src/HOL/List.thy
author kuncar
Tue Jul 31 13:55:39 2012 +0200 (2012-07-31)
changeset 48619 558e4e77ce69
parent 47841 179b5e7c9803
child 48828 441a4eed7823
permissions -rw-r--r--
more set operations expressed by Finite_Set.fold
     1 (*  Title:      HOL/List.thy
     2     Author:     Tobias Nipkow
     3 *)
     4 
     5 header {* The datatype of finite lists *}
     6 
     7 theory List
     8 imports Plain Presburger Code_Numeral Quotient ATP
     9 uses
    10   ("Tools/list_code.ML")
    11   ("Tools/list_to_set_comprehension.ML")
    12 begin
    13 
    14 datatype 'a list =
    15     Nil    ("[]")
    16   | Cons 'a  "'a list"    (infixr "#" 65)
    17 
    18 syntax
    19   -- {* list Enumeration *}
    20   "_list" :: "args => 'a list"    ("[(_)]")
    21 
    22 translations
    23   "[x, xs]" == "x#[xs]"
    24   "[x]" == "x#[]"
    25 
    26 
    27 subsection {* Basic list processing functions *}
    28 
    29 primrec
    30   hd :: "'a list \<Rightarrow> 'a" where
    31   "hd (x # xs) = x"
    32 
    33 primrec
    34   tl :: "'a list \<Rightarrow> 'a list" where
    35     "tl [] = []"
    36   | "tl (x # xs) = xs"
    37 
    38 primrec
    39   last :: "'a list \<Rightarrow> 'a" where
    40   "last (x # xs) = (if xs = [] then x else last xs)"
    41 
    42 primrec
    43   butlast :: "'a list \<Rightarrow> 'a list" where
    44     "butlast []= []"
    45   | "butlast (x # xs) = (if xs = [] then [] else x # butlast xs)"
    46 
    47 primrec
    48   set :: "'a list \<Rightarrow> 'a set" where
    49     "set [] = {}"
    50   | "set (x # xs) = insert x (set xs)"
    51 
    52 definition
    53   coset :: "'a list \<Rightarrow> 'a set" where
    54   [simp]: "coset xs = - set xs"
    55 
    56 primrec
    57   map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
    58     "map f [] = []"
    59   | "map f (x # xs) = f x # map f xs"
    60 
    61 primrec
    62   append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65) where
    63     append_Nil:"[] @ ys = ys"
    64   | append_Cons: "(x#xs) @ ys = x # xs @ ys"
    65 
    66 primrec
    67   rev :: "'a list \<Rightarrow> 'a list" where
    68     "rev [] = []"
    69   | "rev (x # xs) = rev xs @ [x]"
    70 
    71 primrec
    72   filter:: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
    73     "filter P [] = []"
    74   | "filter P (x # xs) = (if P x then x # filter P xs else filter P xs)"
    75 
    76 syntax
    77   -- {* Special syntax for filter *}
    78   "_filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_<-_./ _])")
    79 
    80 translations
    81   "[x<-xs . P]"== "CONST filter (%x. P) xs"
    82 
    83 syntax (xsymbols)
    84   "_filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
    85 syntax (HTML output)
    86   "_filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
    87 
    88 primrec fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b"
    89 where
    90   fold_Nil:  "fold f [] = id"
    91 | fold_Cons: "fold f (x # xs) = fold f xs \<circ> f x" -- {* natural argument order *}
    92 
    93 primrec foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b"
    94 where
    95   foldr_Nil:  "foldr f [] = id"
    96 | foldr_Cons: "foldr f (x # xs) = f x \<circ> foldr f xs" -- {* natural argument order *}
    97 
    98 primrec foldl :: "('b \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"
    99 where
   100   foldl_Nil:  "foldl f a [] = a"
   101 | foldl_Cons: "foldl f a (x # xs) = foldl f (f a x) xs"
   102 
   103 primrec
   104   concat:: "'a list list \<Rightarrow> 'a list" where
   105     "concat [] = []"
   106   | "concat (x # xs) = x @ concat xs"
   107 
   108 definition (in monoid_add)
   109   listsum :: "'a list \<Rightarrow> 'a" where
   110   "listsum xs = foldr plus xs 0"
   111 
   112 primrec
   113   drop:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   114     drop_Nil: "drop n [] = []"
   115   | drop_Cons: "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
   116   -- {*Warning: simpset does not contain this definition, but separate
   117        theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
   118 
   119 primrec
   120   take:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   121     take_Nil:"take n [] = []"
   122   | take_Cons: "take n (x # xs) = (case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # take m xs)"
   123   -- {*Warning: simpset does not contain this definition, but separate
   124        theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
   125 
   126 primrec
   127   nth :: "'a list => nat => 'a" (infixl "!" 100) where
   128   nth_Cons: "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
   129   -- {*Warning: simpset does not contain this definition, but separate
   130        theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
   131 
   132 primrec
   133   list_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
   134     "list_update [] i v = []"
   135   | "list_update (x # xs) i v = (case i of 0 \<Rightarrow> v # xs | Suc j \<Rightarrow> x # list_update xs j v)"
   136 
   137 nonterminal lupdbinds and lupdbind
   138 
   139 syntax
   140   "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
   141   "" :: "lupdbind => lupdbinds"    ("_")
   142   "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
   143   "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
   144 
   145 translations
   146   "_LUpdate xs (_lupdbinds b bs)" == "_LUpdate (_LUpdate xs b) bs"
   147   "xs[i:=x]" == "CONST list_update xs i x"
   148 
   149 primrec
   150   takeWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   151     "takeWhile P [] = []"
   152   | "takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])"
   153 
   154 primrec
   155   dropWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   156     "dropWhile P [] = []"
   157   | "dropWhile P (x # xs) = (if P x then dropWhile P xs else x # xs)"
   158 
   159 primrec
   160   zip :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
   161     "zip xs [] = []"
   162   | zip_Cons: "zip xs (y # ys) = (case xs of [] => [] | z # zs => (z, y) # zip zs ys)"
   163   -- {*Warning: simpset does not contain this definition, but separate
   164        theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
   165 
   166 primrec 
   167   upt :: "nat \<Rightarrow> nat \<Rightarrow> nat list" ("(1[_..</_'])") where
   168     upt_0: "[i..<0] = []"
   169   | upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
   170 
   171 definition
   172   insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   173   "insert x xs = (if x \<in> set xs then xs else x # xs)"
   174 
   175 hide_const (open) insert
   176 hide_fact (open) insert_def
   177 
   178 primrec find :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a option" where
   179   "find _ [] = None"
   180 | "find P (x#xs) = (if P x then Some x else find P xs)"
   181 
   182 hide_const (open) find
   183 
   184 primrec
   185   remove1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   186     "remove1 x [] = []"
   187   | "remove1 x (y # xs) = (if x = y then xs else y # remove1 x xs)"
   188 
   189 primrec
   190   removeAll :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   191     "removeAll x [] = []"
   192   | "removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)"
   193 
   194 primrec
   195   distinct :: "'a list \<Rightarrow> bool" where
   196     "distinct [] \<longleftrightarrow> True"
   197   | "distinct (x # xs) \<longleftrightarrow> x \<notin> set xs \<and> distinct xs"
   198 
   199 primrec
   200   remdups :: "'a list \<Rightarrow> 'a list" where
   201     "remdups [] = []"
   202   | "remdups (x # xs) = (if x \<in> set xs then remdups xs else x # remdups xs)"
   203 
   204 primrec
   205   replicate :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
   206     replicate_0: "replicate 0 x = []"
   207   | replicate_Suc: "replicate (Suc n) x = x # replicate n x"
   208 
   209 text {*
   210   Function @{text size} is overloaded for all datatypes. Users may
   211   refer to the list version as @{text length}. *}
   212 
   213 abbreviation
   214   length :: "'a list \<Rightarrow> nat" where
   215   "length \<equiv> size"
   216 
   217 primrec rotate1 :: "'a list \<Rightarrow> 'a list" where
   218   "rotate1 [] = []" |
   219   "rotate1 (x # xs) = xs @ [x]"
   220 
   221 definition
   222   rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   223   "rotate n = rotate1 ^^ n"
   224 
   225 definition
   226   list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
   227   "list_all2 P xs ys =
   228     (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"
   229 
   230 definition
   231   sublist :: "'a list => nat set => 'a list" where
   232   "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"
   233 
   234 fun splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   235 "splice [] ys = ys" |
   236 "splice xs [] = xs" |
   237 "splice (x#xs) (y#ys) = x # y # splice xs ys"
   238 
   239 text{*
   240 \begin{figure}[htbp]
   241 \fbox{
   242 \begin{tabular}{l}
   243 @{lemma "[a,b]@[c,d] = [a,b,c,d]" by simp}\\
   244 @{lemma "length [a,b,c] = 3" by simp}\\
   245 @{lemma "set [a,b,c] = {a,b,c}" by simp}\\
   246 @{lemma "map f [a,b,c] = [f a, f b, f c]" by simp}\\
   247 @{lemma "rev [a,b,c] = [c,b,a]" by simp}\\
   248 @{lemma "hd [a,b,c,d] = a" by simp}\\
   249 @{lemma "tl [a,b,c,d] = [b,c,d]" by simp}\\
   250 @{lemma "last [a,b,c,d] = d" by simp}\\
   251 @{lemma "butlast [a,b,c,d] = [a,b,c]" by simp}\\
   252 @{lemma[source] "filter (\<lambda>n::nat. n<2) [0,2,1] = [0,1]" by simp}\\
   253 @{lemma "concat [[a,b],[c,d,e],[],[f]] = [a,b,c,d,e,f]" by simp}\\
   254 @{lemma "fold f [a,b,c] x = f c (f b (f a x))" by simp}\\
   255 @{lemma "foldr f [a,b,c] x = f a (f b (f c x))" by simp}\\
   256 @{lemma "foldl f x [a,b,c] = f (f (f x a) b) c" by simp}\\
   257 @{lemma "zip [a,b,c] [x,y,z] = [(a,x),(b,y),(c,z)]" by simp}\\
   258 @{lemma "zip [a,b] [x,y,z] = [(a,x),(b,y)]" by simp}\\
   259 @{lemma "splice [a,b,c] [x,y,z] = [a,x,b,y,c,z]" by simp}\\
   260 @{lemma "splice [a,b,c,d] [x,y] = [a,x,b,y,c,d]" by simp}\\
   261 @{lemma "take 2 [a,b,c,d] = [a,b]" by simp}\\
   262 @{lemma "take 6 [a,b,c,d] = [a,b,c,d]" by simp}\\
   263 @{lemma "drop 2 [a,b,c,d] = [c,d]" by simp}\\
   264 @{lemma "drop 6 [a,b,c,d] = []" by simp}\\
   265 @{lemma "takeWhile (%n::nat. n<3) [1,2,3,0] = [1,2]" by simp}\\
   266 @{lemma "dropWhile (%n::nat. n<3) [1,2,3,0] = [3,0]" by simp}\\
   267 @{lemma "distinct [2,0,1::nat]" by simp}\\
   268 @{lemma "remdups [2,0,2,1::nat,2] = [0,1,2]" by simp}\\
   269 @{lemma "List.insert 2 [0::nat,1,2] = [0,1,2]" by (simp add: List.insert_def)}\\
   270 @{lemma "List.insert 3 [0::nat,1,2] = [3,0,1,2]" by (simp add: List.insert_def)}\\
   271 @{lemma "List.find (%i::int. i>0) [0,0] = None" by simp}\\
   272 @{lemma "List.find (%i::int. i>0) [0,1,0,2] = Some 1" by simp}\\
   273 @{lemma "remove1 2 [2,0,2,1::nat,2] = [0,2,1,2]" by simp}\\
   274 @{lemma "removeAll 2 [2,0,2,1::nat,2] = [0,1]" by simp}\\
   275 @{lemma "nth [a,b,c,d] 2 = c" by simp}\\
   276 @{lemma "[a,b,c,d][2 := x] = [a,b,x,d]" by simp}\\
   277 @{lemma "sublist [a,b,c,d,e] {0,2,3} = [a,c,d]" by (simp add:sublist_def)}\\
   278 @{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by simp}\\
   279 @{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate_def eval_nat_numeral)}\\
   280 @{lemma "replicate 4 a = [a,a,a,a]" by (simp add:eval_nat_numeral)}\\
   281 @{lemma "[2..<5] = [2,3,4]" by (simp add:eval_nat_numeral)}\\
   282 @{lemma "listsum [1,2,3::nat] = 6" by (simp add: listsum_def)}
   283 \end{tabular}}
   284 \caption{Characteristic examples}
   285 \label{fig:Characteristic}
   286 \end{figure}
   287 Figure~\ref{fig:Characteristic} shows characteristic examples
   288 that should give an intuitive understanding of the above functions.
   289 *}
   290 
   291 text{* The following simple sort functions are intended for proofs,
   292 not for efficient implementations. *}
   293 
   294 context linorder
   295 begin
   296 
   297 inductive sorted :: "'a list \<Rightarrow> bool" where
   298   Nil [iff]: "sorted []"
   299 | Cons: "\<forall>y\<in>set xs. x \<le> y \<Longrightarrow> sorted xs \<Longrightarrow> sorted (x # xs)"
   300 
   301 lemma sorted_single [iff]:
   302   "sorted [x]"
   303   by (rule sorted.Cons) auto
   304 
   305 lemma sorted_many:
   306   "x \<le> y \<Longrightarrow> sorted (y # zs) \<Longrightarrow> sorted (x # y # zs)"
   307   by (rule sorted.Cons) (cases "y # zs" rule: sorted.cases, auto)
   308 
   309 lemma sorted_many_eq [simp, code]:
   310   "sorted (x # y # zs) \<longleftrightarrow> x \<le> y \<and> sorted (y # zs)"
   311   by (auto intro: sorted_many elim: sorted.cases)
   312 
   313 lemma [code]:
   314   "sorted [] \<longleftrightarrow> True"
   315   "sorted [x] \<longleftrightarrow> True"
   316   by simp_all
   317 
   318 primrec insort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
   319   "insort_key f x [] = [x]" |
   320   "insort_key f x (y#ys) = (if f x \<le> f y then (x#y#ys) else y#(insort_key f x ys))"
   321 
   322 definition sort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b list \<Rightarrow> 'b list" where
   323   "sort_key f xs = foldr (insort_key f) xs []"
   324 
   325 definition insort_insert_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
   326   "insort_insert_key f x xs = (if f x \<in> f ` set xs then xs else insort_key f x xs)"
   327 
   328 abbreviation "sort \<equiv> sort_key (\<lambda>x. x)"
   329 abbreviation "insort \<equiv> insort_key (\<lambda>x. x)"
   330 abbreviation "insort_insert \<equiv> insort_insert_key (\<lambda>x. x)"
   331 
   332 end
   333 
   334 
   335 subsubsection {* List comprehension *}
   336 
   337 text{* Input syntax for Haskell-like list comprehension notation.
   338 Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
   339 the list of all pairs of distinct elements from @{text xs} and @{text ys}.
   340 The syntax is as in Haskell, except that @{text"|"} becomes a dot
   341 (like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
   342 \verb![e| x <- xs, ...]!.
   343 
   344 The qualifiers after the dot are
   345 \begin{description}
   346 \item[generators] @{text"p \<leftarrow> xs"},
   347  where @{text p} is a pattern and @{text xs} an expression of list type, or
   348 \item[guards] @{text"b"}, where @{text b} is a boolean expression.
   349 %\item[local bindings] @ {text"let x = e"}.
   350 \end{description}
   351 
   352 Just like in Haskell, list comprehension is just a shorthand. To avoid
   353 misunderstandings, the translation into desugared form is not reversed
   354 upon output. Note that the translation of @{text"[e. x \<leftarrow> xs]"} is
   355 optmized to @{term"map (%x. e) xs"}.
   356 
   357 It is easy to write short list comprehensions which stand for complex
   358 expressions. During proofs, they may become unreadable (and
   359 mangled). In such cases it can be advisable to introduce separate
   360 definitions for the list comprehensions in question.  *}
   361 
   362 nonterminal lc_qual and lc_quals
   363 
   364 syntax
   365   "_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
   366   "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ <- _")
   367   "_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
   368   (*"_lc_let" :: "letbinds => lc_qual"  ("let _")*)
   369   "_lc_end" :: "lc_quals" ("]")
   370   "_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals"  (", __")
   371   "_lc_abs" :: "'a => 'b list => 'b list"
   372 
   373 (* These are easier than ML code but cannot express the optimized
   374    translation of [e. p<-xs]
   375 translations
   376   "[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
   377   "_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
   378    => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
   379   "[e. P]" => "if P then [e] else []"
   380   "_listcompr e (_lc_test P) (_lc_quals Q Qs)"
   381    => "if P then (_listcompr e Q Qs) else []"
   382   "_listcompr e (_lc_let b) (_lc_quals Q Qs)"
   383    => "_Let b (_listcompr e Q Qs)"
   384 *)
   385 
   386 syntax (xsymbols)
   387   "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ \<leftarrow> _")
   388 syntax (HTML output)
   389   "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ \<leftarrow> _")
   390 
   391 parse_translation (advanced) {*
   392   let
   393     val NilC = Syntax.const @{const_syntax Nil};
   394     val ConsC = Syntax.const @{const_syntax Cons};
   395     val mapC = Syntax.const @{const_syntax map};
   396     val concatC = Syntax.const @{const_syntax concat};
   397     val IfC = Syntax.const @{const_syntax If};
   398 
   399     fun single x = ConsC $ x $ NilC;
   400 
   401     fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
   402       let
   403         (* FIXME proper name context!? *)
   404         val x =
   405           Free (singleton (Name.variant_list (fold Term.add_free_names [p, e] [])) "x", dummyT);
   406         val e = if opti then single e else e;
   407         val case1 = Syntax.const @{syntax_const "_case1"} $ p $ e;
   408         val case2 =
   409           Syntax.const @{syntax_const "_case1"} $
   410             Syntax.const @{const_syntax dummy_pattern} $ NilC;
   411         val cs = Syntax.const @{syntax_const "_case2"} $ case1 $ case2;
   412       in Syntax_Trans.abs_tr [x, Datatype_Case.case_tr false ctxt [x, cs]] end;
   413 
   414     fun abs_tr ctxt p e opti =
   415       (case Term_Position.strip_positions p of
   416         Free (s, T) =>
   417           let
   418             val thy = Proof_Context.theory_of ctxt;
   419             val s' = Proof_Context.intern_const ctxt s;
   420           in
   421             if Sign.declared_const thy s'
   422             then (pat_tr ctxt p e opti, false)
   423             else (Syntax_Trans.abs_tr [p, e], true)
   424           end
   425       | _ => (pat_tr ctxt p e opti, false));
   426 
   427     fun lc_tr ctxt [e, Const (@{syntax_const "_lc_test"}, _) $ b, qs] =
   428           let
   429             val res =
   430               (case qs of
   431                 Const (@{syntax_const "_lc_end"}, _) => single e
   432               | Const (@{syntax_const "_lc_quals"}, _) $ q $ qs => lc_tr ctxt [e, q, qs]);
   433           in IfC $ b $ res $ NilC end
   434       | lc_tr ctxt
   435             [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
   436               Const(@{syntax_const "_lc_end"}, _)] =
   437           (case abs_tr ctxt p e true of
   438             (f, true) => mapC $ f $ es
   439           | (f, false) => concatC $ (mapC $ f $ es))
   440       | lc_tr ctxt
   441             [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
   442               Const (@{syntax_const "_lc_quals"}, _) $ q $ qs] =
   443           let val e' = lc_tr ctxt [e, q, qs];
   444           in concatC $ (mapC $ (fst (abs_tr ctxt p e' false)) $ es) end;
   445 
   446   in [(@{syntax_const "_listcompr"}, lc_tr)] end
   447 *}
   448 
   449 ML {*
   450   let
   451     val read = Syntax.read_term @{context};
   452     fun check s1 s2 = read s1 aconv read s2 orelse error ("Check failed: " ^ quote s1);
   453   in
   454     check "[(x,y,z). b]" "if b then [(x, y, z)] else []";
   455     check "[(x,y,z). x\<leftarrow>xs]" "map (\<lambda>x. (x, y, z)) xs";
   456     check "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]" "concat (map (\<lambda>x. map (\<lambda>y. e x y) ys) xs)";
   457     check "[(x,y,z). x<a, x>b]" "if x < a then if b < x then [(x, y, z)] else [] else []";
   458     check "[(x,y,z). x\<leftarrow>xs, x>b]" "concat (map (\<lambda>x. if b < x then [(x, y, z)] else []) xs)";
   459     check "[(x,y,z). x<a, x\<leftarrow>xs]" "if x < a then map (\<lambda>x. (x, y, z)) xs else []";
   460     check "[(x,y). Cons True x \<leftarrow> xs]"
   461       "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | True # x \<Rightarrow> [(x, y)] | False # x \<Rightarrow> []) xs)";
   462     check "[(x,y,z). Cons x [] \<leftarrow> xs]"
   463       "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | [x] \<Rightarrow> [(x, y, z)] | x # aa # lista \<Rightarrow> []) xs)";
   464     check "[(x,y,z). x<a, x>b, x=d]"
   465       "if x < a then if b < x then if x = d then [(x, y, z)] else [] else [] else []";
   466     check "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
   467       "if x < a then if b < x then map (\<lambda>y. (x, y, z)) ys else [] else []";
   468     check "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
   469       "if x < a then concat (map (\<lambda>x. if b < y then [(x, y, z)] else []) xs) else []";
   470     check "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
   471       "if x < a then concat (map (\<lambda>x. map (\<lambda>y. (x, y, z)) ys) xs) else []";
   472     check "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
   473       "concat (map (\<lambda>x. if b < x then if y < a then [(x, y, z)] else [] else []) xs)";
   474     check "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
   475       "concat (map (\<lambda>x. if b < x then map (\<lambda>y. (x, y, z)) ys else []) xs)";
   476     check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
   477       "concat (map (\<lambda>x. concat (map (\<lambda>y. if x < y then [(x, y, z)] else []) ys)) xs)";
   478     check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
   479       "concat (map (\<lambda>x. concat (map (\<lambda>y. map (\<lambda>z. (x, y, z)) zs) ys)) xs)"
   480   end;
   481 *}
   482 
   483 (*
   484 term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
   485 *)
   486 
   487 
   488 use "Tools/list_to_set_comprehension.ML"
   489 
   490 simproc_setup list_to_set_comprehension ("set xs") = {* K List_to_Set_Comprehension.simproc *}
   491 
   492 code_datatype set coset
   493 
   494 hide_const (open) coset
   495 
   496 subsubsection {* @{const Nil} and @{const Cons} *}
   497 
   498 lemma not_Cons_self [simp]:
   499   "xs \<noteq> x # xs"
   500 by (induct xs) auto
   501 
   502 lemma not_Cons_self2 [simp]:
   503   "x # xs \<noteq> xs"
   504 by (rule not_Cons_self [symmetric])
   505 
   506 lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
   507 by (induct xs) auto
   508 
   509 lemma length_induct:
   510   "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
   511 by (rule measure_induct [of length]) iprover
   512 
   513 lemma list_nonempty_induct [consumes 1, case_names single cons]:
   514   assumes "xs \<noteq> []"
   515   assumes single: "\<And>x. P [x]"
   516   assumes cons: "\<And>x xs. xs \<noteq> [] \<Longrightarrow> P xs \<Longrightarrow> P (x # xs)"
   517   shows "P xs"
   518 using `xs \<noteq> []` proof (induct xs)
   519   case Nil then show ?case by simp
   520 next
   521   case (Cons x xs) show ?case proof (cases xs)
   522     case Nil with single show ?thesis by simp
   523   next
   524     case Cons then have "xs \<noteq> []" by simp
   525     moreover with Cons.hyps have "P xs" .
   526     ultimately show ?thesis by (rule cons)
   527   qed
   528 qed
   529 
   530 lemma inj_split_Cons: "inj_on (\<lambda>(xs, n). n#xs) X"
   531   by (auto intro!: inj_onI)
   532 
   533 subsubsection {* @{const length} *}
   534 
   535 text {*
   536   Needs to come before @{text "@"} because of theorem @{text
   537   append_eq_append_conv}.
   538 *}
   539 
   540 lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
   541 by (induct xs) auto
   542 
   543 lemma length_map [simp]: "length (map f xs) = length xs"
   544 by (induct xs) auto
   545 
   546 lemma length_rev [simp]: "length (rev xs) = length xs"
   547 by (induct xs) auto
   548 
   549 lemma length_tl [simp]: "length (tl xs) = length xs - 1"
   550 by (cases xs) auto
   551 
   552 lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
   553 by (induct xs) auto
   554 
   555 lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
   556 by (induct xs) auto
   557 
   558 lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
   559 by auto
   560 
   561 lemma length_Suc_conv:
   562 "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
   563 by (induct xs) auto
   564 
   565 lemma Suc_length_conv:
   566 "(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
   567 apply (induct xs, simp, simp)
   568 apply blast
   569 done
   570 
   571 lemma impossible_Cons: "length xs <= length ys ==> xs = x # ys = False"
   572   by (induct xs) auto
   573 
   574 lemma list_induct2 [consumes 1, case_names Nil Cons]:
   575   "length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
   576    (\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
   577    \<Longrightarrow> P xs ys"
   578 proof (induct xs arbitrary: ys)
   579   case Nil then show ?case by simp
   580 next
   581   case (Cons x xs ys) then show ?case by (cases ys) simp_all
   582 qed
   583 
   584 lemma list_induct3 [consumes 2, case_names Nil Cons]:
   585   "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
   586    (\<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))
   587    \<Longrightarrow> P xs ys zs"
   588 proof (induct xs arbitrary: ys zs)
   589   case Nil then show ?case by simp
   590 next
   591   case (Cons x xs ys zs) then show ?case by (cases ys, simp_all)
   592     (cases zs, simp_all)
   593 qed
   594 
   595 lemma list_induct4 [consumes 3, case_names Nil Cons]:
   596   "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow>
   597    P [] [] [] [] \<Longrightarrow> (\<And>x xs y ys z zs w ws. length xs = length ys \<Longrightarrow>
   598    length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow> P xs ys zs ws \<Longrightarrow>
   599    P (x#xs) (y#ys) (z#zs) (w#ws)) \<Longrightarrow> P xs ys zs ws"
   600 proof (induct xs arbitrary: ys zs ws)
   601   case Nil then show ?case by simp
   602 next
   603   case (Cons x xs ys zs ws) then show ?case by ((cases ys, simp_all), (cases zs,simp_all)) (cases ws, simp_all)
   604 qed
   605 
   606 lemma list_induct2': 
   607   "\<lbrakk> P [] [];
   608   \<And>x xs. P (x#xs) [];
   609   \<And>y ys. P [] (y#ys);
   610    \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
   611  \<Longrightarrow> P xs ys"
   612 by (induct xs arbitrary: ys) (case_tac x, auto)+
   613 
   614 lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
   615 by (rule Eq_FalseI) auto
   616 
   617 simproc_setup list_neq ("(xs::'a list) = ys") = {*
   618 (*
   619 Reduces xs=ys to False if xs and ys cannot be of the same length.
   620 This is the case if the atomic sublists of one are a submultiset
   621 of those of the other list and there are fewer Cons's in one than the other.
   622 *)
   623 
   624 let
   625 
   626 fun len (Const(@{const_name Nil},_)) acc = acc
   627   | len (Const(@{const_name Cons},_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
   628   | len (Const(@{const_name append},_) $ xs $ ys) acc = len xs (len ys acc)
   629   | len (Const(@{const_name rev},_) $ xs) acc = len xs acc
   630   | len (Const(@{const_name map},_) $ _ $ xs) acc = len xs acc
   631   | len t (ts,n) = (t::ts,n);
   632 
   633 fun list_neq _ ss ct =
   634   let
   635     val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
   636     val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
   637     fun prove_neq() =
   638       let
   639         val Type(_,listT::_) = eqT;
   640         val size = HOLogic.size_const listT;
   641         val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
   642         val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
   643         val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
   644           (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
   645       in SOME (thm RS @{thm neq_if_length_neq}) end
   646   in
   647     if m < n andalso submultiset (op aconv) (ls,rs) orelse
   648        n < m andalso submultiset (op aconv) (rs,ls)
   649     then prove_neq() else NONE
   650   end;
   651 in list_neq end;
   652 *}
   653 
   654 
   655 subsubsection {* @{text "@"} -- append *}
   656 
   657 lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
   658 by (induct xs) auto
   659 
   660 lemma append_Nil2 [simp]: "xs @ [] = xs"
   661 by (induct xs) auto
   662 
   663 lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
   664 by (induct xs) auto
   665 
   666 lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
   667 by (induct xs) auto
   668 
   669 lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
   670 by (induct xs) auto
   671 
   672 lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
   673 by (induct xs) auto
   674 
   675 lemma append_eq_append_conv [simp, no_atp]:
   676  "length xs = length ys \<or> length us = length vs
   677  ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
   678 apply (induct xs arbitrary: ys)
   679  apply (case_tac ys, simp, force)
   680 apply (case_tac ys, force, simp)
   681 done
   682 
   683 lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
   684   (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
   685 apply (induct xs arbitrary: ys zs ts)
   686  apply fastforce
   687 apply(case_tac zs)
   688  apply simp
   689 apply fastforce
   690 done
   691 
   692 lemma same_append_eq [iff, induct_simp]: "(xs @ ys = xs @ zs) = (ys = zs)"
   693 by simp
   694 
   695 lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
   696 by simp
   697 
   698 lemma append_same_eq [iff, induct_simp]: "(ys @ xs = zs @ xs) = (ys = zs)"
   699 by simp
   700 
   701 lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
   702 using append_same_eq [of _ _ "[]"] by auto
   703 
   704 lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
   705 using append_same_eq [of "[]"] by auto
   706 
   707 lemma hd_Cons_tl [simp,no_atp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
   708 by (induct xs) auto
   709 
   710 lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
   711 by (induct xs) auto
   712 
   713 lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
   714 by (simp add: hd_append split: list.split)
   715 
   716 lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
   717 by (simp split: list.split)
   718 
   719 lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
   720 by (simp add: tl_append split: list.split)
   721 
   722 
   723 lemma Cons_eq_append_conv: "x#xs = ys@zs =
   724  (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
   725 by(cases ys) auto
   726 
   727 lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
   728  (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
   729 by(cases ys) auto
   730 
   731 
   732 text {* Trivial rules for solving @{text "@"}-equations automatically. *}
   733 
   734 lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
   735 by simp
   736 
   737 lemma Cons_eq_appendI:
   738 "[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
   739 by (drule sym) simp
   740 
   741 lemma append_eq_appendI:
   742 "[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
   743 by (drule sym) simp
   744 
   745 
   746 text {*
   747 Simplification procedure for all list equalities.
   748 Currently only tries to rearrange @{text "@"} to see if
   749 - both lists end in a singleton list,
   750 - or both lists end in the same list.
