src/HOL/List.thy
 author blanchet Fri, 18 Oct 2013 10:43:20 +0200 changeset 54147 97a8ff4e4ac9 parent 53954 ccfd22f937be child 54295 45a5523d4a63 permissions -rw-r--r--
killed most "no_atp", to make Sledgehammer more complete

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

header {* The datatype of finite lists *}

theory List
imports Presburger Code_Numeral Quotient ATP Lifting_Set Lifting_Option Lifting_Product
begin

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

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

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

subsection {* Basic list processing functions *}

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

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

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

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

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

definition coset :: "'a list \<Rightarrow> 'a set" where
[simp]: "coset xs = - set xs"

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

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

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

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

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

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

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

primrec 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

primrec those :: "'a option list \<Rightarrow> 'a list option"
where
"those [] = Some []" |
"those (x # xs) = (case x of
None \<Rightarrow> None
| Some y \<Rightarrow> Option.map (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 (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 list_all2 :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool" where
"list_all2 P xs ys =
(length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"

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

primrec 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 "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"} $Syntax.const @{const_syntax dummy_pattern}$ 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
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 thy = Proof_Context.theory_of ctxt
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 $(Const (@{const_name List.Nil}, _))) = t | dest_singleton_list t = raise TERM ("dest_singleton_list", [t]) (* We check that one case returns a singleton list and all other cases return [], and return the index of the one singleton list case *) fun possible_index_of_singleton_case cases = let fun check (i, case_t) s = (case strip_abs_body case_t of (Const (@{const_name List.Nil}, _)) => s | _ => (case s of SOME NONE => SOME (SOME i) | _ => NONE)) in fold_index check cases (SOME NONE) |> the_default NONE end (* returns (case_expr type index chosen_case) option *) fun dest_case case_term = let val (case_const, args) = strip_comb case_term in (case try dest_Const case_const of SOME (c, T) => (case Datatype.info_of_case thy c of SOME _ => (case possible_index_of_singleton_case (fst (split_last args)) of SOME i => let val (Ts, _) = strip_type T val T' = List.last Ts in SOME (List.last args, T', i, nth args i) end | NONE => NONE) | NONE => NONE) | NONE => NONE) end (* returns condition continuing term option *) fun dest_if (Const (@{const_name If}, _)$ 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 info = Datatype.the_info thy (fst (dest_Type T))
in
(* do case distinction *)
Splitter.split_tac [#split info] 1
THEN EVERY (map_index (fn (i', _) =>
(if i' < length (#case_rewrites info) - 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 (#inject info)))))
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}) (#distinct info))))
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_rewrites info))
end
fun make_inner_eqs bound_vs Tis eqs t =
(case dest_case t of
SOME (x, T, i, cont) =>
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_name, _) = nth (the (Datatype.get_constrs thy (fst (dest_Type T)))) i
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 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)

enriched_type map: map

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"

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} *}

declare set.simps [code_post]  --"pretty output"

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 (metis List.set.simps(1) emptyE)
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
qed

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

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

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

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

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

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

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

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

subsubsection {* @{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)"

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
also have "\<dots> = card ?S'" using eq fin
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
also have "\<dots> = card ?S'" using eq fin
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"

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
done

lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
apply (induct xs, simp, simp)
apply safe
apply (metis nat_case_0 nth.simps zero_less_Suc)
apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
apply (case_tac i, simp)
apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff)
done

lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
by(auto simp:set_conv_nth)

lemma 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)"

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

lemma all_nth_imp_all_set:
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"

lemma all_set_conv_all_nth:
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"

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)
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"

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(metis length_0_conv 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"

lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"

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"
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
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 (metis append_butlast_last_id append_is_Nil_conv butlast_snoc last_snoc not_Cons_self)

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)"

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)
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
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
done

lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)"
apply(induct xs arbitrary: n)
apply simp
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_max.inf_absorb1 min_max.inf_absorb2)

lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)"

lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs"

lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)"

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

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)
apply (metis set_drop_subset subset_iff)
done

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

"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
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 (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"

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)}"

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)]"

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"

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"

lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"

lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"

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

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_def zip_rev cong: conj_cong)

