src/HOL/List.thy
author bulwahn
Thu Apr 12 11:01:15 2012 +0200 (2012-04-12)
changeset 47436 d8fad618a67a
parent 47399 b72fa7bf9a10
parent 47433 07f4bf913230
child 47640 62bfba15b212
permissions -rw-r--r--
merged
     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. \<lbrakk>P x y; R xs ys\<rbrakk> \<Longrightarrow> R (x # xs) (y # ys)"
  2265   shows "R xs ys"
  2266 using P
  2267 by (induct xs arbitrary: ys) (auto simp add: list_all2_Cons1 Nil Cons)
  2268 
  2269 lemma list_all2_rev [iff]:
  2270 "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
  2271 by (simp add: list_all2_def zip_rev cong: conj_cong)
  2272 
  2273 lemma list_all2_rev1:
  2274 "list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
  2275 by (subst list_all2_rev [symmetric]) simp
  2276 
  2277 lemma list_all2_append1:
  2278 "list_all2 P (xs @ ys) zs =
  2279 (EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
  2280 list_all2 P xs us \<and> list_all2 P ys vs)"
  2281 apply (simp add: list_all2_def zip_append1)
  2282 apply (rule iffI)
  2283  apply (rule_tac x = "take (length xs) zs" in exI)
  2284  apply (rule_tac x = "drop (length xs) zs" in exI)
  2285  apply (force split: nat_diff_split simp add: min_def, clarify)
  2286 apply (simp add: ball_Un)
  2287 done
  2288 
  2289 lemma list_all2_append2:
  2290 "list_all2 P xs (ys @ zs) =
  2291 (EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
  2292 list_all2 P us ys \<and> list_all2 P vs zs)"
  2293 apply (simp add: list_all2_def zip_append2)
  2294 apply (rule iffI)
  2295  apply (rule_tac x = "take (length ys) xs" in exI)
  2296  apply (rule_tac x = "drop (length ys) xs" in exI)
  2297  apply (force split: nat_diff_split simp add: min_def, clarify)
  2298 apply (simp add: ball_Un)
  2299 done
  2300 
  2301 lemma list_all2_append:
  2302   "length xs = length ys \<Longrightarrow>
  2303   list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
  2304 by (induct rule:list_induct2, simp_all)
  2305 
  2306 lemma list_all2_appendI [intro?, trans]:
  2307   "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
  2308 by (simp add: list_all2_append list_all2_lengthD)
  2309 
  2310 lemma list_all2_conv_all_nth:
  2311 "list_all2 P xs ys =
  2312 (length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
  2313 by (force simp add: list_all2_def set_zip)
  2314 
  2315 lemma list_all2_trans:
  2316   assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
  2317   shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
  2318         (is "!!bs cs. PROP ?Q as bs cs")
  2319 proof (induct as)
  2320   fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
  2321   show "!!cs. PROP ?Q (x # xs) bs cs"
  2322   proof (induct bs)
  2323     fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
  2324     show "PROP ?Q (x # xs) (y # ys) cs"
  2325       by (induct cs) (auto intro: tr I1 I2)
  2326   qed simp
  2327 qed simp
  2328 
  2329 lemma list_all2_all_nthI [intro?]:
  2330   "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
  2331 by (simp add: list_all2_conv_all_nth)
  2332 
  2333 lemma list_all2I:
  2334   "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
  2335 by (simp add: list_all2_def)
  2336 
  2337 lemma list_all2_nthD:
  2338   "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
  2339 by (simp add: list_all2_conv_all_nth)
  2340 
  2341 lemma list_all2_nthD2:
  2342   "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
  2343 by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
  2344 
  2345 lemma list_all2_map1: 
  2346   "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
  2347 by (simp add: list_all2_conv_all_nth)
  2348 
  2349 lemma list_all2_map2: 
  2350   "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
  2351 by (auto simp add: list_all2_conv_all_nth)
  2352 
  2353 lemma list_all2_refl [intro?]:
  2354   "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
  2355 by (simp add: list_all2_conv_all_nth)
  2356 
  2357 lemma list_all2_update_cong:
  2358   "\<lbrakk> list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
  2359 by (cases "i < length ys") (auto simp add: list_all2_conv_all_nth nth_list_update)
  2360 
  2361 lemma list_all2_takeI [simp,intro?]:
  2362   "list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
  2363 apply (induct xs arbitrary: n ys)
  2364  apply simp
  2365 apply (clarsimp simp add: list_all2_Cons1)
  2366 apply (case_tac n)
  2367 apply auto
  2368 done
  2369 
  2370 lemma list_all2_dropI [simp,intro?]:
  2371   "list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
  2372 apply (induct as arbitrary: n bs, simp)
  2373 apply (clarsimp simp add: list_all2_Cons1)
  2374 apply (case_tac n, simp, simp)
  2375 done
  2376 
  2377 lemma list_all2_mono [intro?]:
  2378   "list_all2 P xs ys \<Longrightarrow> (\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys) \<Longrightarrow> list_all2 Q xs ys"
  2379 apply (induct xs arbitrary: ys, simp)
  2380 apply (case_tac ys, auto)
  2381 done
  2382 
  2383 lemma list_all2_eq:
  2384   "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
  2385 by (induct xs ys rule: list_induct2') auto
  2386 
  2387 lemma list_eq_iff_zip_eq:
  2388   "xs = ys \<longleftrightarrow> length xs = length ys \<and> (\<forall>(x,y) \<in> set (zip xs ys). x = y)"
  2389 by(auto simp add: set_zip list_all2_eq list_all2_conv_all_nth cong: conj_cong)
  2390 
  2391 
  2392 subsubsection {* @{const fold} with natural argument order *}
  2393 
  2394 lemma fold_remove1_split:
  2395   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"
  2396     and x: "x \<in> set xs"
  2397   shows "fold f xs = fold f (remove1 x xs) \<circ> f x"
  2398   using assms by (induct xs) (auto simp add: o_assoc [symmetric])
  2399 
  2400 lemma fold_cong [fundef_cong]:
  2401   "a = b \<Longrightarrow> xs = ys \<Longrightarrow> (\<And>x. x \<in> set xs \<Longrightarrow> f x = g x)
  2402     \<Longrightarrow> fold f xs a = fold g ys b"
  2403   by (induct ys arbitrary: a b xs) simp_all
  2404 
  2405 lemma fold_id:
  2406   assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x = id"
  2407   shows "fold f xs = id"
  2408   using assms by (induct xs) simp_all
  2409 
  2410 lemma fold_commute:
  2411   assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
  2412   shows "h \<circ> fold g xs = fold f xs \<circ> h"
  2413   using assms by (induct xs) (simp_all add: fun_eq_iff)
  2414 
  2415 lemma fold_commute_apply:
  2416   assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
  2417   shows "h (fold g xs s) = fold f xs (h s)"
  2418 proof -
  2419   from assms have "h \<circ> fold g xs = fold f xs \<circ> h" by (rule fold_commute)
  2420   then show ?thesis by (simp add: fun_eq_iff)
  2421 qed
  2422 
  2423 lemma fold_invariant: 
  2424   assumes "\<And>x. x \<in> set xs \<Longrightarrow> Q x" and "P s"
  2425     and "\<And>x s. Q x \<Longrightarrow> P s \<Longrightarrow> P (f x s)"
  2426   shows "P (fold f xs s)"
  2427   using assms by (induct xs arbitrary: s) simp_all
  2428 
  2429 lemma fold_append [simp]:
  2430   "fold f (xs @ ys) = fold f ys \<circ> fold f xs"
  2431   by (induct xs) simp_all
  2432 
  2433 lemma fold_map [code_unfold]:
  2434   "fold g (map f xs) = fold (g o f) xs"
  2435   by (induct xs) simp_all
  2436 
  2437 lemma fold_rev:
  2438   assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
  2439   shows "fold f (rev xs) = fold f xs"
  2440 using assms by (induct xs) (simp_all add: fold_commute_apply fun_eq_iff)
  2441 
  2442 lemma fold_Cons_rev:
  2443   "fold Cons xs = append (rev xs)"
  2444   by (induct xs) simp_all
  2445 
  2446 lemma rev_conv_fold [code]:
  2447   "rev xs = fold Cons xs []"
  2448   by (simp add: fold_Cons_rev)
  2449 
  2450 lemma fold_append_concat_rev:
  2451   "fold append xss = append (concat (rev xss))"
  2452   by (induct xss) simp_all
  2453 
  2454 text {* @{const Finite_Set.fold} and @{const fold} *}
  2455 
  2456 lemma (in comp_fun_commute) fold_set_fold_remdups:
  2457   "Finite_Set.fold f y (set xs) = fold f (remdups xs) y"
  2458   by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm insert_absorb)
  2459 
  2460 lemma (in comp_fun_idem) fold_set_fold:
  2461   "Finite_Set.fold f y (set xs) = fold f xs y"
  2462   by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm)
  2463 
  2464 lemma (in ab_semigroup_idem_mult) fold1_set_fold:
  2465   assumes "xs \<noteq> []"
  2466   shows "Finite_Set.fold1 times (set xs) = fold times (tl xs) (hd xs)"
  2467 proof -
  2468   interpret comp_fun_idem times by (fact comp_fun_idem)
  2469   from assms obtain y ys where xs: "xs = y # ys"
  2470     by (cases xs) 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 (insert y (set ys)) = Finite_Set.fold times y (set ys)"
  2477       by (simp only: finite_set fold1_eq_fold_idem)
  2478     with xs show ?thesis by (simp add: fold_set_fold mult_commute)
  2479   qed
  2480 qed
  2481 
  2482 lemma union_set_fold [code]:
  2483   "set xs \<union> A = fold Set.insert xs A"
  2484 proof -
  2485   interpret comp_fun_idem Set.insert
  2486     by (fact comp_fun_idem_insert)
  2487   show ?thesis by (simp add: union_fold_insert fold_set_fold)
  2488 qed
  2489 
  2490 lemma union_coset_filter [code]:
  2491   "List.coset xs \<union> A = List.coset (List.filter (\<lambda>x. x \<notin> A) xs)"
  2492   by auto
  2493 
  2494 lemma minus_set_fold [code]:
  2495   "A - set xs = fold Set.remove xs A"
  2496 proof -
  2497   interpret comp_fun_idem Set.remove
  2498     by (fact comp_fun_idem_remove)
  2499   show ?thesis
  2500     by (simp add: minus_fold_remove [of _ A] fold_set_fold)
  2501 qed
  2502 
  2503 lemma minus_coset_filter [code]:
  2504   "A - List.coset xs = set (List.filter (\<lambda>x. x \<in> A) xs)"
  2505   by auto
  2506 
  2507 lemma inter_set_filter [code]:
  2508   "A \<inter> set xs = set (List.filter (\<lambda>x. x \<in> A) xs)"
  2509   by auto
  2510 
  2511 lemma inter_coset_fold [code]:
  2512   "A \<inter> List.coset xs = fold Set.remove xs A"
  2513   by (simp add: Diff_eq [symmetric] minus_set_fold)
  2514 
  2515 lemma (in lattice) Inf_fin_set_fold:
  2516   "Inf_fin (set (x # xs)) = fold inf xs x"
  2517 proof -
  2518   interpret ab_semigroup_idem_mult "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2519     by (fact ab_semigroup_idem_mult_inf)
  2520   show ?thesis
  2521     by (simp add: Inf_fin_def fold1_set_fold del: set.simps)
  2522 qed
  2523 
  2524 declare Inf_fin_set_fold [code]
  2525 
  2526 lemma (in lattice) Sup_fin_set_fold:
  2527   "Sup_fin (set (x # xs)) = fold sup xs x"
  2528 proof -
  2529   interpret ab_semigroup_idem_mult "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2530     by (fact ab_semigroup_idem_mult_sup)
  2531   show ?thesis
  2532     by (simp add: Sup_fin_def fold1_set_fold del: set.simps)
  2533 qed
  2534 
  2535 declare Sup_fin_set_fold [code]
  2536 
  2537 lemma (in linorder) Min_fin_set_fold:
  2538   "Min (set (x # xs)) = fold min xs x"
  2539 proof -
  2540   interpret ab_semigroup_idem_mult "min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2541     by (fact ab_semigroup_idem_mult_min)
  2542   show ?thesis
  2543     by (simp add: Min_def fold1_set_fold del: set.simps)
  2544 qed
  2545 
  2546 declare Min_fin_set_fold [code]
  2547 
  2548 lemma (in linorder) Max_fin_set_fold:
  2549   "Max (set (x # xs)) = fold max xs x"
  2550 proof -
  2551   interpret ab_semigroup_idem_mult "max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2552     by (fact ab_semigroup_idem_mult_max)
  2553   show ?thesis
  2554     by (simp add: Max_def fold1_set_fold del: set.simps)
  2555 qed
  2556 
  2557 declare Max_fin_set_fold [code]
  2558 
  2559 lemma (in complete_lattice) Inf_set_fold:
  2560   "Inf (set xs) = fold inf xs top"
  2561 proof -
  2562   interpret comp_fun_idem "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2563     by (fact comp_fun_idem_inf)
  2564   show ?thesis by (simp add: Inf_fold_inf fold_set_fold inf_commute)
  2565 qed
  2566 
  2567 declare Inf_set_fold [where 'a = "'a set", code]
  2568 
  2569 lemma (in complete_lattice) Sup_set_fold:
  2570   "Sup (set xs) = fold sup xs bot"
  2571 proof -
  2572   interpret comp_fun_idem "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2573     by (fact comp_fun_idem_sup)
  2574   show ?thesis by (simp add: Sup_fold_sup fold_set_fold sup_commute)
  2575 qed
  2576 
  2577 declare Sup_set_fold [where 'a = "'a set", code]
  2578 
  2579 lemma (in complete_lattice) INF_set_fold:
  2580   "INFI (set xs) f = fold (inf \<circ> f) xs top"
  2581   unfolding INF_def set_map [symmetric] Inf_set_fold fold_map ..
  2582 
  2583 declare INF_set_fold [code]
  2584 
  2585 lemma (in complete_lattice) SUP_set_fold:
  2586   "SUPR (set xs) f = fold (sup \<circ> f) xs bot"
  2587   unfolding SUP_def set_map [symmetric] Sup_set_fold fold_map ..