   751 *}
   752 
   753 simproc_setup list_eq ("(xs::'a list) = ys")  = {*
   754   let
   755     fun last (cons as Const (@{const_name Cons}, _) $ _ $ xs) =
   756           (case xs of Const (@{const_name Nil}, _) => cons | _ => last xs)
   757       | last (Const(@{const_name append},_) $ _ $ ys) = last ys
   758       | last t = t;
   759     
   760     fun list1 (Const(@{const_name Cons},_) $ _ $ Const(@{const_name Nil},_)) = true
   761       | list1 _ = false;
   762     
   763     fun butlast ((cons as Const(@{const_name Cons},_) $ x) $ xs) =
   764           (case xs of Const (@{const_name Nil}, _) => xs | _ => cons $ butlast xs)
   765       | butlast ((app as Const (@{const_name append}, _) $ xs) $ ys) = app $ butlast ys
   766       | butlast xs = Const(@{const_name Nil}, fastype_of xs);
   767     
   768     val rearr_ss =
   769       HOL_basic_ss addsimps [@{thm append_assoc}, @{thm append_Nil}, @{thm append_Cons}];
   770     
   771     fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
   772       let
   773         val lastl = last lhs and lastr = last rhs;
   774         fun rearr conv =
   775           let
   776             val lhs1 = butlast lhs and rhs1 = butlast rhs;
   777             val Type(_,listT::_) = eqT
   778             val appT = [listT,listT] ---> listT
   779             val app = Const(@{const_name append},appT)
   780             val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
   781             val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
   782             val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
   783               (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
   784           in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
   785       in
   786         if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
   787         else if lastl aconv lastr then rearr @{thm append_same_eq}
   788         else NONE
   789       end;
   790   in fn _ => fn ss => fn ct => list_eq ss (term_of ct) end;
   791 *}
   792 
   793 
   794 subsubsection {* @{text map} *}
   795 
   796 lemma hd_map:
   797   "xs \<noteq> [] \<Longrightarrow> hd (map f xs) = f (hd xs)"
   798   by (cases xs) simp_all
   799 
   800 lemma map_tl:
   801   "map f (tl xs) = tl (map f xs)"
   802   by (cases xs) simp_all
   803 
   804 lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
   805 by (induct xs) simp_all
   806 
   807 lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
   808 by (rule ext, induct_tac xs) auto
   809 
   810 lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
   811 by (induct xs) auto
   812 
   813 lemma map_map [simp]: "map f (map g xs) = map (f \<circ> g) xs"
   814 by (induct xs) auto
   815 
   816 lemma map_comp_map[simp]: "((map f) o (map g)) = map(f o g)"
   817 apply(rule ext)
   818 apply(simp)
   819 done
   820 
   821 lemma rev_map: "rev (map f xs) = map f (rev xs)"
   822 by (induct xs) auto
   823 
   824 lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
   825 by (induct xs) auto
   826 
   827 lemma map_cong [fundef_cong]:
   828   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> map f xs = map g ys"
   829   by simp
   830 
   831 lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
   832 by (cases xs) auto
   833 
   834 lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
   835 by (cases xs) auto
   836 
   837 lemma map_eq_Cons_conv:
   838  "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
   839 by (cases xs) auto
   840 
   841 lemma Cons_eq_map_conv:
   842  "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
   843 by (cases ys) auto
   844 
   845 lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
   846 lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
   847 declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
   848 
   849 lemma ex_map_conv:
   850   "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
   851 by(induct ys, auto simp add: Cons_eq_map_conv)
   852 
   853 lemma map_eq_imp_length_eq:
   854   assumes "map f xs = map g ys"
   855   shows "length xs = length ys"
   856 using assms proof (induct ys arbitrary: xs)
   857   case Nil then show ?case by simp
   858 next
   859   case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto
   860   from Cons xs have "map f zs = map g ys" by simp
   861   moreover with Cons have "length zs = length ys" by blast
   862   with xs show ?case by simp
   863 qed
   864   
   865 lemma map_inj_on:
   866  "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
   867   ==> xs = ys"
   868 apply(frule map_eq_imp_length_eq)
   869 apply(rotate_tac -1)
   870 apply(induct rule:list_induct2)
   871  apply simp
   872 apply(simp)
   873 apply (blast intro:sym)
   874 done
   875 
   876 lemma inj_on_map_eq_map:
   877  "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
   878 by(blast dest:map_inj_on)
   879 
   880 lemma map_injective:
   881  "map f xs = map f ys ==> inj f ==> xs = ys"
   882 by (induct ys arbitrary: xs) (auto dest!:injD)
   883 
   884 lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
   885 by(blast dest:map_injective)
   886 
   887 lemma inj_mapI: "inj f ==> inj (map f)"
   888 by (iprover dest: map_injective injD intro: inj_onI)
   889 
   890 lemma inj_mapD: "inj (map f) ==> inj f"
   891 apply (unfold inj_on_def, clarify)
   892 apply (erule_tac x = "[x]" in ballE)
   893  apply (erule_tac x = "[y]" in ballE, simp, blast)
   894 apply blast
   895 done
   896 
   897 lemma inj_map[iff]: "inj (map f) = inj f"
   898 by (blast dest: inj_mapD intro: inj_mapI)
   899 
   900 lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
   901 apply(rule inj_onI)
   902 apply(erule map_inj_on)
   903 apply(blast intro:inj_onI dest:inj_onD)
   904 done
   905 
   906 lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
   907 by (induct xs, auto)
   908 
   909 lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
   910 by (induct xs) auto
   911 
   912 lemma map_fst_zip[simp]:
   913   "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
   914 by (induct rule:list_induct2, simp_all)
   915 
   916 lemma map_snd_zip[simp]:
   917   "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
   918 by (induct rule:list_induct2, simp_all)
   919 
   920 enriched_type map: map
   921 by (simp_all add: id_def)
   922 
   923 declare map.id[simp]
   924 
   925 subsubsection {* @{text rev} *}
   926 
   927 lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
   928 by (induct xs) auto
   929 
   930 lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
   931 by (induct xs) auto
   932 
   933 lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
   934 by auto
   935 
   936 lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
   937 by (induct xs) auto
   938 
   939 lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
   940 by (induct xs) auto
   941 
   942 lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
   943 by (cases xs) auto
   944 
   945 lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
   946 by (cases xs) auto
   947 
   948 lemma rev_is_rev_conv [iff, no_atp]: "(rev xs = rev ys) = (xs = ys)"
   949 apply (induct xs arbitrary: ys, force)
   950 apply (case_tac ys, simp, force)
   951 done
   952 
   953 lemma inj_on_rev[iff]: "inj_on rev A"
   954 by(simp add:inj_on_def)
   955 
   956 lemma rev_induct [case_names Nil snoc]:
   957   "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
   958 apply(simplesubst rev_rev_ident[symmetric])
   959 apply(rule_tac list = "rev xs" in list.induct, simp_all)
   960 done
   961 
   962 lemma rev_exhaust [case_names Nil snoc]:
   963   "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
   964 by (induct xs rule: rev_induct) auto
   965 
   966 lemmas rev_cases = rev_exhaust
   967 
   968 lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
   969 by(rule rev_cases[of xs]) auto
   970 
   971 
   972 subsubsection {* @{text set} *}
   973 
   974 declare set.simps [code_post]  --"pretty output"
   975 
   976 lemma finite_set [iff]: "finite (set xs)"
   977 by (induct xs) auto
   978 
   979 lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
   980 by (induct xs) auto
   981 
   982 lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
   983 by(cases xs) auto
   984 
   985 lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
   986 by auto
   987 
   988 lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
   989 by auto
   990 
   991 lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
   992 by (induct xs) auto
   993 
   994 lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
   995 by(induct xs) auto
   996 
   997 lemma set_rev [simp]: "set (rev xs) = set xs"
   998 by (induct xs) auto
   999 
  1000 lemma set_map [simp]: "set (map f xs) = f`(set xs)"
  1001 by (induct xs) auto
  1002 
  1003 lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
  1004 by (induct xs) auto
  1005 
  1006 lemma set_upt [simp]: "set[i..<j] = {i..<j}"
  1007 by (induct j) auto
  1008 
  1009 
  1010 lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs"
  1011 proof (induct xs)
  1012   case Nil thus ?case by simp
  1013 next
  1014   case Cons thus ?case by (auto intro: Cons_eq_appendI)
  1015 qed
  1016 
  1017 lemma in_set_conv_decomp: "x \<in> set xs \<longleftrightarrow> (\<exists>ys zs. xs = ys @ x # zs)"
  1018   by (auto elim: split_list)
  1019 
  1020 lemma split_list_first: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
  1021 proof (induct xs)
  1022   case Nil thus ?case by simp
  1023 next
  1024   case (Cons a xs)
  1025   show ?case
  1026   proof cases
  1027     assume "x = a" thus ?case using Cons by fastforce
  1028   next
  1029     assume "x \<noteq> a" thus ?case using Cons by(fastforce intro!: Cons_eq_appendI)
  1030   qed
  1031 qed
  1032 
  1033 lemma in_set_conv_decomp_first:
  1034   "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
  1035   by (auto dest!: split_list_first)
  1036 
  1037 lemma split_list_last: "x \<in> set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
  1038 proof (induct xs rule: rev_induct)
  1039   case Nil thus ?case by simp
  1040 next
  1041   case (snoc a xs)
  1042   show ?case
  1043   proof cases
  1044     assume "x = a" thus ?case using snoc by (metis List.set.simps(1) emptyE)
  1045   next
  1046     assume "x \<noteq> a" thus ?case using snoc by fastforce
  1047   qed
  1048 qed
  1049 
  1050 lemma in_set_conv_decomp_last:
  1051   "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
  1052   by (auto dest!: split_list_last)
  1053 
  1054 lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs & P x"
  1055 proof (induct xs)
  1056   case Nil thus ?case by simp
  1057 next
  1058   case Cons thus ?case
  1059     by(simp add:Bex_def)(metis append_Cons append.simps(1))
  1060 qed
  1061 
  1062 lemma split_list_propE:
  1063   assumes "\<exists>x \<in> set xs. P x"
  1064   obtains ys x zs where "xs = ys @ x # zs" and "P x"
  1065 using split_list_prop [OF assms] by blast
  1066 
  1067 lemma split_list_first_prop:
  1068   "\<exists>x \<in> set xs. P x \<Longrightarrow>
  1069    \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)"
  1070 proof (induct xs)
  1071   case Nil thus ?case by simp
  1072 next
  1073   case (Cons x xs)
  1074   show ?case
  1075   proof cases
  1076     assume "P x"
  1077     thus ?thesis by simp (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
  1078   next
  1079     assume "\<not> P x"
  1080     hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
  1081     thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
  1082   qed
  1083 qed
  1084 
  1085 lemma split_list_first_propE:
  1086   assumes "\<exists>x \<in> set xs. P x"
  1087   obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y"
  1088 using split_list_first_prop [OF assms] by blast
  1089 
  1090 lemma split_list_first_prop_iff:
  1091   "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
  1092    (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))"
  1093 by (rule, erule split_list_first_prop) auto
  1094 
  1095 lemma split_list_last_prop:
  1096   "\<exists>x \<in> set xs. P x \<Longrightarrow>
  1097    \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)"
  1098 proof(induct xs rule:rev_induct)
  1099   case Nil thus ?case by simp
  1100 next
  1101   case (snoc x xs)
  1102   show ?case
  1103   proof cases
  1104     assume "P x" thus ?thesis by (metis emptyE set_empty)
  1105   next
  1106     assume "\<not> P x"
  1107     hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
  1108     thus ?thesis using `\<not> P x` snoc(1) by fastforce
  1109   qed
  1110 qed
  1111 
  1112 lemma split_list_last_propE:
  1113   assumes "\<exists>x \<in> set xs. P x"
  1114   obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z"
  1115 using split_list_last_prop [OF assms] by blast
  1116 
  1117 lemma split_list_last_prop_iff:
  1118   "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
  1119    (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
  1120 by (metis split_list_last_prop [where P=P] in_set_conv_decomp)
  1121 
  1122 lemma finite_list: "finite A ==> EX xs. set xs = A"
  1123   by (erule finite_induct)
  1124     (auto simp add: set.simps(2) [symmetric] simp del: set.simps(2))
  1125 
  1126 lemma card_length: "card (set xs) \<le> length xs"
  1127 by (induct xs) (auto simp add: card_insert_if)
  1128 
  1129 lemma set_minus_filter_out:
  1130   "set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)"
  1131   by (induct xs) auto
  1132 
  1133 
  1134 subsubsection {* @{text filter} *}
  1135 
  1136 lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
  1137 by (induct xs) auto
  1138 
  1139 lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
  1140 by (induct xs) simp_all
  1141 
  1142 lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
  1143 by (induct xs) auto
  1144 
  1145 lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
  1146 by (induct xs) (auto simp add: le_SucI)
  1147 
  1148 lemma sum_length_filter_compl:
  1149   "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
  1150 by(induct xs) simp_all
  1151 
  1152 lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
  1153 by (induct xs) auto
  1154 
  1155 lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
  1156 by (induct xs) auto
  1157 
  1158 lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
  1159 by (induct xs) simp_all
  1160 
  1161 lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
  1162 apply (induct xs)
  1163  apply auto
  1164 apply(cut_tac P=P and xs=xs in length_filter_le)
  1165 apply simp
  1166 done
  1167 
  1168 lemma filter_map:
  1169   "filter P (map f xs) = map f (filter (P o f) xs)"
  1170 by (induct xs) simp_all
  1171 
  1172 lemma length_filter_map[simp]:
  1173   "length (filter P (map f xs)) = length(filter (P o f) xs)"
  1174 by (simp add:filter_map)
  1175 
  1176 lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
  1177 by auto
  1178 
  1179 lemma length_filter_less:
  1180   "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
  1181 proof (induct xs)
  1182   case Nil thus ?case by simp
  1183 next
  1184   case (Cons x xs) thus ?case
  1185     apply (auto split:split_if_asm)
  1186     using length_filter_le[of P xs] apply arith
  1187   done
  1188 qed
  1189 
  1190 lemma length_filter_conv_card:
  1191  "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
  1192 proof (induct xs)
  1193   case Nil thus ?case by simp
  1194 next
  1195   case (Cons x xs)
  1196   let ?S = "{i. i < length xs & p(xs!i)}"
  1197   have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
  1198   show ?case (is "?l = card ?S'")
  1199   proof (cases)
  1200     assume "p x"
  1201     hence eq: "?S' = insert 0 (Suc ` ?S)"
  1202       by(auto simp: image_def split:nat.split dest:gr0_implies_Suc)
  1203     have "length (filter p (x # xs)) = Suc(card ?S)"
  1204       using Cons `p x` by simp
  1205     also have "\<dots> = Suc(card(Suc ` ?S))" using fin
  1206       by (simp add: card_image)
  1207     also have "\<dots> = card ?S'" using eq fin
  1208       by (simp add:card_insert_if) (simp add:image_def)
  1209     finally show ?thesis .
  1210   next
  1211     assume "\<not> p x"
  1212     hence eq: "?S' = Suc ` ?S"
  1213       by(auto simp add: image_def split:nat.split elim:lessE)
  1214     have "length (filter p (x # xs)) = card ?S"
  1215       using Cons `\<not> p x` by simp
  1216     also have "\<dots> = card(Suc ` ?S)" using fin
  1217       by (simp add: card_image)
  1218     also have "\<dots> = card ?S'" using eq fin
  1219       by (simp add:card_insert_if)
  1220     finally show ?thesis .
  1221   qed
  1222 qed
  1223 
  1224 lemma Cons_eq_filterD:
  1225  "x#xs = filter P ys \<Longrightarrow>
  1226   \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
  1227   (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
  1228 proof(induct ys)
  1229   case Nil thus ?case by simp
  1230 next
  1231   case (Cons y ys)
  1232   show ?case (is "\<exists>x. ?Q x")
  1233   proof cases
  1234     assume Py: "P y"
  1235     show ?thesis
  1236     proof cases
  1237       assume "x = y"
  1238       with Py Cons.prems have "?Q []" by simp
  1239       then show ?thesis ..
  1240     next
  1241       assume "x \<noteq> y"
  1242       with Py Cons.prems show ?thesis by simp
  1243     qed
  1244   next
  1245     assume "\<not> P y"
  1246     with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastforce
  1247     then have "?Q (y#us)" by simp
  1248     then show ?thesis ..
  1249   qed
  1250 qed
  1251 
  1252 lemma filter_eq_ConsD:
  1253  "filter P ys = x#xs \<Longrightarrow>
  1254   \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
  1255 by(rule Cons_eq_filterD) simp
  1256 
  1257 lemma filter_eq_Cons_iff:
  1258  "(filter P ys = x#xs) =
  1259   (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
  1260 by(auto dest:filter_eq_ConsD)
  1261 
  1262 lemma Cons_eq_filter_iff:
  1263  "(x#xs = filter P ys) =
  1264   (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
  1265 by(auto dest:Cons_eq_filterD)
  1266 
  1267 lemma filter_cong[fundef_cong]:
  1268  "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
  1269 apply simp
  1270 apply(erule thin_rl)
  1271 by (induct ys) simp_all
  1272 
  1273 
  1274 subsubsection {* List partitioning *}
  1275 
  1276 primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where
  1277   "partition P [] = ([], [])"
  1278   | "partition P (x # xs) = 
  1279       (let (yes, no) = partition P xs
  1280       in if P x then (x # yes, no) else (yes, x # no))"
  1281 
  1282 lemma partition_filter1:
  1283     "fst (partition P xs) = filter P xs"
  1284 by (induct xs) (auto simp add: Let_def split_def)
  1285 
  1286 lemma partition_filter2:
  1287     "snd (partition P xs) = filter (Not o P) xs"
  1288 by (induct xs) (auto simp add: Let_def split_def)
  1289 
  1290 lemma partition_P:
  1291   assumes "partition P xs = (yes, no)"
  1292   shows "(\<forall>p \<in> set yes.  P p) \<and> (\<forall>p  \<in> set no. \<not> P p)"
  1293 proof -
  1294   from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
  1295     by simp_all
  1296   then show ?thesis by (simp_all add: partition_filter1 partition_filter2)
  1297 qed
  1298 
  1299 lemma partition_set:
  1300   assumes "partition P xs = (yes, no)"
  1301   shows "set yes \<union> set no = set xs"
  1302 proof -
  1303   from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
  1304     by simp_all
  1305   then show ?thesis by (auto simp add: partition_filter1 partition_filter2) 
  1306 qed
  1307 
  1308 lemma partition_filter_conv[simp]:
  1309   "partition f xs = (filter f xs,filter (Not o f) xs)"
  1310 unfolding partition_filter2[symmetric]
  1311 unfolding partition_filter1[symmetric] by simp
  1312 
  1313 declare partition.simps[simp del]
  1314 
  1315 
  1316 subsubsection {* @{text concat} *}
  1317 
  1318 lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
  1319 by (induct xs) auto
  1320 
  1321 lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
  1322 by (induct xss) auto
  1323 
  1324 lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
  1325 by (induct xss) auto
  1326 
  1327 lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
  1328 by (induct xs) auto
  1329 
  1330 lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs"
  1331 by (induct xs) auto
  1332 
  1333 lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
  1334 by (induct xs) auto
  1335 
  1336 lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
  1337 by (induct xs) auto
  1338 
  1339 lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
  1340 by (induct xs) auto
  1341 
  1342 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)"
  1343 proof (induct xs arbitrary: ys)
  1344   case (Cons x xs ys)
  1345   thus ?case by (cases ys) auto
  1346 qed (auto)
  1347 
  1348 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"
  1349 by (simp add: concat_eq_concat_iff)
  1350 
  1351 
  1352 subsubsection {* @{text nth} *}
  1353 
  1354 lemma nth_Cons_0 [simp, code]: "(x # xs)!0 = x"
  1355 by auto
  1356 
  1357 lemma nth_Cons_Suc [simp, code]: "(x # xs)!(Suc n) = xs!n"
  1358 by auto
  1359 
  1360 declare nth.simps [simp del]
  1361 
  1362 lemma nth_Cons_pos[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
  1363 by(auto simp: Nat.gr0_conv_Suc)
  1364 
  1365 lemma nth_append:
  1366   "(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
  1367 apply (induct xs arbitrary: n, simp)
  1368 apply (case_tac n, auto)
  1369 done
  1370 
  1371 lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
  1372 by (induct xs) auto
  1373 
  1374 lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
  1375 by (induct xs) auto
  1376 
  1377 lemma nth_map [simp]: "n < length xs ==> (map f xs)!n = f(xs!n)"
  1378 apply (induct xs arbitrary: n, simp)
  1379 apply (case_tac n, auto)
  1380 done
  1381 
  1382 lemma nth_tl:
  1383   assumes "n < length (tl x)" shows "tl x ! n = x ! Suc n"
  1384 using assms by (induct x) auto
  1385 
  1386 lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
  1387 by(cases xs) simp_all
  1388 
  1389 
  1390 lemma list_eq_iff_nth_eq:
  1391  "(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
  1392 apply(induct xs arbitrary: ys)
  1393  apply force
  1394 apply(case_tac ys)
  1395  apply simp
  1396 apply(simp add:nth_Cons split:nat.split)apply blast
  1397 done
  1398 
  1399 lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
  1400 apply (induct xs, simp, simp)
  1401 apply safe
  1402 apply (metis nat_case_0 nth.simps zero_less_Suc)
  1403 apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
  1404 apply (case_tac i, simp)
  1405 apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff)
  1406 done
  1407 
  1408 lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
  1409 by(auto simp:set_conv_nth)
  1410 
  1411 lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
  1412 by (auto simp add: set_conv_nth)
  1413 
  1414 lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
  1415 by (auto simp add: set_conv_nth)
  1416 
  1417 lemma all_nth_imp_all_set:
  1418 "[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
  1419 by (auto simp add: set_conv_nth)
  1420 
  1421 lemma all_set_conv_all_nth:
  1422 "(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
  1423 by (auto simp add: set_conv_nth)
  1424 
  1425 lemma rev_nth:
  1426   "n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)"
  1427 proof (induct xs arbitrary: n)
  1428   case Nil thus ?case by simp
  1429 next
  1430   case (Cons x xs)
  1431   hence n: "n < Suc (length xs)" by simp
  1432   moreover
  1433   { assume "n < length xs"
  1434     with n obtain n' where "length xs - n = Suc n'"
  1435       by (cases "length xs - n", auto)
  1436     moreover
  1437     then have "length xs - Suc n = n'" by simp
  1438     ultimately
  1439     have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp
  1440   }
  1441   ultimately
  1442   show ?case by (clarsimp simp add: Cons nth_append)
  1443 qed
  1444 
  1445 lemma Skolem_list_nth:
  1446   "(ALL i<k. EX x. P i x) = (EX xs. size xs = k & (ALL i<k. P i (xs!i)))"
  1447   (is "_ = (EX xs. ?P k xs)")
  1448 proof(induct k)
  1449   case 0 show ?case by simp
  1450 next
  1451   case (Suc k)
  1452   show ?case (is "?L = ?R" is "_ = (EX xs. ?P' xs)")
  1453   proof
  1454     assume "?R" thus "?L" using Suc by auto
  1455   next
  1456     assume "?L"
  1457     with Suc obtain x xs where "?P k xs & P k x" by (metis less_Suc_eq)
  1458     hence "?P'(xs@[x])" by(simp add:nth_append less_Suc_eq)
  1459     thus "?R" ..
  1460   qed
  1461 qed
  1462 
  1463 
  1464 subsubsection {* @{text list_update} *}
  1465 
  1466 lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
  1467 by (induct xs arbitrary: i) (auto split: nat.split)
  1468 
  1469 lemma nth_list_update:
  1470 "i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
  1471 by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
  1472 
  1473 lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
  1474 by (simp add: nth_list_update)
  1475 
  1476 lemma nth_list_update_neq [simp]: "i \<noteq> j ==> xs[i:=x]!j = xs!j"
  1477 by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
  1478 
  1479 lemma list_update_id[simp]: "xs[i := xs!i] = xs"
  1480 by (induct xs arbitrary: i) (simp_all split:nat.splits)
  1481 
  1482 lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
  1483 apply (induct xs arbitrary: i)
  1484  apply simp
  1485 apply (case_tac i)
  1486 apply simp_all
  1487 done
  1488 
  1489 lemma list_update_nonempty[simp]: "xs[k:=x] = [] \<longleftrightarrow> xs=[]"
  1490 by(metis length_0_conv length_list_update)
  1491 
  1492 lemma list_update_same_conv:
  1493 "i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
  1494 by (induct xs arbitrary: i) (auto split: nat.split)
  1495 
  1496 lemma list_update_append1:
  1497  "i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
  1498 apply (induct xs arbitrary: i, simp)
  1499 apply(simp split:nat.split)
  1500 done
  1501 
  1502 lemma list_update_append:
  1503   "(xs @ ys) [n:= x] = 
  1504   (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
  1505 by (induct xs arbitrary: n) (auto split:nat.splits)
  1506 
  1507 lemma list_update_length [simp]:
  1508  "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
  1509 by (induct xs, auto)
  1510 
  1511 lemma map_update: "map f (xs[k:= y]) = (map f xs)[k := f y]"
  1512 by(induct xs arbitrary: k)(auto split:nat.splits)
  1513 
  1514 lemma rev_update:
  1515   "k < length xs \<Longrightarrow> rev (xs[k:= y]) = (rev xs)[length xs - k - 1 := y]"
  1516 by (induct xs arbitrary: k) (auto simp: list_update_append split:nat.splits)
  1517 
  1518 lemma update_zip:
  1519   "(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
  1520 by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split)
  1521 
  1522 lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)"
  1523 by (induct xs arbitrary: i) (auto split: nat.split)
  1524 
  1525 lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
  1526 by (blast dest!: set_update_subset_insert [THEN subsetD])
  1527 
  1528 lemma set_update_memI: "n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
  1529 by (induct xs arbitrary: n) (auto split:nat.splits)
  1530 
  1531 lemma list_update_overwrite[simp]:
  1532   "xs [i := x, i := y] = xs [i := y]"
  1533 apply (induct xs arbitrary: i) apply simp
  1534 apply (case_tac i, simp_all)
  1535 done
  1536 
  1537 lemma list_update_swap:
  1538   "i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]"
  1539 apply (induct xs arbitrary: i i')
  1540 apply simp
  1541 apply (case_tac i, case_tac i')
  1542 apply auto
  1543 apply (case_tac i')
  1544 apply auto
  1545 done
  1546 
  1547 lemma list_update_code [code]:
  1548   "[][i := y] = []"
  1549   "(x # xs)[0 := y] = y # xs"
  1550   "(x # xs)[Suc i := y] = x # xs[i := y]"
  1551   by simp_all
  1552 
  1553 
  1554 subsubsection {* @{text last} and @{text butlast} *}
  1555 
  1556 lemma last_snoc [simp]: "last (xs @ [x]) = x"
  1557 by (induct xs) auto
  1558 
  1559 lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
  1560 by (induct xs) auto
  1561 
  1562 lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
  1563   by simp
  1564 
  1565 lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
  1566   by simp
  1567 
  1568 lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
  1569 by (induct xs) (auto)
  1570 
  1571 lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
  1572 by(simp add:last_append)
  1573 
  1574 lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
  1575 by(simp add:last_append)
  1576 
  1577 lemma last_tl: "xs = [] \<or> tl xs \<noteq> [] \<Longrightarrow>last (tl xs) = last xs"
  1578 by (induct xs) simp_all
  1579 
  1580 lemma butlast_tl: "butlast (tl xs) = tl (butlast xs)"
  1581 by (induct xs) simp_all
  1582 
  1583 lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
  1584 by(rule rev_exhaust[of xs]) simp_all
  1585 
  1586 lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
  1587 by(cases xs) simp_all
  1588 
  1589 lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
  1590 by (induct as) auto
  1591 
  1592 lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
  1593 by (induct xs rule: rev_induct) auto
  1594 
  1595 lemma butlast_append:
  1596   "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
  1597 by (induct xs arbitrary: ys) auto
  1598 
  1599 lemma append_butlast_last_id [simp]:
  1600 "xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
  1601 by (induct xs) auto
  1602 
  1603 lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
  1604 by (induct xs) (auto split: split_if_asm)
  1605 
  1606 lemma in_set_butlast_appendI:
  1607 "x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
  1608 by (auto dest: in_set_butlastD simp add: butlast_append)
  1609 
  1610 lemma last_drop[simp]: "n < length xs \<Longrightarrow> last (drop n xs) = last xs"
  1611 apply (induct xs arbitrary: n)
  1612  apply simp
  1613 apply (auto split:nat.split)
  1614 done
  1615 
  1616 lemma nth_butlast:
  1617   assumes "n < length (butlast xs)" shows "butlast xs ! n = xs ! n"
  1618 proof (cases xs)
  1619   case (Cons y ys)
  1620   moreover from assms have "butlast xs ! n = (butlast xs @ [last xs]) ! n"
  1621     by (simp add: nth_append)
  1622   ultimately show ?thesis using append_butlast_last_id by simp
  1623 qed simp
  1624 
  1625 lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
  1626 by(induct xs)(auto simp:neq_Nil_conv)
  1627 
  1628 lemma butlast_conv_take: "butlast xs = take (length xs - 1) xs"
  1629 by (induct xs, simp, case_tac xs, simp_all)
  1630 
  1631 lemma last_list_update:
  1632   "xs \<noteq> [] \<Longrightarrow> last(xs[k:=x]) = (if k = size xs - 1 then x else last xs)"
  1633 by (auto simp: last_conv_nth)
  1634 
  1635 lemma butlast_list_update:
  1636   "butlast(xs[k:=x]) =
  1637  (if k = size xs - 1 then butlast xs else (butlast xs)[k:=x])"
  1638 apply(cases xs rule:rev_cases)
  1639 apply simp
  1640 apply(simp add:list_update_append split:nat.splits)
  1641 done
  1642 
  1643 lemma last_map:
  1644   "xs \<noteq> [] \<Longrightarrow> last (map f xs) = f (last xs)"
  1645   by (cases xs rule: rev_cases) simp_all
  1646 
  1647 lemma map_butlast:
  1648   "map f (butlast xs) = butlast (map f xs)"
  1649   by (induct xs) simp_all
  1650 
  1651 lemma snoc_eq_iff_butlast:
  1652   "xs @ [x] = ys \<longleftrightarrow> (ys \<noteq> [] & butlast ys = xs & last ys = x)"
  1653 by (metis append_butlast_last_id append_is_Nil_conv butlast_snoc last_snoc not_Cons_self)
  1654 
  1655 
  1656 subsubsection {* @{text take} and @{text drop} *}
  1657 
  1658 lemma take_0 [simp]: "take 0 xs = []"
  1659 by (induct xs) auto
  1660 
  1661 lemma drop_0 [simp]: "drop 0 xs = xs"
  1662 by (induct xs) auto
  1663 
  1664 lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
  1665 by simp
  1666 
  1667 lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
  1668 by simp
  1669 
  1670 declare take_Cons [simp del] and drop_Cons [simp del]
  1671 
  1672 lemma take_1_Cons [simp]: "take 1 (x # xs) = [x]"
  1673   unfolding One_nat_def by simp
  1674 
  1675 lemma drop_1_Cons [simp]: "drop 1 (x # xs) = xs"
  1676   unfolding One_nat_def by simp
  1677 
  1678 lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
  1679 by(clarsimp simp add:neq_Nil_conv)
  1680 
  1681 lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
  1682 by(cases xs, simp_all)
  1683 
  1684 lemma take_tl: "take n (tl xs) = tl (take (Suc n) xs)"
  1685 by (induct xs arbitrary: n) simp_all
  1686 
  1687 lemma drop_tl: "drop n (tl xs) = tl(drop n xs)"
  1688 by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split)
  1689 
  1690 lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)"
  1691 by (cases n, simp, cases xs, auto)
  1692 
  1693 lemma tl_drop: "tl (drop n xs) = drop n (tl xs)"
  1694 by (simp only: drop_tl)
  1695 
  1696 lemma nth_via_drop: "drop n xs = y#ys \<Longrightarrow> xs!n = y"
  1697 apply (induct xs arbitrary: n, simp)
  1698 apply(simp add:drop_Cons nth_Cons split:nat.splits)
  1699 done
  1700 
  1701 lemma take_Suc_conv_app_nth:
  1702   "i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
  1703 apply (induct xs arbitrary: i, simp)
  1704 apply (case_tac i, auto)
  1705 done
  1706 
  1707 lemma drop_Suc_conv_tl:
  1708   "i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
  1709 apply (induct xs arbitrary: i, simp)
  1710 apply (case_tac i, auto)
  1711 done
  1712 
  1713 lemma length_take [simp]: "length (take n xs) = min (length xs) n"
  1714 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1715 
  1716 lemma length_drop [simp]: "length (drop n xs) = (length xs - n)"
  1717 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1718 
  1719 lemma take_all [simp]: "length xs <= n ==> take n xs = xs"
  1720 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1721 
  1722 lemma drop_all [simp]: "length xs <= n ==> drop n xs = []"
  1723 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1724 
  1725 lemma take_append [simp]:
  1726   "take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
  1727 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1728 
  1729 lemma drop_append [simp]:
  1730   "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
  1731 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1732 
  1733 lemma take_take [simp]: "take n (take m xs) = take (min n m) xs"
  1734 apply (induct m arbitrary: xs n, auto)
  1735 apply (case_tac xs, auto)
  1736 apply (case_tac n, auto)
  1737 done
  1738 
  1739 lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs"
  1740 apply (induct m arbitrary: xs, auto)
  1741 apply (case_tac xs, auto)
  1742 done
  1743 
  1744 lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)"
  1745 apply (induct m arbitrary: xs n, auto)
  1746 apply (case_tac xs, auto)
  1747 done
  1748 
  1749 lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)"
  1750 apply(induct xs arbitrary: m n)
  1751  apply simp
  1752 apply(simp add: take_Cons drop_Cons split:nat.split)
  1753 done
  1754 
  1755 lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs"
  1756 apply (induct n arbitrary: xs, auto)
  1757 apply (case_tac xs, auto)
  1758 done
  1759 
  1760 lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])"
  1761 apply(induct xs arbitrary: n)
  1762  apply simp
  1763 apply(simp add:take_Cons split:nat.split)
  1764 done
  1765 
  1766 lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)"
  1767 apply(induct xs arbitrary: n)
  1768 apply simp
  1769 apply(simp add:drop_Cons split:nat.split)
  1770 done
  1771 
  1772 lemma take_map: "take n (map f xs) = map f (take n xs)"
  1773 apply (induct n arbitrary: xs, auto)
  1774 apply (case_tac xs, auto)
  1775 done
  1776 
  1777 lemma drop_map: "drop n (map f xs) = map f (drop n xs)"
  1778 apply (induct n arbitrary: xs, auto)
  1779 apply (case_tac xs, auto)
  1780 done
  1781 
  1782 lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)"
  1783 apply (induct xs arbitrary: i, auto)
  1784 apply (case_tac i, auto)
  1785 done
  1786 
  1787 lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)"
  1788 apply (induct xs arbitrary: i, auto)
  1789 apply (case_tac i, auto)
  1790 done
  1791 
  1792 lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i"
  1793 apply (induct xs arbitrary: i n, auto)
  1794 apply (case_tac n, blast)
  1795 apply (case_tac i, auto)
  1796 done
  1797 
  1798 lemma nth_drop [simp]:
  1799   "n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
  1800 apply (induct n arbitrary: xs i, auto)
  1801 apply (case_tac xs, auto)
  1802 done
  1803 
  1804 lemma butlast_take:
  1805   "n <= length xs ==> butlast (take n xs) = take (n - 1) xs"
  1806 by (simp add: butlast_conv_take min_max.inf_absorb1 min_max.inf_absorb2)
  1807 
  1808 lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)"
  1809 by (simp add: butlast_conv_take drop_take add_ac)
  1810 
  1811 lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs"
  1812 by (simp add: butlast_conv_take min_max.inf_absorb1)
  1813 
  1814 lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)"
  1815 by (simp add: butlast_conv_take drop_take add_ac)
  1816 
  1817 lemma hd_drop_conv_nth: "n < length xs \<Longrightarrow> hd(drop n xs) = xs!n"
  1818 by(simp add: hd_conv_nth)
  1819 
  1820 lemma set_take_subset_set_take:
  1821   "m <= n \<Longrightarrow> set(take m xs) <= set(take n xs)"
  1822 apply (induct xs arbitrary: m n)
  1823 apply simp
  1824 apply (case_tac n)
  1825 apply (auto simp: take_Cons)
  1826 done
  1827 
  1828 lemma set_take_subset: "set(take n xs) \<subseteq> set xs"
  1829 by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split)
  1830 
  1831 lemma set_drop_subset: "set(drop n xs) \<subseteq> set xs"
  1832 by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split)
  1833 
  1834 lemma set_drop_subset_set_drop:
  1835   "m >= n \<Longrightarrow> set(drop m xs) <= set(drop n xs)"
  1836 apply(induct xs arbitrary: m n)
  1837 apply(auto simp:drop_Cons split:nat.split)
  1838 apply (metis set_drop_subset subset_iff)
  1839 done
  1840 
  1841 lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
  1842 using set_take_subset by fast
  1843 
  1844 lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
  1845 using set_drop_subset by fast
  1846 
  1847 lemma append_eq_conv_conj:
  1848   "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
  1849 apply (induct xs arbitrary: zs, simp, clarsimp)
  1850 apply (case_tac zs, auto)
  1851 done
  1852 
  1853 lemma take_add: 
  1854   "take (i+j) xs = take i xs @ take j (drop i xs)"
  1855 apply (induct xs arbitrary: i, auto) 
  1856 apply (case_tac i, simp_all)
  1857 done
  1858 
  1859 lemma append_eq_append_conv_if:
  1860  "(xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
  1861   (if size xs\<^isub>1 \<le> size ys\<^isub>1
  1862    then xs\<^isub>1 = take (size xs\<^isub>1) ys\<^isub>1 \<and> xs\<^isub>2 = drop (size xs\<^isub>1) ys\<^isub>1 @ ys\<^isub>2
  1863    else take (size ys\<^isub>1) xs\<^isub>1 = ys\<^isub>1 \<and> drop (size ys\<^isub>1) xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>2)"
  1864 apply(induct xs\<^isub>1 arbitrary: ys\<^isub>1)
  1865  apply simp
  1866 apply(case_tac ys\<^isub>1)
  1867 apply simp_all
  1868 done
  1869 
  1870 lemma take_hd_drop:
  1871   "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (Suc n) xs"
  1872 apply(induct xs arbitrary: n)
  1873 apply simp
  1874 apply(simp add:drop_Cons split:nat.split)
  1875 done
  1876 
  1877 lemma id_take_nth_drop:
  1878  "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
  1879 proof -
  1880   assume si: "i < length xs"
  1881   hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
  1882   moreover
  1883   from si have "take (Suc i) xs = take i xs @ [xs!i]"
  1884     apply (rule_tac take_Suc_conv_app_nth) by arith
  1885   ultimately show ?thesis by auto
  1886 qed
  1887   
  1888 lemma upd_conv_take_nth_drop:
  1889  "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
  1890 proof -
  1891   assume i: "i < length xs"
  1892   have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
  1893     by(rule arg_cong[OF id_take_nth_drop[OF i]])
  1894   also have "\<dots> = take i xs @ a # drop (Suc i) xs"
  1895     using i by (simp add: list_update_append)
  1896   finally show ?thesis .
  1897 qed
  1898 
  1899 lemma nth_drop':
  1900   "i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs"
  1901 apply (induct i arbitrary: xs)
  1902 apply (simp add: neq_Nil_conv)
  1903 apply (erule exE)+
  1904 apply simp
  1905 apply (case_tac xs)
  1906 apply simp_all
  1907 done
  1908 
  1909 
  1910 subsubsection {* @{text takeWhile} and @{text dropWhile} *}
  1911 
  1912 lemma length_takeWhile_le: "length (takeWhile P xs) \<le> length xs"
  1913   by (induct xs) auto
  1914 
  1915 lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
  1916 by (induct xs) auto
  1917 
  1918 lemma takeWhile_append1 [simp]:
  1919 "[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
  1920 by (induct xs) auto
  1921 
  1922 lemma takeWhile_append2 [simp]:
  1923 "(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
  1924 by (induct xs) auto
  1925 
  1926 lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
  1927 by (induct xs) auto
  1928 
  1929 lemma takeWhile_nth: "j < length (takeWhile P xs) \<Longrightarrow> takeWhile P xs ! j = xs ! j"
  1930 apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
  1931 
  1932 lemma dropWhile_nth: "j < length (dropWhile P xs) \<Longrightarrow> dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))"
  1933 apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
  1934 
  1935 lemma length_dropWhile_le: "length (dropWhile P xs) \<le> length xs"
  1936 by (induct xs) auto
  1937 
  1938 lemma dropWhile_append1 [simp]:
  1939 "[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
  1940 by (induct xs) auto
  1941 
  1942 lemma dropWhile_append2 [simp]:
  1943 "(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
  1944 by (induct xs) auto
  1945 
  1946 lemma dropWhile_append3:
  1947   "\<not> P y \<Longrightarrow>dropWhile P (xs @ y # ys) = dropWhile P xs @ y # ys"
  1948 by (induct xs) auto
  1949 
  1950 lemma dropWhile_last:
  1951   "x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> last (dropWhile P xs) = last xs"
  1952 by (auto simp add: dropWhile_append3 in_set_conv_decomp)
  1953 
  1954 lemma set_dropWhileD: "x \<in> set (dropWhile P xs) \<Longrightarrow> x \<in> set xs"
  1955 by (induct xs) (auto split: split_if_asm)
  1956 
  1957 lemma set_takeWhileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
  1958 by (induct xs) (auto split: split_if_asm)
  1959 
  1960 lemma takeWhile_eq_all_conv[simp]:
  1961  "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
  1962 by(induct xs, auto)
  1963 
  1964 lemma dropWhile_eq_Nil_conv[simp]:
  1965  "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
  1966 by(induct xs, auto)
  1967 
  1968 lemma dropWhile_eq_Cons_conv:
  1969  "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
  1970 by(induct xs, auto)
  1971 
  1972 lemma distinct_takeWhile[simp]: "distinct xs ==> distinct (takeWhile P xs)"
  1973 by (induct xs) (auto dest: set_takeWhileD)
  1974 
  1975 lemma distinct_dropWhile[simp]: "distinct xs ==> distinct (dropWhile P xs)"
  1976 by (induct xs) auto
  1977 
  1978 lemma takeWhile_map: "takeWhile P (map f xs) = map f (takeWhile (P \<circ> f) xs)"
  1979 by (induct xs) auto
  1980 
  1981 lemma dropWhile_map: "dropWhile P (map f xs) = map f (dropWhile (P \<circ> f) xs)"
  1982 by (induct xs) auto
  1983 
  1984 lemma takeWhile_eq_take: "takeWhile P xs = take (length (takeWhile P xs)) xs"
  1985 by (induct xs) auto
  1986 
  1987 lemma dropWhile_eq_drop: "dropWhile P xs = drop (length (takeWhile P xs)) xs"
  1988 by (induct xs) auto
  1989 
  1990 lemma hd_dropWhile:
  1991   "dropWhile P xs \<noteq> [] \<Longrightarrow> \<not> P (hd (dropWhile P xs))"
  1992 using assms by (induct xs) auto
  1993 
  1994 lemma takeWhile_eq_filter:
  1995   assumes "\<And> x. x \<in> set (dropWhile P xs) \<Longrightarrow> \<not> P x"
  1996   shows "takeWhile P xs = filter P xs"
  1997 proof -
  1998   have A: "filter P xs = filter P (takeWhile P xs @ dropWhile P xs)"
  1999     by simp
  2000   have B: "filter P (dropWhile P xs) = []"
  2001     unfolding filter_empty_conv using assms by blast
  2002   have "filter P xs = takeWhile P xs"
  2003     unfolding A filter_append B
  2004     by (auto simp add: filter_id_conv dest: set_takeWhileD)
  2005   thus ?thesis ..
  2006 qed
  2007 
  2008 lemma takeWhile_eq_take_P_nth:
  2009   "\<lbrakk> \<And> i. \<lbrakk> i < n ; i < length xs \<rbrakk> \<Longrightarrow> P (xs ! i) ; n < length xs \<Longrightarrow> \<not> P (xs ! n) \<rbrakk> \<Longrightarrow>
  2010   takeWhile P xs = take n xs"
  2011 proof (induct xs arbitrary: n)
  2012   case (Cons x xs)
  2013   thus ?case
  2014   proof (cases n)
  2015     case (Suc n') note this[simp]
  2016     have "P x" using Cons.prems(1)[of 0] by simp
  2017     moreover have "takeWhile P xs = take n' xs"
  2018     proof (rule Cons.hyps)
  2019       case goal1 thus "P (xs ! i)" using Cons.prems(1)[of "Suc i"] by simp
  2020     next case goal2 thus ?case using Cons by auto
  2021     qed
  2022     ultimately show ?thesis by simp
  2023    qed simp
  2024 qed simp
  2025 
  2026 lemma nth_length_takeWhile:
  2027   "length (takeWhile P xs) < length xs \<Longrightarrow> \<not> P (xs ! length (takeWhile P xs))"
  2028 by (induct xs) auto
  2029 
  2030 lemma length_takeWhile_less_P_nth:
  2031   assumes all: "\<And> i. i < j \<Longrightarrow> P (xs ! i)" and "j \<le> length xs"
  2032   shows "j \<le> length (takeWhile P xs)"
  2033 proof (rule classical)
  2034   assume "\<not> ?thesis"
  2035   hence "length (takeWhile P xs) < length xs" using assms by simp
  2036   thus ?thesis using all `\<not> ?thesis` nth_length_takeWhile[of P xs] by auto
  2037 qed
  2038 
  2039 text{* The following two lemmmas could be generalized to an arbitrary
  2040 property. *}
  2041 
  2042 lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
  2043  takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
  2044 by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
  2045 
  2046 lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
  2047   dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
  2048 apply(induct xs)
  2049  apply simp
  2050 apply auto
  2051 apply(subst dropWhile_append2)
  2052 apply auto
  2053 done
  2054 
  2055 lemma takeWhile_not_last:
  2056  "distinct xs \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
  2057 apply(induct xs)
  2058  apply simp
  2059 apply(case_tac xs)
  2060 apply(auto)
  2061 done
  2062 
  2063 lemma takeWhile_cong [fundef_cong]:
  2064   "[| l = k; !!x. x : set l ==> P x = Q x |] 
  2065   ==> takeWhile P l = takeWhile Q k"
  2066 by (induct k arbitrary: l) (simp_all)
  2067 
  2068 lemma dropWhile_cong [fundef_cong]:
  2069   "[| l = k; !!x. x : set l ==> P x = Q x |] 
  2070   ==> dropWhile P l = dropWhile Q k"
  2071 by (induct k arbitrary: l, simp_all)
  2072 
  2073 
  2074 subsubsection {* @{text zip} *}
  2075 
  2076 lemma zip_Nil [simp]: "zip [] ys = []"
  2077 by (induct ys) auto
  2078 
  2079 lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
  2080 by simp
  2081 
  2082 declare zip_Cons [simp del]
  2083 
  2084 lemma [code]:
  2085   "zip [] ys = []"
  2086   "zip xs [] = []"
  2087   "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
  2088   by (fact zip_Nil zip.simps(1) zip_Cons_Cons)+
  2089 
  2090 lemma zip_Cons1:
  2091  "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
  2092 by(auto split:list.split)
  2093 
  2094 lemma length_zip [simp]:
  2095 "length (zip xs ys) = min (length xs) (length ys)"
  2096 by (induct xs ys rule:list_induct2') auto
  2097 
  2098 lemma zip_obtain_same_length:
  2099   assumes "\<And>zs ws n. length zs = length ws \<Longrightarrow> n = min (length xs) (length ys)
  2100     \<Longrightarrow> zs = take n xs \<Longrightarrow> ws = take n ys \<Longrightarrow> P (zip zs ws)"
  2101   shows "P (zip xs ys)"
  2102 proof -
  2103   let ?n = "min (length xs) (length ys)"
  2104   have "P (zip (take ?n xs) (take ?n ys))"
  2105     by (rule assms) simp_all
  2106   moreover have "zip xs ys = zip (take ?n xs) (take ?n ys)"
  2107   proof (induct xs arbitrary: ys)
  2108     case Nil then show ?case by simp
  2109   next
  2110     case (Cons x xs) then show ?case by (cases ys) simp_all
  2111   qed
  2112   ultimately show ?thesis by simp
  2113 qed
  2114 
  2115 lemma zip_append1:
  2116 "zip (xs @ ys) zs =
  2117 zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
  2118 by (induct xs zs rule:list_induct2') auto
  2119 
  2120 lemma zip_append2:
  2121 "zip xs (ys @ zs) =
  2122 zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
  2123 by (induct xs ys rule:list_induct2') auto
  2124 
  2125 lemma zip_append [simp]:
  2126  "[| length xs = length us |] ==>
  2127 zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
  2128 by (simp add: zip_append1)
  2129 
  2130 lemma zip_rev:
  2131 "length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
  2132 by (induct rule:list_induct2, simp_all)
  2133 
  2134 lemma zip_map_map:
  2135   "zip (map f xs) (map g ys) = map (\<lambda> (x, y). (f x, g y)) (zip xs ys)"
  2136 proof (induct xs arbitrary: ys)
  2137   case (Cons x xs) note Cons_x_xs = Cons.hyps
  2138   show ?case
  2139   proof (cases ys)
  2140     case (Cons y ys')
  2141     show ?thesis unfolding Cons using Cons_x_xs by simp
  2142   qed simp
  2143 qed simp
  2144 
  2145 lemma zip_map1:
  2146   "zip (map f xs) ys = map (\<lambda>(x, y). (f x, y)) (zip xs ys)"
  2147 using zip_map_map[of f xs "\<lambda>x. x" ys] by simp
  2148 
  2149 lemma zip_map2:
  2150   "zip xs (map f ys) = map (\<lambda>(x, y). (x, f y)) (zip xs ys)"
  2151 using zip_map_map[of "\<lambda>x. x" xs f ys] by simp
  2152 
  2153 lemma map_zip_map:
  2154   "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
  2155 unfolding zip_map1 by auto
  2156 
  2157 lemma map_zip_map2:
  2158   "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
  2159 unfolding zip_map2 by auto
  2160 
  2161 text{* Courtesy of Andreas Lochbihler: *}
  2162 lemma zip_same_conv_map: "zip xs xs = map (\<lambda>x. (x, x)) xs"
  2163 by(induct xs) auto
  2164 
  2165 lemma nth_zip [simp]:
  2166 "[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
  2167 apply (induct ys arbitrary: i xs, simp)
  2168 apply (case_tac xs)
  2169  apply (simp_all add: nth.simps split: nat.split)
  2170 done
  2171 
  2172 lemma set_zip:
  2173 "set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
  2174 by(simp add: set_conv_nth cong: rev_conj_cong)
  2175 
  2176 lemma zip_same: "((a,b) \<in> set (zip xs xs)) = (a \<in> set xs \<and> a = b)"
  2177 by(induct xs) auto
  2178 
  2179 lemma zip_update:
  2180   "zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
  2181 by(rule sym, simp add: update_zip)
  2182 
  2183 lemma zip_replicate [simp]:
  2184   "zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
  2185 apply (induct i arbitrary: j, auto)
  2186 apply (case_tac j, auto)
  2187 done
  2188 
  2189 lemma take_zip:
  2190   "take n (zip xs ys) = zip (take n xs) (take n ys)"
  2191 apply (induct n arbitrary: xs ys)
  2192  apply simp
  2193 apply (case_tac xs, simp)
  2194 apply (case_tac ys, simp_all)
  2195 done
  2196 
  2197 lemma drop_zip:
  2198   "drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
  2199 apply (induct n arbitrary: xs ys)
  2200  apply simp
  2201 apply (case_tac xs, simp)
  2202 apply (case_tac ys, simp_all)
  2203 done
  2204 
  2205 lemma zip_takeWhile_fst: "zip (takeWhile P xs) ys = takeWhile (P \<circ> fst) (zip xs ys)"
  2206 proof (induct xs arbitrary: ys)
  2207   case (Cons x xs) thus ?case by (cases ys) auto
  2208 qed simp
  2209 
  2210 lemma zip_takeWhile_snd: "zip xs (takeWhile P ys) = takeWhile (P \<circ> snd) (zip xs ys)"
  2211 proof (induct xs arbitrary: ys)
  2212   case (Cons x xs) thus ?case by (cases ys) auto
  2213 qed simp
  2214 
  2215 lemma set_zip_leftD:
  2216   "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
  2217 by (induct xs ys rule:list_induct2') auto
  2218 
  2219 lemma set_zip_rightD:
  2220   "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
  2221 by (induct xs ys rule:list_induct2') auto
  2222 
  2223 lemma in_set_zipE:
  2224   "(x,y) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
  2225 by(blast dest: set_zip_leftD set_zip_rightD)
  2226 
  2227 lemma zip_map_fst_snd:
  2228   "zip (map fst zs) (map snd zs) = zs"
  2229   by (induct zs) simp_all
  2230 
  2231 lemma zip_eq_conv:
  2232   "length xs = length ys \<Longrightarrow> zip xs ys = zs \<longleftrightarrow> map fst zs = xs \<and> map snd zs = ys"
  2233   by (auto simp add: zip_map_fst_snd)
  2234 
  2235 
  2236 subsubsection {* @{text list_all2} *}
  2237 
  2238 lemma list_all2_lengthD [intro?]: 
  2239   "list_all2 P xs ys ==> length xs = length ys"
  2240 by (simp add: list_all2_def)
  2241 
  2242 lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
  2243 by (simp add: list_all2_def)
  2244 
  2245 lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
  2246 by (simp add: list_all2_def)
  2247 
  2248 lemma list_all2_Cons [iff, code]:
  2249   "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
  2250 by (auto simp add: list_all2_def)
  2251 
  2252 lemma list_all2_Cons1:
  2253 "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
  2254 by (cases ys) auto
  2255 
  2256 lemma list_all2_Cons2:
  2257 "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
  2258 by (cases xs) auto
  2259 
  2260 lemma list_all2_induct
  2261   [consumes 1, case_names Nil Cons, induct set: list_all2]:
  2262   assumes P: "list_all2 P xs ys"
  2263   assumes Nil: "R [] []"
  2264   assumes Cons: "\<And>x xs y ys.
  2265     \<lbrakk>P x y; list_all2 P xs ys; R xs ys\<rbrakk> \<Longrightarrow> R (x # xs) (y # ys)"
  2266   shows "R xs ys"
  2267 using P
  2268 by (induct xs arbitrary: ys) (auto simp add: list_all2_Cons1 Nil Cons)
  2269 
  2270 lemma list_all2_rev [iff]:
  2271 "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
  2272 by (simp add: list_all2_def zip_rev cong: conj_cong)
  2273 
  2274 lemma list_all2_rev1:
  2275 "list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
  2276 by (subst list_all2_rev [symmetric]) simp
  2277 
  2278 lemma list_all2_append1:
  2279 "list_all2 P (xs @ ys) zs =
  2280 (EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
  2281 list_all2 P xs us \<and> list_all2 P ys vs)"
  2282 apply (simp add: list_all2_def zip_append1)
  2283 apply (rule iffI)
  2284  apply (rule_tac x = "take (length xs) zs" in exI)
  2285  apply (rule_tac x = "drop (length xs) zs" in exI)
  2286  apply (force split: nat_diff_split simp add: min_def, clarify)
  2287 apply (simp add: ball_Un)
  2288 done
  2289 
  2290 lemma list_all2_append2:
  2291 "list_all2 P xs (ys @ zs) =
  2292 (EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
  2293 list_all2 P us ys \<and> list_all2 P vs zs)"
  2294 apply (simp add: list_all2_def zip_append2)
  2295 apply (rule iffI)
  2296  apply (rule_tac x = "take (length ys) xs" in exI)
  2297  apply (rule_tac x = "drop (length ys) xs" in exI)
  2298  apply (force split: nat_diff_split simp add: min_def, clarify)
  2299 apply (simp add: ball_Un)
  2300 done
  2301 
  2302 lemma list_all2_append:
  2303   "length xs = length ys \<Longrightarrow>
  2304   list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
  2305 by (induct rule:list_induct2, simp_all)
  2306 
  2307 lemma list_all2_appendI [intro?, trans]:
  2308   "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
  2309 by (simp add: list_all2_append list_all2_lengthD)
  2310 
  2311 lemma list_all2_conv_all_nth:
  2312 "list_all2 P xs ys =
  2313 (length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
  2314 by (force simp add: list_all2_def set_zip)
  2315 
  2316 lemma list_all2_trans:
  2317   assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
  2318   shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
  2319         (is "!!bs cs. PROP ?Q as bs cs")
  2320 proof (induct as)
  2321   fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
  2322   show "!!cs. PROP ?Q (x # xs) bs cs"
  2323   proof (induct bs)
  2324     fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
  2325     show "PROP ?Q (x # xs) (y # ys) cs"
  2326       by (induct cs) (auto intro: tr I1 I2)
  2327   qed simp
  2328 qed simp
  2329 
  2330 lemma list_all2_all_nthI [intro?]:
  2331   "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
  2332 by (simp add: list_all2_conv_all_nth)
  2333 
  2334 lemma list_all2I:
  2335   "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
  2336 by (simp add: list_all2_def)
  2337 
  2338 lemma list_all2_nthD:
  2339   "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
  2340 by (simp add: list_all2_conv_all_nth)
  2341 
  2342 lemma list_all2_nthD2:
  2343   "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
  2344 by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
  2345 
  2346 lemma list_all2_map1: 
  2347   "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
  2348 by (simp add: list_all2_conv_all_nth)
  2349 
  2350 lemma list_all2_map2: 
  2351   "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
  2352 by (auto simp add: list_all2_conv_all_nth)
  2353 
  2354 lemma list_all2_refl [intro?]:
  2355   "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
  2356 by (simp add: list_all2_conv_all_nth)
  2357 
  2358 lemma list_all2_update_cong:
  2359   "\<lbrakk> list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
  2360 by (cases "i < length ys") (auto simp add: list_all2_conv_all_nth nth_list_update)
  2361 
  2362 lemma list_all2_takeI [simp,intro?]:
  2363   "list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
  2364 apply (induct xs arbitrary: n ys)
  2365  apply simp
  2366 apply (clarsimp simp add: list_all2_Cons1)
  2367 apply (case_tac n)
  2368 apply auto
  2369 done
  2370 
  2371 lemma list_all2_dropI [simp,intro?]:
  2372   "list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
  2373 apply (induct as arbitrary: n bs, simp)
  2374 apply (clarsimp simp add: list_all2_Cons1)
  2375 apply (case_tac n, simp, simp)
  2376 done
  2377 
  2378 lemma list_all2_mono [intro?]:
  2379   "list_all2 P xs ys \<Longrightarrow> (\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys) \<Longrightarrow> list_all2 Q xs ys"
  2380 apply (induct xs arbitrary: ys, simp)
  2381 apply (case_tac ys, auto)
  2382 done
  2383 
  2384 lemma list_all2_eq:
  2385   "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
  2386 by (induct xs ys rule: list_induct2') auto
  2387 
  2388 lemma list_eq_iff_zip_eq:
  2389   "xs = ys \<longleftrightarrow> length xs = length ys \<and> (\<forall>(x,y) \<in> set (zip xs ys). x = y)"
  2390 by(auto simp add: set_zip list_all2_eq list_all2_conv_all_nth cong: conj_cong)
  2391 
  2392 
  2393 subsubsection {* @{const fold} with natural argument order *}
  2394 
  2395 lemma fold_remove1_split:
  2396   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"
  2397     and x: "x \<in> set xs"
  2398   shows "fold f xs = fold f (remove1 x xs) \<circ> f x"
  2399   using assms by (induct xs) (auto simp add: o_assoc [symmetric])
  2400 
  2401 lemma fold_cong [fundef_cong]:
  2402   "a = b \<Longrightarrow> xs = ys \<Longrightarrow> (\<And>x. x \<in> set xs \<Longrightarrow> f x = g x)
  2403     \<Longrightarrow> fold f xs a = fold g ys b"
  2404   by (induct ys arbitrary: a b xs) simp_all
  2405 
  2406 lemma fold_id:
  2407   assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x = id"
  2408   shows "fold f xs = id"
  2409   using assms by (induct xs) simp_all
  2410 
  2411 lemma fold_commute:
  2412   assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
  2413   shows "h \<circ> fold g xs = fold f xs \<circ> h"
  2414   using assms by (induct xs) (simp_all add: fun_eq_iff)
  2415 
  2416 lemma fold_commute_apply:
  2417   assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
  2418   shows "h (fold g xs s) = fold f xs (h s)"
  2419 proof -
  2420   from assms have "h \<circ> fold g xs = fold f xs \<circ> h" by (rule fold_commute)
  2421   then show ?thesis by (simp add: fun_eq_iff)
  2422 qed
  2423 
  2424 lemma fold_invariant: 
  2425   assumes "\<And>x. x \<in> set xs \<Longrightarrow> Q x" and "P s"
  2426     and "\<And>x s. Q x \<Longrightarrow> P s \<Longrightarrow> P (f x s)"
  2427   shows "P (fold f xs s)"
  2428   using assms by (induct xs arbitrary: s) simp_all
  2429 
  2430 lemma fold_append [simp]:
  2431   "fold f (xs @ ys) = fold f ys \<circ> fold f xs"
  2432   by (induct xs) simp_all
  2433 
  2434 lemma fold_map [code_unfold]:
  2435   "fold g (map f xs) = fold (g o f) xs"
  2436   by (induct xs) simp_all
  2437 
  2438 lemma fold_rev:
  2439   assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
  2440   shows "fold f (rev xs) = fold f xs"
  2441 using assms by (induct xs) (simp_all add: fold_commute_apply fun_eq_iff)
  2442 
  2443 lemma fold_Cons_rev:
  2444   "fold Cons xs = append (rev xs)"
  2445   by (induct xs) simp_all
  2446 
  2447 lemma rev_conv_fold [code]:
  2448   "rev xs = fold Cons xs []"
  2449   by (simp add: fold_Cons_rev)
  2450 
  2451 lemma fold_append_concat_rev:
  2452   "fold append xss = append (concat (rev xss))"
  2453   by (induct xss) simp_all
  2454 
  2455 text {* @{const Finite_Set.fold} and @{const fold} *}
  2456 
  2457 lemma (in comp_fun_commute) fold_set_fold_remdups:
  2458   "Finite_Set.fold f y (set xs) = fold f (remdups xs) y"
  2459   by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm insert_absorb)
  2460 
  2461 lemma (in ab_semigroup_mult) fold1_distinct_set_fold:
  2462   assumes "xs \<noteq> []"
  2463   assumes d: "distinct xs"
  2464   shows "Finite_Set.fold1 times (set xs) = List.fold times (tl xs) (hd xs)"
  2465 proof -
  2466   interpret comp_fun_commute times by (fact comp_fun_commute)
  2467   from assms obtain y ys where xs: "xs = y # ys"
  2468     by (cases xs) auto
  2469   then have *: "y \<notin> set ys" using assms by simp
  2470   from xs d have **: "remdups ys = ys"  by safe (induct ys, auto)
  2471   show ?thesis
  2472   proof (cases "set ys = {}")
  2473     case True with xs show ?thesis by simp
  2474   next
  2475     case False
  2476     then have "fold1 times (Set.insert y (set ys)) = Finite_Set.fold times y (set ys)"
  2477       by (simp_all add: fold1_eq_fold *)
  2478     with xs show ?thesis
  2479       by (simp add: fold_set_fold_remdups **)
  2480   qed
  2481 qed
  2482 
  2483 lemma (in comp_fun_idem) fold_set_fold:
  2484   "Finite_Set.fold f y (set xs) = fold f xs y"
  2485   by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm)
  2486 
  2487 lemma (in ab_semigroup_idem_mult) fold1_set_fold:
  2488   assumes "xs \<noteq> []"
  2489   shows "Finite_Set.fold1 times (set xs) = fold times (tl xs) (hd xs)"
  2490 proof -
  2491   interpret comp_fun_idem times by (fact comp_fun_idem)
  2492   from assms obtain y ys where xs: "xs = y # ys"
  2493     by (cases xs) auto
  2494   show ?thesis
  2495   proof (cases "set ys = {}")
  2496     case True with xs show ?thesis by simp
  2497   next
  2498     case False
  2499     then have "fold1 times (insert y (set ys)) = Finite_Set.fold times y (set ys)"
  2500       by (simp only: finite_set fold1_eq_fold_idem)
  2501     with xs show ?thesis by (simp add: fold_set_fold mult_commute)
  2502   qed
  2503 qed
  2504 
  2505 lemma union_set_fold [code]:
  2506   "set xs \<union> A = fold Set.insert xs A"
  2507 proof -
  2508   interpret comp_fun_idem Set.insert
  2509     by (fact comp_fun_idem_insert)
  2510   show ?thesis by (simp add: union_fold_insert fold_set_fold)
  2511 qed
  2512 
  2513 lemma union_coset_filter [code]:
  2514   "List.coset xs \<union> A = List.coset (List.filter (\<lambda>x. x \<notin> A) xs)"
  2515   by auto
  2516 
  2517 lemma minus_set_fold [code]:
  2518   "A - set xs = fold Set.remove xs A"
  2519 proof -
  2520   interpret comp_fun_idem Set.remove
  2521     by (fact comp_fun_idem_remove)
  2522   show ?thesis
  2523     by (simp add: minus_fold_remove [of _ A] fold_set_fold)
  2524 qed
  2525 
  2526 lemma minus_coset_filter [code]:
  2527   "A - List.coset xs = set (List.filter (\<lambda>x. x \<in> A) xs)"
  2528   by auto
  2529 
  2530 lemma inter_set_filter [code]:
  2531   "A \<inter> set xs = set (List.filter (\<lambda>x. x \<in> A) xs)"
  2532   by auto
  2533 
  2534 lemma inter_coset_fold [code]:
  2535   "A \<inter> List.coset xs = fold Set.remove xs A"
  2536   by (simp add: Diff_eq [symmetric] minus_set_fold)
  2537 
  2538 lemma (in lattice) Inf_fin_set_fold:
  2539   "Inf_fin (set (x # xs)) = fold inf xs x"
  2540 proof -
  2541   interpret ab_semigroup_idem_mult "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2542     by (fact ab_semigroup_idem_mult_inf)
  2543   show ?thesis
  2544     by (simp add: Inf_fin_def fold1_set_fold del: set.simps)
  2545 qed
  2546 
  2547 declare Inf_fin_set_fold [code]
  2548 
  2549 lemma (in lattice) Sup_fin_set_fold:
  2550   "Sup_fin (set (x # xs)) = fold sup xs x"
  2551 proof -
  2552   interpret ab_semigroup_idem_mult "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2553     by (fact ab_semigroup_idem_mult_sup)
  2554   show ?thesis
  2555     by (simp add: Sup_fin_def fold1_set_fold del: set.simps)
  2556 qed
  2557 
  2558 declare Sup_fin_set_fold [code]
  2559 
  2560 lemma (in linorder) Min_fin_set_fold:
  2561   "Min (set (x # xs)) = fold min xs x"
  2562 proof -
  2563   interpret ab_semigroup_idem_mult "min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2564     by (fact ab_semigroup_idem_mult_min)
  2565   show ?thesis
  2566     by (simp add: Min_def fold1_set_fold del: set.simps)
  2567 qed
  2568 
  2569 declare Min_fin_set_fold [code]
  2570 
  2571 lemma (in linorder) Max_fin_set_fold:
  2572   "Max (set (x # xs)) = fold max xs x"
  2573 proof -
  2574   interpret ab_semigroup_idem_mult "max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2575     by (fact ab_semigroup_idem_mult_max)
  2576   show ?thesis
  2577     by (simp add: Max_def fold1_set_fold del: set.simps)
  2578 qed
  2579 
  2580 declare Max_fin_set_fold [code]
  2581 
  2582 lemma (in complete_lattice) Inf_set_fold:
  2583   "Inf (set xs) = fold inf xs top"
  2584 proof -
  2585   interpret comp_fun_idem "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2586     by (fact comp_fun_idem_inf)
  2587   show ?thesis by (simp add: Inf_fold_inf fold_set_fold inf_commute)
  2588 qed
  2589 
  2590 declare Inf_set_fold [where 'a = "'a set", code]
  2591 
  2592 lemma (in complete_lattice) Sup_set_fold:
  2593   "Sup (set xs) = fold sup xs bot"
  2594 proof -
  2595   interpret comp_fun_idem "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2596     by (fact comp_fun_idem_sup)
  2597   show ?thesis by (simp add: Sup_fold_sup fold_set_fold sup_commute)
  2598 qed
  2599 
  2600 declare Sup_set_fold [where 'a = "'a set", code]
  2601 
  2602 lemma (in complete_lattice) INF_set_fold:
  2603   "INFI (set xs) f = fold (inf \<circ> f) xs top"
  2604   unfolding INF_def set_map [symmetric] Inf_set_fold fold_map ..
  2605 
  2606 declare INF_set_fold [code]
  2607 
  2608 lemma (in complete_lattice) SUP_set_fold:
  2609   "SUPR (set xs) f = fold (sup \<circ> f) xs bot"
  2610   unfolding SUP_def set_map [symmetric] Sup_set_fold fold_map ..
  2611 
  2612 declare SUP_set_fold [code]
  2613 
  2614 subsubsection {* Fold variants: @{const foldr} and @{const foldl} *}
  2615 
  2616 text {* Correspondence *}
  2617 
  2618 lemma foldr_conv_fold [code_abbrev]:
  2619   "foldr f xs = fold f (rev xs)"
  2620   by (induct xs) simp_all
  2621 
  2622 lemma foldl_conv_fold:
  2623   "foldl f s xs = fold (\<lambda>x s. f s x) xs s"
  2624   by (induct xs arbitrary: s) simp_all
  2625 
  2626 lemma foldr_conv_foldl: -- {* The ``Third Duality Theorem'' in Bird \& Wadler: *}
  2627   "foldr f xs a = foldl (\<lambda>x y. f y x) a (rev xs)"
  2628   by (simp add: foldr_conv_fold foldl_conv_fold)
  2629 
  2630 lemma foldl_conv_foldr:
  2631   "foldl f a xs = foldr (\<lambda>x y. f y x) (rev xs) a"
  2632   by (simp add: foldr_conv_fold foldl_conv_fold)
  2633 
  2634 lemma foldr_fold:
  2635   assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
  2636   shows "foldr f xs = fold f xs"
  2637   using assms unfolding foldr_conv_fold by (rule fold_rev)
  2638 
  2639 lemma foldr_cong [fundef_cong]:
  2640   "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"
  2641   by (auto simp add: foldr_conv_fold intro!: fold_cong)
  2642 
  2643 lemma foldl_cong [fundef_cong]:
  2644   "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"
  2645   by (auto simp add: foldl_conv_fold intro!: fold_cong)
  2646 
  2647 lemma foldr_append [simp]:
  2648   "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
  2649   by (simp add: foldr_conv_fold)
  2650 
  2651 lemma foldl_append [simp]:
  2652   "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
  2653   by (simp add: foldl_conv_fold)
  2654 
  2655 lemma foldr_map [code_unfold]:
  2656   "foldr g (map f xs) a = foldr (g o f) xs a"
  2657   by (simp add: foldr_conv_fold fold_map rev_map)
  2658 
  2659 lemma foldl_map [code_unfold]:
  2660   "foldl g a (map f xs) = foldl (\<lambda>a x. g a (f x)) a xs"
  2661   by (simp add: foldl_conv_fold fold_map comp_def)
  2662 
  2663 lemma concat_conv_foldr [code]:
  2664   "concat xss = foldr append xss []"
  2665   by (simp add: fold_append_concat_rev foldr_conv_fold)
  2666 
  2667 
  2668 subsubsection {* @{text upt} *}
  2669 
  2670 lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
  2671 -- {* simp does not terminate! *}
  2672 by (induct j) auto
  2673 
  2674 lemmas upt_rec_numeral[simp] = upt_rec[of "numeral m" "numeral n"] for m n
  2675 
  2676 lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
  2677 by (subst upt_rec) simp
  2678 
  2679 lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
  2680 by(induct j)simp_all
  2681 
  2682 lemma upt_eq_Cons_conv:
  2683  "([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
  2684 apply(induct j arbitrary: x xs)
  2685  apply simp
  2686 apply(clarsimp simp add: append_eq_Cons_conv)
  2687 apply arith
  2688 done
  2689 
  2690 lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
  2691 -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
  2692 by simp
  2693 
  2694 lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
  2695   by (simp add: upt_rec)
  2696 
  2697 lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
  2698 -- {* LOOPS as a simprule, since @{text "j <= j"}. *}
  2699 by (induct k) auto
  2700 
  2701 lemma length_upt [simp]: "length [i..<j] = j - i"
  2702 by (induct j) (auto simp add: Suc_diff_le)
  2703 
  2704 lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
  2705 apply (induct j)
  2706 apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
  2707 done
  2708 
  2709 
  2710 lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
  2711 by(simp add:upt_conv_Cons)
  2712 
  2713 lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
  2714 apply(cases j)
  2715  apply simp
  2716 by(simp add:upt_Suc_append)
  2717 
  2718 lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]"
  2719 apply (induct m arbitrary: i, simp)
  2720 apply (subst upt_rec)
  2721 apply (rule sym)
  2722 apply (subst upt_rec)
  2723 apply (simp del: upt.simps)
  2724 done
  2725 
  2726 lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
  2727 apply(induct j)
  2728 apply auto
  2729 done
  2730 
  2731 lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]"
  2732 by (induct n) auto
  2733 
  2734 lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
  2735 apply (induct n m  arbitrary: i rule: diff_induct)
  2736 prefer 3 apply (subst map_Suc_upt[symmetric])
  2737 apply (auto simp add: less_diff_conv)
  2738 done
  2739 
  2740 lemma nth_take_lemma:
  2741   "k <= length xs ==> k <= length ys ==>
  2742      (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
  2743 apply (atomize, induct k arbitrary: xs ys)
  2744 apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
  2745 txt {* Both lists must be non-empty *}
  2746 apply (case_tac xs, simp)
  2747 apply (case_tac ys, clarify)
  2748  apply (simp (no_asm_use))
  2749 apply clarify
  2750 txt {* prenexing's needed, not miniscoping *}
  2751 apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
  2752 apply blast
  2753 done
  2754 
  2755 lemma nth_equalityI:
  2756  "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
  2757   by (frule nth_take_lemma [OF le_refl eq_imp_le]) simp_all
  2758 
  2759 lemma map_nth:
  2760   "map (\<lambda>i. xs ! i) [0..<length xs] = xs"
  2761   by (rule nth_equalityI, auto)
  2762 
  2763 (* needs nth_equalityI *)
  2764 lemma list_all2_antisym:
  2765   "\<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> 
  2766   \<Longrightarrow> xs = ys"
  2767   apply (simp add: list_all2_conv_all_nth) 
  2768   apply (rule nth_equalityI, blast, simp)
  2769   done
  2770 
  2771 lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
  2772 -- {* The famous take-lemma. *}
  2773 apply (drule_tac x = "max (length xs) (length ys)" in spec)
  2774 apply (simp add: le_max_iff_disj)
  2775 done
  2776 
  2777 
  2778 lemma take_Cons':
  2779      "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
  2780 by (cases n) simp_all
  2781 
  2782 lemma drop_Cons':
  2783      "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
  2784 by (cases n) simp_all
  2785 
  2786 lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
  2787 by (cases n) simp_all
  2788 
  2789 lemma take_Cons_numeral [simp]:
  2790   "take (numeral v) (x # xs) = x # take (numeral v - 1) xs"
  2791 by (simp add: take_Cons')
  2792 
  2793 lemma drop_Cons_numeral [simp]:
  2794   "drop (numeral v) (x # xs) = drop (numeral v - 1) xs"
  2795 by (simp add: drop_Cons')
  2796 
  2797 lemma nth_Cons_numeral [simp]:
  2798   "(x # xs) ! numeral v = xs ! (numeral v - 1)"
  2799 by (simp add: nth_Cons')
  2800 
  2801 
  2802 subsubsection {* @{text upto}: interval-list on @{typ int} *}
  2803 
  2804 (* FIXME make upto tail recursive? *)
  2805 
  2806 function upto :: "int \<Rightarrow> int \<Rightarrow> int list" ("(1[_../_])") where
  2807 "upto i j = (if i \<le> j then i # [i+1..j] else [])"
  2808 by auto
  2809 termination
  2810 by(relation "measure(%(i::int,j). nat(j - i + 1))") auto
  2811 
  2812 declare upto.simps[code, simp del]
  2813 
  2814 lemmas upto_rec_numeral [simp] =
  2815   upto.simps[of "numeral m" "numeral n"]
  2816   upto.simps[of "numeral m" "neg_numeral n"]
  2817   upto.simps[of "neg_numeral m" "numeral n"]
  2818   upto.simps[of "neg_numeral m" "neg_numeral n"] for m n
  2819 
  2820 lemma upto_empty[simp]: "j < i \<Longrightarrow> [i..j] = []"
  2821 by(simp add: upto.simps)
  2822 
  2823 lemma set_upto[simp]: "set[i..j] = {i..j}"
  2824 proof(induct i j rule:upto.induct)
  2825   case (1 i j)
  2826   from this show ?case
  2827     unfolding upto.simps[of i j] simp_from_to[of i j] by auto
  2828 qed
  2829 
  2830 
  2831 subsubsection {* @{text "distinct"} and @{text remdups} *}
  2832 
  2833 lemma distinct_tl:
  2834   "distinct xs \<Longrightarrow> distinct (tl xs)"
  2835   by (cases xs) simp_all
  2836 
  2837 lemma distinct_append [simp]:
  2838 "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
  2839 by (induct xs) auto
  2840 
  2841 lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
  2842 by(induct xs) auto
  2843 
  2844 lemma set_remdups [simp]: "set (remdups xs) = set xs"
  2845 by (induct xs) (auto simp add: insert_absorb)
  2846 
  2847 lemma distinct_remdups [iff]: "distinct (remdups xs)"
  2848 by (induct xs) auto
  2849 
  2850 lemma distinct_remdups_id: "distinct xs ==> remdups xs = xs"
  2851 by (induct xs, auto)
  2852 
  2853 lemma remdups_id_iff_distinct [simp]: "remdups xs = xs \<longleftrightarrow> distinct xs"
  2854 by (metis distinct_remdups distinct_remdups_id)
  2855 
  2856 lemma finite_distinct_list: "finite A \<Longrightarrow> EX xs. set xs = A & distinct xs"
  2857 by (metis distinct_remdups finite_list set_remdups)
  2858 
  2859 lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
  2860 by (induct x, auto)
  2861 
  2862 lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
  2863 by (induct x, auto)
  2864 
  2865 lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
  2866 by (induct xs) auto
  2867 
  2868 lemma length_remdups_eq[iff]:
  2869   "(length (remdups xs) = length xs) = (remdups xs = xs)"
  2870 apply(induct xs)
  2871  apply auto
  2872 apply(subgoal_tac "length (remdups xs) <= length xs")
  2873  apply arith
  2874 apply(rule length_remdups_leq)
  2875 done
  2876 
  2877 lemma remdups_filter: "remdups(filter P xs) = filter P (remdups xs)"
  2878 apply(induct xs)
  2879 apply auto
  2880 done
  2881 
  2882 lemma distinct_map:
  2883   "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
  2884 by (induct xs) auto
  2885 
  2886 
  2887 lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
  2888 by (induct xs) auto
  2889 
  2890 lemma distinct_upt[simp]: "distinct[i..<j]"
  2891 by (induct j) auto
  2892 
  2893 lemma distinct_upto[simp]: "distinct[i..j]"
  2894 apply(induct i j rule:upto.induct)
  2895 apply(subst upto.simps)
  2896 apply(simp)
  2897 done
  2898 
  2899 lemma distinct_take[simp]: "distinct xs \<Longrightarrow> distinct (take i xs)"
  2900 apply(induct xs arbitrary: i)
  2901  apply simp
  2902 apply (case_tac i)
  2903  apply simp_all
  2904 apply(blast dest:in_set_takeD)
  2905 done
  2906 
  2907 lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)"
  2908 apply(induct xs arbitrary: i)
  2909  apply simp
  2910 apply (case_tac i)
  2911  apply simp_all
  2912 done
  2913 
  2914 lemma distinct_list_update:
  2915 assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
  2916 shows "distinct (xs[i:=a])"
  2917 proof (cases "i < length xs")
  2918   case True
  2919   with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
  2920     apply (drule_tac id_take_nth_drop) by simp
  2921   with d True show ?thesis
  2922     apply (simp add: upd_conv_take_nth_drop)
  2923     apply (drule subst [OF id_take_nth_drop]) apply assumption
  2924     apply simp apply (cases "a = xs!i") apply simp by blast
  2925 next
  2926   case False with d show ?thesis by auto
  2927 qed
  2928 
  2929 lemma distinct_concat:
  2930   assumes "distinct xs"
  2931   and "\<And> ys. ys \<in> set xs \<Longrightarrow> distinct ys"
  2932   and "\<And> ys zs. \<lbrakk> ys \<in> set xs ; zs \<in> set xs ; ys \<noteq> zs \<rbrakk> \<Longrightarrow> set ys \<inter> set zs = {}"
  2933   shows "distinct (concat xs)"
  2934   using assms by (induct xs) auto
  2935 
  2936 text {* It is best to avoid this indexed version of distinct, but
  2937 sometimes it is useful. *}
  2938 
  2939 lemma distinct_conv_nth:
  2940 "distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
  2941 apply (induct xs, simp, simp)
  2942 apply (rule iffI, clarsimp)
  2943  apply (case_tac i)
  2944 apply (case_tac j, simp)
  2945 apply (simp add: set_conv_nth)
  2946  apply (case_tac j)
  2947 apply (clarsimp simp add: set_conv_nth, simp)
  2948 apply (rule conjI)
  2949 (*TOO SLOW
  2950 apply (metis Zero_neq_Suc gr0_conv_Suc in_set_conv_nth lessI less_trans_Suc nth_Cons' nth_Cons_Suc)
  2951 *)
  2952  apply (clarsimp simp add: set_conv_nth)
  2953  apply (erule_tac x = 0 in allE, simp)
  2954  apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
  2955 (*TOO SLOW
  2956 apply (metis Suc_Suc_eq lessI less_trans_Suc nth_Cons_Suc)
  2957 *)
  2958 apply (erule_tac x = "Suc i" in allE, simp)
  2959 apply (erule_tac x = "Suc j" in allE, simp)
  2960 done
  2961 
  2962 lemma nth_eq_iff_index_eq:
  2963  "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
  2964 by(auto simp: distinct_conv_nth)
  2965 
  2966 lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
  2967 by (induct xs) auto
  2968 
  2969 lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
  2970 proof (induct xs)
  2971   case Nil thus ?case by simp
  2972 next
  2973   case (Cons x xs)
  2974   show ?case
  2975   proof (cases "x \<in> set xs")
  2976     case False with Cons show ?thesis by simp
  2977   next
  2978     case True with Cons.prems
  2979     have "card (set xs) = Suc (length xs)"
  2980       by (simp add: card_insert_if split: split_if_asm)
  2981     moreover have "card (set xs) \<le> length xs" by (rule card_length)
  2982     ultimately have False by simp
  2983     thus ?thesis ..
  2984   qed
  2985 qed
  2986 
  2987 lemma distinct_length_filter: "distinct xs \<Longrightarrow> length (filter P xs) = card ({x. P x} Int set xs)"
  2988 by (induct xs) (auto)
  2989 
  2990 lemma not_distinct_decomp: "~ distinct ws ==> EX xs ys zs y. ws = xs@[y]@ys@[y]@zs"
  2991 apply (induct n == "length ws" arbitrary:ws) apply simp
  2992 apply(case_tac ws) apply simp
  2993 apply (simp split:split_if_asm)
  2994 apply (metis Cons_eq_appendI eq_Nil_appendI split_list)
  2995 done
  2996 
  2997 lemma not_distinct_conv_prefix:
  2998   defines "dec as xs y ys \<equiv> y \<in> set xs \<and> distinct xs \<and> as = xs @ y # ys"
  2999   shows "\<not>distinct as \<longleftrightarrow> (\<exists>xs y ys. dec as xs y ys)" (is "?L = ?R")
  3000 proof
  3001   assume "?L" then show "?R"
  3002   proof (induct "length as" arbitrary: as rule: less_induct)
  3003     case less
  3004     obtain xs ys zs y where decomp: "as = (xs @ y # ys) @ y # zs"
  3005       using not_distinct_decomp[OF less.prems] by auto
  3006     show ?case
  3007     proof (cases "distinct (xs @ y # ys)")
  3008       case True
  3009       with decomp have "dec as (xs @ y # ys) y zs" by (simp add: dec_def)
  3010       then show ?thesis by blast
  3011     next
  3012       case False
  3013       with less decomp obtain xs' y' ys' where "dec (xs @ y # ys) xs' y' ys'"
  3014         by atomize_elim auto
  3015       with decomp have "dec as xs' y' (ys' @ y # zs)" by (simp add: dec_def)
  3016       then show ?thesis by blast
  3017     qed
  3018   qed
  3019 qed (auto simp: dec_def)
  3020 
  3021 lemma length_remdups_concat:
  3022   "length (remdups (concat xss)) = card (\<Union>xs\<in>set xss. set xs)"
  3023   by (simp add: distinct_card [symmetric])
  3024 
  3025 lemma length_remdups_card_conv: "length(remdups xs) = card(set xs)"
  3026 proof -
  3027   have xs: "concat[xs] = xs" by simp
  3028   from length_remdups_concat[of "[xs]"] show ?thesis unfolding xs by simp
  3029 qed
  3030 
  3031 lemma remdups_remdups:
  3032   "remdups (remdups xs) = remdups xs"
  3033   by (induct xs) simp_all
  3034 
  3035 lemma distinct_butlast:
  3036   assumes "distinct xs"
  3037   shows "distinct (butlast xs)"
  3038 proof (cases "xs = []")
  3039   case False
  3040     from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  3041     with `distinct xs` show ?thesis by simp
  3042 qed (auto)
  3043 
  3044 lemma remdups_map_remdups:
  3045   "remdups (map f (remdups xs)) = remdups (map f xs)"
  3046   by (induct xs) simp_all
  3047 
  3048 lemma distinct_zipI1:
  3049   assumes "distinct xs"
  3050   shows "distinct (zip xs ys)"
  3051 proof (rule zip_obtain_same_length)
  3052   fix xs' :: "'a list" and ys' :: "'b list" and n
  3053   assume "length xs' = length ys'"
  3054   assume "xs' = take n xs"
  3055   with assms have "distinct xs'" by simp
  3056   with `length xs' = length ys'` show "distinct (zip xs' ys')"
  3057     by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
  3058 qed
  3059 
  3060 lemma distinct_zipI2:
  3061   assumes "distinct ys"
  3062   shows "distinct (zip xs ys)"
  3063 proof (rule zip_obtain_same_length)
  3064   fix xs' :: "'b list" and ys' :: "'a list" and n
  3065   assume "length xs' = length ys'"
  3066   assume "ys' = take n ys"
  3067   with assms have "distinct ys'" by simp
  3068   with `length xs' = length ys'` show "distinct (zip xs' ys')"
  3069     by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
  3070 qed
  3071 
  3072 lemma set_take_disj_set_drop_if_distinct:
  3073   "distinct vs \<Longrightarrow> i \<le> j \<Longrightarrow> set (take i vs) \<inter> set (drop j vs) = {}"
  3074 by (auto simp: in_set_conv_nth distinct_conv_nth)
  3075 
  3076 (* The next two lemmas help Sledgehammer. *)
  3077 
  3078 lemma distinct_singleton: "distinct [x]" by simp
  3079 
  3080 lemma distinct_length_2_or_more:
  3081 "distinct (a # b # xs) \<longleftrightarrow> (a \<noteq> b \<and> distinct (a # xs) \<and> distinct (b # xs))"
  3082 by (metis distinct.simps(2) hd.simps hd_in_set list.simps(2) set_ConsD set_rev_mp set_subset_Cons)
  3083 
  3084 subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
  3085 
  3086 lemma (in monoid_add) listsum_simps [simp]:
  3087   "listsum [] = 0"
  3088   "listsum (x # xs) = x + listsum xs"
  3089   by (simp_all add: listsum_def)
  3090 
  3091 lemma (in monoid_add) listsum_append [simp]:
  3092   "listsum (xs @ ys) = listsum xs + listsum ys"
  3093   by (induct xs) (simp_all add: add.assoc)
  3094 
  3095 lemma (in comm_monoid_add) listsum_rev [simp]:
  3096   "listsum (rev xs) = listsum xs"
  3097   by (simp add: listsum_def foldr_fold fold_rev fun_eq_iff add_ac)
  3098 
  3099 lemma (in monoid_add) fold_plus_listsum_rev:
  3100   "fold plus xs = plus (listsum (rev xs))"
  3101 proof
  3102   fix x
  3103   have "fold plus xs x = fold plus xs (x + 0)" by simp
  3104   also have "\<dots> = fold plus (x # xs) 0" by simp
  3105   also have "\<dots> = foldr plus (rev xs @ [x]) 0" by (simp add: foldr_conv_fold)
  3106   also have "\<dots> = listsum (rev xs @ [x])" by (simp add: listsum_def)
  3107   also have "\<dots> = listsum (rev xs) + listsum [x]" by simp
  3108   finally show "fold plus xs x = listsum (rev xs) + x" by simp
  3109 qed
  3110 
  3111 text{* Some syntactic sugar for summing a function over a list: *}
  3112 
  3113 syntax
  3114   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
  3115 syntax (xsymbols)
  3116   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
  3117 syntax (HTML output)
  3118   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
  3119 
  3120 translations -- {* Beware of argument permutation! *}
  3121   "SUM x<-xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  3122   "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  3123 
  3124 lemma (in comm_monoid_add) listsum_map_remove1:
  3125   "x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))"
  3126   by (induct xs) (auto simp add: ac_simps)
  3127 
  3128 lemma (in monoid_add) list_size_conv_listsum:
  3129   "list_size f xs = listsum (map f xs) + size xs"
  3130   by (induct xs) auto
  3131 
  3132 lemma (in monoid_add) length_concat:
  3133   "length (concat xss) = listsum (map length xss)"
  3134   by (induct xss) simp_all
  3135 
  3136 lemma (in monoid_add) listsum_map_filter:
  3137   assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
  3138   shows "listsum (map f (filter P xs)) = listsum (map f xs)"
  3139   using assms by (induct xs) auto
  3140 
  3141 lemma (in monoid_add) distinct_listsum_conv_Setsum:
  3142   "distinct xs \<Longrightarrow> listsum xs = Setsum (set xs)"
  3143   by (induct xs) simp_all
  3144 
  3145 lemma listsum_eq_0_nat_iff_nat [simp]:
  3146   "listsum ns = (0::nat) \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
  3147   by (induct ns) simp_all
  3148 
  3149 lemma member_le_listsum_nat:
  3150   "(n :: nat) \<in> set ns \<Longrightarrow> n \<le> listsum ns"
  3151   by (induct ns) auto
  3152 
  3153 lemma elem_le_listsum_nat:
  3154   "k < size ns \<Longrightarrow> ns ! k \<le> listsum (ns::nat list)"
  3155   by (rule member_le_listsum_nat) simp
  3156 
  3157 lemma listsum_update_nat:
  3158   "k < size ns \<Longrightarrow> listsum (ns[k := (n::nat)]) = listsum ns + n - ns ! k"
  3159 apply(induct ns arbitrary:k)
  3160  apply (auto split:nat.split)
  3161 apply(drule elem_le_listsum_nat)
  3162 apply arith
  3163 done
  3164 
  3165 lemma (in monoid_add) listsum_triv:
  3166   "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
  3167   by (induct xs) (simp_all add: left_distrib)
  3168 
  3169 lemma (in monoid_add) listsum_0 [simp]:
  3170   "(\<Sum>x\<leftarrow>xs. 0) = 0"
  3171   by (induct xs) (simp_all add: left_distrib)
  3172 
  3173 text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
  3174 lemma (in ab_group_add) uminus_listsum_map:
  3175   "- listsum (map f xs) = listsum (map (uminus \<circ> f) xs)"
  3176   by (induct xs) simp_all
  3177 
  3178 lemma (in comm_monoid_add) listsum_addf:
  3179   "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
  3180   by (induct xs) (simp_all add: algebra_simps)
  3181 
  3182 lemma (in ab_group_add) listsum_subtractf:
  3183   "(\<Sum>x\<leftarrow>xs. f x - g x) = listsum (map f xs) - listsum (map g xs)"
  3184   by (induct xs) (simp_all add: algebra_simps)
  3185 
  3186 lemma (in semiring_0) listsum_const_mult:
  3187   "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
  3188   by (induct xs) (simp_all add: algebra_simps)
  3189 
  3190 lemma (in semiring_0) listsum_mult_const:
  3191   "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
  3192   by (induct xs) (simp_all add: algebra_simps)
  3193 
  3194 lemma (in ordered_ab_group_add_abs) listsum_abs:
  3195   "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
  3196   by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])
  3197 
  3198 lemma listsum_mono:
  3199   fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}"
  3200   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)"
  3201   by (induct xs) (simp, simp add: add_mono)
  3202 
  3203 lemma (in monoid_add) listsum_distinct_conv_setsum_set:
  3204   "distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)"
  3205   by (induct xs) simp_all
  3206 
  3207 lemma (in monoid_add) interv_listsum_conv_setsum_set_nat:
  3208   "listsum (map f [m..<n]) = setsum f (set [m..<n])"
  3209   by (simp add: listsum_distinct_conv_setsum_set)
  3210 
  3211 lemma (in monoid_add) interv_listsum_conv_setsum_set_int:
  3212   "listsum (map f [k..l]) = setsum f (set [k..l])"
  3213   by (simp add: listsum_distinct_conv_setsum_set)
  3214 
  3215 text {* General equivalence between @{const listsum} and @{const setsum} *}
  3216 lemma (in monoid_add) listsum_setsum_nth:
  3217   "listsum xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
  3218   using interv_listsum_conv_setsum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth)
  3219 
  3220 
  3221 subsubsection {* @{const insert} *}
  3222 
  3223 lemma in_set_insert [simp]:
  3224   "x \<in> set xs \<Longrightarrow> List.insert x xs = xs"
  3225   by (simp add: List.insert_def)
  3226 
  3227 lemma not_in_set_insert [simp]:
  3228   "x \<notin> set xs \<Longrightarrow> List.insert x xs = x # xs"
  3229   by (simp add: List.insert_def)
  3230 
  3231 lemma insert_Nil [simp]:
  3232   "List.insert x [] = [x]"
  3233   by simp
  3234 
  3235 lemma set_insert [simp]:
  3236   "set (List.insert x xs) = insert x (set xs)"
  3237   by (auto simp add: List.insert_def)
  3238 
  3239 lemma distinct_insert [simp]:
  3240   "distinct xs \<Longrightarrow> distinct (List.insert x xs)"
  3241   by (simp add: List.insert_def)
  3242 
  3243 lemma insert_remdups:
  3244   "List.insert x (remdups xs) = remdups (List.insert x xs)"
  3245   by (simp add: List.insert_def)
  3246 
  3247 
  3248 subsubsection {* @{const List.find} *}
  3249 
  3250 lemma find_None_iff: "List.find P xs = None \<longleftrightarrow> \<not> (\<exists>x. x \<in> set xs \<and> P x)"
  3251 proof (induction xs)
  3252   case Nil thus ?case by simp
  3253 next
  3254   case (Cons x xs) thus ?case by (fastforce split: if_splits)
  3255 qed
  3256 
  3257 lemma find_Some_iff:
  3258   "List.find P xs = Some x \<longleftrightarrow>
  3259   (\<exists>i<length xs. P (xs!i) \<and> x = xs!i \<and> (\<forall>j<i. \<not> P (xs!j)))"
  3260 proof (induction xs)
  3261   case Nil thus ?case by simp
  3262 next
  3263   case (Cons x xs) thus ?case
  3264     by(auto simp: nth_Cons' split: if_splits)
  3265       (metis One_nat_def diff_Suc_1 less_Suc_eq_0_disj)
  3266 qed
  3267 
  3268 lemma find_cong[fundef_cong]:
  3269   assumes "xs = ys" and "\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x" 
  3270   shows "List.find P xs = List.find Q ys"
  3271 proof (cases "List.find P xs")
  3272   case None thus ?thesis by (metis find_None_iff assms)
  3273 next
  3274   case (Some x)
  3275   hence "List.find Q ys = Some x" using assms
  3276     by (auto simp add: find_Some_iff)
  3277   thus ?thesis using Some by auto
  3278 qed
  3279 
  3280 
  3281 subsubsection {* @{const remove1} *}
  3282 
  3283 lemma remove1_append:
  3284   "remove1 x (xs @ ys) =
  3285   (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
  3286 by (induct xs) auto
  3287 
  3288 lemma remove1_commute: "remove1 x (remove1 y zs) = remove1 y (remove1 x zs)"
  3289 by (induct zs) auto
  3290 
  3291 lemma in_set_remove1[simp]:
  3292   "a \<noteq> b \<Longrightarrow> a : set(remove1 b xs) = (a : set xs)"
  3293 apply (induct xs)
  3294 apply auto
  3295 done
  3296 
  3297 lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
  3298 apply(induct xs)
  3299  apply simp
  3300 apply simp
  3301 apply blast
  3302 done
  3303 
  3304 lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
  3305 apply(induct xs)
  3306  apply simp
  3307 apply simp
  3308 apply blast
  3309 done
  3310 
  3311 lemma length_remove1:
  3312   "length(remove1 x xs) = (if x : set xs then length xs - 1 else length xs)"
  3313 apply (induct xs)
  3314  apply (auto dest!:length_pos_if_in_set)
  3315 done
  3316 
  3317 lemma remove1_filter_not[simp]:
  3318   "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
  3319 by(induct xs) auto
  3320 
  3321 lemma filter_remove1:
  3322   "filter Q (remove1 x xs) = remove1 x (filter Q xs)"
  3323 by (induct xs) auto
  3324 
  3325 lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
  3326 apply(insert set_remove1_subset)
  3327 apply fast
  3328 done
  3329 
  3330 lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
  3331 by (induct xs) simp_all
  3332 
  3333 lemma remove1_remdups:
  3334   "distinct xs \<Longrightarrow> remove1 x (remdups xs) = remdups (remove1 x xs)"
  3335   by (induct xs) simp_all
  3336 
  3337 lemma remove1_idem:
  3338   assumes "x \<notin> set xs"
  3339   shows "remove1 x xs = xs"
  3340   using assms by (induct xs) simp_all
  3341 
  3342 
  3343 subsubsection {* @{text removeAll} *}
  3344 
  3345 lemma removeAll_filter_not_eq:
  3346   "removeAll x = filter (\<lambda>y. x \<noteq> y)"
  3347 proof
  3348   fix xs
  3349   show "removeAll x xs = filter (\<lambda>y. x \<noteq> y) xs"
  3350     by (induct xs) auto
  3351 qed
  3352 
  3353 lemma removeAll_append[simp]:
  3354   "removeAll x (xs @ ys) = removeAll x xs @ removeAll x ys"
  3355 by (induct xs) auto
  3356 
  3357 lemma set_removeAll[simp]: "set(removeAll x xs) = set xs - {x}"
  3358 by (induct xs) auto
  3359 
  3360 lemma removeAll_id[simp]: "x \<notin> set xs \<Longrightarrow> removeAll x xs = xs"
  3361 by (induct xs) auto
  3362 
  3363 (* Needs count:: 'a \<Rightarrow> 'a list \<Rightarrow> nat
  3364 lemma length_removeAll:
  3365   "length(removeAll x xs) = length xs - count x xs"
  3366 *)
  3367 
  3368 lemma removeAll_filter_not[simp]:
  3369   "\<not> P x \<Longrightarrow> removeAll x (filter P xs) = filter P xs"
  3370 by(induct xs) auto
  3371 
  3372 lemma distinct_removeAll:
  3373   "distinct xs \<Longrightarrow> distinct (removeAll x xs)"
  3374   by (simp add: removeAll_filter_not_eq)
  3375 
  3376 lemma distinct_remove1_removeAll:
  3377   "distinct xs ==> remove1 x xs = removeAll x xs"
  3378 by (induct xs) simp_all
  3379 
  3380 lemma map_removeAll_inj_on: "inj_on f (insert x (set xs)) \<Longrightarrow>
  3381   map f (removeAll x xs) = removeAll (f x) (map f xs)"
  3382 by (induct xs) (simp_all add:inj_on_def)
  3383 
  3384 lemma map_removeAll_inj: "inj f \<Longrightarrow>
  3385   map f (removeAll x xs) = removeAll (f x) (map f xs)"
  3386 by(metis map_removeAll_inj_on subset_inj_on subset_UNIV)
  3387 
  3388 
  3389 subsubsection {* @{text replicate} *}
  3390 
  3391 lemma length_replicate [simp]: "length (replicate n x) = n"
  3392 by (induct n) auto
  3393 
  3394 lemma Ex_list_of_length: "\<exists>xs. length xs = n"
  3395 by (rule exI[of _ "replicate n undefined"]) simp
  3396 
  3397 lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
  3398 by (induct n) auto
  3399 
  3400 lemma map_replicate_const:
  3401   "map (\<lambda> x. k) lst = replicate (length lst) k"
  3402   by (induct lst) auto
  3403 
  3404 lemma replicate_app_Cons_same:
  3405 "(replicate n x) @ (x # xs) = x # replicate n x @ xs"
  3406 by (induct n) auto
  3407 
  3408 lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
  3409 apply (induct n, simp)
  3410 apply (simp add: replicate_app_Cons_same)
  3411 done
  3412 
  3413 lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
  3414 by (induct n) auto
  3415 
  3416 text{* Courtesy of Matthias Daum: *}
  3417 lemma append_replicate_commute:
  3418   "replicate n x @ replicate k x = replicate k x @ replicate n x"
  3419 apply (simp add: replicate_add [THEN sym])
  3420 apply (simp add: add_commute)
  3421 done
  3422 
  3423 text{* Courtesy of Andreas Lochbihler: *}
  3424 lemma filter_replicate:
  3425   "filter P (replicate n x) = (if P x then replicate n x else [])"
  3426 by(induct n) auto
  3427 
  3428 lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
  3429 by (induct n) auto
  3430 
  3431 lemma tl_replicate [simp]: "tl (replicate n x) = replicate (n - 1) x"
  3432 by (induct n) auto
  3433 
  3434 lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
  3435 by (atomize (full), induct n) auto
  3436 
  3437 lemma nth_replicate[simp]: "i < n ==> (replicate n x)!i = x"
  3438 apply (induct n arbitrary: i, simp)
  3439 apply (simp add: nth_Cons split: nat.split)
  3440 done
  3441 
  3442 text{* Courtesy of Matthias Daum (2 lemmas): *}
  3443 lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
  3444 apply (case_tac "k \<le> i")
  3445  apply  (simp add: min_def)
  3446 apply (drule not_leE)
  3447 apply (simp add: min_def)
  3448 apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
  3449  apply  simp
  3450 apply (simp add: replicate_add [symmetric])
  3451 done
  3452 
  3453 lemma drop_replicate[simp]: "drop i (replicate k x) = replicate (k-i) x"
  3454 apply (induct k arbitrary: i)
  3455  apply simp
  3456 apply clarsimp
  3457 apply (case_tac i)
  3458  apply simp
  3459 apply clarsimp
  3460 done
  3461 
  3462 
  3463 lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
  3464 by (induct n) auto
  3465 
  3466 lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
  3467 by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
  3468 
  3469 lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
  3470 by auto
  3471 
  3472 lemma in_set_replicate[simp]: "(x : set (replicate n y)) = (x = y & n \<noteq> 0)"
  3473 by (simp add: set_replicate_conv_if)
  3474 
  3475 lemma Ball_set_replicate[simp]:
  3476   "(ALL x : set(replicate n a). P x) = (P a | n=0)"
  3477 by(simp add: set_replicate_conv_if)
  3478 
  3479 lemma Bex_set_replicate[simp]:
  3480   "(EX x : set(replicate n a). P x) = (P a & n\<noteq>0)"
  3481 by(simp add: set_replicate_conv_if)
  3482 
  3483 lemma replicate_append_same:
  3484   "replicate i x @ [x] = x # replicate i x"
  3485   by (induct i) simp_all
  3486 
  3487 lemma map_replicate_trivial:
  3488   "map (\<lambda>i. x) [0..<i] = replicate i x"
  3489   by (induct i) (simp_all add: replicate_append_same)
  3490 
  3491 lemma concat_replicate_trivial[simp]:
  3492   "concat (replicate i []) = []"
  3493   by (induct i) (auto simp add: map_replicate_const)
  3494 
  3495 lemma replicate_empty[simp]: "(replicate n x = []) \<longleftrightarrow> n=0"
  3496 by (induct n) auto
  3497 
  3498 lemma empty_replicate[simp]: "([] = replicate n x) \<longleftrightarrow> n=0"
  3499 by (induct n) auto
  3500 
  3501 lemma replicate_eq_replicate[simp]:
  3502   "(replicate m x = replicate n y) \<longleftrightarrow> (m=n & (m\<noteq>0 \<longrightarrow> x=y))"
  3503 apply(induct m arbitrary: n)
  3504  apply simp
  3505 apply(induct_tac n)
  3506 apply auto
  3507 done
  3508 
  3509 lemma replicate_length_filter:
  3510   "replicate (length (filter (\<lambda>y. x = y) xs)) x = filter (\<lambda>y. x = y) xs"
  3511   by (induct xs) auto
  3512 
  3513 lemma comm_append_are_replicate:
  3514   fixes xs ys :: "'a list"
  3515   assumes "xs \<noteq> []" "ys \<noteq> []"
  3516   assumes "xs @ ys = ys @ xs"
  3517   shows "\<exists>m n zs. concat (replicate m zs) = xs \<and> concat (replicate n zs) = ys"
  3518   using assms
  3519 proof (induct "length (xs @ ys)" arbitrary: xs ys rule: less_induct)
  3520   case less
  3521 
  3522   def xs' \<equiv> "if (length xs \<le> length ys) then xs else ys"
  3523     and ys' \<equiv> "if (length xs \<le> length ys) then ys else xs"
  3524   then have
  3525     prems': "length xs' \<le> length ys'"
  3526             "xs' @ ys' = ys' @ xs'"
  3527       and "xs' \<noteq> []"
  3528       and len: "length (xs @ ys) = length (xs' @ ys')"
  3529     using less by (auto intro: less.hyps)
  3530 
  3531   from prems'
  3532   obtain ws where "ys' = xs' @ ws"
  3533     by (auto simp: append_eq_append_conv2)
  3534 
  3535   have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ys'"
  3536   proof (cases "ws = []")
  3537     case True
  3538     then have "concat (replicate 1 xs') = xs'"
  3539       and "concat (replicate 1 xs') = ys'"
  3540       using `ys' = xs' @ ws` by auto
  3541     then show ?thesis by blast
  3542   next
  3543     case False
  3544     from `ys' = xs' @ ws` and `xs' @ ys' = ys' @ xs'`
  3545     have "xs' @ ws = ws @ xs'" by simp
  3546     then have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ws"
  3547       using False and `xs' \<noteq> []` and `ys' = xs' @ ws` and len
  3548       by (intro less.hyps) auto
  3549     then obtain m n zs where "concat (replicate m zs) = xs'"
  3550       and "concat (replicate n zs) = ws" by blast
  3551     moreover
  3552     then have "concat (replicate (m + n) zs) = ys'"
  3553       using `ys' = xs' @ ws`
  3554       by (simp add: replicate_add)
  3555     ultimately
  3556     show ?thesis by blast
  3557   qed
  3558   then show ?case
  3559     using xs'_def ys'_def by metis
  3560 qed
  3561 
  3562 lemma comm_append_is_replicate:
  3563   fixes xs ys :: "'a list"
  3564   assumes "xs \<noteq> []" "ys \<noteq> []"
  3565   assumes "xs @ ys = ys @ xs"
  3566   shows "\<exists>n zs. n > 1 \<and> concat (replicate n zs) = xs @ ys"
  3567 
  3568 proof -
  3569   obtain m n zs where "concat (replicate m zs) = xs"
  3570     and "concat (replicate n zs) = ys"
  3571     using assms by (metis comm_append_are_replicate)
  3572   then have "m + n > 1" and "concat (replicate (m+n) zs) = xs @ ys"
  3573     using `xs \<noteq> []` and `ys \<noteq> []`
  3574     by (auto simp: replicate_add)
  3575   then show ?thesis by blast
  3576 qed
  3577 
  3578 
  3579 subsubsection{*@{text rotate1} and @{text rotate}*}
  3580 
  3581 lemma rotate0[simp]: "rotate 0 = id"
  3582 by(simp add:rotate_def)
  3583 
  3584 lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
  3585 by(simp add:rotate_def)
  3586 
  3587 lemma rotate_add:
  3588   "rotate (m+n) = rotate m o rotate n"
  3589 by(simp add:rotate_def funpow_add)
  3590 
  3591 lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
  3592 by(simp add:rotate_add)
  3593 
  3594 lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
  3595 by(simp add:rotate_def funpow_swap1)
  3596 
  3597 lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
  3598 by(cases xs) simp_all
  3599 
  3600 lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
  3601 apply(induct n)
  3602  apply simp
  3603 apply (simp add:rotate_def)
  3604 done
  3605 
  3606 lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
  3607 by (cases xs) simp_all
  3608 
  3609 lemma rotate_drop_take:
  3610   "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
  3611 apply(induct n)
  3612  apply simp
  3613 apply(simp add:rotate_def)
  3614 apply(cases "xs = []")
  3615  apply (simp)
  3616 apply(case_tac "n mod length xs = 0")
  3617  apply(simp add:mod_Suc)
  3618  apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
  3619 apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
  3620                 take_hd_drop linorder_not_le)
  3621 done
  3622 
  3623 lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
  3624 by(simp add:rotate_drop_take)
  3625 
  3626 lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
  3627 by(simp add:rotate_drop_take)
  3628 
  3629 lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
  3630 by (cases xs) simp_all
  3631 
  3632 lemma length_rotate[simp]: "length(rotate n xs) = length xs"
  3633 by (induct n arbitrary: xs) (simp_all add:rotate_def)
  3634 
  3635 lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
  3636 by (cases xs) auto
  3637 
  3638 lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
  3639 by (induct n) (simp_all add:rotate_def)
  3640 
  3641 lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
  3642 by(simp add:rotate_drop_take take_map drop_map)
  3643 
  3644 lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
  3645 by (cases xs) auto
  3646 
  3647 lemma set_rotate[simp]: "set(rotate n xs) = set xs"
  3648 by (induct n) (simp_all add:rotate_def)
  3649 
  3650 lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
  3651 by (cases xs) auto
  3652 
  3653 lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
  3654 by (induct n) (simp_all add:rotate_def)
  3655 
  3656 lemma rotate_rev:
  3657   "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
  3658 apply(simp add:rotate_drop_take rev_drop rev_take)
  3659 apply(cases "length xs = 0")
  3660  apply simp
  3661 apply(cases "n mod length xs = 0")
  3662  apply simp
  3663 apply(simp add:rotate_drop_take rev_drop rev_take)
  3664 done
  3665 
  3666 lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
  3667 apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
  3668 apply(subgoal_tac "length xs \<noteq> 0")
  3669  prefer 2 apply simp
  3670 using mod_less_divisor[of "length xs" n] by arith
  3671 
  3672 
  3673 subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
  3674 
  3675 lemma sublist_empty [simp]: "sublist xs {} = []"
  3676 by (auto simp add: sublist_def)
  3677 
  3678 lemma sublist_nil [simp]: "sublist [] A = []"
  3679 by (auto simp add: sublist_def)
  3680 
  3681 lemma length_sublist:
  3682   "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
  3683 by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
  3684 
  3685 lemma sublist_shift_lemma_Suc:
  3686   "map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
  3687    map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
  3688 apply(induct xs arbitrary: "is")
  3689  apply simp
  3690 apply (case_tac "is")
  3691  apply simp
  3692 apply simp
  3693 done
  3694 
  3695 lemma sublist_shift_lemma:
  3696      "map fst [p<-zip xs [i..<i + length xs] . snd p : A] =
  3697       map fst [p<-zip xs [0..<length xs] . snd p + i : A]"
  3698 by (induct xs rule: rev_induct) (simp_all add: add_commute)
  3699 
  3700 lemma sublist_append:
  3701      "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
  3702 apply (unfold sublist_def)
  3703 apply (induct l' rule: rev_induct, simp)
  3704 apply (simp add: upt_add_eq_append[of 0] sublist_shift_lemma)
  3705 apply (simp add: add_commute)
  3706 done
  3707 
  3708 lemma sublist_Cons:
  3709 "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
  3710 apply (induct l rule: rev_induct)
  3711  apply (simp add: sublist_def)
  3712 apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
  3713 done
  3714 
  3715 lemma set_sublist: "set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
  3716 apply(induct xs arbitrary: I)
  3717 apply(auto simp: sublist_Cons nth_Cons split:nat.split dest!: gr0_implies_Suc)
  3718 done
  3719 
  3720 lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
  3721 by(auto simp add:set_sublist)
  3722 
  3723 lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
  3724 by(auto simp add:set_sublist)
  3725 
  3726 lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
  3727 by(auto simp add:set_sublist)
  3728 
  3729 lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
  3730 by (simp add: sublist_Cons)
  3731 
  3732 
  3733 lemma distinct_sublistI[simp]: "distinct xs \<Longrightarrow> distinct(sublist xs I)"
  3734 apply(induct xs arbitrary: I)
  3735  apply simp
  3736 apply(auto simp add:sublist_Cons)
  3737 done
  3738 
  3739 
  3740 lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
  3741 apply (induct l rule: rev_induct, simp)
  3742 apply (simp split: nat_diff_split add: sublist_append)
  3743 done
  3744 
  3745 lemma filter_in_sublist:
  3746  "distinct xs \<Longrightarrow> filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
  3747 proof (induct xs arbitrary: s)
  3748   case Nil thus ?case by simp
  3749 next
  3750   case (Cons a xs)
  3751   moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
  3752   ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
  3753 qed
  3754 
  3755 
  3756 subsubsection {* @{const splice} *}
  3757 
  3758 lemma splice_Nil2 [simp, code]: "splice xs [] = xs"
  3759 by (cases xs) simp_all
  3760 
  3761 declare splice.simps(1,3)[code]
  3762 declare splice.simps(2)[simp del]
  3763 
  3764 lemma length_splice[simp]: "length(splice xs ys) = length xs + length ys"
  3765 by (induct xs ys rule: splice.induct) auto
  3766 
  3767 
  3768 subsubsection {* Transpose *}
  3769 
  3770 function transpose where
  3771 "transpose []             = []" |
  3772 "transpose ([]     # xss) = transpose xss" |
  3773 "transpose ((x#xs) # xss) =
  3774   (x # [h. (h#t) \<leftarrow> xss]) # transpose (xs # [t. (h#t) \<leftarrow> xss])"
  3775 by pat_completeness auto
  3776 
  3777 lemma transpose_aux_filter_head:
  3778   "concat (map (list_case [] (\<lambda>h t. [h])) xss) =
  3779   map (\<lambda>xs. hd xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  3780   by (induct xss) (auto split: list.split)
  3781 
  3782 lemma transpose_aux_filter_tail:
  3783   "concat (map (list_case [] (\<lambda>h t. [t])) xss) =
  3784   map (\<lambda>xs. tl xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  3785   by (induct xss) (auto split: list.split)
  3786 
  3787 lemma transpose_aux_max:
  3788   "max (Suc (length xs)) (foldr (\<lambda>xs. max (length xs)) xss 0) =
  3789   Suc (max (length xs) (foldr (\<lambda>x. max (length x - Suc 0)) [ys\<leftarrow>xss . ys\<noteq>[]] 0))"
  3790   (is "max _ ?foldB = Suc (max _ ?foldA)")
  3791 proof (cases "[ys\<leftarrow>xss . ys\<noteq>[]] = []")
  3792   case True
  3793   hence "foldr (\<lambda>xs. max (length xs)) xss 0 = 0"
  3794   proof (induct xss)
  3795     case (Cons x xs)
  3796     moreover hence "x = []" by (cases x) auto
  3797     ultimately show ?case by auto
  3798   qed simp
  3799   thus ?thesis using True by simp
  3800 next
  3801   case False
  3802 
  3803   have foldA: "?foldA = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0 - 1"
  3804     by (induct xss) auto
  3805   have foldB: "?foldB = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0"
  3806     by (induct xss) auto
  3807 
  3808   have "0 < ?foldB"
  3809   proof -
  3810     from False
  3811     obtain z zs where zs: "[ys\<leftarrow>xss . ys \<noteq> []] = z#zs" by (auto simp: neq_Nil_conv)
  3812     hence "z \<in> set ([ys\<leftarrow>xss . ys \<noteq> []])" by auto
  3813     hence "z \<noteq> []" by auto
  3814     thus ?thesis
  3815       unfolding foldB zs
  3816       by (auto simp: max_def intro: less_le_trans)
  3817   qed
  3818   thus ?thesis
  3819     unfolding foldA foldB max_Suc_Suc[symmetric]
  3820     by simp
  3821 qed
  3822 
  3823 termination transpose
  3824   by (relation "measure (\<lambda>xs. foldr (\<lambda>xs. max (length xs)) xs 0 + length xs)")
  3825      (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max less_Suc_eq_le)
  3826 
  3827 lemma transpose_empty: "(transpose xs = []) \<longleftrightarrow> (\<forall>x \<in> set xs. x = [])"
  3828   by (induct rule: transpose.induct) simp_all
  3829 
  3830 lemma length_transpose:
  3831   fixes xs :: "'a list list"
  3832   shows "length (transpose xs) = foldr (\<lambda>xs. max (length xs)) xs 0"
  3833   by (induct rule: transpose.induct)
  3834     (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max
  3835                 max_Suc_Suc[symmetric] simp del: max_Suc_Suc)
  3836 
  3837 lemma nth_transpose:
  3838   fixes xs :: "'a list list"
  3839   assumes "i < length (transpose xs)"
  3840   shows "transpose xs ! i = map (\<lambda>xs. xs ! i) [ys \<leftarrow> xs. i < length ys]"
  3841 using assms proof (induct arbitrary: i rule: transpose.induct)
  3842   case (3 x xs xss)
  3843   def XS == "(x # xs) # xss"
  3844   hence [simp]: "XS \<noteq> []" by auto
  3845   thus ?case
  3846   proof (cases i)
  3847     case 0
  3848     thus ?thesis by (simp add: transpose_aux_filter_head hd_conv_nth)
  3849   next
  3850     case (Suc j)
  3851     have *: "\<And>xss. xs # map tl xss = map tl ((x#xs)#xss)" by simp
  3852     have **: "\<And>xss. (x#xs) # filter (\<lambda>ys. ys \<noteq> []) xss = filter (\<lambda>ys. ys \<noteq> []) ((x#xs)#xss)" by simp
  3853     { fix x have "Suc j < length x \<longleftrightarrow> x \<noteq> [] \<and> j < length x - Suc 0"
  3854       by (cases x) simp_all
  3855     } note *** = this
  3856 
  3857     have j_less: "j < length (transpose (xs # concat (map (list_case [] (\<lambda>h t. [t])) xss)))"
  3858       using "3.prems" by (simp add: transpose_aux_filter_tail length_transpose Suc)
  3859 
  3860     show ?thesis
  3861       unfolding transpose.simps `i = Suc j` nth_Cons_Suc "3.hyps"[OF j_less]
  3862       apply (auto simp: transpose_aux_filter_tail filter_map comp_def length_transpose * ** *** XS_def[symmetric])
  3863       apply (rule_tac y=x in list.exhaust)
  3864       by auto
  3865   qed
  3866 qed simp_all
  3867 
  3868 lemma transpose_map_map:
  3869   "transpose (map (map f) xs) = map (map f) (transpose xs)"
  3870 proof (rule nth_equalityI, safe)
  3871   have [simp]: "length (transpose (map (map f) xs)) = length (transpose xs)"
  3872     by (simp add: length_transpose foldr_map comp_def)
  3873   show "length (transpose (map (map f) xs)) = length (map (map f) (transpose xs))" by simp
  3874 
  3875   fix i assume "i < length (transpose (map (map f) xs))"
  3876   thus "transpose (map (map f) xs) ! i = map (map f) (transpose xs) ! i"
  3877     by (simp add: nth_transpose filter_map comp_def)
  3878 qed
  3879 
  3880 
  3881 subsubsection {* (In)finiteness *}
  3882 
  3883 lemma finite_maxlen:
  3884   "finite (M::'a list set) ==> EX n. ALL s:M. size s < n"
  3885 proof (induct rule: finite.induct)
  3886   case emptyI show ?case by simp
  3887 next
  3888   case (insertI M xs)
  3889   then obtain n where "\<forall>s\<in>M. length s < n" by blast
  3890   hence "ALL s:insert xs M. size s < max n (size xs) + 1" by auto
  3891   thus ?case ..
  3892 qed
  3893 
  3894 lemma lists_length_Suc_eq:
  3895   "{xs. set xs \<subseteq> A \<and> length xs = Suc n} =
  3896     (\<lambda>(xs, n). n#xs) ` ({xs. set xs \<subseteq> A \<and> length xs = n} \<times> A)"
  3897   by (auto simp: length_Suc_conv)
  3898 
  3899 lemma
  3900   assumes "finite A"
  3901   shows finite_lists_length_eq: "finite {xs. set xs \<subseteq> A \<and> length xs = n}"
  3902   and card_lists_length_eq: "card {xs. set xs \<subseteq> A \<and> length xs = n} = (card A)^n"
  3903   using `finite A`
  3904   by (induct n)
  3905      (auto simp: card_image inj_split_Cons lists_length_Suc_eq cong: conj_cong)
  3906 
  3907 lemma finite_lists_length_le:
  3908   assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs \<le> n}"
  3909  (is "finite ?S")
  3910 proof-
  3911   have "?S = (\<Union>n\<in>{0..n}. {xs. set xs \<subseteq> A \<and> length xs = n})" by auto
  3912   thus ?thesis by (auto intro: finite_lists_length_eq[OF `finite A`])
  3913 qed
  3914 
  3915 lemma card_lists_length_le:
  3916   assumes "finite A" shows "card {xs. set xs \<subseteq> A \<and> length xs \<le> n} = (\<Sum>i\<le>n. card A^i)"
  3917 proof -
  3918   have "(\<Sum>i\<le>n. card A^i) = card (\<Union>i\<le>n. {xs. set xs \<subseteq> A \<and> length xs = i})"
  3919     using `finite A`
  3920     by (subst card_UN_disjoint)
  3921        (auto simp add: card_lists_length_eq finite_lists_length_eq)
  3922   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}"
  3923     by auto
  3924   finally show ?thesis by simp
  3925 qed
  3926 
  3927 lemma card_lists_distinct_length_eq:
  3928   assumes "k < card A"
  3929   shows "card {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A} = \<Prod>{card A - k + 1 .. card A}"
  3930 using assms
  3931 proof (induct k)
  3932   case 0
  3933   then have "{xs. length xs = 0 \<and> distinct xs \<and> set xs \<subseteq> A} = {[]}" by auto
  3934   then show ?case by simp
  3935 next
  3936   case (Suc k)
  3937   let "?k_list" = "\<lambda>k xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A"
  3938   have inj_Cons: "\<And>A. inj_on (\<lambda>(xs, n). n # xs) A"  by (rule inj_onI) auto
  3939 
  3940   from Suc have "k < card A" by simp
  3941   moreover have "finite A" using assms by (simp add: card_ge_0_finite)
  3942   moreover have "finite {xs. ?k_list k xs}"
  3943     using finite_lists_length_eq[OF `finite A`, of k]
  3944     by - (rule finite_subset, auto)
  3945   moreover have "\<And>i j. i \<noteq> j \<longrightarrow> {i} \<times> (A - set i) \<inter> {j} \<times> (A - set j) = {}"
  3946     by auto
  3947   moreover have "\<And>i. i \<in>Collect (?k_list k) \<Longrightarrow> card (A - set i) = card A - k"
  3948     by (simp add: card_Diff_subset distinct_card)
  3949   moreover have "{xs. ?k_list (Suc k) xs} =
  3950       (\<lambda>(xs, n). n#xs) ` \<Union>(\<lambda>xs. {xs} \<times> (A - set xs)) ` {xs. ?k_list k xs}"
  3951     by (auto simp: length_Suc_conv)
  3952   moreover
  3953   have "Suc (card A - Suc k) = card A - k" using Suc.prems by simp
  3954   then have "(card A - k) * \<Prod>{Suc (card A - k)..card A} = \<Prod>{Suc (card A - Suc k)..card A}"
  3955     by (subst setprod_insert[symmetric]) (simp add: atLeastAtMost_insertL)+
  3956   ultimately show ?case
  3957     by (simp add: card_image inj_Cons card_UN_disjoint Suc.hyps algebra_simps)
  3958 qed
  3959 
  3960 lemma infinite_UNIV_listI: "~ finite(UNIV::'a list set)"
  3961 apply(rule notI)
  3962 apply(drule finite_maxlen)
  3963 apply (metis UNIV_I length_replicate less_not_refl)
  3964 done
  3965 
  3966 
  3967 subsection {* Sorting *}
  3968 
  3969 text{* Currently it is not shown that @{const sort} returns a
  3970 permutation of its input because the nicest proof is via multisets,
  3971 which are not yet available. Alternatively one could define a function
  3972 that counts the number of occurrences of an element in a list and use
  3973 that instead of multisets to state the correctness property. *}
  3974 
  3975 context linorder
  3976 begin
  3977 
  3978 lemma length_insort [simp]:
  3979   "length (insort_key f x xs) = Suc (length xs)"
  3980   by (induct xs) simp_all
  3981 
  3982 lemma insort_key_left_comm:
  3983   assumes "f x \<noteq> f y"
  3984   shows "insort_key f y (insort_key f x xs) = insort_key f x (insort_key f y xs)"
  3985   by (induct xs) (auto simp add: assms dest: antisym)
  3986 
  3987 lemma insort_left_comm:
  3988   "insort x (insort y xs) = insort y (insort x xs)"
  3989   by (cases "x = y") (auto intro: insort_key_left_comm)
  3990 
  3991 lemma comp_fun_commute_insort:
  3992   "comp_fun_commute insort"
  3993 proof
  3994 qed (simp add: insort_left_comm fun_eq_iff)
  3995 
  3996 lemma sort_key_simps [simp]:
  3997   "sort_key f [] = []"
  3998   "sort_key f (x#xs) = insort_key f x (sort_key f xs)"
  3999   by (simp_all add: sort_key_def)
  4000 
  4001 lemma (in linorder) sort_key_conv_fold:
  4002   assumes "inj_on f (set xs)"
  4003   shows "sort_key f xs = fold (insort_key f) xs []"
  4004 proof -
  4005   have "fold (insort_key f) (rev xs) = fold (insort_key f) xs"
  4006   proof (rule fold_rev, rule ext)
  4007     fix zs
  4008     fix x y
  4009     assume "x \<in> set xs" "y \<in> set xs"
  4010     with assms have *: "f y = f x \<Longrightarrow> y = x" by (auto dest: inj_onD)
  4011     have **: "x = y \<longleftrightarrow> y = x" by auto
  4012     show "(insort_key f y \<circ> insort_key f x) zs = (insort_key f x \<circ> insort_key f y) zs"
  4013       by (induct zs) (auto intro: * simp add: **)
  4014   qed
  4015   then show ?thesis by (simp add: sort_key_def foldr_conv_fold)
  4016 qed
  4017 
  4018 lemma (in linorder) sort_conv_fold:
  4019   "sort xs = fold insort xs []"
  4020   by (rule sort_key_conv_fold) simp
  4021 
  4022 lemma length_sort[simp]: "length (sort_key f xs) = length xs"
  4023 by (induct xs, auto)
  4024 
  4025 lemma sorted_Cons: "sorted (x#xs) = (sorted xs & (ALL y:set xs. x <= y))"
  4026 apply(induct xs arbitrary: x) apply simp
  4027 by simp (blast intro: order_trans)
  4028 
  4029 lemma sorted_tl:
  4030   "sorted xs \<Longrightarrow> sorted (tl xs)"
  4031   by (cases xs) (simp_all add: sorted_Cons)
  4032 
  4033 lemma sorted_append:
  4034   "sorted (xs@ys) = (sorted xs & sorted ys & (\<forall>x \<in> set xs. \<forall>y \<in> set ys. x\<le>y))"
  4035 by (induct xs) (auto simp add:sorted_Cons)
  4036 
  4037 lemma sorted_nth_mono:
  4038   "sorted xs \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!i \<le> xs!j"
  4039 by (induct xs arbitrary: i j) (auto simp:nth_Cons' sorted_Cons)
  4040 
  4041 lemma sorted_rev_nth_mono:
  4042   "sorted (rev xs) \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!j \<le> xs!i"
  4043 using sorted_nth_mono[ of "rev xs" "length xs - j - 1" "length xs - i - 1"]
  4044       rev_nth[of "length xs - i - 1" "xs"] rev_nth[of "length xs - j - 1" "xs"]
  4045 by auto
  4046 
  4047 lemma sorted_nth_monoI:
  4048   "(\<And> i j. \<lbrakk> i \<le> j ; j < length xs \<rbrakk> \<Longrightarrow> xs ! i \<le> xs ! j) \<Longrightarrow> sorted xs"
  4049 proof (induct xs)
  4050   case (Cons x xs)
  4051   have "sorted xs"
  4052   proof (rule Cons.hyps)
  4053     fix i j assume "i \<le> j" and "j < length xs"
  4054     with Cons.prems[of "Suc i" "Suc j"]
  4055     show "xs ! i \<le> xs ! j" by auto
  4056   qed
  4057   moreover
  4058   {
  4059     fix y assume "y \<in> set xs"
  4060     then obtain j where "j < length xs" and "xs ! j = y"
  4061       unfolding in_set_conv_nth by blast
  4062     with Cons.prems[of 0 "Suc j"]
  4063     have "x \<le> y"
  4064       by auto
  4065   }
  4066   ultimately
  4067   show ?case
  4068     unfolding sorted_Cons by auto
  4069 qed simp
  4070 
  4071 lemma sorted_equals_nth_mono:
  4072   "sorted xs = (\<forall>j < length xs. \<forall>i \<le> j. xs ! i \<le> xs ! j)"
  4073 by (auto intro: sorted_nth_monoI sorted_nth_mono)
  4074 
  4075 lemma set_insort: "set(insort_key f x xs) = insert x (set xs)"
  4076 by (induct xs) auto
  4077 
  4078 lemma set_sort[simp]: "set(sort_key f xs) = set xs"
  4079 by (induct xs) (simp_all add:set_insort)
  4080 
  4081 lemma distinct_insort: "distinct (insort_key f x xs) = (x \<notin> set xs \<and> distinct xs)"
  4082 by(induct xs)(auto simp:set_insort)
  4083 
  4084 lemma distinct_sort[simp]: "distinct (sort_key f xs) = distinct xs"
  4085   by (induct xs) (simp_all add: distinct_insort)
  4086 
  4087 lemma sorted_insort_key: "sorted (map f (insort_key f x xs)) = sorted (map f xs)"
  4088   by (induct xs) (auto simp:sorted_Cons set_insort)
  4089 
  4090 lemma sorted_insort: "sorted (insort x xs) = sorted xs"
  4091   using sorted_insort_key [where f="\<lambda>x. x"] by simp
  4092 
  4093 theorem sorted_sort_key [simp]: "sorted (map f (sort_key f xs))"
  4094   by (induct xs) (auto simp:sorted_insort_key)
  4095 
  4096 theorem sorted_sort [simp]: "sorted (sort xs)"
  4097   using sorted_sort_key [where f="\<lambda>x. x"] by simp
  4098 
  4099 lemma sorted_butlast:
  4100   assumes "xs \<noteq> []" and "sorted xs"
  4101   shows "sorted (butlast xs)"
  4102 proof -
  4103   from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  4104   with `sorted xs` show ?thesis by (simp add: sorted_append)
  4105 qed
  4106   
  4107 lemma insort_not_Nil [simp]:
  4108   "insort_key f a xs \<noteq> []"
  4109   by (induct xs) simp_all
  4110 
  4111 lemma insort_is_Cons: "\<forall>x\<in>set xs. f a \<le> f x \<Longrightarrow> insort_key f a xs = a # xs"
  4112 by (cases xs) auto
  4113 
  4114 lemma sorted_sort_id: "sorted xs \<Longrightarrow> sort xs = xs"
  4115   by (induct xs) (auto simp add: sorted_Cons insort_is_Cons)
  4116 
  4117 lemma sorted_map_remove1:
  4118   "sorted (map f xs) \<Longrightarrow> sorted (map f (remove1 x xs))"
  4119   by (induct xs) (auto simp add: sorted_Cons)
  4120 
  4121 lemma sorted_remove1: "sorted xs \<Longrightarrow> sorted (remove1 a xs)"
  4122   using sorted_map_remove1 [of "\<lambda>x. x"] by simp
  4123 
  4124 lemma insort_key_remove1:
  4125   assumes "a \<in> set xs" and "sorted (map f xs)" and "hd (filter (\<lambda>x. f a = f x) xs) = a"
  4126   shows "insort_key f a (remove1 a xs) = xs"
  4127 using assms proof (induct xs)
  4128   case (Cons x xs)
  4129   then show ?case
  4130   proof (cases "x = a")
  4131     case False
  4132     then have "f x \<noteq> f a" using Cons.prems by auto
  4133     then have "f x < f a" using Cons.prems by (auto simp: sorted_Cons)
  4134     with `f x \<noteq> f a` show ?thesis using Cons by (auto simp: sorted_Cons insort_is_Cons)
  4135   qed (auto simp: sorted_Cons insort_is_Cons)
  4136 qed simp
  4137 
  4138 lemma insort_remove1:
  4139   assumes "a \<in> set xs" and "sorted xs"
  4140   shows "insort a (remove1 a xs) = xs"
  4141 proof (rule insort_key_remove1)
  4142   from `a \<in> set xs` show "a \<in> set xs" .
  4143   from `sorted xs` show "sorted (map (\<lambda>x. x) xs)" by simp
  4144   from `a \<in> set xs` have "a \<in> set (filter (op = a) xs)" by auto
  4145   then have "set (filter (op = a) xs) \<noteq> {}" by auto
  4146   then have "filter (op = a) xs \<noteq> []" by (auto simp only: set_empty)
  4147   then have "length (filter (op = a) xs) > 0" by simp
  4148   then obtain n where n: "Suc n = length (filter (op = a) xs)"
  4149     by (cases "length (filter (op = a) xs)") simp_all
  4150   moreover have "replicate (Suc n) a = a # replicate n a"
  4151     by simp
  4152   ultimately show "hd (filter (op = a) xs) = a" by (simp add: replicate_length_filter)
  4153 qed
  4154 
  4155 lemma sorted_remdups[simp]:
  4156   "sorted l \<Longrightarrow> sorted (remdups l)"
  4157 by (induct l) (auto simp: sorted_Cons)
  4158 
  4159 lemma sorted_distinct_set_unique:
  4160 assumes "sorted xs" "distinct xs" "sorted ys" "distinct ys" "set xs = set ys"
  4161 shows "xs = ys"
  4162 proof -
  4163   from assms have 1: "length xs = length ys" by (auto dest!: distinct_card)
  4164   from assms show ?thesis
  4165   proof(induct rule:list_induct2[OF 1])
  4166     case 1 show ?case by simp
  4167   next
  4168     case 2 thus ?case by (simp add:sorted_Cons)
  4169        (metis Diff_insert_absorb antisym insertE insert_iff)
  4170   qed
  4171 qed
  4172 
  4173 lemma map_sorted_distinct_set_unique:
  4174   assumes "inj_on f (set xs \<union> set ys)"
  4175   assumes "sorted (map f xs)" "distinct (map f xs)"
  4176     "sorted (map f ys)" "distinct (map f ys)"
  4177   assumes "set xs = set ys"
  4178   shows "xs = ys"
  4179 proof -
  4180   from assms have "map f xs = map f ys"
  4181     by (simp add: sorted_distinct_set_unique)
  4182   moreover with `inj_on f (set xs \<union> set ys)` show "xs = ys"
  4183     by (blast intro: map_inj_on)
  4184 qed
  4185 
  4186 lemma finite_sorted_distinct_unique:
  4187 shows "finite A \<Longrightarrow> EX! xs. set xs = A & sorted xs & distinct xs"
  4188 apply(drule finite_distinct_list)
  4189 apply clarify
  4190 apply(rule_tac a="sort xs" in ex1I)
  4191 apply (auto simp: sorted_distinct_set_unique)
  4192 done
  4193 
  4194 lemma
  4195   assumes "sorted xs"
  4196   shows sorted_take: "sorted (take n xs)"
  4197   and sorted_drop: "sorted (drop n xs)"
  4198 proof -
  4199   from assms have "sorted (take n xs @ drop n xs)" by simp
  4200   then show "sorted (take n xs)" and "sorted (drop n xs)"
  4201     unfolding sorted_append by simp_all
  4202 qed
  4203 
  4204 lemma sorted_dropWhile: "sorted xs \<Longrightarrow> sorted (dropWhile P xs)"
  4205   by (auto dest: sorted_drop simp add: dropWhile_eq_drop)
  4206 
  4207 lemma sorted_takeWhile: "sorted xs \<Longrightarrow> sorted (takeWhile P xs)"
  4208   by (subst takeWhile_eq_take) (auto dest: sorted_take)
  4209 
  4210 lemma sorted_filter:
  4211   "sorted (map f xs) \<Longrightarrow> sorted (map f (filter P xs))"
  4212   by (induct xs) (simp_all add: sorted_Cons)
  4213 
  4214 lemma foldr_max_sorted:
  4215   assumes "sorted (rev xs)"
  4216   shows "foldr max xs y = (if xs = [] then y else max (xs ! 0) y)"
  4217 using assms proof (induct xs)
  4218   case (Cons x xs)
  4219   moreover hence "sorted (rev xs)" using sorted_append by auto
  4220   ultimately show ?case
  4221     by (cases xs, auto simp add: sorted_append max_def)
  4222 qed simp
  4223 
  4224 lemma filter_equals_takeWhile_sorted_rev:
  4225   assumes sorted: "sorted (rev (map f xs))"
  4226   shows "[x \<leftarrow> xs. t < f x] = takeWhile (\<lambda> x. t < f x) xs"
  4227     (is "filter ?P xs = ?tW")
  4228 proof (rule takeWhile_eq_filter[symmetric])
  4229   let "?dW" = "dropWhile ?P xs"
  4230   fix x assume "x \<in> set ?dW"
  4231   then obtain i where i: "i < length ?dW" and nth_i: "x = ?dW ! i"
  4232     unfolding in_set_conv_nth by auto
  4233   hence "length ?tW + i < length (?tW @ ?dW)"
  4234     unfolding length_append by simp
  4235   hence i': "length (map f ?tW) + i < length (map f xs)" by simp
  4236   have "(map f ?tW @ map f ?dW) ! (length (map f ?tW) + i) \<le>
  4237         (map f ?tW @ map f ?dW) ! (length (map f ?tW) + 0)"
  4238     using sorted_rev_nth_mono[OF sorted _ i', of "length ?tW"]
  4239     unfolding map_append[symmetric] by simp
  4240   hence "f x \<le> f (?dW ! 0)"
  4241     unfolding nth_append_length_plus nth_i
  4242     using i preorder_class.le_less_trans[OF le0 i] by simp
  4243   also have "... \<le> t"
  4244     using hd_dropWhile[of "?P" xs] le0[THEN preorder_class.le_less_trans, OF i]
  4245     using hd_conv_nth[of "?dW"] by simp
  4246   finally show "\<not> t < f x" by simp
  4247 qed
  4248 
  4249 lemma insort_insert_key_triv:
  4250   "f x \<in> f ` set xs \<Longrightarrow> insort_insert_key f x xs = xs"
  4251   by (simp add: insort_insert_key_def)
  4252 
  4253 lemma insort_insert_triv:
  4254   "x \<in> set xs \<Longrightarrow> insort_insert x xs = xs"
  4255   using insort_insert_key_triv [of "\<lambda>x. x"] by simp
  4256 
  4257 lemma insort_insert_insort_key:
  4258   "f x \<notin> f ` set xs \<Longrightarrow> insort_insert_key f x xs = insort_key f x xs"
  4259   by (simp add: insort_insert_key_def)
  4260 
  4261 lemma insort_insert_insort:
  4262   "x \<notin> set xs \<Longrightarrow> insort_insert x xs = insort x xs"
  4263   using insort_insert_insort_key [of "\<lambda>x. x"] by simp
  4264 
  4265 lemma set_insort_insert:
  4266   "set (insort_insert x xs) = insert x (set xs)"
  4267   by (auto simp add: insort_insert_key_def set_insort)
  4268 
  4269 lemma distinct_insort_insert:
  4270   assumes "distinct xs"
  4271   shows "distinct (insort_insert_key f x xs)"
  4272   using assms by (induct xs) (auto simp add: insort_insert_key_def set_insort)
  4273 
  4274 lemma sorted_insort_insert_key:
  4275   assumes "sorted (map f xs)"
  4276   shows "sorted (map f (insort_insert_key f x xs))"
  4277   using assms by (simp add: insort_insert_key_def sorted_insort_key)
  4278 
  4279 lemma sorted_insort_insert:
  4280   assumes "sorted xs"
  4281   shows "sorted (insort_insert x xs)"
  4282   using assms sorted_insort_insert_key [of "\<lambda>x. x"] by simp
  4283 
  4284 lemma filter_insort_triv:
  4285   "\<not> P x \<Longrightarrow> filter P (insort_key f x xs) = filter P xs"
  4286   by (induct xs) simp_all
  4287 
  4288 lemma filter_insort:
  4289   "sorted (map f xs) \<Longrightarrow> P x \<Longrightarrow> filter P (insort_key f x xs) = insort_key f x (filter P xs)"
  4290   using assms by (induct xs)
  4291     (auto simp add: sorted_Cons, subst insort_is_Cons, auto)
  4292 
  4293 lemma filter_sort:
  4294   "filter P (sort_key f xs) = sort_key f (filter P xs)"
  4295   by (induct xs) (simp_all add: filter_insort_triv filter_insort)
  4296 
  4297 lemma sorted_map_same:
  4298   "sorted (map f [x\<leftarrow>xs. f x = g xs])"
  4299 proof (induct xs arbitrary: g)
  4300   case Nil then show ?case by simp
  4301 next
  4302   case (Cons x xs)
  4303   then have "sorted (map f [y\<leftarrow>xs . f y = (\<lambda>xs. f x) xs])" .
  4304   moreover from Cons have "sorted (map f [y\<leftarrow>xs . f y = (g \<circ> Cons x) xs])" .
  4305   ultimately show ?case by (simp_all add: sorted_Cons)
  4306 qed
  4307 
  4308 lemma sorted_same:
  4309   "sorted [x\<leftarrow>xs. x = g xs]"
  4310   using sorted_map_same [of "\<lambda>x. x"] by simp
  4311 
  4312 lemma remove1_insort [simp]:
  4313   "remove1 x (insort x xs) = xs"
  4314   by (induct xs) simp_all
  4315 
  4316 end
  4317 
  4318 lemma sorted_upt[simp]: "sorted[i..<j]"
  4319 by (induct j) (simp_all add:sorted_append)
  4320 
  4321 lemma sorted_upto[simp]: "sorted[i..j]"
  4322 apply(induct i j rule:upto.induct)
  4323 apply(subst upto.simps)
  4324 apply(simp add:sorted_Cons)
  4325 done
  4326 
  4327 
  4328 subsubsection {* @{const transpose} on sorted lists *}
  4329 
  4330 lemma sorted_transpose[simp]:
  4331   shows "sorted (rev (map length (transpose xs)))"
  4332   by (auto simp: sorted_equals_nth_mono rev_nth nth_transpose
  4333     length_filter_conv_card intro: card_mono)
  4334 
  4335 lemma transpose_max_length:
  4336   "foldr (\<lambda>xs. max (length xs)) (transpose xs) 0 = length [x \<leftarrow> xs. x \<noteq> []]"
  4337   (is "?L = ?R")
  4338 proof (cases "transpose xs = []")
  4339   case False
  4340   have "?L = foldr max (map length (transpose xs)) 0"
  4341     by (simp add: foldr_map comp_def)
  4342   also have "... = length (transpose xs ! 0)"
  4343     using False sorted_transpose by (simp add: foldr_max_sorted)
  4344   finally show ?thesis
  4345     using False by (simp add: nth_transpose)
  4346 next
  4347   case True
  4348   hence "[x \<leftarrow> xs. x \<noteq> []] = []"
  4349     by (auto intro!: filter_False simp: transpose_empty)
  4350   thus ?thesis by (simp add: transpose_empty True)
  4351 qed
  4352 
  4353 lemma length_transpose_sorted:
  4354   fixes xs :: "'a list list"
  4355   assumes sorted: "sorted (rev (map length xs))"
  4356   shows "length (transpose xs) = (if xs = [] then 0 else length (xs ! 0))"
  4357 proof (cases "xs = []")
  4358   case False
  4359   thus ?thesis
  4360     using foldr_max_sorted[OF sorted] False
  4361     unfolding length_transpose foldr_map comp_def
  4362     by simp
  4363 qed simp
  4364 
  4365 lemma nth_nth_transpose_sorted[simp]:
  4366   fixes xs :: "'a list list"
  4367   assumes sorted: "sorted (rev (map length xs))"
  4368   and i: "i < length (transpose xs)"
  4369   and j: "j < length [ys \<leftarrow> xs. i < length ys]"
  4370   shows "transpose xs ! i ! j = xs ! j  ! i"
  4371   using j filter_equals_takeWhile_sorted_rev[OF sorted, of i]
  4372     nth_transpose[OF i] nth_map[OF j]
  4373   by (simp add: takeWhile_nth)
  4374 
  4375 lemma transpose_column_length:
  4376   fixes xs :: "'a list list"
  4377   assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  4378   shows "length (filter (\<lambda>ys. i < length ys) (transpose xs)) = length (xs ! i)"
  4379 proof -
  4380   have "xs \<noteq> []" using `i < length xs` by auto
  4381   note filter_equals_takeWhile_sorted_rev[OF sorted, simp]
  4382   { fix j assume "j \<le> i"
  4383     note sorted_rev_nth_mono[OF sorted, of j i, simplified, OF this `i < length xs`]
  4384   } note sortedE = this[consumes 1]
  4385 
  4386   have "{j. j < length (transpose xs) \<and> i < length (transpose xs ! j)}
  4387     = {..< length (xs ! i)}"
  4388   proof safe
  4389     fix j
  4390     assume "j < length (transpose xs)" and "i < length (transpose xs ! j)"
  4391     with this(2) nth_transpose[OF this(1)]
  4392     have "i < length (takeWhile (\<lambda>ys. j < length ys) xs)" by simp
  4393     from nth_mem[OF this] takeWhile_nth[OF this]
  4394     show "j < length (xs ! i)" by (auto dest: set_takeWhileD)
  4395   next
  4396     fix j assume "j < length (xs ! i)"
  4397     thus "j < length (transpose xs)"
  4398       using foldr_max_sorted[OF sorted] `xs \<noteq> []` sortedE[OF le0]
  4399       by (auto simp: length_transpose comp_def foldr_map)
  4400 
  4401     have "Suc i \<le> length (takeWhile (\<lambda>ys. j < length ys) xs)"
  4402       using `i < length xs` `j < length (xs ! i)` less_Suc_eq_le
  4403       by (auto intro!: length_takeWhile_less_P_nth dest!: sortedE)
  4404     with nth_transpose[OF `j < length (transpose xs)`]
  4405     show "i < length (transpose xs ! j)" by simp
  4406   qed
  4407   thus ?thesis by (simp add: length_filter_conv_card)
  4408 qed
  4409 
  4410 lemma transpose_column:
  4411   fixes xs :: "'a list list"
  4412   assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  4413   shows "map (\<lambda>ys. ys ! i) (filter (\<lambda>ys. i < length ys) (transpose xs))
  4414     = xs ! i" (is "?R = _")
  4415 proof (rule nth_equalityI, safe)
  4416   show length: "length ?R = length (xs ! i)"
  4417     using transpose_column_length[OF assms] by simp
  4418 
  4419   fix j assume j: "j < length ?R"
  4420   note * = less_le_trans[OF this, unfolded length_map, OF length_filter_le]
  4421   from j have j_less: "j < length (xs ! i)" using length by simp
  4422   have i_less_tW: "Suc i \<le> length (takeWhile (\<lambda>ys. Suc j \<le> length ys) xs)"
  4423   proof (rule length_takeWhile_less_P_nth)
  4424     show "Suc i \<le> length xs" using `i < length xs` by simp
  4425     fix k assume "k < Suc i"
  4426     hence "k \<le> i" by auto
  4427     with sorted_rev_nth_mono[OF sorted this] `i < length xs`
  4428     have "length (xs ! i) \<le> length (xs ! k)" by simp
  4429     thus "Suc j \<le> length (xs ! k)" using j_less by simp
  4430   qed
  4431   have i_less_filter: "i < length [ys\<leftarrow>xs . j < length ys]"
  4432     unfolding filter_equals_takeWhile_sorted_rev[OF sorted, of j]
  4433     using i_less_tW by (simp_all add: Suc_le_eq)
  4434   from j show "?R ! j = xs ! i ! j"
  4435     unfolding filter_equals_takeWhile_sorted_rev[OF sorted_transpose, of i]
  4436     by (simp add: takeWhile_nth nth_nth_transpose_sorted[OF sorted * i_less_filter])
  4437 qed
  4438 
  4439 lemma transpose_transpose:
  4440   fixes xs :: "'a list list"
  4441   assumes sorted: "sorted (rev (map length xs))"
  4442   shows "transpose (transpose xs) = takeWhile (\<lambda>x. x \<noteq> []) xs" (is "?L = ?R")
  4443 proof -
  4444   have len: "length ?L = length ?R"
  4445     unfolding length_transpose transpose_max_length
  4446     using filter_equals_takeWhile_sorted_rev[OF sorted, of 0]
  4447     by simp
  4448 
  4449   { fix i assume "i < length ?R"
  4450     with less_le_trans[OF _ length_takeWhile_le[of _ xs]]
  4451     have "i < length xs" by simp
  4452   } note * = this
  4453   show ?thesis
  4454     by (rule nth_equalityI)
  4455        (simp_all add: len nth_transpose transpose_column[OF sorted] * takeWhile_nth)
  4456 qed
  4457 
  4458 theorem transpose_rectangle:
  4459   assumes "xs = [] \<Longrightarrow> n = 0"
  4460   assumes rect: "\<And> i. i < length xs \<Longrightarrow> length (xs ! i) = n"
  4461   shows "transpose xs = map (\<lambda> i. map (\<lambda> j. xs ! j ! i) [0..<length xs]) [0..<n]"
  4462     (is "?trans = ?map")
  4463 proof (rule nth_equalityI)
  4464   have "sorted (rev (map length xs))"
  4465     by (auto simp: rev_nth rect intro!: sorted_nth_monoI)
  4466   from foldr_max_sorted[OF this] assms
  4467   show len: "length ?trans = length ?map"
  4468     by (simp_all add: length_transpose foldr_map comp_def)
  4469   moreover
  4470   { fix i assume "i < n" hence "[ys\<leftarrow>xs . i < length ys] = xs"
  4471       using rect by (auto simp: in_set_conv_nth intro!: filter_True) }
  4472   ultimately show "\<forall>i < length ?trans. ?trans ! i = ?map ! i"
  4473     by (auto simp: nth_transpose intro: nth_equalityI)
  4474 qed
  4475 
  4476 
  4477 subsubsection {* @{text sorted_list_of_set} *}
  4478 
  4479 text{* This function maps (finite) linearly ordered sets to sorted
  4480 lists. Warning: in most cases it is not a good idea to convert from
  4481 sets to lists but one should convert in the other direction (via
  4482 @{const set}). *}
  4483 
  4484 context linorder
  4485 begin
  4486 
  4487 definition sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where
  4488   "sorted_list_of_set = Finite_Set.fold insort []"
  4489 
  4490 lemma sorted_list_of_set_empty [simp]:
  4491   "sorted_list_of_set {} = []"
  4492   by (simp add: sorted_list_of_set_def)
  4493 
  4494 lemma sorted_list_of_set_insert [simp]:
  4495   assumes "finite A"
  4496   shows "sorted_list_of_set (insert x A) = insort x (sorted_list_of_set (A - {x}))"
  4497 proof -
  4498   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  4499   from assms show ?thesis
  4500     by (simp add: sorted_list_of_set_def fold_insert_remove)
  4501 qed
  4502 
  4503 lemma sorted_list_of_set [simp]:
  4504   "finite A \<Longrightarrow> set (sorted_list_of_set A) = A \<and> sorted (sorted_list_of_set A) 
  4505     \<and> distinct (sorted_list_of_set A)"
  4506   by (induct A rule: finite_induct) (simp_all add: set_insort sorted_insort distinct_insort)
  4507 
  4508 lemma sorted_list_of_set_sort_remdups [code]:
  4509   "sorted_list_of_set (set xs) = sort (remdups xs)"
  4510 proof -
  4511   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  4512   show ?thesis by (simp add: sorted_list_of_set_def sort_conv_fold fold_set_fold_remdups)
  4513 qed
  4514 
  4515 lemma sorted_list_of_set_remove:
  4516   assumes "finite A"
  4517   shows "sorted_list_of_set (A - {x}) = remove1 x (sorted_list_of_set A)"
  4518 proof (cases "x \<in> A")
  4519   case False with assms have "x \<notin> set (sorted_list_of_set A)" by simp
  4520   with False show ?thesis by (simp add: remove1_idem)
  4521 next
  4522   case True then obtain B where A: "A = insert x B" by (rule Set.set_insert)
  4523   with assms show ?thesis by simp
  4524 qed
  4525 
  4526 end
  4527 
  4528 lemma sorted_list_of_set_range [simp]:
  4529   "sorted_list_of_set {m..<n} = [m..<n]"
  4530   by (rule sorted_distinct_set_unique) simp_all
  4531 
  4532 
  4533 subsubsection {* @{text lists}: the list-forming operator over sets *}
  4534 
  4535 inductive_set
  4536   lists :: "'a set => 'a list set"
  4537   for A :: "'a set"
  4538 where
  4539     Nil [intro!, simp]: "[]: lists A"
  4540   | Cons [intro!, simp, no_atp]: "[| a: A; l: lists A|] ==> a#l : lists A"
  4541 
  4542 inductive_cases listsE [elim!,no_atp]: "x#l : lists A"
  4543 inductive_cases listspE [elim!,no_atp]: "listsp A (x # l)"
  4544 
  4545 inductive_simps listsp_simps[code]:
  4546   "listsp A []"
  4547   "listsp A (x # xs)"
  4548 
  4549 lemma listsp_mono [mono]: "A \<le> B ==> listsp A \<le> listsp B"
  4550 by (rule predicate1I, erule listsp.induct, blast+)
  4551 
  4552 lemmas lists_mono = listsp_mono [to_set]
  4553 
  4554 lemma listsp_infI:
  4555   assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
  4556 by induct blast+
  4557 
  4558 lemmas lists_IntI = listsp_infI [to_set]
  4559 
  4560 lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
  4561 proof (rule mono_inf [where f=listsp, THEN order_antisym])
  4562   show "mono listsp" by (simp add: mono_def listsp_mono)
  4563   show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro!: listsp_infI)
  4564 qed
  4565 
  4566 lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_def inf_bool_def]
  4567 
  4568 lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set]
  4569 
  4570 lemma Cons_in_lists_iff[simp]: "x#xs : lists A \<longleftrightarrow> x:A \<and> xs : lists A"
  4571 by auto
  4572 
  4573 lemma append_in_listsp_conv [iff]:
  4574      "(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
  4575 by (induct xs) auto
  4576 
  4577 lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]
  4578 
  4579 lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
  4580 -- {* eliminate @{text listsp} in favour of @{text set} *}
  4581 by (induct xs) auto
  4582 
  4583 lemmas in_lists_conv_set [code_unfold] = in_listsp_conv_set [to_set]
  4584 
  4585 lemma in_listspD [dest!,no_atp]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
  4586 by (rule in_listsp_conv_set [THEN iffD1])
  4587 
  4588 lemmas in_listsD [dest!,no_atp] = in_listspD [to_set]
  4589 
  4590 lemma in_listspI [intro!,no_atp]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
  4591 by (rule in_listsp_conv_set [THEN iffD2])
  4592 
  4593 lemmas in_listsI [intro!,no_atp] = in_listspI [to_set]
  4594 
  4595 lemma lists_eq_set: "lists A = {xs. set xs <= A}"
  4596 by auto
  4597 
  4598 lemma lists_empty [simp]: "lists {} = {[]}"
  4599 by auto
  4600 
  4601 lemma lists_UNIV [simp]: "lists UNIV = UNIV"
  4602 by auto
  4603 
  4604 
  4605 subsubsection {* Inductive definition for membership *}
  4606 
  4607 inductive ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
  4608 where
  4609     elem:  "ListMem x (x # xs)"
  4610   | insert:  "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"
  4611 
  4612 lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
  4613 apply (rule iffI)
  4614  apply (induct set: ListMem)
  4615   apply auto
  4616 apply (induct xs)
  4617  apply (auto intro: ListMem.intros)
  4618 done
  4619 
  4620 
  4621 subsubsection {* Lists as Cartesian products *}
  4622 
  4623 text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
  4624 @{term A} and tail drawn from @{term Xs}.*}
  4625 
  4626 definition
  4627   set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set" where
  4628   "set_Cons A XS = {z. \<exists>x xs. z = x # xs \<and> x \<in> A \<and> xs \<in> XS}"
  4629 
  4630 lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
  4631 by (auto simp add: set_Cons_def)
  4632 
  4633 text{*Yields the set of lists, all of the same length as the argument and
  4634 with elements drawn from the corresponding element of the argument.*}
  4635 
  4636 primrec
  4637   listset :: "'a set list \<Rightarrow> 'a list set" where
  4638      "listset [] = {[]}"
  4639   |  "listset (A # As) = set_Cons A (listset As)"
  4640 
  4641 
  4642 subsection {* Relations on Lists *}
  4643 
  4644 subsubsection {* Length Lexicographic Ordering *}
  4645 
  4646 text{*These orderings preserve well-foundedness: shorter lists 
  4647   precede longer lists. These ordering are not used in dictionaries.*}
  4648         
  4649 primrec -- {*The lexicographic ordering for lists of the specified length*}
  4650   lexn :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> ('a list \<times> 'a list) set" where
  4651     "lexn r 0 = {}"
  4652   | "lexn r (Suc n) = (map_pair (%(x, xs). x#xs) (%(x, xs). x#xs) ` (r <*lex*> lexn r n)) Int
  4653       {(xs, ys). length xs = Suc n \<and> length ys = Suc n}"
  4654 
  4655 definition
  4656   lex :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4657   "lex r = (\<Union>n. lexn r n)" -- {*Holds only between lists of the same length*}
  4658 
  4659 definition
  4660   lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" where
  4661   "lenlex r = inv_image (less_than <*lex*> lex r) (\<lambda>xs. (length xs, xs))"
  4662         -- {*Compares lists by their length and then lexicographically*}
  4663 
  4664 lemma wf_lexn: "wf r ==> wf (lexn r n)"
  4665 apply (induct n, simp, simp)
  4666 apply(rule wf_subset)
  4667  prefer 2 apply (rule Int_lower1)
  4668 apply(rule wf_map_pair_image)
  4669  prefer 2 apply (rule inj_onI, auto)
  4670 done
  4671 
  4672 lemma lexn_length:
  4673   "(xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
  4674 by (induct n arbitrary: xs ys) auto
  4675 
  4676 lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
  4677 apply (unfold lex_def)
  4678 apply (rule wf_UN)
  4679 apply (blast intro: wf_lexn, clarify)
  4680 apply (rename_tac m n)
  4681 apply (subgoal_tac "m \<noteq> n")
  4682  prefer 2 apply blast
  4683 apply (blast dest: lexn_length not_sym)
  4684 done
  4685 
  4686 lemma lexn_conv:
  4687   "lexn r n =
  4688     {(xs,ys). length xs = n \<and> length ys = n \<and>
  4689     (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
  4690 apply (induct n, simp)
  4691 apply (simp add: image_Collect lex_prod_def, safe, blast)
  4692  apply (rule_tac x = "ab # xys" in exI, simp)
  4693 apply (case_tac xys, simp_all, blast)
  4694 done
  4695 
  4696 lemma lex_conv:
  4697   "lex r =
  4698     {(xs,ys). length xs = length ys \<and>
  4699     (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
  4700 by (force simp add: lex_def lexn_conv)
  4701 
  4702 lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
  4703 by (unfold lenlex_def) blast
  4704 
  4705 lemma lenlex_conv:
  4706     "lenlex r = {(xs,ys). length xs < length ys |
  4707                  length xs = length ys \<and> (xs, ys) : lex r}"
  4708 by (simp add: lenlex_def Id_on_def lex_prod_def inv_image_def)
  4709 
  4710 lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
  4711 by (simp add: lex_conv)
  4712 
  4713 lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
  4714 by (simp add:lex_conv)
  4715 
  4716 lemma Cons_in_lex [simp]:
  4717     "((x # xs, y # ys) : lex r) =
  4718       ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
  4719 apply (simp add: lex_conv)
  4720 apply (rule iffI)
  4721  prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
  4722 apply (case_tac xys, simp, simp)
  4723 apply blast
  4724 done
  4725 
  4726 
  4727 subsubsection {* Lexicographic Ordering *}
  4728 
  4729 text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
  4730     This ordering does \emph{not} preserve well-foundedness.
  4731      Author: N. Voelker, March 2005. *} 
  4732 
  4733 definition
  4734   lexord :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4735   "lexord r = {(x,y ). \<exists> a v. y = x @ a # v \<or>
  4736             (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
  4737 
  4738 lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
  4739 by (unfold lexord_def, induct_tac y, auto) 
  4740 
  4741 lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
  4742 by (unfold lexord_def, induct_tac x, auto)
  4743 
  4744 lemma lexord_cons_cons[simp]:
  4745      "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
  4746   apply (unfold lexord_def, safe, simp_all)
  4747   apply (case_tac u, simp, simp)
  4748   apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
  4749   apply (erule_tac x="b # u" in allE)
  4750   by force
  4751 
  4752 lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
  4753 
  4754 lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
  4755 by (induct_tac x, auto)  
  4756 
  4757 lemma lexord_append_left_rightI:
  4758      "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
  4759 by (induct_tac u, auto)
  4760 
  4761 lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
  4762 by (induct x, auto)
  4763 
  4764 lemma lexord_append_leftD:
  4765      "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
  4766 by (erule rev_mp, induct_tac x, auto)
  4767 
  4768 lemma lexord_take_index_conv: 
  4769    "((x,y) : lexord r) = 
  4770     ((length x < length y \<and> take (length x) y = x) \<or> 
  4771      (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
  4772   apply (unfold lexord_def Let_def, clarsimp) 
  4773   apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
  4774   apply auto 
  4775   apply (rule_tac x="hd (drop (length x) y)" in exI)
  4776   apply (rule_tac x="tl (drop (length x) y)" in exI)
  4777   apply (erule subst, simp add: min_def) 
  4778   apply (rule_tac x ="length u" in exI, simp) 
  4779   apply (rule_tac x ="take i x" in exI) 
  4780   apply (rule_tac x ="x ! i" in exI) 
  4781   apply (rule_tac x ="y ! i" in exI, safe) 
  4782   apply (rule_tac x="drop (Suc i) x" in exI)
  4783   apply (drule sym, simp add: drop_Suc_conv_tl) 
  4784   apply (rule_tac x="drop (Suc i) y" in exI)
  4785   by (simp add: drop_Suc_conv_tl) 
  4786 
  4787 -- {* lexord is extension of partial ordering List.lex *} 
  4788 lemma lexord_lex: "(x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
  4789   apply (rule_tac x = y in spec) 
  4790   apply (induct_tac x, clarsimp) 
  4791   by (clarify, case_tac x, simp, force)
  4792 
  4793 lemma lexord_irreflexive: "ALL x. (x,x) \<notin> r \<Longrightarrow> (xs,xs) \<notin> lexord r"
  4794 by (induct xs) auto
  4795 
  4796 text{* By Ren\'e Thiemann: *}
  4797 lemma lexord_partial_trans: 
  4798   "(\<And>x y z. x \<in> set xs \<Longrightarrow> (x,y) \<in> r \<Longrightarrow> (y,z) \<in> r \<Longrightarrow> (x,z) \<in> r)
  4799    \<Longrightarrow>  (xs,ys) \<in> lexord r  \<Longrightarrow>  (ys,zs) \<in> lexord r \<Longrightarrow>  (xs,zs) \<in> lexord r"
  4800 proof (induct xs arbitrary: ys zs)
  4801   case Nil
  4802   from Nil(3) show ?case unfolding lexord_def by (cases zs, auto)
  4803 next
  4804   case (Cons x xs yys zzs)
  4805   from Cons(3) obtain y ys where yys: "yys = y # ys" unfolding lexord_def
  4806     by (cases yys, auto)
  4807   note Cons = Cons[unfolded yys]
  4808   from Cons(3) have one: "(x,y) \<in> r \<or> x = y \<and> (xs,ys) \<in> lexord r" by auto
  4809   from Cons(4) obtain z zs where zzs: "zzs = z # zs" unfolding lexord_def
  4810     by (cases zzs, auto)
  4811   note Cons = Cons[unfolded zzs]
  4812   from Cons(4) have two: "(y,z) \<in> r \<or> y = z \<and> (ys,zs) \<in> lexord r" by auto
  4813   {
  4814     assume "(xs,ys) \<in> lexord r" and "(ys,zs) \<in> lexord r"
  4815     from Cons(1)[OF _ this] Cons(2)
  4816     have "(xs,zs) \<in> lexord r" by auto
  4817   } note ind1 = this
  4818   {
  4819     assume "(x,y) \<in> r" and "(y,z) \<in> r"
  4820     from Cons(2)[OF _ this] have "(x,z) \<in> r" by auto
  4821   } note ind2 = this
  4822   from one two ind1 ind2
  4823   have "(x,z) \<in> r \<or> x = z \<and> (xs,zs) \<in> lexord r" by blast
  4824   thus ?case unfolding zzs by auto
  4825 qed
  4826 
  4827 lemma lexord_trans: 
  4828     "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
  4829 by(auto simp: trans_def intro:lexord_partial_trans)
  4830 
  4831 lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
  4832 by (rule transI, drule lexord_trans, blast) 
  4833 
  4834 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"
  4835   apply (rule_tac x = y in spec) 
  4836   apply (induct_tac x, rule allI) 
  4837   apply (case_tac x, simp, simp) 
  4838   apply (rule allI, case_tac x, simp, simp) 
  4839   by blast
  4840 
  4841 
  4842 subsubsection {* Lexicographic combination of measure functions *}
  4843 
  4844 text {* These are useful for termination proofs *}
  4845 
  4846 definition
  4847   "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"
  4848 
  4849 lemma wf_measures[simp]: "wf (measures fs)"
  4850 unfolding measures_def
  4851 by blast
  4852 
  4853 lemma in_measures[simp]: 
  4854   "(x, y) \<in> measures [] = False"
  4855   "(x, y) \<in> measures (f # fs)
  4856          = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"  
  4857 unfolding measures_def
  4858 by auto
  4859 
  4860 lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
  4861 by simp
  4862 
  4863 lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
  4864 by auto
  4865 
  4866 
  4867 subsubsection {* Lifting Relations to Lists: one element *}
  4868 
  4869 definition listrel1 :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4870 "listrel1 r = {(xs,ys).
  4871    \<exists>us z z' vs. xs = us @ z # vs \<and> (z,z') \<in> r \<and> ys = us @ z' # vs}"
  4872 
  4873 lemma listrel1I:
  4874   "\<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow>
  4875   (xs, ys) \<in> listrel1 r"
  4876 unfolding listrel1_def by auto
  4877 
  4878 lemma listrel1E:
  4879   "\<lbrakk> (xs, ys) \<in> listrel1 r;
  4880      !!x y us vs. \<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow> P
  4881    \<rbrakk> \<Longrightarrow> P"
  4882 unfolding listrel1_def by auto
  4883 
  4884 lemma not_Nil_listrel1 [iff]: "([], xs) \<notin> listrel1 r"
  4885 unfolding listrel1_def by blast
  4886 
  4887 lemma not_listrel1_Nil [iff]: "(xs, []) \<notin> listrel1 r"
  4888 unfolding listrel1_def by blast
  4889 
  4890 lemma Cons_listrel1_Cons [iff]:
  4891   "(x # xs, y # ys) \<in> listrel1 r \<longleftrightarrow>
  4892    (x,y) \<in> r \<and> xs = ys \<or> x = y \<and> (xs, ys) \<in> listrel1 r"
  4893 by (simp add: listrel1_def Cons_eq_append_conv) (blast)
  4894 
  4895 lemma listrel1I1: "(x,y) \<in> r \<Longrightarrow> (x # xs, y # xs) \<in> listrel1 r"
  4896 by (metis Cons_listrel1_Cons)
  4897 
  4898 lemma listrel1I2: "(xs, ys) \<in> listrel1 r \<Longrightarrow> (x # xs, x # ys) \<in> listrel1 r"
  4899 by (metis Cons_listrel1_Cons)
  4900 
  4901 lemma append_listrel1I:
  4902   "(xs, ys) \<in> listrel1 r \<and> us = vs \<or> xs = ys \<and> (us, vs) \<in> listrel1 r
  4903     \<Longrightarrow> (xs @ us, ys @ vs) \<in> listrel1 r"
  4904 unfolding listrel1_def
  4905 by auto (blast intro: append_eq_appendI)+
  4906 
  4907 lemma Cons_listrel1E1[elim!]:
  4908   assumes "(x # xs, ys) \<in> listrel1 r"
  4909     and "\<And>y. ys = y # xs \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
  4910     and "\<And>zs. ys = x # zs \<Longrightarrow> (xs, zs) \<in> listrel1 r \<Longrightarrow> R"
  4911   shows R
  4912 using assms by (cases ys) blast+
  4913 
  4914 lemma Cons_listrel1E2[elim!]:
  4915   assumes "(xs, y # ys) \<in> listrel1 r"
  4916     and "\<And>x. xs = x # ys \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
  4917     and "\<And>zs. xs = y # zs \<Longrightarrow> (zs, ys) \<in> listrel1 r \<Longrightarrow> R"
  4918   shows R
  4919 using assms by (cases xs) blast+
  4920 
  4921 lemma snoc_listrel1_snoc_iff:
  4922   "(xs @ [x], ys @ [y]) \<in> listrel1 r
  4923     \<longleftrightarrow> (xs, ys) \<in> listrel1 r \<and> x = y \<or> xs = ys \<and> (x,y) \<in> r" (is "?L \<longleftrightarrow> ?R")
  4924 proof
  4925   assume ?L thus ?R
  4926     by (fastforce simp: listrel1_def snoc_eq_iff_butlast butlast_append)
  4927 next
  4928   assume ?R then show ?L unfolding listrel1_def by force
  4929 qed
  4930 
  4931 lemma listrel1_eq_len: "(xs,ys) \<in> listrel1 r \<Longrightarrow> length xs = length ys"
  4932 unfolding listrel1_def by auto
  4933 
  4934 lemma listrel1_mono:
  4935   "r \<subseteq> s \<Longrightarrow> listrel1 r \<subseteq> listrel1 s"
  4936 unfolding listrel1_def by blast
  4937 
  4938 
  4939 lemma listrel1_converse: "listrel1 (r^-1) = (listrel1 r)^-1"
  4940 unfolding listrel1_def by blast
  4941 
  4942 lemma in_listrel1_converse:
  4943   "(x,y) : listrel1 (r^-1) \<longleftrightarrow> (x,y) : (listrel1 r)^-1"
  4944 unfolding listrel1_def by blast
  4945 
  4946 lemma listrel1_iff_update:
  4947   "(xs,ys) \<in> (listrel1 r)
  4948    \<longleftrightarrow> (\<exists>y n. (xs ! n, y) \<in> r \<and> n < length xs \<and> ys = xs[n:=y])" (is "?L \<longleftrightarrow> ?R")
  4949 proof
  4950   assume "?L"
  4951   then obtain x y u v where "xs = u @ x # v"  "ys = u @ y # v"  "(x,y) \<in> r"
  4952     unfolding listrel1_def by auto
  4953   then have "ys = xs[length u := y]" and "length u < length xs"
  4954     and "(xs ! length u, y) \<in> r" by auto
  4955   then show "?R" by auto
  4956 next
  4957   assume "?R"
  4958   then obtain x y n where "(xs!n, y) \<in> r" "n < size xs" "ys = xs[n:=y]" "x = xs!n"
  4959     by auto
  4960   then obtain u v where "xs = u @ x # v" and "ys = u @ y # v" and "(x, y) \<in> r"
  4961     by (auto intro: upd_conv_take_nth_drop id_take_nth_drop)
  4962   then show "?L" by (auto simp: listrel1_def)
  4963 qed
  4964 
  4965 
  4966 text{* Accessible part and wellfoundedness: *}
  4967 
  4968 lemma Cons_acc_listrel1I [intro!]:
  4969   "x \<in> acc r \<Longrightarrow> xs \<in> acc (listrel1 r) \<Longrightarrow> (x # xs) \<in> acc (listrel1 r)"
  4970 apply (induct arbitrary: xs set: acc)
  4971 apply (erule thin_rl)
  4972 apply (erule acc_induct)