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

lemma list_all2_append1:
"list_all2 P (xs @ ys) zs =
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
list_all2 P xs us \<and> list_all2 P ys vs)"
apply (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)
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 (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)
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)"

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

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

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

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

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

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"

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

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

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 (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 (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 []"

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:
"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

declare Inf_fin.set_eq_fold [code]
declare Sup_fin.set_eq_fold [code]
declare Min.set_eq_fold [code]
declare Max.set_eq_fold [code]

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:
"INFI (set xs) f = fold (inf \<circ> f) xs top"
unfolding INF_def set_map [symmetric] Inf_set_fold fold_map ..

declare INF_set_fold [code]

lemma (in complete_lattice) SUP_set_fold:
"SUPR (set xs) f = fold (sup \<circ> f) xs bot"
unfolding SUP_def set_map [symmetric] Sup_set_fold 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)"

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

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)"

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

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 []"

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 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]"

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"

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

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 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])
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 (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)
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"

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

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

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" "neg_numeral n"]
upto.simps[of "neg_numeral m" "numeral n"]
upto.simps[of "neg_numeral m" "neg_numeral n"] for m n

lemma upto_empty[simp]: "j < i \<Longrightarrow> [i..j] = []"

lemma upto_rec1: "i \<le> j \<Longrightarrow> [i..j] = i#[i+1..j]"

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] simp_from_to[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))"

lemma upto_code[code]: "[i..j] = upto_aux i j []"

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 (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 (case_tac j)
apply (clarsimp simp add: set_conv_nth, simp)
apply (rule conjI)
(*TOO SLOW
apply (metis Zero_neq_Suc gr0_conv_Suc in_set_conv_nth lessI less_trans_Suc nth_Cons' nth_Cons_Suc)
*)
apply (erule_tac x = 0 in allE, simp)
apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
(*TOO SLOW
apply (metis Suc_Suc_eq lessI less_trans_Suc nth_Cons_Suc)
*)
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)"

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 (metis distinct.simps(2) hd.simps hd_in_set list.simps(2) set_ConsD set_rev_mp set_subset_Cons)

(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)

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

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)

by (induct xs rule: remdups_adj.induct, simp_all)

by (induct xs rule: remdups_adj.induct, simp_all)

by (induct xs rule: remdups_adj.induct, auto)

by (induct xs rule: remdups_adj.induct, simp_all)

"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)

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

assumes "inj f"

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

"listsum [] = 0"
"listsum (x # xs) = x + listsum xs"

"listsum (xs @ ys) = listsum xs + listsum ys"

"listsum (rev xs) = listsum xs"

"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)"

"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)

"list_size f xs = listsum (map f xs) + size xs"
by (induct xs) auto

"length (concat xss) = listsum (map length xss)"
by (induct xss) simp_all

"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

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

"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

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

"(\<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: *}
"- listsum (map f xs) = listsum (map (uminus \<circ> f) xs)"
by (induct xs) simp_all

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

"(\<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)

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

lemma listsum_mono:
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)"

"distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)"
by (induct xs) simp_all

"listsum (map f [m..<n]) = setsum f (set [m..<n])"

"listsum (map f [k..l]) = setsum f (set [k..l])"

text {* General equivalence between @{const listsum} and @{const setsum} *}
"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"

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

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

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

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

lemma insert_remdups:
"List.insert x (remdups xs) = remdups (List.insert x xs)"

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
by(auto simp: nth_Cons' split: if_splits)
(metis One_nat_def diff_Suc_1 less_Suc_eq_0_disj)
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
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 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)"

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)"

lemma map_removeAll_inj: "inj f \<Longrightarrow>
map f (removeAll x xs) = removeAll (f x) (map f xs)"
by(metis map_removeAll_inj_on subset_inj_on 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)
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"
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 (drule not_leE)
apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
apply  simp
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)"

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

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

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
with * show ?thesis by blast
qed
then show ?case
using xs'_def ys'_def by metis
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 assms by (metis comm_append_are_replicate)
then have "m + n > 1" and "concat (replicate (m+n) zs) = xs @ ys"
using xs \<noteq> [] and ys \<noteq> []
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")
done

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

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

lemma map_snd_enumerate [simp]:
"map snd (enumerate n xs) = xs"

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)

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)"

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

lemma rotate0[simp]: "rotate 0 = id"

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

"rotate (m+n) = rotate m o rotate n"

lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"

lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"

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
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(cases "xs = []")
apply (simp)
apply(case_tac "n mod length xs = 0")
take_hd_drop linorder_not_le)
done

lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"

lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"

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"

lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"

lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
by (cases xs) auto

lemma set_rotate[simp]: "set(rotate n xs) = set xs"

lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
by (cases xs) auto

lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"

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

lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
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 {} = []"

lemma sublist_nil [simp]: "sublist [] A = []"

lemma length_sublist:
"length(sublist xs I) = card{i. i < length xs \<and> i : I}"

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]"

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)
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 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"

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

lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"

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

lemma distinct_sublistI[simp]: "distinct xs \<Longrightarrow> distinct(sublist xs I)"
apply(induct xs arbitrary: I)
apply simp
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"
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))"
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)"
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

"concat (map (list_case [] (\<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 (list_case [] (\<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
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 (list_case [] (\<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_tac y=x in 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)
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"
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 (metis UNIV_I length_replicate less_not_refl)
done

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

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

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"

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)

"sorted xs \<Longrightarrow> sorted (remdups_adj xs)"

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"
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"

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"

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]"

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

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}). *}

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

sublocale linorder < 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

context linorder
begin

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

lemma sorted_list_of_set_insert [simp]:
assumes "finite A"
shows "sorted_list_of_set (insert x A) = insort x (sorted_list_of_set (A - {x}))"
using assms 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 = {}"
using assms 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 (fA) = 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: map.simps simp add: map.simps[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"

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_pair (%(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_pair_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"

lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"

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 (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)

-- {* 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

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 (metis Cons_listrel1_Cons)

lemma listrel1I2: "(xs, ys) \<in> listrel1 r \<Longrightarrow> (x # xs, x # ys) \<in> listrel1 r"
by (metis Cons_listrel1_Cons)

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> acc r \<Longrightarrow> xs \<in> acc (listrel1 r) \<Longrightarrow> (x # xs) \<in> acc (listrel1 r)"
apply (induct arbitrary: xs set: acc)
apply (erule thin_rl)
apply (erule acc_induct)
apply (rule accI)
apply (blast)
done

lemma lists_accD: "xs \<in> lists (acc r) \<Longrightarrow> xs \<in> 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> acc (listrel1 r) \<Longrightarrow> xs \<in> lists (acc r)"
apply (induct set: 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(metis wf_acc_iff in_lists_conv_set lists_accI lists_accD Cons_in_lists_iff)

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 (erule listrel.induct)
apply (blast intro: listrel.intros)+
done

lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)"
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+

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"

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"

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"

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

definition list_all :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
list_all_iff [code_abbrev]: "list_all P xs \<longleftrightarrow> Ball (set xs) P"

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 [simp, code]:
"list_all P (x # xs) \<longleftrightarrow> P x \<and> list_all P xs"
"list_all P [] \<longleftrightarrow> True"

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

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"

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

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

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

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

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

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"

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"

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"

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"

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])"

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)"

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])"

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"

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

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 [] = []"

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"

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"

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"

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)"

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"

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)"

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"

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

lemma length_code [code]: "length = gen_length 0"

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: string -> string -> 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 nil' cons' t =
let
fun dest_cons (IConst { name = c, ... } $t1 $ t2) =
if c = cons'
then SOME (t1, t2)
else NONE
| dest_cons _ = NONE;
val (ts, t') = Code_Thingol.unfoldr dest_cons t;
in case t'
of IConst { name = c, ... } => if c = nil' then SOME ts else NONE
| _ => 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
);

let
fun pretty literals [nil', cons'] pr thm vars fxy [(t1, _), (t2, _)] =
case Option.map (cons t1) (implode_list nil' cons' 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, ([@{const_name Nil}, @{const_name Cons}], pretty))))]))
end

end;
*}

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

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 (Scala) infixl 7 "++"
| constant map \<rightharpoonup>
| constant filter \<rightharpoonup>
| constant concat \<rightharpoonup>
| constant List.maps \<rightharpoonup>
| constant rev \<rightharpoonup>
| constant zip \<rightharpoonup>
| constant List.null \<rightharpoonup>
| constant takeWhile \<rightharpoonup>
| constant dropWhile \<rightharpoonup>
| constant list_all \<rightharpoonup>
| constant list_ex \<rightharpoonup>

subsubsection {* Implementation of sets by lists *}

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

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"

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)"

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"

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

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)"

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

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]"

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

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)"

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

lemma wf_set [code]:
"wf (set xs) = acyclic (set xs)"

subsection {* Setup for Lifting/Transfer *}

subsubsection {* Relator and predicator properties *}

lemma list_all2_eq'[relator_eq]:
"list_all2 (op =) = (op =)"
by (rule ext)+ (simp add: list_all2_eq)

lemma list_all2_mono'[relator_mono]:
assumes "A \<le> B"
shows "(list_all2 A) \<le> (list_all2 B)"
using assms by (auto intro: list_all2_mono)

lemma list_all2_OO[relator_distr]: "list_all2 A OO list_all2 B = list_all2 (A OO B)"
proof (intro ext iffI)
fix xs ys
assume "list_all2 (A OO B) xs ys"
thus "(list_all2 A OO list_all2 B) xs ys"
unfolding OO_def
by (induct, simp, simp add: list_all2_Cons1 list_all2_Cons2, fast)
next
fix xs ys
assume "(list_all2 A OO list_all2 B) xs ys"
then obtain zs where "list_all2 A xs zs" and "list_all2 B zs ys" ..
thus "list_all2 (A OO B) xs ys"
by (induct arbitrary: ys, simp, clarsimp simp add: list_all2_Cons1, fast)
qed

lemma Domainp_list[relator_domain]:
assumes "Domainp A = P"
shows "Domainp (list_all2 A) = (list_all P)"
proof -
{
fix x
have *: "\<And>x. (\<exists>y. A x y) = P x" using assms unfolding Domainp_iff by blast
have "(\<exists>y. (list_all2 A x y)) = list_all P x"
by (induction x) (simp_all add: * list_all2_Cons1)
}
then show ?thesis
unfolding Domainp_iff[abs_def]
by (auto iff: fun_eq_iff)
qed

lemma reflp_list_all2[reflexivity_rule]:
assumes "reflp R"
shows "reflp (list_all2 R)"
proof (rule reflpI)
from assms have *: "\<And>xs. R xs xs" by (rule reflpE)
fix xs
show "list_all2 R xs xs"
by (induct xs) (simp_all add: *)
qed

lemma left_total_list_all2[reflexivity_rule]:
"left_total R \<Longrightarrow> left_total (list_all2 R)"
unfolding left_total_def
apply safe
apply (rename_tac xs, induct_tac xs, simp, simp add: list_all2_Cons1)
done

lemma left_unique_list_all2 [reflexivity_rule]:
"left_unique R \<Longrightarrow> left_unique (list_all2 R)"
unfolding left_unique_def
apply (subst (2) all_comm, subst (1) all_comm)
apply (rule allI, rename_tac zs, induct_tac zs)
done

lemma right_total_list_all2 [transfer_rule]:
"right_total R \<Longrightarrow> right_total (list_all2 R)"
unfolding right_total_def
by (rule allI, induct_tac y, simp, simp add: list_all2_Cons2)

lemma right_unique_list_all2 [transfer_rule]:
"right_unique R \<Longrightarrow> right_unique (list_all2 R)"
unfolding right_unique_def
apply (rule allI, rename_tac xs, induct_tac xs)
done

lemma bi_total_list_all2 [transfer_rule]:
"bi_total A \<Longrightarrow> bi_total (list_all2 A)"
unfolding bi_total_def
apply safe
apply (rename_tac xs, induct_tac xs, simp, simp add: list_all2_Cons1)
apply (rename_tac ys, induct_tac ys, simp, simp add: list_all2_Cons2)
done

lemma bi_unique_list_all2 [transfer_rule]:
"bi_unique A \<Longrightarrow> bi_unique (list_all2 A)"
unfolding bi_unique_def
apply (rule conjI)
apply (rule allI, rename_tac xs, induct_tac xs)
apply (simp, force simp add: list_all2_Cons1)
apply (subst (2) all_comm, subst (1) all_comm)
apply (rule allI, rename_tac xs, induct_tac xs)
apply (simp, force simp add: list_all2_Cons2)
done

lemma list_invariant_commute [invariant_commute]:
"list_all2 (Lifting.invariant P) = Lifting.invariant (list_all P)"
apply (simp add: fun_eq_iff list_all2_def list_all_iff Lifting.invariant_def Ball_def)
apply (intro allI)
apply (induct_tac rule: list_induct2')
apply simp_all
apply fastforce
done

subsubsection {* Quotient theorem for the Lifting package *}

lemma Quotient_list[quot_map]:
assumes "Quotient R Abs Rep T"
shows "Quotient (list_all2 R) (map Abs) (map Rep) (list_all2 T)"
proof (unfold Quotient_alt_def, intro conjI allI impI)
from assms have 1: "\<And>x y. T x y \<Longrightarrow> Abs x = y"
unfolding Quotient_alt_def by simp
fix xs ys assume "list_all2 T xs ys" thus "map Abs xs = ys"
by (induct, simp, simp add: 1)
next
from assms have 2: "\<And>x. T (Rep x) x"
unfolding Quotient_alt_def by simp
fix xs show "list_all2 T (map Rep xs) xs"
by (induct xs, simp, simp add: 2)
next
from assms have 3: "\<And>x y. R x y \<longleftrightarrow> T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y"
unfolding Quotient_alt_def by simp
fix xs ys show "list_all2 R xs ys \<longleftrightarrow> list_all2 T xs (map Abs xs) \<and>
list_all2 T ys (map Abs ys) \<and> map Abs xs = map Abs ys"
by (induct xs ys rule: list_induct2', simp_all, metis 3)
qed

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 fun_rel_def by simp

lemma list_case_transfer [transfer_rule]:
"(B ===> (A ===> list_all2 A ===> B) ===> list_all2 A ===> B)
list_case list_case"
unfolding fun_rel_def by (simp split: list.split)

lemma list_rec_transfer [transfer_rule]:
"(B ===> (A ===> list_all2 A ===> B ===> B) ===> list_all2 A ===> B)
list_rec list_rec"
unfolding fun_rel_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 by transfer_prover

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

lemma set_transfer [transfer_rule]:
"(list_all2 A ===> set_rel A) set set"
unfolding set_def by transfer_prover

lemma map_transfer [transfer_rule]:
"((A ===> B) ===> list_all2 A ===> list_all2 B) map map"
unfolding List.map_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 (prod_rel A B)) zip zip"
unfolding zip_def by transfer_prover

lemma product_transfer [transfer_rule]:
"(list_all2 A ===> list_all2 B ===> list_all2 (prod_rel 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 ===> option_rel 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

assumes [transfer_rule]: "bi_unique A"
proof (rule fun_relI, 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"

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_def [abs_def])
apply (subst (3) list_all2_def [abs_def])
apply transfer_prover
done

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

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

lemma lists_transfer [transfer_rule]:
"(set_rel A ===> set_rel (list_all2 A)) lists lists"
apply (rule fun_relI, rule set_relI)
apply (erule lists.induct, simp)
apply (simp only: set_rel_def list_all2_Cons1, metis lists.Cons)
apply (erule lists.induct, simp)
apply (simp only: set_rel_def list_all2_Cons2, metis lists.Cons)
done

lemma set_Cons_transfer [transfer_rule]:
"(set_rel A ===> set_rel (list_all2 A) ===> set_rel (list_all2 A))
set_Cons set_Cons"
unfolding fun_rel_def set_rel_def set_Cons_def
apply safe
done

lemma listset_transfer [transfer_rule]:
"(list_all2 (set_rel A) ===> set_rel (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 fun_rel_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 fun_relI, erule list_all2_induct, simp add: fun_rel_def, simp)
apply (rule fun_relI)
apply (erule_tac xs=x in list_all2_induct, simp, simp add: fun_rel_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