src/HOL/List.thy

Add filter_remove1

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

header {* The datatype of finite lists *}

theory List
imports Plain Quotient Presburger Code_Numeral Sledgehammer Recdef
uses ("Tools/list_code.ML")
begin

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

syntax
  -- {* list Enumeration *}
  "_list" :: "args => 'a list"    ("[(_)]")

translations
  "[x, xs]" == "x#[xs]"
  "[x]" == "x#[]"


subsection {* Basic list processing functions *}

primrec
  hd :: "'a list \<Rightarrow> 'a" where
  "hd (x # xs) = x"

primrec
  tl :: "'a list \<Rightarrow> 'a list" where
    "tl [] = []"
  | "tl (x # xs) = xs"

primrec
  last :: "'a list \<Rightarrow> 'a" where
  "last (x # xs) = (if xs = [] then x else last xs)"

primrec
  butlast :: "'a list \<Rightarrow> 'a list" where
    "butlast []= []"
  | "butlast (x # xs) = (if xs = [] then [] else x # butlast xs)"

primrec
  set :: "'a list \<Rightarrow> 'a set" where
    "set [] = {}"
  | "set (x # xs) = insert x (set xs)"

primrec
  map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
    "map f [] = []"
  | "map f (x # xs) = f x # map f xs"

primrec
  append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65) where
    append_Nil:"[] @ ys = ys"
  | append_Cons: "(x#xs) @ ys = x # xs @ ys"

primrec
  rev :: "'a list \<Rightarrow> 'a list" where
    "rev [] = []"
  | "rev (x # xs) = rev xs @ [x]"

primrec
  filter:: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
    "filter P [] = []"
  | "filter P (x # xs) = (if P x then x # filter P xs else filter P xs)"

syntax
  -- {* Special syntax for filter *}
  "_filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_<-_./ _])")

translations
  "[x<-xs . P]"== "CONST filter (%x. P) xs"

syntax (xsymbols)
  "_filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
syntax (HTML output)
  "_filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")

primrec
  foldl :: "('b \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b" where
    foldl_Nil: "foldl f a [] = a"
  | foldl_Cons: "foldl f a (x # xs) = foldl f (f a x) xs"

primrec
  foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where
    "foldr f [] a = a"
  | "foldr f (x # xs) a = f x (foldr f xs a)"

primrec
  concat:: "'a list list \<Rightarrow> 'a list" where
    "concat [] = []"
  | "concat (x # xs) = x @ concat xs"

primrec (in monoid_add)
  listsum :: "'a list \<Rightarrow> 'a" where
    "listsum [] = 0"
  | "listsum (x # xs) = x + listsum xs"

primrec
  drop:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
    drop_Nil: "drop n [] = []"
  | drop_Cons: "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
  -- {*Warning: simpset does not contain this definition, but separate
       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}

primrec
  take:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
    take_Nil:"take n [] = []"
  | take_Cons: "take n (x # xs) = (case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # take m xs)"
  -- {*Warning: simpset does not contain this definition, but separate
       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}

primrec
  nth :: "'a list => nat => 'a" (infixl "!" 100) where
  nth_Cons: "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
  -- {*Warning: simpset does not contain this definition, but separate
       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}

primrec
  list_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
    "list_update [] i v = []"
  | "list_update (x # xs) i v = (case i of 0 \<Rightarrow> v # xs | Suc j \<Rightarrow> x # list_update xs j v)"

nonterminals lupdbinds lupdbind

syntax
  "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
  "" :: "lupdbind => lupdbinds"    ("_")
  "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
  "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)

translations
  "_LUpdate xs (_lupdbinds b bs)" == "_LUpdate (_LUpdate xs b) bs"
  "xs[i:=x]" == "CONST list_update xs i x"

primrec
  takeWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
    "takeWhile P [] = []"
  | "takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])"

primrec
  dropWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
    "dropWhile P [] = []"
  | "dropWhile P (x # xs) = (if P x then dropWhile P xs else x # xs)"

primrec
  zip :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
    "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 :: "nat \<Rightarrow> nat \<Rightarrow> nat list" ("(1[_..</_'])") where
    upt_0: "[i..<0] = []"
  | upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"

primrec
  distinct :: "'a list \<Rightarrow> bool" where
    "distinct [] \<longleftrightarrow> True"
  | "distinct (x # xs) \<longleftrightarrow> x \<notin> set xs \<and> distinct xs"

