src/HOL/List.thy

more operations and lemmas

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

header {* The datatype of finite lists *}

theory List
imports Presburger Code_Numeral Quotient Lifting_Set Lifting_Option Lifting_Product
begin

datatype_new (set: 'a) list (map: map rel: list_all2) =
    =: Nil (defaults tl: "[]")  ("[]")
  | Cons (hd: 'a) (tl: "'a list")  (infixr "#" 65)
datatype_compat list

lemma [case_names Nil Cons, cases type: list]:
  -- {* for backward compatibility -- names of variables differ *}
  "(y = [] \<Longrightarrow> P) \<Longrightarrow> (\<And>a list. y = a # list \<Longrightarrow> P) \<Longrightarrow> P"
by (rule list.exhaust)

lemma [case_names Nil Cons, induct type: list]:
  -- {* for backward compatibility -- names of variables differ *}
  "P [] \<Longrightarrow> (\<And>a list. P list \<Longrightarrow> P (a # list)) \<Longrightarrow> P list"
by (rule list.induct)

text {* Compatibility: *}

setup {* Sign.mandatory_path "list" *}

lemmas inducts = list.induct
lemmas recs = list.rec
lemmas cases = list.case

setup {* Sign.parent_path *}

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

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


subsection {* Basic list processing functions *}

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)"

declare list.set[simp del, code del]

lemma set_simps[simp, code, code_post]:
  "set [] = {}"
  "set (x # xs) = insert x (set xs)"
by (simp_all add: list.set)

lemma set_rec: "set xs = rec_list {} (\<lambda>x _. insert x) xs"
  by (induct xs) auto

definition coset :: "'a list \<Rightarrow> 'a set" where
[simp]: "coset xs = - set 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 fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where
fold_Nil:  "fold f [] = id" |
fold_Cons: "fold f (x # xs) = fold f xs \<circ> f x"

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

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

definition (in monoid_add) listsum :: "'a list \<Rightarrow> 'a" where
"listsum xs = foldr plus xs 0"

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)"

nonterminal lupdbinds and 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 product :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
"product [] _ = []" |
"product (x#xs) ys = map (Pair x) ys @ product xs ys"

hide_const (open) product

primrec product_lists :: "'a list list \<Rightarrow> 'a list list" where
"product_lists [] = [[]]" |
"product_lists (xs # xss) = concat (map (\<lambda>x. map (Cons x) (product_lists xss)) xs)"

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 [])"

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 find :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a option" where
"find _ [] = None" |
"find P (x#xs) = (if P x then Some x else find P xs)"

hide_const (open) find

definition
   "extract" :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> ('a list * 'a * 'a list) option"
where "extract P xs =
  (case dropWhile (Not o P) xs of
     [] \<Rightarrow> None |
     y#ys \<Rightarrow> Some(takeWhile (Not o P) xs, y, ys))"

hide_const (open) "extract"

primrec those :: "'a option list \<Rightarrow> 'a list option"
where
"those [] = Some []" |
"those (x # xs) = (case x of
  None \<Rightarrow> None
| Some y \<Rightarrow> map_option (Cons y) (those xs))"

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 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)"

fun remdups_adj :: "'a list \<Rightarrow> 'a list" where
"remdups_adj [] = []" |
"remdups_adj [x] = [x]" |
"remdups_adj (x # y # xs) = (if x = y then remdups_adj (x # xs) else x # remdups_adj (y # 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 enumerate :: "nat \<Rightarrow> 'a list \<Rightarrow> (nat \<times> 'a) list" where
enumerate_eq_zip: "enumerate n xs = zip [n..<n + length xs] xs"

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

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

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 sublists :: "'a list \<Rightarrow> 'a list list" where
"sublists [] = [[]]" |
"sublists (x#xs) = (let xss = sublists xs in map (Cons x) xss @ xss)"

primrec n_lists :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list list" where
"n_lists 0 xs = [[]]" |
"n_lists (Suc n) xs = concat (map (\<lambda>ys. map (\<lambda>y. y # ys) xs) (n_lists n xs))"

hide_const (open) n_lists

fun splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
"splice [] ys = ys" |
"splice xs [] = xs" |
"splice (x#xs) (y#ys) = x # y # splice xs ys"

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 "fold f [a,b,c] x = f c (f b (f a x))" by simp}\\
@{lemma "foldr f [a,b,c] x = f a (f b (f c x))" by simp}\\
@{lemma "foldl f x [a,b,c] = f (f (f x a) b) c" 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 "enumerate 3 [a,b,c] = [(3,a),(4,b),(5,c)]" by normalization}\\
@{lemma "List.product [a,b] [c,d] = [(a, c), (a, d), (b, c), (b, d)]" by simp}\\
@{lemma "product_lists [[a,b], [c], [d,e]] = [[a,c,d], [a,c,e], [b,c,d], [b,c,e]]" 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 "remdups_adj [2,2,3,1,1::nat,2,1] = [2,3,1,2,1]" 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 "List.find (%i::int. i>0) [0,0] = None" by simp}\\
@{lemma "List.find (%i::int. i>0) [0,1,0,2] = Some 1" by simp}\\
@{lemma "List.extract (%i::int. i>0) [0,0] = None" by(simp add: extract_def)}\\
@{lemma "List.extract (%i::int. i>0) [0,1,0,2] = Some([0], 1, [0,2])" by(simp add: extract_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 "sublists [a,b] = [[a, b], [a], [b], []]" by simp}\\
@{lemma "List.n_lists 2 [a,b,c] = [[a, a], [b, a], [c, a], [a, b], [b, b], [c, b], [a, c], [b, c], [c, c]]" by (simp add: eval_nat_numeral)}\\
@{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by simp}\\
@{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate_def eval_nat_numeral)}\\
@{lemma "replicate 4 a = [a,a,a,a]" by (simp add:eval_nat_numeral)}\\
@{lemma "[2..<5] = [2,3,4]" by (simp add:eval_nat_numeral)}\\
@{lemma "listsum [1,2,3::nat] = 6" by (simp add: listsum_def)}
\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

inductive sorted :: "'a list \<Rightarrow> bool" where
  Nil [iff]: "sorted []"
| Cons: "\<forall>y\<in>set xs. x \<le> y \<Longrightarrow> sorted xs \<Longrightarrow> sorted (x # xs)"

lemma sorted_single [iff]:
  "sorted [x]"
  by (rule sorted.Cons) auto

lemma sorted_many:
  "x \<le> y \<Longrightarrow> sorted (y # zs) \<Longrightarrow> sorted (x # y # zs)"
  by (rule sorted.Cons) (cases "y # zs" rule: sorted.cases, auto)

lemma sorted_many_eq [simp, code]:
  "sorted (x # y # zs) \<longleftrightarrow> x \<le> y \<and> sorted (y # zs)"
  by (auto intro: sorted_many elim: sorted.cases)

lemma [code]:
  "sorted [] \<longleftrightarrow> True"
  "sorted [x] \<longleftrightarrow> True"
  by simp_all

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 []"

definition insort_insert_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
"insort_insert_key f x xs =
  (if f x \<in> f ` set xs then xs else insort_key f x xs)"

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

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.  *}

nonterminal lc_qual and 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 {*
  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 single x = ConsC $ x $ NilC;

    fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
      let
        (* FIXME proper name context!? *)
        val x =
          Free (singleton (Name.variant_list (fold Term.add_free_names [p, e] [])) "x", dummyT);
        val e = if opti then single e else e;
        val case1 = Syntax.const @{syntax_const "_case1"} $ p $ e;
        val case2 =
          Syntax.const @{syntax_const "_case1"} $
            Syntax.const @{const_syntax Pure.dummy_pattern} $ NilC;
        val cs = Syntax.const @{syntax_const "_case2"} $ case1 $ case2;
      in Syntax_Trans.abs_tr [x, Case_Translation.case_tr false ctxt [x, cs]] end;

    fun abs_tr ctxt p e opti =
      (case Term_Position.strip_positions p of
        Free (s, T) =>
          let
            val thy = Proof_Context.theory_of ctxt;
            val s' = Proof_Context.intern_const ctxt s;
          in
            if Sign.declared_const thy s'
            then (pat_tr ctxt p e opti, false)
            else (Syntax_Trans.abs_tr [p, e], true)
          end
      | _ => (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"}, _) => single 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
*}

ML_val {*
  let
    val read = Syntax.read_term @{context};
    fun check s1 s2 = read s1 aconv read s2 orelse error ("Check failed: " ^ quote s1);
  in
    check "[(x,y,z). b]" "if b then [(x, y, z)] else []";
    check "[(x,y,z). x\<leftarrow>xs]" "map (\<lambda>x. (x, y, z)) xs";
    check "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]" "concat (map (\<lambda>x. map (\<lambda>y. e x y) ys) xs)";
    check "[(x,y,z). x<a, x>b]" "if x < a then if b < x then [(x, y, z)] else [] else []";
    check "[(x,y,z). x\<leftarrow>xs, x>b]" "concat (map (\<lambda>x. if b < x then [(x, y, z)] else []) xs)";
    check "[(x,y,z). x<a, x\<leftarrow>xs]" "if x < a then map (\<lambda>x. (x, y, z)) xs else []";
    check "[(x,y). Cons True x \<leftarrow> xs]"
      "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | True # x \<Rightarrow> [(x, y)] | False # x \<Rightarrow> []) xs)";
    check "[(x,y,z). Cons x [] \<leftarrow> xs]"
      "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | [x] \<Rightarrow> [(x, y, z)] | x # aa # lista \<Rightarrow> []) xs)";
    check "[(x,y,z). x<a, x>b, x=d]"
      "if x < a then if b < x then if x = d then [(x, y, z)] else [] else [] else []";
    check "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
      "if x < a then if b < x then map (\<lambda>y. (x, y, z)) ys else [] else []";
    check "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
      "if x < a then concat (map (\<lambda>x. if b < y then [(x, y, z)] else []) xs) else []";
    check "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
      "if x < a then concat (map (\<lambda>x. map (\<lambda>y. (x, y, z)) ys) xs) else []";
    check "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
      "concat (map (\<lambda>x. if b < x then if y < a then [(x, y, z)] else [] else []) xs)";
    check "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
      "concat (map (\<lambda>x. if b < x then map (\<lambda>y. (x, y, z)) ys else []) xs)";
    check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
      "concat (map (\<lambda>x. concat (map (\<lambda>y. if x < y then [(x, y, z)] else []) ys)) xs)";
    check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
      "concat (map (\<lambda>x. concat (map (\<lambda>y. map (\<lambda>z. (x, y, z)) zs) ys)) xs)"
  end;
*}

(*
term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
*)


ML {*
(* Simproc for rewriting list comprehensions applied to List.set to set
   comprehension. *)

signature LIST_TO_SET_COMPREHENSION =
sig
  val simproc : Proof.context -> cterm -> thm option
end

structure List_to_Set_Comprehension : LIST_TO_SET_COMPREHENSION =
struct

(* conversion *)

fun all_exists_conv cv ctxt ct =
  (case Thm.term_of ct of
    Const (@{const_name HOL.Ex}, _) $ Abs _ =>
      Conv.arg_conv (Conv.abs_conv (all_exists_conv cv o #2) ctxt) ct
  | _ => cv ctxt ct)

fun all_but_last_exists_conv cv ctxt ct =
  (case Thm.term_of ct of
    Const (@{const_name HOL.Ex}, _) $ Abs (_, _, Const (@{const_name HOL.Ex}, _) $ _) =>
      Conv.arg_conv (Conv.abs_conv (all_but_last_exists_conv cv o #2) ctxt) ct
  | _ => cv ctxt ct)

fun Collect_conv cv ctxt ct =
  (case Thm.term_of ct of
    Const (@{const_name Set.Collect}, _) $ Abs _ => Conv.arg_conv (Conv.abs_conv cv ctxt) ct
  | _ => raise CTERM ("Collect_conv", [ct]))

fun rewr_conv' th = Conv.rewr_conv (mk_meta_eq th)

fun conjunct_assoc_conv ct =
  Conv.try_conv
    (rewr_conv' @{thm conj_assoc} then_conv HOLogic.conj_conv Conv.all_conv conjunct_assoc_conv) ct

fun right_hand_set_comprehension_conv conv ctxt =
  HOLogic.Trueprop_conv (HOLogic.eq_conv Conv.all_conv
    (Collect_conv (all_exists_conv conv o #2) ctxt))


(* term abstraction of list comprehension patterns *)

datatype termlets = If | Case of (typ * int)

fun simproc ctxt redex =
  let
    val set_Nil_I = @{thm trans} OF [@{thm set_simps(1)}, @{thm empty_def}]
    val set_singleton = @{lemma "set [a] = {x. x = a}" by simp}
    val inst_Collect_mem_eq = @{lemma "set A = {x. x : set A}" by simp}
    val del_refl_eq = @{lemma "(t = t & P) == P" by simp}
    fun mk_set T = Const (@{const_name List.set}, HOLogic.listT T --> HOLogic.mk_setT T)
    fun dest_set (Const (@{const_name List.set}, _) $ xs) = xs
    fun dest_singleton_list (Const (@{const_name List.Cons}, _)
          $ t $ (Const (@{const_name List.Nil}, _))) = t
      | dest_singleton_list t = raise TERM ("dest_singleton_list", [t])
    (* We check that one case returns a singleton list and all other cases
       return [], and return the index of the one singleton list case *)
    fun possible_index_of_singleton_case cases =
      let
        fun check (i, case_t) s =
          (case strip_abs_body case_t of
            (Const (@{const_name List.Nil}, _)) => s
          | _ => (case s of SOME NONE => SOME (SOME i) | _ => NONE))
      in
        fold_index check cases (SOME NONE) |> the_default NONE
      end
    (* returns (case_expr type index chosen_case constr_name) option  *)
    fun dest_case case_term =
      let
        val (case_const, args) = strip_comb case_term
      in
        (case try dest_Const case_const of
          SOME (c, T) =>
            (case Ctr_Sugar.ctr_sugar_of_case ctxt c of
              SOME {ctrs, ...} =>
                (case possible_index_of_singleton_case (fst (split_last args)) of
                  SOME i =>
                    let
                      val constr_names = map (fst o dest_Const) ctrs
                      val (Ts, _) = strip_type T
                      val T' = List.last Ts
                    in SOME (List.last args, T', i, nth args i, nth constr_names i) end
                | NONE => NONE)
            | NONE => NONE)
        | NONE => NONE)
      end
    (* returns condition continuing term option *)
    fun dest_if (Const (@{const_name If}, _) $ cond $ then_t $ Const (@{const_name Nil}, _)) =
          SOME (cond, then_t)
      | dest_if _ = NONE
    fun tac _ [] = rtac set_singleton 1 ORELSE rtac inst_Collect_mem_eq 1
      | tac ctxt (If :: cont) =
          Splitter.split_tac [@{thm split_if}] 1
          THEN rtac @{thm conjI} 1
          THEN rtac @{thm impI} 1
          THEN Subgoal.FOCUS (fn {prems, context, ...} =>
            CONVERSION (right_hand_set_comprehension_conv (K
              (HOLogic.conj_conv (Conv.rewr_conv (List.last prems RS @{thm Eq_TrueI})) Conv.all_conv
               then_conv
               rewr_conv' @{lemma "(True & P) = P" by simp})) context) 1) ctxt 1
          THEN tac ctxt cont
          THEN rtac @{thm impI} 1
          THEN Subgoal.FOCUS (fn {prems, context, ...} =>
              CONVERSION (right_hand_set_comprehension_conv (K
                (HOLogic.conj_conv (Conv.rewr_conv (List.last prems RS @{thm Eq_FalseI})) Conv.all_conv
                 then_conv rewr_conv' @{lemma "(False & P) = False" by simp})) context) 1) ctxt 1
          THEN rtac set_Nil_I 1
      | tac ctxt (Case (T, i) :: cont) =
          let
            val SOME {injects, distincts, case_thms, split, ...} =
              Ctr_Sugar.ctr_sugar_of ctxt (fst (dest_Type T))
          in
            (* do case distinction *)
            Splitter.split_tac [split] 1
            THEN EVERY (map_index (fn (i', _) =>
              (if i' < length case_thms - 1 then rtac @{thm conjI} 1 else all_tac)
              THEN REPEAT_DETERM (rtac @{thm allI} 1)
              THEN rtac @{thm impI} 1
              THEN (if i' = i then
                (* continue recursively *)
                Subgoal.FOCUS (fn {prems, context, ...} =>
                  CONVERSION (Thm.eta_conversion then_conv right_hand_set_comprehension_conv (K
                      ((HOLogic.conj_conv
                        (HOLogic.eq_conv Conv.all_conv (rewr_conv' (List.last prems)) then_conv
                          (Conv.try_conv (Conv.rewrs_conv (map mk_meta_eq injects))))
                        Conv.all_conv)
                        then_conv (Conv.try_conv (Conv.rewr_conv del_refl_eq))
                        then_conv conjunct_assoc_conv)) context
                    then_conv (HOLogic.Trueprop_conv (HOLogic.eq_conv Conv.all_conv (Collect_conv (fn (_, ctxt) =>
                      Conv.repeat_conv
                        (all_but_last_exists_conv
                          (K (rewr_conv'
                            @{lemma "(EX x. x = t & P x) = P t" by simp})) ctxt)) context)))) 1) ctxt 1
                THEN tac ctxt cont
              else
                Subgoal.FOCUS (fn {prems, context, ...} =>
                  CONVERSION
                    (right_hand_set_comprehension_conv (K
                      (HOLogic.conj_conv
                        ((HOLogic.eq_conv Conv.all_conv
                          (rewr_conv' (List.last prems))) then_conv
                          (Conv.rewrs_conv (map (fn th => th RS @{thm Eq_FalseI}) distincts)))
                        Conv.all_conv then_conv
                        (rewr_conv' @{lemma "(False & P) = False" by simp}))) context then_conv
                      HOLogic.Trueprop_conv
                        (HOLogic.eq_conv Conv.all_conv
                          (Collect_conv (fn (_, ctxt) =>
                            Conv.repeat_conv
                              (Conv.bottom_conv
                                (K (rewr_conv'
                                  @{lemma "(EX x. P) = P" by simp})) ctxt)) context))) 1) ctxt 1
                THEN rtac set_Nil_I 1)) case_thms)
          end
    fun make_inner_eqs bound_vs Tis eqs t =
      (case dest_case t of
        SOME (x, T, i, cont, constr_name) =>
          let
            val (vs, body) = strip_abs (Envir.eta_long (map snd bound_vs) cont)
            val x' = incr_boundvars (length vs) x
            val eqs' = map (incr_boundvars (length vs)) eqs
            val constr_t =
              list_comb
                (Const (constr_name, map snd vs ---> T), map Bound (((length vs) - 1) downto 0))
            val constr_eq = Const (@{const_name HOL.eq}, T --> T --> @{typ bool}) $ constr_t $ x'
          in
            make_inner_eqs (rev vs @ bound_vs) (Case (T, i) :: Tis) (constr_eq :: eqs') body
          end
      | NONE =>
          (case dest_if t of
            SOME (condition, cont) => make_inner_eqs bound_vs (If :: Tis) (condition :: eqs) cont
          | NONE =>
            if eqs = [] then NONE (* no rewriting, nothing to be done *)
            else
              let
                val Type (@{type_name List.list}, [rT]) = fastype_of1 (map snd bound_vs, t)
                val pat_eq =
                  (case try dest_singleton_list t of
                    SOME t' =>
                      Const (@{const_name HOL.eq}, rT --> rT --> @{typ bool}) $
                        Bound (length bound_vs) $ t'
                  | NONE =>
                      Const (@{const_name Set.member}, rT --> HOLogic.mk_setT rT --> @{typ bool}) $
                        Bound (length bound_vs) $ (mk_set rT $ t))
                val reverse_bounds = curry subst_bounds
                  ((map Bound ((length bound_vs - 1) downto 0)) @ [Bound (length bound_vs)])
                val eqs' = map reverse_bounds eqs
                val pat_eq' = reverse_bounds pat_eq
                val inner_t =
                  fold (fn (_, T) => fn t => HOLogic.exists_const T $ absdummy T t)
                    (rev bound_vs) (fold (curry HOLogic.mk_conj) eqs' pat_eq')
                val lhs = term_of redex
                val rhs = HOLogic.mk_Collect ("x", rT, inner_t)
                val rewrite_rule_t = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
              in
                SOME
                  ((Goal.prove ctxt [] [] rewrite_rule_t
                    (fn {context, ...} => tac context (rev Tis))) RS @{thm eq_reflection})
              end))
  in
    make_inner_eqs [] [] [] (dest_set (term_of redex))
  end

end
*}

simproc_setup list_to_set_comprehension ("set xs") = {* K List_to_Set_Comprehension.simproc *}

code_datatype set coset

hide_const (open) coset


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

lemma not_Cons_self [simp]:
  "xs \<noteq> x # xs"
by (induct xs) auto

lemma not_Cons_self2 [simp]:
  "x # xs \<noteq> xs"
by (rule not_Cons_self [symmetric])

lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
by (induct xs) auto

lemma tl_Nil: "tl xs = [] \<longleftrightarrow> xs = [] \<or> (EX x. xs = [x])"
by (cases xs) auto

lemma Nil_tl: "[] = tl xs \<longleftrightarrow> xs = [] \<or> (EX x. xs = [x])"
by (cases xs) auto

lemma length_induct:
  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
by (fact measure_induct)

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
    show ?thesis
    proof (rule cons)
      from Cons show "xs \<noteq> []" by simp
      with Cons.hyps show "P xs" .
    qed
  qed
qed

lemma inj_split_Cons: "inj_on (\<lambda>(xs, n). n#xs) X"
  by (auto intro!: inj_onI)


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 list_all2_iff:
  "list_all2 P xs ys \<longleftrightarrow> length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)"
by (induct xs ys rule: list_induct2') 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);

val ss = simpset_of @{context};

fun list_neq ctxt 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 ctxt [] [] neq_len
          (K (simp_tac (put_simpset ss ctxt) 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 K 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]:
 "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 fastforce
apply(case_tac zs)
 apply simp
apply fastforce
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]: "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.
*}

simproc_setup list_eq ("(xs::'a list) = ys")  = {*
  let
    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 =
      simpset_of (put_simpset HOL_basic_ss @{context}
        addsimps [@{thm append_assoc}, @{thm append_Nil}, @{thm append_Cons}]);
    
    fun list_eq ctxt (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 ctxt [] [] eq
              (K (simp_tac (put_simpset rearr_ss ctxt) 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 fn _ => fn ctxt => fn ct => list_eq ctxt (term_of ct) end;
*}


subsubsection {* @{const map} *}

lemma hd_map:
  "xs \<noteq> [] \<Longrightarrow> hd (map f xs) = f (hd xs)"
  by (cases xs) simp_all

lemma map_tl:
  "map f (tl xs) = tl (map f xs)"
  by (cases xs) simp_all

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

functor map: map
by (simp_all add: id_def)

declare map.id [simp]


subsubsection {* @{const 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 {* @{const 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) auto


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 fastforce
  next
    assume "x \<noteq> a" thus ?case using Cons by(fastforce 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 \<in> 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 (auto intro!: exI)
  next
    assume "x \<noteq> a" thus ?case using snoc by fastforce
  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"
    hence "x # xs = [] @ x # xs \<and> P x \<and> (\<forall>y\<in>set []. \<not> P y)" by simp
    thus ?thesis by fast
  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 (auto intro!: exI)
  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 fastforce
  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 rule (erule split_list_last_prop, auto)


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 {* @{const 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)
    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)
    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 fastforce
    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]:
 "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 {* @{const 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

lemma concat_eq_concat_iff: "\<forall>(x, y) \<in> set (zip xs ys). length x = length y ==> length xs = length ys ==> (concat xs = concat ys) = (xs = ys)"
proof (induct xs arbitrary: ys)
  case (Cons x xs ys)
  thus ?case by (cases ys) auto
qed (auto)

lemma concat_injective: "concat xs = concat ys ==> length xs = length ys ==> \<forall>(x, y) \<in> set (zip xs ys). length x = length y ==> xs = ys"
by (simp add: concat_eq_concat_iff)


subsubsection {* @{const 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_Cons_pos[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
by(auto simp: Nat.gr0_conv_Suc)

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 nth_tl:
  assumes "n < length (tl x)" shows "tl x ! n = x ! Suc n"
using assms by (induct x) auto

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(1) 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(2) 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 nth_equal_first_eq:
  assumes "x \<notin> set xs"
  assumes "n \<le> length xs"
  shows "(x # xs) ! n = x \<longleftrightarrow> n = 0" (is "?lhs \<longleftrightarrow> ?rhs")
proof
  assume ?lhs
  show ?rhs
  proof (rule ccontr)
    assume "n \<noteq> 0"
    then have "n > 0" by simp
    with `?lhs` have "xs ! (n - 1) = x" by simp
    moreover from `n > 0` `n \<le> length xs` have "n - 1 < length xs" by simp
    ultimately have "\<exists>i<length xs. xs ! i = x" by auto
    with `x \<notin> set xs` in_set_conv_nth [of x xs] show False by simp
  qed
next
  assume ?rhs then show ?lhs by simp
qed

lemma nth_non_equal_first_eq:
  assumes "x \<noteq> y"
  shows "(x # xs) ! n = y \<longleftrightarrow> xs ! (n - 1) = y \<and> n > 0" (is "?lhs \<longleftrightarrow> ?rhs")
proof
  assume "?lhs" with assms have "n > 0" by (cases n) simp_all
  with `?lhs` show ?rhs by simp
next
  assume "?rhs" then show "?lhs" by simp
qed

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)

lemma rev_nth:
  "n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)"
proof (induct xs arbitrary: n)
  case Nil thus ?case by simp
next
  case (Cons x xs)
  hence n: "n < Suc (length xs)" by simp
  moreover
  { assume "n < length xs"
    with n obtain n' where n': "length xs - n = Suc n'"
      by (cases "length xs - n", auto)
    moreover
    from n' have "length xs - Suc n = n'" by simp
    ultimately
    have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp
  }
  ultimately
  show ?case by (clarsimp simp add: Cons nth_append)
qed

lemma Skolem_list_nth:
  "(ALL i<k. EX x. P i x) = (EX xs. size xs = k & (ALL i<k. P i (xs!i)))"
  (is "_ = (EX xs. ?P k xs)")
proof(induct k)
  case 0 show ?case by simp
next
  case (Suc k)
  show ?case (is "?L = ?R" is "_ = (EX xs. ?P' xs)")
  proof
    assume "?R" thus "?L" using Suc by auto
  next
    assume "?L"
    with Suc obtain x xs where "?P k xs & P k x" by (metis less_Suc_eq)
    hence "?P'(xs@[x])" by(simp add:nth_append less_Suc_eq)
    thus "?R" ..
  qed
qed


subsubsection {* @{const list_update} *}

lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
by (induct xs arbitrary: i) (auto split: nat.split)

lemma nth_list_update:
"i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
by (induct xs arbitrary: i j) (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 \<noteq> j ==> xs[i:=x]!j = xs!j"
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)

lemma list_update_id[simp]: "xs[i := xs!i] = xs"
by (induct xs arbitrary: i) (simp_all split:nat.splits)

lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
apply (induct xs arbitrary: i)
 apply simp
apply (case_tac i)
apply simp_all
done

lemma list_update_nonempty[simp]: "xs[k:=x] = [] \<longleftrightarrow> xs=[]"
by (simp only: length_0_conv[symmetric] length_list_update)

lemma list_update_same_conv:
"i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
by (induct xs arbitrary: i) (auto split: nat.split)

lemma list_update_append1:
 "i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
apply (induct xs arbitrary: i, simp)
apply(simp split:nat.split)
done

lemma list_update_append:
  "(xs @ ys) [n:= x] = 
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
by (induct xs arbitrary: n) (auto split:nat.splits)

lemma list_update_length [simp]:
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
by (induct xs, auto)

lemma map_update: "map f (xs[k:= y]) = (map f xs)[k := f y]"
by(induct xs arbitrary: k)(auto split:nat.splits)

lemma rev_update:
  "k < length xs \<Longrightarrow> rev (xs[k:= y]) = (rev xs)[length xs - k - 1 := y]"
by (induct xs arbitrary: k) (auto simp: list_update_append split:nat.splits)

lemma update_zip:
  "(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split)

lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)"
by (induct xs arbitrary: i) (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 < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
by (induct xs arbitrary: n) (auto split:nat.splits)

lemma list_update_overwrite[simp]:
  "xs [i := x, i := y] = xs [i := y]"
apply (induct xs arbitrary: i) apply simp
apply (case_tac i, simp_all)
done

lemma list_update_swap:
  "i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]"
apply (induct xs arbitrary: i i')
apply simp
apply (case_tac i, case_tac i')
apply auto
apply (case_tac i')
apply auto
done

lemma list_update_code [code]:
  "[][i := y] = []"
  "(x # xs)[0 := y] = y # xs"
  "(x # xs)[Suc i := y] = x # xs[i := y]"
  by simp_all


subsubsection {* @{const last} and @{const 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

lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
  by simp

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 last_tl: "xs = [] \<or> tl xs \<noteq> [] \<Longrightarrow>last (tl xs) = last xs"
by (induct xs) simp_all

lemma butlast_tl: "butlast (tl xs) = tl (butlast xs)"
by (induct xs) simp_all

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:
  "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
by (induct xs arbitrary: ys) 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 < length xs \<Longrightarrow> last (drop n xs) = last xs"
apply (induct xs arbitrary: n)
 apply simp
apply (auto split:nat.split)
done

lemma nth_butlast:
  assumes "n < length (butlast xs)" shows "butlast xs ! n = xs ! n"
proof (cases xs)
  case (Cons y ys)
  moreover from assms have "butlast xs ! n = (butlast xs @ [last xs]) ! n"
    by (simp add: nth_append)
  ultimately show ?thesis using append_butlast_last_id by simp
qed simp

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

lemma butlast_conv_take: "butlast xs = take (length xs - 1) xs"
by (induct xs, simp, case_tac xs, simp_all)

lemma last_list_update:
  "xs \<noteq> [] \<Longrightarrow> last(xs[k:=x]) = (if k = size xs - 1 then x else last xs)"
by (auto simp: last_conv_nth)

lemma butlast_list_update:
  "butlast(xs[k:=x]) =
 (if k = size xs - 1 then butlast xs else (butlast xs)[k:=x])"
apply(cases xs rule:rev_cases)
apply simp
apply(simp add:list_update_append split:nat.splits)
done

lemma last_map:
  "xs \<noteq> [] \<Longrightarrow> last (map f xs) = f (last xs)"
  by (cases xs rule: rev_cases) simp_all

lemma map_butlast:
  "map f (butlast xs) = butlast (map f xs)"
  by (induct xs) simp_all

lemma snoc_eq_iff_butlast:
  "xs @ [x] = ys \<longleftrightarrow> (ys \<noteq> [] & butlast ys = xs & last ys = x)"
by fastforce


subsubsection {* @{const take} and @{const 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_1_Cons [simp]: "take 1 (x # xs) = [x]"
  unfolding One_nat_def by simp

lemma drop_1_Cons [simp]: "drop 1 (x # xs) = xs"
  unfolding One_nat_def by simp

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 take_tl: "take n (tl xs) = tl (take (Suc n) xs)"
by (induct xs arbitrary: n) simp_all

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

lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)"
by (cases n, simp, cases xs, auto)

lemma tl_drop: "tl (drop n xs) = drop n (tl xs)"
by (simp only: drop_tl)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

lemma butlast_take:
  "n <= length xs ==> butlast (take n xs) = take (n - 1) xs"
by (simp add: butlast_conv_take min.absorb1 min.absorb2)

lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)"
by (simp add: butlast_conv_take drop_take add_ac)

lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs"
by (simp add: butlast_conv_take min.absorb1)

lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)"
by (simp add: butlast_conv_take drop_take add_ac)

lemma hd_drop_conv_nth: "n < length xs \<Longrightarrow> hd(drop n xs) = xs!n"
by(simp add: hd_conv_nth)

lemma set_take_subset_set_take:
  "m <= n \<Longrightarrow> set(take m xs) <= set(take n xs)"
apply (induct xs arbitrary: m n)
apply simp
apply (case_tac n)
apply (auto simp: take_Cons)
done

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

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

lemma set_drop_subset_set_drop:
  "m >= n \<Longrightarrow> set(drop m xs) <= set(drop n xs)"
apply(induct xs arbitrary: m n)
apply(auto simp:drop_Cons split:nat.split)
by (metis set_drop_subset subset_iff)

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

lemma take_add: 
  "take (i+j) xs = take i xs @ take j (drop i xs)"
apply (induct xs arbitrary: i, auto) 
apply (case_tac i, simp_all)
done

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

lemma take_hd_drop:
  "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (Suc n) xs"
apply(induct xs arbitrary: n)
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

lemma nth_drop':
  "i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs"
apply (induct i arbitrary: xs)
apply (simp add: neq_Nil_conv)
apply (erule exE)+
apply simp
apply (case_tac xs)
apply simp_all
done


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

lemma length_takeWhile_le: "length (takeWhile P xs) \<le> length xs"
  by (induct xs) auto

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 takeWhile_nth: "j < length (takeWhile P xs) \<Longrightarrow> takeWhile P xs ! j = xs ! j"
apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto

lemma dropWhile_nth: "j < length (dropWhile P xs) \<Longrightarrow> dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))"
apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto

lemma length_dropWhile_le: "length (dropWhile P xs) \<le> length 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 dropWhile_append3:
  "\<not> P y \<Longrightarrow>dropWhile P (xs @ y # ys) = dropWhile P xs @ y # ys"
by (induct xs) auto

lemma dropWhile_last:
  "x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> last (dropWhile P xs) = last xs"
by (auto simp add: dropWhile_append3 in_set_conv_decomp)

lemma set_dropWhileD: "x \<in> set (dropWhile P xs) \<Longrightarrow> x \<in> set xs"
by (induct xs) (auto split: split_if_asm)

lemma set_takeWhileD: "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)

lemma distinct_takeWhile[simp]: "distinct xs ==> distinct (takeWhile P xs)"
by (induct xs) (auto dest: set_takeWhileD)

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

lemma takeWhile_map: "takeWhile P (map f xs) = map f (takeWhile (P \<circ> f) xs)"
by (induct xs) auto

lemma dropWhile_map: "dropWhile P (map f xs) = map f (dropWhile (P \<circ> f) xs)"
by (induct xs) auto

lemma takeWhile_eq_take: "takeWhile P xs = take (length (takeWhile P xs)) xs"
by (induct xs) auto

lemma dropWhile_eq_drop: "dropWhile P xs = drop (length (takeWhile P xs)) xs"
by (induct xs) auto

lemma hd_dropWhile:
  "dropWhile P xs \<noteq> [] \<Longrightarrow> \<not> P (hd (dropWhile P xs))"
using assms by (induct xs) auto

lemma takeWhile_eq_filter:
  assumes "\<And> x. x \<in> set (dropWhile P xs) \<Longrightarrow> \<not> P x"
  shows "takeWhile P xs = filter P xs"
proof -
  have A: "filter P xs = filter P (takeWhile P xs @ dropWhile P xs)"
    by simp
  have B: "filter P (dropWhile P xs) = []"
    unfolding filter_empty_conv using assms by blast
  have "filter P xs = takeWhile P xs"
    unfolding A filter_append B
    by (auto simp add: filter_id_conv dest: set_takeWhileD)
  thus ?thesis ..
qed

lemma takeWhile_eq_take_P_nth:
  "\<lbrakk> \<And> i. \<lbrakk> i < n ; i < length xs \<rbrakk> \<Longrightarrow> P (xs ! i) ; n < length xs \<Longrightarrow> \<not> P (xs ! n) \<rbrakk> \<Longrightarrow>
  takeWhile P xs = take n xs"
proof (induct xs arbitrary: n)
  case (Cons x xs)
  thus ?case
  proof (cases n)
    case (Suc n') note this[simp]
    have "P x" using Cons.prems(1)[of 0] by simp
    moreover have "takeWhile P xs = take n' xs"
    proof (rule Cons.hyps)
      case goal1 thus "P (xs ! i)" using Cons.prems(1)[of "Suc i"] by simp
    next case goal2 thus ?case using Cons by auto
    qed
    ultimately show ?thesis by simp
   qed simp
qed simp

lemma nth_length_takeWhile:
  "length (takeWhile P xs) < length xs \<Longrightarrow> \<not> P (xs ! length (takeWhile P xs))"
by (induct xs) auto

lemma length_takeWhile_less_P_nth:
  assumes all: "\<And> i. i < j \<Longrightarrow> P (xs ! i)" and "j \<le> length xs"
  shows "j \<le> length (takeWhile P xs)"
proof (rule classical)
  assume "\<not> ?thesis"
  hence "length (takeWhile P xs) < length xs" using assms by simp
  thus ?thesis using all `\<not> ?thesis` nth_length_takeWhile[of P xs] by auto
qed

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:
 "distinct xs \<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]:
  "[| 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]:
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
  ==> dropWhile P l = dropWhile Q k"
by (induct k arbitrary: l, simp_all)

lemma takeWhile_idem [simp]:
  "takeWhile P (takeWhile P xs) = takeWhile P xs"
  by (induct xs) auto

lemma dropWhile_idem [simp]:
  "dropWhile P (dropWhile P xs) = dropWhile P xs"
  by (induct xs) auto


subsubsection {* @{const 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 [code]:
  "zip [] ys = []"
  "zip xs [] = []"
  "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
  by (fact zip_Nil zip.simps(1) zip_Cons_Cons)+

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_obtain_same_length:
  assumes "\<And>zs ws n. length zs = length ws \<Longrightarrow> n = min (length xs) (length ys)
    \<Longrightarrow> zs = take n xs \<Longrightarrow> ws = take n ys \<Longrightarrow> P (zip zs ws)"
  shows "P (zip xs ys)"
proof -
  let ?n = "min (length xs) (length ys)"
  have "P (zip (take ?n xs) (take ?n ys))"
    by (rule assms) simp_all
  moreover have "zip xs ys = zip (take ?n xs) (take ?n ys)"
  proof (induct xs arbitrary: ys)
    case Nil then show ?case by simp
  next
    case (Cons x xs) then show ?case by (cases ys) simp_all
  qed
  ultimately show ?thesis by simp
qed

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 |] ==>
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 zip_map_map:
  "zip (map f xs) (map g ys) = map (\<lambda> (x, y). (f x, g y)) (zip xs ys)"
proof (induct xs arbitrary: ys)
  case (Cons x xs) note Cons_x_xs = Cons.hyps
  show ?case
  proof (cases ys)
    case (Cons y ys')
    show ?thesis unfolding Cons using Cons_x_xs by simp
  qed simp
qed simp

lemma zip_map1:
  "zip (map f xs) ys = map (\<lambda>(x, y). (f x, y)) (zip xs ys)"
using zip_map_map[of f xs "\<lambda>x. x" ys] by simp

lemma zip_map2:
  "zip xs (map f ys) = map (\<lambda>(x, y). (x, f y)) (zip xs ys)"
using zip_map_map[of "\<lambda>x. x" xs f ys] by simp

lemma map_zip_map:
  "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
unfolding zip_map1 by auto

lemma map_zip_map2:
  "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
unfolding zip_map2 by auto

text{* Courtesy of Andreas Lochbihler: *}
lemma zip_same_conv_map: "zip xs xs = map (\<lambda>x. (x, x)) xs"
by(induct xs) auto

lemma nth_zip [simp]:
"[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
apply (induct ys arbitrary: i xs, 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_same: "((a,b) \<in> set (zip xs xs)) = (a \<in> set xs \<and> a = b)"
by(induct xs) auto

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

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

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

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

lemma zip_takeWhile_fst: "zip (takeWhile P xs) ys = takeWhile (P \<circ> fst) (zip xs ys)"
proof (induct xs arbitrary: ys)
  case (Cons x xs) thus ?case by (cases ys) auto
qed simp

lemma zip_takeWhile_snd: "zip xs (takeWhile P ys) = takeWhile (P \<circ> snd) (zip xs ys)"
proof (induct xs arbitrary: ys)
  case (Cons x xs) thus ?case by (cases ys) auto
qed simp

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

lemma in_set_zipE:
  "(x,y) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
by(blast dest: set_zip_leftD set_zip_rightD)

lemma zip_map_fst_snd:
  "zip (map fst zs) (map snd zs) = zs"
  by (induct zs) simp_all

lemma zip_eq_conv:
  "length xs = length ys \<Longrightarrow> zip xs ys = zs \<longleftrightarrow> map fst zs = xs \<and> map snd zs = ys"
  by (auto simp add: zip_map_fst_snd)

lemma in_set_zip:
  "p \<in> set (zip xs ys) \<longleftrightarrow> (\<exists>n. xs ! n = fst p \<and> ys ! n = snd p
    \<and> n < length xs \<and> n < length ys)"
  by (cases p) (auto simp add: set_zip)

lemma pair_list_eqI:
  assumes "map fst xs = map fst ys" and "map snd xs = map snd ys"
  shows "xs = ys"
proof -
  from assms(1) have "length xs = length ys" by (rule map_eq_imp_length_eq)
  from this assms show ?thesis
    by (induct xs ys rule: list_induct2) (simp_all add: prod_eqI)
qed


subsubsection {* @{const list_all2} *}

lemma list_all2_lengthD [intro?]: 
  "list_all2 P xs ys ==> length xs = length ys"
by (simp add: list_all2_iff)

lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
by (simp add: list_all2_iff)

lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
by (simp add: list_all2_iff)

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_iff)

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_induct
  [consumes 1, case_names Nil Cons, induct set: list_all2]:
  assumes P: "list_all2 P xs ys"
  assumes Nil: "R [] []"
  assumes Cons: "\<And>x xs y ys.
    \<lbrakk>P x y; list_all2 P xs ys; R xs ys\<rbrakk> \<Longrightarrow> R (x # xs) (y # ys)"
  shows "R xs ys"
using P
by (induct xs arbitrary: ys) (auto simp add: list_all2_Cons1 Nil Cons)

lemma list_all2_rev [iff]:
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
by (simp add: list_all2_iff 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_iff 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_iff 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_iff 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_iff)

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> list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
by (cases "i < length ys") (auto simp add: list_all2_conv_all_nth nth_list_update)

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

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

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

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

lemma list_eq_iff_zip_eq:
  "xs = ys \<longleftrightarrow> length xs = length ys \<and> (\<forall>(x,y) \<in> set (zip xs ys). x = y)"
by(auto simp add: set_zip list_all2_eq list_all2_conv_all_nth cong: conj_cong)


subsubsection {* @{const List.product} and @{const product_lists} *}

lemma product_list_set:
  "set (List.product xs ys) = set xs \<times> set ys"
  by (induct xs) auto

lemma length_product [simp]:
  "length (List.product xs ys) = length xs * length ys"
  by (induct xs) simp_all

lemma product_nth:
  assumes "n < length xs * length ys"
  shows "List.product xs ys ! n = (xs ! (n div length ys), ys ! (n mod length ys))"
using assms proof (induct xs arbitrary: n)
  case Nil then show ?case by simp
next
  case (Cons x xs n)
  then have "length ys > 0" by auto
  with Cons show ?case
    by (auto simp add: nth_append not_less le_mod_geq le_div_geq)
qed

lemma in_set_product_lists_length: 
  "xs \<in> set (product_lists xss) \<Longrightarrow> length xs = length xss"
  by (induct xss arbitrary: xs) auto

lemma product_lists_set:
  "set (product_lists xss) = {xs. list_all2 (\<lambda>x ys. x \<in> set ys) xs xss}" (is "?L = Collect ?R")
proof (intro equalityI subsetI, unfold mem_Collect_eq)
  fix xs assume "xs \<in> ?L"
  then have "length xs = length xss" by (rule in_set_product_lists_length)
  from this `xs \<in> ?L` show "?R xs" by (induct xs xss rule: list_induct2) auto
next
  fix xs assume "?R xs"
  then show "xs \<in> ?L" by induct auto
qed


subsubsection {* @{const fold} with natural argument order *}

lemma fold_simps [code]: -- {* eta-expanded variant for generated code -- enables tail-recursion optimisation in Scala *}
  "fold f [] s = s"
  "fold f (x # xs) s = fold f xs (f x s)" 
  by simp_all

lemma fold_remove1_split:
  assumes f: "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
    and x: "x \<in> set xs"
  shows "fold f xs = fold f (remove1 x xs) \<circ> f x"
  using assms by (induct xs) (auto simp add: comp_assoc)

lemma fold_cong [fundef_cong]:
  "a = b \<Longrightarrow> xs = ys \<Longrightarrow> (\<And>x. x \<in> set xs \<Longrightarrow> f x = g x)
    \<Longrightarrow> fold f xs a = fold g ys b"
  by (induct ys arbitrary: a b xs) simp_all

lemma fold_id:
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x = id"
  shows "fold f xs = id"
  using assms by (induct xs) simp_all

lemma fold_commute:
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
  shows "h \<circ> fold g xs = fold f xs \<circ> h"
  using assms by (induct xs) (simp_all add: fun_eq_iff)

lemma fold_commute_apply:
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
  shows "h (fold g xs s) = fold f xs (h s)"
proof -
  from assms have "h \<circ> fold g xs = fold f xs \<circ> h" by (rule fold_commute)
  then show ?thesis by (simp add: fun_eq_iff)
qed

lemma fold_invariant: 
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> Q x" and "P s"
    and "\<And>x s. Q x \<Longrightarrow> P s \<Longrightarrow> P (f x s)"
  shows "P (fold f xs s)"
  using assms by (induct xs arbitrary: s) simp_all

lemma fold_append [simp]:
  "fold f (xs @ ys) = fold f ys \<circ> fold f xs"
  by (induct xs) simp_all

lemma fold_map [code_unfold]:
  "fold g (map f xs) = fold (g o f) xs"
  by (induct xs) simp_all

lemma fold_rev:
  assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
  shows "fold f (rev xs) = fold f xs"
using assms by (induct xs) (simp_all add: fold_commute_apply fun_eq_iff)

lemma fold_Cons_rev:
  "fold Cons xs = append (rev xs)"
  by (induct xs) simp_all

lemma rev_conv_fold [code]:
  "rev xs = fold Cons xs []"
  by (simp add: fold_Cons_rev)

lemma fold_append_concat_rev:
  "fold append xss = append (concat (rev xss))"
  by (induct xss) simp_all

text {* @{const Finite_Set.fold} and @{const fold} *}

lemma (in comp_fun_commute) fold_set_fold_remdups:
  "Finite_Set.fold f y (set xs) = fold f (remdups xs) y"
  by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_left_comm insert_absorb)

lemma (in comp_fun_idem) fold_set_fold:
  "Finite_Set.fold f y (set xs) = fold f xs y"
  by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_left_comm)

lemma union_set_fold [code]:
  "set xs \<union> A = fold Set.insert xs A"
proof -
  interpret comp_fun_idem Set.insert
    by (fact comp_fun_idem_insert)
  show ?thesis by (simp add: union_fold_insert fold_set_fold)
qed

lemma union_coset_filter [code]:
  "List.coset xs \<union> A = List.coset (List.filter (\<lambda>x. x \<notin> A) xs)"
  by auto

lemma minus_set_fold [code]:
  "A - set xs = fold Set.remove xs A"
proof -
  interpret comp_fun_idem Set.remove
    by (fact comp_fun_idem_remove)
  show ?thesis
    by (simp add: minus_fold_remove [of _ A] fold_set_fold)
qed

lemma minus_coset_filter [code]:
  "A - List.coset xs = set (List.filter (\<lambda>x. x \<in> A) xs)"
  by auto

lemma inter_set_filter [code]:
  "A \<inter> set xs = set (List.filter (\<lambda>x. x \<in> A) xs)"
  by auto

lemma inter_coset_fold [code]:
  "A \<inter> List.coset xs = fold Set.remove xs A"
  by (simp add: Diff_eq [symmetric] minus_set_fold)

lemma (in semilattice_set) set_eq_fold [code]:
  "F (set (x # xs)) = fold f xs x"
proof -
  interpret comp_fun_idem f
    by default (simp_all add: fun_eq_iff left_commute)
  show ?thesis by (simp add: eq_fold fold_set_fold)
qed

lemma (in complete_lattice) Inf_set_fold:
  "Inf (set xs) = fold inf xs top"
proof -
  interpret comp_fun_idem "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
    by (fact comp_fun_idem_inf)
  show ?thesis by (simp add: Inf_fold_inf fold_set_fold inf_commute)
qed

declare Inf_set_fold [where 'a = "'a set", code]

lemma (in complete_lattice) Sup_set_fold:
  "Sup (set xs) = fold sup xs bot"
proof -
  interpret comp_fun_idem "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
    by (fact comp_fun_idem_sup)
  show ?thesis by (simp add: Sup_fold_sup fold_set_fold sup_commute)
qed

declare Sup_set_fold [where 'a = "'a set", code]

lemma (in complete_lattice) INF_set_fold:
  "INFIMUM (set xs) f = fold (inf \<circ> f) xs top"
  using Inf_set_fold [of "map f xs "] by (simp add: fold_map)

declare INF_set_fold [code]

lemma (in complete_lattice) SUP_set_fold:
  "SUPREMUM (set xs) f = fold (sup \<circ> f) xs bot"
  using Sup_set_fold [of "map f xs "] by (simp add: fold_map)

declare SUP_set_fold [code]


subsubsection {* Fold variants: @{const foldr} and @{const foldl} *}

text {* Correspondence *}

lemma foldr_conv_fold [code_abbrev]:
  "foldr f xs = fold f (rev xs)"
  by (induct xs) simp_all

lemma foldl_conv_fold:
  "foldl f s xs = fold (\<lambda>x s. f s x) xs s"
  by (induct xs arbitrary: s) simp_all

lemma foldr_conv_foldl: -- {* The ``Third Duality Theorem'' in Bird \& Wadler: *}
  "foldr f xs a = foldl (\<lambda>x y. f y x) a (rev xs)"
  by (simp add: foldr_conv_fold foldl_conv_fold)

lemma foldl_conv_foldr:
  "foldl f a xs = foldr (\<lambda>x y. f y x) (rev xs) a"
  by (simp add: foldr_conv_fold foldl_conv_fold)

lemma foldr_fold:
  assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
  shows "foldr f xs = fold f xs"
  using assms unfolding foldr_conv_fold by (rule fold_rev)

lemma foldr_cong [fundef_cong]:
  "a = b \<Longrightarrow> l = k \<Longrightarrow> (\<And>a x. x \<in> set l \<Longrightarrow> f x a = g x a) \<Longrightarrow> foldr f l a = foldr g k b"
  by (auto simp add: foldr_conv_fold intro!: fold_cong)

lemma foldl_cong [fundef_cong]:
  "a = b \<Longrightarrow> l = k \<Longrightarrow> (\<And>a x. x \<in> set l \<Longrightarrow> f a x = g a x) \<Longrightarrow> foldl f a l = foldl g b k"
  by (auto simp add: foldl_conv_fold intro!: fold_cong)

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

lemma foldl_append [simp]:
  "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
  by (simp add: foldl_conv_fold)

lemma foldr_map [code_unfold]:
  "foldr g (map f xs) a = foldr (g o f) xs a"
  by (simp add: foldr_conv_fold fold_map rev_map)

lemma foldl_map [code_unfold]:
  "foldl g a (map f xs) = foldl (\<lambda>a x. g a (f x)) a xs"
  by (simp add: foldl_conv_fold fold_map comp_def)

lemma concat_conv_foldr [code]:
  "concat xss = foldr append xss []"
  by (simp add: fold_append_concat_rev foldr_conv_fold)


subsubsection {* @{const upt} *}

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

lemmas upt_rec_numeral[simp] = upt_rec[of "numeral m" "numeral n"] for m n

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:
 "([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
apply(induct j arbitrary: x xs)
 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]"
  by (simp add: upt_rec)

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+m <= n ==> take m [i..<n] = [i..<i+m]"
apply (induct m arbitrary: i, 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..<Suc n]"
by (induct n) auto

lemma map_add_upt: "map (\<lambda>i. i + n) [0..<m] = [n..<m + n]"
  by (induct m) simp_all

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

lemma map_decr_upt:
  "map (\<lambda>n. n - Suc 0) [Suc m..<Suc n] = [m..<n]"
  by (induct n) simp_all

lemma nth_take_lemma:
  "k <= length xs ==> k <= length ys ==>
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
apply (atomize, induct k arbitrary: xs ys)
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"
  by (frule nth_take_lemma [OF le_refl eq_imp_le]) simp_all

lemma map_nth:
  "map (\<lambda>i. xs ! i) [0..<length xs] = xs"
  by (rule nth_equalityI, auto)

(* 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)
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

lemma take_Cons_numeral [simp]:
  "take (numeral v) (x # xs) = x # take (numeral v - 1) xs"
by (simp add: take_Cons')

lemma drop_Cons_numeral [simp]:
  "drop (numeral v) (x # xs) = drop (numeral v - 1) xs"
by (simp add: drop_Cons')

lemma nth_Cons_numeral [simp]:
  "(x # xs) ! numeral v = xs ! (numeral v - 1)"
by (simp add: nth_Cons')


subsubsection {* @{text upto}: interval-list on @{typ int} *}

function upto :: "int \<Rightarrow> int \<Rightarrow> int list" ("(1[_../_])") where
  "upto i j = (if i \<le> j then i # [i+1..j] else [])"
by auto
termination
by(relation "measure(%(i::int,j). nat(j - i + 1))") auto

declare upto.simps[simp del]

lemmas upto_rec_numeral [simp] =
  upto.simps[of "numeral m" "numeral n"]
  upto.simps[of "numeral m" "- numeral n"]
  upto.simps[of "- numeral m" "numeral n"]
  upto.simps[of "- numeral m" "- numeral n"] for m n

lemma upto_empty[simp]: "j < i \<Longrightarrow> [i..j] = []"
by(simp add: upto.simps)

lemma upto_rec1: "i \<le> j \<Longrightarrow> [i..j] = i#[i+1..j]"
by(simp add: upto.simps)

lemma upto_rec2: "i \<le> j \<Longrightarrow> [i..j] = [i..j - 1]@[j]"
proof(induct "nat(j-i)" arbitrary: i j)
  case 0 thus ?case by(simp add: upto.simps)
next
  case (Suc n)
  hence "n = nat (j - (i + 1))" "i < j" by linarith+
  from this(2) Suc.hyps(1)[OF this(1)] Suc(2,3) upto_rec1 show ?case by simp
qed

lemma set_upto[simp]: "set[i..j] = {i..j}"
proof(induct i j rule:upto.induct)
  case (1 i j)
  from this show ?case
    unfolding upto.simps[of i j] by auto
qed

text{* Tail recursive version for code generation: *}

definition upto_aux :: "int \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> int list" where
  "upto_aux i j js = [i..j] @ js"

lemma upto_aux_rec [code]:
  "upto_aux i j js = (if j<i then js else upto_aux i (j - 1) (j#js))"
  by (simp add: upto_aux_def upto_rec2)

lemma upto_code[code]: "[i..j] = upto_aux i j []"
by(simp add: upto_aux_def)


subsubsection {* @{const distinct} and @{const remdups} and @{const remdups_adj} *}

lemma distinct_tl:
  "distinct xs \<Longrightarrow> distinct (tl xs)"
  by (cases xs) simp_all

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 distinct_remdups_id: "distinct xs ==> remdups xs = xs"
by (induct xs, auto)

lemma remdups_id_iff_distinct [simp]: "remdups xs = xs \<longleftrightarrow> distinct xs"
by (metis distinct_remdups distinct_remdups_id)

lemma finite_distinct_list: "finite A \<Longrightarrow> EX xs. set xs = A & distinct xs"
by (metis distinct_remdups finite_list set_remdups)

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 remdups_filter: "remdups(filter P xs) = filter P (remdups xs)"
apply(induct xs)
apply auto
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_upto[simp]: "distinct[i..j]"
apply(induct i j rule:upto.induct)
apply(subst upto.simps)
apply(simp)
done

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

lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)"
apply(induct xs arbitrary: i)
 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

lemma distinct_concat:
  assumes "distinct xs"
  and "\<And> ys. ys \<in> set xs \<Longrightarrow> distinct ys"
  and "\<And> ys zs. \<lbrakk> ys \<in> set xs ; zs \<in> set xs ; ys \<noteq> zs \<rbrakk> \<Longrightarrow> set ys \<inter> set zs = {}"
  shows "distinct (concat xs)"
  using assms by (induct xs) auto

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

lemma distinct_conv_nth:
"distinct xs = (\<forall>i < 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 distinct_length_filter: "distinct xs \<Longrightarrow> length (filter P xs) = card ({x. P x} Int set xs)"
by (induct xs) (auto)

lemma not_distinct_decomp: "~ distinct ws ==> EX xs ys zs y. ws = xs@[y]@ys@[y]@zs"
apply (induct n == "length ws" arbitrary:ws) apply simp
apply(case_tac ws) apply simp
apply (simp split:split_if_asm)
apply (metis Cons_eq_appendI eq_Nil_appendI split_list)
done

lemma not_distinct_conv_prefix:
  defines "dec as xs y ys \<equiv> y \<in> set xs \<and> distinct xs \<and> as = xs @ y # ys"
  shows "\<not>distinct as \<longleftrightarrow> (\<exists>xs y ys. dec as xs y ys)" (is "?L = ?R")
proof
  assume "?L" then show "?R"
  proof (induct "length as" arbitrary: as rule: less_induct)
    case less
    obtain xs ys zs y where decomp: "as = (xs @ y # ys) @ y # zs"
      using not_distinct_decomp[OF less.prems] by auto
    show ?case
    proof (cases "distinct (xs @ y # ys)")
      case True
      with decomp have "dec as (xs @ y # ys) y zs" by (simp add: dec_def)
      then show ?thesis by blast
    next
      case False
      with less decomp obtain xs' y' ys' where "dec (xs @ y # ys) xs' y' ys'"
        by atomize_elim auto
      with decomp have "dec as xs' y' (ys' @ y # zs)" by (simp add: dec_def)
      then show ?thesis by blast
    qed
  qed
qed (auto simp: dec_def)

lemma distinct_product:
  assumes "distinct xs" and "distinct ys"
  shows "distinct (List.product xs ys)"
  using assms by (induct xs)
    (auto intro: inj_onI simp add: product_list_set distinct_map)

lemma distinct_product_lists:
  assumes "\<forall>xs \<in> set xss. distinct xs"
  shows "distinct (product_lists xss)"
using assms proof (induction xss)
  case (Cons xs xss) note * = this
  then show ?case
  proof (cases "product_lists xss")
    case Nil then show ?thesis by (induct xs) simp_all
  next
    case (Cons ps pss) with * show ?thesis 
      by (auto intro!: inj_onI distinct_concat simp add: distinct_map)
  qed
qed simp

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

lemma length_remdups_card_conv: "length(remdups xs) = card(set xs)"
proof -
  have xs: "concat[xs] = xs" by simp
  from length_remdups_concat[of "[xs]"] show ?thesis unfolding xs by simp
qed

lemma remdups_remdups:
  "remdups (remdups xs) = remdups xs"
  by (induct xs) simp_all

lemma distinct_butlast:
  assumes "distinct xs"
  shows "distinct (butlast xs)"
proof (cases "xs = []")
  case False
    from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
    with `distinct xs` show ?thesis by simp
qed (auto)

lemma remdups_map_remdups:
  "remdups (map f (remdups xs)) = remdups (map f xs)"
  by (induct xs) simp_all

lemma distinct_zipI1:
  assumes "distinct xs"
  shows "distinct (zip xs ys)"
proof (rule zip_obtain_same_length)
  fix xs' :: "'a list" and ys' :: "'b list" and n
  assume "length xs' = length ys'"
  assume "xs' = take n xs"
  with assms have "distinct xs'" by simp
  with `length xs' = length ys'` show "distinct (zip xs' ys')"
    by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
qed

lemma distinct_zipI2:
  assumes "distinct ys"
  shows "distinct (zip xs ys)"
proof (rule zip_obtain_same_length)
  fix xs' :: "'b list" and ys' :: "'a list" and n
  assume "length xs' = length ys'"
  assume "ys' = take n ys"
  with assms have "distinct ys'" by simp
  with `length xs' = length ys'` show "distinct (zip xs' ys')"
    by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
qed

lemma set_take_disj_set_drop_if_distinct:
  "distinct vs \<Longrightarrow> i \<le> j \<Longrightarrow> set (take i vs) \<inter> set (drop j vs) = {}"
by (auto simp: in_set_conv_nth distinct_conv_nth)

(* The next two lemmas help Sledgehammer. *)

lemma distinct_singleton: "distinct [x]" by simp

lemma distinct_length_2_or_more:
"distinct (a # b # xs) \<longleftrightarrow> (a \<noteq> b \<and> distinct (a # xs) \<and> distinct (b # xs))"
by force

lemma remdups_adj_Cons: "remdups_adj (x # xs) =
  (case remdups_adj xs of [] \<Rightarrow> [x] | y # xs \<Rightarrow> if x = y then y # xs else x # y # xs)"
  by (induct xs arbitrary: x) (auto split: list.splits)

lemma remdups_adj_append_two: 
  "remdups_adj (xs @ [x,y]) = remdups_adj (xs @ [x]) @ (if x = y then [] else [y])"
  by (induct xs rule: remdups_adj.induct, simp_all)

lemma remdups_adj_rev[simp]: "remdups_adj (rev xs) = rev (remdups_adj xs)"
  by (induct xs rule: remdups_adj.induct, simp_all add: remdups_adj_append_two)

lemma remdups_adj_length[simp]: "length (remdups_adj xs) \<le> length xs"
  by (induct xs rule: remdups_adj.induct, auto)

lemma remdups_adj_length_ge1[simp]: "xs \<noteq> [] \<Longrightarrow> length (remdups_adj xs) \<ge> Suc 0"
  by (induct xs rule: remdups_adj.induct, simp_all)

lemma remdups_adj_Nil_iff[simp]: "remdups_adj xs = [] \<longleftrightarrow> xs = []"
  by (induct xs rule: remdups_adj.induct, simp_all)

lemma remdups_adj_set[simp]: "set (remdups_adj xs) = set xs"
  by (induct xs rule: remdups_adj.induct, simp_all)

lemma remdups_adj_Cons_alt[simp]: "x # tl (remdups_adj (x # xs)) = remdups_adj (x # xs)"
    by (induct xs rule: remdups_adj.induct, auto)

lemma remdups_adj_distinct: "distinct xs \<Longrightarrow> remdups_adj xs = xs"
    by (induct xs rule: remdups_adj.induct, simp_all)

lemma remdups_adj_append: 
  "remdups_adj (xs\<^sub>1 @ x # xs\<^sub>2) = remdups_adj (xs\<^sub>1 @ [x]) @ tl (remdups_adj (x # xs\<^sub>2))"
  by (induct xs\<^sub>1 rule: remdups_adj.induct, simp_all)

lemma remdups_adj_singleton:
  "remdups_adj xs = [x] \<Longrightarrow> xs = replicate (length xs) x"
  by (induct xs rule: remdups_adj.induct, auto split: split_if_asm)

lemma remdups_adj_map_injective:
  assumes "inj f"
  shows "remdups_adj (map f xs) = map f (remdups_adj xs)"
  by (induct xs rule: remdups_adj.induct, 
      auto simp add: injD[OF assms])


subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}

lemma (in monoid_add) listsum_simps [simp]:
  "listsum [] = 0"
  "listsum (x # xs) = x + listsum xs"
  by (simp_all add: listsum_def)

lemma (in monoid_add) listsum_append [simp]:
  "listsum (xs @ ys) = listsum xs + listsum ys"
  by (induct xs) (simp_all add: add.assoc)

lemma (in comm_monoid_add) listsum_rev [simp]:
  "listsum (rev xs) = listsum xs"
  by (simp add: listsum_def foldr_fold fold_rev fun_eq_iff add_ac)

lemma (in monoid_add) fold_plus_listsum_rev:
  "fold plus xs = plus (listsum (rev xs))"
proof
  fix x
  have "fold plus xs x = fold plus xs (x + 0)" by simp
  also have "\<dots> = fold plus (x # xs) 0" by simp
  also have "\<dots> = foldr plus (rev xs @ [x]) 0" by (simp add: foldr_conv_fold)
  also have "\<dots> = listsum (rev xs @ [x])" by (simp add: listsum_def)
  also have "\<dots> = listsum (rev xs) + listsum [x]" by simp
  finally show "fold plus xs x = listsum (rev xs) + x" by simp
qed

text{* Some syntactic sugar for summing a function over a list: *}

syntax
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
syntax (xsymbols)
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
syntax (HTML output)
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)

translations -- {* Beware of argument permutation! *}
  "SUM x<-xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"

lemma (in comm_monoid_add) listsum_map_remove1:
  "x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))"
  by (induct xs) (auto simp add: ac_simps)

lemma (in monoid_add) list_size_conv_listsum:
  "list_size f xs = listsum (map f xs) + size xs"
  by (induct xs) auto

lemma (in monoid_add) length_concat:
  "length (concat xss) = listsum (map length xss)"
  by (induct xss) simp_all

lemma (in monoid_add) length_product_lists:
  "length (product_lists xss) = foldr op * (map length xss) 1"
proof (induct xss)
  case (Cons xs xss) then show ?case by (induct xs) (auto simp: length_concat o_def)
qed simp

lemma (in monoid_add) listsum_map_filter:
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
  shows "listsum (map f (filter P xs)) = listsum (map f xs)"
  using assms by (induct xs) auto

lemma (in comm_monoid_add) distinct_listsum_conv_Setsum:
  "distinct xs \<Longrightarrow> listsum xs = Setsum (set xs)"
  by (induct xs) simp_all

lemma listsum_eq_0_nat_iff_nat [simp]:
  "listsum ns = (0::nat) \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
  by (induct ns) simp_all

lemma member_le_listsum_nat:
  "(n :: nat) \<in> set ns \<Longrightarrow> n \<le> listsum ns"
  by (induct ns) auto

lemma elem_le_listsum_nat:
  "k < size ns \<Longrightarrow> ns ! k \<le> listsum (ns::nat list)"
  by (rule member_le_listsum_nat) simp

lemma listsum_update_nat:
  "k < size ns \<Longrightarrow> listsum (ns[k := (n::nat)]) = listsum ns + n - ns ! k"
apply(induct ns arbitrary:k)
 apply (auto split:nat.split)
apply(drule elem_le_listsum_nat)
apply arith
done

lemma (in monoid_add) listsum_triv:
  "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
  by (induct xs) (simp_all add: distrib_right)

lemma (in monoid_add) listsum_0 [simp]:
  "(\<Sum>x\<leftarrow>xs. 0) = 0"
  by (induct xs) (simp_all add: distrib_right)

text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
lemma (in ab_group_add) uminus_listsum_map:
  "- listsum (map f xs) = listsum (map (uminus \<circ> f) xs)"
  by (induct xs) simp_all

lemma (in comm_monoid_add) listsum_addf:
  "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
  by (induct xs) (simp_all add: algebra_simps)

lemma (in ab_group_add) listsum_subtractf:
  "(\<Sum>x\<leftarrow>xs. f x - g x) = listsum (map f xs) - listsum (map g xs)"
  by (induct xs) (simp_all add: algebra_simps)

lemma (in semiring_0) listsum_const_mult:
  "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
  by (induct xs) (simp_all add: algebra_simps)

lemma (in semiring_0) listsum_mult_const:
  "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
  by (induct xs) (simp_all add: algebra_simps)

lemma (in ordered_ab_group_add_abs) listsum_abs:
  "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
  by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])

lemma listsum_mono:
  fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}"
  shows "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (\<Sum>x\<leftarrow>xs. f x) \<le> (\<Sum>x\<leftarrow>xs. g x)"
  by (induct xs) (simp, simp add: add_mono)

lemma (in monoid_add) listsum_distinct_conv_setsum_set:
  "distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)"
  by (induct xs) simp_all

lemma (in monoid_add) interv_listsum_conv_setsum_set_nat:
  "listsum (map f [m..<n]) = setsum f (set [m..<n])"
  by (simp add: listsum_distinct_conv_setsum_set)

lemma (in monoid_add) interv_listsum_conv_setsum_set_int:
  "listsum (map f [k..l]) = setsum f (set [k..l])"
  by (simp add: listsum_distinct_conv_setsum_set)

text {* General equivalence between @{const listsum} and @{const setsum} *}
lemma (in monoid_add) listsum_setsum_nth:
  "listsum xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
  using interv_listsum_conv_setsum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth)


subsubsection {* @{const insert} *}

lemma in_set_insert [simp]:
  "x \<in> set xs \<Longrightarrow> List.insert x xs = xs"
  by (simp add: List.insert_def)

lemma not_in_set_insert [simp]:
  "x \<notin> set xs \<Longrightarrow> List.insert x xs = x # xs"
  by (simp add: List.insert_def)

lemma insert_Nil [simp]:
  "List.insert x [] = [x]"
  by simp

lemma set_insert [simp]:
  "set (List.insert x xs) = insert x (set xs)"
  by (auto simp add: List.insert_def)

lemma distinct_insert [simp]:
  "distinct xs \<Longrightarrow> distinct (List.insert x xs)"
  by (simp add: List.insert_def)

lemma insert_remdups:
  "List.insert x (remdups xs) = remdups (List.insert x xs)"
  by (simp add: List.insert_def)


subsubsection {* @{const List.find} *}

lemma find_None_iff: "List.find P xs = None \<longleftrightarrow> \<not> (\<exists>x. x \<in> set xs \<and> P x)"
proof (induction xs)
  case Nil thus ?case by simp
next
  case (Cons x xs) thus ?case by (fastforce split: if_splits)
qed

lemma find_Some_iff:
  "List.find P xs = Some x \<longleftrightarrow>
  (\<exists>i<length xs. P (xs!i) \<and> x = xs!i \<and> (\<forall>j<i. \<not> P (xs!j)))"
proof (induction xs)
  case Nil thus ?case by simp
next
  case (Cons x xs) thus ?case
    apply(auto simp: nth_Cons' split: if_splits)
    using diff_Suc_1[unfolded One_nat_def] less_Suc_eq_0_disj by fastforce
qed

lemma find_cong[fundef_cong]:
  assumes "xs = ys" and "\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x" 
  shows "List.find P xs = List.find Q ys"
proof (cases "List.find P xs")
  case None thus ?thesis by (metis find_None_iff assms)
next
  case (Some x)
  hence "List.find Q ys = Some x" using assms
    by (auto simp add: find_Some_iff)
  thus ?thesis using Some by auto
qed

lemma find_dropWhile:
  "List.find P xs = (case dropWhile (Not \<circ> P) xs
   of [] \<Rightarrow> None
    | x # _ \<Rightarrow> Some x)"
  by (induct xs) simp_all


subsubsection {* @{const List.extract} *}

lemma extract_None_iff: "List.extract P xs = None \<longleftrightarrow> \<not> (\<exists> x\<in>set xs. P x)"
by(auto simp: extract_def dropWhile_eq_Cons_conv split: list.splits)
  (metis in_set_conv_decomp)

lemma extract_SomeE:
 "List.extract P xs = Some (ys, y, zs) \<Longrightarrow>
  xs = ys @ y # zs \<and> P y \<and> \<not> (\<exists> y \<in> set ys. P y)" 
by(auto simp: extract_def dropWhile_eq_Cons_conv split: list.splits)

lemma extract_Some_iff:
  "List.extract P xs = Some (ys, y, zs) \<longleftrightarrow>
   xs = ys @ y # zs \<and> P y \<and> \<not> (\<exists> y \<in> set ys. P y)" 
by(auto simp: extract_def dropWhile_eq_Cons_conv dest: set_takeWhileD split: list.splits)

lemma extract_Nil_code[code]: "List.extract P [] = None"
by(simp add: extract_def)

lemma extract_Cons_code[code]:
  "List.extract P (x # xs) = (if P x then Some ([], x, xs) else
   (case List.extract P xs of
      None \<Rightarrow> None |
      Some (ys, y, zs) \<Rightarrow> Some (x#ys, y, zs)))"
by(auto simp add: extract_def comp_def split: list.splits)
  (metis dropWhile_eq_Nil_conv list.distinct(1))


subsubsection {* @{const 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 remove1_commute: "remove1 x (remove1 y zs) = remove1 y (remove1 x zs)"
by (induct zs) auto

lemma in_set_remove1[simp]:
  "a \<noteq> b \<Longrightarrow> a : set(remove1 b xs) = (a : set xs)"
apply (induct xs)
apply auto
done

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 length_remove1:
  "length(remove1 x xs) = (if x : set xs then length xs - 1 else length xs)"
apply (induct xs)
 apply (auto dest!:length_pos_if_in_set)
done

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

lemma filter_remove1:
  "filter Q (remove1 x xs) = remove1 x (filter Q 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

lemma remove1_remdups:
  "distinct xs \<Longrightarrow> remove1 x (remdups xs) = remdups (remove1 x xs)"
  by (induct xs) simp_all

lemma remove1_idem:
  assumes "x \<notin> set xs"
  shows "remove1 x xs = xs"
  using assms by (induct xs) simp_all


subsubsection {* @{const removeAll} *}

lemma removeAll_filter_not_eq:
  "removeAll x = filter (\<lambda>y. x \<noteq> y)"
proof
  fix xs
  show "removeAll x xs = filter (\<lambda>y. x \<noteq> y) xs"
    by (induct xs) auto
qed

lemma removeAll_append[simp]:
  "removeAll x (xs @ ys) = removeAll x xs @ removeAll x ys"
by (induct xs) auto

lemma set_removeAll[simp]: "set(removeAll x xs) = set xs - {x}"
by (induct xs) auto

lemma removeAll_id[simp]: "x \<notin> set xs \<Longrightarrow> removeAll x xs = xs"
by (induct xs) auto

(* Needs count:: 'a \<Rightarrow> 'a list \<Rightarrow> nat
lemma length_removeAll:
  "length(removeAll x xs) = length xs - count x xs"
*)

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

lemma distinct_removeAll:
  "distinct xs \<Longrightarrow> distinct (removeAll x xs)"
  by (simp add: removeAll_filter_not_eq)

lemma distinct_remove1_removeAll:
  "distinct xs ==> remove1 x xs = removeAll x xs"
by (induct xs) simp_all

lemma map_removeAll_inj_on: "inj_on f (insert x (set xs)) \<Longrightarrow>
  map f (removeAll x xs) = removeAll (f x) (map f xs)"
by (induct xs) (simp_all add:inj_on_def)

lemma map_removeAll_inj: "inj f \<Longrightarrow>
  map f (removeAll x xs) = removeAll (f x) (map f xs)"
by (rule map_removeAll_inj_on, erule subset_inj_on, rule subset_UNIV)


subsubsection {* @{const replicate} *}

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

lemma Ex_list_of_length: "\<exists>xs. length xs = n"
by (rule exI[of _ "replicate n undefined"]) simp

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

lemma map_replicate_const:
  "map (\<lambda> x. k) lst = replicate (length lst) k"
  by (induct lst) 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

text{* Courtesy of Andreas Lochbihler: *}
lemma filter_replicate:
  "filter P (replicate n x) = (if P x then replicate n x else [])"
by(induct n) auto

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

lemma tl_replicate [simp]: "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 < n ==> (replicate n x)!i = x"
apply (induct n arbitrary: i, 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]: "drop i (replicate k x) = replicate (k-i) x"
apply (induct k arbitrary: i)
 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_replicate[simp]: "(x : set (replicate n y)) = (x = y & n \<noteq> 0)"
by (simp add: set_replicate_conv_if)

lemma Ball_set_replicate[simp]:
  "(ALL x : set(replicate n a). P x) = (P a | n=0)"
by(simp add: set_replicate_conv_if)

lemma Bex_set_replicate[simp]:
  "(EX x : set(replicate n a). P x) = (P a & n\<noteq>0)"
by(simp add: set_replicate_conv_if)

lemma replicate_append_same:
  "replicate i x @ [x] = x # replicate i x"
  by (induct i) simp_all

lemma map_replicate_trivial:
  "map (\<lambda>i. x) [0..<i] = replicate i x"
  by (induct i) (simp_all add: replicate_append_same)

lemma concat_replicate_trivial[simp]:
  "concat (replicate i []) = []"
  by (induct i) (auto simp add: map_replicate_const)

lemma replicate_empty[simp]: "(replicate n x = []) \<longleftrightarrow> n=0"
by (induct n) auto

lemma empty_replicate[simp]: "([] = replicate n x) \<longleftrightarrow> n=0"
by (induct n) auto

lemma replicate_eq_replicate[simp]:
  "(replicate m x = replicate n y) \<longleftrightarrow> (m=n & (m\<noteq>0 \<longrightarrow> x=y))"
apply(induct m arbitrary: n)
 apply simp
apply(induct_tac n)
apply auto
done

lemma replicate_length_filter:
  "replicate (length (filter (\<lambda>y. x = y) xs)) x = filter (\<lambda>y. x = y) xs"
  by (induct xs) auto

lemma comm_append_are_replicate:
  fixes xs ys :: "'a list"
  assumes "xs \<noteq> []" "ys \<noteq> []"
  assumes "xs @ ys = ys @ xs"
  shows "\<exists>m n zs. concat (replicate m zs) = xs \<and> concat (replicate n zs) = ys"
  using assms
proof (induct "length (xs @ ys)" arbitrary: xs ys rule: less_induct)
  case less

  def xs' \<equiv> "if (length xs \<le> length ys) then xs else ys"
    and ys' \<equiv> "if (length xs \<le> length ys) then ys else xs"
  then have
    prems': "length xs' \<le> length ys'"
            "xs' @ ys' = ys' @ xs'"
      and "xs' \<noteq> []"
      and len: "length (xs @ ys) = length (xs' @ ys')"
    using less by (auto intro: less.hyps)

  from prems'
  obtain ws where "ys' = xs' @ ws"
    by (auto simp: append_eq_append_conv2)

  have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ys'"
  proof (cases "ws = []")
    case True
    then have "concat (replicate 1 xs') = xs'"
      and "concat (replicate 1 xs') = ys'"
      using `ys' = xs' @ ws` by auto
    then show ?thesis by blast
  next
    case False
    from `ys' = xs' @ ws` and `xs' @ ys' = ys' @ xs'`
    have "xs' @ ws = ws @ xs'" by simp
    then have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ws"
      using False and `xs' \<noteq> []` and `ys' = xs' @ ws` and len
      by (intro less.hyps) auto
    then obtain m n zs where *: "concat (replicate m zs) = xs'"
      and "concat (replicate n zs) = ws" by blast
    then have "concat (replicate (m + n) zs) = ys'"
      using `ys' = xs' @ ws`
      by (simp add: replicate_add)
    with * show ?thesis by blast
  qed
  then show ?case
    using xs'_def ys'_def by meson
qed

lemma comm_append_is_replicate:
  fixes xs ys :: "'a list"
  assumes "xs \<noteq> []" "ys \<noteq> []"
  assumes "xs @ ys = ys @ xs"
  shows "\<exists>n zs. n > 1 \<and> concat (replicate n zs) = xs @ ys"

proof -
  obtain m n zs where "concat (replicate m zs) = xs"
    and "concat (replicate n zs) = ys"
    using comm_append_are_replicate[of xs ys, OF assms] by blast
  then have "m + n > 1" and "concat (replicate (m+n) zs) = xs @ ys"
    using `xs \<noteq> []` and `ys \<noteq> []`
    by (auto simp: replicate_add)
  then show ?thesis by blast
qed

lemma Cons_replicate_eq:
  "x # xs = replicate n y \<longleftrightarrow> x = y \<and> n > 0 \<and> xs = replicate (n - 1) x"
  by (induct n) auto

lemma replicate_length_same:
  "(\<forall>y\<in>set xs. y = x) \<Longrightarrow> replicate (length xs) x = xs"
  by (induct xs) simp_all

lemma foldr_replicate [simp]:
  "foldr f (replicate n x) = f x ^^ n"
  by (induct n) (simp_all)

lemma fold_replicate [simp]:
  "fold f (replicate n x) = f x ^^ n"
  by (subst foldr_fold [symmetric]) simp_all


subsubsection {* @{const enumerate} *}

lemma enumerate_simps [simp, code]:
  "enumerate n [] = []"
  "enumerate n (x # xs) = (n, x) # enumerate (Suc n) xs"
  apply (auto simp add: enumerate_eq_zip not_le)
  apply (cases "n < n + length xs")
  apply (auto simp add: upt_conv_Cons)
  done

lemma length_enumerate [simp]:
  "length (enumerate n xs) = length xs"
  by (simp add: enumerate_eq_zip)

lemma map_fst_enumerate [simp]:
  "map fst (enumerate n xs) = [n..<n + length xs]"
  by (simp add: enumerate_eq_zip)

lemma map_snd_enumerate [simp]:
  "map snd (enumerate n xs) = xs"
  by (simp add: enumerate_eq_zip)
  
lemma in_set_enumerate_eq:
  "p \<in> set (enumerate n xs) \<longleftrightarrow> n \<le> fst p \<and> fst p < length xs + n \<and> nth xs (fst p - n) = snd p"
proof -
  { fix m
    assume "n \<le> m"
    moreover assume "m < length xs + n"
    ultimately have "[n..<n + length xs] ! (m - n) = m \<and>
      xs ! (m - n) = xs ! (m - n) \<and> m - n < length xs" by auto
    then have "\<exists>q. [n..<n + length xs] ! q = m \<and>
        xs ! q = xs ! (m - n) \<and> q < length xs" ..
  } then show ?thesis by (cases p) (auto simp add: enumerate_eq_zip in_set_zip)
qed

lemma nth_enumerate_eq:
  assumes "m < length xs"
  shows "enumerate n xs ! m = (n + m, xs ! m)"
  using assms by (simp add: enumerate_eq_zip)

lemma enumerate_replicate_eq:
  "enumerate n (replicate m a) = map (\<lambda>q. (q, a)) [n..<n + m]"
  by (rule pair_list_eqI)
    (simp_all add: enumerate_eq_zip comp_def map_replicate_const)

lemma enumerate_Suc_eq:
  "enumerate (Suc n) xs = map (apfst Suc) (enumerate n xs)"
  by (rule pair_list_eqI)
    (simp_all add: not_le, simp del: map_map [simp del] add: map_Suc_upt map_map [symmetric])

lemma distinct_enumerate [simp]:
  "distinct (enumerate n xs)"
  by (simp add: enumerate_eq_zip distinct_zipI1)


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

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 (cases xs) simp_all

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 (cases xs) simp_all

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

lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
by (cases xs) auto

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 (cases xs) auto

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 (cases xs) auto

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 {* @{const sublist} --- a generalization of @{const 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:
  "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 arbitrary: "is")
 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] 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: "set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
apply(induct xs arbitrary: I)
apply(auto simp: sublist_Cons nth_Cons split:nat.split dest!: gr0_implies_Suc)
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]: "distinct xs \<Longrightarrow> distinct(sublist xs I)"
apply(induct xs arbitrary: I)
 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:
 "distinct xs \<Longrightarrow> filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
proof (induct xs arbitrary: s)
  case Nil thus ?case by simp
next
  case (Cons a xs)
  then have "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
  with Cons show ?case by(simp add: sublist_Cons cong:filter_cong)
qed


subsubsection {* @{const sublists} and @{const List.n_lists} *}

lemma length_sublists:
  "length (sublists xs) = 2 ^ length xs"
  by (induct xs) (simp_all add: Let_def)

lemma sublists_powset:
  "set ` set (sublists xs) = Pow (set xs)"
proof -
  have aux: "\<And>x A. set ` Cons x ` A = insert x ` set ` A"
    by (auto simp add: image_def)
  have "set (map set (sublists xs)) = Pow (set xs)"
    by (induct xs)
      (simp_all add: aux Let_def Pow_insert Un_commute comp_def del: map_map)
  then show ?thesis by simp
qed

lemma distinct_set_sublists:
  assumes "distinct xs"
  shows "distinct (map set (sublists xs))"
proof (rule card_distinct)
  have "finite (set xs)" by rule
  then have "card (Pow (set xs)) = 2 ^ card (set xs)" by (rule card_Pow)
  with assms distinct_card [of xs]
    have "card (Pow (set xs)) = 2 ^ length xs" by simp
  then show "card (set (map set (sublists xs))) = length (map set (sublists xs))"
    by (simp add: sublists_powset length_sublists)
qed

lemma n_lists_Nil [simp]: "List.n_lists n [] = (if n = 0 then [[]] else [])"
  by (induct n) simp_all

lemma length_n_lists: "length (List.n_lists n xs) = length xs ^ n"
  by (induct n) (auto simp add: length_concat o_def listsum_triv)

lemma length_n_lists_elem: "ys \<in> set (List.n_lists n xs) \<Longrightarrow> length ys = n"
  by (induct n arbitrary: ys) auto

lemma set_n_lists: "set (List.n_lists n xs) = {ys. length ys = n \<and> set ys \<subseteq> set xs}"
proof (rule set_eqI)
  fix ys :: "'a list"
  show "ys \<in> set (List.n_lists n xs) \<longleftrightarrow> ys \<in> {ys. length ys = n \<and> set ys \<subseteq> set xs}"
  proof -
    have "ys \<in> set (List.n_lists n xs) \<Longrightarrow> length ys = n"
      by (induct n arbitrary: ys) auto
    moreover have "\<And>x. ys \<in> set (List.n_lists n xs) \<Longrightarrow> x \<in> set ys \<Longrightarrow> x \<in> set xs"
      by (induct n arbitrary: ys) auto
    moreover have "set ys \<subseteq> set xs \<Longrightarrow> ys \<in> set (List.n_lists (length ys) xs)"
      by (induct ys) auto
    ultimately show ?thesis by auto
  qed
qed

lemma distinct_n_lists:
  assumes "distinct xs"
  shows "distinct (List.n_lists n xs)"
proof (rule card_distinct)
  from assms have card_length: "card (set xs) = length xs" by (rule distinct_card)
  have "card (set (List.n_lists n xs)) = card (set xs) ^ n"
  proof (induct n)
    case 0 then show ?case by simp
  next
    case (Suc n)
    moreover have "card (\<Union>ys\<in>set (List.n_lists n xs). (\<lambda>y. y # ys) ` set xs)
      = (\<Sum>ys\<in>set (List.n_lists n xs). card ((\<lambda>y. y # ys) ` set xs))"
      by (rule card_UN_disjoint) auto
    moreover have "\<And>ys. card ((\<lambda>y. y # ys) ` set xs) = card (set xs)"
      by (rule card_image) (simp add: inj_on_def)
    ultimately show ?case by auto
  qed
  also have "\<dots> = length xs ^ n" by (simp add: card_length)
  finally show "card (set (List.n_lists n xs)) = length (List.n_lists n xs)"
    by (simp add: length_n_lists)
qed


subsubsection {* @{const splice} *}

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

declare splice.simps(1,3)[code]
declare splice.simps(2)[simp del]

lemma length_splice[simp]: "length(splice xs ys) = length xs + length ys"
by (induct xs ys rule: splice.induct) auto


subsubsection {* Transpose *}

function transpose where
"transpose []             = []" |
"transpose ([]     # xss) = transpose xss" |
"transpose ((x#xs) # xss) =
  (x # [h. (h#t) \<leftarrow> xss]) # transpose (xs # [t. (h#t) \<leftarrow> xss])"
by pat_completeness auto

lemma transpose_aux_filter_head:
  "concat (map (case_list [] (\<lambda>h t. [h])) xss) =
  map (\<lambda>xs. hd xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  by (induct xss) (auto split: list.split)

lemma transpose_aux_filter_tail:
  "concat (map (case_list [] (\<lambda>h t. [t])) xss) =
  map (\<lambda>xs. tl xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  by (induct xss) (auto split: list.split)

lemma transpose_aux_max:
  "max (Suc (length xs)) (foldr (\<lambda>xs. max (length xs)) xss 0) =
  Suc (max (length xs) (foldr (\<lambda>x. max (length x - Suc 0)) [ys\<leftarrow>xss . ys\<noteq>[]] 0))"
  (is "max _ ?foldB = Suc (max _ ?foldA)")
proof (cases "[ys\<leftarrow>xss . ys\<noteq>[]] = []")
  case True
  hence "foldr (\<lambda>xs. max (length xs)) xss 0 = 0"
  proof (induct xss)
    case (Cons x xs)
    then have "x = []" by (cases x) auto
    with Cons show ?case by auto
  qed simp
  thus ?thesis using True by simp
next
  case False

  have foldA: "?foldA = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0 - 1"
    by (induct xss) auto
  have foldB: "?foldB = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0"
    by (induct xss) auto

  have "0 < ?foldB"
  proof -
    from False
    obtain z zs where zs: "[ys\<leftarrow>xss . ys \<noteq> []] = z#zs" by (auto simp: neq_Nil_conv)
    hence "z \<in> set ([ys\<leftarrow>xss . ys \<noteq> []])" by auto
    hence "z \<noteq> []" by auto
    thus ?thesis
      unfolding foldB zs
      by (auto simp: max_def intro: less_le_trans)
  qed
  thus ?thesis
    unfolding foldA foldB max_Suc_Suc[symmetric]
    by simp
qed

termination transpose
  by (relation "measure (\<lambda>xs. foldr (\<lambda>xs. max (length xs)) xs 0 + length xs)")
     (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max less_Suc_eq_le)

lemma transpose_empty: "(transpose xs = []) \<longleftrightarrow> (\<forall>x \<in> set xs. x = [])"
  by (induct rule: transpose.induct) simp_all

lemma length_transpose:
  fixes xs :: "'a list list"
  shows "length (transpose xs) = foldr (\<lambda>xs. max (length xs)) xs 0"
  by (induct rule: transpose.induct)
    (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max
                max_Suc_Suc[symmetric] simp del: max_Suc_Suc)

lemma nth_transpose:
  fixes xs :: "'a list list"
  assumes "i < length (transpose xs)"
  shows "transpose xs ! i = map (\<lambda>xs. xs ! i) [ys \<leftarrow> xs. i < length ys]"
using assms proof (induct arbitrary: i rule: transpose.induct)
  case (3 x xs xss)
  def XS == "(x # xs) # xss"
  hence [simp]: "XS \<noteq> []" by auto
  thus ?case
  proof (cases i)
    case 0
    thus ?thesis by (simp add: transpose_aux_filter_head hd_conv_nth)
  next
    case (Suc j)
    have *: "\<And>xss. xs # map tl xss = map tl ((x#xs)#xss)" by simp
    have **: "\<And>xss. (x#xs) # filter (\<lambda>ys. ys \<noteq> []) xss = filter (\<lambda>ys. ys \<noteq> []) ((x#xs)#xss)" by simp
    { fix x have "Suc j < length x \<longleftrightarrow> x \<noteq> [] \<and> j < length x - Suc 0"
      by (cases x) simp_all
    } note *** = this

    have j_less: "j < length (transpose (xs # concat (map (case_list [] (\<lambda>h t. [t])) xss)))"
      using "3.prems" by (simp add: transpose_aux_filter_tail length_transpose Suc)

    show ?thesis
      unfolding transpose.simps `i = Suc j` nth_Cons_Suc "3.hyps"[OF j_less]
      apply (auto simp: transpose_aux_filter_tail filter_map comp_def length_transpose * ** *** XS_def[symmetric])
      apply (rule list.exhaust)
      by auto
  qed
qed simp_all

lemma transpose_map_map:
  "transpose (map (map f) xs) = map (map f) (transpose xs)"
proof (rule nth_equalityI, safe)
  have [simp]: "length (transpose (map (map f) xs)) = length (transpose xs)"
    by (simp add: length_transpose foldr_map comp_def)
  show "length (transpose (map (map f) xs)) = length (map (map f) (transpose xs))" by simp

  fix i assume "i < length (transpose (map (map f) xs))"
  thus "transpose (map (map f) xs) ! i = map (map f) (transpose xs) ! i"
    by (simp add: nth_transpose filter_map comp_def)
qed


subsubsection {* (In)finiteness *}

lemma finite_maxlen:
  "finite (M::'a list set) ==> EX n. ALL s:M. size s < n"
proof (induct rule: finite.induct)
  case emptyI show ?case by simp
next
  case (insertI M xs)
  then obtain n where "\<forall>s\<in>M. length s < n" by blast
  hence "ALL s:insert xs M. size s < max n (size xs) + 1" by auto
  thus ?case ..
qed

lemma lists_length_Suc_eq:
  "{xs. set xs \<subseteq> A \<and> length xs = Suc n} =
    (\<lambda>(xs, n). n#xs) ` ({xs. set xs \<subseteq> A \<and> length xs = n} \<times> A)"
  by (auto simp: length_Suc_conv)

lemma
  assumes "finite A"
  shows finite_lists_length_eq: "finite {xs. set xs \<subseteq> A \<and> length xs = n}"
  and card_lists_length_eq: "card {xs. set xs \<subseteq> A \<and> length xs = n} = (card A)^n"
  using `finite A`
  by (induct n)
     (auto simp: card_image inj_split_Cons lists_length_Suc_eq cong: conj_cong)

lemma finite_lists_length_le:
  assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs \<le> n}"
 (is "finite ?S")
proof-
  have "?S = (\<Union>n\<in>{0..n}. {xs. set xs \<subseteq> A \<and> length xs = n})" by auto
  thus ?thesis by (auto intro!: finite_lists_length_eq[OF `finite A`] simp only:)
qed

lemma card_lists_length_le:
  assumes "finite A" shows "card {xs. set xs \<subseteq> A \<and> length xs \<le> n} = (\<Sum>i\<le>n. card A^i)"
proof -
  have "(\<Sum>i\<le>n. card A^i) = card (\<Union>i\<le>n. {xs. set xs \<subseteq> A \<and> length xs = i})"
    using `finite A`
    by (subst card_UN_disjoint)
       (auto simp add: card_lists_length_eq finite_lists_length_eq)
  also have "(\<Union>i\<le>n. {xs. set xs \<subseteq> A \<and> length xs = i}) = {xs. set xs \<subseteq> A \<and> length xs \<le> n}"
    by auto
  finally show ?thesis by simp
qed

lemma card_lists_distinct_length_eq:
  assumes "k < card A"
  shows "card {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A} = \<Prod>{card A - k + 1 .. card A}"
using assms
proof (induct k)
  case 0
  then have "{xs. length xs = 0 \<and> distinct xs \<and> set xs \<subseteq> A} = {[]}" by auto
  then show ?case by simp
next
  case (Suc k)
  let "?k_list" = "\<lambda>k xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A"
  have inj_Cons: "\<And>A. inj_on (\<lambda>(xs, n). n # xs) A"  by (rule inj_onI) auto

  from Suc have "k < card A" by simp
  moreover have "finite A" using assms by (simp add: card_ge_0_finite)
  moreover have "finite {xs. ?k_list k xs}"
    using finite_lists_length_eq[OF `finite A`, of k]
    by - (rule finite_subset, auto)
  moreover have "\<And>i j. i \<noteq> j \<longrightarrow> {i} \<times> (A - set i) \<inter> {j} \<times> (A - set j) = {}"
    by auto
  moreover have "\<And>i. i \<in>Collect (?k_list k) \<Longrightarrow> card (A - set i) = card A - k"
    by (simp add: card_Diff_subset distinct_card)
  moreover have "{xs. ?k_list (Suc k) xs} =
      (\<lambda>(xs, n). n#xs) ` \<Union>((\<lambda>xs. {xs} \<times> (A - set xs)) ` {xs. ?k_list k xs})"
    by (auto simp: length_Suc_conv)
  moreover
  have "Suc (card A - Suc k) = card A - k" using Suc.prems by simp
  then have "(card A - k) * \<Prod>{Suc (card A - k)..card A} = \<Prod>{Suc (card A - Suc k)..card A}"
    by (subst setprod_insert[symmetric]) (simp add: atLeastAtMost_insertL)+
  ultimately show ?case
    by (simp add: card_image inj_Cons card_UN_disjoint Suc.hyps algebra_simps)
qed

lemma infinite_UNIV_listI: "~ finite(UNIV::'a list set)"
apply (rule notI)
apply (drule finite_maxlen)
apply clarsimp
apply (erule_tac x = "replicate n undefined" in allE)
by simp


subsection {* Sorting *}

text{* Currently it is not shown that @{const sort} returns a
permutation of its input because the nicest proof is via multisets,
which are not yet available. Alternatively one could define a function
that counts the number of occurrences of an element in a list and use
that instead of multisets to state the correctness property. *}

context linorder
begin

lemma set_insort_key:
  "set (insort_key f x xs) = insert x (set xs)"
  by (induct xs) auto

lemma length_insort [simp]:
  "length (insort_key f x xs) = Suc (length xs)"
  by (induct xs) simp_all

lemma insort_key_left_comm:
  assumes "f x \<noteq> f y"
  shows "insort_key f y (insort_key f x xs) = insort_key f x (insort_key f y xs)"
  by (induct xs) (auto simp add: assms dest: antisym)

lemma insort_left_comm:
  "insort x (insort y xs) = insort y (insort x xs)"
  by (cases "x = y") (auto intro: insort_key_left_comm)

lemma comp_fun_commute_insort:
  "comp_fun_commute insort"
proof
qed (simp add: insort_left_comm fun_eq_iff)

lemma sort_key_simps [simp]:
  "sort_key f [] = []"
  "sort_key f (x#xs) = insort_key f x (sort_key f xs)"
  by (simp_all add: sort_key_def)

lemma (in linorder) sort_key_conv_fold:
  assumes "inj_on f (set xs)"
  shows "sort_key f xs = fold (insort_key f) xs []"
proof -
  have "fold (insort_key f) (rev xs) = fold (insort_key f) xs"
  proof (rule fold_rev, rule ext)
    fix zs
    fix x y
    assume "x \<in> set xs" "y \<in> set xs"
    with assms have *: "f y = f x \<Longrightarrow> y = x" by (auto dest: inj_onD)
    have **: "x = y \<longleftrightarrow> y = x" by auto
    show "(insort_key f y \<circ> insort_key f x) zs = (insort_key f x \<circ> insort_key f y) zs"
      by (induct zs) (auto intro: * simp add: **)
  qed
  then show ?thesis by (simp add: sort_key_def foldr_conv_fold)
qed

lemma (in linorder) sort_conv_fold:
  "sort xs = fold insort xs []"
  by (rule sort_key_conv_fold) simp

lemma length_sort[simp]: "length (sort_key f xs) = length xs"
by (induct xs, auto)

lemma sorted_Cons: "sorted (x#xs) = (sorted xs & (ALL y:set xs. x <= y))"
apply(induct xs arbitrary: x) apply simp
by simp (blast intro: order_trans)

lemma sorted_tl:
  "sorted xs \<Longrightarrow> sorted (tl xs)"
  by (cases xs) (simp_all add: sorted_Cons)

lemma sorted_append:
  "sorted (xs@ys) = (sorted xs & sorted ys & (\<forall>x \<in> set xs. \<forall>y \<in> set ys. x\<le>y))"
by (induct xs) (auto simp add:sorted_Cons)

lemma sorted_nth_mono:
  "sorted xs \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!i \<le> xs!j"
by (induct xs arbitrary: i j) (auto simp:nth_Cons' sorted_Cons)

lemma sorted_rev_nth_mono:
  "sorted (rev xs) \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!j \<le> xs!i"
using sorted_nth_mono[ of "rev xs" "length xs - j - 1" "length xs - i - 1"]
      rev_nth[of "length xs - i - 1" "xs"] rev_nth[of "length xs - j - 1" "xs"]
by auto

lemma sorted_nth_monoI:
  "(\<And> i j. \<lbrakk> i \<le> j ; j < length xs \<rbrakk> \<Longrightarrow> xs ! i \<le> xs ! j) \<Longrightarrow> sorted xs"
proof (induct xs)
  case (Cons x xs)
  have "sorted xs"
  proof (rule Cons.hyps)
    fix i j assume "i \<le> j" and "j < length xs"
    with Cons.prems[of "Suc i" "Suc j"]
    show "xs ! i \<le> xs ! j" by auto
  qed
  moreover
  {
    fix y assume "y \<in> set xs"
    then obtain j where "j < length xs" and "xs ! j = y"
      unfolding in_set_conv_nth by blast
    with Cons.prems[of 0 "Suc j"]
    have "x \<le> y"
      by auto
  }
  ultimately
  show ?case
    unfolding sorted_Cons by auto
qed simp

lemma sorted_equals_nth_mono:
  "sorted xs = (\<forall>j < length xs. \<forall>i \<le> j. xs ! i \<le> xs ! j)"
by (auto intro: sorted_nth_monoI sorted_nth_mono)

lemma set_insort: "set(insort_key f x xs) = insert x (set xs)"
by (induct xs) auto

lemma set_sort[simp]: "set(sort_key f xs) = set xs"
by (induct xs) (simp_all add:set_insort)

lemma distinct_insort: "distinct (insort_key f x xs) = (x \<notin> set xs \<and> distinct xs)"
by(induct xs)(auto simp:set_insort)

lemma distinct_sort[simp]: "distinct (sort_key f xs) = distinct xs"
  by (induct xs) (simp_all add: distinct_insort)

lemma sorted_insort_key: "sorted (map f (insort_key f x xs)) = sorted (map f xs)"
  by (induct xs) (auto simp:sorted_Cons set_insort)

lemma sorted_insort: "sorted (insort x xs) = sorted xs"
  using sorted_insort_key [where f="\<lambda>x. x"] by simp

theorem sorted_sort_key [simp]: "sorted (map f (sort_key f xs))"
  by (induct xs) (auto simp:sorted_insort_key)

theorem sorted_sort [simp]: "sorted (sort xs)"
  using sorted_sort_key [where f="\<lambda>x. x"] by simp

lemma sorted_butlast:
  assumes "xs \<noteq> []" and "sorted xs"
  shows "sorted (butlast xs)"
proof -
  from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  with `sorted xs` show ?thesis by (simp add: sorted_append)
qed
  
lemma insort_not_Nil [simp]:
  "insort_key f a xs \<noteq> []"
  by (induct xs) simp_all

lemma insort_is_Cons: "\<forall>x\<in>set xs. f a \<le> f x \<Longrightarrow> insort_key f a xs = a # xs"
by (cases xs) auto

lemma sorted_sort_id: "sorted xs \<Longrightarrow> sort xs = xs"
  by (induct xs) (auto simp add: sorted_Cons insort_is_Cons)

lemma sorted_map_remove1:
  "sorted (map f xs) \<Longrightarrow> sorted (map f (remove1 x xs))"
  by (induct xs) (auto simp add: sorted_Cons)

lemma sorted_remove1: "sorted xs \<Longrightarrow> sorted (remove1 a xs)"
  using sorted_map_remove1 [of "\<lambda>x. x"] by simp

lemma insort_key_remove1:
  assumes "a \<in> set xs" and "sorted (map f xs)" and "hd (filter (\<lambda>x. f a = f x) xs) = a"
  shows "insort_key f a (remove1 a xs) = xs"
using assms proof (induct xs)
  case (Cons x xs)
  then show ?case
  proof (cases "x = a")
    case False
    then have "f x \<noteq> f a" using Cons.prems by auto
    then have "f x < f a" using Cons.prems by (auto simp: sorted_Cons)
    with `f x \<noteq> f a` show ?thesis using Cons by (auto simp: sorted_Cons insort_is_Cons)
  qed (auto simp: sorted_Cons insort_is_Cons)
qed simp

lemma insort_remove1:
  assumes "a \<in> set xs" and "sorted xs"
  shows "insort a (remove1 a xs) = xs"
proof (rule insort_key_remove1)
  from `a \<in> set xs` show "a \<in> set xs" .
  from `sorted xs` show "sorted (map (\<lambda>x. x) xs)" by simp
  from `a \<in> set xs` have "a \<in> set (filter (op = a) xs)" by auto
  then have "set (filter (op = a) xs) \<noteq> {}" by auto
  then have "filter (op = a) xs \<noteq> []" by (auto simp only: set_empty)
  then have "length (filter (op = a) xs) > 0" by simp
  then obtain n where n: "Suc n = length (filter (op = a) xs)"
    by (cases "length (filter (op = a) xs)") simp_all
  moreover have "replicate (Suc n) a = a # replicate n a"
    by simp
  ultimately show "hd (filter (op = a) xs) = a" by (simp add: replicate_length_filter)
qed

lemma sorted_remdups[simp]:
  "sorted l \<Longrightarrow> sorted (remdups l)"
by (induct l) (auto simp: sorted_Cons)

lemma sorted_remdups_adj[simp]:
  "sorted xs \<Longrightarrow> sorted (remdups_adj xs)"
by (induct xs rule: remdups_adj.induct, simp_all split: split_if_asm add: sorted_Cons)

lemma sorted_distinct_set_unique:
assumes "sorted xs" "distinct xs" "sorted ys" "distinct ys" "set xs = set ys"
shows "xs = ys"
proof -
  from assms have 1: "length xs = length ys" by (auto dest!: distinct_card)
  from assms show ?thesis
  proof(induct rule:list_induct2[OF 1])
    case 1 show ?case by simp
  next
    case 2 thus ?case by (simp add: sorted_Cons)
       (metis Diff_insert_absorb antisym insertE insert_iff)
  qed
qed

lemma map_sorted_distinct_set_unique:
  assumes "inj_on f (set xs \<union> set ys)"
  assumes "sorted (map f xs)" "distinct (map f xs)"
    "sorted (map f ys)" "distinct (map f ys)"
  assumes "set xs = set ys"
  shows "xs = ys"
proof -
  from assms have "map f xs = map f ys"
    by (simp add: sorted_distinct_set_unique)
  with `inj_on f (set xs \<union> set ys)` show "xs = ys"
    by (blast intro: map_inj_on)
qed

lemma finite_sorted_distinct_unique:
shows "finite A \<Longrightarrow> EX! xs. set xs = A & sorted xs & distinct xs"
apply(drule finite_distinct_list)
apply clarify
apply(rule_tac a="sort xs" in ex1I)
apply (auto simp: sorted_distinct_set_unique)
done

lemma
  assumes "sorted xs"
  shows sorted_take: "sorted (take n xs)"
  and sorted_drop: "sorted (drop n xs)"
proof -
  from assms have "sorted (take n xs @ drop n xs)" by simp
  then show "sorted (take n xs)" and "sorted (drop n xs)"
    unfolding sorted_append by simp_all
qed

lemma sorted_dropWhile: "sorted xs \<Longrightarrow> sorted (dropWhile P xs)"
  by (auto dest: sorted_drop simp add: dropWhile_eq_drop)

lemma sorted_takeWhile: "sorted xs \<Longrightarrow> sorted (takeWhile P xs)"
  by (subst takeWhile_eq_take) (auto dest: sorted_take)

lemma sorted_filter:
  "sorted (map f xs) \<Longrightarrow> sorted (map f (filter P xs))"
  by (induct xs) (simp_all add: sorted_Cons)

lemma foldr_max_sorted:
  assumes "sorted (rev xs)"
  shows "foldr max xs y = (if xs = [] then y else max (xs ! 0) y)"
  using assms
proof (induct xs)
  case (Cons x xs)
  then have "sorted (rev xs)" using sorted_append by auto
  with Cons show ?case
    by (cases xs) (auto simp add: sorted_append max_def)
qed simp

lemma filter_equals_takeWhile_sorted_rev:
  assumes sorted: "sorted (rev (map f xs))"
  shows "[x \<leftarrow> xs. t < f x] = takeWhile (\<lambda> x. t < f x) xs"
    (is "filter ?P xs = ?tW")
proof (rule takeWhile_eq_filter[symmetric])
  let "?dW" = "dropWhile ?P xs"
  fix x assume "x \<in> set ?dW"
  then obtain i where i: "i < length ?dW" and nth_i: "x = ?dW ! i"
    unfolding in_set_conv_nth by auto
  hence "length ?tW + i < length (?tW @ ?dW)"
    unfolding length_append by simp
  hence i': "length (map f ?tW) + i < length (map f xs)" by simp
  have "(map f ?tW @ map f ?dW) ! (length (map f ?tW) + i) \<le>
        (map f ?tW @ map f ?dW) ! (length (map f ?tW) + 0)"
    using sorted_rev_nth_mono[OF sorted _ i', of "length ?tW"]
    unfolding map_append[symmetric] by simp
  hence "f x \<le> f (?dW ! 0)"
    unfolding nth_append_length_plus nth_i
    using i preorder_class.le_less_trans[OF le0 i] by simp
  also have "... \<le> t"
    using hd_dropWhile[of "?P" xs] le0[THEN preorder_class.le_less_trans, OF i]
    using hd_conv_nth[of "?dW"] by simp
  finally show "\<not> t < f x" by simp
qed

lemma insort_insert_key_triv:
  "f x \<in> f ` set xs \<Longrightarrow> insort_insert_key f x xs = xs"
  by (simp add: insort_insert_key_def)

lemma insort_insert_triv:
  "x \<in> set xs \<Longrightarrow> insort_insert x xs = xs"
  using insort_insert_key_triv [of "\<lambda>x. x"] by simp

lemma insort_insert_insort_key:
  "f x \<notin> f ` set xs \<Longrightarrow> insort_insert_key f x xs = insort_key f x xs"
  by (simp add: insort_insert_key_def)

lemma insort_insert_insort:
  "x \<notin> set xs \<Longrightarrow> insort_insert x xs = insort x xs"
  using insort_insert_insort_key [of "\<lambda>x. x"] by simp

lemma set_insort_insert:
  "set (insort_insert x xs) = insert x (set xs)"
  by (auto simp add: insort_insert_key_def set_insort)

lemma distinct_insort_insert:
  assumes "distinct xs"
  shows "distinct (insort_insert_key f x xs)"
  using assms by (induct xs) (auto simp add: insort_insert_key_def set_insort)

lemma sorted_insort_insert_key:
  assumes "sorted (map f xs)"
  shows "sorted (map f (insort_insert_key f x xs))"
  using assms by (simp add: insort_insert_key_def sorted_insort_key)

lemma sorted_insort_insert:
  assumes "sorted xs"
  shows "sorted (insort_insert x xs)"
  using assms sorted_insort_insert_key [of "\<lambda>x. x"] by simp

lemma filter_insort_triv:
  "\<not> P x \<Longrightarrow> filter P (insort_key f x xs) = filter P xs"
  by (induct xs) simp_all

lemma filter_insort:
  "sorted (map f xs) \<Longrightarrow> P x \<Longrightarrow> filter P (insort_key f x xs) = insort_key f x (filter P xs)"
  using assms by (induct xs)
    (auto simp add: sorted_Cons, subst insort_is_Cons, auto)

lemma filter_sort:
  "filter P (sort_key f xs) = sort_key f (filter P xs)"
  by (induct xs) (simp_all add: filter_insort_triv filter_insort)

lemma sorted_map_same:
  "sorted (map f [x\<leftarrow>xs. f x = g xs])"
proof (induct xs arbitrary: g)
  case Nil then show ?case by simp
next
  case (Cons x xs)
  then have "sorted (map f [y\<leftarrow>xs . f y = (\<lambda>xs. f x) xs])" .
  moreover from Cons have "sorted (map f [y\<leftarrow>xs . f y = (g \<circ> Cons x) xs])" .
  ultimately show ?case by (simp_all add: sorted_Cons)
qed

lemma sorted_same:
  "sorted [x\<leftarrow>xs. x = g xs]"
  using sorted_map_same [of "\<lambda>x. x"] by simp

lemma remove1_insort [simp]:
  "remove1 x (insort x xs) = xs"
  by (induct xs) simp_all

end

lemma sorted_upt[simp]: "sorted[i..<j]"
by (induct j) (simp_all add:sorted_append)

lemma sorted_upto[simp]: "sorted[i..j]"
apply(induct i j rule:upto.induct)
apply(subst upto.simps)
apply(simp add:sorted_Cons)
done

lemma sorted_find_Min:
  assumes "sorted xs"
  assumes "\<exists>x \<in> set xs. P x"
  shows "List.find P xs = Some (Min {x\<in>set xs. P x})"
using assms proof (induct xs rule: sorted.induct)
  case Nil then show ?case by simp
next
  case (Cons xs x) show ?case proof (cases "P x")
    case True with Cons show ?thesis by (auto intro: Min_eqI [symmetric])
  next
    case False then have "{y. (y = x \<or> y \<in> set xs) \<and> P y} = {y \<in> set xs. P y}"
      by auto
    with Cons False show ?thesis by simp_all
  qed
qed


subsubsection {* @{const transpose} on sorted lists *}

lemma sorted_transpose[simp]:
  shows "sorted (rev (map length (transpose xs)))"
  by (auto simp: sorted_equals_nth_mono rev_nth nth_transpose
    length_filter_conv_card intro: card_mono)

lemma transpose_max_length:
  "foldr (\<lambda>xs. max (length xs)) (transpose xs) 0 = length [x \<leftarrow> xs. x \<noteq> []]"
  (is "?L = ?R")
proof (cases "transpose xs = []")
  case False
  have "?L = foldr max (map length (transpose xs)) 0"
    by (simp add: foldr_map comp_def)
  also have "... = length (transpose xs ! 0)"
    using False sorted_transpose by (simp add: foldr_max_sorted)
  finally show ?thesis
    using False by (simp add: nth_transpose)
next
  case True
  hence "[x \<leftarrow> xs. x \<noteq> []] = []"
    by (auto intro!: filter_False simp: transpose_empty)
  thus ?thesis by (simp add: transpose_empty True)
qed

lemma length_transpose_sorted:
  fixes xs :: "'a list list"
  assumes sorted: "sorted (rev (map length xs))"
  shows "length (transpose xs) = (if xs = [] then 0 else length (xs ! 0))"
proof (cases "xs = []")
  case False
  thus ?thesis
    using foldr_max_sorted[OF sorted] False
    unfolding length_transpose foldr_map comp_def
    by simp
qed simp

lemma nth_nth_transpose_sorted[simp]:
  fixes xs :: "'a list list"
  assumes sorted: "sorted (rev (map length xs))"
  and i: "i < length (transpose xs)"
  and j: "j < length [ys \<leftarrow> xs. i < length ys]"
  shows "transpose xs ! i ! j = xs ! j  ! i"
  using j filter_equals_takeWhile_sorted_rev[OF sorted, of i]
    nth_transpose[OF i] nth_map[OF j]
  by (simp add: takeWhile_nth)

lemma transpose_column_length:
  fixes xs :: "'a list list"
  assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  shows "length (filter (\<lambda>ys. i < length ys) (transpose xs)) = length (xs ! i)"
proof -
  have "xs \<noteq> []" using `i < length xs` by auto
  note filter_equals_takeWhile_sorted_rev[OF sorted, simp]
  { fix j assume "j \<le> i"
    note sorted_rev_nth_mono[OF sorted, of j i, simplified, OF this `i < length xs`]
  } note sortedE = this[consumes 1]

  have "{j. j < length (transpose xs) \<and> i < length (transpose xs ! j)}
    = {..< length (xs ! i)}"
  proof safe
    fix j
    assume "j < length (transpose xs)" and "i < length (transpose xs ! j)"
    with this(2) nth_transpose[OF this(1)]
    have "i < length (takeWhile (\<lambda>ys. j < length ys) xs)" by simp
    from nth_mem[OF this] takeWhile_nth[OF this]
    show "j < length (xs ! i)" by (auto dest: set_takeWhileD)
  next
    fix j assume "j < length (xs ! i)"
    thus "j < length (transpose xs)"
      using foldr_max_sorted[OF sorted] `xs \<noteq> []` sortedE[OF le0]
      by (auto simp: length_transpose comp_def foldr_map)

    have "Suc i \<le> length (takeWhile (\<lambda>ys. j < length ys) xs)"
      using `i < length xs` `j < length (xs ! i)` less_Suc_eq_le
      by (auto intro!: length_takeWhile_less_P_nth dest!: sortedE)
    with nth_transpose[OF `j < length (transpose xs)`]
    show "i < length (transpose xs ! j)" by simp
  qed
  thus ?thesis by (simp add: length_filter_conv_card)
qed

lemma transpose_column:
  fixes xs :: "'a list list"
  assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  shows "map (\<lambda>ys. ys ! i) (filter (\<lambda>ys. i < length ys) (transpose xs))
    = xs ! i" (is "?R = _")
proof (rule nth_equalityI, safe)
  show length: "length ?R = length (xs ! i)"
    using transpose_column_length[OF assms] by simp

  fix j assume j: "j < length ?R"
  note * = less_le_trans[OF this, unfolded length_map, OF length_filter_le]
  from j have j_less: "j < length (xs ! i)" using length by simp
  have i_less_tW: "Suc i \<le> length (takeWhile (\<lambda>ys. Suc j \<le> length ys) xs)"
  proof (rule length_takeWhile_less_P_nth)
    show "Suc i \<le> length xs" using `i < length xs` by simp
    fix k assume "k < Suc i"
    hence "k \<le> i" by auto
    with sorted_rev_nth_mono[OF sorted this] `i < length xs`
    have "length (xs ! i) \<le> length (xs ! k)" by simp
    thus "Suc j \<le> length (xs ! k)" using j_less by simp
  qed
  have i_less_filter: "i < length [ys\<leftarrow>xs . j < length ys]"
    unfolding filter_equals_takeWhile_sorted_rev[OF sorted, of j]
    using i_less_tW by (simp_all add: Suc_le_eq)
  from j show "?R ! j = xs ! i ! j"
    unfolding filter_equals_takeWhile_sorted_rev[OF sorted_transpose, of i]
    by (simp add: takeWhile_nth nth_nth_transpose_sorted[OF sorted * i_less_filter])
qed

lemma transpose_transpose:
  fixes xs :: "'a list list"
  assumes sorted: "sorted (rev (map length xs))"
  shows "transpose (transpose xs) = takeWhile (\<lambda>x. x \<noteq> []) xs" (is "?L = ?R")
proof -
  have len: "length ?L = length ?R"
    unfolding length_transpose transpose_max_length
    using filter_equals_takeWhile_sorted_rev[OF sorted, of 0]
    by simp

  { fix i assume "i < length ?R"
    with less_le_trans[OF _ length_takeWhile_le[of _ xs]]
    have "i < length xs" by simp
  } note * = this
  show ?thesis
    by (rule nth_equalityI)
       (simp_all add: len nth_transpose transpose_column[OF sorted] * takeWhile_nth)
qed

theorem transpose_rectangle:
  assumes "xs = [] \<Longrightarrow> n = 0"
  assumes rect: "\<And> i. i < length xs \<Longrightarrow> length (xs ! i) = n"
  shows "transpose xs = map (\<lambda> i. map (\<lambda> j. xs ! j ! i) [0..<length xs]) [0..<n]"
    (is "?trans = ?map")
proof (rule nth_equalityI)
  have "sorted (rev (map length xs))"
    by (auto simp: rev_nth rect intro!: sorted_nth_monoI)
  from foldr_max_sorted[OF this] assms
  show len: "length ?trans = length ?map"
    by (simp_all add: length_transpose foldr_map comp_def)
  moreover
  { fix i assume "i < n" hence "[ys\<leftarrow>xs . i < length ys] = xs"
      using rect by (auto simp: in_set_conv_nth intro!: filter_True) }
  ultimately show "\<forall>i < length ?trans. ?trans ! i = ?map ! i"
    by (auto simp: nth_transpose intro: nth_equalityI)
qed


subsubsection {* @{text sorted_list_of_set} *}

text{* This function maps (finite) linearly ordered sets to sorted
lists. Warning: in most cases it is not a good idea to convert from
sets to lists but one should convert in the other direction (via
@{const set}). *}

subsubsection {* @{text sorted_list_of_set} *}

text{* This function maps (finite) linearly ordered sets to sorted
lists. Warning: in most cases it is not a good idea to convert from
sets to lists but one should convert in the other direction (via
@{const set}). *}

context linorder
begin

definition sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where
  "sorted_list_of_set = folding.F insort []"

sublocale sorted_list_of_set!: folding insort Nil
where
  "folding.F insort [] = sorted_list_of_set"
proof -
  interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  show "folding insort" by default (fact comp_fun_commute)
  show "folding.F insort [] = sorted_list_of_set" by (simp only: sorted_list_of_set_def)
qed

lemma sorted_list_of_set_empty:
  "sorted_list_of_set {} = []"
  by (fact sorted_list_of_set.empty)

lemma sorted_list_of_set_insert [simp]:
  "finite A \<Longrightarrow> sorted_list_of_set (insert x A) = insort x (sorted_list_of_set (A - {x}))"
  by (fact sorted_list_of_set.insert_remove)

lemma sorted_list_of_set_eq_Nil_iff [simp]:
  "finite A \<Longrightarrow> sorted_list_of_set A = [] \<longleftrightarrow> A = {}"
  by (auto simp: sorted_list_of_set.remove)

lemma sorted_list_of_set [simp]:
  "finite A \<Longrightarrow> set (sorted_list_of_set A) = A \<and> sorted (sorted_list_of_set A) 
    \<and> distinct (sorted_list_of_set A)"
  by (induct A rule: finite_induct) (simp_all add: set_insort sorted_insort distinct_insort)

lemma distinct_sorted_list_of_set:
  "distinct (sorted_list_of_set A)"
  using sorted_list_of_set by (cases "finite A") auto

lemma sorted_list_of_set_sort_remdups [code]:
  "sorted_list_of_set (set xs) = sort (remdups xs)"
proof -
  interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  show ?thesis by (simp add: sorted_list_of_set.eq_fold sort_conv_fold fold_set_fold_remdups)
qed

lemma sorted_list_of_set_remove:
  assumes "finite A"
  shows "sorted_list_of_set (A - {x}) = remove1 x (sorted_list_of_set A)"
proof (cases "x \<in> A")
  case False with assms have "x \<notin> set (sorted_list_of_set A)" by simp
  with False show ?thesis by (simp add: remove1_idem)
next
  case True then obtain B where A: "A = insert x B" by (rule Set.set_insert)
  with assms show ?thesis by simp
qed

end

lemma sorted_list_of_set_range [simp]:
  "sorted_list_of_set {m..<n} = [m..<n]"
  by (rule sorted_distinct_set_unique) simp_all


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

inductive_set
  lists :: "'a set => 'a list set"
  for A :: "'a set"
where
    Nil [intro!, simp]: "[]: lists A"
  | Cons [intro!, simp]: "[| a: A; l: lists A|] ==> a#l : lists A"

inductive_cases listsE [elim!]: "x#l : lists A"
inductive_cases listspE [elim!]: "listsp A (x # l)"

inductive_simps listsp_simps[code]:
  "listsp A []"
  "listsp A (x # xs)"

lemma listsp_mono [mono]: "A \<le> B ==> listsp A \<le> listsp B"
by (rule predicate1I, erule listsp.induct, blast+)

lemmas lists_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_def inf_bool_def]

lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set]

lemma Cons_in_lists_iff[simp]: "x#xs : lists A \<longleftrightarrow> x:A \<and> xs : lists A"
by auto

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 [code_unfold] = 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_eq_set: "lists A = {xs. set xs <= A}"
by auto

lemma lists_empty [simp]: "lists {} = {[]}"
by auto

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

lemma lists_image: "lists (f`A) = map f ` lists A"
proof -
  { fix xs have "\<forall>x\<in>set xs. x \<in> f ` A \<Longrightarrow> xs \<in> map f ` lists A"
      by (induct xs) (auto simp del: list.map simp add: list.map[symmetric] intro!: imageI) }
  then show ?thesis by auto
qed

subsubsection {* Inductive definition for membership *}

inductive 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}.*}

definition set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set" where
"set_Cons A XS = {z. \<exists>x xs. z = x # xs \<and> x \<in> A \<and> 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.*}

primrec listset :: "'a set list \<Rightarrow> 'a list set" where
"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.*}
        
primrec -- {*The lexicographic ordering for lists of the specified length*}
  lexn :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> ('a list \<times> 'a list) set" where
"lexn r 0 = {}" |
"lexn r (Suc n) =
  (map_prod (%(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}"

definition lex :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
"lex r = (\<Union>n. lexn r n)" -- {*Holds only between lists of the same length*}

definition lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" where
"lenlex r = inv_image (less_than <*lex*> lex r) (\<lambda>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_map_prod_image)
 prefer 2 apply (rule inj_onI, auto)
done

lemma lexn_length:
  "(xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
by (induct n arbitrary: xs ys) 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 Id_on_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. *} 

definition lexord :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
"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: "ALL x. (x,x) \<notin> r \<Longrightarrow> (xs,xs) \<notin> lexord r"
by (induct xs) auto

text{* By Ren\'e Thiemann: *}
lemma lexord_partial_trans: 
  "(\<And>x y z. x \<in> set xs \<Longrightarrow> (x,y) \<in> r \<Longrightarrow> (y,z) \<in> r \<Longrightarrow> (x,z) \<in> r)
   \<Longrightarrow>  (xs,ys) \<in> lexord r  \<Longrightarrow>  (ys,zs) \<in> lexord r \<Longrightarrow>  (xs,zs) \<in> lexord r"
proof (induct xs arbitrary: ys zs)
  case Nil
  from Nil(3) show ?case unfolding lexord_def by (cases zs, auto)
next
  case (Cons x xs yys zzs)
  from Cons(3) obtain y ys where yys: "yys = y # ys" unfolding lexord_def
    by (cases yys, auto)
  note Cons = Cons[unfolded yys]
  from Cons(3) have one: "(x,y) \<in> r \<or> x = y \<and> (xs,ys) \<in> lexord r" by auto
  from Cons(4) obtain z zs where zzs: "zzs = z # zs" unfolding lexord_def
    by (cases zzs, auto)
  note Cons = Cons[unfolded zzs]
  from Cons(4) have two: "(y,z) \<in> r \<or> y = z \<and> (ys,zs) \<in> lexord r" by auto
  {
    assume "(xs,ys) \<in> lexord r" and "(ys,zs) \<in> lexord r"
    from Cons(1)[OF _ this] Cons(2)
    have "(xs,zs) \<in> lexord r" by auto
  } note ind1 = this
  {
    assume "(x,y) \<in> r" and "(y,z) \<in> r"
    from Cons(2)[OF _ this] have "(x,z) \<in> r" by auto
  } note ind2 = this
  from one two ind1 ind2
  have "(x,z) \<in> r \<or> x = z \<and> (xs,zs) \<in> lexord r" by blast
  thus ?case unfolding zzs by auto
qed

lemma lexord_trans: 
    "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
by(auto simp: trans_def intro:lexord_partial_trans)

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

lemma lexord_irrefl:
  "irrefl R \<Longrightarrow> irrefl (lexord R)"
  by (simp add: irrefl_def lexord_irreflexive)
  
lemma lexord_asym:
  assumes "asym R"
  shows "asym (lexord R)"
proof
  from assms obtain "irrefl R" by (blast elim: asym.cases)
  then show "irrefl (lexord R)" by (rule lexord_irrefl)
next
  fix xs ys
  assume "(xs, ys) \<in> lexord R"
  then show "(ys, xs) \<notin> lexord R"
  proof (induct xs arbitrary: ys)
    case Nil
    then show ?case by simp
  next
    case (Cons x xs)
    then obtain z zs where ys: "ys = z # zs" by (cases ys) auto
    with assms Cons show ?case by (auto elim: asym.cases)
  qed
qed
   
lemma lexord_asymmetric:
  assumes "asym R"
  assumes hyp: "(a, b) \<in> lexord R"
  shows "(b, a) \<notin> lexord R"
proof -
  from `asym R` have "asym (lexord R)" by (rule lexord_asym)
  then show ?thesis by (rule asym.cases) (auto simp add: hyp)
qed


text {*
  Predicate version of lexicographic order integrated with Isabelle's order type classes.
  Author: Andreas Lochbihler
*}

context ord begin

inductive lexordp :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
where
  Nil: "lexordp [] (y # ys)"
| Cons: "x < y \<Longrightarrow> lexordp (x # xs) (y # ys)"
| Cons_eq:
  "\<lbrakk> \<not> x < y; \<not> y < x; lexordp xs ys \<rbrakk> \<Longrightarrow> lexordp (x # xs) (y # ys)"

lemma lexordp_simps [simp]:
  "lexordp [] ys = (ys \<noteq> [])"
  "lexordp xs [] = False"
  "lexordp (x # xs) (y # ys) \<longleftrightarrow> x < y \<or> \<not> y < x \<and> lexordp xs ys"
by(subst lexordp.simps, fastforce simp add: neq_Nil_conv)+

inductive lexordp_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
  Nil: "lexordp_eq [] ys"
| Cons: "x < y \<Longrightarrow> lexordp_eq (x # xs) (y # ys)"
| Cons_eq: "\<lbrakk> \<not> x < y; \<not> y < x; lexordp_eq xs ys \<rbrakk> \<Longrightarrow> lexordp_eq (x # xs) (y # ys)"

lemma lexordp_eq_simps [simp]:
  "lexordp_eq [] ys = True"
  "lexordp_eq xs [] \<longleftrightarrow> xs = []"
  "lexordp_eq (x # xs) [] = False"
  "lexordp_eq (x # xs) (y # ys) \<longleftrightarrow> x < y \<or> \<not> y < x \<and> lexordp_eq xs ys"
by(subst lexordp_eq.simps, fastforce)+

lemma lexordp_append_rightI: "ys \<noteq> Nil \<Longrightarrow> lexordp xs (xs @ ys)"
by(induct xs)(auto simp add: neq_Nil_conv)

lemma lexordp_append_left_rightI: "x < y \<Longrightarrow> lexordp (us @ x # xs) (us @ y # ys)"
by(induct us) auto

lemma lexordp_eq_refl: "lexordp_eq xs xs"
by(induct xs) simp_all

lemma lexordp_append_leftI: "lexordp us vs \<Longrightarrow> lexordp (xs @ us) (xs @ vs)"
by(induct xs) auto

lemma lexordp_append_leftD: "\<lbrakk> lexordp (xs @ us) (xs @ vs); \<forall>a. \<not> a < a \<rbrakk> \<Longrightarrow> lexordp us vs"
by(induct xs) auto

lemma lexordp_irreflexive: 
  assumes irrefl: "\<forall>x. \<not> x < x"
  shows "\<not> lexordp xs xs"
proof
  assume "lexordp xs xs"
  thus False by(induct xs ys\<equiv>xs)(simp_all add: irrefl)
qed

lemma lexordp_into_lexordp_eq:
  assumes "lexordp xs ys"
  shows "lexordp_eq xs ys"
using assms by induct simp_all

end

declare ord.lexordp_simps [simp, code]
declare ord.lexordp_eq_simps [code, simp]

lemma lexord_code [code, code_unfold]: "lexordp = ord.lexordp less"
unfolding lexordp_def ord.lexordp_def ..

context order begin

lemma lexordp_antisym:
  assumes "lexordp xs ys" "lexordp ys xs"
  shows False
using assms by induct auto

lemma lexordp_irreflexive': "\<not> lexordp xs xs"
by(rule lexordp_irreflexive) simp

end

context linorder begin

lemma lexordp_cases [consumes 1, case_names Nil Cons Cons_eq, cases pred: lexordp]:
  assumes "lexordp xs ys"
  obtains (Nil) y ys' where "xs = []" "ys = y # ys'"
  | (Cons) x xs' y ys' where "xs = x # xs'" "ys = y # ys'" "x < y"
  | (Cons_eq) x xs' ys' where "xs = x # xs'" "ys = x # ys'" "lexordp xs' ys'"
using assms by cases (fastforce simp add: not_less_iff_gr_or_eq)+

lemma lexordp_induct [consumes 1, case_names Nil Cons Cons_eq, induct pred: lexordp]:
  assumes major: "lexordp xs ys"
  and Nil: "\<And>y ys. P [] (y # ys)"
  and Cons: "\<And>x xs y ys. x < y \<Longrightarrow> P (x # xs) (y # ys)"
  and Cons_eq: "\<And>x xs ys. \<lbrakk> lexordp xs ys; P xs ys \<rbrakk> \<Longrightarrow> P (x # xs) (x # ys)"
  shows "P xs ys"
using major by induct (simp_all add: Nil Cons not_less_iff_gr_or_eq Cons_eq)

lemma lexordp_iff:
  "lexordp xs ys \<longleftrightarrow> (\<exists>x vs. ys = xs @ x # vs) \<or> (\<exists>us a b vs ws. a < b \<and> xs = us @ a # vs \<and> ys = us @ b # ws)"
  (is "?lhs = ?rhs")
proof
  assume ?lhs thus ?rhs
  proof induct
    case Cons_eq thus ?case by simp (metis append.simps(2))
  qed(fastforce intro: disjI2 del: disjCI intro: exI[where x="[]"])+
next
  assume ?rhs thus ?lhs
    by(auto intro: lexordp_append_leftI[where us="[]", simplified] lexordp_append_leftI)
qed

lemma lexordp_conv_lexord:
  "lexordp xs ys \<longleftrightarrow> (xs, ys) \<in> lexord {(x, y). x < y}"
by(simp add: lexordp_iff lexord_def)

lemma lexordp_eq_antisym: 
  assumes "lexordp_eq xs ys" "lexordp_eq ys xs" 
  shows "xs = ys"
using assms by induct simp_all

lemma lexordp_eq_trans:
  assumes "lexordp_eq xs ys" and "lexordp_eq ys zs"
  shows "lexordp_eq xs zs"
using assms
apply(induct arbitrary: zs)
apply(case_tac [2-3] zs)
apply auto
done

lemma lexordp_trans:
  assumes "lexordp xs ys" "lexordp ys zs"
  shows "lexordp xs zs"
using assms
apply(induct arbitrary: zs)
apply(case_tac [2-3] zs)
apply auto
done

lemma lexordp_linear: "lexordp xs ys \<or> xs = ys \<or> lexordp ys xs"
proof(induct xs arbitrary: ys)
  case Nil thus ?case by(cases ys) simp_all
next
  case Cons thus ?case by(cases ys) auto
qed

lemma lexordp_conv_lexordp_eq: "lexordp xs ys \<longleftrightarrow> lexordp_eq xs ys \<and> \<not> lexordp_eq ys xs"
  (is "?lhs \<longleftrightarrow> ?rhs")
proof
  assume ?lhs
  moreover hence "\<not> lexordp_eq ys xs" by induct simp_all
  ultimately show ?rhs by(simp add: lexordp_into_lexordp_eq)
next
  assume ?rhs
  hence "lexordp_eq xs ys" "\<not> lexordp_eq ys xs" by simp_all
  thus ?lhs by induct simp_all
qed

lemma lexordp_eq_conv_lexord: "lexordp_eq xs ys \<longleftrightarrow> xs = ys \<or> lexordp xs ys"
by(auto simp add: lexordp_conv_lexordp_eq lexordp_eq_refl dest: lexordp_eq_antisym)

lemma lexordp_eq_linear: "lexordp_eq xs ys \<or> lexordp_eq ys xs"
apply(induct xs arbitrary: ys)
apply(case_tac [!] ys)
apply auto
done

lemma lexordp_linorder: "class.linorder lexordp_eq lexordp"
by unfold_locales(auto simp add: lexordp_conv_lexordp_eq lexordp_eq_refl lexordp_eq_antisym intro: lexordp_eq_trans del: disjCI intro: lexordp_eq_linear)

end

subsubsection {* 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[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


subsubsection {* Lifting Relations to Lists: one element *}

definition listrel1 :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
"listrel1 r = {(xs,ys).
   \<exists>us z z' vs. xs = us @ z # vs \<and> (z,z') \<in> r \<and> ys = us @ z' # vs}"

lemma listrel1I:
  "\<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow>
  (xs, ys) \<in> listrel1 r"
unfolding listrel1_def by auto

lemma listrel1E:
  "\<lbrakk> (xs, ys) \<in> listrel1 r;
     !!x y us vs. \<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow> P
   \<rbrakk> \<Longrightarrow> P"
unfolding listrel1_def by auto

lemma not_Nil_listrel1 [iff]: "([], xs) \<notin> listrel1 r"
unfolding listrel1_def by blast

lemma not_listrel1_Nil [iff]: "(xs, []) \<notin> listrel1 r"
unfolding listrel1_def by blast

lemma Cons_listrel1_Cons [iff]:
  "(x # xs, y # ys) \<in> listrel1 r \<longleftrightarrow>
   (x,y) \<in> r \<and> xs = ys \<or> x = y \<and> (xs, ys) \<in> listrel1 r"
by (simp add: listrel1_def Cons_eq_append_conv) (blast)

lemma listrel1I1: "(x,y) \<in> r \<Longrightarrow> (x # xs, y # xs) \<in> listrel1 r"
by fast

lemma listrel1I2: "(xs, ys) \<in> listrel1 r \<Longrightarrow> (x # xs, x # ys) \<in> listrel1 r"
by fast

lemma append_listrel1I:
  "(xs, ys) \<in> listrel1 r \<and> us = vs \<or> xs = ys \<and> (us, vs) \<in> listrel1 r
    \<Longrightarrow> (xs @ us, ys @ vs) \<in> listrel1 r"
unfolding listrel1_def
by auto (blast intro: append_eq_appendI)+

lemma Cons_listrel1E1[elim!]:
  assumes "(x # xs, ys) \<in> listrel1 r"
    and "\<And>y. ys = y # xs \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
    and "\<And>zs. ys = x # zs \<Longrightarrow> (xs, zs) \<in> listrel1 r \<Longrightarrow> R"
  shows R
using assms by (cases ys) blast+

lemma Cons_listrel1E2[elim!]:
  assumes "(xs, y # ys) \<in> listrel1 r"
    and "\<And>x. xs = x # ys \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
    and "\<And>zs. xs = y # zs \<Longrightarrow> (zs, ys) \<in> listrel1 r \<Longrightarrow> R"
  shows R
using assms by (cases xs) blast+

lemma snoc_listrel1_snoc_iff:
  "(xs @ [x], ys @ [y]) \<in> listrel1 r
    \<longleftrightarrow> (xs, ys) \<in> listrel1 r \<and> x = y \<or> xs = ys \<and> (x,y) \<in> r" (is "?L \<longleftrightarrow> ?R")
proof
  assume ?L thus ?R
    by (fastforce simp: listrel1_def snoc_eq_iff_butlast butlast_append)
next
  assume ?R then show ?L unfolding listrel1_def by force
qed

lemma listrel1_eq_len: "(xs,ys) \<in> listrel1 r \<Longrightarrow> length xs = length ys"
unfolding listrel1_def by auto

lemma listrel1_mono:
  "r \<subseteq> s \<Longrightarrow> listrel1 r \<subseteq> listrel1 s"
unfolding listrel1_def by blast


lemma listrel1_converse: "listrel1 (r^-1) = (listrel1 r)^-1"
unfolding listrel1_def by blast

lemma in_listrel1_converse:
  "(x,y) : listrel1 (r^-1) \<longleftrightarrow> (x,y) : (listrel1 r)^-1"
unfolding listrel1_def by blast

lemma listrel1_iff_update:
  "(xs,ys) \<in> (listrel1 r)
   \<longleftrightarrow> (\<exists>y n. (xs ! n, y) \<in> r \<and> n < length xs \<and> ys = xs[n:=y])" (is "?L \<longleftrightarrow> ?R")
proof
  assume "?L"
  then obtain x y u v where "xs = u @ x # v"  "ys = u @ y # v"  "(x,y) \<in> r"
    unfolding listrel1_def by auto
  then have "ys = xs[length u := y]" and "length u < length xs"
    and "(xs ! length u, y) \<in> r" by auto
  then show "?R" by auto
next
  assume "?R"
  then obtain x y n where "(xs!n, y) \<in> r" "n < size xs" "ys = xs[n:=y]" "x = xs!n"
    by auto
  then obtain u v where "xs = u @ x # v" and "ys = u @ y # v" and "(x, y) \<in> r"
    by (auto intro: upd_conv_take_nth_drop id_take_nth_drop)
  then show "?L" by (auto simp: listrel1_def)
qed


text{* Accessible part and wellfoundedness: *}

lemma Cons_acc_listrel1I [intro!]:
  "x \<in> Wellfounded.acc r \<Longrightarrow> xs \<in> Wellfounded.acc (listrel1 r) \<Longrightarrow> (x # xs) \<in> Wellfounded.acc (listrel1 r)"
apply (induct arbitrary: xs set: Wellfounded.acc)
apply (erule thin_rl)
apply (erule acc_induct)
apply (rule accI)
apply (blast)
done

lemma lists_accD: "xs \<in> lists (Wellfounded.acc r) \<Longrightarrow> xs \<in> Wellfounded.acc (listrel1 r)"
apply (induct set: lists)
 apply (rule accI)
 apply simp
apply (rule accI)
apply (fast dest: acc_downward)
done

lemma lists_accI: "xs \<in> Wellfounded.acc (listrel1 r) \<Longrightarrow> xs \<in> lists (Wellfounded.acc r)"
apply (induct set: Wellfounded.acc)
apply clarify
apply (rule accI)
apply (fastforce dest!: in_set_conv_decomp[THEN iffD1] simp: listrel1_def)
done

lemma wf_listrel1_iff[simp]: "wf(listrel1 r) = wf r"
by (auto simp: wf_acc_iff
      intro: lists_accD lists_accI[THEN Cons_in_lists_iff[THEN iffD1, THEN conjunct1]])

subsubsection {* Lifting Relations to Lists: all elements *}

inductive_set
  listrel :: "('a \<times> 'b) set \<Rightarrow> ('a list \<times> 'b list) set"
  for r :: "('a \<times> 'b) set"
where
    Nil:  "([],[]) \<in> listrel r"
  | Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"

inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"


lemma listrel_eq_len:  "(xs, ys) \<in> listrel r \<Longrightarrow> length xs = length ys"
by(induct rule: listrel.induct) auto

lemma listrel_iff_zip [code_unfold]: "(xs,ys) : listrel r \<longleftrightarrow>
  length xs = length ys & (\<forall>(x,y) \<in> set(zip xs ys). (x,y) \<in> r)" (is "?L \<longleftrightarrow> ?R")
proof
  assume ?L thus ?R by induct (auto intro: listrel_eq_len)
next
  assume ?R thus ?L
    apply (clarify)
    by (induct rule: list_induct2) (auto intro: listrel.intros)
qed

lemma listrel_iff_nth: "(xs,ys) : listrel r \<longleftrightarrow>
  length xs = length ys & (\<forall>n < length xs. (xs!n, ys!n) \<in> r)" (is "?L \<longleftrightarrow> ?R")
by (auto simp add: all_set_conv_all_nth listrel_iff_zip)


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_on: "refl_on A r \<Longrightarrow> refl_on (lists A) (listrel r)" 
apply (simp add: refl_on_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_on listrel_sym listrel_trans) 

lemma listrel_rtrancl_refl[iff]: "(xs,xs) : listrel(r^*)"
using listrel_refl_on[of UNIV, OF refl_rtrancl]
by(auto simp: refl_on_def)

lemma listrel_rtrancl_trans:
  "\<lbrakk> (xs,ys) : listrel(r^*);  (ys,zs) : listrel(r^*) \<rbrakk>
  \<Longrightarrow> (xs,zs) : listrel(r^*)"
by (metis listrel_trans trans_def trans_rtrancl)


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)

text {* Relating @{term listrel1}, @{term listrel} and closures: *}

lemma listrel1_rtrancl_subset_rtrancl_listrel1:
  "listrel1 (r^*) \<subseteq> (listrel1 r)^*"
proof (rule subrelI)
  fix xs ys assume 1: "(xs,ys) \<in> listrel1 (r^*)"
  { fix x y us vs
    have "(x,y) : r^* \<Longrightarrow> (us @ x # vs, us @ y # vs) : (listrel1 r)^*"
    proof(induct rule: rtrancl.induct)
      case rtrancl_refl show ?case by simp
    next
      case rtrancl_into_rtrancl thus ?case
        by (metis listrel1I rtrancl.rtrancl_into_rtrancl)
    qed }
  thus "(xs,ys) \<in> (listrel1 r)^*" using 1 by(blast elim: listrel1E)
qed

lemma rtrancl_listrel1_eq_len: "(x,y) \<in> (listrel1 r)^* \<Longrightarrow> length x = length y"
by (induct rule: rtrancl.induct) (auto intro: listrel1_eq_len)

lemma rtrancl_listrel1_ConsI1:
  "(xs,ys) : (listrel1 r)^* \<Longrightarrow> (x#xs,x#ys) : (listrel1 r)^*"
apply(induct rule: rtrancl.induct)
 apply simp
by (metis listrel1I2 rtrancl.rtrancl_into_rtrancl)

lemma rtrancl_listrel1_ConsI2:
  "(x,y) \<in> r^* \<Longrightarrow> (xs, ys) \<in> (listrel1 r)^*
  \<Longrightarrow> (x # xs, y # ys) \<in> (listrel1 r)^*"
  by (blast intro: rtrancl_trans rtrancl_listrel1_ConsI1 
    subsetD[OF listrel1_rtrancl_subset_rtrancl_listrel1 listrel1I1])

lemma listrel1_subset_listrel:
  "r \<subseteq> r' \<Longrightarrow> refl r' \<Longrightarrow> listrel1 r \<subseteq> listrel(r')"
by(auto elim!: listrel1E simp add: listrel_iff_zip set_zip refl_on_def)

lemma listrel_reflcl_if_listrel1:
  "(xs,ys) : listrel1 r \<Longrightarrow> (xs,ys) : listrel(r^*)"
by(erule listrel1E)(auto simp add: listrel_iff_zip set_zip)

lemma listrel_rtrancl_eq_rtrancl_listrel1: "listrel (r^*) = (listrel1 r)^*"
proof
  { fix x y assume "(x,y) \<in> listrel (r^*)"
    then have "(x,y) \<in> (listrel1 r)^*"
    by induct (auto intro: rtrancl_listrel1_ConsI2) }
  then show "listrel (r^*) \<subseteq> (listrel1 r)^*"
    by (rule subrelI)
next
  show "listrel (r^*) \<supseteq> (listrel1 r)^*"
  proof(rule subrelI)
    fix xs ys assume "(xs,ys) \<in> (listrel1 r)^*"
    then show "(xs,ys) \<in> listrel (r^*)"
    proof induct
      case base show ?case by(auto simp add: listrel_iff_zip set_zip)
    next
      case (step ys zs)
      thus ?case by (metis listrel_reflcl_if_listrel1 listrel_rtrancl_trans)
    qed
  qed
qed

lemma rtrancl_listrel1_if_listrel:
  "(xs,ys) : listrel r \<Longrightarrow> (xs,ys) : (listrel1 r)^*"
by(metis listrel_rtrancl_eq_rtrancl_listrel1 subsetD[OF listrel_mono] r_into_rtrancl subsetI)

lemma listrel_subset_rtrancl_listrel1: "listrel r \<subseteq> (listrel1 r)^*"
by(fast intro:rtrancl_listrel1_if_listrel)


subsection {* Size function *}

lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (list_size f)"
by (rule is_measure_trivial)

lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (option_size f)"
by (rule is_measure_trivial)

lemma list_size_estimation[termination_simp]: 
  "x \<in> set xs \<Longrightarrow> y < f x \<Longrightarrow> y < list_size f xs"
by (induct xs) auto

lemma list_size_estimation'[termination_simp]: 
  "x \<in> set xs \<Longrightarrow> y \<le> f x \<Longrightarrow> y \<le> list_size f xs"
by (induct xs) auto

lemma list_size_map[simp]: "list_size f (map g xs) = list_size (f o g) xs"
by (induct xs) auto

lemma list_size_append[simp]: "list_size f (xs @ ys) = list_size f xs + list_size f ys"
by (induct xs, auto)

lemma list_size_pointwise[termination_simp]: 
  "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> list_size f xs \<le> list_size g xs"
by (induct xs) force+


subsection {* Monad operation *}

definition bind :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b list) \<Rightarrow> 'b list" where
"bind xs f = concat (map f xs)"

hide_const (open) bind

lemma bind_simps [simp]:
  "List.bind [] f = []"
  "List.bind (x # xs) f = f x @ List.bind xs f"
  by (simp_all add: bind_def)


subsection {* Transfer *}

definition embed_list :: "nat list \<Rightarrow> int list" where
"embed_list l = map int l"

definition nat_list :: "int list \<Rightarrow> bool" where
"nat_list l = nat_set (set l)"

definition return_list :: "int list \<Rightarrow> nat list" where
"return_list l = map nat l"

lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow>
    embed_list (return_list l) = l"
  unfolding embed_list_def return_list_def nat_list_def nat_set_def
  apply (induct l)
  apply auto
done

lemma transfer_nat_int_list_functions:
  "l @ m = return_list (embed_list l @ embed_list m)"
  "[] = return_list []"
  unfolding return_list_def embed_list_def
  apply auto
  apply (induct l, auto)
  apply (induct m, auto)
done

(*
lemma transfer_nat_int_fold1: "fold f l x =
    fold (%x. f (nat x)) (embed_list l) x";
*)


subsection {* Code generation *}

text{* Optional tail recursive version of @{const map}. Can avoid
stack overflow in some target languages. *}

fun map_tailrec_rev ::  "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'b list" where
"map_tailrec_rev f [] bs = bs" |
"map_tailrec_rev f (a#as) bs = map_tailrec_rev f as (f a # bs)"

lemma map_tailrec_rev:
  "map_tailrec_rev f as bs = rev(map f as) @ bs"
by(induction as arbitrary: bs) simp_all

definition map_tailrec :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
"map_tailrec f as = rev (map_tailrec_rev f as [])"

text{* Code equation: *}
lemma map_eq_map_tailrec: "map = map_tailrec"
by(simp add: fun_eq_iff map_tailrec_def map_tailrec_rev)


subsubsection {* Counterparts for set-related operations *}

definition member :: "'a list \<Rightarrow> 'a \<Rightarrow> bool" where
[code_abbrev]: "member xs x \<longleftrightarrow> x \<in> set xs"

text {*
  Use @{text member} only for generating executable code.  Otherwise use
  @{prop "x \<in> set xs"} instead --- it is much easier to reason about.
*}

lemma member_rec [code]:
  "member (x # xs) y \<longleftrightarrow> x = y \<or> member xs y"
  "member [] y \<longleftrightarrow> False"
  by (auto simp add: member_def)

lemma in_set_member (* FIXME delete candidate *):
  "x \<in> set xs \<longleftrightarrow> member xs x"
  by (simp add: member_def)

abbreviation "list_all == pred_list"

lemma list_all_iff [code_abbrev]: "list_all P xs \<longleftrightarrow> Ball (set xs) P"
unfolding pred_list_def ..

definition list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
list_ex_iff [code_abbrev]: "list_ex P xs \<longleftrightarrow> Bex (set xs) P"

definition list_ex1 :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
list_ex1_iff [code_abbrev]: "list_ex1 P xs \<longleftrightarrow> (\<exists>! x. x \<in> set xs \<and> P x)"

text {*
  Usually you should prefer @{text "\<forall>x\<in>set xs"}, @{text "\<exists>x\<in>set xs"}
  and @{text "\<exists>!x. x\<in>set xs \<and> _"} over @{const list_all}, @{const list_ex}
  and @{const list_ex1} in specifications.
*}

lemma list_all_simps [code]:
  "list_all P (x # xs) \<longleftrightarrow> P x \<and> list_all P xs"
  "list_all P [] \<longleftrightarrow> True"
  by (simp_all add: list_all_iff)

lemma list_ex_simps [simp, code]:
  "list_ex P (x # xs) \<longleftrightarrow> P x \<or> list_ex P xs"
  "list_ex P [] \<longleftrightarrow> False"
  by (simp_all add: list_ex_iff)

lemma list_ex1_simps [simp, code]:
  "list_ex1 P [] = False"
  "list_ex1 P (x # xs) = (if P x then list_all (\<lambda>y. \<not> P y \<or> x = y) xs else list_ex1 P xs)"
  by (auto simp add: list_ex1_iff list_all_iff)

lemma Ball_set_list_all: (* FIXME delete candidate *)
  "Ball (set xs) P \<longleftrightarrow> list_all P xs"
  by (simp add: list_all_iff)

lemma Bex_set_list_ex: (* FIXME delete candidate *)
  "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
  by (simp add: list_ex_iff)

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

lemma list_ex_append [simp]:
  "list_ex P (xs @ ys) \<longleftrightarrow> list_ex P xs \<or> list_ex P ys"
  by (auto simp add: list_ex_iff)

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

lemma list_ex_rev [simp]:
  "list_ex P (rev xs) \<longleftrightarrow> list_ex P xs"
  by (simp add: list_ex_iff)

lemma list_all_length:
  "list_all P xs \<longleftrightarrow> (\<forall>n < length xs. P (xs ! n))"
  by (auto simp add: list_all_iff set_conv_nth)

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

lemma list_all_cong [fundef_cong]:
  "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_all f xs = list_all g ys"
  by (simp add: list_all_iff)

lemma list_ex_cong [fundef_cong]:
  "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_ex f xs = list_ex g ys"
by (simp add: list_ex_iff)

definition can_select :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" where
[code_abbrev]: "can_select P A = (\<exists>!x\<in>A. P x)"

lemma can_select_set_list_ex1 [code]:
  "can_select P (set A) = list_ex1 P A"
  by (simp add: list_ex1_iff can_select_def)


text {* Executable checks for relations on sets *}

definition listrel1p :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
"listrel1p r xs ys = ((xs, ys) \<in> listrel1 {(x, y). r x y})"

lemma [code_unfold]:
  "(xs, ys) \<in> listrel1 r = listrel1p (\<lambda>x y. (x, y) \<in> r) xs ys"
unfolding listrel1p_def by auto

lemma [code]:
  "listrel1p r [] xs = False"
  "listrel1p r xs [] =  False"
  "listrel1p r (x # xs) (y # ys) \<longleftrightarrow>
     r x y \<and> xs = ys \<or> x = y \<and> listrel1p r xs ys"
by (simp add: listrel1p_def)+

definition
  lexordp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
  "lexordp r xs ys = ((xs, ys) \<in> lexord {(x, y). r x y})"

lemma [code_unfold]:
  "(xs, ys) \<in> lexord r = lexordp (\<lambda>x y. (x, y) \<in> r) xs ys"
unfolding lexordp_def by auto

lemma [code]:
  "lexordp r xs [] = False"
  "lexordp r [] (y#ys) = True"
  "lexordp r (x # xs) (y # ys) = (r x y | (x = y & lexordp r xs ys))"
unfolding lexordp_def by auto

text {* Bounded quantification and summation over nats. *}

lemma atMost_upto [code_unfold]:
  "{..n} = set [0..<Suc n]"
  by auto

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

lemma greaterThanLessThan_upt [code_unfold]:
  "{n<..<m} = set [Suc n..<m]"
  by auto

lemmas atLeastLessThan_upt [code_unfold] = set_upt [symmetric]

lemma greaterThanAtMost_upt [code_unfold]:
  "{n<..m} = set [Suc n..<Suc m]"
  by auto

lemma atLeastAtMost_upt [code_unfold]:
  "{n..m} = set [n..<Suc 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

lemma setsum_set_upt_conv_listsum_nat [code_unfold]:
  "setsum f (set [m..<n]) = listsum (map f [m..<n])"
  by (simp add: interv_listsum_conv_setsum_set_nat)

text{* Bounded @{text LEAST} operator: *}

definition "Bleast S P = (LEAST x. x \<in> S \<and> P x)"

definition "abort_Bleast S P = (LEAST x. x \<in> S \<and> P x)"

declare [[code abort: abort_Bleast]]

lemma Bleast_code [code]:
 "Bleast (set xs) P = (case filter P (sort xs) of
    x#xs \<Rightarrow> x |
    [] \<Rightarrow> abort_Bleast (set xs) P)"
proof (cases "filter P (sort xs)")
  case Nil thus ?thesis by (simp add: Bleast_def abort_Bleast_def)
next
  case (Cons x ys)
  have "(LEAST x. x \<in> set xs \<and> P x) = x"
  proof (rule Least_equality)
    show "x \<in> set xs \<and> P x"
      by (metis Cons Cons_eq_filter_iff in_set_conv_decomp set_sort)
    next
      fix y assume "y : set xs \<and> P y"
      hence "y : set (filter P xs)" by auto
      thus "x \<le> y"
        by (metis Cons eq_iff filter_sort set_ConsD set_sort sorted_Cons sorted_sort)
  qed
  thus ?thesis using Cons by (simp add: Bleast_def)
qed

declare Bleast_def[symmetric, code_unfold]

text {* Summation over ints. *}

lemma greaterThanLessThan_upto [code_unfold]:
  "{i<..<j::int} = set [i+1..j - 1]"
by auto

lemma atLeastLessThan_upto [code_unfold]:
  "{i..<j::int} = set [i..j - 1]"
by auto

lemma greaterThanAtMost_upto [code_unfold]:
  "{i<..j::int} = set [i+1..j]"
by auto

lemmas atLeastAtMost_upto [code_unfold] = set_upto [symmetric]

lemma setsum_set_upto_conv_listsum_int [code_unfold]:
  "setsum f (set [i..j::int]) = listsum (map f [i..j])"
  by (simp add: interv_listsum_conv_setsum_set_int)


subsubsection {* Optimizing by rewriting *}

definition null :: "'a list \<Rightarrow> bool" where
  [code_abbrev]: "null xs \<longleftrightarrow> xs = []"

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

lemma null_rec [code]:
  "null (x # xs) \<longleftrightarrow> False"
  "null [] \<longleftrightarrow> True"
  by (simp_all add: null_def)

lemma eq_Nil_null: (* FIXME delete candidate *)
  "xs = [] \<longleftrightarrow> null xs"
  by (simp add: null_def)

lemma equal_Nil_null [code_unfold]:
  "HOL.equal xs [] \<longleftrightarrow> null xs"
  "HOL.equal [] = null"
  by (auto simp add: equal null_def)

definition maps :: "('a \<Rightarrow> 'b list) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  [code_abbrev]: "maps f xs = concat (map f xs)"

definition map_filter :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  [code_post]: "map_filter f xs = map (the \<circ> f) (filter (\<lambda>x. f x \<noteq> None) xs)"

text {*
  Operations @{const maps} and @{const map_filter} avoid
  intermediate lists on execution -- do not use for proving.
*}

lemma maps_simps [code]:
  "maps f (x # xs) = f x @ maps f xs"
  "maps f [] = []"
  by (simp_all add: maps_def)

lemma map_filter_simps [code]:
  "map_filter f (x # xs) = (case f x of None \<Rightarrow> map_filter f xs | Some y \<Rightarrow> y # map_filter f xs)"
  "map_filter f [] = []"
  by (simp_all add: map_filter_def split: option.split)

lemma concat_map_maps: (* FIXME delete candidate *)
  "concat (map f xs) = maps f xs"
  by (simp add: maps_def)

lemma map_filter_map_filter [code_unfold]:
  "map f (filter P xs) = map_filter (\<lambda>x. if P x then Some (f x) else None) xs"
  by (simp add: map_filter_def)

text {* Optimized code for @{text"\<forall>i\<in>{a..b::int}"} and @{text"\<forall>n:{a..<b::nat}"}
and similiarly for @{text"\<exists>"}. *}

definition all_interval_nat :: "(nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
  "all_interval_nat P i j \<longleftrightarrow> (\<forall>n \<in> {i..<j}. P n)"

lemma [code]:
  "all_interval_nat P i j \<longleftrightarrow> i \<ge> j \<or> P i \<and> all_interval_nat P (Suc i) j"
proof -
  have *: "\<And>n. P i \<Longrightarrow> \<forall>n\<in>{Suc i..<j}. P n \<Longrightarrow> i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n"
  proof -
    fix n
    assume "P i" "\<forall>n\<in>{Suc i..<j}. P n" "i \<le> n" "n < j"
    then show "P n" by (cases "n = i") simp_all
  qed
  show ?thesis by (auto simp add: all_interval_nat_def intro: *)
qed

lemma list_all_iff_all_interval_nat [code_unfold]:
  "list_all P [i..<j] \<longleftrightarrow> all_interval_nat P i j"
  by (simp add: list_all_iff all_interval_nat_def)

lemma list_ex_iff_not_all_inverval_nat [code_unfold]:
  "list_ex P [i..<j] \<longleftrightarrow> \<not> (all_interval_nat (Not \<circ> P) i j)"
  by (simp add: list_ex_iff all_interval_nat_def)

definition all_interval_int :: "(int \<Rightarrow> bool) \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool" where
  "all_interval_int P i j \<longleftrightarrow> (\<forall>k \<in> {i..j}. P k)"

lemma [code]:
  "all_interval_int P i j \<longleftrightarrow> i > j \<or> P i \<and> all_interval_int P (i + 1) j"
proof -
  have *: "\<And>k. P i \<Longrightarrow> \<forall>k\<in>{i+1..j}. P k \<Longrightarrow> i \<le> k \<Longrightarrow> k \<le> j \<Longrightarrow> P k"
  proof -
    fix k
    assume "P i" "\<forall>k\<in>{i+1..j}. P k" "i \<le> k" "k \<le> j"
    then show "P k" by (cases "k = i") simp_all
  qed
  show ?thesis by (auto simp add: all_interval_int_def intro: *)
qed

lemma list_all_iff_all_interval_int [code_unfold]:
  "list_all P [i..j] \<longleftrightarrow> all_interval_int P i j"
  by (simp add: list_all_iff all_interval_int_def)

lemma list_ex_iff_not_all_inverval_int [code_unfold]:
  "list_ex P [i..j] \<longleftrightarrow> \<not> (all_interval_int (Not \<circ> P) i j)"
  by (simp add: list_ex_iff all_interval_int_def)

text {* optimized code (tail-recursive) for @{term length} *}

definition gen_length :: "nat \<Rightarrow> 'a list \<Rightarrow> nat"
where "gen_length n xs = n + length xs"

lemma gen_length_code [code]:
  "gen_length n [] = n"
  "gen_length n (x # xs) = gen_length (Suc n) xs"
by(simp_all add: gen_length_def)

declare list.size(3-4)[code del]

lemma length_code [code]: "length = gen_length 0"
by(simp add: gen_length_def fun_eq_iff)

hide_const (open) member null maps map_filter all_interval_nat all_interval_int gen_length


subsubsection {* Pretty lists *}

ML {*
(* Code generation for list literals. *)

signature LIST_CODE =
sig
  val implode_list: Code_Thingol.iterm -> Code_Thingol.iterm list option
  val default_list: int * string
    -> (Code_Printer.fixity -> Code_Thingol.iterm -> Pretty.T)
    -> Code_Printer.fixity -> Code_Thingol.iterm -> Code_Thingol.iterm -> Pretty.T
  val add_literal_list: string -> theory -> theory
end;

structure List_Code : LIST_CODE =
struct

open Basic_Code_Thingol;

fun implode_list t =
  let
    fun dest_cons (IConst { sym = Code_Symbol.Constant @{const_name Cons}, ... } `$ t1 `$ t2) = SOME (t1, t2)
      | dest_cons _ = NONE;
    val (ts, t') = Code_Thingol.unfoldr dest_cons t;
  in case t'
   of IConst { sym = Code_Symbol.Constant @{const_name Nil}, ... } => SOME ts
    | _ => NONE
  end;

fun default_list (target_fxy, target_cons) pr fxy t1 t2 =
  Code_Printer.brackify_infix (target_fxy, Code_Printer.R) fxy (
    pr (Code_Printer.INFX (target_fxy, Code_Printer.X)) t1,
    Code_Printer.str target_cons,
    pr (Code_Printer.INFX (target_fxy, Code_Printer.R)) t2
  );

fun add_literal_list target =
  let
    fun pretty literals pr _ vars fxy [(t1, _), (t2, _)] =
      case Option.map (cons t1) (implode_list t2)
       of SOME ts =>
            Code_Printer.literal_list literals (map (pr vars Code_Printer.NOBR) ts)
        | NONE =>
            default_list (Code_Printer.infix_cons literals) (pr vars) fxy t1 t2;
  in
    Code_Target.set_printings (Code_Symbol.Constant (@{const_name Cons},
      [(target, SOME (Code_Printer.complex_const_syntax (2, pretty)))]))
  end

end;
*}

code_printing
  type_constructor list \<rightharpoonup>
    (SML) "_ list"
    and (OCaml) "_ list"
    and (Haskell) "![(_)]"
    and (Scala) "List[(_)]"
| constant Nil \<rightharpoonup>
    (SML) "[]"
    and (OCaml) "[]"
    and (Haskell) "[]"
    and (Scala) "!Nil"
| class_instance list :: equal \<rightharpoonup>
    (Haskell) -
| constant "HOL.equal :: 'a list \<Rightarrow> 'a list \<Rightarrow> bool" \<rightharpoonup>
    (Haskell) infix 4 "=="

setup {* fold (List_Code.add_literal_list) ["SML", "OCaml", "Haskell", "Scala"] *}

code_reserved SML
  list

code_reserved OCaml
  list


subsubsection {* Use convenient predefined operations *}

code_printing
  constant "op @" \<rightharpoonup>
    (SML) infixr 7 "@"
    and (OCaml) infixr 6 "@"
    and (Haskell) infixr 5 "++"
    and (Scala) infixl 7 "++"
| constant map \<rightharpoonup>
    (Haskell) "map"
| constant filter \<rightharpoonup>
    (Haskell) "filter"
| constant concat \<rightharpoonup>
    (Haskell) "concat"
| constant List.maps \<rightharpoonup>
    (Haskell) "concatMap"
| constant rev \<rightharpoonup>
    (Haskell) "reverse"
| constant zip \<rightharpoonup>
    (Haskell) "zip"
| constant List.null \<rightharpoonup>
    (Haskell) "null"
| constant takeWhile \<rightharpoonup>
    (Haskell) "takeWhile"
| constant dropWhile \<rightharpoonup>
    (Haskell) "dropWhile"
| constant list_all \<rightharpoonup>
    (Haskell) "all"
| constant list_ex \<rightharpoonup>
    (Haskell) "any"


subsubsection {* Implementation of sets by lists *}

lemma is_empty_set [code]:
  "Set.is_empty (set xs) \<longleftrightarrow> List.null xs"
  by (simp add: Set.is_empty_def null_def)

lemma empty_set [code]:
  "{} = set []"
  by simp

lemma UNIV_coset [code]:
  "UNIV = List.coset []"
  by simp

lemma compl_set [code]:
  "- set xs = List.coset xs"
  by simp

lemma compl_coset [code]:
  "- List.coset xs = set xs"
  by simp

lemma [code]:
  "x \<in> set xs \<longleftrightarrow> List.member xs x"
  "x \<in> List.coset xs \<longleftrightarrow> \<not> List.member xs x"
  by (simp_all add: member_def)

lemma insert_code [code]:
  "insert x (set xs) = set (List.insert x xs)"
  "insert x (List.coset xs) = List.coset (removeAll x xs)"
  by simp_all

lemma remove_code [code]:
  "Set.remove x (set xs) = set (removeAll x xs)"
  "Set.remove x (List.coset xs) = List.coset (List.insert x xs)"
  by (simp_all add: remove_def Compl_insert)

lemma filter_set [code]:
  "Set.filter P (set xs) = set (filter P xs)"
  by auto

lemma image_set [code]:
  "image f (set xs) = set (map f xs)"
  by simp

lemma subset_code [code]:
  "set xs \<le> B \<longleftrightarrow> (\<forall>x\<in>set xs. x \<in> B)"
  "A \<le> List.coset ys \<longleftrightarrow> (\<forall>y\<in>set ys. y \<notin> A)"
  "List.coset [] \<le> set [] \<longleftrightarrow> False"
  by auto

text {* A frequent case – avoid intermediate sets *}
lemma [code_unfold]:
  "set xs \<subseteq> set ys \<longleftrightarrow> list_all (\<lambda>x. x \<in> set ys) xs"
  by (auto simp: list_all_iff)

lemma Ball_set [code]:
  "Ball (set xs) P \<longleftrightarrow> list_all P xs"
  by (simp add: list_all_iff)

lemma Bex_set [code]:
  "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
  by (simp add: list_ex_iff)

lemma card_set [code]:
  "card (set xs) = length (remdups xs)"
proof -
  have "card (set (remdups xs)) = length (remdups xs)"
    by (rule distinct_card) simp
  then show ?thesis by simp
qed

lemma the_elem_set [code]:
  "the_elem (set [x]) = x"
  by simp

lemma Pow_set [code]:
  "Pow (set []) = {{}}"
  "Pow (set (x # xs)) = (let A = Pow (set xs) in A \<union> insert x ` A)"
  by (simp_all add: Pow_insert Let_def)

lemma setsum_code [code]:
  "setsum f (set xs) = listsum (map f (remdups xs))"
by (simp add: listsum_distinct_conv_setsum_set)

definition map_project :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a set \<Rightarrow> 'b set" where
  "map_project f A = {b. \<exists> a \<in> A. f a = Some b}"

lemma [code]:
  "map_project f (set xs) = set (List.map_filter f xs)"
  by (auto simp add: map_project_def map_filter_def image_def)

hide_const (open) map_project


text {* Operations on relations *}

lemma product_code [code]:
  "Product_Type.product (set xs) (set ys) = set [(x, y). x \<leftarrow> xs, y \<leftarrow> ys]"
  by (auto simp add: Product_Type.product_def)

lemma Id_on_set [code]:
  "Id_on (set xs) = set [(x, x). x \<leftarrow> xs]"
  by (auto simp add: Id_on_def)

lemma [code]:
  "R `` S = List.map_project (%(x, y). if x : S then Some y else None) R"
unfolding map_project_def by (auto split: prod.split split_if_asm)

lemma trancl_set_ntrancl [code]:
  "trancl (set xs) = ntrancl (card (set xs) - 1) (set xs)"
  by (simp add: finite_trancl_ntranl)

lemma set_relcomp [code]:
  "set xys O set yzs = set ([(fst xy, snd yz). xy \<leftarrow> xys, yz \<leftarrow> yzs, snd xy = fst yz])"
  by (auto simp add: Bex_def)

lemma wf_set [code]:
  "wf (set xs) = acyclic (set xs)"
  by (simp add: wf_iff_acyclic_if_finite)


subsection {* Setup for Lifting/Transfer *}

subsubsection {* Transfer rules for the Transfer package *}

context
begin
interpretation lifting_syntax .

lemma Nil_transfer [transfer_rule]: "(list_all2 A) [] []"
  by simp

lemma Cons_transfer [transfer_rule]:
  "(A ===> list_all2 A ===> list_all2 A) Cons Cons"
  unfolding rel_fun_def by simp

lemma case_list_transfer [transfer_rule]:
  "(B ===> (A ===> list_all2 A ===> B) ===> list_all2 A ===> B)
    case_list case_list"
  unfolding rel_fun_def by (simp split: list.split)

lemma rec_list_transfer [transfer_rule]:
  "(B ===> (A ===> list_all2 A ===> B ===> B) ===> list_all2 A ===> B)
    rec_list rec_list"
  unfolding rel_fun_def by (clarify, erule list_all2_induct, simp_all)

lemma tl_transfer [transfer_rule]:
  "(list_all2 A ===> list_all2 A) tl tl"
  unfolding tl_def[abs_def] by transfer_prover

lemma butlast_transfer [transfer_rule]:
  "(list_all2 A ===> list_all2 A) butlast butlast"
  by (rule rel_funI, erule list_all2_induct, auto)

lemma set_transfer [transfer_rule]:
  "(list_all2 A ===> rel_set A) set set"
  unfolding set_rec[abs_def] by transfer_prover

lemma map_rec: "map f xs = rec_list Nil (%x _ y. Cons (f x) y) xs"
  by (induct xs) auto

lemma map_transfer [transfer_rule]:
  "((A ===> B) ===> list_all2 A ===> list_all2 B) map map"
  unfolding map_rec[abs_def] by transfer_prover

lemma append_transfer [transfer_rule]:
  "(list_all2 A ===> list_all2 A ===> list_all2 A) append append"
  unfolding List.append_def by transfer_prover

lemma rev_transfer [transfer_rule]:
  "(list_all2 A ===> list_all2 A) rev rev"
  unfolding List.rev_def by transfer_prover

lemma filter_transfer [transfer_rule]:
  "((A ===> op =) ===> list_all2 A ===> list_all2 A) filter filter"
  unfolding List.filter_def by transfer_prover

lemma fold_transfer [transfer_rule]:
  "((A ===> B ===> B) ===> list_all2 A ===> B ===> B) fold fold"
  unfolding List.fold_def by transfer_prover

lemma foldr_transfer [transfer_rule]:
  "((A ===> B ===> B) ===> list_all2 A ===> B ===> B) foldr foldr"
  unfolding List.foldr_def by transfer_prover

lemma foldl_transfer [transfer_rule]:
  "((B ===> A ===> B) ===> B ===> list_all2 A ===> B) foldl foldl"
  unfolding List.foldl_def by transfer_prover

lemma concat_transfer [transfer_rule]:
  "(list_all2 (list_all2 A) ===> list_all2 A) concat concat"
  unfolding List.concat_def by transfer_prover

lemma drop_transfer [transfer_rule]:
  "(op = ===> list_all2 A ===> list_all2 A) drop drop"
  unfolding List.drop_def by transfer_prover

lemma take_transfer [transfer_rule]:
  "(op = ===> list_all2 A ===> list_all2 A) take take"
  unfolding List.take_def by transfer_prover

lemma list_update_transfer [transfer_rule]:
  "(list_all2 A ===> op = ===> A ===> list_all2 A) list_update list_update"
  unfolding list_update_def by transfer_prover

lemma takeWhile_transfer [transfer_rule]:
  "((A ===> op =) ===> list_all2 A ===> list_all2 A) takeWhile takeWhile"
  unfolding takeWhile_def by transfer_prover

lemma dropWhile_transfer [transfer_rule]:
  "((A ===> op =) ===> list_all2 A ===> list_all2 A) dropWhile dropWhile"
  unfolding dropWhile_def by transfer_prover

lemma zip_transfer [transfer_rule]:
  "(list_all2 A ===> list_all2 B ===> list_all2 (rel_prod A B)) zip zip"
  unfolding zip_def by transfer_prover

lemma product_transfer [transfer_rule]:
  "(list_all2 A ===> list_all2 B ===> list_all2 (rel_prod A B)) List.product List.product"
  unfolding List.product_def by transfer_prover

lemma product_lists_transfer [transfer_rule]:
  "(list_all2 (list_all2 A) ===> list_all2 (list_all2 A)) product_lists product_lists"
  unfolding product_lists_def by transfer_prover

lemma insert_transfer [transfer_rule]:
  assumes [transfer_rule]: "bi_unique A"
  shows "(A ===> list_all2 A ===> list_all2 A) List.insert List.insert"
  unfolding List.insert_def [abs_def] by transfer_prover

lemma find_transfer [transfer_rule]:
  "((A ===> op =) ===> list_all2 A ===> rel_option A) List.find List.find"
  unfolding List.find_def by transfer_prover

lemma remove1_transfer [transfer_rule]:
  assumes [transfer_rule]: "bi_unique A"
  shows "(A ===> list_all2 A ===> list_all2 A) remove1 remove1"
  unfolding remove1_def by transfer_prover

lemma removeAll_transfer [transfer_rule]:
  assumes [transfer_rule]: "bi_unique A"
  shows "(A ===> list_all2 A ===> list_all2 A) removeAll removeAll"
  unfolding removeAll_def by transfer_prover

lemma distinct_transfer [transfer_rule]:
  assumes [transfer_rule]: "bi_unique A"
  shows "(list_all2 A ===> op =) distinct distinct"
  unfolding distinct_def by transfer_prover

lemma remdups_transfer [transfer_rule]:
  assumes [transfer_rule]: "bi_unique A"
  shows "(list_all2 A ===> list_all2 A) remdups remdups"
  unfolding remdups_def by transfer_prover

lemma remdups_adj_transfer [transfer_rule]:
  assumes [transfer_rule]: "bi_unique A"
  shows "(list_all2 A ===> list_all2 A) remdups_adj remdups_adj"
  proof (rule rel_funI, erule list_all2_induct)
  qed (auto simp: remdups_adj_Cons assms[unfolded bi_unique_def] split: list.splits)

lemma replicate_transfer [transfer_rule]:
  "(op = ===> A ===> list_all2 A) replicate replicate"
  unfolding replicate_def by transfer_prover

lemma length_transfer [transfer_rule]:
  "(list_all2 A ===> op =) length length"
  unfolding list_size_overloaded_def by transfer_prover

lemma rotate1_transfer [transfer_rule]:
  "(list_all2 A ===> list_all2 A) rotate1 rotate1"
  unfolding rotate1_def by transfer_prover

lemma rotate_transfer [transfer_rule]:
  "(op = ===> list_all2 A ===> list_all2 A) rotate rotate"
  unfolding rotate_def [abs_def] by transfer_prover

lemma list_all2_transfer [transfer_rule]:
  "((A ===> B ===> op =) ===> list_all2 A ===> list_all2 B ===> op =)
    list_all2 list_all2"
  apply (subst (4) list_all2_iff [abs_def])
  apply (subst (3) list_all2_iff [abs_def])
  apply transfer_prover
  done

lemma sublist_transfer [transfer_rule]:
  "(list_all2 A ===> rel_set (op =) ===> list_all2 A) sublist sublist"
  unfolding sublist_def [abs_def] by transfer_prover

lemma partition_transfer [transfer_rule]:
  "((A ===> op =) ===> list_all2 A ===> rel_prod (list_all2 A) (list_all2 A))
    partition partition"
  unfolding partition_def by transfer_prover

lemma lists_transfer [transfer_rule]:
  "(rel_set A ===> rel_set (list_all2 A)) lists lists"
  apply (rule rel_funI, rule rel_setI)
  apply (erule lists.induct, simp)
  apply (simp only: rel_set_def list_all2_Cons1, metis lists.Cons)
  apply (erule lists.induct, simp)
  apply (simp only: rel_set_def list_all2_Cons2, metis lists.Cons)
  done

lemma set_Cons_transfer [transfer_rule]:
  "(rel_set A ===> rel_set (list_all2 A) ===> rel_set (list_all2 A))
    set_Cons set_Cons"
  unfolding rel_fun_def rel_set_def set_Cons_def
  apply safe
  apply (simp add: list_all2_Cons1, fast)
  apply (simp add: list_all2_Cons2, fast)
  done

lemma listset_transfer [transfer_rule]:
  "(list_all2 (rel_set A) ===> rel_set (list_all2 A)) listset listset"
  unfolding listset_def by transfer_prover

lemma null_transfer [transfer_rule]:
  "(list_all2 A ===> op =) List.null List.null"
  unfolding rel_fun_def List.null_def by auto

lemma list_all_transfer [transfer_rule]:
  "((A ===> op =) ===> list_all2 A ===> op =) list_all list_all"
  unfolding list_all_iff [abs_def] by transfer_prover

lemma list_ex_transfer [transfer_rule]:
  "((A ===> op =) ===> list_all2 A ===> op =) list_ex list_ex"
  unfolding list_ex_iff [abs_def] by transfer_prover

lemma splice_transfer [transfer_rule]:
  "(list_all2 A ===> list_all2 A ===> list_all2 A) splice splice"
  apply (rule rel_funI, erule list_all2_induct, simp add: rel_fun_def, simp)
  apply (rule rel_funI)
  apply (erule_tac xs=x in list_all2_induct, simp, simp add: rel_fun_def)
  done

lemma listsum_transfer[transfer_rule]:
  assumes [transfer_rule]: "A 0 0"
  assumes [transfer_rule]: "(A ===> A ===> A) op + op +"
  shows "(list_all2 A ===> A) listsum listsum"
  unfolding listsum_def[abs_def]
  by transfer_prover

end

end