ys);
   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
 \<Longrightarrow> P xs ys"
by (induct xs arbitrary: ys) (case_tac x, auto)+

lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
apply(rule Eq_FalseI)
by auto

(*
Reduces xs=ys to False if xs and ys cannot be of the same length.
This is the case if the atomic sublists of one are a submultiset
of those of the other list and there are fewer Cons's in one than the other.
*)
ML_setup {*
local

fun len (Const("List.list.Nil",_)) acc = acc
  | len (Const("List.list.Cons",_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
  | len (Const("List.op @",_) $ xs $ ys) acc = len xs (len ys acc)
  | len (Const("List.rev",_) $ xs) acc = len xs acc
  | len (Const("List.map",_) $ _ $ xs) acc = len xs acc
  | len t (ts,n) = (t::ts,n);

fun list_eq ss (Const(_,eqT) $ lhs $ rhs) =
  let
    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
    fun prove_neq() =
      let
        val Type(_,listT::_) = eqT;
        val size = Const("Nat.size", listT --> HOLogic.natT);
        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
          (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
      in SOME (thm RS @{thm neq_if_length_neq}) end
  in
    if m < n andalso gen_submultiset (op aconv) (ls,rs) orelse
       n < m andalso gen_submultiset (op aconv) (rs,ls)
    then prove_neq() else NONE
  end;

in

val list_neq_simproc =
  Simplifier.simproc @{theory} "list_neq" ["(xs::'a list) = ys"] (K list_eq);

end;

Addsimprocs [list_neq_simproc];
*}


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

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 [@{thm append_assoc},
  @{thm append_Nil}, @{thm append_Cons}];

fun list_eq 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 = Goal.prove (Simplifier.the_context ss) [] [] eq
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;

  in
    if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
    else if lastl aconv lastr then rearr @{thm append_same_eq}
    else NONE
  end;

in

val list_eq_simproc =
  Simplifier.simproc @{theory} "list_eq" ["(xs::'a list) = ys"] (K 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 [fundef_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:
 "(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:
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
by (cases ys) auto

lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]

lemma ex_map_conv:
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
by(induct ys, auto simp add: Cons_eq_map_conv)

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 (iprover 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]: "(rev xs = rev ys) = (xs = ys)"
apply (induct xs arbitrary: ys, 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

lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
by(rule rev_cases[of xs]) auto


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[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
by(cases xs) 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_allpairs[simp]:
 "set(allpairs xs ys) = {(x,y). x : set xs & y : set ys}"
by(induct xs) auto

lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
by (induct xs) auto

lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
apply (induct j, simp_all)
apply (erule ssubst, auto)
done

lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
proof (induct xs)
  case Nil show ?case by simp
  case (Cons a xs)
  show ?case
  proof 
    assume "x \<in> set (a # xs)"
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
      by (simp, blast intro: Cons_eq_appendI)
  next
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
    show "x \<in> set (a # xs)" 
      by (cases ys, auto simp add: eq)
  qed
qed

lemma in_set_conv_decomp_first:
 "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
proof (induct xs)
  case Nil show ?case by simp
next
  case (Cons a xs)
  show ?case
  proof cases
    assume "x = a" thus ?case using Cons by force
  next
    assume "x \<noteq> a"
    show ?case
    proof
      assume "x \<in> set (a # xs)"
      from prems show "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
	by(fastsimp intro!: Cons_eq_appendI)
    next
      assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
      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
qed

lemmas split_list       = in_set_conv_decomp[THEN iffD1, standard]
lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard]


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 sum_length_filter_compl:
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
by(induct xs) simp_all

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 Cons_eq_filterD:
 "x#xs = filter P ys \<Longrightarrow>
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
proof(induct ys)
  case Nil thus ?case by simp
next
  case (Cons y ys)
  show ?case (is "\<exists>x. ?Q x")
  proof cases
    assume Py: "P y"
    show ?thesis
    proof cases
      assume xy: "x = y"
      show ?thesis
      proof from Py xy Cons(2) show "?Q []" by simp qed
    next
      assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp
    qed
  next
    assume Py: "\<not> P y"
    with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp
    show ?thesis (is "? us. ?Q us")
    proof show "?Q (y#us)" using 1 by simp qed
  qed
qed

lemma filter_eq_ConsD:
 "filter P ys = x#xs \<Longrightarrow>
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
by(rule Cons_eq_filterD) simp

lemma filter_eq_Cons_iff:
 "(filter P ys = x#xs) =
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
by(auto dest:filter_eq_ConsD)

lemma Cons_eq_filter_iff:
 "(x#xs = filter P ys) =
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
by(auto dest:Cons_eq_filterD)

lemma filter_cong[fundef_cong, recdef_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 [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
by (induct xss) auto

lemma Nil_eq_concat_conv [simp]: "([] = 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 hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
by(cases xs) simp_all


lemma list_eq_iff_nth_eq:
 "!!ys. (xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
apply(induct xs)
 apply simp apply blast
apply(case_tac ys)
 apply simp
apply(simp add:nth_Cons split:nat.split)apply blast
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 hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
by(rule rev_exhaust[of xs]) simp_all

lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
by(cases xs) simp_all

lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
by (induct as) 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)

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

lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
by(induct xs)(auto simp:neq_Nil_conv)

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

declare take_Cons [simp del] and drop_Cons [simp del]

lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
by(clarsimp simp add:neq_Nil_conv)

lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
by(cases xs, simp_all)

lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)

lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
apply (induct xs, simp)
apply(simp add:drop_Cons nth_Cons split:nat.splits)
done

lemma take_Suc_conv_app_nth:
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
apply (induct xs, simp)
apply (case_tac i, auto)
done

lemma drop_Suc_conv_tl:
  "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
apply (induct xs, simp)
apply (case_tac i, auto)
done

lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
by (induct n) (auto, case_tac xs, auto)

lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
by (induct n) (auto, case_tac xs, auto)

lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
by (induct n) (auto, case_tac xs, auto)

lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
by (induct n) (auto, case_tac xs, auto)

lemma take_append [simp]:
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
by (induct n) (auto, case_tac xs, auto)

lemma drop_append [simp]:
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
by (induct n) (auto, case_tac xs, auto)

lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
apply (induct m, auto)
apply (case_tac xs, auto)
apply (case_tac n, auto)
done

lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
apply (induct m, auto)
apply (case_tac xs, auto)
done

lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
apply (induct m, auto)
apply (case_tac xs, auto)
done

lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"
apply(induct xs)
 apply simp
apply(simp add: take_Cons drop_Cons split:nat.split)
done

lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
apply (induct n, auto)
apply (case_tac xs, auto)
done

lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"
apply(induct xs)
 apply simp
apply(simp add:take_Cons split:nat.split)
done

lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"
apply(induct xs)
apply simp
apply(simp add:drop_Cons split:nat.split)
done

lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
apply (induct n, auto)
apply (case_tac xs, auto)
done

lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
apply (induct n, auto)
apply (case_tac xs, auto)
done

lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
apply (induct xs, auto)
apply (case_tac i, auto)
done

lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
apply (induct xs, auto)
apply (case_tac i, auto)
done

lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
apply (induct xs, auto)
apply (case_tac n, blast)
apply (case_tac i, auto)
done

lemma nth_drop [simp]:
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
apply (induct n, auto)
apply (case_tac xs, auto)
done

lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
by(simp add: hd_conv_nth)

lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
by(induct xs)(auto simp:take_Cons split:nat.split)

lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
by(induct xs)(auto simp:drop_Cons split:nat.split)

lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
using set_take_subset by fast

lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
using set_drop_subset by fast

lemma append_eq_conv_conj:
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
apply (induct xs, simp, clarsimp)
apply (case_tac zs, auto)
done

lemma take_add [rule_format]: 
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
apply (induct xs, auto) 
apply (case_tac i, simp_all) 
done

lemma append_eq_append_conv_if:
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
   then xs\<^isub>1 = take (size xs\<^isub>1) ys\<^isub>1 \<and> xs\<^isub>2 = drop (size xs\<^isub>1) ys\<^isub>1 @ ys\<^isub>2
   else take (size ys\<^isub>1) xs\<^isub>1 = ys\<^isub>1 \<and> drop (size ys\<^isub>1) xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>2)"
apply(induct xs\<^isub>1)
 apply simp
apply(case_tac ys\<^isub>1)
apply simp_all
done

lemma take_hd_drop:
  "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
apply(induct xs)
apply simp
apply(simp add:drop_Cons split:nat.split)
done

lemma id_take_nth_drop:
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
proof -
  assume si: "i < length xs"
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
  moreover
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
    apply (rule_tac take_Suc_conv_app_nth) by arith
  ultimately show ?thesis by auto
qed
  
lemma upd_conv_take_nth_drop:
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
proof -
  assume i: "i < length xs"
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
    using i by (simp add: list_update_append)
  finally show ?thesis .
qed


subsubsection {* @{text takeWhile} and @{text dropWhile} *}

lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
by (induct xs) auto

lemma takeWhile_append1 [simp]:
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
by (induct xs) auto

lemma takeWhile_append2 [simp]:
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
by (induct xs) auto

lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
by (induct xs) auto

lemma dropWhile_append1 [simp]:
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
by (induct xs) auto

lemma dropWhile_append2 [simp]:
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
by (induct xs) auto

lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
by (induct xs) (auto split: split_if_asm)

lemma takeWhile_eq_all_conv[simp]:
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
by(induct xs, auto)

lemma dropWhile_eq_Nil_conv[simp]:
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
by(induct xs, auto)

lemma dropWhile_eq_Cons_conv:
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
by(induct xs, auto)

text{* The following two lemmmas could be generalized to an arbitrary
property. *}

lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])

lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
apply(induct xs)
 apply simp
apply auto
apply(subst dropWhile_append2)
apply auto
done

lemma takeWhile_not_last:
 "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
apply(induct xs)
 apply simp
apply(case_tac xs)
apply(auto)
done

lemma takeWhile_cong [fundef_cong, recdef_cong]:
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
  ==> takeWhile P l = takeWhile Q k"
  by (induct k arbitrary: l) (simp_all)

lemma dropWhile_cong [fundef_cong, recdef_cong]:
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
  ==> dropWhile P l = dropWhile Q k"
  by (induct k arbitrary: l, simp_all)


subsubsection {* @{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 zip_Cons1:
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
by(auto split:list.split)

lemma length_zip [simp]:
"length (zip xs ys) = min (length xs) (length ys)"
by (induct xs ys rule:list_induct2') auto

lemma zip_append1:
"zip (xs @ ys) zs =
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
by (induct xs zs rule:list_induct2') auto

lemma zip_append2:
"zip xs (ys @ zs) =
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
by (induct xs ys rule:list_induct2') auto

lemma zip_append [simp]:
 "[| length xs = length us; length ys = length vs |] ==>
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
by (simp add: zip_append1)

lemma zip_rev:
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
by (induct rule:list_induct2, simp_all)

lemma nth_zip [simp]:
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
apply (induct ys, simp)
apply (case_tac xs)
 apply (simp_all add: nth.simps split: nat.split)
done

lemma set_zip:
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
by (simp add: set_conv_nth cong: rev_conj_cong)

lemma zip_update:
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
by (rule sym, simp add: update_zip)

lemma zip_replicate [simp]:
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
apply (induct i, auto)
apply (case_tac j, auto)
done

lemma take_zip:
 "!!xs ys. take n (zip xs ys) = zip (take n xs) (take n ys)"
apply (induct n)
 apply simp
apply (case_tac xs, simp)
apply (case_tac ys, simp_all)
done

lemma drop_zip:
 "!!xs ys. drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
apply (induct n)
 apply simp
apply (case_tac xs, simp)
apply (case_tac ys, simp_all)
done

lemma set_zip_leftD:
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
by (induct xs ys rule:list_induct2') auto

lemma set_zip_rightD:
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
by (induct xs ys rule:list_induct2') auto

subsubsection {* @{text list_all2} *}

lemma list_all2_lengthD [intro?]: 
  "list_all2 P xs ys ==> length xs = length ys"
  by (simp add: list_all2_def)

lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
  by (simp add: list_all2_def)

lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
  by (simp add: list_all2_def)

lemma list_all2_Cons [iff, code]:
  "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
  by (auto simp add: list_all2_def)

lemma list_all2_Cons1:
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
by (cases ys) auto

lemma list_all2_Cons2:
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
by (cases xs) auto

lemma list_all2_rev [iff]:
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
by (simp add: list_all2_def zip_rev cong: conj_cong)

lemma list_all2_rev1:
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
by (subst list_all2_rev [symmetric]) simp

lemma list_all2_append1:
"list_all2 P (xs @ ys) zs =
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
list_all2 P xs us \<and> list_all2 P ys vs)"
apply (simp add: list_all2_def zip_append1)
apply (rule iffI)
 apply (rule_tac x = "take (length xs) zs" in exI)
 apply (rule_tac x = "drop (length xs) zs" in exI)
 apply (force split: nat_diff_split simp add: min_def, clarify)
apply (simp add: ball_Un)
done

lemma list_all2_append2:
"list_all2 P xs (ys @ zs) =
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
list_all2 P us ys \<and> list_all2 P vs zs)"
apply (simp add: list_all2_def zip_append2)
apply (rule iffI)
 apply (rule_tac x = "take (length ys) xs" in exI)
 apply (rule_tac x = "drop (length ys) xs" in exI)
 apply (force split: nat_diff_split simp add: min_def, clarify)
apply (simp add: ball_Un)
done

lemma list_all2_append:
  "length xs = length ys \<Longrightarrow>
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
by (induct rule:list_induct2, simp_all)

lemma list_all2_appendI [intro?, trans]:
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
  by (simp add: list_all2_append list_all2_lengthD)

lemma list_all2_conv_all_nth:
"list_all2 P xs ys =
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
by (force simp add: list_all2_def set_zip)

lemma list_all2_trans:
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
        (is "!!bs cs. PROP ?Q as bs cs")
proof (induct as)
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
  show "!!cs. PROP ?Q (x # xs) bs cs"
  proof (induct bs)
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
    show "PROP ?Q (x # xs) (y # ys) cs"
      by (induct cs) (auto intro: tr I1 I2)
  qed simp
qed simp

lemma list_all2_all_nthI [intro?]:
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
  by (simp add: list_all2_conv_all_nth)

lemma list_all2I:
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
  by (simp add: list_all2_def)

lemma list_all2_nthD:
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
  by (simp add: list_all2_conv_all_nth)

lemma list_all2_nthD2:
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
  by (frule list_all2_lengthD) (auto intro: list_all2_nthD)

lemma list_all2_map1: 
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
  by (simp add: list_all2_conv_all_nth)

lemma list_all2_map2: 
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
  by (auto simp add: list_all2_conv_all_nth)

lemma list_all2_refl [intro?]:
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
  by (simp add: list_all2_conv_all_nth)

lemma list_all2_update_cong:
  "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
  by (simp add: list_all2_conv_all_nth nth_list_update)

lemma list_all2_update_cong2:
  "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
  by (simp add: list_all2_lengthD list_all2_update_cong)

lemma list_all2_takeI [simp,intro?]:
  "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
  apply (induct xs)
   apply simp
  apply (clarsimp simp add: list_all2_Cons1)
  apply (case_tac n)
  apply auto
  done

lemma list_all2_dropI [simp,intro?]:
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
  apply (induct as, simp)
  apply (clarsimp simp add: list_all2_Cons1)
  apply (case_tac n, simp, simp)
  done

lemma list_all2_mono [intro?]:
  "\<And>y. list_all2 P x y \<Longrightarrow> (\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> list_all2 Q x y"
  apply (induct x, simp)
  apply (case_tac y, auto)
  done

lemma list_all2_eq:
  "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
  by (induct xs ys rule: list_induct2') auto


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

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

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

lemma foldl_cong [fundef_cong, recdef_cong]:
  "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] 
  ==> foldl f a l = foldl g b k"
  by (induct k arbitrary: a b l) simp_all

lemma foldr_cong [fundef_cong, recdef_cong]:
  "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] 
  ==> foldr f l a = foldr g k b"
  by (induct k arbitrary: a b l) simp_all

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

lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])

text {*
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
difficult to use because it requires an additional transitivity step.
*}

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

lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
by (force intro: start_le_sum simp add: in_set_conv_decomp)

lemma sum_eq_0_conv [iff]:
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
by (induct ns) auto


subsubsection {* @{text upto} *}

lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
-- {* simp does not terminate! *}
by (induct j) auto

lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
by (subst upt_rec) simp

lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
by(induct j)simp_all

lemma upt_eq_Cons_conv:
 "!!x xs. ([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
apply(induct j)
 apply simp
apply(clarsimp simp add: append_eq_Cons_conv)
apply arith
done

lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
by simp

lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
apply(rule trans)
apply(subst upt_rec)
 prefer 2 apply (rule refl, simp)
done

lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
by (induct k) auto

lemma length_upt [simp]: "length [i..<j] = j - i"
by (induct j) (auto simp add: Suc_diff_le)

lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
apply (induct j)
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
done


lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
by(simp add:upt_conv_Cons)

lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
apply(cases j)
 apply simp
by(simp add:upt_Suc_append)

lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..<n] = [i..<i+m]"
apply (induct m, simp)
apply (subst upt_rec)
apply (rule sym)
apply (subst upt_rec)
apply (simp del: upt.simps)
done

lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
apply(induct j)
apply auto
done

lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..n]"
by (induct n) auto

lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
apply (induct n m rule: diff_induct)
prefer 3 apply (subst map_Suc_upt[symmetric])
apply (auto simp add: less_diff_conv nth_upt)
done

lemma nth_take_lemma:
  "!!xs ys. k <= length xs ==> k <= length ys ==>
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
apply (atomize, induct k)
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
txt {* Both lists must be non-empty *}
apply (case_tac xs, simp)
apply (case_tac ys, clarify)
 apply (simp (no_asm_use))
apply clarify
txt {* prenexing's needed, not miniscoping *}
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
apply blast
done

lemma nth_equalityI:
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
apply (simp_all add: take_all)
done

(* needs nth_equalityI *)
lemma list_all2_antisym:
  "\<lbrakk> (\<And>x y. \<lbrakk>P x y; Q y x\<rbrakk> \<Longrightarrow> x = y); list_all2 P xs ys; list_all2 Q ys xs \<rbrakk> 
  \<Longrightarrow> xs = ys"
  apply (simp add: list_all2_conv_all_nth) 
  apply (rule nth_equalityI, blast, simp)
  done

lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
-- {* The famous take-lemma. *}
apply (drule_tac x = "max (length xs) (length ys)" in spec)
apply (simp add: le_max_iff_disj take_all)
done


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 take_Cons_number_of = take_Cons'[of "number_of v",standard]
lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard]
lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard]

declare take_Cons_number_of [simp] 
        drop_Cons_number_of [simp] 
        nth_Cons_number_of [simp] 


subsubsection {* @{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 distinct_rev[simp]: "distinct(rev xs) = distinct xs"
by(induct xs) auto

lemma set_remdups [simp]: "set (remdups xs) = set xs"
by (induct xs) (auto simp add: insert_absorb)

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

lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
  by (induct x, auto) 

lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
  by (induct x, auto)

lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
by (induct xs) auto

lemma length_remdups_eq[iff]:
  "(length (remdups xs) = length xs) = (remdups xs = xs)"
apply(induct xs)
 apply auto
apply(subgoal_tac "length (remdups xs) <= length xs")
 apply arith
apply(rule length_remdups_leq)
done


lemma distinct_map:
  "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
by (induct xs) auto


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

lemma distinct_upt[simp]: "distinct[i..<j]"
by (induct j) auto

lemma distinct_take[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (take i xs)"
apply(induct xs)
 apply simp
apply (case_tac i)
 apply simp_all
apply(blast dest:in_set_takeD)
done

lemma distinct_drop[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (drop i xs)"
apply(induct xs)
 apply simp
apply (case_tac i)
 apply simp_all
done

lemma distinct_list_update:
assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
shows "distinct (xs[i:=a])"
proof (cases "i < length xs")
  case True
  with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
    apply (drule_tac id_take_nth_drop) by simp
  with d True show ?thesis
    apply (simp add: upd_conv_take_nth_drop)
    apply (drule subst [OF id_take_nth_drop]) apply assumption
    apply simp apply (cases "a = xs!i") apply simp by blast
next
  case False with d show ?thesis by auto
qed


text {* It is best to avoid this indexed version of distinct, but
sometimes it is useful. *}

lemma distinct_conv_nth:
"distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
apply (induct xs, simp, simp)
apply (rule iffI, clarsimp)
 apply (case_tac i)
apply (case_tac j, simp)
apply (simp add: set_conv_nth)
 apply (case_tac j)
apply (clarsimp simp add: set_conv_nth, simp)
apply (rule conjI)
 apply (clarsimp simp add: set_conv_nth)
 apply (erule_tac x = 0 in allE, simp)
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
apply (erule_tac x = "Suc i" in allE, simp)
apply (erule_tac x = "Suc j" in allE, simp)
done

lemma nth_eq_iff_index_eq:
 "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
by(auto simp: distinct_conv_nth)

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

lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
proof (induct xs)
  case Nil thus ?case by simp
next
  case (Cons x xs)
  show ?case
  proof (cases "x \<in> set xs")
    case False with Cons show ?thesis by simp
  next
    case True with Cons.prems
    have "card (set xs) = Suc (length xs)" 
      by (simp add: card_insert_if split: split_if_asm)
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
    ultimately have False by simp
    thus ?thesis ..
  qed
qed


lemma length_remdups_concat:
 "length(remdups(concat xss)) = card(\<Union>xs \<in> set xss. set xs)"
by(simp add: distinct_card[symmetric])


subsubsection {* @{text remove1} *}

lemma remove1_append:
  "remove1 x (xs @ ys) =
  (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
by (induct xs) auto

lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
apply(induct xs)
 apply simp
apply simp
apply blast
done

lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
apply(induct xs)
 apply simp
apply simp
apply blast
done

lemma remove1_filter_not[simp]:
  "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
by(induct xs) auto

lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
apply(insert set_remove1_subset)
apply fast
done

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


subsubsection {* @{text replicate} *}

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

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

lemma replicate_app_Cons_same:
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
by (induct n) auto

lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
apply (induct n, simp)
apply (simp add: replicate_app_Cons_same)
done

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

text{* Courtesy of Matthias Daum: *}
lemma append_replicate_commute:
  "replicate n x @ replicate k x = replicate k x @ replicate n x"
apply (simp add: replicate_add [THEN sym])
apply (simp add: add_commute)
done

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

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

lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
by (atomize (full), induct n) auto

lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
apply (induct n, simp)
apply (simp add: nth_Cons split: nat.split)
done

text{* Courtesy of Matthias Daum (2 lemmas): *}
lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
apply (case_tac "k \<le> i")
 apply  (simp add: min_def)
apply (drule not_leE)
apply (simp add: min_def)
apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
 apply  simp
apply (simp add: replicate_add [symmetric])
done

lemma drop_replicate[simp]: "!!i. drop i (replicate k x) = replicate (k-i) x"
apply (induct k)
 apply simp
apply clarsimp
apply (case_tac i)
 apply simp
apply clarsimp
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)


subsubsection{*@{text rotate1} and @{text rotate}*}

lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
by(simp add:rotate1_def)

lemma rotate0[simp]: "rotate 0 = id"
by(simp add:rotate_def)

lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
by(simp add:rotate_def)

lemma rotate_add:
  "rotate (m+n) = rotate m o rotate n"
by(simp add:rotate_def funpow_add)

lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
by(simp add:rotate_add)

lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
by(simp add:rotate_def funpow_swap1)

lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
by(cases xs) simp_all

lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
apply(induct n)
 apply simp
apply (simp add:rotate_def)
done

lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
by(simp add:rotate1_def split:list.split)

lemma rotate_drop_take:
  "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
apply(induct n)
 apply simp
apply(simp add:rotate_def)
apply(cases "xs = []")
 apply (simp)
apply(case_tac "n mod length xs = 0")
 apply(simp add:mod_Suc)
 apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
                take_hd_drop linorder_not_le)
done

lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
by(simp add:rotate_drop_take)

lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
by(simp add:rotate_drop_take)

lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
by(simp add:rotate1_def split:list.split)

lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs"
by (induct n) (simp_all add:rotate_def)

lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
by(simp add:rotate1_def split:list.split) blast

lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
by (induct n) (simp_all add:rotate_def)

lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
by(simp add:rotate_drop_take take_map drop_map)

lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
by(simp add:rotate1_def split:list.split)

lemma set_rotate[simp]: "set(rotate n xs) = set xs"
by (induct n) (simp_all add:rotate_def)

lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
by(simp add:rotate1_def split:list.split)

lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
by (induct n) (simp_all add:rotate_def)

lemma rotate_rev:
  "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
apply(simp add:rotate_drop_take rev_drop rev_take)
apply(cases "length xs = 0")
 apply simp
apply(cases "n mod length xs = 0")
 apply simp
apply(simp add:rotate_drop_take rev_drop rev_take)
done

lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
apply(subgoal_tac "length xs \<noteq> 0")
 prefer 2 apply simp
using mod_less_divisor[of "length xs" n] by arith


subsubsection {* @{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 length_sublist:
  "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
by(simp add: sublist_def length_filter_conv_card cong:conj_cong)

lemma sublist_shift_lemma_Suc:
  "!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
         map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
apply(induct xs)
 apply simp
apply (case_tac "is")
 apply simp
apply simp
done

lemma sublist_shift_lemma:
     "map fst [p:zip xs [i..<i + length xs] . snd p : A] =
      map fst [p:zip xs [0..<length xs] . snd p + i : A]"
by (induct xs rule: rev_induct) (simp_all add: add_commute)

lemma sublist_append:
     "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
apply (unfold sublist_def)
apply (induct l' rule: rev_induct, simp)
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
apply (simp add: add_commute)
done

lemma sublist_Cons:
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
apply (induct l rule: rev_induct)
 apply (simp add: sublist_def)
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
done

lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
apply(induct xs)
 apply simp
apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE)
 apply(erule lessE)
  apply auto
apply(erule lessE)
apply auto
done

lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
by(auto simp add:set_sublist)

lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
by(auto simp add:set_sublist)

lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
by(auto simp add:set_sublist)

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


lemma distinct_sublistI[simp]: "!!I. distinct xs \<Longrightarrow> distinct(sublist xs I)"
apply(induct xs)
 apply simp
apply(auto simp add:sublist_Cons)
done


lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
apply (induct l rule: rev_induct, simp)
apply (simp split: nat_diff_split add: sublist_append)
done

lemma filter_in_sublist: "\<And>s. distinct xs \<Longrightarrow>
  filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
proof (induct xs)
  case Nil thus ?case by simp
next
  case (Cons a xs)
  moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
  ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
qed


subsubsection {* @{const splice} *}

lemma splice_Nil2 [simp, code]:
 "splice xs [] = xs"
by (cases xs) simp_all

lemma splice_Cons_Cons [simp, code]:
 "splice (x#xs) (y#ys) = x # y # splice xs ys"
by simp

declare splice.simps(2) [simp del, code del]

lemma length_splice[simp]: "!!ys. length(splice xs ys) = length xs + length ys"
apply(induct xs) apply simp
apply(case_tac ys)
 apply auto
done


subsubsection {* @{const allpairs} *}

lemma allpairs_append:
 "allpairs (xs @ ys) zs = allpairs xs zs @ allpairs ys zs"
by(induct xs) auto


subsubsection{*Sets of Lists*}

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

inductive2
  listsp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
  for A :: "'a \<Rightarrow> bool"
where
    Nil [intro!]: "listsp A []"
  | Cons [intro!]: "[| A a; listsp A l |] ==> listsp A (a # l)"

constdefs
  lists :: "'a set => 'a list set"
  "lists A == Collect (listsp (member A))"

lemma listsp_lists_eq [pred_set_conv]: "listsp (member A) = member (lists A)"
  by (simp add: lists_def)

lemmas lists_intros [intro!] = listsp.intros [to_set]

lemmas lists_induct [consumes 1, case_names Nil Cons, induct set: lists] =
  listsp.induct [to_set]

inductive_cases2 listspE [elim!]: "listsp A (x # l)"

lemmas listsE [elim!] = listspE [to_set]

lemma listsp_mono [mono2]: "A \<le> B ==> listsp A \<le> listsp B"
  by (clarify, erule listsp.induct, blast+)

lemmas lists_mono [mono] = listsp_mono [to_set]

lemma listsp_infI:
  assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
  by induct blast+

lemmas lists_IntI = listsp_infI [to_set]

lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
proof (rule mono_inf [where f=listsp, THEN order_antisym])
  show "mono listsp" by (simp add: mono_def listsp_mono)
  show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro: listsp_infI)
qed

lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_eq inf_bool_eq]

lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set]

lemma append_in_listsp_conv [iff]:
     "(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
by (induct xs) auto

lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]

lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
-- {* eliminate @{text listsp} in favour of @{text set} *}
by (induct xs) auto

lemmas in_lists_conv_set = in_listsp_conv_set [to_set]

lemma in_listspD [dest!]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
by (rule in_listsp_conv_set [THEN iffD1])

lemmas in_listsD [dest!] = in_listspD [to_set]

lemma in_listspI [intro!]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
by (rule in_listsp_conv_set [THEN iffD2])

lemmas in_listsI [intro!] = in_listspI [to_set]

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



subsubsection{* Inductive definition for membership *}

inductive2 ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
where
    elem:  "ListMem x (x # xs)"
  | insert:  "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"

lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
apply (rule iffI)
 apply (induct set: ListMem)
  apply auto
apply (induct xs)
 apply (auto intro: ListMem.intros)
done



subsubsection{*Lists as Cartesian products*}

text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
@{term A} and tail drawn from @{term Xs}.*}

constdefs
  set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
  "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"

lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
by (auto simp add: set_Cons_def)

text{*Yields the set of lists, all of the same length as the argument and
with elements drawn from the corresponding element of the argument.*}

consts  listset :: "'a set list \<Rightarrow> 'a list set"
primrec
   "listset []    = {[]}"
   "listset(A#As) = set_Cons A (listset As)"


subsection{*Relations on Lists*}

subsubsection {* Length Lexicographic Ordering *}

text{*These orderings preserve well-foundedness: shorter lists 
  precede longer lists. These ordering are not used in dictionaries.*}

consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
        --{*The lexicographic ordering for lists of the specified length*}
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"
        --{*Holds only between lists of the same length*}

  lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
    "lenlex r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
        --{*Compares lists by their length and then lexicographically*}


lemma wf_lexn: "wf r ==> wf (lexn r n)"
apply (induct n, simp, simp)
apply(rule wf_subset)
 prefer 2 apply (rule Int_lower1)
apply(rule wf_prod_fun_image)
 prefer 2 apply (rule inj_onI, auto)
done

lemma lexn_length:
     "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
by (induct n) auto

lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
apply (unfold lex_def)
apply (rule wf_UN)
apply (blast intro: wf_lexn, clarify)
apply (rename_tac m n)
apply (subgoal_tac "m \<noteq> n")
 prefer 2 apply blast
apply (blast dest: lexn_length not_sym)
done

lemma lexn_conv:
  "lexn r n =
    {(xs,ys). length xs = n \<and> length ys = n \<and>
    (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
apply (induct n, simp)
apply (simp add: image_Collect lex_prod_def, safe, blast)
 apply (rule_tac x = "ab # xys" in exI, simp)
apply (case_tac xys, simp_all, blast)
done

lemma lex_conv:
  "lex r =
    {(xs,ys). length xs = length ys \<and>
    (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
by (force simp add: lex_def lexn_conv)

lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
by (unfold lenlex_def) blast

lemma lenlex_conv:
    "lenlex r = {(xs,ys). length xs < length ys |
                 length xs = length ys \<and> (xs, ys) : lex r}"
by (simp add: lenlex_def diag_def lex_prod_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 [simp]:
    "((x # xs, y # ys) : lex r) =
      ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
apply (simp add: lex_conv)
apply (rule iffI)
 prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
apply (case_tac xys, simp, simp)
apply blast
done


subsubsection {* Lexicographic Ordering *}

text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
    This ordering does \emph{not} preserve well-foundedness.
     Author: N. Voelker, March 2005. *} 

constdefs 
  lexord :: "('a * 'a)set \<Rightarrow> ('a list * 'a list) set" 
  "lexord  r == {(x,y). \<exists> a v. y = x @ a # v \<or> 
            (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"

lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
  by (unfold lexord_def, induct_tac y, auto) 

lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
  by (unfold lexord_def, induct_tac x, auto)

lemma lexord_cons_cons[simp]:
     "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
  apply (unfold lexord_def, safe, simp_all)
  apply (case_tac u, simp, simp)
  apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
  apply (erule_tac x="b # u" in allE)
  by force

lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons

lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
  by (induct_tac x, auto)  

lemma lexord_append_left_rightI:
     "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
  by (induct_tac u, auto)

lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
  by (induct x, auto)

lemma lexord_append_leftD:
     "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
  by (erule rev_mp, induct_tac x, auto)

lemma lexord_take_index_conv: 
   "((x,y) : lexord r) = 
    ((length x < length y \<and> take (length x) y = x) \<or> 
     (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
  apply (unfold lexord_def Let_def, clarsimp) 
  apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
  apply auto 
  apply (rule_tac x="hd (drop (length x) y)" in exI)
  apply (rule_tac x="tl (drop (length x) y)" in exI)
  apply (erule subst, simp add: min_def) 
  apply (rule_tac x ="length u" in exI, simp) 
  apply (rule_tac x ="take i x" in exI) 
  apply (rule_tac x ="x ! i" in exI) 
  apply (rule_tac x ="y ! i" in exI, safe) 
  apply (rule_tac x="drop (Suc i) x" in exI)
  apply (drule sym, simp add: drop_Suc_conv_tl) 
  apply (rule_tac x="drop (Suc i) y" in exI)
  by (simp add: drop_Suc_conv_tl) 

-- {* lexord is extension of partial ordering List.lex *} 
lemma lexord_lex: " (x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
  apply (rule_tac x = y in spec) 
  apply (induct_tac x, clarsimp) 
  by (clarify, case_tac x, simp, force)

lemma lexord_irreflexive: "(! x. (x,x) \<notin> r) \<Longrightarrow> (y,y) \<notin> lexord r"
  by (induct y, auto)

lemma lexord_trans: 
    "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
   apply (erule rev_mp)+
   apply (rule_tac x = x in spec) 
  apply (rule_tac x = z in spec) 
  apply ( induct_tac y, simp, clarify)
  apply (case_tac xa, erule ssubst) 
  apply (erule allE, erule allE) -- {* avoid simp recursion *} 
  apply (case_tac x, simp, simp) 
  apply (case_tac x, erule allE, erule allE, simp) 
  apply (erule_tac x = listb in allE) 
  apply (erule_tac x = lista in allE, simp)
  apply (unfold trans_def)
  by blast

lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
  by (rule transI, drule lexord_trans, blast) 

lemma lexord_linear: "(! a b. (a,b)\<in> r | a = b | (b,a) \<in> r) \<Longrightarrow> (x,y) : lexord r | x = y | (y,x) : lexord r"
  apply (rule_tac x = y in spec) 
  apply (induct_tac x, rule allI) 
  apply (case_tac x, simp, simp) 
  apply (rule allI, case_tac x, simp, simp) 
  by blast


subsection {* Lexicographic combination of measure functions *}

text {* These are useful for termination proofs *}

definition
  "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"

lemma wf_measures[recdef_wf, simp]: "wf (measures fs)"
  unfolding measures_def
  by blast

lemma in_measures[simp]: 
  "(x, y) \<in> measures [] = False"
  "(x, y) \<in> measures (f # fs)
         = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"  
  unfolding measures_def
  by auto

lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
  by simp

lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
  by auto

(* install the lexicographic_order method and the "fun" command *)
use "Tools/function_package/lexicographic_order.ML"
use "Tools/function_package/fundef_datatype.ML"
setup LexicographicOrder.setup
setup FundefDatatype.setup


subsubsection{*Lifting a Relation on List Elements to the Lists*}

inductive2
  list_all2' :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool"
  for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
where
    Nil:  "list_all2' r [] []"
  | Cons: "[| r x y; list_all2' r xs ys |] ==> list_all2' r (x#xs) (y#ys)"

constdefs
  listrel :: "('a * 'b) set => ('a list * 'b list) set"
  "listrel r == Collect2 (list_all2' (member2 r))"

lemma list_all2_listrel_eq [pred_set_conv]:
  "list_all2' (member2 r) = member2 (listrel r)"
  by (simp add: listrel_def)

lemmas listrel_induct [consumes 1, case_names Nil Cons, induct set: listrel] =
  list_all2'.induct [to_set]

lemmas listrel_intros = list_all2'.intros [to_set]

inductive_cases2 listrel_Nil1 [to_set, elim!]: "list_all2' r [] xs"
inductive_cases2 listrel_Nil2 [to_set, elim!]: "list_all2' r xs []"
inductive_cases2 listrel_Cons1 [to_set, elim!]: "list_all2' r  (y#ys) xs"
inductive_cases2 listrel_Cons2 [to_set, elim!]: "list_all2' r xs (y#ys)"


lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
apply clarify  
apply (erule listrel_induct)
apply (blast intro: listrel_intros)+
done

lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
apply clarify 
apply (erule listrel_induct, auto) 
done

lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" 
apply (simp add: refl_def listrel_subset Ball_def)
apply (rule allI) 
apply (induct_tac x) 
apply (auto intro: listrel_intros)
done

lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
apply (auto simp add: sym_def)
apply (erule listrel_induct) 
apply (blast intro: listrel_intros)+
done

lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
apply (simp add: trans_def)
apply (intro allI) 
apply (rule impI) 
apply (erule listrel_induct) 
apply (blast intro: listrel_intros)+
done

theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) 

lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
by (blast intro: listrel_intros)

lemma listrel_Cons:
     "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})";
by (auto simp add: set_Cons_def intro: listrel_intros) 


subsection{*Miscellany*}

subsubsection {* Characters and strings *}

datatype nibble =
    Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
  | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF

datatype char = Char nibble nibble
  -- "Note: canonical order of character encoding coincides with standard term ordering"

types string = "char list"

syntax
  "_Char" :: "xstr => char"    ("CHR _")
  "_String" :: "xstr => string"    ("_")

setup StringSyntax.setup


subsection {* Code generator *}

subsubsection {* Setup *}

types_code
  "list" ("_ list")
attach (term_of) {*
fun term_of_list f T = HOLogic.mk_list T o map f;
*}
attach (test) {*
fun gen_list' aG i j = frequency
  [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] ()
and gen_list aG i = gen_list' aG i i;
*}
  "char" ("string")
attach (term_of) {*
val term_of_char = HOLogic.mk_char;
*}
attach (test) {*
fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z")));
*}

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

code_type list
  (SML "_ list")
  (OCaml "_ list")
  (Haskell "![_]")

code_reserved SML
  list

code_reserved OCaml
  list

code_const Nil
  (SML "[]")
  (OCaml "[]")
  (Haskell "[]")

setup {*
  fold (fn target => CodegenSerializer.add_pretty_list target
    @{const_name Nil} @{const_name Cons}
  ) ["SML", "OCaml", "Haskell"]
*}

code_instance list :: eq
  (Haskell -)

code_const "op = \<Colon> 'a\<Colon>eq list \<Rightarrow> 'a list \<Rightarrow> bool"
  (Haskell infixl 4 "==")

setup {*
let

fun list_codegen thy defs gr dep thyname b t =
  let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy defs dep thyname false)
    (gr, HOLogic.dest_list t)
  in SOME (gr', Pretty.list "[" "]" ps) end handle TERM _ => NONE;

fun char_codegen thy defs gr dep thyname b t =
  case Option.map chr (try HOLogic.dest_char t) of
      SOME c => SOME (gr, Pretty.quote (Pretty.str (ML_Syntax.print_char c)))
    | NONE => NONE;

in
  Codegen.add_codegen "list_codegen" list_codegen
  #> Codegen.add_codegen "char_codegen" char_codegen
end;
*}


subsubsection {* Generation of efficient code *}

consts
  memberl :: "'a \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "mem" 55)
  null:: "'a list \<Rightarrow> bool"
  list_inter :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
  list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
  list_all :: "('a \<Rightarrow> bool) \<Rightarrow> ('a list \<Rightarrow> bool)"
  itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
  filtermap :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list"
  map_filter :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list"

primrec
  "x mem [] = False"
  "x mem (y#ys) = (x = y \<or> x mem ys)"

primrec
  "null [] = True"
  "null (x#xs) = False"

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)"

text {*
  Only use @{text mem} for generating executable code.  Otherwise use
  @{prop "x : set xs"} instead --- it is much easier to reason about.
  The same is true for @{const list_all} and @{const list_ex}: write
  @{text "\<forall>x\<in>set xs"} and @{text "\<exists>x\<in>set xs"} instead because the HOL
  quantifiers are aleady known to the automatic provers. In fact, the
  declarations in the code subsection make sure that @{text "\<in>"},
  @{text "\<forall>x\<in>set xs"} and @{text "\<exists>x\<in>set xs"} are implemented
  efficiently.

  Efficient emptyness check is implemented by @{const null}.

  The functions @{const itrev}, @{const filtermap} and @{const
  map_filter} are just there to generate efficient code. Do not use
  them for modelling and proving.
*}

lemma mem_iff [normal post]:
  "x mem xs \<longleftrightarrow> x \<in> set xs"
  by (induct xs) auto

lemmas in_set_code [code unfold] = mem_iff [symmetric]

lemma empty_null [code inline]:
  "xs = [] \<longleftrightarrow> null xs"
  by (cases xs) simp_all

lemmas null_empty [normal post] =
  empty_null [symmetric]

lemma list_inter_conv:
  "set (list_inter xs ys) = set xs \<inter> set ys"
  by (induct xs) auto

lemma list_all_iff [normal post]:
  "list_all P xs \<longleftrightarrow> (\<forall>x \<in> set xs. P x)"
  by (induct xs) auto

lemmas list_ball_code [code unfold] = list_all_iff [symmetric]

lemma list_all_append [simp]:
  "list_all P (xs @ ys) \<longleftrightarrow> (list_all P xs \<and> list_all P ys)"
  by (induct xs) auto

lemma list_all_rev [simp]:
  "list_all P (rev xs) \<longleftrightarrow> list_all P xs"
  by (simp add: list_all_iff)

lemma list_all_length:
  "list_all P xs \<longleftrightarrow> (\<forall>n < length xs. P (xs ! n))"
  unfolding list_all_iff by (auto intro: all_nth_imp_all_set)

lemma list_ex_iff [normal post]:
  "list_ex P xs \<longleftrightarrow> (\<exists>x \<in> set xs. P x)"
  by (induct xs) simp_all

lemmas list_bex_code [code unfold] =
  list_ex_iff [symmetric]

lemma list_ex_length:
  "list_ex P xs \<longleftrightarrow> (\<exists>n < length xs. P (xs ! n))"
  unfolding list_ex_iff set_conv_nth by auto

lemma itrev [simp]:
  "itrev xs ys = rev xs @ ys"
  by (induct xs arbitrary: ys) simp_all

lemma filtermap_conv:
   "filtermap f xs = map (\<lambda>x. the (f x)) (filter (\<lambda>x. f x \<noteq> None) xs)"
  by (induct xs) (simp_all split: option.split) 

lemma map_filter_conv [simp]:
  "map_filter f P xs = map f (filter P xs)"
  by (induct xs) auto

lemma rev_code [code func]:
  "rev xs = itrev xs []"
  by simp


text {* code for bounded quantification over nats *}

lemma atMost_upto [code unfold]:
  "{..n} = set [0..n]"
  by auto

lemma atLeast_upt [code unfold]:
  "{..<n} = set [0..<n]"
  by auto

lemma greaterThanLessThan_upd [code unfold]:
  "{n<..<m} = set [Suc n..<m]"
  by auto

lemma atLeastLessThan_upd [code unfold]:
  "{n..<m} = set [n..<m]"
  by auto

lemma greaterThanAtMost_upto [code unfold]:
  "{n<..m} = set [Suc n..m]"
  by auto

lemma atLeastAtMost_upto [code unfold]:
  "{n..m} = set [n..m]"
  by auto

lemma all_nat_less_eq [code unfold]:
  "(\<forall>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..<n}. P m)"
  by auto

lemma ex_nat_less_eq [code unfold]:
  "(\<exists>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..<n}. P m)"
  by auto

lemma all_nat_less [code unfold]:
  "(\<forall>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..n}. P m)"
  by auto

lemma ex_nat_less [code unfold]:
  "(\<exists>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..n}. P m)"
  by auto

end