primrec
  remdups :: "'a list \<Rightarrow> 'a list" where
    "remdups [] = []"
  | "remdups (x # xs) = (if x \<in> set xs then remdups xs else x # remdups xs)"

definition
  insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
  "insert x xs = (if x \<in> set xs then xs else x # xs)"

hide_const (open) insert
hide_fact (open) insert_def

primrec
  remove1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
    "remove1 x [] = []"
  | "remove1 x (y # xs) = (if x = y then xs else y # remove1 x xs)"

primrec
  removeAll :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
    "removeAll x [] = []"
  | "removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)"

primrec
  replicate :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
    replicate_0: "replicate 0 x = []"
  | replicate_Suc: "replicate (Suc n) x = x # replicate n x"

text {*
  Function @{text size} is overloaded for all datatypes. Users may
  refer to the list version as @{text length}. *}

abbreviation
  length :: "'a list \<Rightarrow> nat" where
  "length \<equiv> size"

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 :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
    "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. *}

text{*
\begin{figure}[htbp]
\fbox{
\begin{tabular}{l}
@{lemma "[a,b]@[c,d] = [a,b,c,d]" by simp}\\
@{lemma "length [a,b,c] = 3" by simp}\\
@{lemma "set [a,b,c] = {a,b,c}" by simp}\\
@{lemma "map f [a,b,c] = [f a, f b, f c]" by simp}\\
@{lemma "rev [a,b,c] = [c,b,a]" by simp}\\
@{lemma "hd [a,b,c,d] = a" by simp}\\
@{lemma "tl [a,b,c,d] = [b,c,d]" by simp}\\
@{lemma "last [a,b,c,d] = d" by simp}\\
@{lemma "butlast [a,b,c,d] = [a,b,c]" by simp}\\
@{lemma[source] "filter (\<lambda>n::nat. n<2) [0,2,1] = [0,1]" by simp}\\
@{lemma "concat [[a,b],[c,d,e],[],[f]] = [a,b,c,d,e,f]" by simp}\\
@{lemma "foldl f x [a,b,c] = f (f (f x a) b) c" by simp}\\
@{lemma "foldr f [a,b,c] x = f a (f b (f c x))" by simp}\\
@{lemma "zip [a,b,c] [x,y,z] = [(a,x),(b,y),(c,z)]" by simp}\\
@{lemma "zip [a,b] [x,y,z] = [(a,x),(b,y)]" by simp}\\
@{lemma "splice [a,b,c] [x,y,z] = [a,x,b,y,c,z]" by simp}\\
@{lemma "splice [a,b,c,d] [x,y] = [a,x,b,y,c,d]" by simp}\\
@{lemma "take 2 [a,b,c,d] = [a,b]" by simp}\\
@{lemma "take 6 [a,b,c,d] = [a,b,c,d]" by simp}\\
@{lemma "drop 2 [a,b,c,d] = [c,d]" by simp}\\
@{lemma "drop 6 [a,b,c,d] = []" by simp}\\
@{lemma "takeWhile (%n::nat. n<3) [1,2,3,0] = [1,2]" by simp}\\
@{lemma "dropWhile (%n::nat. n<3) [1,2,3,0] = [3,0]" by simp}\\
@{lemma "distinct [2,0,1::nat]" by simp}\\
@{lemma "remdups [2,0,2,1::nat,2] = [0,1,2]" by simp}\\
@{lemma "List.insert 2 [0::nat,1,2] = [0,1,2]" by (simp add: List.insert_def)}\\
@{lemma "List.insert 3 [0::nat,1,2] = [3,0,1,2]" by (simp add: List.insert_def)}\\
@{lemma "remove1 2 [2,0,2,1::nat,2] = [0,2,1,2]" by simp}\\
@{lemma "removeAll 2 [2,0,2,1::nat,2] = [0,1]" by simp}\\
@{lemma "nth [a,b,c,d] 2 = c" by simp}\\
@{lemma "[a,b,c,d][2 := x] = [a,b,x,d]" by simp}\\
@{lemma "sublist [a,b,c,d,e] {0,2,3} = [a,c,d]" by (simp add:sublist_def)}\\
@{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by (simp add:rotate1_def)}\\
@{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate1_def rotate_def nat_number')}\\
@{lemma "replicate 4 a = [a,a,a,a]" by (simp add:nat_number')}\\
@{lemma "[2..<5] = [2,3,4]" by (simp add:nat_number')}\\
@{lemma "listsum [1,2,3::nat] = 6" by simp}
\end{tabular}}
\caption{Characteristic examples}
\label{fig:Characteristic}
\end{figure}
Figure~\ref{fig:Characteristic} shows characteristic examples
that should give an intuitive understanding of the above functions.
*}