  2588 
  2589 declare SUP_set_fold [code]
  2590 
  2591 subsubsection {* Fold variants: @{const foldr} and @{const foldl} *}
  2592 
  2593 text {* Correspondence *}
  2594 
  2595 lemma foldr_conv_fold [code_abbrev]:
  2596   "foldr f xs = fold f (rev xs)"
  2597   by (induct xs) simp_all
  2598 
  2599 lemma foldl_conv_fold:
  2600   "foldl f s xs = fold (\<lambda>x s. f s x) xs s"
  2601   by (induct xs arbitrary: s) simp_all
  2602 
  2603 lemma foldr_conv_foldl: -- {* The ``Third Duality Theorem'' in Bird \& Wadler: *}
  2604   "foldr f xs a = foldl (\<lambda>x y. f y x) a (rev xs)"
  2605   by (simp add: foldr_conv_fold foldl_conv_fold)
  2606 
  2607 lemma foldl_conv_foldr:
  2608   "foldl f a xs = foldr (\<lambda>x y. f y x) (rev xs) a"
  2609   by (simp add: foldr_conv_fold foldl_conv_fold)
  2610 
  2611 lemma foldr_fold:
  2612   assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
  2613   shows "foldr f xs = fold f xs"
  2614   using assms unfolding foldr_conv_fold by (rule fold_rev)
  2615 
  2616 lemma foldr_cong [fundef_cong]:
  2617   "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"
  2618   by (auto simp add: foldr_conv_fold intro!: fold_cong)
  2619 
  2620 lemma foldl_cong [fundef_cong]:
  2621   "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"
  2622   by (auto simp add: foldl_conv_fold intro!: fold_cong)
  2623 
  2624 lemma foldr_append [simp]:
  2625   "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
  2626   by (simp add: foldr_conv_fold)
  2627 
  2628 lemma foldl_append [simp]:
  2629   "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
  2630   by (simp add: foldl_conv_fold)
  2631 
  2632 lemma foldr_map [code_unfold]:
  2633   "foldr g (map f xs) a = foldr (g o f) xs a"
  2634   by (simp add: foldr_conv_fold fold_map rev_map)
  2635 
  2636 lemma foldl_map [code_unfold]:
  2637   "foldl g a (map f xs) = foldl (\<lambda>a x. g a (f x)) a xs"
  2638   by (simp add: foldl_conv_fold fold_map comp_def)
  2639 
  2640 lemma concat_conv_foldr [code]:
  2641   "concat xss = foldr append xss []"
  2642   by (simp add: fold_append_concat_rev foldr_conv_fold)
  2643 
  2644 
  2645 subsubsection {* @{text upt} *}
  2646 
  2647 lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
  2648 -- {* simp does not terminate! *}
  2649 by (induct j) auto
  2650 
  2651 lemmas upt_rec_numeral[simp] = upt_rec[of "numeral m" "numeral n"] for m n
  2652 
  2653 lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
  2654 by (subst upt_rec) simp
  2655 
  2656 lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
  2657 by(induct j)simp_all
  2658 
  2659 lemma upt_eq_Cons_conv:
  2660  "([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
  2661 apply(induct j arbitrary: x xs)
  2662  apply simp
  2663 apply(clarsimp simp add: append_eq_Cons_conv)
  2664 apply arith
  2665 done
  2666 
  2667 lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
  2668 -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
  2669 by simp
  2670 
  2671 lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
  2672   by (simp add: upt_rec)
  2673 
  2674 lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
  2675 -- {* LOOPS as a simprule, since @{text "j <= j"}. *}
  2676 by (induct k) auto
  2677 
  2678 lemma length_upt [simp]: "length [i..<j] = j - i"
  2679 by (induct j) (auto simp add: Suc_diff_le)
  2680 
  2681 lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
  2682 apply (induct j)
  2683 apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
  2684 done
  2685 
  2686 
  2687 lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
  2688 by(simp add:upt_conv_Cons)
  2689 
  2690 lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
  2691 apply(cases j)
  2692  apply simp
  2693 by(simp add:upt_Suc_append)
  2694 
  2695 lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]"
  2696 apply (induct m arbitrary: i, simp)
  2697 apply (subst upt_rec)
  2698 apply (rule sym)
  2699 apply (subst upt_rec)
  2700 apply (simp del: upt.simps)
  2701 done
  2702 
  2703 lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
  2704 apply(induct j)
  2705 apply auto
  2706 done
  2707 
  2708 lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]"
  2709 by (induct n) auto
  2710 
  2711 lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
  2712 apply (induct n m  arbitrary: i rule: diff_induct)
  2713 prefer 3 apply (subst map_Suc_upt[symmetric])
  2714 apply (auto simp add: less_diff_conv)
  2715 done
  2716 
  2717 lemma nth_take_lemma:
  2718   "k <= length xs ==> k <= length ys ==>
  2719      (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
  2720 apply (atomize, induct k arbitrary: xs ys)
  2721 apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
  2722 txt {* Both lists must be non-empty *}
  2723 apply (case_tac xs, simp)
  2724 apply (case_tac ys, clarify)
  2725  apply (simp (no_asm_use))
  2726 apply clarify
  2727 txt {* prenexing's needed, not miniscoping *}
  2728 apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
  2729 apply blast
  2730 done
  2731 
  2732 lemma nth_equalityI:
  2733  "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
  2734   by (frule nth_take_lemma [OF le_refl eq_imp_le]) simp_all
  2735 
  2736 lemma map_nth:
  2737   "map (\<lambda>i. xs ! i) [0..<length xs] = xs"
  2738   by (rule nth_equalityI, auto)
  2739 
  2740 (* needs nth_equalityI *)
  2741 lemma list_all2_antisym:
  2742   "\<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> 
  2743   \<Longrightarrow> xs = ys"
  2744   apply (simp add: list_all2_conv_all_nth) 
  2745   apply (rule nth_equalityI, blast, simp)
  2746   done
  2747 
  2748 lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
  2749 -- {* The famous take-lemma. *}
  2750 apply (drule_tac x = "max (length xs) (length ys)" in spec)
  2751 apply (simp add: le_max_iff_disj)
  2752 done
  2753 
  2754 
  2755 lemma take_Cons':
  2756      "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
  2757 by (cases n) simp_all
  2758 
  2759 lemma drop_Cons':
  2760      "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
  2761 by (cases n) simp_all
  2762 
  2763 lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
  2764 by (cases n) simp_all
  2765 
  2766 lemma take_Cons_numeral [simp]:
  2767   "take (numeral v) (x # xs) = x # take (numeral v - 1) xs"
  2768 by (simp add: take_Cons')
  2769 
  2770 lemma drop_Cons_numeral [simp]:
  2771   "drop (numeral v) (x # xs) = drop (numeral v - 1) xs"
  2772 by (simp add: drop_Cons')
  2773 
  2774 lemma nth_Cons_numeral [simp]:
  2775   "(x # xs) ! numeral v = xs ! (numeral v - 1)"
  2776 by (simp add: nth_Cons')
  2777 
  2778 
  2779 subsubsection {* @{text upto}: interval-list on @{typ int} *}
  2780 
  2781 (* FIXME make upto tail recursive? *)
  2782 
  2783 function upto :: "int \<Rightarrow> int \<Rightarrow> int list" ("(1[_../_])") where
  2784 "upto i j = (if i \<le> j then i # [i+1..j] else [])"
  2785 by auto
  2786 termination
  2787 by(relation "measure(%(i::int,j). nat(j - i + 1))") auto
  2788 
  2789 declare upto.simps[code, simp del]
  2790 
  2791 lemmas upto_rec_numeral [simp] =
  2792   upto.simps[of "numeral m" "numeral n"]
  2793   upto.simps[of "numeral m" "neg_numeral n"]
  2794   upto.simps[of "neg_numeral m" "numeral n"]
  2795   upto.simps[of "neg_numeral m" "neg_numeral n"] for m n
  2796 
  2797 lemma upto_empty[simp]: "j < i \<Longrightarrow> [i..j] = []"
  2798 by(simp add: upto.simps)
  2799 
  2800 lemma set_upto[simp]: "set[i..j] = {i..j}"
  2801 proof(induct i j rule:upto.induct)
  2802   case (1 i j)
  2803   from this show ?case
  2804     unfolding upto.simps[of i j] simp_from_to[of i j] by auto
  2805 qed
  2806 
  2807 
  2808 subsubsection {* @{text "distinct"} and @{text remdups} *}
  2809 
  2810 lemma distinct_tl:
  2811   "distinct xs \<Longrightarrow> distinct (tl xs)"
  2812   by (cases xs) simp_all
  2813 
  2814 lemma distinct_append [simp]:
  2815 "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
  2816 by (induct xs) auto
  2817 
  2818 lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
  2819 by(induct xs) auto
  2820 
  2821 lemma set_remdups [simp]: "set (remdups xs) = set xs"
  2822 by (induct xs) (auto simp add: insert_absorb)
  2823 
  2824 lemma distinct_remdups [iff]: "distinct (remdups xs)"
  2825 by (induct xs) auto
  2826 
  2827 lemma distinct_remdups_id: "distinct xs ==> remdups xs = xs"
  2828 by (induct xs, auto)
  2829 
  2830 lemma remdups_id_iff_distinct [simp]: "remdups xs = xs \<longleftrightarrow> distinct xs"
  2831 by (metis distinct_remdups distinct_remdups_id)
  2832 
  2833 lemma finite_distinct_list: "finite A \<Longrightarrow> EX xs. set xs = A & distinct xs"
  2834 by (metis distinct_remdups finite_list set_remdups)
  2835 
  2836 lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
  2837 by (induct x, auto)
  2838 
  2839 lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
  2840 by (induct x, auto)
  2841 
  2842 lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
  2843 by (induct xs) auto
  2844 
  2845 lemma length_remdups_eq[iff]:
  2846   "(length (remdups xs) = length xs) = (remdups xs = xs)"
  2847 apply(induct xs)
  2848  apply auto
  2849 apply(subgoal_tac "length (remdups xs) <= length xs")
  2850  apply arith
  2851 apply(rule length_remdups_leq)
  2852 done
  2853 
  2854 lemma remdups_filter: "remdups(filter P xs) = filter P (remdups xs)"
  2855 apply(induct xs)
  2856 apply auto
  2857 done
  2858 
  2859 lemma distinct_map:
  2860   "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
  2861 by (induct xs) auto
  2862 
  2863 
  2864 lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
  2865 by (induct xs) auto
  2866 
  2867 lemma distinct_upt[simp]: "distinct[i..<j]"
  2868 by (induct j) auto
  2869 
  2870 lemma distinct_upto[simp]: "distinct[i..j]"
  2871 apply(induct i j rule:upto.induct)
  2872 apply(subst upto.simps)
  2873 apply(simp)
  2874 done
  2875 
  2876 lemma distinct_take[simp]: "distinct xs \<Longrightarrow> distinct (take i xs)"
  2877 apply(induct xs arbitrary: i)
  2878  apply simp
  2879 apply (case_tac i)
  2880  apply simp_all
  2881 apply(blast dest:in_set_takeD)
  2882 done
  2883 
  2884 lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)"
  2885 apply(induct xs arbitrary: i)
  2886  apply simp
  2887 apply (case_tac i)
  2888  apply simp_all
  2889 done
  2890 
  2891 lemma distinct_list_update:
  2892 assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
  2893 shows "distinct (xs[i:=a])"
  2894 proof (cases "i < length xs")
  2895   case True
  2896   with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
  2897     apply (drule_tac id_take_nth_drop) by simp
  2898   with d True show ?thesis
  2899     apply (simp add: upd_conv_take_nth_drop)
  2900     apply (drule subst [OF id_take_nth_drop]) apply assumption
  2901     apply simp apply (cases "a = xs!i") apply simp by blast
  2902 next
  2903   case False with d show ?thesis by auto
  2904 qed
  2905 
  2906 lemma distinct_concat:
  2907   assumes "distinct xs"
  2908   and "\<And> ys. ys \<in> set xs \<Longrightarrow> distinct ys"
  2909   and "\<And> ys zs. \<lbrakk> ys \<in> set xs ; zs \<in> set xs ; ys \<noteq> zs \<rbrakk> \<Longrightarrow> set ys \<inter> set zs = {}"
  2910   shows "distinct (concat xs)"
  2911   using assms by (induct xs) auto
  2912 
  2913 text {* It is best to avoid this indexed version of distinct, but
  2914 sometimes it is useful. *}
  2915 
  2916 lemma distinct_conv_nth:
  2917 "distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
  2918 apply (induct xs, simp, simp)
  2919 apply (rule iffI, clarsimp)
  2920  apply (case_tac i)
  2921 apply (case_tac j, simp)
  2922 apply (simp add: set_conv_nth)
  2923  apply (case_tac j)
  2924 apply (clarsimp simp add: set_conv_nth, simp)
  2925 apply (rule conjI)
  2926 (*TOO SLOW
  2927 apply (metis Zero_neq_Suc gr0_conv_Suc in_set_conv_nth lessI less_trans_Suc nth_Cons' nth_Cons_Suc)
  2928 *)
  2929  apply (clarsimp simp add: set_conv_nth)
  2930  apply (erule_tac x = 0 in allE, simp)
  2931  apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
  2932 (*TOO SLOW
  2933 apply (metis Suc_Suc_eq lessI less_trans_Suc nth_Cons_Suc)
  2934 *)
  2935 apply (erule_tac x = "Suc i" in allE, simp)
  2936 apply (erule_tac x = "Suc j" in allE, simp)
  2937 done
  2938 
  2939 lemma nth_eq_iff_index_eq:
  2940  "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
  2941 by(auto simp: distinct_conv_nth)
  2942 
  2943 lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
  2944 by (induct xs) auto
  2945 
  2946 lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
  2947 proof (induct xs)
  2948   case Nil thus ?case by simp
  2949 next
  2950   case (Cons x xs)
  2951   show ?case
  2952   proof (cases "x \<in> set xs")
  2953     case False with Cons show ?thesis by simp
  2954   next
  2955     case True with Cons.prems
  2956     have "card (set xs) = Suc (length xs)"
  2957       by (simp add: card_insert_if split: split_if_asm)
  2958     moreover have "card (set xs) \<le> length xs" by (rule card_length)
  2959     ultimately have False by simp
  2960     thus ?thesis ..
  2961   qed
  2962 qed
  2963 
  2964 lemma distinct_length_filter: "distinct xs \<Longrightarrow> length (filter P xs) = card ({x. P x} Int set xs)"
  2965 by (induct xs) (auto)
  2966 
  2967 lemma not_distinct_decomp: "~ distinct ws ==> EX xs ys zs y. ws = xs@[y]@ys@[y]@zs"
  2968 apply (induct n == "length ws" arbitrary:ws) apply simp
  2969 apply(case_tac ws) apply simp
  2970 apply (simp split:split_if_asm)
  2971 apply (metis Cons_eq_appendI eq_Nil_appendI split_list)
  2972 done
  2973 
  2974 lemma not_distinct_conv_prefix:
  2975   defines "dec as xs y ys \<equiv> y \<in> set xs \<and> distinct xs \<and> as = xs @ y # ys"
  2976   shows "\<not>distinct as \<longleftrightarrow> (\<exists>xs y ys. dec as xs y ys)" (is "?L = ?R")
  2977 proof
  2978   assume "?L" then show "?R"
  2979   proof (induct "length as" arbitrary: as rule: less_induct)
  2980     case less
  2981     obtain xs ys zs y where decomp: "as = (xs @ y # ys) @ y # zs"
  2982       using not_distinct_decomp[OF less.prems] by auto
  2983     show ?case
  2984     proof (cases "distinct (xs @ y # ys)")
  2985       case True
  2986       with decomp have "dec as (xs @ y # ys) y zs" by (simp add: dec_def)
  2987       then show ?thesis by blast
  2988     next
  2989       case False
  2990       with less decomp obtain xs' y' ys' where "dec (xs @ y # ys) xs' y' ys'"
  2991         by atomize_elim auto
  2992       with decomp have "dec as xs' y' (ys' @ y # zs)" by (simp add: dec_def)
  2993       then show ?thesis by blast
  2994     qed
  2995   qed
  2996 qed (auto simp: dec_def)
  2997 
  2998 lemma length_remdups_concat:
  2999   "length (remdups (concat xss)) = card (\<Union>xs\<in>set xss. set xs)"
  3000   by (simp add: distinct_card [symmetric])
  3001 
  3002 lemma length_remdups_card_conv: "length(remdups xs) = card(set xs)"
  3003 proof -
  3004   have xs: "concat[xs] = xs" by simp
  3005   from length_remdups_concat[of "[xs]"] show ?thesis unfolding xs by simp
  3006 qed
  3007 
  3008 lemma remdups_remdups:
  3009   "remdups (remdups xs) = remdups xs"
  3010   by (induct xs) simp_all
  3011 
  3012 lemma distinct_butlast:
  3013   assumes "distinct xs"
  3014   shows "distinct (butlast xs)"
  3015 proof (cases "xs = []")
  3016   case False
  3017     from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  3018     with `distinct xs` show ?thesis by simp
  3019 qed (auto)
  3020 
  3021 lemma remdups_map_remdups:
  3022   "remdups (map f (remdups xs)) = remdups (map f xs)"
  3023   by (induct xs) simp_all
  3024 
  3025 lemma distinct_zipI1:
  3026   assumes "distinct xs"
  3027   shows "distinct (zip xs ys)"
  3028 proof (rule zip_obtain_same_length)
  3029   fix xs' :: "'a list" and ys' :: "'b list" and n
  3030   assume "length xs' = length ys'"
  3031   assume "xs' = take n xs"
  3032   with assms have "distinct xs'" by simp
  3033   with `length xs' = length ys'` show "distinct (zip xs' ys')"
  3034     by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
  3035 qed
  3036 
  3037 lemma distinct_zipI2:
  3038   assumes "distinct ys"
  3039   shows "distinct (zip xs ys)"
  3040 proof (rule zip_obtain_same_length)
  3041   fix xs' :: "'b list" and ys' :: "'a list" and n
  3042   assume "length xs' = length ys'"
  3043   assume "ys' = take n ys"
  3044   with assms have "distinct ys'" by simp
  3045   with `length xs' = length ys'` show "distinct (zip xs' ys')"
  3046     by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
  3047 qed
  3048 
  3049 lemma set_take_disj_set_drop_if_distinct:
  3050   "distinct vs \<Longrightarrow> i \<le> j \<Longrightarrow> set (take i vs) \<inter> set (drop j vs) = {}"
  3051 by (auto simp: in_set_conv_nth distinct_conv_nth)
  3052 
  3053 (* The next two lemmas help Sledgehammer. *)
  3054 
  3055 lemma distinct_singleton: "distinct [x]" by simp
  3056 
  3057 lemma distinct_length_2_or_more:
  3058 "distinct (a # b # xs) \<longleftrightarrow> (a \<noteq> b \<and> distinct (a # xs) \<and> distinct (b # xs))"
  3059 by (metis distinct.simps(2) hd.simps hd_in_set list.simps(2) set_ConsD set_rev_mp set_subset_Cons)
  3060 
  3061 subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
  3062 
  3063 lemma (in monoid_add) listsum_simps [simp]:
  3064   "listsum [] = 0"
  3065   "listsum (x # xs) = x + listsum xs"
  3066   by (simp_all add: listsum_def)
  3067 
  3068 lemma (in monoid_add) listsum_append [simp]:
  3069   "listsum (xs @ ys) = listsum xs + listsum ys"
  3070   by (induct xs) (simp_all add: add.assoc)
  3071 
  3072 lemma (in comm_monoid_add) listsum_rev [simp]:
  3073   "listsum (rev xs) = listsum xs"
  3074   by (simp add: listsum_def foldr_fold fold_rev fun_eq_iff add_ac)
  3075 
  3076 lemma (in monoid_add) fold_plus_listsum_rev:
  3077   "fold plus xs = plus (listsum (rev xs))"
  3078 proof
  3079   fix x
  3080   have "fold plus xs x = fold plus xs (x + 0)" by simp
  3081   also have "\<dots> = fold plus (x # xs) 0" by simp
  3082   also have "\<dots> = foldr plus (rev xs @ [x]) 0" by (simp add: foldr_conv_fold)
  3083   also have "\<dots> = listsum (rev xs @ [x])" by (simp add: listsum_def)
  3084   also have "\<dots> = listsum (rev xs) + listsum [x]" by simp
  3085   finally show "fold plus xs x = listsum (rev xs) + x" by simp
  3086 qed
  3087 
  3088 text{* Some syntactic sugar for summing a function over a list: *}
  3089 
  3090 syntax
  3091   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
  3092 syntax (xsymbols)
  3093   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
  3094 syntax (HTML output)
  3095   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
  3096 
  3097 translations -- {* Beware of argument permutation! *}
  3098   "SUM x<-xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  3099   "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  3100 
  3101 lemma (in comm_monoid_add) listsum_map_remove1:
  3102   "x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))"
  3103   by (induct xs) (auto simp add: ac_simps)
  3104 
  3105 lemma (in monoid_add) list_size_conv_listsum:
  3106   "list_size f xs = listsum (map f xs) + size xs"
  3107   by (induct xs) auto
  3108 
  3109 lemma (in monoid_add) length_concat:
  3110   "length (concat xss) = listsum (map length xss)"
  3111   by (induct xss) simp_all
  3112 
  3113 lemma (in monoid_add) listsum_map_filter:
  3114   assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
  3115   shows "listsum (map f (filter P xs)) = listsum (map f xs)"