  4973 apply (rule accI)
  4974 apply (blast)
  4975 done
  4976 
  4977 lemma lists_accD: "xs \<in> lists (acc r) \<Longrightarrow> xs \<in> acc (listrel1 r)"
  4978 apply (induct set: lists)
  4979  apply (rule accI)
  4980  apply simp
  4981 apply (rule accI)
  4982 apply (fast dest: acc_downward)
  4983 done
  4984 
  4985 lemma lists_accI: "xs \<in> acc (listrel1 r) \<Longrightarrow> xs \<in> lists (acc r)"
  4986 apply (induct set: acc)
  4987 apply clarify
  4988 apply (rule accI)
  4989 apply (fastforce dest!: in_set_conv_decomp[THEN iffD1] simp: listrel1_def)
  4990 done
  4991 
  4992 lemma wf_listrel1_iff[simp]: "wf(listrel1 r) = wf r"
  4993 by(metis wf_acc_iff in_lists_conv_set lists_accI lists_accD Cons_in_lists_iff)
  4994 
  4995 
  4996 subsubsection {* Lifting Relations to Lists: all elements *}
  4997 
  4998 inductive_set
  4999   listrel :: "('a \<times> 'b) set \<Rightarrow> ('a list \<times> 'b list) set"
  5000   for r :: "('a \<times> 'b) set"
  5001 where
  5002     Nil:  "([],[]) \<in> listrel r"
  5003   | Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
  5004 
  5005 inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
  5006 inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
  5007 inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
  5008 inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
  5009 
  5010 
  5011 lemma listrel_eq_len:  "(xs, ys) \<in> listrel r \<Longrightarrow> length xs = length ys"
  5012 by(induct rule: listrel.induct) auto
  5013 
  5014 lemma listrel_iff_zip [code_unfold]: "(xs,ys) : listrel r \<longleftrightarrow>
  5015   length xs = length ys & (\<forall>(x,y) \<in> set(zip xs ys). (x,y) \<in> r)" (is "?L \<longleftrightarrow> ?R")
  5016 proof
  5017   assume ?L thus ?R by induct (auto intro: listrel_eq_len)
  5018 next
  5019   assume ?R thus ?L
  5020     apply (clarify)
  5021     by (induct rule: list_induct2) (auto intro: listrel.intros)
  5022 qed
  5023 
  5024 lemma listrel_iff_nth: "(xs,ys) : listrel r \<longleftrightarrow>
  5025   length xs = length ys & (\<forall>n < length xs. (xs!n, ys!n) \<in> r)" (is "?L \<longleftrightarrow> ?R")
  5026 by (auto simp add: all_set_conv_all_nth listrel_iff_zip)
  5027 
  5028 
  5029 lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
  5030 apply clarify  
  5031 apply (erule listrel.induct)
  5032 apply (blast intro: listrel.intros)+
  5033 done
  5034 
  5035 lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
  5036 apply clarify 
  5037 apply (erule listrel.induct, auto) 
  5038 done
  5039 
  5040 lemma listrel_refl_on: "refl_on A r \<Longrightarrow> refl_on (lists A) (listrel r)" 
  5041 apply (simp add: refl_on_def listrel_subset Ball_def)
  5042 apply (rule allI) 
  5043 apply (induct_tac x) 
  5044 apply (auto intro: listrel.intros)
  5045 done
  5046 
  5047 lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
  5048 apply (auto simp add: sym_def)
  5049 apply (erule listrel.induct) 
  5050 apply (blast intro: listrel.intros)+
  5051 done
  5052 
  5053 lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
  5054 apply (simp add: trans_def)
  5055 apply (intro allI) 
  5056 apply (rule impI) 
  5057 apply (erule listrel.induct) 
  5058 apply (blast intro: listrel.intros)+
  5059 done
  5060 
  5061 theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
  5062 by (simp add: equiv_def listrel_refl_on listrel_sym listrel_trans) 
  5063 
  5064 lemma listrel_rtrancl_refl[iff]: "(xs,xs) : listrel(r^*)"
  5065 using listrel_refl_on[of UNIV, OF refl_rtrancl]
  5066 by(auto simp: refl_on_def)
  5067 
  5068 lemma listrel_rtrancl_trans:
  5069   "\<lbrakk> (xs,ys) : listrel(r^*);  (ys,zs) : listrel(r^*) \<rbrakk>
  5070   \<Longrightarrow> (xs,zs) : listrel(r^*)"
  5071 by (metis listrel_trans trans_def trans_rtrancl)
  5072 
  5073 
  5074 lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
  5075 by (blast intro: listrel.intros)
  5076 
  5077 lemma listrel_Cons:
  5078      "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})"
  5079 by (auto simp add: set_Cons_def intro: listrel.intros)
  5080 
  5081 text {* Relating @{term listrel1}, @{term listrel} and closures: *}
  5082 
  5083 lemma listrel1_rtrancl_subset_rtrancl_listrel1:
  5084   "listrel1 (r^*) \<subseteq> (listrel1 r)^*"
  5085 proof (rule subrelI)
  5086   fix xs ys assume 1: "(xs,ys) \<in> listrel1 (r^*)"
  5087   { fix x y us vs
  5088     have "(x,y) : r^* \<Longrightarrow> (us @ x # vs, us @ y # vs) : (listrel1 r)^*"
  5089     proof(induct rule: rtrancl.induct)
  5090       case rtrancl_refl show ?case by simp
  5091     next
  5092       case rtrancl_into_rtrancl thus ?case
  5093         by (metis listrel1I rtrancl.rtrancl_into_rtrancl)
  5094     qed }
  5095   thus "(xs,ys) \<in> (listrel1 r)^*" using 1 by(blast elim: listrel1E)
  5096 qed
  5097 
  5098 lemma rtrancl_listrel1_eq_len: "(x,y) \<in> (listrel1 r)^* \<Longrightarrow> length x = length y"
  5099 by (induct rule: rtrancl.induct) (auto intro: listrel1_eq_len)
  5100 
  5101 lemma rtrancl_listrel1_ConsI1:
  5102   "(xs,ys) : (listrel1 r)^* \<Longrightarrow> (x#xs,x#ys) : (listrel1 r)^*"
  5103 apply(induct rule: rtrancl.induct)
  5104  apply simp
  5105 by (metis listrel1I2 rtrancl.rtrancl_into_rtrancl)
  5106 
  5107 lemma rtrancl_listrel1_ConsI2:
  5108   "(x,y) \<in> r^* \<Longrightarrow> (xs, ys) \<in> (listrel1 r)^*
  5109   \<Longrightarrow> (x # xs, y # ys) \<in> (listrel1 r)^*"
  5110   by (blast intro: rtrancl_trans rtrancl_listrel1_ConsI1 
  5111     subsetD[OF listrel1_rtrancl_subset_rtrancl_listrel1 listrel1I1])
  5112 
  5113 lemma listrel1_subset_listrel:
  5114   "r \<subseteq> r' \<Longrightarrow> refl r' \<Longrightarrow> listrel1 r \<subseteq> listrel(r')"
  5115 by(auto elim!: listrel1E simp add: listrel_iff_zip set_zip refl_on_def)
  5116 
  5117 lemma listrel_reflcl_if_listrel1:
  5118   "(xs,ys) : listrel1 r \<Longrightarrow> (xs,ys) : listrel(r^*)"
  5119 by(erule listrel1E)(auto simp add: listrel_iff_zip set_zip)
  5120 
  5121 lemma listrel_rtrancl_eq_rtrancl_listrel1: "listrel (r^*) = (listrel1 r)^*"
  5122 proof
  5123   { fix x y assume "(x,y) \<in> listrel (r^*)"
  5124     then have "(x,y) \<in> (listrel1 r)^*"
  5125     by induct (auto intro: rtrancl_listrel1_ConsI2) }
  5126   then show "listrel (r^*) \<subseteq> (listrel1 r)^*"
  5127     by (rule subrelI)
  5128 next
  5129   show "listrel (r^*) \<supseteq> (listrel1 r)^*"
  5130   proof(rule subrelI)
  5131     fix xs ys assume "(xs,ys) \<in> (listrel1 r)^*"
  5132     then show "(xs,ys) \<in> listrel (r^*)"
  5133     proof induct
  5134       case base show ?case by(auto simp add: listrel_iff_zip set_zip)
  5135     next
  5136       case (step ys zs)
  5137       thus ?case  by (metis listrel_reflcl_if_listrel1 listrel_rtrancl_trans)
  5138     qed
  5139   qed
  5140 qed
  5141 
  5142 lemma rtrancl_listrel1_if_listrel:
  5143   "(xs,ys) : listrel r \<Longrightarrow> (xs,ys) : (listrel1 r)^*"
  5144 by(metis listrel_rtrancl_eq_rtrancl_listrel1 subsetD[OF listrel_mono] r_into_rtrancl subsetI)
  5145 
  5146 lemma listrel_subset_rtrancl_listrel1: "listrel r \<subseteq> (listrel1 r)^*"
  5147 by(fast intro:rtrancl_listrel1_if_listrel)
  5148 
  5149 
  5150 subsection {* Size function *}
  5151 
  5152 lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (list_size f)"
  5153 by (rule is_measure_trivial)
  5154 
  5155 lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (option_size f)"
  5156 by (rule is_measure_trivial)
  5157 
  5158 lemma list_size_estimation[termination_simp]: 
  5159   "x \<in> set xs \<Longrightarrow> y < f x \<Longrightarrow> y < list_size f xs"
  5160 by (induct xs) auto
  5161 
  5162 lemma list_size_estimation'[termination_simp]: 
  5163   "x \<in> set xs \<Longrightarrow> y \<le> f x \<Longrightarrow> y \<le> list_size f xs"
  5164 by (induct xs) auto
  5165 
  5166 lemma list_size_map[simp]: "list_size f (map g xs) = list_size (f o g) xs"
  5167 by (induct xs) auto
  5168 
  5169 lemma list_size_append[simp]: "list_size f (xs @ ys) = list_size f xs + list_size f ys"
  5170 by (induct xs, auto)
  5171 
  5172 lemma list_size_pointwise[termination_simp]: 
  5173   "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> list_size f xs \<le> list_size g xs"
  5174 by (induct xs) force+
  5175 
  5176 
  5177 subsection {* Monad operation *}
  5178 
  5179 definition bind :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b list) \<Rightarrow> 'b list" where
  5180   "bind xs f = concat (map f xs)"
  5181 
  5182 hide_const (open) bind
  5183 
  5184 lemma bind_simps [simp]:
  5185   "List.bind [] f = []"
  5186   "List.bind (x # xs) f = f x @ List.bind xs f"
  5187   by (simp_all add: bind_def)
  5188 
  5189 
  5190 subsection {* Transfer *}
  5191 
  5192 definition
  5193   embed_list :: "nat list \<Rightarrow> int list"
  5194 where
  5195   "embed_list l = map int l"
  5196 
  5197 definition
  5198   nat_list :: "int list \<Rightarrow> bool"
  5199 where
  5200   "nat_list l = nat_set (set l)"
  5201 
  5202 definition
  5203   return_list :: "int list \<Rightarrow> nat list"
  5204 where
  5205   "return_list l = map nat l"
  5206 
  5207 lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow>
  5208     embed_list (return_list l) = l"
  5209   unfolding embed_list_def return_list_def nat_list_def nat_set_def
  5210   apply (induct l)
  5211   apply auto
  5212 done
  5213 
  5214 lemma transfer_nat_int_list_functions:
  5215   "l @ m = return_list (embed_list l @ embed_list m)"
  5216   "[] = return_list []"
  5217   unfolding return_list_def embed_list_def
  5218   apply auto
  5219   apply (induct l, auto)
  5220   apply (induct m, auto)
  5221 done
  5222 
  5223 (*
  5224 lemma transfer_nat_int_fold1: "fold f l x =
  5225     fold (%x. f (nat x)) (embed_list l) x";
  5226 *)
  5227 
  5228 
  5229 subsection {* Code generation *}
  5230 
  5231 subsubsection {* Counterparts for set-related operations *}
  5232 
  5233 definition member :: "'a list \<Rightarrow> 'a \<Rightarrow> bool" where
  5234   [code_abbrev]: "member xs x \<longleftrightarrow> x \<in> set xs"
  5235 
  5236 text {*
  5237   Use @{text member} only for generating executable code.  Otherwise use
  5238   @{prop "x \<in> set xs"} instead --- it is much easier to reason about.
  5239 *}
  5240 
  5241 lemma member_rec [code]:
  5242   "member (x # xs) y \<longleftrightarrow> x = y \<or> member xs y"
  5243   "member [] y \<longleftrightarrow> False"
  5244   by (auto simp add: member_def)
  5245 
  5246 lemma in_set_member (* FIXME delete candidate *):
  5247   "x \<in> set xs \<longleftrightarrow> member xs x"
  5248   by (simp add: member_def)
  5249 
  5250 definition list_all :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5251   list_all_iff [code_abbrev]: "list_all P xs \<longleftrightarrow> Ball (set xs) P"
  5252 
  5253 definition list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5254   list_ex_iff [code_abbrev]: "list_ex P xs \<longleftrightarrow> Bex (set xs) P"
  5255 
  5256 definition list_ex1 :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5257   list_ex1_iff [code_abbrev]: "list_ex1 P xs \<longleftrightarrow> (\<exists>! x. x \<in> set xs \<and> P x)"
  5258 
  5259 text {*
  5260   Usually you should prefer @{text "\<forall>x\<in>set xs"}, @{text "\<exists>x\<in>set xs"}
  5261   and @{text "\<exists>!x. x\<in>set xs \<and> _"} over @{const list_all}, @{const list_ex}
  5262   and @{const list_ex1} in specifications.
  5263 *}
  5264 
  5265 lemma list_all_simps [simp, code]:
  5266   "list_all P (x # xs) \<longleftrightarrow> P x \<and> list_all P xs"
  5267   "list_all P [] \<longleftrightarrow> True"
  5268   by (simp_all add: list_all_iff)
  5269 
  5270 lemma list_ex_simps [simp, code]:
  5271   "list_ex P (x # xs) \<longleftrightarrow> P x \<or> list_ex P xs"
  5272   "list_ex P [] \<longleftrightarrow> False"
  5273   by (simp_all add: list_ex_iff)
  5274 
  5275 lemma list_ex1_simps [simp, code]:
  5276   "list_ex1 P [] = False"
  5277   "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)"
  5278   by (auto simp add: list_ex1_iff list_all_iff)
  5279 
  5280 lemma Ball_set_list_all: (* FIXME delete candidate *)
  5281   "Ball (set xs) P \<longleftrightarrow> list_all P xs"
  5282   by (simp add: list_all_iff)
  5283 
  5284 lemma Bex_set_list_ex: (* FIXME delete candidate *)
  5285   "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
  5286   by (simp add: list_ex_iff)
  5287 
  5288 lemma list_all_append [simp]:
  5289   "list_all P (xs @ ys) \<longleftrightarrow> list_all P xs \<and> list_all P ys"
  5290   by (auto simp add: list_all_iff)
  5291 
  5292 lemma list_ex_append [simp]:
  5293   "list_ex P (xs @ ys) \<longleftrightarrow> list_ex P xs \<or> list_ex P ys"
  5294   by (auto simp add: list_ex_iff)
  5295 
  5296 lemma list_all_rev [simp]:
  5297   "list_all P (rev xs) \<longleftrightarrow> list_all P xs"
  5298   by (simp add: list_all_iff)
  5299 
  5300 lemma list_ex_rev [simp]:
  5301   "list_ex P (rev xs) \<longleftrightarrow> list_ex P xs"
  5302   by (simp add: list_ex_iff)
  5303 
  5304 lemma list_all_length:
  5305   "list_all P xs \<longleftrightarrow> (\<forall>n < length xs. P (xs ! n))"
  5306   by (auto simp add: list_all_iff set_conv_nth)
  5307 
  5308 lemma list_ex_length:
  5309   "list_ex P xs \<longleftrightarrow> (\<exists>n < length xs. P (xs ! n))"
  5310   by (auto simp add: list_ex_iff set_conv_nth)
  5311 
  5312 lemma list_all_cong [fundef_cong]:
  5313   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_all f xs = list_all g ys"
  5314   by (simp add: list_all_iff)
  5315 
  5316 lemma list_ex_cong [fundef_cong]:
  5317   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_ex f xs = list_ex g ys"
  5318 by (simp add: list_ex_iff)
  5319 
  5320 text {* Executable checks for relations on sets *}
  5321 
  5322 definition listrel1p :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
  5323 "listrel1p r xs ys = ((xs, ys) \<in> listrel1 {(x, y). r x y})"
  5324 
  5325 lemma [code_unfold]:
  5326   "(xs, ys) \<in> listrel1 r = listrel1p (\<lambda>x y. (x, y) \<in> r) xs ys"
  5327 unfolding listrel1p_def by auto
  5328 
  5329 lemma [code]:
  5330   "listrel1p r [] xs = False"
  5331   "listrel1p r xs [] =  False"
  5332   "listrel1p r (x # xs) (y # ys) \<longleftrightarrow>
  5333      r x y \<and> xs = ys \<or> x = y \<and> listrel1p r xs ys"
  5334 by (simp add: listrel1p_def)+
  5335 
  5336 definition
  5337   lexordp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
  5338   "lexordp r xs ys = ((xs, ys) \<in> lexord {(x, y). r x y})"
  5339 
  5340 lemma [code_unfold]:
  5341   "(xs, ys) \<in> lexord r = lexordp (\<lambda>x y. (x, y) \<in> r) xs ys"
  5342 unfolding lexordp_def by auto
  5343 
  5344 lemma [code]:
  5345   "lexordp r xs [] = False"
  5346   "lexordp r [] (y#ys) = True"
  5347   "lexordp r (x # xs) (y # ys) = (r x y | (x = y & lexordp r xs ys))"
  5348 unfolding lexordp_def by auto
  5349 
  5350 text {* Bounded quantification and summation over nats. *}
  5351 
  5352 lemma atMost_upto [code_unfold]:
  5353   "{..n} = set [0..<Suc n]"
  5354   by auto
  5355 
  5356 lemma atLeast_upt [code_unfold]:
  5357   "{..<n} = set [0..<n]"
  5358   by auto
  5359 
  5360 lemma greaterThanLessThan_upt [code_unfold]:
  5361   "{n<..<m} = set [Suc n..<m]"
  5362   by auto
  5363 
  5364 lemmas atLeastLessThan_upt [code_unfold] = set_upt [symmetric]
  5365 
  5366 lemma greaterThanAtMost_upt [code_unfold]:
  5367   "{n<..m} = set [Suc n..<Suc m]"
  5368   by auto
  5369 
  5370 lemma atLeastAtMost_upt [code_unfold]:
  5371   "{n..m} = set [n..<Suc m]"
  5372   by auto
  5373 
  5374 lemma all_nat_less_eq [code_unfold]:
  5375   "(\<forall>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..<n}. P m)"
  5376   by auto
  5377 
  5378 lemma ex_nat_less_eq [code_unfold]:
  5379   "(\<exists>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..<n}. P m)"
  5380   by auto
  5381 
  5382 lemma all_nat_less [code_unfold]:
  5383   "(\<forall>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..n}. P m)"
  5384   by auto
  5385 
  5386 lemma ex_nat_less [code_unfold]:
  5387   "(\<exists>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..n}. P m)"
  5388   by auto
  5389 
  5390 lemma setsum_set_upt_conv_listsum_nat [code_unfold]:
  5391   "setsum f (set [m..<n]) = listsum (map f [m..<n])"
  5392   by (simp add: interv_listsum_conv_setsum_set_nat)
  5393 
  5394 text {* Summation over ints. *}
  5395 
  5396 lemma greaterThanLessThan_upto [code_unfold]:
  5397   "{i<..<j::int} = set [i+1..j - 1]"
  5398 by auto
  5399 
  5400 lemma atLeastLessThan_upto [code_unfold]:
  5401   "{i..<j::int} = set [i..j - 1]"
  5402 by auto
  5403 
  5404 lemma greaterThanAtMost_upto [code_unfold]:
  5405   "{i<..j::int} = set [i+1..j]"
  5406 by auto
  5407 
  5408 lemmas atLeastAtMost_upto [code_unfold] = set_upto [symmetric]
  5409 
  5410 lemma setsum_set_upto_conv_listsum_int [code_unfold]:
  5411   "setsum f (set [i..j::int]) = listsum (map f [i..j])"
  5412   by (simp add: interv_listsum_conv_setsum_set_int)
  5413 
  5414 
  5415 subsubsection {* Optimizing by rewriting *}
  5416 
  5417 definition null :: "'a list \<Rightarrow> bool" where
  5418   [code_abbrev]: "null xs \<longleftrightarrow> xs = []"
  5419 
  5420 text {*
  5421   Efficient emptyness check is implemented by @{const null}.
  5422 *}
  5423 
  5424 lemma null_rec [code]:
  5425   "null (x # xs) \<longleftrightarrow> False"
  5426   "null [] \<longleftrightarrow> True"
  5427   by (simp_all add: null_def)
  5428 
  5429 lemma eq_Nil_null: (* FIXME delete candidate *)
  5430   "xs = [] \<longleftrightarrow> null xs"
  5431   by (simp add: null_def)
  5432 
  5433 lemma equal_Nil_null [code_unfold]:
  5434   "HOL.equal xs [] \<longleftrightarrow> null xs"
  5435   by (simp add: equal eq_Nil_null)
  5436 
  5437 definition maps :: "('a \<Rightarrow> 'b list) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  5438   [code_abbrev]: "maps f xs = concat (map f xs)"
  5439 
  5440 definition map_filter :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  5441   [code_post]: "map_filter f xs = map (the \<circ> f) (filter (\<lambda>x. f x \<noteq> None) xs)"
  5442 
  5443 text {*
  5444   Operations @{const maps} and @{const map_filter} avoid
  5445   intermediate lists on execution -- do not use for proving.
  5446 *}
  5447 
  5448 lemma maps_simps [code]:
  5449   "maps f (x # xs) = f x @ maps f xs"
  5450   "maps f [] = []"
  5451   by (simp_all add: maps_def)
  5452 
  5453 lemma map_filter_simps [code]:
  5454   "map_filter f (x # xs) = (case f x of None \<Rightarrow> map_filter f xs | Some y \<Rightarrow> y # map_filter f xs)"
  5455   "map_filter f [] = []"
  5456   by (simp_all add: map_filter_def split: option.split)
  5457 
  5458 lemma concat_map_maps: (* FIXME delete candidate *)
  5459   "concat (map f xs) = maps f xs"
  5460   by (simp add: maps_def)
  5461 
  5462 lemma map_filter_map_filter [code_unfold]:
  5463   "map f (filter P xs) = map_filter (\<lambda>x. if P x then Some (f x) else None) xs"
  5464   by (simp add: map_filter_def)
  5465 
  5466 text {* Optimized code for @{text"\<forall>i\<in>{a..b::int}"} and @{text"\<forall>n:{a..<b::nat}"}
  5467 and similiarly for @{text"\<exists>"}. *}
  5468 
  5469 definition all_interval_nat :: "(nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
  5470   "all_interval_nat P i j \<longleftrightarrow> (\<forall>n \<in> {i..<j}. P n)"
  5471 
  5472 lemma [code]:
  5473   "all_interval_nat P i j \<longleftrightarrow> i \<ge> j \<or> P i \<and> all_interval_nat P (Suc i) j"
  5474 proof -
  5475   have *: "\<And>n. P i \<Longrightarrow> \<forall>n\<in>{Suc i..<j}. P n \<Longrightarrow> i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n"
  5476   proof -
  5477     fix n
  5478     assume "P i" "\<forall>n\<in>{Suc i..<j}. P n" "i \<le> n" "n < j"
  5479     then show "P n" by (cases "n = i") simp_all
  5480   qed
  5481   show ?thesis by (auto simp add: all_interval_nat_def intro: *)
  5482 qed
  5483 
  5484 lemma list_all_iff_all_interval_nat [code_unfold]:
  5485   "list_all P [i..<j] \<longleftrightarrow> all_interval_nat P i j"
  5486   by (simp add: list_all_iff all_interval_nat_def)
  5487 
  5488 lemma list_ex_iff_not_all_inverval_nat [code_unfold]:
  5489   "list_ex P [i..<j] \<longleftrightarrow> \<not> (all_interval_nat (Not \<circ> P) i j)"
  5490   by (simp add: list_ex_iff all_interval_nat_def)
  5491 
  5492 definition all_interval_int :: "(int \<Rightarrow> bool) \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool" where
  5493   "all_interval_int P i j \<longleftrightarrow> (\<forall>k \<in> {i..j}. P k)"
  5494 
  5495 lemma [code]:
  5496   "all_interval_int P i j \<longleftrightarrow> i > j \<or> P i \<and> all_interval_int P (i + 1) j"
  5497 proof -
  5498   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"
  5499   proof -
  5500     fix k
  5501     assume "P i" "\<forall>k\<in>{i+1..j}. P k" "i \<le> k" "k \<le> j"
  5502     then show "P k" by (cases "k = i") simp_all
  5503   qed
  5504   show ?thesis by (auto simp add: all_interval_int_def intro: *)
  5505 qed
  5506 
  5507 lemma list_all_iff_all_interval_int [code_unfold]:
  5508   "list_all P [i..j] \<longleftrightarrow> all_interval_int P i j"
  5509   by (simp add: list_all_iff all_interval_int_def)
  5510 
  5511 lemma list_ex_iff_not_all_inverval_int [code_unfold]:
  5512   "list_ex P [i..j] \<longleftrightarrow> \<not> (all_interval_int (Not \<circ> P) i j)"
  5513   by (simp add: list_ex_iff all_interval_int_def)
  5514 
  5515 hide_const (open) member null maps map_filter all_interval_nat all_interval_int
  5516 
  5517 
  5518 subsubsection {* Pretty lists *}
  5519 
  5520 use "Tools/list_code.ML"
  5521 
  5522 code_type list
  5523   (SML "_ list")
  5524   (OCaml "_ list")
  5525   (Haskell "![(_)]")
  5526   (Scala "List[(_)]")
  5527 
  5528 code_const Nil
  5529   (SML "[]")
  5530   (OCaml "[]")
  5531   (Haskell "[]")
  5532   (Scala "!Nil")
  5533 
  5534 code_instance list :: equal
  5535   (Haskell -)
  5536 
  5537 code_const "HOL.equal \<Colon> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
  5538   (Haskell infix 4 "==")
  5539 
  5540 code_reserved SML
  5541   list
  5542 
  5543 code_reserved OCaml
  5544   list
  5545 
  5546 setup {* fold (List_Code.add_literal_list) ["SML", "OCaml", "Haskell", "Scala"] *}
  5547 
  5548 
  5549 subsubsection {* Use convenient predefined operations *}
  5550 
  5551 code_const "op @"
  5552   (SML infixr 7 "@")
  5553   (OCaml infixr 6 "@")
  5554   (Haskell infixr 5 "++")
  5555   (Scala infixl 7 "++")
  5556 
  5557 code_const map
  5558   (Haskell "map")
  5559 
  5560 code_const filter
  5561   (Haskell "filter")
  5562 
  5563 code_const concat
  5564   (Haskell "concat")
  5565 
  5566 code_const List.maps
  5567   (Haskell "concatMap")
  5568 
  5569 code_const rev
  5570   (Haskell "reverse")
  5571 
  5572 code_const zip
  5573   (Haskell "zip")
  5574 
  5575 code_const List.null
  5576   (Haskell "null")
  5577 
  5578 code_const takeWhile
  5579   (Haskell "takeWhile")
  5580 
  5581 code_const dropWhile
  5582   (Haskell "dropWhile")
  5583 
  5584 code_const list_all
  5585   (Haskell "all")
  5586 
  5587 code_const list_ex
  5588   (Haskell "any")
  5589 
  5590 
  5591 subsubsection {* Implementation of sets by lists *}
  5592 
  5593 lemma is_empty_set [code]:
  5594   "Set.is_empty (set xs) \<longleftrightarrow> List.null xs"
  5595   by (simp add: Set.is_empty_def null_def)
  5596 
  5597 lemma empty_set [code]:
  5598   "{} = set []"
  5599   by simp
  5600 
  5601 lemma UNIV_coset [code]:
  5602   "UNIV = List.coset []"
  5603   by simp
  5604 
  5605 lemma compl_set [code]:
  5606   "- set xs = List.coset xs"
  5607   by simp
  5608 
  5609 lemma compl_coset [code]:
  5610   "- List.coset xs = set xs"
  5611   by simp
  5612 
  5613 lemma [code]:
  5614   "x \<in> set xs \<longleftrightarrow> List.member xs x"
  5615   "x \<in> List.coset xs \<longleftrightarrow> \<not> List.member xs x"
  5616   by (simp_all add: member_def)
  5617 
  5618 lemma insert_code [code]:
  5619   "insert x (set xs) = set (List.insert x xs)"
  5620   "insert x (List.coset xs) = List.coset (removeAll x xs)"
  5621   by simp_all
  5622 
  5623 lemma remove_code [code]:
  5624   "Set.remove x (set xs) = set (removeAll x xs)"
  5625   "Set.remove x (List.coset xs) = List.coset (List.insert x xs)"
  5626   by (simp_all add: remove_def Compl_insert)
  5627 
  5628 lemma project_set [code]:
  5629   "Set.project P (set xs) = set (filter P xs)"
  5630   by auto
  5631 
  5632 lemma image_set [code]:
  5633   "image f (set xs) = set (map f xs)"
  5634   by simp
  5635 
  5636 lemma subset_code [code]:
  5637   "set xs \<le> B \<longleftrightarrow> (\<forall>x\<in>set xs. x \<in> B)"
  5638   "A \<le> List.coset ys \<longleftrightarrow> (\<forall>y\<in>set ys. y \<notin> A)"
  5639   "List.coset [] \<le> set [] \<longleftrightarrow> False"
  5640   by auto
  5641 
  5642 text {* A frequent case – avoid intermediate sets *}
  5643 lemma [code_unfold]:
  5644   "set xs \<subseteq> set ys \<longleftrightarrow> list_all (\<lambda>x. x \<in> set ys) xs"
  5645   by (auto simp: list_all_iff)
  5646 
  5647 lemma Ball_set [code]:
  5648   "Ball (set xs) P \<longleftrightarrow> list_all P xs"
  5649   by (simp add: list_all_iff)
  5650 
  5651 lemma Bex_set [code]:
  5652   "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
  5653   by (simp add: list_ex_iff)
  5654 
  5655 lemma card_set [code]:
  5656   "card (set xs) = length (remdups xs)"
  5657 proof -
  5658   have "card (set (remdups xs)) = length (remdups xs)"
  5659     by (rule distinct_card) simp
  5660   then show ?thesis by simp
  5661 qed
  5662 
  5663 lemma the_elem_set [code]:
  5664   "the_elem (set [x]) = x"
  5665   by simp
  5666 
  5667 lemma Pow_set [code]:
  5668   "Pow (set []) = {{}}"
  5669   "Pow (set (x # xs)) = (let A = Pow (set xs) in A \<union> insert x ` A)"
  5670   by (simp_all add: Pow_insert Let_def)
  5671 
  5672 lemma setsum_code [code]:
  5673   "setsum f (set xs) = listsum (map f (remdups xs))"
  5674 by (simp add: listsum_distinct_conv_setsum_set)
  5675 
  5676 definition map_project :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a set \<Rightarrow> 'b set" where
  5677   "map_project f A = {b. \<exists> a \<in> A. f a = Some b}"
  5678 
  5679 lemma [code]:
  5680   "map_project f (set xs) = set (List.map_filter f xs)"
  5681   by (auto simp add: map_project_def map_filter_def image_def)
  5682 
  5683 hide_const (open) map_project
  5684 
  5685 text {* Operations on relations *}
  5686 
  5687 lemma product_code [code]:
  5688   "Product_Type.product (set xs) (set ys) = set [(x, y). x \<leftarrow> xs, y \<leftarrow> ys]"
  5689   by (auto simp add: Product_Type.product_def)
  5690 
  5691 lemma Id_on_set [code]:
  5692   "Id_on (set xs) = set [(x, x). x \<leftarrow> xs]"
  5693   by (auto simp add: Id_on_def)
  5694 
  5695 lemma [code]:
  5696   "R `` S = List.map_project (%(x, y). if x : S then Some y else None) R"
  5697 unfolding map_project_def by (auto split: prod.split split_if_asm)
  5698 
  5699 lemma trancl_set_ntrancl [code]:
  5700   "trancl (set xs) = ntrancl (card (set xs) - 1) (set xs)"
  5701   by (simp add: finite_trancl_ntranl)
  5702 
  5703 lemma set_relcomp [code]:
  5704   "set xys O set yzs = set ([(fst xy, snd yz). xy \<leftarrow> xys, yz \<leftarrow> yzs, snd xy = fst yz])"
  5705   by (auto simp add: Bex_def)
  5706 
  5707 lemma wf_set [code]:
  5708   "wf (set xs) = acyclic (set xs)"
  5709   by (simp add: wf_iff_acyclic_if_finite)
  5710 
  5711 end
  5712