text{* The following simple sort functions are intended for proofs,
not for efficient implementations. *}

context linorder
begin

fun sorted :: "'a list \<Rightarrow> bool" where
"sorted [] \<longleftrightarrow> True" |
"sorted [x] \<longleftrightarrow> True" |
"sorted (x#y#zs) \<longleftrightarrow> x <= y \<and> sorted (y#zs)"

primrec insort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
"insort_key f x [] = [x]" |
"insort_key f x (y#ys) = (if f x \<le> f y then (x#y#ys) else y#(insort_key f x ys))"

definition sort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b list \<Rightarrow> 'b list" where
"sort_key f xs = foldr (insort_key f) xs []"

abbreviation "sort \<equiv> sort_key (\<lambda>x. x)"
abbreviation "insort \<equiv> insort_key (\<lambda>x. x)"

definition insort_insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
  "insort_insert x xs = (if x \<in> set xs then xs else insort x xs)"

end


subsubsection {* List comprehension *}

text{* Input syntax for Haskell-like list comprehension notation.
Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
the list of all pairs of distinct elements from @{text xs} and @{text ys}.
The syntax is as in Haskell, except that @{text"|"} becomes a dot
(like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
\verb![e| x <- xs, ...]!.

The qualifiers after the dot are
\begin{description}
\item[generators] @{text"p \<leftarrow> xs"},
 where @{text p} is a pattern and @{text xs} an expression of list type, or
\item[guards] @{text"b"}, where @{text b} is a boolean expression.
%\item[local bindings] @ {text"let x = e"}.
\end{description}

Just like in Haskell, list comprehension is just a shorthand. To avoid
misunderstandings, the translation into desugared form is not reversed
upon output. Note that the translation of @{text"[e. x \<leftarrow> xs]"} is
optmized to @{term"map (%x. e) xs"}.

It is easy to write short list comprehensions which stand for complex
expressions. During proofs, they may become unreadable (and
mangled). In such cases it can be advisable to introduce separate
definitions for the list comprehensions in question.  *}

(*
Proper theorem proving support would be nice. For example, if
@{text"set[f x y. x \<leftarrow> xs, y \<leftarrow> ys, P x y]"}
produced something like
@{term"{z. EX x: set xs. EX y:set ys. P x y \<and> z = f x y}"}.
*)

nonterminals lc_qual lc_quals

syntax
"_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ <- _")
"_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
(*"_lc_let" :: "letbinds => lc_qual"  ("let _")*)
"_lc_end" :: "lc_quals" ("]")
"_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals" (", __")
"_lc_abs" :: "'a => 'b list => 'b list"

(* These are easier than ML code but cannot express the optimized
   translation of [e. p<-xs]
translations
"[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
"_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
 => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
"[e. P]" => "if P then [e] else []"
"_listcompr e (_lc_test P) (_lc_quals Q Qs)"
 => "if P then (_listcompr e Q Qs) else []"
"_listcompr e (_lc_let b) (_lc_quals Q Qs)"
 => "_Let b (_listcompr e Q Qs)"
*)

syntax (xsymbols)
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
syntax (HTML output)
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")

parse_translation (advanced) {*
let
  val NilC = Syntax.const @{const_syntax Nil};
  val ConsC = Syntax.const @{const_syntax Cons};
  val mapC = Syntax.const @{const_syntax map};
  val concatC = Syntax.const @{const_syntax concat};
  val IfC = Syntax.const @{const_syntax If};

  fun singl x = ConsC $ x $ NilC;

  fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
    let
      val x = Free (Name.variant (fold Term.add_free_names [p, e] []) "x", dummyT);
      val e = if opti then singl e else e;
      val case1 = Syntax.const @{syntax_const "_case1"} $ p $ e;
      val case2 =
        Syntax.const @{syntax_const "_case1"} $
          Syntax.const @{const_syntax dummy_pattern} $ NilC;
      val cs = Syntax.const @{syntax_const "_case2"} $ case1 $ case2;
      val ft = Datatype_Case.case_tr false Datatype.info_of_constr ctxt [x, cs];
    in lambda x ft end;