  3116   using assms by (induct xs) auto
  3117 
  3118 lemma (in monoid_add) distinct_listsum_conv_Setsum:
  3119   "distinct xs \<Longrightarrow> listsum xs = Setsum (set xs)"
  3120   by (induct xs) simp_all
  3121 
  3122 lemma listsum_eq_0_nat_iff_nat [simp]:
  3123   "listsum ns = (0::nat) \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
  3124   by (induct ns) simp_all
  3125 
  3126 lemma member_le_listsum_nat:
  3127   "(n :: nat) \<in> set ns \<Longrightarrow> n \<le> listsum ns"
  3128   by (induct ns) auto
  3129 
  3130 lemma elem_le_listsum_nat:
  3131   "k < size ns \<Longrightarrow> ns ! k \<le> listsum (ns::nat list)"
  3132   by (rule member_le_listsum_nat) simp
  3133 
  3134 lemma listsum_update_nat:
  3135   "k < size ns \<Longrightarrow> listsum (ns[k := (n::nat)]) = listsum ns + n - ns ! k"
  3136 apply(induct ns arbitrary:k)
  3137  apply (auto split:nat.split)
  3138 apply(drule elem_le_listsum_nat)
  3139 apply arith
  3140 done
  3141 
  3142 lemma (in monoid_add) listsum_triv:
  3143   "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
  3144   by (induct xs) (simp_all add: left_distrib)
  3145 
  3146 lemma (in monoid_add) listsum_0 [simp]:
  3147   "(\<Sum>x\<leftarrow>xs. 0) = 0"
  3148   by (induct xs) (simp_all add: left_distrib)
  3149 
  3150 text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
  3151 lemma (in ab_group_add) uminus_listsum_map:
  3152   "- listsum (map f xs) = listsum (map (uminus \<circ> f) xs)"
  3153   by (induct xs) simp_all
  3154 
  3155 lemma (in comm_monoid_add) listsum_addf:
  3156   "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
  3157   by (induct xs) (simp_all add: algebra_simps)
  3158 
  3159 lemma (in ab_group_add) listsum_subtractf:
  3160   "(\<Sum>x\<leftarrow>xs. f x - g x) = listsum (map f xs) - listsum (map g xs)"
  3161   by (induct xs) (simp_all add: algebra_simps)
  3162 
  3163 lemma (in semiring_0) listsum_const_mult:
  3164   "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
  3165   by (induct xs) (simp_all add: algebra_simps)
  3166 
  3167 lemma (in semiring_0) listsum_mult_const:
  3168   "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
  3169   by (induct xs) (simp_all add: algebra_simps)
  3170 
  3171 lemma (in ordered_ab_group_add_abs) listsum_abs:
  3172   "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
  3173   by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])
  3174 
  3175 lemma listsum_mono:
  3176   fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}"
  3177   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)"
  3178   by (induct xs) (simp, simp add: add_mono)
  3179 
  3180 lemma (in monoid_add) listsum_distinct_conv_setsum_set:
  3181   "distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)"
  3182   by (induct xs) simp_all
  3183 
  3184 lemma (in monoid_add) interv_listsum_conv_setsum_set_nat:
  3185   "listsum (map f [m..<n]) = setsum f (set [m..<n])"
  3186   by (simp add: listsum_distinct_conv_setsum_set)
  3187 
  3188 lemma (in monoid_add) interv_listsum_conv_setsum_set_int:
  3189   "listsum (map f [k..l]) = setsum f (set [k..l])"
  3190   by (simp add: listsum_distinct_conv_setsum_set)
  3191 
  3192 text {* General equivalence between @{const listsum} and @{const setsum} *}
  3193 lemma (in monoid_add) listsum_setsum_nth:
  3194   "listsum xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
  3195   using interv_listsum_conv_setsum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth)
  3196 
  3197 
  3198 subsubsection {* @{const insert} *}
  3199 
  3200 lemma in_set_insert [simp]:
  3201   "x \<in> set xs \<Longrightarrow> List.insert x xs = xs"
  3202   by (simp add: List.insert_def)
  3203 
  3204 lemma not_in_set_insert [simp]:
  3205   "x \<notin> set xs \<Longrightarrow> List.insert x xs = x # xs"
  3206   by (simp add: List.insert_def)
  3207 
  3208 lemma insert_Nil [simp]:
  3209   "List.insert x [] = [x]"
  3210   by simp
  3211 
  3212 lemma set_insert [simp]:
  3213   "set (List.insert x xs) = insert x (set xs)"
  3214   by (auto simp add: List.insert_def)
  3215 
  3216 lemma distinct_insert [simp]:
  3217   "distinct xs \<Longrightarrow> distinct (List.insert x xs)"
  3218   by (simp add: List.insert_def)
  3219 
  3220 lemma insert_remdups:
  3221   "List.insert x (remdups xs) = remdups (List.insert x xs)"
  3222   by (simp add: List.insert_def)
  3223 
  3224 
  3225 subsubsection {* @{const List.find} *}
  3226 
  3227 lemma find_None_iff: "List.find P xs = None \<longleftrightarrow> \<not> (\<exists>x. x \<in> set xs \<and> P x)"
  3228 proof (induction xs)
  3229   case Nil thus ?case by simp
  3230 next
  3231   case (Cons x xs) thus ?case by (fastforce split: if_splits)
  3232 qed
  3233 
  3234 lemma find_Some_iff:
  3235   "List.find P xs = Some x \<longleftrightarrow>
  3236   (\<exists>i<length xs. P (xs!i) \<and> x = xs!i \<and> (\<forall>j<i. \<not> P (xs!j)))"
  3237 proof (induction xs)
  3238   case Nil thus ?case by simp
  3239 next
  3240   case (Cons x xs) thus ?case
  3241     by(auto simp: nth_Cons' split: if_splits)
  3242       (metis One_nat_def diff_Suc_1 less_Suc_eq_0_disj)
  3243 qed
  3244 
  3245 lemma find_cong[fundef_cong]:
  3246   assumes "xs = ys" and "\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x" 
  3247   shows "List.find P xs = List.find Q ys"
  3248 proof (cases "List.find P xs")
  3249   case None thus ?thesis by (metis find_None_iff assms)
  3250 next
  3251   case (Some x)
  3252   hence "List.find Q ys = Some x" using assms
  3253     by (auto simp add: find_Some_iff)
  3254   thus ?thesis using Some by auto
  3255 qed
  3256 
  3257 
  3258 subsubsection {* @{const remove1} *}
  3259 
  3260 lemma remove1_append:
  3261   "remove1 x (xs @ ys) =
  3262   (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
  3263 by (induct xs) auto
  3264 
  3265 lemma remove1_commute: "remove1 x (remove1 y zs) = remove1 y (remove1 x zs)"
  3266 by (induct zs) auto
  3267 
  3268 lemma in_set_remove1[simp]:
  3269   "a \<noteq> b \<Longrightarrow> a : set(remove1 b xs) = (a : set xs)"
  3270 apply (induct xs)
  3271 apply auto
  3272 done
  3273 
  3274 lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
  3275 apply(induct xs)
  3276  apply simp
  3277 apply simp
  3278 apply blast
  3279 done
  3280 
  3281 lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
  3282 apply(induct xs)
  3283  apply simp
  3284 apply simp
  3285 apply blast
  3286 done
  3287 
  3288 lemma length_remove1:
  3289   "length(remove1 x xs) = (if x : set xs then length xs - 1 else length xs)"
  3290 apply (induct xs)
  3291  apply (auto dest!:length_pos_if_in_set)
  3292 done
  3293 
  3294 lemma remove1_filter_not[simp]:
  3295   "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
  3296 by(induct xs) auto
  3297 
  3298 lemma filter_remove1:
  3299   "filter Q (remove1 x xs) = remove1 x (filter Q xs)"
  3300 by (induct xs) auto
  3301 
  3302 lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
  3303 apply(insert set_remove1_subset)
  3304 apply fast
  3305 done
  3306 
  3307 lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
  3308 by (induct xs) simp_all
  3309 
  3310 lemma remove1_remdups:
  3311   "distinct xs \<Longrightarrow> remove1 x (remdups xs) = remdups (remove1 x xs)"
  3312   by (induct xs) simp_all
  3313 
  3314 lemma remove1_idem:
  3315   assumes "x \<notin> set xs"
  3316   shows "remove1 x xs = xs"
  3317   using assms by (induct xs) simp_all
  3318 
  3319 
  3320 subsubsection {* @{text removeAll} *}
  3321 
  3322 lemma removeAll_filter_not_eq:
  3323   "removeAll x = filter (\<lambda>y. x \<noteq> y)"
  3324 proof
  3325   fix xs
  3326   show "removeAll x xs = filter (\<lambda>y. x \<noteq> y) xs"
  3327     by (induct xs) auto
  3328 qed
  3329 
  3330 lemma removeAll_append[simp]:
  3331   "removeAll x (xs @ ys) = removeAll x xs @ removeAll x ys"
  3332 by (induct xs) auto
  3333 
  3334 lemma set_removeAll[simp]: "set(removeAll x xs) = set xs - {x}"
  3335 by (induct xs) auto
  3336 
  3337 lemma removeAll_id[simp]: "x \<notin> set xs \<Longrightarrow> removeAll x xs = xs"
  3338 by (induct xs) auto
  3339 
  3340 (* Needs count:: 'a \<Rightarrow> 'a list \<Rightarrow> nat
  3341 lemma length_removeAll:
  3342   "length(removeAll x xs) = length xs - count x xs"
  3343 *)
  3344 
  3345 lemma removeAll_filter_not[simp]:
  3346   "\<not> P x \<Longrightarrow> removeAll x (filter P xs) = filter P xs"
  3347 by(induct xs) auto
  3348 
  3349 lemma distinct_removeAll:
  3350   "distinct xs \<Longrightarrow> distinct (removeAll x xs)"
  3351   by (simp add: removeAll_filter_not_eq)
  3352 
  3353 lemma distinct_remove1_removeAll:
  3354   "distinct xs ==> remove1 x xs = removeAll x xs"
  3355 by (induct xs) simp_all
  3356 
  3357 lemma map_removeAll_inj_on: "inj_on f (insert x (set xs)) \<Longrightarrow>
  3358   map f (removeAll x xs) = removeAll (f x) (map f xs)"
  3359 by (induct xs) (simp_all add:inj_on_def)
  3360 
  3361 lemma map_removeAll_inj: "inj f \<Longrightarrow>
  3362   map f (removeAll x xs) = removeAll (f x) (map f xs)"
  3363 by(metis map_removeAll_inj_on subset_inj_on subset_UNIV)
  3364 
  3365 
  3366 subsubsection {* @{text replicate} *}
  3367 
  3368 lemma length_replicate [simp]: "length (replicate n x) = n"
  3369 by (induct n) auto
  3370 
  3371 lemma Ex_list_of_length: "\<exists>xs. length xs = n"
  3372 by (rule exI[of _ "replicate n undefined"]) simp
  3373 
  3374 lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
  3375 by (induct n) auto
  3376 
  3377 lemma map_replicate_const:
  3378   "map (\<lambda> x. k) lst = replicate (length lst) k"
  3379   by (induct lst) auto
  3380 
  3381 lemma replicate_app_Cons_same:
  3382 "(replicate n x) @ (x # xs) = x # replicate n x @ xs"
  3383 by (induct n) auto
  3384 
  3385 lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
  3386 apply (induct n, simp)
  3387 apply (simp add: replicate_app_Cons_same)
  3388 done
  3389 
  3390 lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
  3391 by (induct n) auto
  3392 
  3393 text{* Courtesy of Matthias Daum: *}
  3394 lemma append_replicate_commute:
  3395   "replicate n x @ replicate k x = replicate k x @ replicate n x"
  3396 apply (simp add: replicate_add [THEN sym])
  3397 apply (simp add: add_commute)
  3398 done
  3399 
  3400 text{* Courtesy of Andreas Lochbihler: *}
  3401 lemma filter_replicate:
  3402   "filter P (replicate n x) = (if P x then replicate n x else [])"
  3403 by(induct n) auto
  3404 
  3405 lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
  3406 by (induct n) auto
  3407 
  3408 lemma tl_replicate [simp]: "tl (replicate n x) = replicate (n - 1) x"
  3409 by (induct n) auto
  3410 
  3411 lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
  3412 by (atomize (full), induct n) auto
  3413 
  3414 lemma nth_replicate[simp]: "i < n ==> (replicate n x)!i = x"
  3415 apply (induct n arbitrary: i, simp)
  3416 apply (simp add: nth_Cons split: nat.split)
  3417 done
  3418 
  3419 text{* Courtesy of Matthias Daum (2 lemmas): *}
  3420 lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
  3421 apply (case_tac "k \<le> i")
  3422  apply  (simp add: min_def)
  3423 apply (drule not_leE)
  3424 apply (simp add: min_def)
  3425 apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
  3426  apply  simp
  3427 apply (simp add: replicate_add [symmetric])
  3428 done
  3429 
  3430 lemma drop_replicate[simp]: "drop i (replicate k x) = replicate (k-i) x"
  3431 apply (induct k arbitrary: i)
  3432  apply simp
  3433 apply clarsimp
  3434 apply (case_tac i)
  3435  apply simp
  3436 apply clarsimp
  3437 done
  3438 
  3439 
  3440 lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
  3441 by (induct n) auto
  3442 
  3443 lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
  3444 by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
  3445 
  3446 lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
  3447 by auto
  3448 
  3449 lemma in_set_replicate[simp]: "(x : set (replicate n y)) = (x = y & n \<noteq> 0)"
  3450 by (simp add: set_replicate_conv_if)
  3451 
  3452 lemma Ball_set_replicate[simp]:
  3453   "(ALL x : set(replicate n a). P x) = (P a | n=0)"
  3454 by(simp add: set_replicate_conv_if)
  3455 
  3456 lemma Bex_set_replicate[simp]:
  3457   "(EX x : set(replicate n a). P x) = (P a & n\<noteq>0)"
  3458 by(simp add: set_replicate_conv_if)
  3459 
  3460 lemma replicate_append_same:
  3461   "replicate i x @ [x] = x # replicate i x"
  3462   by (induct i) simp_all
  3463 
  3464 lemma map_replicate_trivial:
  3465   "map (\<lambda>i. x) [0..<i] = replicate i x"
  3466   by (induct i) (simp_all add: replicate_append_same)
  3467 
  3468 lemma concat_replicate_trivial[simp]:
  3469   "concat (replicate i []) = []"
  3470   by (induct i) (auto simp add: map_replicate_const)
  3471 
  3472 lemma replicate_empty[simp]: "(replicate n x = []) \<longleftrightarrow> n=0"
  3473 by (induct n) auto
  3474 
  3475 lemma empty_replicate[simp]: "([] = replicate n x) \<longleftrightarrow> n=0"
  3476 by (induct n) auto
  3477 
  3478 lemma replicate_eq_replicate[simp]:
  3479   "(replicate m x = replicate n y) \<longleftrightarrow> (m=n & (m\<noteq>0 \<longrightarrow> x=y))"
  3480 apply(induct m arbitrary: n)
  3481  apply simp
  3482 apply(induct_tac n)
  3483 apply auto
  3484 done
  3485 
  3486 lemma replicate_length_filter:
  3487   "replicate (length (filter (\<lambda>y. x = y) xs)) x = filter (\<lambda>y. x = y) xs"
  3488   by (induct xs) auto
  3489 
  3490 lemma comm_append_are_replicate:
  3491   fixes xs ys :: "'a list"
  3492   assumes "xs \<noteq> []" "ys \<noteq> []"
  3493   assumes "xs @ ys = ys @ xs"
  3494   shows "\<exists>m n zs. concat (replicate m zs) = xs \<and> concat (replicate n zs) = ys"
  3495   using assms
  3496 proof (induct "length (xs @ ys)" arbitrary: xs ys rule: less_induct)
  3497   case less
  3498 
  3499   def xs' \<equiv> "if (length xs \<le> length ys) then xs else ys"
  3500     and ys' \<equiv> "if (length xs \<le> length ys) then ys else xs"
  3501   then have
  3502     prems': "length xs' \<le> length ys'"
  3503             "xs' @ ys' = ys' @ xs'"
  3504       and "xs' \<noteq> []"
  3505       and len: "length (xs @ ys) = length (xs' @ ys')"
  3506     using less by (auto intro: less.hyps)
  3507 
  3508   from prems'
  3509   obtain ws where "ys' = xs' @ ws"
  3510     by (auto simp: append_eq_append_conv2)
  3511 
  3512   have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ys'"
  3513   proof (cases "ws = []")
  3514     case True
  3515     then have "concat (replicate 1 xs') = xs'"
  3516       and "concat (replicate 1 xs') = ys'"
  3517       using `ys' = xs' @ ws` by auto
  3518     then show ?thesis by blast
  3519   next
  3520     case False
  3521     from `ys' = xs' @ ws` and `xs' @ ys' = ys' @ xs'`
  3522     have "xs' @ ws = ws @ xs'" by simp
  3523     then have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ws"
  3524       using False and `xs' \<noteq> []` and `ys' = xs' @ ws` and len
  3525       by (intro less.hyps) auto
  3526     then obtain m n zs where "concat (replicate m zs) = xs'"
  3527       and "concat (replicate n zs) = ws" by blast
  3528     moreover
  3529     then have "concat (replicate (m + n) zs) = ys'"
  3530       using `ys' = xs' @ ws`
  3531       by (simp add: replicate_add)
  3532     ultimately
  3533     show ?thesis by blast
  3534   qed
  3535   then show ?case
  3536     using xs'_def ys'_def by metis
  3537 qed
  3538 
  3539 lemma comm_append_is_replicate:
  3540   fixes xs ys :: "'a list"
  3541   assumes "xs \<noteq> []" "ys \<noteq> []"
  3542   assumes "xs @ ys = ys @ xs"
  3543   shows "\<exists>n zs. n > 1 \<and> concat (replicate n zs) = xs @ ys"
  3544 
  3545 proof -
  3546   obtain m n zs where "concat (replicate m zs) = xs"
  3547     and "concat (replicate n zs) = ys"
  3548     using assms by (metis comm_append_are_replicate)
  3549   then have "m + n > 1" and "concat (replicate (m+n) zs) = xs @ ys"
  3550     using `xs \<noteq> []` and `ys \<noteq> []`
  3551     by (auto simp: replicate_add)
  3552   then show ?thesis by blast
  3553 qed
  3554 
  3555 
  3556 subsubsection{*@{text rotate1} and @{text rotate}*}
  3557 
  3558 lemma rotate0[simp]: "rotate 0 = id"
  3559 by(simp add:rotate_def)
  3560 
  3561 lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
  3562 by(simp add:rotate_def)
  3563 
  3564 lemma rotate_add:
  3565   "rotate (m+n) = rotate m o rotate n"
  3566 by(simp add:rotate_def funpow_add)
  3567 
  3568 lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
  3569 by(simp add:rotate_add)
  3570 
  3571 lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
  3572 by(simp add:rotate_def funpow_swap1)
  3573 
  3574 lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
  3575 by(cases xs) simp_all
  3576 
  3577 lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
  3578 apply(induct n)
  3579  apply simp
  3580 apply (simp add:rotate_def)
  3581 done
  3582 
  3583 lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
  3584 by (cases xs) simp_all
  3585 
  3586 lemma rotate_drop_take:
  3587   "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
  3588 apply(induct n)
  3589  apply simp
  3590 apply(simp add:rotate_def)
  3591 apply(cases "xs = []")
  3592  apply (simp)
  3593 apply(case_tac "n mod length xs = 0")
  3594  apply(simp add:mod_Suc)
  3595  apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
  3596 apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
  3597                 take_hd_drop linorder_not_le)
  3598 done
  3599 
  3600 lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
  3601 by(simp add:rotate_drop_take)
  3602 
  3603 lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
  3604 by(simp add:rotate_drop_take)
  3605 
  3606 lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
  3607 by (cases xs) simp_all
  3608 
  3609 lemma length_rotate[simp]: "length(rotate n xs) = length xs"
  3610 by (induct n arbitrary: xs) (simp_all add:rotate_def)
  3611 
  3612 lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
  3613 by (cases xs) auto
  3614 
  3615 lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
  3616 by (induct n) (simp_all add:rotate_def)
  3617 
  3618 lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
  3619 by(simp add:rotate_drop_take take_map drop_map)
  3620 
  3621 lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
  3622 by (cases xs) auto
  3623 
  3624 lemma set_rotate[simp]: "set(rotate n xs) = set xs"
  3625 by (induct n) (simp_all add:rotate_def)
  3626 
  3627 lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
  3628 by (cases xs) auto
  3629 
  3630 lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
  3631 by (induct n) (simp_all add:rotate_def)
  3632 
  3633 lemma rotate_rev:
  3634   "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
  3635 apply(simp add:rotate_drop_take rev_drop rev_take)
  3636 apply(cases "length xs = 0")
  3637  apply simp
  3638 apply(cases "n mod length xs = 0")
  3639  apply simp
  3640 apply(simp add:rotate_drop_take rev_drop rev_take)
  3641 done
  3642 
  3643 lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
  3644 apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
  3645 apply(subgoal_tac "length xs \<noteq> 0")
  3646  prefer 2 apply simp
  3647 using mod_less_divisor[of "length xs" n] by arith
  3648 
  3649 
  3650 subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
  3651 
  3652 lemma sublist_empty [simp]: "sublist xs {} = []"
  3653 by (auto simp add: sublist_def)
  3654 
  3655 lemma sublist_nil [simp]: "sublist [] A = []"
  3656 by (auto simp add: sublist_def)
  3657 
  3658 lemma length_sublist:
  3659   "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
  3660 by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
  3661 
  3662 lemma sublist_shift_lemma_Suc:
  3663   "map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
  3664    map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
  3665 apply(induct xs arbitrary: "is")
  3666  apply simp
  3667 apply (case_tac "is")
  3668  apply simp
  3669 apply simp
  3670 done
  3671 
  3672 lemma sublist_shift_lemma:
  3673      "map fst [p<-zip xs [i..<i + length xs] . snd p : A] =
  3674       map fst [p<-zip xs [0..<length xs] . snd p + i : A]"
  3675 by (induct xs rule: rev_induct) (simp_all add: add_commute)
  3676 
  3677 lemma sublist_append:
  3678      "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
  3679 apply (unfold sublist_def)
  3680 apply (induct l' rule: rev_induct, simp)
  3681 apply (simp add: upt_add_eq_append[of 0] sublist_shift_lemma)
  3682 apply (simp add: add_commute)
  3683 done
  3684 
  3685 lemma sublist_Cons:
  3686 "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
  3687 apply (induct l rule: rev_induct)
  3688  apply (simp add: sublist_def)
  3689 apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
  3690 done
  3691 
  3692 lemma set_sublist: "set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
  3693 apply(induct xs arbitrary: I)
  3694 apply(auto simp: sublist_Cons nth_Cons split:nat.split dest!: gr0_implies_Suc)
  3695 done
  3696 
  3697 lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
  3698 by(auto simp add:set_sublist)
  3699 
  3700 lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
  3701 by(auto simp add:set_sublist)
  3702 
  3703 lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
  3704 by(auto simp add:set_sublist)
  3705 
  3706 lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
  3707 by (simp add: sublist_Cons)
  3708 
  3709 
  3710 lemma distinct_sublistI[simp]: "distinct xs \<Longrightarrow> distinct(sublist xs I)"
  3711 apply(induct xs arbitrary: I)
  3712  apply simp
  3713 apply(auto simp add:sublist_Cons)
  3714 done
  3715 
  3716 
  3717 lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
  3718 apply (induct l rule: rev_induct, simp)
  3719 apply (simp split: nat_diff_split add: sublist_append)
  3720 done
  3721 
  3722 lemma filter_in_sublist:
  3723  "distinct xs \<Longrightarrow> filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
  3724 proof (induct xs arbitrary: s)
  3725   case Nil thus ?case by simp
  3726 next
  3727   case (Cons a xs)
  3728   moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
  3729   ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
  3730 qed
  3731 
  3732 
  3733 subsubsection {* @{const splice} *}
  3734 
  3735 lemma splice_Nil2 [simp, code]: "splice xs [] = xs"
  3736 by (cases xs) simp_all
  3737 
  3738 declare splice.simps(1,3)[code]
  3739 declare splice.simps(2)[simp del]
  3740 
  3741 lemma length_splice[simp]: "length(splice xs ys) = length xs + length ys"
  3742 by (induct xs ys rule: splice.induct) auto
  3743 
  3744 
  3745 subsubsection {* Transpose *}
  3746 
  3747 function transpose where
  3748 "transpose []             = []" |
  3749 "transpose ([]     # xss) = transpose xss" |
  3750 "transpose ((x#xs) # xss) =
  3751   (x # [h. (h#t) \<leftarrow> xss]) # transpose (xs # [t. (h#t) \<leftarrow> xss])"
  3752 by pat_completeness auto
  3753 
  3754 lemma transpose_aux_filter_head:
  3755   "concat (map (list_case [] (\<lambda>h t. [h])) xss) =
  3756   map (\<lambda>xs. hd xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  3757   by (induct xss) (auto split: list.split)
  3758 
  3759 lemma transpose_aux_filter_tail:
  3760   "concat (map (list_case [] (\<lambda>h t. [t])) xss) =
  3761   map (\<lambda>xs. tl xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  3762   by (induct xss) (auto split: list.split)
  3763 
  3764 lemma transpose_aux_max:
  3765   "max (Suc (length xs)) (foldr (\<lambda>xs. max (length xs)) xss 0) =
  3766   Suc (max (length xs) (foldr (\<lambda>x. max (length x - Suc 0)) [ys\<leftarrow>xss . ys\<noteq>[]] 0))"
  3767   (is "max _ ?foldB = Suc (max _ ?foldA)")
  3768 proof (cases "[ys\<leftarrow>xss . ys\<noteq>[]] = []")
  3769   case True
  3770   hence "foldr (\<lambda>xs. max (length xs)) xss 0 = 0"
  3771   proof (induct xss)
  3772     case (Cons x xs)
  3773     moreover hence "x = []" by (cases x) auto
  3774     ultimately show ?case by auto
  3775   qed simp
  3776   thus ?thesis using True by simp
  3777 next
  3778   case False
  3779 
  3780   have foldA: "?foldA = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0 - 1"
  3781     by (induct xss) auto
  3782   have foldB: "?foldB = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0"
  3783     by (induct xss) auto
  3784 
  3785   have "0 < ?foldB"
  3786   proof -
  3787     from False
  3788     obtain z zs where zs: "[ys\<leftarrow>xss . ys \<noteq> []] = z#zs" by (auto simp: neq_Nil_conv)
  3789     hence "z \<in> set ([ys\<leftarrow>xss . ys \<noteq> []])" by auto
  3790     hence "z \<noteq> []" by auto
  3791     thus ?thesis
  3792       unfolding foldB zs
  3793       by (auto simp: max_def intro: less_le_trans)
  3794   qed
  3795   thus ?thesis
  3796     unfolding foldA foldB max_Suc_Suc[symmetric]
  3797     by simp
  3798 qed
  3799 
  3800 termination transpose
  3801   by (relation "measure (\<lambda>xs. foldr (\<lambda>xs. max (length xs)) xs 0 + length xs)")
  3802      (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max less_Suc_eq_le)
  3803 
  3804 lemma transpose_empty: "(transpose xs = []) \<longleftrightarrow> (\<forall>x \<in> set xs. x = [])"
  3805   by (induct rule: transpose.induct) simp_all
  3806 
  3807 lemma length_transpose:
  3808   fixes xs :: "'a list list"
  3809   shows "length (transpose xs) = foldr (\<lambda>xs. max (length xs)) xs 0"
  3810   by (induct rule: transpose.induct)
  3811     (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max
  3812                 max_Suc_Suc[symmetric] simp del: max_Suc_Suc)
  3813 
  3814 lemma nth_transpose:
  3815   fixes xs :: "'a list list"
  3816   assumes "i < length (transpose xs)"
  3817   shows "transpose xs ! i = map (\<lambda>xs. xs ! i) [ys \<leftarrow> xs. i < length ys]"
  3818 using assms proof (induct arbitrary: i rule: transpose.induct)
  3819   case (3 x xs xss)
  3820   def XS == "(x # xs) # xss"
  3821   hence [simp]: "XS \<noteq> []" by auto
  3822   thus ?case
  3823   proof (cases i)
  3824     case 0
  3825     thus ?thesis by (simp add: transpose_aux_filter_head hd_conv_nth)
  3826   next
  3827     case (Suc j)
  3828     have *: "\<And>xss. xs # map tl xss = map tl ((x#xs)#xss)" by simp
  3829     have **: "\<And>xss. (x#xs) # filter (\<lambda>ys. ys \<noteq> []) xss = filter (\<lambda>ys. ys \<noteq> []) ((x#xs)#xss)" by simp
  3830     { fix x have "Suc j < length x \<longleftrightarrow> x \<noteq> [] \<and> j < length x - Suc 0"
  3831       by (cases x) simp_all
  3832     } note *** = this
  3833 
  3834     have j_less: "j < length (transpose (xs # concat (map (list_case [] (\<lambda>h t. [t])) xss)))"
  3835       using "3.prems" by (simp add: transpose_aux_filter_tail length_transpose Suc)
  3836 
  3837     show ?thesis
  3838       unfolding transpose.simps `i = Suc j` nth_Cons_Suc "3.hyps"[OF j_less]
  3839       apply (auto simp: transpose_aux_filter_tail filter_map comp_def length_transpose * ** *** XS_def[symmetric])
  3840       apply (rule_tac y=x in list.exhaust)
  3841       by auto
  3842   qed
  3843 qed simp_all
  3844 
  3845 lemma transpose_map_map:
  3846   "transpose (map (map f) xs) = map (map f) (transpose xs)"
  3847 proof (rule nth_equalityI, safe)
  3848   have [simp]: "length (transpose (map (map f) xs)) = length (transpose xs)"
  3849     by (simp add: length_transpose foldr_map comp_def)
  3850   show "length (transpose (map (map f) xs)) = length (map (map f) (transpose xs))" by simp
  3851 
  3852   fix i assume "i < length (transpose (map (map f) xs))"
  3853   thus "transpose (map (map f) xs) ! i = map (map f) (transpose xs) ! i"
  3854     by (simp add: nth_transpose filter_map comp_def)
  3855 qed
  3856 
  3857 
  3858 subsubsection {* (In)finiteness *}
  3859 
  3860 lemma finite_maxlen:
  3861   "finite (M::'a list set) ==> EX n. ALL s:M. size s < n"
  3862 proof (induct rule: finite.induct)
  3863   case emptyI show ?case by simp
  3864 next
  3865   case (insertI M xs)
  3866   then obtain n where "\<forall>s\<in>M. length s < n" by blast
  3867   hence "ALL s:insert xs M. size s < max n (size xs) + 1" by auto
  3868   thus ?case ..
  3869 qed
  3870 
  3871 lemma lists_length_Suc_eq:
  3872   "{xs. set xs \<subseteq> A \<and> length xs = Suc n} =
  3873     (\<lambda>(xs, n). n#xs) ` ({xs. set xs \<subseteq> A \<and> length xs = n} \<times> A)"
  3874   by (auto simp: length_Suc_conv)
  3875 
  3876 lemma
  3877   assumes "finite A"
  3878   shows finite_lists_length_eq: "finite {xs. set xs \<subseteq> A \<and> length xs = n}"
  3879   and card_lists_length_eq: "card {xs. set xs \<subseteq> A \<and> length xs = n} = (card A)^n"
  3880   using `finite A`
  3881   by (induct n)
  3882      (auto simp: card_image inj_split_Cons lists_length_Suc_eq cong: conj_cong)
  3883 
  3884 lemma finite_lists_length_le:
  3885   assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs \<le> n}"
  3886  (is "finite ?S")
  3887 proof-
  3888   have "?S = (\<Union>n\<in>{0..n}. {xs. set xs \<subseteq> A \<and> length xs = n})" by auto
  3889   thus ?thesis by (auto intro: finite_lists_length_eq[OF `finite A`])
  3890 qed
  3891 
  3892 lemma card_lists_length_le:
  3893   assumes "finite A" shows "card {xs. set xs \<subseteq> A \<and> length xs \<le> n} = (\<Sum>i\<le>n. card A^i)"
  3894 proof -
  3895   have "(\<Sum>i\<le>n. card A^i) = card (\<Union>i\<le>n. {xs. set xs \<subseteq> A \<and> length xs = i})"
  3896     using `finite A`
  3897     by (subst card_UN_disjoint)
  3898        (auto simp add: card_lists_length_eq finite_lists_length_eq)
  3899   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}"
  3900     by auto
  3901   finally show ?thesis by simp
  3902 qed
  3903 
  3904 lemma card_lists_distinct_length_eq:
  3905   assumes "k < card A"
  3906   shows "card {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A} = \<Prod>{card A - k + 1 .. card A}"
  3907 using assms
  3908 proof (induct k)
  3909   case 0
  3910   then have "{xs. length xs = 0 \<and> distinct xs \<and> set xs \<subseteq> A} = {[]}" by auto
  3911   then show ?case by simp
  3912 next
  3913   case (Suc k)
  3914   let "?k_list" = "\<lambda>k xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A"
  3915   have inj_Cons: "\<And>A. inj_on (\<lambda>(xs, n). n # xs) A"  by (rule inj_onI) auto
  3916 
  3917   from Suc have "k < card A" by simp
  3918   moreover have "finite A" using assms by (simp add: card_ge_0_finite)
  3919   moreover have "finite {xs. ?k_list k xs}"
  3920     using finite_lists_length_eq[OF `finite A`, of k]
  3921     by - (rule finite_subset, auto)
  3922   moreover have "\<And>i j. i \<noteq> j \<longrightarrow> {i} \<times> (A - set i) \<inter> {j} \<times> (A - set j) = {}"
  3923     by auto
  3924   moreover have "\<And>i. i \<in>Collect (?k_list k) \<Longrightarrow> card (A - set i) = card A - k"
  3925     by (simp add: card_Diff_subset distinct_card)
  3926   moreover have "{xs. ?k_list (Suc k) xs} =
  3927       (\<lambda>(xs, n). n#xs) ` \<Union>(\<lambda>xs. {xs} \<times> (A - set xs)) ` {xs. ?k_list k xs}"
  3928     by (auto simp: length_Suc_conv)
  3929   moreover
  3930   have "Suc (card A - Suc k) = card A - k" using Suc.prems by simp
  3931   then have "(card A - k) * \<Prod>{Suc (card A - k)..card A} = \<Prod>{Suc (card A - Suc k)..card A}"
  3932     by (subst setprod_insert[symmetric]) (simp add: atLeastAtMost_insertL)+
  3933   ultimately show ?case
  3934     by (simp add: card_image inj_Cons card_UN_disjoint Suc.hyps algebra_simps)
  3935 qed
  3936 
  3937 lemma infinite_UNIV_listI: "~ finite(UNIV::'a list set)"
  3938 apply(rule notI)
  3939 apply(drule finite_maxlen)
  3940 apply (metis UNIV_I length_replicate less_not_refl)
  3941 done
  3942 
  3943 
  3944 subsection {* Sorting *}
  3945 
  3946 text{* Currently it is not shown that @{const sort} returns a
  3947 permutation of its input because the nicest proof is via multisets,
  3948 which are not yet available. Alternatively one could define a function
  3949 that counts the number of occurrences of an element in a list and use
  3950 that instead of multisets to state the correctness property. *}
  3951 
  3952 context linorder
  3953 begin
  3954 
  3955 lemma length_insort [simp]:
  3956   "length (insort_key f x xs) = Suc (length xs)"
  3957   by (induct xs) simp_all
  3958 
  3959 lemma insort_key_left_comm:
  3960   assumes "f x \<noteq> f y"
  3961   shows "insort_key f y (insort_key f x xs) = insort_key f x (insort_key f y xs)"
  3962   by (induct xs) (auto simp add: assms dest: antisym)
  3963 
  3964 lemma insort_left_comm:
  3965   "insort x (insort y xs) = insort y (insort x xs)"
  3966   by (cases "x = y") (auto intro: insort_key_left_comm)
  3967 
  3968 lemma comp_fun_commute_insort:
  3969   "comp_fun_commute insort"
  3970 proof
  3971 qed (simp add: insort_left_comm fun_eq_iff)
  3972 
  3973 lemma sort_key_simps [simp]:
  3974   "sort_key f [] = []"
  3975   "sort_key f (x#xs) = insort_key f x (sort_key f xs)"
  3976   by (simp_all add: sort_key_def)
  3977 
  3978 lemma (in linorder) sort_key_conv_fold:
  3979   assumes "inj_on f (set xs)"
  3980   shows "sort_key f xs = fold (insort_key f) xs []"
  3981 proof -
  3982   have "fold (insort_key f) (rev xs) = fold (insort_key f) xs"
  3983   proof (rule fold_rev, rule ext)
  3984     fix zs
  3985     fix x y
  3986     assume "x \<in> set xs" "y \<in> set xs"
  3987     with assms have *: "f y = f x \<Longrightarrow> y = x" by (auto dest: inj_onD)
  3988     have **: "x = y \<longleftrightarrow> y = x" by auto
  3989     show "(insort_key f y \<circ> insort_key f x) zs = (insort_key f x \<circ> insort_key f y) zs"
  3990       by (induct zs) (auto intro: * simp add: **)
  3991   qed
  3992   then show ?thesis by (simp add: sort_key_def foldr_conv_fold)
  3993 qed
  3994 
  3995 lemma (in linorder) sort_conv_fold:
  3996   "sort xs = fold insort xs []"
  3997   by (rule sort_key_conv_fold) simp
  3998 
  3999 lemma length_sort[simp]: "length (sort_key f xs) = length xs"
  4000 by (induct xs, auto)
  4001 
  4002 lemma sorted_Cons: "sorted (x#xs) = (sorted xs & (ALL y:set xs. x <= y))"
  4003 apply(induct xs arbitrary: x) apply simp
  4004 by simp (blast intro: order_trans)
  4005 
  4006 lemma sorted_tl:
  4007   "sorted xs \<Longrightarrow> sorted (tl xs)"
  4008   by (cases xs) (simp_all add: sorted_Cons)
  4009 
  4010 lemma sorted_append:
  4011   "sorted (xs@ys) = (sorted xs & sorted ys & (\<forall>x \<in> set xs. \<forall>y \<in> set ys. x\<le>y))"
  4012 by (induct xs) (auto simp add:sorted_Cons)
  4013 
  4014 lemma sorted_nth_mono:
  4015   "sorted xs \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!i \<le> xs!j"
  4016 by (induct xs arbitrary: i j) (auto simp:nth_Cons' sorted_Cons)
  4017 
  4018 lemma sorted_rev_nth_mono:
  4019   "sorted (rev xs) \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!j \<le> xs!i"
  4020 using sorted_nth_mono[ of "rev xs" "length xs - j - 1" "length xs - i - 1"]
  4021       rev_nth[of "length xs - i - 1" "xs"] rev_nth[of "length xs - j - 1" "xs"]
  4022 by auto
  4023 
  4024 lemma sorted_nth_monoI:
  4025   "(\<And> i j. \<lbrakk> i \<le> j ; j < length xs \<rbrakk> \<Longrightarrow> xs ! i \<le> xs ! j) \<Longrightarrow> sorted xs"
  4026 proof (induct xs)
  4027   case (Cons x xs)
  4028   have "sorted xs"
  4029   proof (rule Cons.hyps)
  4030     fix i j assume "i \<le> j" and "j < length xs"
  4031     with Cons.prems[of "Suc i" "Suc j"]
  4032     show "xs ! i \<le> xs ! j" by auto
  4033   qed
  4034   moreover
  4035   {
  4036     fix y assume "y \<in> set xs"
  4037     then obtain j where "j < length xs" and "xs ! j = y"
  4038       unfolding in_set_conv_nth by blast
  4039     with Cons.prems[of 0 "Suc j"]
  4040     have "x \<le> y"
  4041       by auto
  4042   }
  4043   ultimately
  4044   show ?case
  4045     unfolding sorted_Cons by auto
  4046 qed simp
  4047 
  4048 lemma sorted_equals_nth_mono:
  4049   "sorted xs = (\<forall>j < length xs. \<forall>i \<le> j. xs ! i \<le> xs ! j)"
  4050 by (auto intro: sorted_nth_monoI sorted_nth_mono)
  4051 
  4052 lemma set_insort: "set(insort_key f x xs) = insert x (set xs)"
  4053 by (induct xs) auto
  4054 
  4055 lemma set_sort[simp]: "set(sort_key f xs) = set xs"
  4056 by (induct xs) (simp_all add:set_insort)
  4057 
  4058 lemma distinct_insort: "distinct (insort_key f x xs) = (x \<notin> set xs \<and> distinct xs)"
  4059 by(induct xs)(auto simp:set_insort)
  4060 
  4061 lemma distinct_sort[simp]: "distinct (sort_key f xs) = distinct xs"
  4062   by (induct xs) (simp_all add: distinct_insort)
  4063 
  4064 lemma sorted_insort_key: "sorted (map f (insort_key f x xs)) = sorted (map f xs)"
  4065   by (induct xs) (auto simp:sorted_Cons set_insort)
  4066 
  4067 lemma sorted_insort: "sorted (insort x xs) = sorted xs"
  4068   using sorted_insort_key [where f="\<lambda>x. x"] by simp
  4069 
  4070 theorem sorted_sort_key [simp]: "sorted (map f (sort_key f xs))"
  4071   by (induct xs) (auto simp:sorted_insort_key)
  4072 
  4073 theorem sorted_sort [simp]: "sorted (sort xs)"
  4074   using sorted_sort_key [where f="\<lambda>x. x"] by simp
  4075 
  4076 lemma sorted_butlast:
  4077   assumes "xs \<noteq> []" and "sorted xs"
  4078   shows "sorted (butlast xs)"
  4079 proof -
  4080   from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  4081   with `sorted xs` show ?thesis by (simp add: sorted_append)
  4082 qed
  4083   
  4084 lemma insort_not_Nil [simp]:
  4085   "insort_key f a xs \<noteq> []"
  4086   by (induct xs) simp_all
  4087 
  4088 lemma insort_is_Cons: "\<forall>x\<in>set xs. f a \<le> f x \<Longrightarrow> insort_key f a xs = a # xs"
  4089 by (cases xs) auto
  4090 
  4091 lemma sorted_sort_id: "sorted xs \<Longrightarrow> sort xs = xs"
  4092   by (induct xs) (auto simp add: sorted_Cons insort_is_Cons)
  4093 
  4094 lemma sorted_map_remove1:
  4095   "sorted (map f xs) \<Longrightarrow> sorted (map f (remove1 x xs))"
  4096   by (induct xs) (auto simp add: sorted_Cons)
  4097 
  4098 lemma sorted_remove1: "sorted xs \<Longrightarrow> sorted (remove1 a xs)"
  4099   using sorted_map_remove1 [of "\<lambda>x. x"] by simp
  4100 
  4101 lemma insort_key_remove1:
  4102   assumes "a \<in> set xs" and "sorted (map f xs)" and "hd (filter (\<lambda>x. f a = f x) xs) = a"
  4103   shows "insort_key f a (remove1 a xs) = xs"
  4104 using assms proof (induct xs)
  4105   case (Cons x xs)
  4106   then show ?case
  4107   proof (cases "x = a")
  4108     case False
  4109     then have "f x \<noteq> f a" using Cons.prems by auto
  4110     then have "f x < f a" using Cons.prems by (auto simp: sorted_Cons)
  4111     with `f x \<noteq> f a` show ?thesis using Cons by (auto simp: sorted_Cons insort_is_Cons)
  4112   qed (auto simp: sorted_Cons insort_is_Cons)
  4113 qed simp
  4114 
  4115 lemma insort_remove1:
  4116   assumes "a \<in> set xs" and "sorted xs"
  4117   shows "insort a (remove1 a xs) = xs"
  4118 proof (rule insort_key_remove1)
  4119   from `a \<in> set xs` show "a \<in> set xs" .
  4120   from `sorted xs` show "sorted (map (\<lambda>x. x) xs)" by simp
  4121   from `a \<in> set xs` have "a \<in> set (filter (op = a) xs)" by auto
  4122   then have "set (filter (op = a) xs) \<noteq> {}" by auto
  4123   then have "filter (op = a) xs \<noteq> []" by (auto simp only: set_empty)
  4124   then have "length (filter (op = a) xs) > 0" by simp
  4125   then obtain n where n: "Suc n = length (filter (op = a) xs)"
  4126     by (cases "length (filter (op = a) xs)") simp_all
  4127   moreover have "replicate (Suc n) a = a # replicate n a"
  4128     by simp
  4129   ultimately show "hd (filter (op = a) xs) = a" by (simp add: replicate_length_filter)
  4130 qed
  4131 
  4132 lemma sorted_remdups[simp]:
  4133   "sorted l \<Longrightarrow> sorted (remdups l)"
  4134 by (induct l) (auto simp: sorted_Cons)
  4135 
  4136 lemma sorted_distinct_set_unique:
  4137 assumes "sorted xs" "distinct xs" "sorted ys" "distinct ys" "set xs = set ys"
  4138 shows "xs = ys"
  4139 proof -
  4140   from assms have 1: "length xs = length ys" by (auto dest!: distinct_card)
  4141   from assms show ?thesis
  4142   proof(induct rule:list_induct2[OF 1])
  4143     case 1 show ?case by simp
  4144   next
  4145     case 2 thus ?case by (simp add:sorted_Cons)
  4146        (metis Diff_insert_absorb antisym insertE insert_iff)
  4147   qed
  4148 qed
  4149 
  4150 lemma map_sorted_distinct_set_unique:
  4151   assumes "inj_on f (set xs \<union> set ys)"
  4152   assumes "sorted (map f xs)" "distinct (map f xs)"
  4153     "sorted (map f ys)" "distinct (map f ys)"
  4154   assumes "set xs = set ys"
  4155   shows "xs = ys"
  4156 proof -
  4157   from assms have "map f xs = map f ys"
  4158     by (simp add: sorted_distinct_set_unique)
  4159   moreover with `inj_on f (set xs \<union> set ys)` show "xs = ys"
  4160     by (blast intro: map_inj_on)
  4161 qed
  4162 
  4163 lemma finite_sorted_distinct_unique:
  4164 shows "finite A \<Longrightarrow> EX! xs. set xs = A & sorted xs & distinct xs"
  4165 apply(drule finite_distinct_list)
  4166 apply clarify
  4167 apply(rule_tac a="sort xs" in ex1I)
  4168 apply (auto simp: sorted_distinct_set_unique)
  4169 done
  4170 
  4171 lemma
  4172   assumes "sorted xs"
  4173   shows sorted_take: "sorted (take n xs)"
  4174   and sorted_drop: "sorted (drop n xs)"
  4175 proof -
  4176   from assms have "sorted (take n xs @ drop n xs)" by simp
  4177   then show "sorted (take n xs)" and "sorted (drop n xs)"
  4178     unfolding sorted_append by simp_all
  4179 qed
  4180 
  4181 lemma sorted_dropWhile: "sorted xs \<Longrightarrow> sorted (dropWhile P xs)"
  4182   by (auto dest: sorted_drop simp add: dropWhile_eq_drop)
  4183 
  4184 lemma sorted_takeWhile: "sorted xs \<Longrightarrow> sorted (takeWhile P xs)"
  4185   by (subst takeWhile_eq_take) (auto dest: sorted_take)
  4186 
  4187 lemma sorted_filter:
  4188   "sorted (map f xs) \<Longrightarrow> sorted (map f (filter P xs))"
  4189   by (induct xs) (simp_all add: sorted_Cons)
  4190 
  4191 lemma foldr_max_sorted:
  4192   assumes "sorted (rev xs)"
  4193   shows "foldr max xs y = (if xs = [] then y else max (xs ! 0) y)"
  4194 using assms proof (induct xs)
  4195   case (Cons x xs)
  4196   moreover hence "sorted (rev xs)" using sorted_append by auto
  4197   ultimately show ?case
  4198     by (cases xs, auto simp add: sorted_append max_def)
  4199 qed simp
  4200 
  4201 lemma filter_equals_takeWhile_sorted_rev:
  4202   assumes sorted: "sorted (rev (map f xs))"
  4203   shows "[x \<leftarrow> xs. t < f x] = takeWhile (\<lambda> x. t < f x) xs"
  4204     (is "filter ?P xs = ?tW")
  4205 proof (rule takeWhile_eq_filter[symmetric])
  4206   let "?dW" = "dropWhile ?P xs"
  4207   fix x assume "x \<in> set ?dW"
  4208   then obtain i where i: "i < length ?dW" and nth_i: "x = ?dW ! i"
  4209     unfolding in_set_conv_nth by auto
  4210   hence "length ?tW + i < length (?tW @ ?dW)"
  4211     unfolding length_append by simp
  4212   hence i': "length (map f ?tW) + i < length (map f xs)" by simp
  4213   have "(map f ?tW @ map f ?dW) ! (length (map f ?tW) + i) \<le>
  4214         (map f ?tW @ map f ?dW) ! (length (map f ?tW) + 0)"
  4215     using sorted_rev_nth_mono[OF sorted _ i', of "length ?tW"]
  4216     unfolding map_append[symmetric] by simp
  4217   hence "f x \<le> f (?dW ! 0)"
  4218     unfolding nth_append_length_plus nth_i
  4219     using i preorder_class.le_less_trans[OF le0 i] by simp
  4220   also have "... \<le> t"
  4221     using hd_dropWhile[of "?P" xs] le0[THEN preorder_class.le_less_trans, OF i]
  4222     using hd_conv_nth[of "?dW"] by simp
  4223   finally show "\<not> t < f x" by simp
  4224 qed
  4225 
  4226 lemma insort_insert_key_triv:
  4227   "f x \<in> f ` set xs \<Longrightarrow> insort_insert_key f x xs = xs"
  4228   by (simp add: insort_insert_key_def)
  4229 
  4230 lemma insort_insert_triv:
  4231   "x \<in> set xs \<Longrightarrow> insort_insert x xs = xs"
  4232   using insort_insert_key_triv [of "\<lambda>x. x"] by simp
  4233 
  4234 lemma insort_insert_insort_key:
  4235   "f x \<notin> f ` set xs \<Longrightarrow> insort_insert_key f x xs = insort_key f x xs"
  4236   by (simp add: insort_insert_key_def)
  4237 
  4238 lemma insort_insert_insort:
  4239   "x \<notin> set xs \<Longrightarrow> insort_insert x xs = insort x xs"
  4240   using insort_insert_insort_key [of "\<lambda>x. x"] by simp
  4241 
  4242 lemma set_insort_insert:
  4243   "set (insort_insert x xs) = insert x (set xs)"
  4244   by (auto simp add: insort_insert_key_def set_insort)
  4245 
  4246 lemma distinct_insort_insert:
  4247   assumes "distinct xs"
  4248   shows "distinct (insort_insert_key f x xs)"
  4249   using assms by (induct xs) (auto simp add: insort_insert_key_def set_insort)
  4250 
  4251 lemma sorted_insort_insert_key:
  4252   assumes "sorted (map f xs)"
  4253   shows "sorted (map f (insort_insert_key f x xs))"
  4254   using assms by (simp add: insort_insert_key_def sorted_insort_key)
  4255 
  4256 lemma sorted_insort_insert:
  4257   assumes "sorted xs"
  4258   shows "sorted (insort_insert x xs)"
  4259   using assms sorted_insort_insert_key [of "\<lambda>x. x"] by simp
  4260 
  4261 lemma filter_insort_triv:
  4262   "\<not> P x \<Longrightarrow> filter P (insort_key f x xs) = filter P xs"
  4263   by (induct xs) simp_all
  4264 
  4265 lemma filter_insort:
  4266   "sorted (map f xs) \<Longrightarrow> P x \<Longrightarrow> filter P (insort_key f x xs) = insort_key f x (filter P xs)"
  4267   using assms by (induct xs)
  4268     (auto simp add: sorted_Cons, subst insort_is_Cons, auto)
  4269 
  4270 lemma filter_sort:
  4271   "filter P (sort_key f xs) = sort_key f (filter P xs)"
  4272   by (induct xs) (simp_all add: filter_insort_triv filter_insort)
  4273 
  4274 lemma sorted_map_same:
  4275   "sorted (map f [x\<leftarrow>xs. f x = g xs])"
  4276 proof (induct xs arbitrary: g)
  4277   case Nil then show ?case by simp
  4278 next
  4279   case (Cons x xs)
  4280   then have "sorted (map f [y\<leftarrow>xs . f y = (\<lambda>xs. f x) xs])" .
  4281   moreover from Cons have "sorted (map f [y\<leftarrow>xs . f y = (g \<circ> Cons x) xs])" .
  4282   ultimately show ?case by (simp_all add: sorted_Cons)
  4283 qed
  4284 
  4285 lemma sorted_same:
  4286   "sorted [x\<leftarrow>xs. x = g xs]"
  4287   using sorted_map_same [of "\<lambda>x. x"] by simp
  4288 
  4289 lemma remove1_insort [simp]:
  4290   "remove1 x (insort x xs) = xs"
  4291   by (induct xs) simp_all
  4292 
  4293 end
  4294 
  4295 lemma sorted_upt[simp]: "sorted[i..<j]"
  4296 by (induct j) (simp_all add:sorted_append)
  4297 
  4298 lemma sorted_upto[simp]: "sorted[i..j]"
  4299 apply(induct i j rule:upto.induct)
  4300 apply(subst upto.simps)
  4301 apply(simp add:sorted_Cons)
  4302 done
  4303 
  4304 
  4305 subsubsection {* @{const transpose} on sorted lists *}
  4306 
  4307 lemma sorted_transpose[simp]:
  4308   shows "sorted (rev (map length (transpose xs)))"
  4309   by (auto simp: sorted_equals_nth_mono rev_nth nth_transpose
  4310     length_filter_conv_card intro: card_mono)
  4311 
  4312 lemma transpose_max_length:
  4313   "foldr (\<lambda>xs. max (length xs)) (transpose xs) 0 = length [x \<leftarrow> xs. x \<noteq> []]"
  4314   (is "?L = ?R")
  4315 proof (cases "transpose xs = []")
  4316   case False
  4317   have "?L = foldr max (map length (transpose xs)) 0"
  4318     by (simp add: foldr_map comp_def)
  4319   also have "... = length (transpose xs ! 0)"
  4320     using False sorted_transpose by (simp add: foldr_max_sorted)
  4321   finally show ?thesis
  4322     using False by (simp add: nth_transpose)
  4323 next
  4324   case True
  4325   hence "[x \<leftarrow> xs. x \<noteq> []] = []"
  4326     by (auto intro!: filter_False simp: transpose_empty)
  4327   thus ?thesis by (simp add: transpose_empty True)
  4328 qed
  4329 
  4330 lemma length_transpose_sorted:
  4331   fixes xs :: "'a list list"
  4332   assumes sorted: "sorted (rev (map length xs))"
  4333   shows "length (transpose xs) = (if xs = [] then 0 else length (xs ! 0))"
  4334 proof (cases "xs = []")
  4335   case False
  4336   thus ?thesis
  4337     using foldr_max_sorted[OF sorted] False
  4338     unfolding length_transpose foldr_map comp_def
  4339     by simp
  4340 qed simp
  4341 
  4342 lemma nth_nth_transpose_sorted[simp]:
  4343   fixes xs :: "'a list list"
  4344   assumes sorted: "sorted (rev (map length xs))"
  4345   and i: "i < length (transpose xs)"
  4346   and j: "j < length [ys \<leftarrow> xs. i < length ys]"
  4347   shows "transpose xs ! i ! j = xs ! j  ! i"
  4348   using j filter_equals_takeWhile_sorted_rev[OF sorted, of i]
  4349     nth_transpose[OF i] nth_map[OF j]
  4350   by (simp add: takeWhile_nth)
  4351 
  4352 lemma transpose_column_length:
  4353   fixes xs :: "'a list list"
  4354   assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  4355   shows "length (filter (\<lambda>ys. i < length ys) (transpose xs)) = length (xs ! i)"
  4356 proof -
  4357   have "xs \<noteq> []" using `i < length xs` by auto
  4358   note filter_equals_takeWhile_sorted_rev[OF sorted, simp]
  4359   { fix j assume "j \<le> i"
  4360     note sorted_rev_nth_mono[OF sorted, of j i, simplified, OF this `i < length xs`]
  4361   } note sortedE = this[consumes 1]
  4362 
  4363   have "{j. j < length (transpose xs) \<and> i < length (transpose xs ! j)}
  4364     = {..< length (xs ! i)}"
  4365   proof safe
  4366     fix j
  4367     assume "j < length (transpose xs)" and "i < length (transpose xs ! j)"
  4368     with this(2) nth_transpose[OF this(1)]
  4369     have "i < length (takeWhile (\<lambda>ys. j < length ys) xs)" by simp
  4370     from nth_mem[OF this] takeWhile_nth[OF this]
  4371     show "j < length (xs ! i)" by (auto dest: set_takeWhileD)
  4372   next
  4373     fix j assume "j < length (xs ! i)"
  4374     thus "j < length (transpose xs)"
  4375       using foldr_max_sorted[OF sorted] `xs \<noteq> []` sortedE[OF le0]
  4376       by (auto simp: length_transpose comp_def foldr_map)
  4377 
  4378     have "Suc i \<le> length (takeWhile (\<lambda>ys. j < length ys) xs)"
  4379       using `i < length xs` `j < length (xs ! i)` less_Suc_eq_le
  4380       by (auto intro!: length_takeWhile_less_P_nth dest!: sortedE)
  4381     with nth_transpose[OF `j < length (transpose xs)`]
  4382     show "i < length (transpose xs ! j)" by simp
  4383   qed
  4384   thus ?thesis by (simp add: length_filter_conv_card)
  4385 qed
  4386 
  4387 lemma transpose_column:
  4388   fixes xs :: "'a list list"
  4389   assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  4390   shows "map (\<lambda>ys. ys ! i) (filter (\<lambda>ys. i < length ys) (transpose xs))
  4391     = xs ! i" (is "?R = _")
  4392 proof (rule nth_equalityI, safe)
  4393   show length: "length ?R = length (xs ! i)"
  4394     using transpose_column_length[OF assms] by simp
  4395 
  4396   fix j assume j: "j < length ?R"
  4397   note * = less_le_trans[OF this, unfolded length_map, OF length_filter_le]
  4398   from j have j_less: "j < length (xs ! i)" using length by simp
  4399   have i_less_tW: "Suc i \<le> length (takeWhile (\<lambda>ys. Suc j \<le> length ys) xs)"
  4400   proof (rule length_takeWhile_less_P_nth)
  4401     show "Suc i \<le> length xs" using `i < length xs` by simp
  4402     fix k assume "k < Suc i"
  4403     hence "k \<le> i" by auto
  4404     with sorted_rev_nth_mono[OF sorted this] `i < length xs`
  4405     have "length (xs ! i) \<le> length (xs ! k)" by simp
  4406     thus "Suc j \<le> length (xs ! k)" using j_less by simp
  4407   qed
  4408   have i_less_filter: "i < length [ys\<leftarrow>xs . j < length ys]"
  4409     unfolding filter_equals_takeWhile_sorted_rev[OF sorted, of j]
  4410     using i_less_tW by (simp_all add: Suc_le_eq)
  4411   from j show "?R ! j = xs ! i ! j"
  4412     unfolding filter_equals_takeWhile_sorted_rev[OF sorted_transpose, of i]
  4413     by (simp add: takeWhile_nth nth_nth_transpose_sorted[OF sorted * i_less_filter])
  4414 qed
  4415 
  4416 lemma transpose_transpose:
  4417   fixes xs :: "'a list list"
  4418   assumes sorted: "sorted (rev (map length xs))"
  4419   shows "transpose (transpose xs) = takeWhile (\<lambda>x. x \<noteq> []) xs" (is "?L = ?R")
  4420 proof -
  4421   have len: "length ?L = length ?R"
  4422     unfolding length_transpose transpose_max_length
  4423     using filter_equals_takeWhile_sorted_rev[OF sorted, of 0]
  4424     by simp
  4425 
  4426   { fix i assume "i < length ?R"
  4427     with less_le_trans[OF _ length_takeWhile_le[of _ xs]]
  4428     have "i < length xs" by simp
  4429   } note * = this
  4430   show ?thesis
  4431     by (rule nth_equalityI)
  4432        (simp_all add: len nth_transpose transpose_column[OF sorted] * takeWhile_nth)
  4433 qed
  4434 
  4435 theorem transpose_rectangle:
  4436   assumes "xs = [] \<Longrightarrow> n = 0"
  4437   assumes rect: "\<And> i. i < length xs \<Longrightarrow> length (xs ! i) = n"
  4438   shows "transpose xs = map (\<lambda> i. map (\<lambda> j. xs ! j ! i) [0..<length xs]) [0..<n]"
  4439     (is "?trans = ?map")
  4440 proof (rule nth_equalityI)
  4441   have "sorted (rev (map length xs))"
  4442     by (auto simp: rev_nth rect intro!: sorted_nth_monoI)
  4443   from foldr_max_sorted[OF this] assms
  4444   show len: "length ?trans = length ?map"
  4445     by (simp_all add: length_transpose foldr_map comp_def)
  4446   moreover
  4447   { fix i assume "i < n" hence "[ys\<leftarrow>xs . i < length ys] = xs"
  4448       using rect by (auto simp: in_set_conv_nth intro!: filter_True) }
  4449   ultimately show "\<forall>i < length ?trans. ?trans ! i = ?map ! i"
  4450     by (auto simp: nth_transpose intro: nth_equalityI)
  4451 qed
  4452 
  4453 
  4454 subsubsection {* @{text sorted_list_of_set} *}
  4455 
  4456 text{* This function maps (finite) linearly ordered sets to sorted
  4457 lists. Warning: in most cases it is not a good idea to convert from
  4458 sets to lists but one should convert in the other direction (via
  4459 @{const set}). *}
  4460 
  4461 context linorder
  4462 begin
  4463 
  4464 definition sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where
  4465   "sorted_list_of_set = Finite_Set.fold insort []"
  4466 
  4467 lemma sorted_list_of_set_empty [simp]:
  4468   "sorted_list_of_set {} = []"
  4469   by (simp add: sorted_list_of_set_def)
  4470 
  4471 lemma sorted_list_of_set_insert [simp]:
  4472   assumes "finite A"
  4473   shows "sorted_list_of_set (insert x A) = insort x (sorted_list_of_set (A - {x}))"
  4474 proof -
  4475   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  4476   from assms show ?thesis
  4477     by (simp add: sorted_list_of_set_def fold_insert_remove)
  4478 qed
  4479 
  4480 lemma sorted_list_of_set [simp]:
  4481   "finite A \<Longrightarrow> set (sorted_list_of_set A) = A \<and> sorted (sorted_list_of_set A) 
  4482     \<and> distinct (sorted_list_of_set A)"
  4483   by (induct A rule: finite_induct) (simp_all add: set_insort sorted_insort distinct_insort)
  4484 
  4485 lemma sorted_list_of_set_sort_remdups:
  4486   "sorted_list_of_set (set xs) = sort (remdups xs)"
  4487 proof -
  4488   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  4489   show ?thesis by (simp add: sorted_list_of_set_def sort_conv_fold fold_set_fold_remdups)
  4490 qed
  4491 
  4492 lemma sorted_list_of_set_remove:
  4493   assumes "finite A"
  4494   shows "sorted_list_of_set (A - {x}) = remove1 x (sorted_list_of_set A)"
  4495 proof (cases "x \<in> A")
  4496   case False with assms have "x \<notin> set (sorted_list_of_set A)" by simp
  4497   with False show ?thesis by (simp add: remove1_idem)
  4498 next
  4499   case True then obtain B where A: "A = insert x B" by (rule Set.set_insert)
  4500   with assms show ?thesis by simp
  4501 qed
  4502 
  4503 end
  4504 
  4505 lemma sorted_list_of_set_range [simp]:
  4506   "sorted_list_of_set {m..<n} = [m..<n]"
  4507   by (rule sorted_distinct_set_unique) simp_all
  4508 
  4509 
  4510 subsubsection {* @{text lists}: the list-forming operator over sets *}
  4511 
  4512 inductive_set
  4513   lists :: "'a set => 'a list set"
  4514   for A :: "'a set"
  4515 where
  4516     Nil [intro!, simp]: "[]: lists A"
  4517   | Cons [intro!, simp, no_atp]: "[| a: A; l: lists A|] ==> a#l : lists A"
  4518 
  4519 inductive_cases listsE [elim!,no_atp]: "x#l : lists A"
  4520 inductive_cases listspE [elim!,no_atp]: "listsp A (x # l)"
  4521 
  4522 inductive_simps listsp_simps[code]:
  4523   "listsp A []"
  4524   "listsp A (x # xs)"
  4525 
  4526 lemma listsp_mono [mono]: "A \<le> B ==> listsp A \<le> listsp B"
  4527 by (rule predicate1I, erule listsp.induct, blast+)
  4528 
  4529 lemmas lists_mono = listsp_mono [to_set]
  4530 
  4531 lemma listsp_infI:
  4532   assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
  4533 by induct blast+
  4534 
  4535 lemmas lists_IntI = listsp_infI [to_set]
  4536 
  4537 lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
  4538 proof (rule mono_inf [where f=listsp, THEN order_antisym])
  4539   show "mono listsp" by (simp add: mono_def listsp_mono)
  4540   show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro!: listsp_infI)
  4541 qed
  4542 
  4543 lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_def inf_bool_def]
  4544 
  4545 lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set]
  4546 
  4547 lemma Cons_in_lists_iff[simp]: "x#xs : lists A \<longleftrightarrow> x:A \<and> xs : lists A"
  4548 by auto
  4549 
  4550 lemma append_in_listsp_conv [iff]:
  4551      "(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
  4552 by (induct xs) auto
  4553 
  4554 lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]
  4555 
  4556 lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
  4557 -- {* eliminate @{text listsp} in favour of @{text set} *}
  4558 by (induct xs) auto
  4559 
  4560 lemmas in_lists_conv_set [code_unfold] = in_listsp_conv_set [to_set]
  4561 
  4562 lemma in_listspD [dest!,no_atp]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
  4563 by (rule in_listsp_conv_set [THEN iffD1])
  4564 
  4565 lemmas in_listsD [dest!,no_atp] = in_listspD [to_set]
  4566 
  4567 lemma in_listspI [intro!,no_atp]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
  4568 by (rule in_listsp_conv_set [THEN iffD2])
  4569 
  4570 lemmas in_listsI [intro!,no_atp] = in_listspI [to_set]
  4571 
  4572 lemma lists_eq_set: "lists A = {xs. set xs <= A}"
  4573 by auto
  4574 
  4575 lemma lists_empty [simp]: "lists {} = {[]}"
  4576 by auto
  4577 
  4578 lemma lists_UNIV [simp]: "lists UNIV = UNIV"
  4579 by auto
  4580 
  4581 
  4582 subsubsection {* Inductive definition for membership *}
  4583 
  4584 inductive ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
  4585 where
  4586     elem:  "ListMem x (x # xs)"
  4587   | insert:  "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"
  4588 
  4589 lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
  4590 apply (rule iffI)
  4591  apply (induct set: ListMem)
  4592   apply auto
  4593 apply (induct xs)
  4594  apply (auto intro: ListMem.intros)
  4595 done
  4596 
  4597 
  4598 subsubsection {* Lists as Cartesian products *}
  4599 
  4600 text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
  4601 @{term A} and tail drawn from @{term Xs}.*}
  4602 
  4603 definition
  4604   set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set" where
  4605   "set_Cons A XS = {z. \<exists>x xs. z = x # xs \<and> x \<in> A \<and> xs \<in> XS}"
  4606 
  4607 lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
  4608 by (auto simp add: set_Cons_def)
  4609 
  4610 text{*Yields the set of lists, all of the same length as the argument and
  4611 with elements drawn from the corresponding element of the argument.*}
  4612 
  4613 primrec
  4614   listset :: "'a set list \<Rightarrow> 'a list set" where
  4615      "listset [] = {[]}"
  4616   |  "listset (A # As) = set_Cons A (listset As)"
  4617 
  4618 
  4619 subsection {* Relations on Lists *}
  4620 
  4621 subsubsection {* Length Lexicographic Ordering *}
  4622 
  4623 text{*These orderings preserve well-foundedness: shorter lists 
  4624   precede longer lists. These ordering are not used in dictionaries.*}
  4625         
  4626 primrec -- {*The lexicographic ordering for lists of the specified length*}
  4627   lexn :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> ('a list \<times> 'a list) set" where
  4628     "lexn r 0 = {}"
  4629   | "lexn r (Suc n) = (map_pair (%(x, xs). x#xs) (%(x, xs). x#xs) ` (r <*lex*> lexn r n)) Int
  4630       {(xs, ys). length xs = Suc n \<and> length ys = Suc n}"
  4631 
  4632 definition
  4633   lex :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4634   "lex r = (\<Union>n. lexn r n)" -- {*Holds only between lists of the same length*}
  4635 
  4636 definition
  4637   lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" where
  4638   "lenlex r = inv_image (less_than <*lex*> lex r) (\<lambda>xs. (length xs, xs))"
  4639         -- {*Compares lists by their length and then lexicographically*}
  4640 
  4641 lemma wf_lexn: "wf r ==> wf (lexn r n)"
  4642 apply (induct n, simp, simp)
  4643 apply(rule wf_subset)
  4644  prefer 2 apply (rule Int_lower1)
  4645 apply(rule wf_map_pair_image)
  4646  prefer 2 apply (rule inj_onI, auto)
  4647 done
  4648 
  4649 lemma lexn_length:
  4650   "(xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
  4651 by (induct n arbitrary: xs ys) auto
  4652 
  4653 lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
  4654 apply (unfold lex_def)
  4655 apply (rule wf_UN)
  4656 apply (blast intro: wf_lexn, clarify)
  4657 apply (rename_tac m n)
  4658 apply (subgoal_tac "m \<noteq> n")
  4659  prefer 2 apply blast
  4660 apply (blast dest: lexn_length not_sym)
  4661 done
  4662 
  4663 lemma lexn_conv:
  4664   "lexn r n =
  4665     {(xs,ys). length xs = n \<and> length ys = n \<and>
  4666     (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
  4667 apply (induct n, simp)
  4668 apply (simp add: image_Collect lex_prod_def, safe, blast)
  4669  apply (rule_tac x = "ab # xys" in exI, simp)
  4670 apply (case_tac xys, simp_all, blast)
  4671 done
  4672 
  4673 lemma lex_conv:
  4674   "lex r =
  4675     {(xs,ys). length xs = length ys \<and>
  4676     (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
  4677 by (force simp add: lex_def lexn_conv)
  4678 
  4679 lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
  4680 by (unfold lenlex_def) blast
  4681 
  4682 lemma lenlex_conv:
  4683     "lenlex r = {(xs,ys). length xs < length ys |
  4684                  length xs = length ys \<and> (xs, ys) : lex r}"
  4685 by (simp add: lenlex_def Id_on_def lex_prod_def inv_image_def)
  4686 
  4687 lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
  4688 by (simp add: lex_conv)
  4689 
  4690 lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
  4691 by (simp add:lex_conv)
  4692 
  4693 lemma Cons_in_lex [simp]:
  4694     "((x # xs, y # ys) : lex r) =
  4695       ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
  4696 apply (simp add: lex_conv)
  4697 apply (rule iffI)
  4698  prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
  4699 apply (case_tac xys, simp, simp)
  4700 apply blast
  4701 done
  4702 
  4703 
  4704 subsubsection {* Lexicographic Ordering *}
  4705 
  4706 text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
  4707     This ordering does \emph{not} preserve well-foundedness.
  4708      Author: N. Voelker, March 2005. *} 
  4709 
  4710 definition
  4711   lexord :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4712   "lexord r = {(x,y ). \<exists> a v. y = x @ a # v \<or>
  4713             (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
  4714 
  4715 lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
  4716 by (unfold lexord_def, induct_tac y, auto) 
  4717 
  4718 lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
  4719 by (unfold lexord_def, induct_tac x, auto)
  4720 
  4721 lemma lexord_cons_cons[simp]:
  4722      "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
  4723   apply (unfold lexord_def, safe, simp_all)
  4724   apply (case_tac u, simp, simp)
  4725   apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
  4726   apply (erule_tac x="b # u" in allE)
  4727   by force
  4728 
  4729 lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
  4730 
  4731 lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
  4732 by (induct_tac x, auto)  
  4733 
  4734 lemma lexord_append_left_rightI:
  4735      "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
  4736 by (induct_tac u, auto)
  4737 
  4738 lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
  4739 by (induct x, auto)
  4740 
  4741 lemma lexord_append_leftD:
  4742      "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
  4743 by (erule rev_mp, induct_tac x, auto)
  4744 
  4745 lemma lexord_take_index_conv: 
  4746    "((x,y) : lexord r) = 
  4747     ((length x < length y \<and> take (length x) y = x) \<or> 
  4748      (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
  4749   apply (unfold lexord_def Let_def, clarsimp) 
  4750   apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
  4751   apply auto 
  4752   apply (rule_tac x="hd (drop (length x) y)" in exI)
  4753   apply (rule_tac x="tl (drop (length x) y)" in exI)
  4754   apply (erule subst, simp add: min_def) 
  4755   apply (rule_tac x ="length u" in exI, simp) 
  4756   apply (rule_tac x ="take i x" in exI) 
  4757   apply (rule_tac x ="x ! i" in exI) 
  4758   apply (rule_tac x ="y ! i" in exI, safe) 
  4759   apply (rule_tac x="drop (Suc i) x" in exI)
  4760   apply (drule sym, simp add: drop_Suc_conv_tl) 
  4761   apply (rule_tac x="drop (Suc i) y" in exI)
  4762   by (simp add: drop_Suc_conv_tl) 
  4763 
  4764 -- {* lexord is extension of partial ordering List.lex *} 
  4765 lemma lexord_lex: "(x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
  4766   apply (rule_tac x = y in spec) 
  4767   apply (induct_tac x, clarsimp) 
  4768   by (clarify, case_tac x, simp, force)
  4769 
  4770 lemma lexord_irreflexive: "ALL x. (x,x) \<notin> r \<Longrightarrow> (xs,xs) \<notin> lexord r"
  4771 by (induct xs) auto
  4772 
  4773 text{* By Ren\'e Thiemann: *}
  4774 lemma lexord_partial_trans: 
  4775   "(\<And>x y z. x \<in> set xs \<Longrightarrow> (x,y) \<in> r \<Longrightarrow> (y,z) \<in> r \<Longrightarrow> (x,z) \<in> r)
  4776    \<Longrightarrow>  (xs,ys) \<in> lexord r  \<Longrightarrow>  (ys,zs) \<in> lexord r \<Longrightarrow>  (xs,zs) \<in> lexord r"
  4777 proof (induct xs arbitrary: ys zs)
  4778   case Nil
  4779   from Nil(3) show ?case unfolding lexord_def by (cases zs, auto)
  4780 next
  4781   case (Cons x xs yys zzs)
  4782   from Cons(3) obtain y ys where yys: "yys = y # ys" unfolding lexord_def
  4783     by (cases yys, auto)
  4784   note Cons = Cons[unfolded yys]
  4785   from Cons(3) have one: "(x,y) \<in> r \<or> x = y \<and> (xs,ys) \<in> lexord r" by auto
  4786   from Cons(4) obtain z zs where zzs: "zzs = z # zs" unfolding lexord_def
  4787     by (cases zzs, auto)
  4788   note Cons = Cons[unfolded zzs]
  4789   from Cons(4) have two: "(y,z) \<in> r \<or> y = z \<and> (ys,zs) \<in> lexord r" by auto
  4790   {
  4791     assume "(xs,ys) \<in> lexord r" and "(ys,zs) \<in> lexord r"
  4792     from Cons(1)[OF _ this] Cons(2)
  4793     have "(xs,zs) \<in> lexord r" by auto
  4794   } note ind1 = this
  4795   {
  4796     assume "(x,y) \<in> r" and "(y,z) \<in> r"
  4797     from Cons(2)[OF _ this] have "(x,z) \<in> r" by auto
  4798   } note ind2 = this
  4799   from one two ind1 ind2
  4800   have "(x,z) \<in> r \<or> x = z \<and> (xs,zs) \<in> lexord r" by blast
  4801   thus ?case unfolding zzs by auto
  4802 qed
  4803 
  4804 lemma lexord_trans: 
  4805     "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
  4806 by(auto simp: trans_def intro:lexord_partial_trans)
  4807 
  4808 lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
  4809 by (rule transI, drule lexord_trans, blast) 
  4810 
  4811 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"
  4812   apply (rule_tac x = y in spec) 
  4813   apply (induct_tac x, rule allI) 
  4814   apply (case_tac x, simp, simp) 
  4815   apply (rule allI, case_tac x, simp, simp) 
  4816   by blast
  4817 
  4818 
  4819 subsubsection {* Lexicographic combination of measure functions *}
  4820 
  4821 text {* These are useful for termination proofs *}
  4822 
  4823 definition
  4824   "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"
  4825 
  4826 lemma wf_measures[simp]: "wf (measures fs)"
  4827 unfolding measures_def
  4828 by blast
  4829 
  4830 lemma in_measures[simp]: 
  4831   "(x, y) \<in> measures [] = False"
  4832   "(x, y) \<in> measures (f # fs)
  4833          = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"  
  4834 unfolding measures_def
  4835 by auto
  4836 
  4837 lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
  4838 by simp
  4839 
  4840 lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
  4841 by auto
  4842 
  4843 
  4844 subsubsection {* Lifting Relations to Lists: one element *}
  4845 
  4846 definition listrel1 :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4847 "listrel1 r = {(xs,ys).
  4848    \<exists>us z z' vs. xs = us @ z # vs \<and> (z,z') \<in> r \<and> ys = us @ z' # vs}"
  4849 
  4850 lemma listrel1I:
  4851   "\<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow>
  4852   (xs, ys) \<in> listrel1 r"
  4853 unfolding listrel1_def by auto
  4854 
  4855 lemma listrel1E:
  4856   "\<lbrakk> (xs, ys) \<in> listrel1 r;
  4857      !!x y us vs. \<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow> P
  4858    \<rbrakk> \<Longrightarrow> P"
  4859 unfolding listrel1_def by auto
  4860 
  4861 lemma not_Nil_listrel1 [iff]: "([], xs) \<notin> listrel1 r"
  4862 unfolding listrel1_def by blast
  4863 
  4864 lemma not_listrel1_Nil [iff]: "(xs, []) \<notin> listrel1 r"
  4865 unfolding listrel1_def by blast
  4866 
  4867 lemma Cons_listrel1_Cons [iff]:
  4868   "(x # xs, y # ys) \<in> listrel1 r \<longleftrightarrow>
  4869    (x,y) \<in> r \<and> xs = ys \<or> x = y \<and> (xs, ys) \<in> listrel1 r"
  4870 by (simp add: listrel1_def Cons_eq_append_conv) (blast)
  4871 
  4872 lemma listrel1I1: "(x,y) \<in> r \<Longrightarrow> (x # xs, y # xs) \<in> listrel1 r"
  4873 by (metis Cons_listrel1_Cons)
  4874 
  4875 lemma listrel1I2: "(xs, ys) \<in> listrel1 r \<Longrightarrow> (x # xs, x # ys) \<in> listrel1 r"
  4876 by (metis Cons_listrel1_Cons)
  4877 
  4878 lemma append_listrel1I:
  4879   "(xs, ys) \<in> listrel1 r \<and> us = vs \<or> xs = ys \<and> (us, vs) \<in> listrel1 r
  4880     \<Longrightarrow> (xs @ us, ys @ vs) \<in> listrel1 r"
  4881 unfolding listrel1_def
  4882 by auto (blast intro: append_eq_appendI)+
  4883 
  4884 lemma Cons_listrel1E1[elim!]:
  4885   assumes "(x # xs, ys) \<in> listrel1 r"
  4886     and "\<And>y. ys = y # xs \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
  4887     and "\<And>zs. ys = x # zs \<Longrightarrow> (xs, zs) \<in> listrel1 r \<Longrightarrow> R"
  4888   shows R
  4889 using assms by (cases ys) blast+
  4890 
  4891 lemma Cons_listrel1E2[elim!]:
  4892   assumes "(xs, y # ys) \<in> listrel1 r"
  4893     and "\<And>x. xs = x # ys \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
  4894     and "\<And>zs. xs = y # zs \<Longrightarrow> (zs, ys) \<in> listrel1 r \<Longrightarrow> R"
  4895   shows R
  4896 using assms by (cases xs) blast+
  4897 
  4898 lemma snoc_listrel1_snoc_iff:
  4899   "(xs @ [x], ys @ [y]) \<in> listrel1 r
  4900     \<longleftrightarrow> (xs, ys) \<in> listrel1 r \<and> x = y \<or> xs = ys \<and> (x,y) \<in> r" (is "?L \<longleftrightarrow> ?R")
  4901 proof
  4902   assume ?L thus ?R
  4903     by (fastforce simp: listrel1_def snoc_eq_iff_butlast butlast_append)
  4904 next
  4905   assume ?R then show ?L unfolding listrel1_def by force
  4906 qed
  4907 
  4908 lemma listrel1_eq_len: "(xs,ys) \<in> listrel1 r \<Longrightarrow> length xs = length ys"
  4909 unfolding listrel1_def by auto
  4910 
  4911 lemma listrel1_mono:
  4912   "r \<subseteq> s \<Longrightarrow> listrel1 r \<subseteq> listrel1 s"
  4913 unfolding listrel1_def by blast
  4914 
  4915 
  4916 lemma listrel1_converse: "listrel1 (r^-1) = (listrel1 r)^-1"
  4917 unfolding listrel1_def by blast
  4918 
  4919 lemma in_listrel1_converse:
  4920   "(x,y) : listrel1 (r^-1) \<longleftrightarrow> (x,y) : (listrel1 r)^-1"
  4921 unfolding listrel1_def by blast
  4922 
  4923 lemma listrel1_iff_update:
  4924   "(xs,ys) \<in> (listrel1 r)
  4925    \<longleftrightarrow> (\<exists>y n. (xs ! n, y) \<in> r \<and> n < length xs \<and> ys = xs[n:=y])" (is "?L \<longleftrightarrow> ?R")
  4926 proof
  4927   assume "?L"
  4928   then obtain x y u v where "xs = u @ x # v"  "ys = u @ y # v"  "(x,y) \<in> r"
  4929     unfolding listrel1_def by auto
  4930   then have "ys = xs[length u := y]" and "length u < length xs"
  4931     and "(xs ! length u, y) \<in> r" by auto
  4932   then show "?R" by auto
  4933 next
  4934   assume "?R"
  4935   then obtain x y n where "(xs!n, y) \<in> r" "n < size xs" "ys = xs[n:=y]" "x = xs!n"
  4936     by auto
  4937   then obtain u v where "xs = u @ x # v" and "ys = u @ y # v" and "(x, y) \<in> r"
  4938     by (auto intro: upd_conv_take_nth_drop id_take_nth_drop)
  4939   then show "?L" by (auto simp: listrel1_def)
  4940 qed
  4941 
  4942 
  4943 text{* Accessible part and wellfoundedness: *}
  4944 
  4945 lemma Cons_acc_listrel1I [intro!]:
  4946   "x \<in> acc r \<Longrightarrow> xs \<in> acc (listrel1 r) \<Longrightarrow> (x # xs) \<in> acc (listrel1 r)"
  4947 apply (induct arbitrary: xs set: acc)
  4948 apply (erule thin_rl)
  4949 apply (erule acc_induct)
  4950 apply (rule accI)
  4951 apply (blast)
  4952 done
  4953 
  4954 lemma lists_accD: "xs \<in> lists (acc r) \<Longrightarrow> xs \<in> acc (listrel1 r)"
  4955 apply (induct set: lists)
  4956  apply (rule accI)
  4957  apply simp
  4958 apply (rule accI)
  4959 apply (fast dest: acc_downward)
  4960 done
  4961 
  4962 lemma lists_accI: "xs \<in> acc (listrel1 r) \<Longrightarrow> xs \<in> lists (acc r)"
  4963 apply (induct set: acc)
  4964 apply clarify
  4965 apply (rule accI)
  4966 apply (fastforce dest!: in_set_conv_decomp[THEN iffD1] simp: listrel1_def)
  4967 done
  4968 
  4969 lemma wf_listrel1_iff[simp]: "wf(listrel1 r) = wf r"
  4970 by(metis wf_acc_iff in_lists_conv_set lists_accI lists_accD Cons_in_lists_iff)
  4971 
  4972 
  4973 subsubsection {* Lifting Relations to Lists: all elements *}
  4974 
  4975 inductive_set
  4976   listrel :: "('a \<times> 'b) set \<Rightarrow> ('a list \<times> 'b list) set"
  4977   for r :: "('a \<times> 'b) set"
  4978 where
  4979     Nil:  "([],[]) \<in> listrel r"
  4980   | Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
  4981 
  4982 inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
  4983 inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
  4984 inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
  4985 inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
  4986 
  4987 
  4988 lemma listrel_eq_len:  "(xs, ys) \<in> listrel r \<Longrightarrow> length xs = length ys"
  4989 by(induct rule: listrel.induct) auto
  4990 
  4991 lemma listrel_iff_zip [code_unfold]: "(xs,ys) : listrel r \<longleftrightarrow>
  4992   length xs = length ys & (\<forall>(x,y) \<in> set(zip xs ys). (x,y) \<in> r)" (is "?L \<longleftrightarrow> ?R")
  4993 proof
  4994   assume ?L thus ?R by induct (auto intro: listrel_eq_len)
  4995 next
  4996   assume ?R thus ?L
  4997     apply (clarify)
  4998     by (induct rule: list_induct2) (auto intro: listrel.intros)
  4999 qed
  5000 
  5001 lemma listrel_iff_nth: "(xs,ys) : listrel r \<longleftrightarrow>
  5002   length xs = length ys & (\<forall>n < length xs. (xs!n, ys!n) \<in> r)" (is "?L \<longleftrightarrow> ?R")
  5003 by (auto simp add: all_set_conv_all_nth listrel_iff_zip)
  5004 
  5005 
  5006 lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
  5007 apply clarify  
  5008 apply (erule listrel.induct)
  5009 apply (blast intro: listrel.intros)+
  5010 done
  5011 
  5012 lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
  5013 apply clarify 
  5014 apply (erule listrel.induct, auto) 
  5015 done
  5016 
  5017 lemma listrel_refl_on: "refl_on A r \<Longrightarrow> refl_on (lists A) (listrel r)" 
  5018 apply (simp add: refl_on_def listrel_subset Ball_def)
  5019 apply (rule allI) 
  5020 apply (induct_tac x) 
  5021 apply (auto intro: listrel.intros)
  5022 done
  5023 
  5024 lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
  5025 apply (auto simp add: sym_def)
  5026 apply (erule listrel.induct) 
  5027 apply (blast intro: listrel.intros)+
  5028 done
  5029 
  5030 lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
  5031 apply (simp add: trans_def)
  5032 apply (intro allI) 
  5033 apply (rule impI) 
  5034 apply (erule listrel.induct) 
  5035 apply (blast intro: listrel.intros)+
  5036 done
  5037 
  5038 theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
  5039 by (simp add: equiv_def listrel_refl_on listrel_sym listrel_trans) 
  5040 
  5041 lemma listrel_rtrancl_refl[iff]: "(xs,xs) : listrel(r^*)"
  5042 using listrel_refl_on[of UNIV, OF refl_rtrancl]
  5043 by(auto simp: refl_on_def)
  5044 
  5045 lemma listrel_rtrancl_trans:
  5046   "\<lbrakk> (xs,ys) : listrel(r^*);  (ys,zs) : listrel(r^*) \<rbrakk>
  5047   \<Longrightarrow> (xs,zs) : listrel(r^*)"
  5048 by (metis listrel_trans trans_def trans_rtrancl)
  5049 
  5050 
  5051 lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
  5052 by (blast intro: listrel.intros)
  5053 
  5054 lemma listrel_Cons:
  5055      "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})"
  5056 by (auto simp add: set_Cons_def intro: listrel.intros)
  5057 
  5058 text {* Relating @{term listrel1}, @{term listrel} and closures: *}
  5059 
  5060 lemma listrel1_rtrancl_subset_rtrancl_listrel1:
  5061   "listrel1 (r^*) \<subseteq> (listrel1 r)^*"
  5062 proof (rule subrelI)
  5063   fix xs ys assume 1: "(xs,ys) \<in> listrel1 (r^*)"
  5064   { fix x y us vs
  5065     have "(x,y) : r^* \<Longrightarrow> (us @ x # vs, us @ y # vs) : (listrel1 r)^*"
  5066     proof(induct rule: rtrancl.induct)
  5067       case rtrancl_refl show ?case by simp
  5068     next
  5069       case rtrancl_into_rtrancl thus ?case
  5070         by (metis listrel1I rtrancl.rtrancl_into_rtrancl)
  5071     qed }
  5072   thus "(xs,ys) \<in> (listrel1 r)^*" using 1 by(blast elim: listrel1E)
  5073 qed
  5074 
  5075 lemma rtrancl_listrel1_eq_len: "(x,y) \<in> (listrel1 r)^* \<Longrightarrow> length x = length y"
  5076 by (induct rule: rtrancl.induct) (auto intro: listrel1_eq_len)
  5077 
  5078 lemma rtrancl_listrel1_ConsI1:
  5079   "(xs,ys) : (listrel1 r)^* \<Longrightarrow> (x#xs,x#ys) : (listrel1 r)^*"
  5080 apply(induct rule: rtrancl.induct)
  5081  apply simp
  5082 by (metis listrel1I2 rtrancl.rtrancl_into_rtrancl)
  5083 
  5084 lemma rtrancl_listrel1_ConsI2:
  5085   "(x,y) \<in> r^* \<Longrightarrow> (xs, ys) \<in> (listrel1 r)^*
  5086   \<Longrightarrow> (x # xs, y # ys) \<in> (listrel1 r)^*"
  5087   by (blast intro: rtrancl_trans rtrancl_listrel1_ConsI1 
  5088     subsetD[OF listrel1_rtrancl_subset_rtrancl_listrel1 listrel1I1])
  5089 
  5090 lemma listrel1_subset_listrel:
  5091   "r \<subseteq> r' \<Longrightarrow> refl r' \<Longrightarrow> listrel1 r \<subseteq> listrel(r')"
  5092 by(auto elim!: listrel1E simp add: listrel_iff_zip set_zip refl_on_def)
  5093 
  5094 lemma listrel_reflcl_if_listrel1:
  5095   "(xs,ys) : listrel1 r \<Longrightarrow> (xs,ys) : listrel(r^*)"
  5096 by(erule listrel1E)(auto simp add: listrel_iff_zip set_zip)
  5097 
  5098 lemma listrel_rtrancl_eq_rtrancl_listrel1: "listrel (r^*) = (listrel1 r)^*"
  5099 proof
  5100   { fix x y assume "(x,y) \<in> listrel (r^*)"
  5101     then have "(x,y) \<in> (listrel1 r)^*"
  5102     by induct (auto intro: rtrancl_listrel1_ConsI2) }
  5103   then show "listrel (r^*) \<subseteq> (listrel1 r)^*"
  5104     by (rule subrelI)
  5105 next
  5106   show "listrel (r^*) \<supseteq> (listrel1 r)^*"
  5107   proof(rule subrelI)
  5108     fix xs ys assume "(xs,ys) \<in> (listrel1 r)^*"
  5109     then show "(xs,ys) \<in> listrel (r^*)"
  5110     proof induct
  5111       case base show ?case by(auto simp add: listrel_iff_zip set_zip)
  5112     next
  5113       case (step ys zs)
  5114       thus ?case  by (metis listrel_reflcl_if_listrel1 listrel_rtrancl_trans)
  5115     qed
  5116   qed
  5117 qed
  5118 
  5119 lemma rtrancl_listrel1_if_listrel:
  5120   "(xs,ys) : listrel r \<Longrightarrow> (xs,ys) : (listrel1 r)^*"
  5121 by(metis listrel_rtrancl_eq_rtrancl_listrel1 subsetD[OF listrel_mono] r_into_rtrancl subsetI)
  5122 
  5123 lemma listrel_subset_rtrancl_listrel1: "listrel r \<subseteq> (listrel1 r)^*"
  5124 by(fast intro:rtrancl_listrel1_if_listrel)
  5125 
  5126 
  5127 subsection {* Size function *}
  5128 
  5129 lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (list_size f)"
  5130 by (rule is_measure_trivial)
  5131 
  5132 lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (option_size f)"
  5133 by (rule is_measure_trivial)
  5134 
  5135 lemma list_size_estimation[termination_simp]: 
  5136   "x \<in> set xs \<Longrightarrow> y < f x \<Longrightarrow> y < list_size f xs"
  5137 by (induct xs) auto
  5138 
  5139 lemma list_size_estimation'[termination_simp]: 
  5140   "x \<in> set xs \<Longrightarrow> y \<le> f x \<Longrightarrow> y \<le> list_size f xs"
  5141 by (induct xs) auto
  5142 
  5143 lemma list_size_map[simp]: "list_size f (map g xs) = list_size (f o g) xs"
  5144 by (induct xs) auto
  5145 
  5146 lemma list_size_append[simp]: "list_size f (xs @ ys) = list_size f xs + list_size f ys"
  5147 by (induct xs, auto)
  5148 
  5149 lemma list_size_pointwise[termination_simp]: 
  5150   "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> list_size f xs \<le> list_size g xs"
  5151 by (induct xs) force+
  5152 
  5153 
  5154 subsection {* Monad operation *}
  5155 
  5156 definition bind :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b list) \<Rightarrow> 'b list" where
  5157   "bind xs f = concat (map f xs)"
  5158 
  5159 hide_const (open) bind
  5160 
  5161 lemma bind_simps [simp]:
  5162   "List.bind [] f = []"
  5163   "List.bind (x # xs) f = f x @ List.bind xs f"
  5164   by (simp_all add: bind_def)
  5165 
  5166 
  5167 subsection {* Transfer *}
  5168 
  5169 definition
  5170   embed_list :: "nat list \<Rightarrow> int list"
  5171 where
  5172   "embed_list l = map int l"
  5173 
  5174 definition
  5175   nat_list :: "int list \<Rightarrow> bool"
  5176 where
  5177   "nat_list l = nat_set (set l)"
  5178 
  5179 definition
  5180   return_list :: "int list \<Rightarrow> nat list"
  5181 where
  5182   "return_list l = map nat l"
  5183 
  5184 lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow>
  5185     embed_list (return_list l) = l"
  5186   unfolding embed_list_def return_list_def nat_list_def nat_set_def
  5187   apply (induct l)
  5188   apply auto
  5189 done
  5190 
  5191 lemma transfer_nat_int_list_functions:
  5192   "l @ m = return_list (embed_list l @ embed_list m)"
  5193   "[] = return_list []"
  5194   unfolding return_list_def embed_list_def
  5195   apply auto
  5196   apply (induct l, auto)
  5197   apply (induct m, auto)
  5198 done
  5199 
  5200 (*
  5201 lemma transfer_nat_int_fold1: "fold f l x =
  5202     fold (%x. f (nat x)) (embed_list l) x";
  5203 *)
  5204 
  5205 
  5206 subsection {* Code generation *}
  5207 
  5208 subsubsection {* Counterparts for set-related operations *}
  5209 
  5210 definition member :: "'a list \<Rightarrow> 'a \<Rightarrow> bool" where
  5211   [code_abbrev]: "member xs x \<longleftrightarrow> x \<in> set xs"
  5212 
  5213 text {*
  5214   Use @{text member} only for generating executable code.  Otherwise use
  5215   @{prop "x \<in> set xs"} instead --- it is much easier to reason about.
  5216 *}
  5217 
  5218 lemma member_rec [code]:
  5219   "member (x # xs) y \<longleftrightarrow> x = y \<or> member xs y"
  5220   "member [] y \<longleftrightarrow> False"
  5221   by (auto simp add: member_def)
  5222 
  5223 lemma in_set_member (* FIXME delete candidate *):
  5224   "x \<in> set xs \<longleftrightarrow> member xs x"
  5225   by (simp add: member_def)
  5226 
  5227 definition list_all :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5228   list_all_iff [code_abbrev]: "list_all P xs \<longleftrightarrow> Ball (set xs) P"
  5229 
  5230 definition list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5231   list_ex_iff [code_abbrev]: "list_ex P xs \<longleftrightarrow> Bex (set xs) P"
  5232 
  5233 definition list_ex1 :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5234   list_ex1_iff [code_abbrev]: "list_ex1 P xs \<longleftrightarrow> (\<exists>! x. x \<in> set xs \<and> P x)"
  5235 
  5236 text {*
  5237   Usually you should prefer @{text "\<forall>x\<in>set xs"}, @{text "\<exists>x\<in>set xs"}
  5238   and @{text "\<exists>!x. x\<in>set xs \<and> _"} over @{const list_all}, @{const list_ex}
  5239   and @{const list_ex1} in specifications.
  5240 *}
  5241 
  5242 lemma list_all_simps [simp, code]:
  5243   "list_all P (x # xs) \<longleftrightarrow> P x \<and> list_all P xs"
  5244   "list_all P [] \<longleftrightarrow> True"
  5245   by (simp_all add: list_all_iff)
  5246 
  5247 lemma list_ex_simps [simp, code]:
  5248   "list_ex P (x # xs) \<longleftrightarrow> P x \<or> list_ex P xs"
  5249   "list_ex P [] \<longleftrightarrow> False"
  5250   by (simp_all add: list_ex_iff)
  5251 
  5252 lemma list_ex1_simps [simp, code]:
  5253   "list_ex1 P [] = False"
  5254   "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)"
  5255   by (auto simp add: list_ex1_iff list_all_iff)
  5256 
  5257 lemma Ball_set_list_all: (* FIXME delete candidate *)
  5258   "Ball (set xs) P \<longleftrightarrow> list_all P xs"
  5259   by (simp add: list_all_iff)
  5260 
  5261 lemma Bex_set_list_ex: (* FIXME delete candidate *)
  5262   "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
  5263   by (simp add: list_ex_iff)
  5264 
  5265 lemma list_all_append [simp]:
  5266   "list_all P (xs @ ys) \<longleftrightarrow> list_all P xs \<and> list_all P ys"
  5267   by (auto simp add: list_all_iff)
  5268 
  5269 lemma list_ex_append [simp]:
  5270   "list_ex P (xs @ ys) \<longleftrightarrow> list_ex P xs \<or> list_ex P ys"
  5271   by (auto simp add: list_ex_iff)
  5272 
  5273 lemma list_all_rev [simp]:
  5274   "list_all P (rev xs) \<longleftrightarrow> list_all P xs"
  5275   by (simp add: list_all_iff)
  5276 
  5277 lemma list_ex_rev [simp]:
  5278   "list_ex P (rev xs) \<longleftrightarrow> list_ex P xs"
  5279   by (simp add: list_ex_iff)
  5280 
  5281 lemma list_all_length:
  5282   "list_all P xs \<longleftrightarrow> (\<forall>n < length xs. P (xs ! n))"
  5283   by (auto simp add: list_all_iff set_conv_nth)
  5284 
  5285 lemma list_ex_length:
  5286   "list_ex P xs \<longleftrightarrow> (\<exists>n < length xs. P (xs ! n))"
  5287   by (auto simp add: list_ex_iff set_conv_nth)
  5288 
  5289 lemma list_all_cong [fundef_cong]:
  5290   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_all f xs = list_all g ys"
  5291   by (simp add: list_all_iff)
  5292 
  5293 lemma list_ex_cong [fundef_cong]:
  5294   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_ex f xs = list_ex g ys"
  5295 by (simp add: list_ex_iff)
  5296 
  5297 text {* Executable checks for relations on sets *}
  5298 
  5299 definition listrel1p :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
  5300 "listrel1p r xs ys = ((xs, ys) \<in> listrel1 {(x, y). r x y})"
  5301 
  5302 lemma [code_unfold]:
  5303   "(xs, ys) \<in> listrel1 r = listrel1p (\<lambda>x y. (x, y) \<in> r) xs ys"
  5304 unfolding listrel1p_def by auto
  5305 
  5306 lemma [code]:
  5307   "listrel1p r [] xs = False"
  5308   "listrel1p r xs [] =  False"
  5309   "listrel1p r (x # xs) (y # ys) \<longleftrightarrow>
  5310      r x y \<and> xs = ys \<or> x = y \<and> listrel1p r xs ys"
  5311 by (simp add: listrel1p_def)+
  5312 
  5313 definition
  5314   lexordp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
  5315   "lexordp r xs ys = ((xs, ys) \<in> lexord {(x, y). r x y})"
  5316 
  5317 lemma [code_unfold]:
  5318   "(xs, ys) \<in> lexord r = lexordp (\<lambda>x y. (x, y) \<in> r) xs ys"
  5319 unfolding lexordp_def by auto
  5320 
  5321 lemma [code]:
  5322   "lexordp r xs [] = False"
  5323   "lexordp r [] (y#ys) = True"
  5324   "lexordp r (x # xs) (y # ys) = (r x y | (x = y & lexordp r xs ys))"
  5325 unfolding lexordp_def by auto
  5326 
  5327 text {* Bounded quantification and summation over nats. *}
  5328 
  5329 lemma atMost_upto [code_unfold]:
  5330   "{..n} = set [0..<Suc n]"
  5331   by auto
  5332 
  5333 lemma atLeast_upt [code_unfold]:
  5334   "{..<n} = set [0..<n]"
  5335   by auto
  5336 
  5337 lemma greaterThanLessThan_upt [code_unfold]:
  5338   "{n<..<m} = set [Suc n..<m]"
  5339   by auto
  5340 
  5341 lemmas atLeastLessThan_upt [code_unfold] = set_upt [symmetric]
  5342 
  5343 lemma greaterThanAtMost_upt [code_unfold]:
  5344   "{n<..m} = set [Suc n..<Suc m]"
  5345   by auto
  5346 
  5347 lemma atLeastAtMost_upt [code_unfold]:
  5348   "{n..m} = set [n..<Suc m]"
  5349   by auto
  5350 
  5351 lemma all_nat_less_eq [code_unfold]:
  5352   "(\<forall>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..<n}. P m)"
  5353   by auto
  5354 
  5355 lemma ex_nat_less_eq [code_unfold]:
  5356   "(\<exists>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..<n}. P m)"
  5357   by auto
  5358 
  5359 lemma all_nat_less [code_unfold]:
  5360   "(\<forall>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..n}. P m)"
  5361   by auto
  5362 
  5363 lemma ex_nat_less [code_unfold]:
  5364   "(\<exists>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..n}. P m)"
  5365   by auto
  5366 
  5367 lemma setsum_set_upt_conv_listsum_nat [code_unfold]:
  5368   "setsum f (set [m..<n]) = listsum (map f [m..<n])"
  5369   by (simp add: interv_listsum_conv_setsum_set_nat)
  5370 
  5371 text {* Summation over ints. *}
  5372 
  5373 lemma greaterThanLessThan_upto [code_unfold]:
  5374   "{i<..<j::int} = set [i+1..j - 1]"
  5375 by auto
  5376 
  5377 lemma atLeastLessThan_upto [code_unfold]:
  5378   "{i..<j::int} = set [i..j - 1]"
  5379 by auto
  5380 
  5381 lemma greaterThanAtMost_upto [code_unfold]:
  5382   "{i<..j::int} = set [i+1..j]"
  5383 by auto
  5384 
  5385 lemmas atLeastAtMost_upto [code_unfold] = set_upto [symmetric]
  5386 
  5387 lemma setsum_set_upto_conv_listsum_int [code_unfold]:
  5388   "setsum f (set [i..j::int]) = listsum (map f [i..j])"
  5389   by (simp add: interv_listsum_conv_setsum_set_int)
  5390 
  5391 
  5392 subsubsection {* Optimizing by rewriting *}
  5393 
  5394 definition null :: "'a list \<Rightarrow> bool" where
  5395   [code_abbrev]: "null xs \<longleftrightarrow> xs = []"
  5396 
  5397 text {*
  5398   Efficient emptyness check is implemented by @{const null}.
  5399 *}
  5400 
  5401 lemma null_rec [code]:
  5402   "null (x # xs) \<longleftrightarrow> False"
  5403   "null [] \<longleftrightarrow> True"
  5404   by (simp_all add: null_def)
  5405 
  5406 lemma eq_Nil_null: (* FIXME delete candidate *)
  5407   "xs = [] \<longleftrightarrow> null xs"
  5408   by (simp add: null_def)
  5409 
  5410 lemma equal_Nil_null [code_unfold]:
  5411   "HOL.equal xs [] \<longleftrightarrow> null xs"
  5412   by (simp add: equal eq_Nil_null)
  5413 
  5414 definition maps :: "('a \<Rightarrow> 'b list) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  5415   [code_abbrev]: "maps f xs = concat (map f xs)"
  5416 
  5417 definition map_filter :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  5418   [code_post]: "map_filter f xs = map (the \<circ> f) (filter (\<lambda>x. f x \<noteq> None) xs)"
  5419 
  5420 text {*
  5421   Operations @{const maps} and @{const map_filter} avoid
  5422   intermediate lists on execution -- do not use for proving.
  5423 *}
  5424 
  5425 lemma maps_simps [code]:
  5426   "maps f (x # xs) = f x @ maps f xs"
  5427   "maps f [] = []"
  5428   by (simp_all add: maps_def)
  5429 
  5430 lemma map_filter_simps [code]:
  5431   "map_filter f (x # xs) = (case f x of None \<Rightarrow> map_filter f xs | Some y \<Rightarrow> y # map_filter f xs)"
  5432   "map_filter f [] = []"
  5433   by (simp_all add: map_filter_def split: option.split)
  5434 
  5435 lemma concat_map_maps: (* FIXME delete candidate *)
  5436   "concat (map f xs) = maps f xs"
  5437   by (simp add: maps_def)
  5438 
  5439 lemma map_filter_map_filter [code_unfold]:
  5440   "map f (filter P xs) = map_filter (\<lambda>x. if P x then Some (f x) else None) xs"
  5441   by (simp add: map_filter_def)
  5442 
  5443 text {* Optimized code for @{text"\<forall>i\<in>{a..b::int}"} and @{text"\<forall>n:{a..<b::nat}"}
  5444 and similiarly for @{text"\<exists>"}. *}
  5445 
  5446 definition all_interval_nat :: "(nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
  5447   "all_interval_nat P i j \<longleftrightarrow> (\<forall>n \<in> {i..<j}. P n)"
  5448 
  5449 lemma [code]:
  5450   "all_interval_nat P i j \<longleftrightarrow> i \<ge> j \<or> P i \<and> all_interval_nat P (Suc i) j"
  5451 proof -
  5452   have *: "\<And>n. P i \<Longrightarrow> \<forall>n\<in>{Suc i..<j}. P n \<Longrightarrow> i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n"
  5453   proof -
  5454     fix n
  5455     assume "P i" "\<forall>n\<in>{Suc i..<j}. P n" "i \<le> n" "n < j"
  5456     then show "P n" by (cases "n = i") simp_all
  5457   qed
  5458   show ?thesis by (auto simp add: all_interval_nat_def intro: *)
  5459 qed
  5460 
  5461 lemma list_all_iff_all_interval_nat [code_unfold]:
  5462   "list_all P [i..<j] \<longleftrightarrow> all_interval_nat P i j"
  5463   by (simp add: list_all_iff all_interval_nat_def)
  5464 
  5465 lemma list_ex_iff_not_all_inverval_nat [code_unfold]:
  5466   "list_ex P [i..<j] \<longleftrightarrow> \<not> (all_interval_nat (Not \<circ> P) i j)"
  5467   by (simp add: list_ex_iff all_interval_nat_def)
  5468 
  5469 definition all_interval_int :: "(int \<Rightarrow> bool) \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool" where
  5470   "all_interval_int P i j \<longleftrightarrow> (\<forall>k \<in> {i..j}. P k)"
  5471 
  5472 lemma [code]:
  5473   "all_interval_int P i j \<longleftrightarrow> i > j \<or> P i \<and> all_interval_int P (i + 1) j"
  5474 proof -
  5475   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"
  5476   proof -
  5477     fix k
  5478     assume "P i" "\<forall>k\<in>{i+1..j}. P k" "i \<le> k" "k \<le> j"
  5479     then show "P k" by (cases "k = i") simp_all
  5480   qed
  5481   show ?thesis by (auto simp add: all_interval_int_def intro: *)
  5482 qed
  5483 
  5484 lemma list_all_iff_all_interval_int [code_unfold]:
  5485   "list_all P [i..j] \<longleftrightarrow> all_interval_int P i j"
  5486   by (simp add: list_all_iff all_interval_int_def)
  5487 
  5488 lemma list_ex_iff_not_all_inverval_int [code_unfold]:
  5489   "list_ex P [i..j] \<longleftrightarrow> \<not> (all_interval_int (Not \<circ> P) i j)"
  5490   by (simp add: list_ex_iff all_interval_int_def)
  5491 
  5492 hide_const (open) member null maps map_filter all_interval_nat all_interval_int
  5493 
  5494 
  5495 subsubsection {* Pretty lists *}
  5496 
  5497 use "Tools/list_code.ML"
  5498 
  5499 code_type list
  5500   (SML "_ list")
  5501   (OCaml "_ list")
  5502   (Haskell "![(_)]")
  5503   (Scala "List[(_)]")
  5504 
  5505 code_const Nil
  5506   (SML "[]")
  5507   (OCaml "[]")
  5508   (Haskell "[]")
  5509   (Scala "!Nil")
  5510 
  5511 code_instance list :: equal
  5512   (Haskell -)
  5513 
  5514 code_const "HOL.equal \<Colon> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
  5515   (Haskell infix 4 "==")
  5516 
  5517 code_reserved SML
  5518   list
  5519 
  5520 code_reserved OCaml
  5521   list
  5522 
  5523 setup {* fold (List_Code.add_literal_list) ["SML", "OCaml", "Haskell", "Scala"] *}
  5524 
  5525 
  5526 subsubsection {* Use convenient predefined operations *}
  5527 
  5528 code_const "op @"
  5529   (SML infixr 7 "@")
  5530   (OCaml infixr 6 "@")
  5531   (Haskell infixr 5 "++")
  5532   (Scala infixl 7 "++")
  5533 
  5534 code_const map
  5535   (Haskell "map")
  5536 
  5537 code_const filter
  5538   (Haskell "filter")
  5539 
  5540 code_const concat
  5541   (Haskell "concat")
  5542 
  5543 code_const List.maps
  5544   (Haskell "concatMap")
  5545 
  5546 code_const rev
  5547   (Haskell "reverse")
  5548 
  5549 code_const zip
  5550   (Haskell "zip")
  5551 
  5552 code_const List.null
  5553   (Haskell "null")
  5554 
  5555 code_const takeWhile
  5556   (Haskell "takeWhile")
  5557 
  5558 code_const dropWhile
  5559   (Haskell "dropWhile")
  5560 
  5561 code_const list_all
  5562   (Haskell "all")
  5563 
  5564 code_const list_ex
  5565   (Haskell "any")
  5566 
  5567 
  5568 subsubsection {* Implementation of sets by lists *}
  5569 
  5570 lemma is_empty_set [code]:
  5571   "Set.is_empty (set xs) \<longleftrightarrow> List.null xs"
  5572   by (simp add: Set.is_empty_def null_def)
  5573 
  5574 lemma empty_set [code]:
  5575   "{} = set []"
  5576   by simp
  5577 
  5578 lemma UNIV_coset [code]:
  5579   "UNIV = List.coset []"
  5580   by simp
  5581 
  5582 lemma compl_set [code]:
  5583   "- set xs = List.coset xs"
  5584   by simp
  5585 
  5586 lemma compl_coset [code]:
  5587   "- List.coset xs = set xs"
  5588   by simp
  5589 
  5590 lemma [code]:
  5591   "x \<in> set xs \<longleftrightarrow> List.member xs x"
  5592   "x \<in> List.coset xs \<longleftrightarrow> \<not> List.member xs x"
  5593   by (simp_all add: member_def)
  5594 
  5595 lemma insert_code [code]:
  5596   "insert x (set xs) = set (List.insert x xs)"
  5597   "insert x (List.coset xs) = List.coset (removeAll x xs)"
  5598   by simp_all
  5599 
  5600 lemma remove_code [code]:
  5601   "Set.remove x (set xs) = set (removeAll x xs)"
  5602   "Set.remove x (List.coset xs) = List.coset (List.insert x xs)"
  5603   by (simp_all add: remove_def Compl_insert)
  5604 
  5605 lemma project_set [code]:
  5606   "Set.project P (set xs) = set (filter P xs)"
  5607   by auto
  5608 
  5609 lemma image_set [code]:
  5610   "image f (set xs) = set (map f xs)"
  5611   by simp
  5612 
  5613 lemma subset_code [code]:
  5614   "set xs \<le> B \<longleftrightarrow> (\<forall>x\<in>set xs. x \<in> B)"
  5615   "A \<le> List.coset ys \<longleftrightarrow> (\<forall>y\<in>set ys. y \<notin> A)"
  5616   "List.coset [] \<le> set [] \<longleftrightarrow> False"
  5617   by auto
  5618 
  5619 text {* A frequent case – avoid intermediate sets *}
  5620 lemma [code_unfold]:
  5621   "set xs \<subseteq> set ys \<longleftrightarrow> list_all (\<lambda>x. x \<in> set ys) xs"
  5622   by (auto simp: list_all_iff)
  5623 
  5624 lemma Ball_set [code]:
  5625   "Ball (set xs) P \<longleftrightarrow> list_all P xs"
  5626   by (simp add: list_all_iff)
  5627 
  5628 lemma Bex_set [code]:
  5629   "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
  5630   by (simp add: list_ex_iff)
  5631 
  5632 lemma card_set [code]:
  5633   "card (set xs) = length (remdups xs)"
  5634 proof -
  5635   have "card (set (remdups xs)) = length (remdups xs)"
  5636     by (rule distinct_card) simp
  5637   then show ?thesis by simp
  5638 qed
  5639 
  5640 lemma the_elem_set [code]:
  5641   "the_elem (set [x]) = x"
  5642   by simp
  5643 
  5644 lemma Pow_set [code]:
  5645   "Pow (set []) = {{}}"
  5646   "Pow (set (x # xs)) = (let A = Pow (set xs) in A \<union> insert x ` A)"
  5647   by (simp_all add: Pow_insert Let_def)
  5648 
  5649 lemma setsum_code [code]:
  5650   "setsum f (set xs) = listsum (map f (remdups xs))"
  5651 by (simp add: listsum_distinct_conv_setsum_set)
  5652 
  5653 definition map_project :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a set \<Rightarrow> 'b set" where
  5654   "map_project f A = {b. \<exists> a \<in> A. f a = Some b}"
  5655 
  5656 lemma [code]:
  5657   "map_project f (set xs) = set (List.map_filter f xs)"
  5658   by (auto simp add: map_project_def map_filter_def image_def)
  5659 
  5660 hide_const (open) map_project
  5661 
  5662 text {* Operations on relations *}
  5663 
  5664 lemma product_code [code]:
  5665   "Product_Type.product (set xs) (set ys) = set [(x, y). x \<leftarrow> xs, y \<leftarrow> ys]"
  5666   by (auto simp add: Product_Type.product_def)
  5667 
  5668 lemma Id_on_set [code]:
  5669   "Id_on (set xs) = set [(x, x). x \<leftarrow> xs]"
  5670   by (auto simp add: Id_on_def)
  5671 
  5672 lemma [code]:
  5673   "R `` S = List.map_project (%(x, y). if x : S then Some y else None) R"
  5674 unfolding map_project_def by (auto split: prod.split split_if_asm)
  5675 
  5676 lemma trancl_set_ntrancl [code]:
  5677   "trancl (set xs) = ntrancl (card (set xs) - 1) (set xs)"
  5678   by (simp add: finite_trancl_ntranl)
  5679 
  5680 lemma set_relcomp [code]:
  5681   "set xys O set yzs = set ([(fst xy, snd yz). xy \<leftarrow> xys, yz \<leftarrow> yzs, snd xy = fst yz])"
  5682   by (auto simp add: Bex_def)
  5683 
  5684 lemma wf_set [code]:
  5685   "wf (set xs) = acyclic (set xs)"
  5686   by (simp add: wf_iff_acyclic_if_finite)
  5687 
  5688 end
  5689