  fun abs_tr ctxt (p as Free (s, T)) e opti =
        let
          val thy = ProofContext.theory_of ctxt;
          val s' = Sign.intern_const thy s;
        in
          if Sign.declared_const thy s'
          then (pat_tr ctxt p e opti, false)
          else (lambda p e, true)
        end
    | abs_tr ctxt p e opti = (pat_tr ctxt p e opti, false);

  fun lc_tr ctxt [e, Const (@{syntax_const "_lc_test"}, _) $ b, qs] =
        let
          val res =
            (case qs of
              Const (@{syntax_const "_lc_end"}, _) => singl e
            | Const (@{syntax_const "_lc_quals"}, _) $ q $ qs => lc_tr ctxt [e, q, qs]);
        in IfC $ b $ res $ NilC end
    | lc_tr ctxt
          [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
            Const(@{syntax_const "_lc_end"}, _)] =
        (case abs_tr ctxt p e true of
          (f, true) => mapC $ f $ es
        | (f, false) => concatC $ (mapC $ f $ es))
    | lc_tr ctxt
          [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
            Const (@{syntax_const "_lc_quals"}, _) $ q $ qs] =
        let val e' = lc_tr ctxt [e, q, qs];
        in concatC $ (mapC $ (fst (abs_tr ctxt p e' false)) $ es) end;

in [(@{syntax_const "_listcompr"}, lc_tr)] end
*}

term "[(x,y,z). b]"
term "[(x,y,z). x\<leftarrow>xs]"
term "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]"
term "[(x,y,z). x<a, x>b]"
term "[(x,y,z). x\<leftarrow>xs, x>b]"
term "[(x,y,z). x<a, x\<leftarrow>xs]"
term "[(x,y). Cons True x \<leftarrow> xs]"
term "[(x,y,z). Cons x [] \<leftarrow> xs]"
term "[(x,y,z). x<a, x>b, x=d]"
term "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
term "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
term "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
term "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
term "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
(*
term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
*)


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

lemma list_nonempty_induct [consumes 1, case_names single cons]:
  assumes "xs \<noteq> []"
  assumes single: "\<And>x. P [x]"
  assumes cons: "\<And>x xs. xs \<noteq> [] \<Longrightarrow> P xs \<Longrightarrow> P (x # xs)"
  shows "P xs"
using `xs \<noteq> []` proof (induct xs)
  case Nil then show ?case by simp
next
  case (Cons x xs) show ?case proof (cases xs)
    case Nil with single show ?thesis by simp
  next
    case Cons then have "xs \<noteq> []" by simp
    moreover with Cons.hyps have "P xs" .
    ultimately show ?thesis by (rule cons)
  qed
qed


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_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_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
by 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: "length xs <= length ys ==> xs = x # ys = False"
  by (induct xs) auto

lemma list_induct2 [consumes 1, case_names Nil Cons]:
  "length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
   (\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
   \<Longrightarrow> P xs ys"
proof (induct xs arbitrary: ys)
  case Nil then show ?case by simp
next
  case (Cons x xs ys) then show ?case by (cases ys) simp_all
qed

lemma list_induct3 [consumes 2, case_names Nil Cons]:
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
   (\<And>x xs y ys z zs. length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P xs ys zs \<Longrightarrow> P (x#xs) (y#ys) (z#zs))
   \<Longrightarrow> P xs ys zs"
proof (induct xs arbitrary: ys zs)
  case Nil then show ?case by simp
next
  case (Cons x xs ys zs) then show ?case by (cases ys, simp_all)
    (cases zs, simp_all)
qed

lemma list_induct4 [consumes 3, case_names Nil Cons]:
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow>
   P [] [] [] [] \<Longrightarrow> (\<And>x xs y ys z zs w ws. length xs = length ys \<Longrightarrow>
   length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow> P xs ys zs ws \<Longrightarrow>
   P (x#xs) (y#ys) (z#zs) (w#ws)) \<Longrightarrow> P xs ys zs ws"
proof (induct xs arbitrary: ys zs ws)
  case Nil then show ?case by simp
next
  case (Cons x xs ys zs ws) then show ?case by ((cases ys, simp_all), (cases zs,simp_all)) (cases ws, simp_all)
qed

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"
by (rule Eq_FalseI) auto

simproc_setup list_neq ("(xs::'a list) = ys") = {*
(*
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.
*)

let

fun len (Const(@{const_name Nil},_)) acc = acc
  | len (Const(@{const_name Cons},_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
  | len (Const(@{const_name append},_) $ xs $ ys) acc = len xs (len ys acc)
  | len (Const(@{const_name rev},_) $ xs) acc = len xs acc
  | len (Const(@{const_name map},_) $ _ $ xs) acc = len xs acc
  | len t (ts,n) = (t::ts,n);

fun list_neq _ ss ct =
  let
    val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
    fun prove_neq() =
      let
        val Type(_,listT::_) = eqT;
        val size = HOLogic.size_const listT;
        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 submultiset (op aconv) (ls,rs) orelse
       n < m andalso submultiset (op aconv) (rs,ls)
    then prove_neq() else NONE
  end;
in list_neq end;
*}


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, no_atp]:
 "length xs = length ys \<or> length us = length vs
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
apply (induct xs arbitrary: ys)
 apply (case_tac ys, simp, force)
apply (case_tac ys, force, simp)
done

lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
  (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
apply (induct xs arbitrary: ys zs ts)
 apply fastsimp
apply(case_tac zs)
 apply simp
apply fastsimp
done

lemma same_append_eq [iff, induct_simp]: "(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, induct_simp]: "(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,no_atp]: "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 {*
local

fun last (cons as Const(@{const_name Cons},_) $ _ $ xs) =
  (case xs of Const(@{const_name Nil},_) => cons | _ => last xs)
  | last (Const(@{const_name append},_) $ _ $ ys) = last ys
  | last t = t;

fun list1 (Const(@{const_name Cons},_) $ _ $ Const(@{const_name Nil},_)) = true
  | list1 _ = false;

fun butlast ((cons as Const(@{const_name Cons},_) $ x) $ xs) =
  (case xs of Const(@{const_name Nil},_) => xs | _ => cons $ butlast xs)
  | butlast ((app as Const(@{const_name append},_) $ xs) $ ys) = app $ butlast ys
  | butlast xs = Const(@{const_name 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(@{const_name append},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_global @{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_map [simp]: "map f (map g xs) = map (f \<circ> g) xs"
by (induct xs) auto

lemma map_comp_map[simp]: "((map f) o (map g)) = map(f o g)"
apply(rule ext)
apply(simp)
done

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:
  assumes "map f xs = map g ys"
  shows "length xs = length ys"
using assms proof (induct ys arbitrary: xs)
  case Nil then show ?case by simp
next
  case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto
  from Cons xs have "map f zs = map g ys" by simp
  moreover with Cons have "length zs = length ys" by blast
  with xs show ?case by simp
qed
  
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:
 "map f xs = map f ys ==> inj f ==> xs = ys"
by (induct ys arbitrary: xs) (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

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

lemma set_upt [simp]: "set[i..<j] = {i..<j}"
by (induct j) (simp_all add: atLeastLessThanSuc)


lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs"
proof (induct xs)
  case Nil thus ?case by simp
next
  case Cons thus ?case by (auto intro: Cons_eq_appendI)
qed

lemma in_set_conv_decomp: "x \<in> set xs \<longleftrightarrow> (\<exists>ys zs. xs = ys @ x # zs)"
  by (auto elim: split_list)

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

lemma in_set_conv_decomp_first:
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
  by (auto dest!: split_list_first)

lemma split_list_last: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
proof (induct xs rule:rev_induct)
  case Nil thus ?case by simp
next
  case (snoc a xs)
  show ?case
  proof cases
    assume "x = a" thus ?case using snoc by simp (metis ex_in_conv set_empty2)
  next
    assume "x \<noteq> a" thus ?case using snoc by fastsimp
  qed
qed

lemma in_set_conv_decomp_last:
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
  by (auto dest!: split_list_last)

lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs & P x"
proof (induct xs)
  case Nil thus ?case by simp
next
  case Cons thus ?case
    by(simp add:Bex_def)(metis append_Cons append.simps(1))
qed

lemma split_list_propE:
  assumes "\<exists>x \<in> set xs. P x"
  obtains ys x zs where "xs = ys @ x # zs" and "P x"
using split_list_prop [OF assms] by blast

lemma split_list_first_prop:
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)"
proof (induct xs)
  case Nil thus ?case by simp
next
  case (Cons x xs)
  show ?case
  proof cases
    assume "P x"
    thus ?thesis by simp
      (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
  next
    assume "\<not> P x"
    hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
    thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
  qed
qed

lemma split_list_first_propE:
  assumes "\<exists>x \<in> set xs. P x"
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y"
using split_list_first_prop [OF assms] by blast

lemma split_list_first_prop_iff:
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))"
by (rule, erule split_list_first_prop) auto

lemma split_list_last_prop:
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)"
proof(induct xs rule:rev_induct)
  case Nil thus ?case by simp
next
  case (snoc x xs)
  show ?case
  proof cases
    assume "P x" thus ?thesis by (metis emptyE set_empty)
  next
    assume "\<not> P x"
    hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
    thus ?thesis using `\<not> P x` snoc(1) by fastsimp
  qed
qed

lemma split_list_last_propE:
  assumes "\<exists>x \<in> set xs. P x"
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z"
using split_list_last_prop [OF assms] by blast

lemma split_list_last_prop_iff:
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
by (metis split_list_last_prop [where P=P] in_set_conv_decomp)

lemma finite_list: "finite A ==> EX xs. set xs = A"
  by (erule finite_induct)
    (auto simp add: set.simps(2) [symmetric] simp del: set.simps(2))

lemma card_length: "card (set xs) \<le> length xs"
by (induct xs) (auto simp add: card_insert_if)

lemma set_minus_filter_out:
  "set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)"
  by (induct xs) auto


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: image_def split:nat.split dest:gr0_implies_Suc)
    have "length (filter p (x # xs)) = Suc(card ?S)"
      using Cons `p x` 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: image_def split:nat.split elim:lessE)
    have "length (filter p (x # xs)) = card ?S"
      using Cons `\<not> p x` 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 "x = y"
      with Py Cons.prems have "?Q []" by simp
      then show ?thesis ..
    next
      assume "x \<noteq> y"
      with Py Cons.prems show ?thesis by simp
    qed
  next
    assume "\<not> P y"
    with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastsimp
    then have "?Q (y#us)" by simp
    then show ?thesis ..
  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 {* List partitioning *}

primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where
  "partition P [] = ([], [])"
  | "partition P (x # xs) = 
      (let (yes, no) = partition P xs
      in if P x then (x # yes, no) else (yes, x # no))"

lemma partition_filter1:
    "fst (partition P xs) = filter P xs"
by (induct xs) (auto simp add: Let_def split_def)

lemma partition_filter2:
    "snd (partition P xs) = filter (Not o P) xs"
by (induct xs) (auto simp add: Let_def split_def)

lemma partition_P:
  assumes "partition P xs = (yes, no)"
  shows "(\<forall>p \<in> set yes.  P p) \<and> (\<forall>p  \<in> set no. \<not> P p)"
proof -
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
    by simp_all
  then show ?thesis by (simp_all add: partition_filter1 partition_filter2)
qed

lemma partition_set:
  assumes "partition P xs = (yes, no)"
  shows "set yes \<union> set no = set xs"
proof -
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
    by simp_all
  then show ?thesis by (auto simp add: partition_filter1 partition_filter2) 
qed

lemma partition_filter_conv[simp]:
  "partition f xs = (filter f xs,filter (Not o f) xs)"
unfolding partition_filter2[symmetric]
unfolding partition_filter1[symmetric] by simp

declare partition.simps[simp del]


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) = (UN x:set xs. set x)"
by (induct xs) auto

lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f 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, code]: "(x # xs)!0 = x"
by auto

lemma nth_Cons_Suc [simp, code]: "(x # xs)!(Suc n) = xs!n"
by auto

declare nth.simps [simp del]

lemma nth_append:
  "(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
apply (induct xs arbitrary: n, 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 < length xs ==> (map f xs)!n = f(xs!n)"
apply (induct xs arbitrary: n, 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:
 "(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
apply(induct xs arbitrary: ys)
 apply force
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 (metis nat_case_0 nth.simps zero_less_Suc)
apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
apply (case_tac i, simp)
apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff)
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: