src/HOL/List.thy
author wenzelm
Thu May 24 17:25:53 2012 +0200 (2012-05-24)
changeset 47988 e4b69e10b990
parent 47841 179b5e7c9803
child 48619 558e4e77ce69
permissions -rw-r--r--
tuned proofs;
     1 (*  Title:      HOL/List.thy
     2     Author:     Tobias Nipkow
     3 *)
     4 
     5 header {* The datatype of finite lists *}
     6 
     7 theory List
     8 imports Plain Presburger Code_Numeral Quotient ATP
     9 uses
    10   ("Tools/list_code.ML")
    11   ("Tools/list_to_set_comprehension.ML")
    12 begin
    13 
    14 datatype 'a list =
    15     Nil    ("[]")
    16   | Cons 'a  "'a list"    (infixr "#" 65)
    17 
    18 syntax
    19   -- {* list Enumeration *}
    20   "_list" :: "args => 'a list"    ("[(_)]")
    21 
    22 translations
    23   "[x, xs]" == "x#[xs]"
    24   "[x]" == "x#[]"
    25 
    26 
    27 subsection {* Basic list processing functions *}
    28 
    29 primrec
    30   hd :: "'a list \<Rightarrow> 'a" where
    31   "hd (x # xs) = x"
    32 
    33 primrec
    34   tl :: "'a list \<Rightarrow> 'a list" where
    35     "tl [] = []"
    36   | "tl (x # xs) = xs"
    37 
    38 primrec
    39   last :: "'a list \<Rightarrow> 'a" where
    40   "last (x # xs) = (if xs = [] then x else last xs)"
    41 
    42 primrec
    43   butlast :: "'a list \<Rightarrow> 'a list" where
    44     "butlast []= []"
    45   | "butlast (x # xs) = (if xs = [] then [] else x # butlast xs)"
    46 
    47 primrec
    48   set :: "'a list \<Rightarrow> 'a set" where
    49     "set [] = {}"
    50   | "set (x # xs) = insert x (set xs)"
    51 
    52 definition
    53   coset :: "'a list \<Rightarrow> 'a set" where
    54   [simp]: "coset xs = - set xs"
    55 
    56 primrec
    57   map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
    58     "map f [] = []"
    59   | "map f (x # xs) = f x # map f xs"
    60 
    61 primrec
    62   append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65) where
    63     append_Nil:"[] @ ys = ys"
    64   | append_Cons: "(x#xs) @ ys = x # xs @ ys"
    65 
    66 primrec
    67   rev :: "'a list \<Rightarrow> 'a list" where
    68     "rev [] = []"
    69   | "rev (x # xs) = rev xs @ [x]"
    70 
    71 primrec
    72   filter:: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
    73     "filter P [] = []"
    74   | "filter P (x # xs) = (if P x then x # filter P xs else filter P xs)"
    75 
    76 syntax
    77   -- {* Special syntax for filter *}
    78   "_filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_<-_./ _])")
    79 
    80 translations
    81   "[x<-xs . P]"== "CONST filter (%x. P) xs"
    82 
    83 syntax (xsymbols)
    84   "_filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
    85 syntax (HTML output)
    86   "_filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
    87 
    88 primrec fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b"
    89 where
    90   fold_Nil:  "fold f [] = id"
    91 | fold_Cons: "fold f (x # xs) = fold f xs \<circ> f x" -- {* natural argument order *}
    92 
    93 primrec foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b"
    94 where
    95   foldr_Nil:  "foldr f [] = id"
    96 | foldr_Cons: "foldr f (x # xs) = f x \<circ> foldr f xs" -- {* natural argument order *}
    97 
    98 primrec foldl :: "('b \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"
    99 where
   100   foldl_Nil:  "foldl f a [] = a"
   101 | foldl_Cons: "foldl f a (x # xs) = foldl f (f a x) xs"
   102 
   103 primrec
   104   concat:: "'a list list \<Rightarrow> 'a list" where
   105     "concat [] = []"
   106   | "concat (x # xs) = x @ concat xs"
   107 
   108 definition (in monoid_add)
   109   listsum :: "'a list \<Rightarrow> 'a" where
   110   "listsum xs = foldr plus xs 0"
   111 
   112 primrec
   113   drop:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   114     drop_Nil: "drop n [] = []"
   115   | drop_Cons: "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
   116   -- {*Warning: simpset does not contain this definition, but separate
   117        theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
   118 
   119 primrec
   120   take:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   121     take_Nil:"take n [] = []"
   122   | take_Cons: "take n (x # xs) = (case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # take m xs)"
   123   -- {*Warning: simpset does not contain this definition, but separate
   124        theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
   125 
   126 primrec
   127   nth :: "'a list => nat => 'a" (infixl "!" 100) where
   128   nth_Cons: "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
   129   -- {*Warning: simpset does not contain this definition, but separate
   130        theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
   131 
   132 primrec
   133   list_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
   134     "list_update [] i v = []"
   135   | "list_update (x # xs) i v = (case i of 0 \<Rightarrow> v # xs | Suc j \<Rightarrow> x # list_update xs j v)"
   136 
   137 nonterminal lupdbinds and lupdbind
   138 
   139 syntax
   140   "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
   141   "" :: "lupdbind => lupdbinds"    ("_")
   142   "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
   143   "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
   144 
   145 translations
   146   "_LUpdate xs (_lupdbinds b bs)" == "_LUpdate (_LUpdate xs b) bs"
   147   "xs[i:=x]" == "CONST list_update xs i x"
   148 
   149 primrec
   150   takeWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   151     "takeWhile P [] = []"
   152   | "takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])"
   153 
   154 primrec
   155   dropWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   156     "dropWhile P [] = []"
   157   | "dropWhile P (x # xs) = (if P x then dropWhile P xs else x # xs)"
   158 
   159 primrec
   160   zip :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
   161     "zip xs [] = []"
   162   | zip_Cons: "zip xs (y # ys) = (case xs of [] => [] | z # zs => (z, y) # zip zs ys)"
   163   -- {*Warning: simpset does not contain this definition, but separate
   164        theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
   165 
   166 primrec 
   167   upt :: "nat \<Rightarrow> nat \<Rightarrow> nat list" ("(1[_..</_'])") where
   168     upt_0: "[i..<0] = []"
   169   | upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
   170 
   171 definition
   172   insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   173   "insert x xs = (if x \<in> set xs then xs else x # xs)"
   174 
   175 hide_const (open) insert
   176 hide_fact (open) insert_def
   177 
   178 primrec find :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a option" where
   179   "find _ [] = None"
   180 | "find P (x#xs) = (if P x then Some x else find P xs)"
   181 
   182 hide_const (open) find
   183 
   184 primrec
   185   remove1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   186     "remove1 x [] = []"
   187   | "remove1 x (y # xs) = (if x = y then xs else y # remove1 x xs)"
   188 
   189 primrec
   190   removeAll :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   191     "removeAll x [] = []"
   192   | "removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)"
   193 
   194 primrec
   195   distinct :: "'a list \<Rightarrow> bool" where
   196     "distinct [] \<longleftrightarrow> True"
   197   | "distinct (x # xs) \<longleftrightarrow> x \<notin> set xs \<and> distinct xs"
   198 
   199 primrec
   200   remdups :: "'a list \<Rightarrow> 'a list" where
   201     "remdups [] = []"
   202   | "remdups (x # xs) = (if x \<in> set xs then remdups xs else x # remdups xs)"
   203 
   204 primrec
   205   replicate :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
   206     replicate_0: "replicate 0 x = []"
   207   | replicate_Suc: "replicate (Suc n) x = x # replicate n x"
   208 
   209 text {*
   210   Function @{text size} is overloaded for all datatypes. Users may
   211   refer to the list version as @{text length}. *}
   212 
   213 abbreviation
   214   length :: "'a list \<Rightarrow> nat" where
   215   "length \<equiv> size"
   216 
   217 primrec rotate1 :: "'a list \<Rightarrow> 'a list" where
   218   "rotate1 [] = []" |
   219   "rotate1 (x # xs) = xs @ [x]"
   220 
   221 definition
   222   rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   223   "rotate n = rotate1 ^^ n"
   224 
   225 definition
   226   list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
   227   "list_all2 P xs ys =
   228     (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"
   229 
   230 definition
   231   sublist :: "'a list => nat set => 'a list" where
   232   "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"
   233 
   234 fun splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   235 "splice [] ys = ys" |
   236 "splice xs [] = xs" |
   237 "splice (x#xs) (y#ys) = x # y # splice xs ys"
   238 
   239 text{*
   240 \begin{figure}[htbp]
   241 \fbox{
   242 \begin{tabular}{l}
   243 @{lemma "[a,b]@[c,d] = [a,b,c,d]" by simp}\\
   244 @{lemma "length [a,b,c] = 3" by simp}\\
   245 @{lemma "set [a,b,c] = {a,b,c}" by simp}\\
   246 @{lemma "map f [a,b,c] = [f a, f b, f c]" by simp}\\
   247 @{lemma "rev [a,b,c] = [c,b,a]" by simp}\\
   248 @{lemma "hd [a,b,c,d] = a" by simp}\\
   249 @{lemma "tl [a,b,c,d] = [b,c,d]" by simp}\\
   250 @{lemma "last [a,b,c,d] = d" by simp}\\
   251 @{lemma "butlast [a,b,c,d] = [a,b,c]" by simp}\\
   252 @{lemma[source] "filter (\<lambda>n::nat. n<2) [0,2,1] = [0,1]" by simp}\\
   253 @{lemma "concat [[a,b],[c,d,e],[],[f]] = [a,b,c,d,e,f]" by simp}\\
   254 @{lemma "fold f [a,b,c] x = f c (f b (f a x))" by simp}\\
   255 @{lemma "foldr f [a,b,c] x = f a (f b (f c x))" by simp}\\
   256 @{lemma "foldl f x [a,b,c] = f (f (f x a) b) c" by simp}\\
   257 @{lemma "zip [a,b,c] [x,y,z] = [(a,x),(b,y),(c,z)]" by simp}\\
   258 @{lemma "zip [a,b] [x,y,z] = [(a,x),(b,y)]" by simp}\\
   259 @{lemma "splice [a,b,c] [x,y,z] = [a,x,b,y,c,z]" by simp}\\
   260 @{lemma "splice [a,b,c,d] [x,y] = [a,x,b,y,c,d]" by simp}\\
   261 @{lemma "take 2 [a,b,c,d] = [a,b]" by simp}\\
   262 @{lemma "take 6 [a,b,c,d] = [a,b,c,d]" by simp}\\
   263 @{lemma "drop 2 [a,b,c,d] = [c,d]" by simp}\\
   264 @{lemma "drop 6 [a,b,c,d] = []" by simp}\\
   265 @{lemma "takeWhile (%n::nat. n<3) [1,2,3,0] = [1,2]" by simp}\\
   266 @{lemma "dropWhile (%n::nat. n<3) [1,2,3,0] = [3,0]" by simp}\\
   267 @{lemma "distinct [2,0,1::nat]" by simp}\\
   268 @{lemma "remdups [2,0,2,1::nat,2] = [0,1,2]" by simp}\\
   269 @{lemma "List.insert 2 [0::nat,1,2] = [0,1,2]" by (simp add: List.insert_def)}\\
   270 @{lemma "List.insert 3 [0::nat,1,2] = [3,0,1,2]" by (simp add: List.insert_def)}\\
   271 @{lemma "List.find (%i::int. i>0) [0,0] = None" by simp}\\
   272 @{lemma "List.find (%i::int. i>0) [0,1,0,2] = Some 1" by simp}\\
   273 @{lemma "remove1 2 [2,0,2,1::nat,2] = [0,2,1,2]" by simp}\\
   274 @{lemma "removeAll 2 [2,0,2,1::nat,2] = [0,1]" by simp}\\
   275 @{lemma "nth [a,b,c,d] 2 = c" by simp}\\
   276 @{lemma "[a,b,c,d][2 := x] = [a,b,x,d]" by simp}\\
   277 @{lemma "sublist [a,b,c,d,e] {0,2,3} = [a,c,d]" by (simp add:sublist_def)}\\
   278 @{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by simp}\\
   279 @{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate_def eval_nat_numeral)}\\
   280 @{lemma "replicate 4 a = [a,a,a,a]" by (simp add:eval_nat_numeral)}\\
   281 @{lemma "[2..<5] = [2,3,4]" by (simp add:eval_nat_numeral)}\\
   282 @{lemma "listsum [1,2,3::nat] = 6" by (simp add: listsum_def)}
   283 \end{tabular}}
   284 \caption{Characteristic examples}
   285 \label{fig:Characteristic}
   286 \end{figure}
   287 Figure~\ref{fig:Characteristic} shows characteristic examples
   288 that should give an intuitive understanding of the above functions.
   289 *}
   290 
   291 text{* The following simple sort functions are intended for proofs,
   292 not for efficient implementations. *}
   293 
   294 context linorder
   295 begin
   296 
   297 inductive sorted :: "'a list \<Rightarrow> bool" where
   298   Nil [iff]: "sorted []"
   299 | Cons: "\<forall>y\<in>set xs. x \<le> y \<Longrightarrow> sorted xs \<Longrightarrow> sorted (x # xs)"
   300 
   301 lemma sorted_single [iff]:
   302   "sorted [x]"
   303   by (rule sorted.Cons) auto
   304 
   305 lemma sorted_many:
   306   "x \<le> y \<Longrightarrow> sorted (y # zs) \<Longrightarrow> sorted (x # y # zs)"
   307   by (rule sorted.Cons) (cases "y # zs" rule: sorted.cases, auto)
   308 
   309 lemma sorted_many_eq [simp, code]:
   310   "sorted (x # y # zs) \<longleftrightarrow> x \<le> y \<and> sorted (y # zs)"
   311   by (auto intro: sorted_many elim: sorted.cases)
   312 
   313 lemma [code]:
   314   "sorted [] \<longleftrightarrow> True"
   315   "sorted [x] \<longleftrightarrow> True"
   316   by simp_all
   317 
   318 primrec insort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
   319   "insort_key f x [] = [x]" |
   320   "insort_key f x (y#ys) = (if f x \<le> f y then (x#y#ys) else y#(insort_key f x ys))"
   321 
   322 definition sort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b list \<Rightarrow> 'b list" where
   323   "sort_key f xs = foldr (insort_key f) xs []"
   324 
   325 definition insort_insert_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
   326   "insort_insert_key f x xs = (if f x \<in> f ` set xs then xs else insort_key f x xs)"
   327 
   328 abbreviation "sort \<equiv> sort_key (\<lambda>x. x)"
   329 abbreviation "insort \<equiv> insort_key (\<lambda>x. x)"
   330 abbreviation "insort_insert \<equiv> insort_insert_key (\<lambda>x. x)"
   331 
   332 end
   333 
   334 
   335 subsubsection {* List comprehension *}
   336 
   337 text{* Input syntax for Haskell-like list comprehension notation.
   338 Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
   339 the list of all pairs of distinct elements from @{text xs} and @{text ys}.
   340 The syntax is as in Haskell, except that @{text"|"} becomes a dot
   341 (like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
   342 \verb![e| x <- xs, ...]!.
   343 
   344 The qualifiers after the dot are
   345 \begin{description}
   346 \item[generators] @{text"p \<leftarrow> xs"},
   347  where @{text p} is a pattern and @{text xs} an expression of list type, or
   348 \item[guards] @{text"b"}, where @{text b} is a boolean expression.
   349 %\item[local bindings] @ {text"let x = e"}.
   350 \end{description}
   351 
   352 Just like in Haskell, list comprehension is just a shorthand. To avoid
   353 misunderstandings, the translation into desugared form is not reversed
   354 upon output. Note that the translation of @{text"[e. x \<leftarrow> xs]"} is
   355 optmized to @{term"map (%x. e) xs"}.
   356 
   357 It is easy to write short list comprehensions which stand for complex
   358 expressions. During proofs, they may become unreadable (and
   359 mangled). In such cases it can be advisable to introduce separate
   360 definitions for the list comprehensions in question.  *}
   361 
   362 nonterminal lc_qual and lc_quals
   363 
   364 syntax
   365   "_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
   366   "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ <- _")
   367   "_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
   368   (*"_lc_let" :: "letbinds => lc_qual"  ("let _")*)
   369   "_lc_end" :: "lc_quals" ("]")
   370   "_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals"  (", __")
   371   "_lc_abs" :: "'a => 'b list => 'b list"
   372 
   373 (* These are easier than ML code but cannot express the optimized
   374    translation of [e. p<-xs]
   375 translations
   376   "[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
   377   "_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
   378    => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
   379   "[e. P]" => "if P then [e] else []"
   380   "_listcompr e (_lc_test P) (_lc_quals Q Qs)"
   381    => "if P then (_listcompr e Q Qs) else []"
   382   "_listcompr e (_lc_let b) (_lc_quals Q Qs)"
   383    => "_Let b (_listcompr e Q Qs)"
   384 *)
   385 
   386 syntax (xsymbols)
   387   "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ \<leftarrow> _")
   388 syntax (HTML output)
   389   "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ \<leftarrow> _")
   390 
   391 parse_translation (advanced) {*
   392   let
   393     val NilC = Syntax.const @{const_syntax Nil};
   394     val ConsC = Syntax.const @{const_syntax Cons};
   395     val mapC = Syntax.const @{const_syntax map};
   396     val concatC = Syntax.const @{const_syntax concat};
   397     val IfC = Syntax.const @{const_syntax If};
   398 
   399     fun single x = ConsC $ x $ NilC;
   400 
   401     fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
   402       let
   403         (* FIXME proper name context!? *)
   404         val x =
   405           Free (singleton (Name.variant_list (fold Term.add_free_names [p, e] [])) "x", dummyT);
   406         val e = if opti then single e else e;
   407         val case1 = Syntax.const @{syntax_const "_case1"} $ p $ e;
   408         val case2 =
   409           Syntax.const @{syntax_const "_case1"} $
   410             Syntax.const @{const_syntax dummy_pattern} $ NilC;
   411         val cs = Syntax.const @{syntax_const "_case2"} $ case1 $ case2;
   412       in Syntax_Trans.abs_tr [x, Datatype_Case.case_tr false ctxt [x, cs]] end;
   413 
   414     fun abs_tr ctxt p e opti =
   415       (case Term_Position.strip_positions p of
   416         Free (s, T) =>
   417           let
   418             val thy = Proof_Context.theory_of ctxt;
   419             val s' = Proof_Context.intern_const ctxt s;
   420           in
   421             if Sign.declared_const thy s'
   422             then (pat_tr ctxt p e opti, false)
   423             else (Syntax_Trans.abs_tr [p, e], true)
   424           end
   425       | _ => (pat_tr ctxt p e opti, false));
   426 
   427     fun lc_tr ctxt [e, Const (@{syntax_const "_lc_test"}, _) $ b, qs] =
   428           let
   429             val res =
   430               (case qs of
   431                 Const (@{syntax_const "_lc_end"}, _) => single e
   432               | Const (@{syntax_const "_lc_quals"}, _) $ q $ qs => lc_tr ctxt [e, q, qs]);
   433           in IfC $ b $ res $ NilC end
   434       | lc_tr ctxt
   435             [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
   436               Const(@{syntax_const "_lc_end"}, _)] =
   437           (case abs_tr ctxt p e true of
   438             (f, true) => mapC $ f $ es
   439           | (f, false) => concatC $ (mapC $ f $ es))
   440       | lc_tr ctxt
   441             [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
   442               Const (@{syntax_const "_lc_quals"}, _) $ q $ qs] =
   443           let val e' = lc_tr ctxt [e, q, qs];
   444           in concatC $ (mapC $ (fst (abs_tr ctxt p e' false)) $ es) end;
   445 
   446   in [(@{syntax_const "_listcompr"}, lc_tr)] end
   447 *}
   448 
   449 ML {*
   450   let
   451     val read = Syntax.read_term @{context};
   452     fun check s1 s2 = read s1 aconv read s2 orelse error ("Check failed: " ^ quote s1);
   453   in
   454     check "[(x,y,z). b]" "if b then [(x, y, z)] else []";
   455     check "[(x,y,z). x\<leftarrow>xs]" "map (\<lambda>x. (x, y, z)) xs";
   456     check "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]" "concat (map (\<lambda>x. map (\<lambda>y. e x y) ys) xs)";
   457     check "[(x,y,z). x<a, x>b]" "if x < a then if b < x then [(x, y, z)] else [] else []";
   458     check "[(x,y,z). x\<leftarrow>xs, x>b]" "concat (map (\<lambda>x. if b < x then [(x, y, z)] else []) xs)";
   459     check "[(x,y,z). x<a, x\<leftarrow>xs]" "if x < a then map (\<lambda>x. (x, y, z)) xs else []";
   460     check "[(x,y). Cons True x \<leftarrow> xs]"
   461       "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | True # x \<Rightarrow> [(x, y)] | False # x \<Rightarrow> []) xs)";
   462     check "[(x,y,z). Cons x [] \<leftarrow> xs]"
   463       "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | [x] \<Rightarrow> [(x, y, z)] | x # aa # lista \<Rightarrow> []) xs)";
   464     check "[(x,y,z). x<a, x>b, x=d]"
   465       "if x < a then if b < x then if x = d then [(x, y, z)] else [] else [] else []";
   466     check "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
   467       "if x < a then if b < x then map (\<lambda>y. (x, y, z)) ys else [] else []";
   468     check "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
   469       "if x < a then concat (map (\<lambda>x. if b < y then [(x, y, z)] else []) xs) else []";
   470     check "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
   471       "if x < a then concat (map (\<lambda>x. map (\<lambda>y. (x, y, z)) ys) xs) else []";
   472     check "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
   473       "concat (map (\<lambda>x. if b < x then if y < a then [(x, y, z)] else [] else []) xs)";
   474     check "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
   475       "concat (map (\<lambda>x. if b < x then map (\<lambda>y. (x, y, z)) ys else []) xs)";
   476     check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
   477       "concat (map (\<lambda>x. concat (map (\<lambda>y. if x < y then [(x, y, z)] else []) ys)) xs)";
   478     check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
   479       "concat (map (\<lambda>x. concat (map (\<lambda>y. map (\<lambda>z. (x, y, z)) zs) ys)) xs)"
   480   end;
   481 *}
   482 
   483 (*
   484 term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
   485 *)
   486 
   487 
   488 use "Tools/list_to_set_comprehension.ML"
   489 
   490 simproc_setup list_to_set_comprehension ("set xs") = {* K List_to_Set_Comprehension.simproc *}
   491 
   492 code_datatype set coset
   493 
   494 hide_const (open) coset
   495 
   496 subsubsection {* @{const Nil} and @{const Cons} *}
   497 
   498 lemma not_Cons_self [simp]:
   499   "xs \<noteq> x # xs"
   500 by (induct xs) auto
   501 
   502 lemma not_Cons_self2 [simp]:
   503   "x # xs \<noteq> xs"
   504 by (rule not_Cons_self [symmetric])
   505 
   506 lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
   507 by (induct xs) auto
   508 
   509 lemma length_induct:
   510   "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
   511 by (rule measure_induct [of length]) iprover
   512 
   513 lemma list_nonempty_induct [consumes 1, case_names single cons]:
   514   assumes "xs \<noteq> []"
   515   assumes single: "\<And>x. P [x]"
   516   assumes cons: "\<And>x xs. xs \<noteq> [] \<Longrightarrow> P xs \<Longrightarrow> P (x # xs)"
   517   shows "P xs"
   518 using `xs \<noteq> []` proof (induct xs)
   519   case Nil then show ?case by simp
   520 next
   521   case (Cons x xs) show ?case proof (cases xs)
   522     case Nil with single show ?thesis by simp
   523   next
   524     case Cons then have "xs \<noteq> []" by simp
   525     moreover with Cons.hyps have "P xs" .
   526     ultimately show ?thesis by (rule cons)
   527   qed
   528 qed
   529 
   530 lemma inj_split_Cons: "inj_on (\<lambda>(xs, n). n#xs) X"
   531   by (auto intro!: inj_onI)
   532 
   533 subsubsection {* @{const length} *}
   534 
   535 text {*
   536   Needs to come before @{text "@"} because of theorem @{text
   537   append_eq_append_conv}.
   538 *}
   539 
   540 lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
   541 by (induct xs) auto
   542 
   543 lemma length_map [simp]: "length (map f xs) = length xs"
   544 by (induct xs) auto
   545 
   546 lemma length_rev [simp]: "length (rev xs) = length xs"
   547 by (induct xs) auto
   548 
   549 lemma length_tl [simp]: "length (tl xs) = length xs - 1"
   550 by (cases xs) auto
   551 
   552 lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
   553 by (induct xs) auto
   554 
   555 lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
   556 by (induct xs) auto
   557 
   558 lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
   559 by auto
   560 
   561 lemma length_Suc_conv:
   562 "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
   563 by (induct xs) auto
   564 
   565 lemma Suc_length_conv:
   566 "(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
   567 apply (induct xs, simp, simp)
   568 apply blast
   569 done
   570 
   571 lemma impossible_Cons: "length xs <= length ys ==> xs = x # ys = False"
   572   by (induct xs) auto
   573 
   574 lemma list_induct2 [consumes 1, case_names Nil Cons]:
   575   "length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
   576    (\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
   577    \<Longrightarrow> P xs ys"
   578 proof (induct xs arbitrary: ys)
   579   case Nil then show ?case by simp
   580 next
   581   case (Cons x xs ys) then show ?case by (cases ys) simp_all
   582 qed
   583 
   584 lemma list_induct3 [consumes 2, case_names Nil Cons]:
   585   "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
   586    (\<And>x xs y ys z zs. length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P xs ys zs \<Longrightarrow> P (x#xs) (y#ys) (z#zs))
   587    \<Longrightarrow> P xs ys zs"
   588 proof (induct xs arbitrary: ys zs)
   589   case Nil then show ?case by simp
   590 next
   591   case (Cons x xs ys zs) then show ?case by (cases ys, simp_all)
   592     (cases zs, simp_all)
   593 qed
   594 
   595 lemma list_induct4 [consumes 3, case_names Nil Cons]:
   596   "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow>
   597    P [] [] [] [] \<Longrightarrow> (\<And>x xs y ys z zs w ws. length xs = length ys \<Longrightarrow>
   598    length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow> P xs ys zs ws \<Longrightarrow>
   599    P (x#xs) (y#ys) (z#zs) (w#ws)) \<Longrightarrow> P xs ys zs ws"
   600 proof (induct xs arbitrary: ys zs ws)
   601   case Nil then show ?case by simp
   602 next
   603   case (Cons x xs ys zs ws) then show ?case by ((cases ys, simp_all), (cases zs,simp_all)) (cases ws, simp_all)
   604 qed
   605 
   606 lemma list_induct2': 
   607   "\<lbrakk> P [] [];
   608   \<And>x xs. P (x#xs) [];
   609   \<And>y ys. P [] (y#ys);
   610    \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
   611  \<Longrightarrow> P xs ys"
   612 by (induct xs arbitrary: ys) (case_tac x, auto)+
   613 
   614 lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
   615 by (rule Eq_FalseI) auto
   616 
   617 simproc_setup list_neq ("(xs::'a list) = ys") = {*
   618 (*
   619 Reduces xs=ys to False if xs and ys cannot be of the same length.
   620 This is the case if the atomic sublists of one are a submultiset
   621 of those of the other list and there are fewer Cons's in one than the other.
   622 *)
   623 
   624 let
   625 
   626 fun len (Const(@{const_name Nil},_)) acc = acc
   627   | len (Const(@{const_name Cons},_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
   628   | len (Const(@{const_name append},_) $ xs $ ys) acc = len xs (len ys acc)
   629   | len (Const(@{const_name rev},_) $ xs) acc = len xs acc
   630   | len (Const(@{const_name map},_) $ _ $ xs) acc = len xs acc
   631   | len t (ts,n) = (t::ts,n);
   632 
   633 fun list_neq _ ss ct =
   634   let
   635     val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
   636     val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
   637     fun prove_neq() =
   638       let
   639         val Type(_,listT::_) = eqT;
   640         val size = HOLogic.size_const listT;
   641         val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
   642         val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
   643         val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
   644           (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
   645       in SOME (thm RS @{thm neq_if_length_neq}) end
   646   in
   647     if m < n andalso submultiset (op aconv) (ls,rs) orelse
   648        n < m andalso submultiset (op aconv) (rs,ls)
   649     then prove_neq() else NONE
   650   end;
   651 in list_neq end;
   652 *}
   653 
   654 
   655 subsubsection {* @{text "@"} -- append *}
   656 
   657 lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
   658 by (induct xs) auto
   659 
   660 lemma append_Nil2 [simp]: "xs @ [] = xs"
   661 by (induct xs) auto
   662 
   663 lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
   664 by (induct xs) auto
   665 
   666 lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
   667 by (induct xs) auto
   668 
   669 lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
   670 by (induct xs) auto
   671 
   672 lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
   673 by (induct xs) auto
   674 
   675 lemma append_eq_append_conv [simp, no_atp]:
   676  "length xs = length ys \<or> length us = length vs
   677  ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
   678 apply (induct xs arbitrary: ys)
   679  apply (case_tac ys, simp, force)
   680 apply (case_tac ys, force, simp)
   681 done
   682 
   683 lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
   684   (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
   685 apply (induct xs arbitrary: ys zs ts)
   686  apply fastforce
   687 apply(case_tac zs)
   688  apply simp
   689 apply fastforce
   690 done
   691 
   692 lemma same_append_eq [iff, induct_simp]: "(xs @ ys = xs @ zs) = (ys = zs)"
   693 by simp
   694 
   695 lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
   696 by simp
   697 
   698 lemma append_same_eq [iff, induct_simp]: "(ys @ xs = zs @ xs) = (ys = zs)"
   699 by simp
   700 
   701 lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
   702 using append_same_eq [of _ _ "[]"] by auto
   703 
   704 lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
   705 using append_same_eq [of "[]"] by auto
   706 
   707 lemma hd_Cons_tl [simp,no_atp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
   708 by (induct xs) auto
   709 
   710 lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
   711 by (induct xs) auto
   712 
   713 lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
   714 by (simp add: hd_append split: list.split)
   715 
   716 lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
   717 by (simp split: list.split)
   718 
   719 lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
   720 by (simp add: tl_append split: list.split)
   721 
   722 
   723 lemma Cons_eq_append_conv: "x#xs = ys@zs =
   724  (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
   725 by(cases ys) auto
   726 
   727 lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
   728  (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
   729 by(cases ys) auto
   730 
   731 
   732 text {* Trivial rules for solving @{text "@"}-equations automatically. *}
   733 
   734 lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
   735 by simp
   736 
   737 lemma Cons_eq_appendI:
   738 "[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
   739 by (drule sym) simp
   740 
   741 lemma append_eq_appendI:
   742 "[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
   743 by (drule sym) simp
   744 
   745 
   746 text {*
   747 Simplification procedure for all list equalities.
   748 Currently only tries to rearrange @{text "@"} to see if
   749 - both lists end in a singleton list,
   750 - or both lists end in the same list.
   751 *}
   752 
   753 simproc_setup list_eq ("(xs::'a list) = ys")  = {*
   754   let
   755     fun last (cons as Const (@{const_name Cons}, _) $ _ $ xs) =
   756           (case xs of Const (@{const_name Nil}, _) => cons | _ => last xs)
   757       | last (Const(@{const_name append},_) $ _ $ ys) = last ys
   758       | last t = t;
   759     
   760     fun list1 (Const(@{const_name Cons},_) $ _ $ Const(@{const_name Nil},_)) = true
   761       | list1 _ = false;
   762     
   763     fun butlast ((cons as Const(@{const_name Cons},_) $ x) $ xs) =
   764           (case xs of Const (@{const_name Nil}, _) => xs | _ => cons $ butlast xs)
   765       | butlast ((app as Const (@{const_name append}, _) $ xs) $ ys) = app $ butlast ys
   766       | butlast xs = Const(@{const_name Nil}, fastype_of xs);
   767     
   768     val rearr_ss =
   769       HOL_basic_ss addsimps [@{thm append_assoc}, @{thm append_Nil}, @{thm append_Cons}];
   770     
   771     fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
   772       let
   773         val lastl = last lhs and lastr = last rhs;
   774         fun rearr conv =
   775           let
   776             val lhs1 = butlast lhs and rhs1 = butlast rhs;
   777             val Type(_,listT::_) = eqT
   778             val appT = [listT,listT] ---> listT
   779             val app = Const(@{const_name append},appT)
   780             val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
   781             val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
   782             val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
   783               (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
   784           in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
   785       in
   786         if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
   787         else if lastl aconv lastr then rearr @{thm append_same_eq}
   788         else NONE
   789       end;
   790   in fn _ => fn ss => fn ct => list_eq ss (term_of ct) end;
   791 *}
   792 
   793 
   794 subsubsection {* @{text map} *}
   795 
   796 lemma hd_map:
   797   "xs \<noteq> [] \<Longrightarrow> hd (map f xs) = f (hd xs)"
   798   by (cases xs) simp_all
   799 
   800 lemma map_tl:
   801   "map f (tl xs) = tl (map f xs)"
   802   by (cases xs) simp_all
   803 
   804 lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
   805 by (induct xs) simp_all
   806 
   807 lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
   808 by (rule ext, induct_tac xs) auto
   809 
   810 lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
   811 by (induct xs) auto
   812 
   813 lemma map_map [simp]: "map f (map g xs) = map (f \<circ> g) xs"
   814 by (induct xs) auto
   815 
   816 lemma map_comp_map[simp]: "((map f) o (map g)) = map(f o g)"
   817 apply(rule ext)
   818 apply(simp)
   819 done
   820 
   821 lemma rev_map: "rev (map f xs) = map f (rev xs)"
   822 by (induct xs) auto
   823 
   824 lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
   825 by (induct xs) auto
   826 
   827 lemma map_cong [fundef_cong]:
   828   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> map f xs = map g ys"
   829   by simp
   830 
   831 lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
   832 by (cases xs) auto
   833 
   834 lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
   835 by (cases xs) auto
   836 
   837 lemma map_eq_Cons_conv:
   838  "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
   839 by (cases xs) auto
   840 
   841 lemma Cons_eq_map_conv:
   842  "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
   843 by (cases ys) auto
   844 
   845 lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
   846 lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
   847 declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
   848 
   849 lemma ex_map_conv:
   850   "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
   851 by(induct ys, auto simp add: Cons_eq_map_conv)
   852 
   853 lemma map_eq_imp_length_eq:
   854   assumes "map f xs = map g ys"
   855   shows "length xs = length ys"
   856 using assms proof (induct ys arbitrary: xs)
   857   case Nil then show ?case by simp
   858 next
   859   case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto
   860   from Cons xs have "map f zs = map g ys" by simp
   861   moreover with Cons have "length zs = length ys" by blast
   862   with xs show ?case by simp
   863 qed
   864   
   865 lemma map_inj_on:
   866  "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
   867   ==> xs = ys"
   868 apply(frule map_eq_imp_length_eq)
   869 apply(rotate_tac -1)
   870 apply(induct rule:list_induct2)
   871  apply simp
   872 apply(simp)
   873 apply (blast intro:sym)
   874 done
   875 
   876 lemma inj_on_map_eq_map:
   877  "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
   878 by(blast dest:map_inj_on)
   879 
   880 lemma map_injective:
   881  "map f xs = map f ys ==> inj f ==> xs = ys"
   882 by (induct ys arbitrary: xs) (auto dest!:injD)
   883 
   884 lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
   885 by(blast dest:map_injective)
   886 
   887 lemma inj_mapI: "inj f ==> inj (map f)"
   888 by (iprover dest: map_injective injD intro: inj_onI)
   889 
   890 lemma inj_mapD: "inj (map f) ==> inj f"
   891 apply (unfold inj_on_def, clarify)
   892 apply (erule_tac x = "[x]" in ballE)
   893  apply (erule_tac x = "[y]" in ballE, simp, blast)
   894 apply blast
   895 done
   896 
   897 lemma inj_map[iff]: "inj (map f) = inj f"
   898 by (blast dest: inj_mapD intro: inj_mapI)
   899 
   900 lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
   901 apply(rule inj_onI)
   902 apply(erule map_inj_on)
   903 apply(blast intro:inj_onI dest:inj_onD)
   904 done
   905 
   906 lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
   907 by (induct xs, auto)
   908 
   909 lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
   910 by (induct xs) auto
   911 
   912 lemma map_fst_zip[simp]:
   913   "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
   914 by (induct rule:list_induct2, simp_all)
   915 
   916 lemma map_snd_zip[simp]:
   917   "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
   918 by (induct rule:list_induct2, simp_all)
   919 
   920 enriched_type map: map
   921 by (simp_all add: id_def)
   922 
   923 declare map.id[simp]
   924 
   925 subsubsection {* @{text rev} *}
   926 
   927 lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
   928 by (induct xs) auto
   929 
   930 lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
   931 by (induct xs) auto
   932 
   933 lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
   934 by auto
   935 
   936 lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
   937 by (induct xs) auto
   938 
   939 lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
   940 by (induct xs) auto
   941 
   942 lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
   943 by (cases xs) auto
   944 
   945 lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
   946 by (cases xs) auto
   947 
   948 lemma rev_is_rev_conv [iff, no_atp]: "(rev xs = rev ys) = (xs = ys)"
   949 apply (induct xs arbitrary: ys, force)
   950 apply (case_tac ys, simp, force)
   951 done
   952 
   953 lemma inj_on_rev[iff]: "inj_on rev A"
   954 by(simp add:inj_on_def)
   955 
   956 lemma rev_induct [case_names Nil snoc]:
   957   "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
   958 apply(simplesubst rev_rev_ident[symmetric])
   959 apply(rule_tac list = "rev xs" in list.induct, simp_all)
   960 done
   961 
   962 lemma rev_exhaust [case_names Nil snoc]:
   963   "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
   964 by (induct xs rule: rev_induct) auto
   965 
   966 lemmas rev_cases = rev_exhaust
   967 
   968 lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
   969 by(rule rev_cases[of xs]) auto
   970 
   971 
   972 subsubsection {* @{text set} *}
   973 
   974 declare set.simps [code_post]  --"pretty output"
   975 
   976 lemma finite_set [iff]: "finite (set xs)"
   977 by (induct xs) auto
   978 
   979 lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
   980 by (induct xs) auto
   981 
   982 lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
   983 by(cases xs) auto
   984 
   985 lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
   986 by auto
   987 
   988 lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
   989 by auto
   990 
   991 lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
   992 by (induct xs) auto
   993 
   994 lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
   995 by(induct xs) auto
   996 
   997 lemma set_rev [simp]: "set (rev xs) = set xs"
   998 by (induct xs) auto
   999 
  1000 lemma set_map [simp]: "set (map f xs) = f`(set xs)"
  1001 by (induct xs) auto
  1002 
  1003 lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
  1004 by (induct xs) auto
  1005 
  1006 lemma set_upt [simp]: "set[i..<j] = {i..<j}"
  1007 by (induct j) auto
  1008 
  1009 
  1010 lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs"
  1011 proof (induct xs)
  1012   case Nil thus ?case by simp
  1013 next
  1014   case Cons thus ?case by (auto intro: Cons_eq_appendI)
  1015 qed
  1016 
  1017 lemma in_set_conv_decomp: "x \<in> set xs \<longleftrightarrow> (\<exists>ys zs. xs = ys @ x # zs)"
  1018   by (auto elim: split_list)
  1019 
  1020 lemma split_list_first: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
  1021 proof (induct xs)
  1022   case Nil thus ?case by simp
  1023 next
  1024   case (Cons a xs)
  1025   show ?case
  1026   proof cases
  1027     assume "x = a" thus ?case using Cons by fastforce
  1028   next
  1029     assume "x \<noteq> a" thus ?case using Cons by(fastforce intro!: Cons_eq_appendI)
  1030   qed
  1031 qed
  1032 
  1033 lemma in_set_conv_decomp_first:
  1034   "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
  1035   by (auto dest!: split_list_first)
  1036 
  1037 lemma split_list_last: "x \<in> set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
  1038 proof (induct xs rule: rev_induct)
  1039   case Nil thus ?case by simp
  1040 next
  1041   case (snoc a xs)
  1042   show ?case
  1043   proof cases
  1044     assume "x = a" thus ?case using snoc by (metis List.set.simps(1) emptyE)
  1045   next
  1046     assume "x \<noteq> a" thus ?case using snoc by fastforce
  1047   qed
  1048 qed
  1049 
  1050 lemma in_set_conv_decomp_last:
  1051   "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
  1052   by (auto dest!: split_list_last)
  1053 
  1054 lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs & P x"
  1055 proof (induct xs)
  1056   case Nil thus ?case by simp
  1057 next
  1058   case Cons thus ?case
  1059     by(simp add:Bex_def)(metis append_Cons append.simps(1))
  1060 qed
  1061 
  1062 lemma split_list_propE:
  1063   assumes "\<exists>x \<in> set xs. P x"
  1064   obtains ys x zs where "xs = ys @ x # zs" and "P x"
  1065 using split_list_prop [OF assms] by blast
  1066 
  1067 lemma split_list_first_prop:
  1068   "\<exists>x \<in> set xs. P x \<Longrightarrow>
  1069    \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)"
  1070 proof (induct xs)
  1071   case Nil thus ?case by simp
  1072 next
  1073   case (Cons x xs)
  1074   show ?case
  1075   proof cases
  1076     assume "P x"
  1077     thus ?thesis by simp (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
  1078   next
  1079     assume "\<not> P x"
  1080     hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
  1081     thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
  1082   qed
  1083 qed
  1084 
  1085 lemma split_list_first_propE:
  1086   assumes "\<exists>x \<in> set xs. P x"
  1087   obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y"
  1088 using split_list_first_prop [OF assms] by blast
  1089 
  1090 lemma split_list_first_prop_iff:
  1091   "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
  1092    (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))"
  1093 by (rule, erule split_list_first_prop) auto
  1094 
  1095 lemma split_list_last_prop:
  1096   "\<exists>x \<in> set xs. P x \<Longrightarrow>
  1097    \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)"
  1098 proof(induct xs rule:rev_induct)
  1099   case Nil thus ?case by simp
  1100 next
  1101   case (snoc x xs)
  1102   show ?case
  1103   proof cases
  1104     assume "P x" thus ?thesis by (metis emptyE set_empty)
  1105   next
  1106     assume "\<not> P x"
  1107     hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
  1108     thus ?thesis using `\<not> P x` snoc(1) by fastforce
  1109   qed
  1110 qed
  1111 
  1112 lemma split_list_last_propE:
  1113   assumes "\<exists>x \<in> set xs. P x"
  1114   obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z"
  1115 using split_list_last_prop [OF assms] by blast
  1116 
  1117 lemma split_list_last_prop_iff:
  1118   "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
  1119    (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
  1120 by (metis split_list_last_prop [where P=P] in_set_conv_decomp)
  1121 
  1122 lemma finite_list: "finite A ==> EX xs. set xs = A"
  1123   by (erule finite_induct)
  1124     (auto simp add: set.simps(2) [symmetric] simp del: set.simps(2))
  1125 
  1126 lemma card_length: "card (set xs) \<le> length xs"
  1127 by (induct xs) (auto simp add: card_insert_if)
  1128 
  1129 lemma set_minus_filter_out:
  1130   "set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)"
  1131   by (induct xs) auto
  1132 
  1133 
  1134 subsubsection {* @{text filter} *}
  1135 
  1136 lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
  1137 by (induct xs) auto
  1138 
  1139 lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
  1140 by (induct xs) simp_all
  1141 
  1142 lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
  1143 by (induct xs) auto
  1144 
  1145 lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
  1146 by (induct xs) (auto simp add: le_SucI)
  1147 
  1148 lemma sum_length_filter_compl:
  1149   "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
  1150 by(induct xs) simp_all
  1151 
  1152 lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
  1153 by (induct xs) auto
  1154 
  1155 lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
  1156 by (induct xs) auto
  1157 
  1158 lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
  1159 by (induct xs) simp_all
  1160 
  1161 lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
  1162 apply (induct xs)
  1163  apply auto
  1164 apply(cut_tac P=P and xs=xs in length_filter_le)
  1165 apply simp
  1166 done
  1167 
  1168 lemma filter_map:
  1169   "filter P (map f xs) = map f (filter (P o f) xs)"
  1170 by (induct xs) simp_all
  1171 
  1172 lemma length_filter_map[simp]:
  1173   "length (filter P (map f xs)) = length(filter (P o f) xs)"
  1174 by (simp add:filter_map)
  1175 
  1176 lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
  1177 by auto
  1178 
  1179 lemma length_filter_less:
  1180   "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
  1181 proof (induct xs)
  1182   case Nil thus ?case by simp
  1183 next
  1184   case (Cons x xs) thus ?case
  1185     apply (auto split:split_if_asm)
  1186     using length_filter_le[of P xs] apply arith
  1187   done
  1188 qed
  1189 
  1190 lemma length_filter_conv_card:
  1191  "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
  1192 proof (induct xs)
  1193   case Nil thus ?case by simp
  1194 next
  1195   case (Cons x xs)
  1196   let ?S = "{i. i < length xs & p(xs!i)}"
  1197   have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
  1198   show ?case (is "?l = card ?S'")
  1199   proof (cases)
  1200     assume "p x"
  1201     hence eq: "?S' = insert 0 (Suc ` ?S)"
  1202       by(auto simp: image_def split:nat.split dest:gr0_implies_Suc)
  1203     have "length (filter p (x # xs)) = Suc(card ?S)"
  1204       using Cons `p x` by simp
  1205     also have "\<dots> = Suc(card(Suc ` ?S))" using fin
  1206       by (simp add: card_image)
  1207     also have "\<dots> = card ?S'" using eq fin
  1208       by (simp add:card_insert_if) (simp add:image_def)
  1209     finally show ?thesis .
  1210   next
  1211     assume "\<not> p x"
  1212     hence eq: "?S' = Suc ` ?S"
  1213       by(auto simp add: image_def split:nat.split elim:lessE)
  1214     have "length (filter p (x # xs)) = card ?S"
  1215       using Cons `\<not> p x` by simp
  1216     also have "\<dots> = card(Suc ` ?S)" using fin
  1217       by (simp add: card_image)
  1218     also have "\<dots> = card ?S'" using eq fin
  1219       by (simp add:card_insert_if)
  1220     finally show ?thesis .
  1221   qed
  1222 qed
  1223 
  1224 lemma Cons_eq_filterD:
  1225  "x#xs = filter P ys \<Longrightarrow>
  1226   \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
  1227   (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
  1228 proof(induct ys)
  1229   case Nil thus ?case by simp
  1230 next
  1231   case (Cons y ys)
  1232   show ?case (is "\<exists>x. ?Q x")
  1233   proof cases
  1234     assume Py: "P y"
  1235     show ?thesis
  1236     proof cases
  1237       assume "x = y"
  1238       with Py Cons.prems have "?Q []" by simp
  1239       then show ?thesis ..
  1240     next
  1241       assume "x \<noteq> y"
  1242       with Py Cons.prems show ?thesis by simp
  1243     qed
  1244   next
  1245     assume "\<not> P y"
  1246     with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastforce
  1247     then have "?Q (y#us)" by simp
  1248     then show ?thesis ..
  1249   qed
  1250 qed
  1251 
  1252 lemma filter_eq_ConsD:
  1253  "filter P ys = x#xs \<Longrightarrow>
  1254   \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
  1255 by(rule Cons_eq_filterD) simp
  1256 
  1257 lemma filter_eq_Cons_iff:
  1258  "(filter P ys = x#xs) =
  1259   (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
  1260 by(auto dest:filter_eq_ConsD)
  1261 
  1262 lemma Cons_eq_filter_iff:
  1263  "(x#xs = filter P ys) =
  1264   (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
  1265 by(auto dest:Cons_eq_filterD)
  1266 
  1267 lemma filter_cong[fundef_cong]:
  1268  "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
  1269 apply simp
  1270 apply(erule thin_rl)
  1271 by (induct ys) simp_all
  1272 
  1273 
  1274 subsubsection {* List partitioning *}
  1275 
  1276 primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where
  1277   "partition P [] = ([], [])"
  1278   | "partition P (x # xs) = 
  1279       (let (yes, no) = partition P xs
  1280       in if P x then (x # yes, no) else (yes, x # no))"
  1281 
  1282 lemma partition_filter1:
  1283     "fst (partition P xs) = filter P xs"
  1284 by (induct xs) (auto simp add: Let_def split_def)
  1285 
  1286 lemma partition_filter2:
  1287     "snd (partition P xs) = filter (Not o P) xs"
  1288 by (induct xs) (auto simp add: Let_def split_def)
  1289 
  1290 lemma partition_P:
  1291   assumes "partition P xs = (yes, no)"
  1292   shows "(\<forall>p \<in> set yes.  P p) \<and> (\<forall>p  \<in> set no. \<not> P p)"
  1293 proof -
  1294   from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
  1295     by simp_all
  1296   then show ?thesis by (simp_all add: partition_filter1 partition_filter2)
  1297 qed
  1298 
  1299 lemma partition_set:
  1300   assumes "partition P xs = (yes, no)"
  1301   shows "set yes \<union> set no = set xs"
  1302 proof -
  1303   from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
  1304     by simp_all
  1305   then show ?thesis by (auto simp add: partition_filter1 partition_filter2) 
  1306 qed
  1307 
  1308 lemma partition_filter_conv[simp]:
  1309   "partition f xs = (filter f xs,filter (Not o f) xs)"
  1310 unfolding partition_filter2[symmetric]
  1311 unfolding partition_filter1[symmetric] by simp
  1312 
  1313 declare partition.simps[simp del]
  1314 
  1315 
  1316 subsubsection {* @{text concat} *}
  1317 
  1318 lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
  1319 by (induct xs) auto
  1320 
  1321 lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
  1322 by (induct xss) auto
  1323 
  1324 lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
  1325 by (induct xss) auto
  1326 
  1327 lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
  1328 by (induct xs) auto
  1329 
  1330 lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs"
  1331 by (induct xs) auto
  1332 
  1333 lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
  1334 by (induct xs) auto
  1335 
  1336 lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
  1337 by (induct xs) auto
  1338 
  1339 lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
  1340 by (induct xs) auto
  1341 
  1342 lemma concat_eq_concat_iff: "\<forall>(x, y) \<in> set (zip xs ys). length x = length y ==> length xs = length ys ==> (concat xs = concat ys) = (xs = ys)"
  1343 proof (induct xs arbitrary: ys)
  1344   case (Cons x xs ys)
  1345   thus ?case by (cases ys) auto
  1346 qed (auto)
  1347 
  1348 lemma concat_injective: "concat xs = concat ys ==> length xs = length ys ==> \<forall>(x, y) \<in> set (zip xs ys). length x = length y ==> xs = ys"
  1349 by (simp add: concat_eq_concat_iff)
  1350 
  1351 
  1352 subsubsection {* @{text nth} *}
  1353 
  1354 lemma nth_Cons_0 [simp, code]: "(x # xs)!0 = x"
  1355 by auto
  1356 
  1357 lemma nth_Cons_Suc [simp, code]: "(x # xs)!(Suc n) = xs!n"
  1358 by auto
  1359 
  1360 declare nth.simps [simp del]
  1361 
  1362 lemma nth_Cons_pos[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
  1363 by(auto simp: Nat.gr0_conv_Suc)
  1364 
  1365 lemma nth_append:
  1366   "(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
  1367 apply (induct xs arbitrary: n, simp)
  1368 apply (case_tac n, auto)
  1369 done
  1370 
  1371 lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
  1372 by (induct xs) auto
  1373 
  1374 lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
  1375 by (induct xs) auto
  1376 
  1377 lemma nth_map [simp]: "n < length xs ==> (map f xs)!n = f(xs!n)"
  1378 apply (induct xs arbitrary: n, simp)
  1379 apply (case_tac n, auto)
  1380 done
  1381 
  1382 lemma nth_tl:
  1383   assumes "n < length (tl x)" shows "tl x ! n = x ! Suc n"
  1384 using assms by (induct x) auto
  1385 
  1386 lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
  1387 by(cases xs) simp_all
  1388 
  1389 
  1390 lemma list_eq_iff_nth_eq:
  1391  "(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
  1392 apply(induct xs arbitrary: ys)
  1393  apply force
  1394 apply(case_tac ys)
  1395  apply simp
  1396 apply(simp add:nth_Cons split:nat.split)apply blast
  1397 done
  1398 
  1399 lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
  1400 apply (induct xs, simp, simp)
  1401 apply safe
  1402 apply (metis nat_case_0 nth.simps zero_less_Suc)
  1403 apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
  1404 apply (case_tac i, simp)
  1405 apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff)
  1406 done
  1407 
  1408 lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
  1409 by(auto simp:set_conv_nth)
  1410 
  1411 lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
  1412 by (auto simp add: set_conv_nth)
  1413 
  1414 lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
  1415 by (auto simp add: set_conv_nth)
  1416 
  1417 lemma all_nth_imp_all_set:
  1418 "[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
  1419 by (auto simp add: set_conv_nth)
  1420 
  1421 lemma all_set_conv_all_nth:
  1422 "(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
  1423 by (auto simp add: set_conv_nth)
  1424 
  1425 lemma rev_nth:
  1426   "n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)"
  1427 proof (induct xs arbitrary: n)
  1428   case Nil thus ?case by simp
  1429 next
  1430   case (Cons x xs)
  1431   hence n: "n < Suc (length xs)" by simp
  1432   moreover
  1433   { assume "n < length xs"
  1434     with n obtain n' where "length xs - n = Suc n'"
  1435       by (cases "length xs - n", auto)
  1436     moreover
  1437     then have "length xs - Suc n = n'" by simp
  1438     ultimately
  1439     have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp
  1440   }
  1441   ultimately
  1442   show ?case by (clarsimp simp add: Cons nth_append)
  1443 qed
  1444 
  1445 lemma Skolem_list_nth:
  1446   "(ALL i<k. EX x. P i x) = (EX xs. size xs = k & (ALL i<k. P i (xs!i)))"
  1447   (is "_ = (EX xs. ?P k xs)")
  1448 proof(induct k)
  1449   case 0 show ?case by simp
  1450 next
  1451   case (Suc k)
  1452   show ?case (is "?L = ?R" is "_ = (EX xs. ?P' xs)")
  1453   proof
  1454     assume "?R" thus "?L" using Suc by auto
  1455   next
  1456     assume "?L"
  1457     with Suc obtain x xs where "?P k xs & P k x" by (metis less_Suc_eq)
  1458     hence "?P'(xs@[x])" by(simp add:nth_append less_Suc_eq)
  1459     thus "?R" ..
  1460   qed
  1461 qed
  1462 
  1463 
  1464 subsubsection {* @{text list_update} *}
  1465 
  1466 lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
  1467 by (induct xs arbitrary: i) (auto split: nat.split)
  1468 
  1469 lemma nth_list_update:
  1470 "i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
  1471 by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
  1472 
  1473 lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
  1474 by (simp add: nth_list_update)
  1475 
  1476 lemma nth_list_update_neq [simp]: "i \<noteq> j ==> xs[i:=x]!j = xs!j"
  1477 by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
  1478 
  1479 lemma list_update_id[simp]: "xs[i := xs!i] = xs"
  1480 by (induct xs arbitrary: i) (simp_all split:nat.splits)
  1481 
  1482 lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
  1483 apply (induct xs arbitrary: i)
  1484  apply simp
  1485 apply (case_tac i)
  1486 apply simp_all
  1487 done
  1488 
  1489 lemma list_update_nonempty[simp]: "xs[k:=x] = [] \<longleftrightarrow> xs=[]"
  1490 by(metis length_0_conv length_list_update)
  1491 
  1492 lemma list_update_same_conv:
  1493 "i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
  1494 by (induct xs arbitrary: i) (auto split: nat.split)
  1495 
  1496 lemma list_update_append1:
  1497  "i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
  1498 apply (induct xs arbitrary: i, simp)
  1499 apply(simp split:nat.split)
  1500 done
  1501 
  1502 lemma list_update_append:
  1503   "(xs @ ys) [n:= x] = 
  1504   (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
  1505 by (induct xs arbitrary: n) (auto split:nat.splits)
  1506 
  1507 lemma list_update_length [simp]:
  1508  "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
  1509 by (induct xs, auto)
  1510 
  1511 lemma map_update: "map f (xs[k:= y]) = (map f xs)[k := f y]"
  1512 by(induct xs arbitrary: k)(auto split:nat.splits)
  1513 
  1514 lemma rev_update:
  1515   "k < length xs \<Longrightarrow> rev (xs[k:= y]) = (rev xs)[length xs - k - 1 := y]"
  1516 by (induct xs arbitrary: k) (auto simp: list_update_append split:nat.splits)
  1517 
  1518 lemma update_zip:
  1519   "(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
  1520 by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split)
  1521 
  1522 lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)"
  1523 by (induct xs arbitrary: i) (auto split: nat.split)
  1524 
  1525 lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
  1526 by (blast dest!: set_update_subset_insert [THEN subsetD])
  1527 
  1528 lemma set_update_memI: "n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
  1529 by (induct xs arbitrary: n) (auto split:nat.splits)
  1530 
  1531 lemma list_update_overwrite[simp]:
  1532   "xs [i := x, i := y] = xs [i := y]"
  1533 apply (induct xs arbitrary: i) apply simp
  1534 apply (case_tac i, simp_all)
  1535 done
  1536 
  1537 lemma list_update_swap:
  1538   "i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]"
  1539 apply (induct xs arbitrary: i i')
  1540 apply simp
  1541 apply (case_tac i, case_tac i')
  1542 apply auto
  1543 apply (case_tac i')
  1544 apply auto
  1545 done
  1546 
  1547 lemma list_update_code [code]:
  1548   "[][i := y] = []"
  1549   "(x # xs)[0 := y] = y # xs"
  1550   "(x # xs)[Suc i := y] = x # xs[i := y]"
  1551   by simp_all
  1552 
  1553 
  1554 subsubsection {* @{text last} and @{text butlast} *}
  1555 
  1556 lemma last_snoc [simp]: "last (xs @ [x]) = x"
  1557 by (induct xs) auto
  1558 
  1559 lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
  1560 by (induct xs) auto
  1561 
  1562 lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
  1563   by simp
  1564 
  1565 lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
  1566   by simp
  1567 
  1568 lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
  1569 by (induct xs) (auto)
  1570 
  1571 lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
  1572 by(simp add:last_append)
  1573 
  1574 lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
  1575 by(simp add:last_append)
  1576 
  1577 lemma last_tl: "xs = [] \<or> tl xs \<noteq> [] \<Longrightarrow>last (tl xs) = last xs"
  1578 by (induct xs) simp_all
  1579 
  1580 lemma butlast_tl: "butlast (tl xs) = tl (butlast xs)"
  1581 by (induct xs) simp_all
  1582 
  1583 lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
  1584 by(rule rev_exhaust[of xs]) simp_all
  1585 
  1586 lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
  1587 by(cases xs) simp_all
  1588 
  1589 lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
  1590 by (induct as) auto
  1591 
  1592 lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
  1593 by (induct xs rule: rev_induct) auto
  1594 
  1595 lemma butlast_append:
  1596   "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
  1597 by (induct xs arbitrary: ys) auto
  1598 
  1599 lemma append_butlast_last_id [simp]:
  1600 "xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
  1601 by (induct xs) auto
  1602 
  1603 lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
  1604 by (induct xs) (auto split: split_if_asm)
  1605 
  1606 lemma in_set_butlast_appendI:
  1607 "x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
  1608 by (auto dest: in_set_butlastD simp add: butlast_append)
  1609 
  1610 lemma last_drop[simp]: "n < length xs \<Longrightarrow> last (drop n xs) = last xs"
  1611 apply (induct xs arbitrary: n)
  1612  apply simp
  1613 apply (auto split:nat.split)
  1614 done
  1615 
  1616 lemma nth_butlast:
  1617   assumes "n < length (butlast xs)" shows "butlast xs ! n = xs ! n"
  1618 proof (cases xs)
  1619   case (Cons y ys)
  1620   moreover from assms have "butlast xs ! n = (butlast xs @ [last xs]) ! n"
  1621     by (simp add: nth_append)
  1622   ultimately show ?thesis using append_butlast_last_id by simp
  1623 qed simp
  1624 
  1625 lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
  1626 by(induct xs)(auto simp:neq_Nil_conv)
  1627 
  1628 lemma butlast_conv_take: "butlast xs = take (length xs - 1) xs"
  1629 by (induct xs, simp, case_tac xs, simp_all)
  1630 
  1631 lemma last_list_update:
  1632   "xs \<noteq> [] \<Longrightarrow> last(xs[k:=x]) = (if k = size xs - 1 then x else last xs)"
  1633 by (auto simp: last_conv_nth)
  1634 
  1635 lemma butlast_list_update:
  1636   "butlast(xs[k:=x]) =
  1637  (if k = size xs - 1 then butlast xs else (butlast xs)[k:=x])"
  1638 apply(cases xs rule:rev_cases)
  1639 apply simp
  1640 apply(simp add:list_update_append split:nat.splits)
  1641 done
  1642 
  1643 lemma last_map:
  1644   "xs \<noteq> [] \<Longrightarrow> last (map f xs) = f (last xs)"
  1645   by (cases xs rule: rev_cases) simp_all
  1646 
  1647 lemma map_butlast:
  1648   "map f (butlast xs) = butlast (map f xs)"
  1649   by (induct xs) simp_all
  1650 
  1651 lemma snoc_eq_iff_butlast:
  1652   "xs @ [x] = ys \<longleftrightarrow> (ys \<noteq> [] & butlast ys = xs & last ys = x)"
  1653 by (metis append_butlast_last_id append_is_Nil_conv butlast_snoc last_snoc not_Cons_self)
  1654 
  1655 
  1656 subsubsection {* @{text take} and @{text drop} *}
  1657 
  1658 lemma take_0 [simp]: "take 0 xs = []"
  1659 by (induct xs) auto
  1660 
  1661 lemma drop_0 [simp]: "drop 0 xs = xs"
  1662 by (induct xs) auto
  1663 
  1664 lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
  1665 by simp
  1666 
  1667 lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
  1668 by simp
  1669 
  1670 declare take_Cons [simp del] and drop_Cons [simp del]
  1671 
  1672 lemma take_1_Cons [simp]: "take 1 (x # xs) = [x]"
  1673   unfolding One_nat_def by simp
  1674 
  1675 lemma drop_1_Cons [simp]: "drop 1 (x # xs) = xs"
  1676   unfolding One_nat_def by simp
  1677 
  1678 lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
  1679 by(clarsimp simp add:neq_Nil_conv)
  1680 
  1681 lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
  1682 by(cases xs, simp_all)
  1683 
  1684 lemma take_tl: "take n (tl xs) = tl (take (Suc n) xs)"
  1685 by (induct xs arbitrary: n) simp_all
  1686 
  1687 lemma drop_tl: "drop n (tl xs) = tl(drop n xs)"
  1688 by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split)
  1689 
  1690 lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)"
  1691 by (cases n, simp, cases xs, auto)
  1692 
  1693 lemma tl_drop: "tl (drop n xs) = drop n (tl xs)"
  1694 by (simp only: drop_tl)
  1695 
  1696 lemma nth_via_drop: "drop n xs = y#ys \<Longrightarrow> xs!n = y"
  1697 apply (induct xs arbitrary: n, simp)
  1698 apply(simp add:drop_Cons nth_Cons split:nat.splits)
  1699 done
  1700 
  1701 lemma take_Suc_conv_app_nth:
  1702   "i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
  1703 apply (induct xs arbitrary: i, simp)
  1704 apply (case_tac i, auto)
  1705 done
  1706 
  1707 lemma drop_Suc_conv_tl:
  1708   "i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
  1709 apply (induct xs arbitrary: i, simp)
  1710 apply (case_tac i, auto)
  1711 done
  1712 
  1713 lemma length_take [simp]: "length (take n xs) = min (length xs) n"
  1714 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1715 
  1716 lemma length_drop [simp]: "length (drop n xs) = (length xs - n)"
  1717 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1718 
  1719 lemma take_all [simp]: "length xs <= n ==> take n xs = xs"
  1720 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1721 
  1722 lemma drop_all [simp]: "length xs <= n ==> drop n xs = []"
  1723 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1724 
  1725 lemma take_append [simp]:
  1726   "take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
  1727 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1728 
  1729 lemma drop_append [simp]:
  1730   "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
  1731 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1732 
  1733 lemma take_take [simp]: "take n (take m xs) = take (min n m) xs"
  1734 apply (induct m arbitrary: xs n, auto)
  1735 apply (case_tac xs, auto)
  1736 apply (case_tac n, auto)
  1737 done
  1738 
  1739 lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs"
  1740 apply (induct m arbitrary: xs, auto)
  1741 apply (case_tac xs, auto)
  1742 done
  1743 
  1744 lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)"
  1745 apply (induct m arbitrary: xs n, auto)
  1746 apply (case_tac xs, auto)
  1747 done
  1748 
  1749 lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)"
  1750 apply(induct xs arbitrary: m n)
  1751  apply simp
  1752 apply(simp add: take_Cons drop_Cons split:nat.split)
  1753 done
  1754 
  1755 lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs"
  1756 apply (induct n arbitrary: xs, auto)
  1757 apply (case_tac xs, auto)
  1758 done
  1759 
  1760 lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])"
  1761 apply(induct xs arbitrary: n)
  1762  apply simp
  1763 apply(simp add:take_Cons split:nat.split)
  1764 done
  1765 
  1766 lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)"
  1767 apply(induct xs arbitrary: n)
  1768 apply simp
  1769 apply(simp add:drop_Cons split:nat.split)
  1770 done
  1771 
  1772 lemma take_map: "take n (map f xs) = map f (take n xs)"
  1773 apply (induct n arbitrary: xs, auto)
  1774 apply (case_tac xs, auto)
  1775 done
  1776 
  1777 lemma drop_map: "drop n (map f xs) = map f (drop n xs)"
  1778 apply (induct n arbitrary: xs, auto)
  1779 apply (case_tac xs, auto)
  1780 done
  1781 
  1782 lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)"
  1783 apply (induct xs arbitrary: i, auto)
  1784 apply (case_tac i, auto)
  1785 done
  1786 
  1787 lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)"
  1788 apply (induct xs arbitrary: i, auto)
  1789 apply (case_tac i, auto)
  1790 done
  1791 
  1792 lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i"
  1793 apply (induct xs arbitrary: i n, auto)
  1794 apply (case_tac n, blast)
  1795 apply (case_tac i, auto)
  1796 done
  1797 
  1798 lemma nth_drop [simp]:
  1799   "n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
  1800 apply (induct n arbitrary: xs i, auto)
  1801 apply (case_tac xs, auto)
  1802 done
  1803 
  1804 lemma butlast_take:
  1805   "n <= length xs ==> butlast (take n xs) = take (n - 1) xs"
  1806 by (simp add: butlast_conv_take min_max.inf_absorb1 min_max.inf_absorb2)
  1807 
  1808 lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)"
  1809 by (simp add: butlast_conv_take drop_take add_ac)
  1810 
  1811 lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs"
  1812 by (simp add: butlast_conv_take min_max.inf_absorb1)
  1813 
  1814 lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)"
  1815 by (simp add: butlast_conv_take drop_take add_ac)
  1816 
  1817 lemma hd_drop_conv_nth: "n < length xs \<Longrightarrow> hd(drop n xs) = xs!n"
  1818 by(simp add: hd_conv_nth)
  1819 
  1820 lemma set_take_subset_set_take:
  1821   "m <= n \<Longrightarrow> set(take m xs) <= set(take n xs)"
  1822 apply (induct xs arbitrary: m n)
  1823 apply simp
  1824 apply (case_tac n)
  1825 apply (auto simp: take_Cons)
  1826 done
  1827 
  1828 lemma set_take_subset: "set(take n xs) \<subseteq> set xs"
  1829 by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split)
  1830 
  1831 lemma set_drop_subset: "set(drop n xs) \<subseteq> set xs"
  1832 by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split)
  1833 
  1834 lemma set_drop_subset_set_drop:
  1835   "m >= n \<Longrightarrow> set(drop m xs) <= set(drop n xs)"
  1836 apply(induct xs arbitrary: m n)
  1837 apply(auto simp:drop_Cons split:nat.split)
  1838 apply (metis set_drop_subset subset_iff)
  1839 done
  1840 
  1841 lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
  1842 using set_take_subset by fast
  1843 
  1844 lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
  1845 using set_drop_subset by fast
  1846 
  1847 lemma append_eq_conv_conj:
  1848   "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
  1849 apply (induct xs arbitrary: zs, simp, clarsimp)
  1850 apply (case_tac zs, auto)
  1851 done
  1852 
  1853 lemma take_add: 
  1854   "take (i+j) xs = take i xs @ take j (drop i xs)"
  1855 apply (induct xs arbitrary: i, auto) 
  1856 apply (case_tac i, simp_all)
  1857 done
  1858 
  1859 lemma append_eq_append_conv_if:
  1860  "(xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
  1861   (if size xs\<^isub>1 \<le> size ys\<^isub>1
  1862    then xs\<^isub>1 = take (size xs\<^isub>1) ys\<^isub>1 \<and> xs\<^isub>2 = drop (size xs\<^isub>1) ys\<^isub>1 @ ys\<^isub>2
  1863    else take (size ys\<^isub>1) xs\<^isub>1 = ys\<^isub>1 \<and> drop (size ys\<^isub>1) xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>2)"
  1864 apply(induct xs\<^isub>1 arbitrary: ys\<^isub>1)
  1865  apply simp
  1866 apply(case_tac ys\<^isub>1)
  1867 apply simp_all
  1868 done
  1869 
  1870 lemma take_hd_drop:
  1871   "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (Suc n) xs"
  1872 apply(induct xs arbitrary: n)
  1873 apply simp
  1874 apply(simp add:drop_Cons split:nat.split)
  1875 done
  1876 
  1877 lemma id_take_nth_drop:
  1878  "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
  1879 proof -
  1880   assume si: "i < length xs"
  1881   hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
  1882   moreover
  1883   from si have "take (Suc i) xs = take i xs @ [xs!i]"
  1884     apply (rule_tac take_Suc_conv_app_nth) by arith
  1885   ultimately show ?thesis by auto
  1886 qed
  1887   
  1888 lemma upd_conv_take_nth_drop:
  1889  "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
  1890 proof -
  1891   assume i: "i < length xs"
  1892   have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
  1893     by(rule arg_cong[OF id_take_nth_drop[OF i]])
  1894   also have "\<dots> = take i xs @ a # drop (Suc i) xs"
  1895     using i by (simp add: list_update_append)
  1896   finally show ?thesis .
  1897 qed
  1898 
  1899 lemma nth_drop':
  1900   "i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs"
  1901 apply (induct i arbitrary: xs)
  1902 apply (simp add: neq_Nil_conv)
  1903 apply (erule exE)+
  1904 apply simp
  1905 apply (case_tac xs)
  1906 apply simp_all
  1907 done
  1908 
  1909 
  1910 subsubsection {* @{text takeWhile} and @{text dropWhile} *}
  1911 
  1912 lemma length_takeWhile_le: "length (takeWhile P xs) \<le> length xs"
  1913   by (induct xs) auto
  1914 
  1915 lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
  1916 by (induct xs) auto
  1917 
  1918 lemma takeWhile_append1 [simp]:
  1919 "[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
  1920 by (induct xs) auto
  1921 
  1922 lemma takeWhile_append2 [simp]:
  1923 "(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
  1924 by (induct xs) auto
  1925 
  1926 lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
  1927 by (induct xs) auto
  1928 
  1929 lemma takeWhile_nth: "j < length (takeWhile P xs) \<Longrightarrow> takeWhile P xs ! j = xs ! j"
  1930 apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
  1931 
  1932 lemma dropWhile_nth: "j < length (dropWhile P xs) \<Longrightarrow> dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))"
  1933 apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
  1934 
  1935 lemma length_dropWhile_le: "length (dropWhile P xs) \<le> length xs"
  1936 by (induct xs) auto
  1937 
  1938 lemma dropWhile_append1 [simp]:
  1939 "[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
  1940 by (induct xs) auto
  1941 
  1942 lemma dropWhile_append2 [simp]:
  1943 "(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
  1944 by (induct xs) auto
  1945 
  1946 lemma dropWhile_append3:
  1947   "\<not> P y \<Longrightarrow>dropWhile P (xs @ y # ys) = dropWhile P xs @ y # ys"
  1948 by (induct xs) auto
  1949 
  1950 lemma dropWhile_last:
  1951   "x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> last (dropWhile P xs) = last xs"
  1952 by (auto simp add: dropWhile_append3 in_set_conv_decomp)
  1953 
  1954 lemma set_dropWhileD: "x \<in> set (dropWhile P xs) \<Longrightarrow> x \<in> set xs"
  1955 by (induct xs) (auto split: split_if_asm)
  1956 
  1957 lemma set_takeWhileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
  1958 by (induct xs) (auto split: split_if_asm)
  1959 
  1960 lemma takeWhile_eq_all_conv[simp]:
  1961  "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
  1962 by(induct xs, auto)
  1963 
  1964 lemma dropWhile_eq_Nil_conv[simp]:
  1965  "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
  1966 by(induct xs, auto)
  1967 
  1968 lemma dropWhile_eq_Cons_conv:
  1969  "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
  1970 by(induct xs, auto)
  1971 
  1972 lemma distinct_takeWhile[simp]: "distinct xs ==> distinct (takeWhile P xs)"
  1973 by (induct xs) (auto dest: set_takeWhileD)
  1974 
  1975 lemma distinct_dropWhile[simp]: "distinct xs ==> distinct (dropWhile P xs)"
  1976 by (induct xs) auto
  1977 
  1978 lemma takeWhile_map: "takeWhile P (map f xs) = map f (takeWhile (P \<circ> f) xs)"
  1979 by (induct xs) auto
  1980 
  1981 lemma dropWhile_map: "dropWhile P (map f xs) = map f (dropWhile (P \<circ> f) xs)"
  1982 by (induct xs) auto
  1983 
  1984 lemma takeWhile_eq_take: "takeWhile P xs = take (length (takeWhile P xs)) xs"
  1985 by (induct xs) auto
  1986 
  1987 lemma dropWhile_eq_drop: "dropWhile P xs = drop (length (takeWhile P xs)) xs"
  1988 by (induct xs) auto
  1989 
  1990 lemma hd_dropWhile:
  1991   "dropWhile P xs \<noteq> [] \<Longrightarrow> \<not> P (hd (dropWhile P xs))"
  1992 using assms by (induct xs) auto
  1993 
  1994 lemma takeWhile_eq_filter:
  1995   assumes "\<And> x. x \<in> set (dropWhile P xs) \<Longrightarrow> \<not> P x"
  1996   shows "takeWhile P xs = filter P xs"
  1997 proof -
  1998   have A: "filter P xs = filter P (takeWhile P xs @ dropWhile P xs)"
  1999     by simp
  2000   have B: "filter P (dropWhile P xs) = []"
  2001     unfolding filter_empty_conv using assms by blast
  2002   have "filter P xs = takeWhile P xs"
  2003     unfolding A filter_append B
  2004     by (auto simp add: filter_id_conv dest: set_takeWhileD)
  2005   thus ?thesis ..
  2006 qed
  2007 
  2008 lemma takeWhile_eq_take_P_nth:
  2009   "\<lbrakk> \<And> i. \<lbrakk> i < n ; i < length xs \<rbrakk> \<Longrightarrow> P (xs ! i) ; n < length xs \<Longrightarrow> \<not> P (xs ! n) \<rbrakk> \<Longrightarrow>
  2010   takeWhile P xs = take n xs"
  2011 proof (induct xs arbitrary: n)
  2012   case (Cons x xs)
  2013   thus ?case
  2014   proof (cases n)
  2015     case (Suc n') note this[simp]
  2016     have "P x" using Cons.prems(1)[of 0] by simp
  2017     moreover have "takeWhile P xs = take n' xs"
  2018     proof (rule Cons.hyps)
  2019       case goal1 thus "P (xs ! i)" using Cons.prems(1)[of "Suc i"] by simp
  2020     next case goal2 thus ?case using Cons by auto
  2021     qed
  2022     ultimately show ?thesis by simp
  2023    qed simp
  2024 qed simp
  2025 
  2026 lemma nth_length_takeWhile:
  2027   "length (takeWhile P xs) < length xs \<Longrightarrow> \<not> P (xs ! length (takeWhile P xs))"
  2028 by (induct xs) auto
  2029 
  2030 lemma length_takeWhile_less_P_nth:
  2031   assumes all: "\<And> i. i < j \<Longrightarrow> P (xs ! i)" and "j \<le> length xs"
  2032   shows "j \<le> length (takeWhile P xs)"
  2033 proof (rule classical)
  2034   assume "\<not> ?thesis"
  2035   hence "length (takeWhile P xs) < length xs" using assms by simp
  2036   thus ?thesis using all `\<not> ?thesis` nth_length_takeWhile[of P xs] by auto
  2037 qed
  2038 
  2039 text{* The following two lemmmas could be generalized to an arbitrary
  2040 property. *}
  2041 
  2042 lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
  2043  takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
  2044 by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
  2045 
  2046 lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
  2047   dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
  2048 apply(induct xs)
  2049  apply simp
  2050 apply auto
  2051 apply(subst dropWhile_append2)
  2052 apply auto
  2053 done
  2054 
  2055 lemma takeWhile_not_last:
  2056  "distinct xs \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
  2057 apply(induct xs)
  2058  apply simp
  2059 apply(case_tac xs)
  2060 apply(auto)
  2061 done
  2062 
  2063 lemma takeWhile_cong [fundef_cong]:
  2064   "[| l = k; !!x. x : set l ==> P x = Q x |] 
  2065   ==> takeWhile P l = takeWhile Q k"
  2066 by (induct k arbitrary: l) (simp_all)
  2067 
  2068 lemma dropWhile_cong [fundef_cong]:
  2069   "[| l = k; !!x. x : set l ==> P x = Q x |] 
  2070   ==> dropWhile P l = dropWhile Q k"
  2071 by (induct k arbitrary: l, simp_all)
  2072 
  2073 
  2074 subsubsection {* @{text zip} *}
  2075 
  2076 lemma zip_Nil [simp]: "zip [] ys = []"
  2077 by (induct ys) auto
  2078 
  2079 lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
  2080 by simp
  2081 
  2082 declare zip_Cons [simp del]
  2083 
  2084 lemma [code]:
  2085   "zip [] ys = []"
  2086   "zip xs [] = []"
  2087   "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
  2088   by (fact zip_Nil zip.simps(1) zip_Cons_Cons)+
  2089 
  2090 lemma zip_Cons1:
  2091  "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
  2092 by(auto split:list.split)
  2093 
  2094 lemma length_zip [simp]:
  2095 "length (zip xs ys) = min (length xs) (length ys)"
  2096 by (induct xs ys rule:list_induct2') auto
  2097 
  2098 lemma zip_obtain_same_length:
  2099   assumes "\<And>zs ws n. length zs = length ws \<Longrightarrow> n = min (length xs) (length ys)
  2100     \<Longrightarrow> zs = take n xs \<Longrightarrow> ws = take n ys \<Longrightarrow> P (zip zs ws)"
  2101   shows "P (zip xs ys)"
  2102 proof -
  2103   let ?n = "min (length xs) (length ys)"
  2104   have "P (zip (take ?n xs) (take ?n ys))"
  2105     by (rule assms) simp_all
  2106   moreover have "zip xs ys = zip (take ?n xs) (take ?n ys)"
  2107   proof (induct xs arbitrary: ys)
  2108     case Nil then show ?case by simp
  2109   next
  2110     case (Cons x xs) then show ?case by (cases ys) simp_all
  2111   qed
  2112   ultimately show ?thesis by simp
  2113 qed
  2114 
  2115 lemma zip_append1:
  2116 "zip (xs @ ys) zs =
  2117 zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
  2118 by (induct xs zs rule:list_induct2') auto
  2119 
  2120 lemma zip_append2:
  2121 "zip xs (ys @ zs) =
  2122 zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
  2123 by (induct xs ys rule:list_induct2') auto
  2124 
  2125 lemma zip_append [simp]:
  2126  "[| length xs = length us |] ==>
  2127 zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
  2128 by (simp add: zip_append1)
  2129 
  2130 lemma zip_rev:
  2131 "length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
  2132 by (induct rule:list_induct2, simp_all)
  2133 
  2134 lemma zip_map_map:
  2135   "zip (map f xs) (map g ys) = map (\<lambda> (x, y). (f x, g y)) (zip xs ys)"
  2136 proof (induct xs arbitrary: ys)
  2137   case (Cons x xs) note Cons_x_xs = Cons.hyps
  2138   show ?case
  2139   proof (cases ys)
  2140     case (Cons y ys')
  2141     show ?thesis unfolding Cons using Cons_x_xs by simp
  2142   qed simp
  2143 qed simp
  2144 
  2145 lemma zip_map1:
  2146   "zip (map f xs) ys = map (\<lambda>(x, y). (f x, y)) (zip xs ys)"
  2147 using zip_map_map[of f xs "\<lambda>x. x" ys] by simp
  2148 
  2149 lemma zip_map2:
  2150   "zip xs (map f ys) = map (\<lambda>(x, y). (x, f y)) (zip xs ys)"
  2151 using zip_map_map[of "\<lambda>x. x" xs f ys] by simp
  2152 
  2153 lemma map_zip_map:
  2154   "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
  2155 unfolding zip_map1 by auto
  2156 
  2157 lemma map_zip_map2:
  2158   "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
  2159 unfolding zip_map2 by auto
  2160 
  2161 text{* Courtesy of Andreas Lochbihler: *}
  2162 lemma zip_same_conv_map: "zip xs xs = map (\<lambda>x. (x, x)) xs"
  2163 by(induct xs) auto
  2164 
  2165 lemma nth_zip [simp]:
  2166 "[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
  2167 apply (induct ys arbitrary: i xs, simp)
  2168 apply (case_tac xs)
  2169  apply (simp_all add: nth.simps split: nat.split)
  2170 done
  2171 
  2172 lemma set_zip:
  2173 "set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
  2174 by(simp add: set_conv_nth cong: rev_conj_cong)
  2175 
  2176 lemma zip_same: "((a,b) \<in> set (zip xs xs)) = (a \<in> set xs \<and> a = b)"
  2177 by(induct xs) auto
  2178 
  2179 lemma zip_update:
  2180   "zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
  2181 by(rule sym, simp add: update_zip)
  2182 
  2183 lemma zip_replicate [simp]:
  2184   "zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
  2185 apply (induct i arbitrary: j, auto)
  2186 apply (case_tac j, auto)
  2187 done
  2188 
  2189 lemma take_zip:
  2190   "take n (zip xs ys) = zip (take n xs) (take n ys)"
  2191 apply (induct n arbitrary: xs ys)
  2192  apply simp
  2193 apply (case_tac xs, simp)
  2194 apply (case_tac ys, simp_all)
  2195 done
  2196 
  2197 lemma drop_zip:
  2198   "drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
  2199 apply (induct n arbitrary: xs ys)
  2200  apply simp
  2201 apply (case_tac xs, simp)
  2202 apply (case_tac ys, simp_all)
  2203 done
  2204 
  2205 lemma zip_takeWhile_fst: "zip (takeWhile P xs) ys = takeWhile (P \<circ> fst) (zip xs ys)"
  2206 proof (induct xs arbitrary: ys)
  2207   case (Cons x xs) thus ?case by (cases ys) auto
  2208 qed simp
  2209 
  2210 lemma zip_takeWhile_snd: "zip xs (takeWhile P ys) = takeWhile (P \<circ> snd) (zip xs ys)"
  2211 proof (induct xs arbitrary: ys)
  2212   case (Cons x xs) thus ?case by (cases ys) auto
  2213 qed simp
  2214 
  2215 lemma set_zip_leftD:
  2216   "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
  2217 by (induct xs ys rule:list_induct2') auto
  2218 
  2219 lemma set_zip_rightD:
  2220   "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
  2221 by (induct xs ys rule:list_induct2') auto
  2222 
  2223 lemma in_set_zipE:
  2224   "(x,y) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
  2225 by(blast dest: set_zip_leftD set_zip_rightD)
  2226 
  2227 lemma zip_map_fst_snd:
  2228   "zip (map fst zs) (map snd zs) = zs"
  2229   by (induct zs) simp_all
  2230 
  2231 lemma zip_eq_conv:
  2232   "length xs = length ys \<Longrightarrow> zip xs ys = zs \<longleftrightarrow> map fst zs = xs \<and> map snd zs = ys"
  2233   by (auto simp add: zip_map_fst_snd)
  2234 
  2235 
  2236 subsubsection {* @{text list_all2} *}
  2237 
  2238 lemma list_all2_lengthD [intro?]: 
  2239   "list_all2 P xs ys ==> length xs = length ys"
  2240 by (simp add: list_all2_def)
  2241 
  2242 lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
  2243 by (simp add: list_all2_def)
  2244 
  2245 lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
  2246 by (simp add: list_all2_def)
  2247 
  2248 lemma list_all2_Cons [iff, code]:
  2249   "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
  2250 by (auto simp add: list_all2_def)
  2251 
  2252 lemma list_all2_Cons1:
  2253 "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
  2254 by (cases ys) auto
  2255 
  2256 lemma list_all2_Cons2:
  2257 "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
  2258 by (cases xs) auto
  2259 
  2260 lemma list_all2_induct
  2261   [consumes 1, case_names Nil Cons, induct set: list_all2]:
  2262   assumes P: "list_all2 P xs ys"
  2263   assumes Nil: "R [] []"
  2264   assumes Cons: "\<And>x xs y ys.
  2265     \<lbrakk>P x y; list_all2 P xs ys; R xs ys\<rbrakk> \<Longrightarrow> R (x # xs) (y # ys)"
  2266   shows "R xs ys"
  2267 using P
  2268 by (induct xs arbitrary: ys) (auto simp add: list_all2_Cons1 Nil Cons)
  2269 
  2270 lemma list_all2_rev [iff]:
  2271 "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
  2272 by (simp add: list_all2_def zip_rev cong: conj_cong)
  2273 
  2274 lemma list_all2_rev1:
  2275 "list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
  2276 by (subst list_all2_rev [symmetric]) simp
  2277 
  2278 lemma list_all2_append1:
  2279 "list_all2 P (xs @ ys) zs =
  2280 (EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
  2281 list_all2 P xs us \<and> list_all2 P ys vs)"
  2282 apply (simp add: list_all2_def zip_append1)
  2283 apply (rule iffI)
  2284  apply (rule_tac x = "take (length xs) zs" in exI)
  2285  apply (rule_tac x = "drop (length xs) zs" in exI)
  2286  apply (force split: nat_diff_split simp add: min_def, clarify)
  2287 apply (simp add: ball_Un)
  2288 done
  2289 
  2290 lemma list_all2_append2:
  2291 "list_all2 P xs (ys @ zs) =
  2292 (EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
  2293 list_all2 P us ys \<and> list_all2 P vs zs)"
  2294 apply (simp add: list_all2_def zip_append2)
  2295 apply (rule iffI)
  2296  apply (rule_tac x = "take (length ys) xs" in exI)
  2297  apply (rule_tac x = "drop (length ys) xs" in exI)
  2298  apply (force split: nat_diff_split simp add: min_def, clarify)
  2299 apply (simp add: ball_Un)
  2300 done
  2301 
  2302 lemma list_all2_append:
  2303   "length xs = length ys \<Longrightarrow>
  2304   list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
  2305 by (induct rule:list_induct2, simp_all)
  2306 
  2307 lemma list_all2_appendI [intro?, trans]:
  2308   "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
  2309 by (simp add: list_all2_append list_all2_lengthD)
  2310 
  2311 lemma list_all2_conv_all_nth:
  2312 "list_all2 P xs ys =
  2313 (length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
  2314 by (force simp add: list_all2_def set_zip)
  2315 
  2316 lemma list_all2_trans:
  2317   assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
  2318   shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
  2319         (is "!!bs cs. PROP ?Q as bs cs")
  2320 proof (induct as)
  2321   fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
  2322   show "!!cs. PROP ?Q (x # xs) bs cs"
  2323   proof (induct bs)
  2324     fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
  2325     show "PROP ?Q (x # xs) (y # ys) cs"
  2326       by (induct cs) (auto intro: tr I1 I2)
  2327   qed simp
  2328 qed simp
  2329 
  2330 lemma list_all2_all_nthI [intro?]:
  2331   "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
  2332 by (simp add: list_all2_conv_all_nth)
  2333 
  2334 lemma list_all2I:
  2335   "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
  2336 by (simp add: list_all2_def)
  2337 
  2338 lemma list_all2_nthD:
  2339   "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
  2340 by (simp add: list_all2_conv_all_nth)
  2341 
  2342 lemma list_all2_nthD2:
  2343   "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
  2344 by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
  2345 
  2346 lemma list_all2_map1: 
  2347   "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
  2348 by (simp add: list_all2_conv_all_nth)
  2349 
  2350 lemma list_all2_map2: 
  2351   "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
  2352 by (auto simp add: list_all2_conv_all_nth)
  2353 
  2354 lemma list_all2_refl [intro?]:
  2355   "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
  2356 by (simp add: list_all2_conv_all_nth)
  2357 
  2358 lemma list_all2_update_cong:
  2359   "\<lbrakk> list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
  2360 by (cases "i < length ys") (auto simp add: list_all2_conv_all_nth nth_list_update)
  2361 
  2362 lemma list_all2_takeI [simp,intro?]:
  2363   "list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
  2364 apply (induct xs arbitrary: n ys)
  2365  apply simp
  2366 apply (clarsimp simp add: list_all2_Cons1)
  2367 apply (case_tac n)
  2368 apply auto
  2369 done
  2370 
  2371 lemma list_all2_dropI [simp,intro?]:
  2372   "list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
  2373 apply (induct as arbitrary: n bs, simp)
  2374 apply (clarsimp simp add: list_all2_Cons1)
  2375 apply (case_tac n, simp, simp)
  2376 done
  2377 
  2378 lemma list_all2_mono [intro?]:
  2379   "list_all2 P xs ys \<Longrightarrow> (\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys) \<Longrightarrow> list_all2 Q xs ys"
  2380 apply (induct xs arbitrary: ys, simp)
  2381 apply (case_tac ys, auto)
  2382 done
  2383 
  2384 lemma list_all2_eq:
  2385   "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
  2386 by (induct xs ys rule: list_induct2') auto
  2387 
  2388 lemma list_eq_iff_zip_eq:
  2389   "xs = ys \<longleftrightarrow> length xs = length ys \<and> (\<forall>(x,y) \<in> set (zip xs ys). x = y)"
  2390 by(auto simp add: set_zip list_all2_eq list_all2_conv_all_nth cong: conj_cong)
  2391 
  2392 
  2393 subsubsection {* @{const fold} with natural argument order *}
  2394 
  2395 lemma fold_remove1_split:
  2396   assumes f: "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
  2397     and x: "x \<in> set xs"
  2398   shows "fold f xs = fold f (remove1 x xs) \<circ> f x"
  2399   using assms by (induct xs) (auto simp add: o_assoc [symmetric])
  2400 
  2401 lemma fold_cong [fundef_cong]:
  2402   "a = b \<Longrightarrow> xs = ys \<Longrightarrow> (\<And>x. x \<in> set xs \<Longrightarrow> f x = g x)
  2403     \<Longrightarrow> fold f xs a = fold g ys b"
  2404   by (induct ys arbitrary: a b xs) simp_all
  2405 
  2406 lemma fold_id:
  2407   assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x = id"
  2408   shows "fold f xs = id"
  2409   using assms by (induct xs) simp_all
  2410 
  2411 lemma fold_commute:
  2412   assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
  2413   shows "h \<circ> fold g xs = fold f xs \<circ> h"
  2414   using assms by (induct xs) (simp_all add: fun_eq_iff)
  2415 
  2416 lemma fold_commute_apply:
  2417   assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
  2418   shows "h (fold g xs s) = fold f xs (h s)"
  2419 proof -
  2420   from assms have "h \<circ> fold g xs = fold f xs \<circ> h" by (rule fold_commute)
  2421   then show ?thesis by (simp add: fun_eq_iff)
  2422 qed
  2423 
  2424 lemma fold_invariant: 
  2425   assumes "\<And>x. x \<in> set xs \<Longrightarrow> Q x" and "P s"
  2426     and "\<And>x s. Q x \<Longrightarrow> P s \<Longrightarrow> P (f x s)"
  2427   shows "P (fold f xs s)"
  2428   using assms by (induct xs arbitrary: s) simp_all
  2429 
  2430 lemma fold_append [simp]:
  2431   "fold f (xs @ ys) = fold f ys \<circ> fold f xs"
  2432   by (induct xs) simp_all
  2433 
  2434 lemma fold_map [code_unfold]:
  2435   "fold g (map f xs) = fold (g o f) xs"
  2436   by (induct xs) simp_all
  2437 
  2438 lemma fold_rev:
  2439   assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
  2440   shows "fold f (rev xs) = fold f xs"
  2441 using assms by (induct xs) (simp_all add: fold_commute_apply fun_eq_iff)
  2442 
  2443 lemma fold_Cons_rev:
  2444   "fold Cons xs = append (rev xs)"
  2445   by (induct xs) simp_all
  2446 
  2447 lemma rev_conv_fold [code]:
  2448   "rev xs = fold Cons xs []"
  2449   by (simp add: fold_Cons_rev)
  2450 
  2451 lemma fold_append_concat_rev:
  2452   "fold append xss = append (concat (rev xss))"
  2453   by (induct xss) simp_all
  2454 
  2455 text {* @{const Finite_Set.fold} and @{const fold} *}
  2456 
  2457 lemma (in comp_fun_commute) fold_set_fold_remdups:
  2458   "Finite_Set.fold f y (set xs) = fold f (remdups xs) y"
  2459   by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm insert_absorb)
  2460 
  2461 lemma (in comp_fun_idem) fold_set_fold:
  2462   "Finite_Set.fold f y (set xs) = fold f xs y"
  2463   by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm)
  2464 
  2465 lemma (in ab_semigroup_idem_mult) fold1_set_fold:
  2466   assumes "xs \<noteq> []"
  2467   shows "Finite_Set.fold1 times (set xs) = fold times (tl xs) (hd xs)"
  2468 proof -
  2469   interpret comp_fun_idem times by (fact comp_fun_idem)
  2470   from assms obtain y ys where xs: "xs = y # ys"
  2471     by (cases xs) auto
  2472   show ?thesis
  2473   proof (cases "set ys = {}")
  2474     case True with xs show ?thesis by simp
  2475   next
  2476     case False
  2477     then have "fold1 times (insert y (set ys)) = Finite_Set.fold times y (set ys)"
  2478       by (simp only: finite_set fold1_eq_fold_idem)
  2479     with xs show ?thesis by (simp add: fold_set_fold mult_commute)
  2480   qed
  2481 qed
  2482 
  2483 lemma union_set_fold [code]:
  2484   "set xs \<union> A = fold Set.insert xs A"
  2485 proof -
  2486   interpret comp_fun_idem Set.insert
  2487     by (fact comp_fun_idem_insert)
  2488   show ?thesis by (simp add: union_fold_insert fold_set_fold)
  2489 qed
  2490 
  2491 lemma union_coset_filter [code]:
  2492   "List.coset xs \<union> A = List.coset (List.filter (\<lambda>x. x \<notin> A) xs)"
  2493   by auto
  2494 
  2495 lemma minus_set_fold [code]:
  2496   "A - set xs = fold Set.remove xs A"
  2497 proof -
  2498   interpret comp_fun_idem Set.remove
  2499     by (fact comp_fun_idem_remove)
  2500   show ?thesis
  2501     by (simp add: minus_fold_remove [of _ A] fold_set_fold)
  2502 qed
  2503 
  2504 lemma minus_coset_filter [code]:
  2505   "A - List.coset xs = set (List.filter (\<lambda>x. x \<in> A) xs)"
  2506   by auto
  2507 
  2508 lemma inter_set_filter [code]:
  2509   "A \<inter> set xs = set (List.filter (\<lambda>x. x \<in> A) xs)"
  2510   by auto
  2511 
  2512 lemma inter_coset_fold [code]:
  2513   "A \<inter> List.coset xs = fold Set.remove xs A"
  2514   by (simp add: Diff_eq [symmetric] minus_set_fold)
  2515 
  2516 lemma (in lattice) Inf_fin_set_fold:
  2517   "Inf_fin (set (x # xs)) = fold inf xs x"
  2518 proof -
  2519   interpret ab_semigroup_idem_mult "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2520     by (fact ab_semigroup_idem_mult_inf)
  2521   show ?thesis
  2522     by (simp add: Inf_fin_def fold1_set_fold del: set.simps)
  2523 qed
  2524 
  2525 declare Inf_fin_set_fold [code]
  2526 
  2527 lemma (in lattice) Sup_fin_set_fold:
  2528   "Sup_fin (set (x # xs)) = fold sup xs x"
  2529 proof -
  2530   interpret ab_semigroup_idem_mult "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2531     by (fact ab_semigroup_idem_mult_sup)
  2532   show ?thesis
  2533     by (simp add: Sup_fin_def fold1_set_fold del: set.simps)
  2534 qed
  2535 
  2536 declare Sup_fin_set_fold [code]
  2537 
  2538 lemma (in linorder) Min_fin_set_fold:
  2539   "Min (set (x # xs)) = fold min xs x"
  2540 proof -
  2541   interpret ab_semigroup_idem_mult "min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2542     by (fact ab_semigroup_idem_mult_min)
  2543   show ?thesis
  2544     by (simp add: Min_def fold1_set_fold del: set.simps)
  2545 qed
  2546 
  2547 declare Min_fin_set_fold [code]
  2548 
  2549 lemma (in linorder) Max_fin_set_fold:
  2550   "Max (set (x # xs)) = fold max xs x"
  2551 proof -
  2552   interpret ab_semigroup_idem_mult "max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2553     by (fact ab_semigroup_idem_mult_max)
  2554   show ?thesis
  2555     by (simp add: Max_def fold1_set_fold del: set.simps)
  2556 qed
  2557 
  2558 declare Max_fin_set_fold [code]
  2559 
  2560 lemma (in complete_lattice) Inf_set_fold:
  2561   "Inf (set xs) = fold inf xs top"
  2562 proof -
  2563   interpret comp_fun_idem "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2564     by (fact comp_fun_idem_inf)
  2565   show ?thesis by (simp add: Inf_fold_inf fold_set_fold inf_commute)
  2566 qed
  2567 
  2568 declare Inf_set_fold [where 'a = "'a set", code]
  2569 
  2570 lemma (in complete_lattice) Sup_set_fold:
  2571   "Sup (set xs) = fold sup xs bot"
  2572 proof -
  2573   interpret comp_fun_idem "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2574     by (fact comp_fun_idem_sup)
  2575   show ?thesis by (simp add: Sup_fold_sup fold_set_fold sup_commute)
  2576 qed
  2577 
  2578 declare Sup_set_fold [where 'a = "'a set", code]
  2579 
  2580 lemma (in complete_lattice) INF_set_fold:
  2581   "INFI (set xs) f = fold (inf \<circ> f) xs top"
  2582   unfolding INF_def set_map [symmetric] Inf_set_fold fold_map ..
  2583 
  2584 declare INF_set_fold [code]
  2585 
  2586 lemma (in complete_lattice) SUP_set_fold:
  2587   "SUPR (set xs) f = fold (sup \<circ> f) xs bot"
  2588   unfolding SUP_def set_map [symmetric] Sup_set_fold fold_map ..
  2589 
  2590 declare SUP_set_fold [code]
  2591 
  2592 subsubsection {* Fold variants: @{const foldr} and @{const foldl} *}
  2593 
  2594 text {* Correspondence *}
  2595 
  2596 lemma foldr_conv_fold [code_abbrev]:
  2597   "foldr f xs = fold f (rev xs)"
  2598   by (induct xs) simp_all
  2599 
  2600 lemma foldl_conv_fold:
  2601   "foldl f s xs = fold (\<lambda>x s. f s x) xs s"
  2602   by (induct xs arbitrary: s) simp_all
  2603 
  2604 lemma foldr_conv_foldl: -- {* The ``Third Duality Theorem'' in Bird \& Wadler: *}
  2605   "foldr f xs a = foldl (\<lambda>x y. f y x) a (rev xs)"
  2606   by (simp add: foldr_conv_fold foldl_conv_fold)
  2607 
  2608 lemma foldl_conv_foldr:
  2609   "foldl f a xs = foldr (\<lambda>x y. f y x) (rev xs) a"
  2610   by (simp add: foldr_conv_fold foldl_conv_fold)
  2611 
  2612 lemma foldr_fold:
  2613   assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
  2614   shows "foldr f xs = fold f xs"
  2615   using assms unfolding foldr_conv_fold by (rule fold_rev)
  2616 
  2617 lemma foldr_cong [fundef_cong]:
  2618   "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"
  2619   by (auto simp add: foldr_conv_fold intro!: fold_cong)
  2620 
  2621 lemma foldl_cong [fundef_cong]:
  2622   "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"
  2623   by (auto simp add: foldl_conv_fold intro!: fold_cong)
  2624 
  2625 lemma foldr_append [simp]:
  2626   "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
  2627   by (simp add: foldr_conv_fold)
  2628 
  2629 lemma foldl_append [simp]:
  2630   "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
  2631   by (simp add: foldl_conv_fold)
  2632 
  2633 lemma foldr_map [code_unfold]:
  2634   "foldr g (map f xs) a = foldr (g o f) xs a"
  2635   by (simp add: foldr_conv_fold fold_map rev_map)
  2636 
  2637 lemma foldl_map [code_unfold]:
  2638   "foldl g a (map f xs) = foldl (\<lambda>a x. g a (f x)) a xs"
  2639   by (simp add: foldl_conv_fold fold_map comp_def)
  2640 
  2641 lemma concat_conv_foldr [code]:
  2642   "concat xss = foldr append xss []"
  2643   by (simp add: fold_append_concat_rev foldr_conv_fold)
  2644 
  2645 
  2646 subsubsection {* @{text upt} *}
  2647 
  2648 lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
  2649 -- {* simp does not terminate! *}
  2650 by (induct j) auto
  2651 
  2652 lemmas upt_rec_numeral[simp] = upt_rec[of "numeral m" "numeral n"] for m n
  2653 
  2654 lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
  2655 by (subst upt_rec) simp
  2656 
  2657 lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
  2658 by(induct j)simp_all
  2659 
  2660 lemma upt_eq_Cons_conv:
  2661  "([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
  2662 apply(induct j arbitrary: x xs)
  2663  apply simp
  2664 apply(clarsimp simp add: append_eq_Cons_conv)
  2665 apply arith
  2666 done
  2667 
  2668 lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
  2669 -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
  2670 by simp
  2671 
  2672 lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
  2673   by (simp add: upt_rec)
  2674 
  2675 lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
  2676 -- {* LOOPS as a simprule, since @{text "j <= j"}. *}
  2677 by (induct k) auto
  2678 
  2679 lemma length_upt [simp]: "length [i..<j] = j - i"
  2680 by (induct j) (auto simp add: Suc_diff_le)
  2681 
  2682 lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
  2683 apply (induct j)
  2684 apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
  2685 done
  2686 
  2687 
  2688 lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
  2689 by(simp add:upt_conv_Cons)
  2690 
  2691 lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
  2692 apply(cases j)
  2693  apply simp
  2694 by(simp add:upt_Suc_append)
  2695 
  2696 lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]"
  2697 apply (induct m arbitrary: i, simp)
  2698 apply (subst upt_rec)
  2699 apply (rule sym)
  2700 apply (subst upt_rec)
  2701 apply (simp del: upt.simps)
  2702 done
  2703 
  2704 lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
  2705 apply(induct j)
  2706 apply auto
  2707 done
  2708 
  2709 lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]"
  2710 by (induct n) auto
  2711 
  2712 lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
  2713 apply (induct n m  arbitrary: i rule: diff_induct)
  2714 prefer 3 apply (subst map_Suc_upt[symmetric])
  2715 apply (auto simp add: less_diff_conv)
  2716 done
  2717 
  2718 lemma nth_take_lemma:
  2719   "k <= length xs ==> k <= length ys ==>
  2720      (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
  2721 apply (atomize, induct k arbitrary: xs ys)
  2722 apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
  2723 txt {* Both lists must be non-empty *}
  2724 apply (case_tac xs, simp)
  2725 apply (case_tac ys, clarify)
  2726  apply (simp (no_asm_use))
  2727 apply clarify
  2728 txt {* prenexing's needed, not miniscoping *}
  2729 apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
  2730 apply blast
  2731 done
  2732 
  2733 lemma nth_equalityI:
  2734  "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
  2735   by (frule nth_take_lemma [OF le_refl eq_imp_le]) simp_all
  2736 
  2737 lemma map_nth:
  2738   "map (\<lambda>i. xs ! i) [0..<length xs] = xs"
  2739   by (rule nth_equalityI, auto)
  2740 
  2741 (* needs nth_equalityI *)
  2742 lemma list_all2_antisym:
  2743   "\<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> 
  2744   \<Longrightarrow> xs = ys"
  2745   apply (simp add: list_all2_conv_all_nth) 
  2746   apply (rule nth_equalityI, blast, simp)
  2747   done
  2748 
  2749 lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
  2750 -- {* The famous take-lemma. *}
  2751 apply (drule_tac x = "max (length xs) (length ys)" in spec)
  2752 apply (simp add: le_max_iff_disj)
  2753 done
  2754 
  2755 
  2756 lemma take_Cons':
  2757      "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
  2758 by (cases n) simp_all
  2759 
  2760 lemma drop_Cons':
  2761      "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
  2762 by (cases n) simp_all
  2763 
  2764 lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
  2765 by (cases n) simp_all
  2766 
  2767 lemma take_Cons_numeral [simp]:
  2768   "take (numeral v) (x # xs) = x # take (numeral v - 1) xs"
  2769 by (simp add: take_Cons')
  2770 
  2771 lemma drop_Cons_numeral [simp]:
  2772   "drop (numeral v) (x # xs) = drop (numeral v - 1) xs"
  2773 by (simp add: drop_Cons')
  2774 
  2775 lemma nth_Cons_numeral [simp]:
  2776   "(x # xs) ! numeral v = xs ! (numeral v - 1)"
  2777 by (simp add: nth_Cons')
  2778 
  2779 
  2780 subsubsection {* @{text upto}: interval-list on @{typ int} *}
  2781 
  2782 (* FIXME make upto tail recursive? *)
  2783 
  2784 function upto :: "int \<Rightarrow> int \<Rightarrow> int list" ("(1[_../_])") where
  2785 "upto i j = (if i \<le> j then i # [i+1..j] else [])"
  2786 by auto
  2787 termination
  2788 by(relation "measure(%(i::int,j). nat(j - i + 1))") auto
  2789 
  2790 declare upto.simps[code, simp del]
  2791 
  2792 lemmas upto_rec_numeral [simp] =
  2793   upto.simps[of "numeral m" "numeral n"]
  2794   upto.simps[of "numeral m" "neg_numeral n"]
  2795   upto.simps[of "neg_numeral m" "numeral n"]
  2796   upto.simps[of "neg_numeral m" "neg_numeral n"] for m n
  2797 
  2798 lemma upto_empty[simp]: "j < i \<Longrightarrow> [i..j] = []"
  2799 by(simp add: upto.simps)
  2800 
  2801 lemma set_upto[simp]: "set[i..j] = {i..j}"
  2802 proof(induct i j rule:upto.induct)
  2803   case (1 i j)
  2804   from this show ?case
  2805     unfolding upto.simps[of i j] simp_from_to[of i j] by auto
  2806 qed
  2807 
  2808 
  2809 subsubsection {* @{text "distinct"} and @{text remdups} *}
  2810 
  2811 lemma distinct_tl:
  2812   "distinct xs \<Longrightarrow> distinct (tl xs)"
  2813   by (cases xs) simp_all
  2814 
  2815 lemma distinct_append [simp]:
  2816 "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
  2817 by (induct xs) auto
  2818 
  2819 lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
  2820 by(induct xs) auto
  2821 
  2822 lemma set_remdups [simp]: "set (remdups xs) = set xs"
  2823 by (induct xs) (auto simp add: insert_absorb)
  2824 
  2825 lemma distinct_remdups [iff]: "distinct (remdups xs)"
  2826 by (induct xs) auto
  2827 
  2828 lemma distinct_remdups_id: "distinct xs ==> remdups xs = xs"
  2829 by (induct xs, auto)
  2830 
  2831 lemma remdups_id_iff_distinct [simp]: "remdups xs = xs \<longleftrightarrow> distinct xs"
  2832 by (metis distinct_remdups distinct_remdups_id)
  2833 
  2834 lemma finite_distinct_list: "finite A \<Longrightarrow> EX xs. set xs = A & distinct xs"
  2835 by (metis distinct_remdups finite_list set_remdups)
  2836 
  2837 lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
  2838 by (induct x, auto)
  2839 
  2840 lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
  2841 by (induct x, auto)
  2842 
  2843 lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
  2844 by (induct xs) auto
  2845 
  2846 lemma length_remdups_eq[iff]:
  2847   "(length (remdups xs) = length xs) = (remdups xs = xs)"
  2848 apply(induct xs)
  2849  apply auto
  2850 apply(subgoal_tac "length (remdups xs) <= length xs")
  2851  apply arith
  2852 apply(rule length_remdups_leq)
  2853 done
  2854 
  2855 lemma remdups_filter: "remdups(filter P xs) = filter P (remdups xs)"
  2856 apply(induct xs)
  2857 apply auto
  2858 done
  2859 
  2860 lemma distinct_map:
  2861   "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
  2862 by (induct xs) auto
  2863 
  2864 
  2865 lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
  2866 by (induct xs) auto
  2867 
  2868 lemma distinct_upt[simp]: "distinct[i..<j]"
  2869 by (induct j) auto
  2870 
  2871 lemma distinct_upto[simp]: "distinct[i..j]"
  2872 apply(induct i j rule:upto.induct)
  2873 apply(subst upto.simps)
  2874 apply(simp)
  2875 done
  2876 
  2877 lemma distinct_take[simp]: "distinct xs \<Longrightarrow> distinct (take i xs)"
  2878 apply(induct xs arbitrary: i)
  2879  apply simp
  2880 apply (case_tac i)
  2881  apply simp_all
  2882 apply(blast dest:in_set_takeD)
  2883 done
  2884 
  2885 lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)"
  2886 apply(induct xs arbitrary: i)
  2887  apply simp
  2888 apply (case_tac i)
  2889  apply simp_all
  2890 done
  2891 
  2892 lemma distinct_list_update:
  2893 assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
  2894 shows "distinct (xs[i:=a])"
  2895 proof (cases "i < length xs")
  2896   case True
  2897   with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
  2898     apply (drule_tac id_take_nth_drop) by simp
  2899   with d True show ?thesis
  2900     apply (simp add: upd_conv_take_nth_drop)
  2901     apply (drule subst [OF id_take_nth_drop]) apply assumption
  2902     apply simp apply (cases "a = xs!i") apply simp by blast
  2903 next
  2904   case False with d show ?thesis by auto
  2905 qed
  2906 
  2907 lemma distinct_concat:
  2908   assumes "distinct xs"
  2909   and "\<And> ys. ys \<in> set xs \<Longrightarrow> distinct ys"
  2910   and "\<And> ys zs. \<lbrakk> ys \<in> set xs ; zs \<in> set xs ; ys \<noteq> zs \<rbrakk> \<Longrightarrow> set ys \<inter> set zs = {}"
  2911   shows "distinct (concat xs)"
  2912   using assms by (induct xs) auto
  2913 
  2914 text {* It is best to avoid this indexed version of distinct, but
  2915 sometimes it is useful. *}
  2916 
  2917 lemma distinct_conv_nth:
  2918 "distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
  2919 apply (induct xs, simp, simp)
  2920 apply (rule iffI, clarsimp)
  2921  apply (case_tac i)
  2922 apply (case_tac j, simp)
  2923 apply (simp add: set_conv_nth)
  2924  apply (case_tac j)
  2925 apply (clarsimp simp add: set_conv_nth, simp)
  2926 apply (rule conjI)
  2927 (*TOO SLOW
  2928 apply (metis Zero_neq_Suc gr0_conv_Suc in_set_conv_nth lessI less_trans_Suc nth_Cons' nth_Cons_Suc)
  2929 *)
  2930  apply (clarsimp simp add: set_conv_nth)
  2931  apply (erule_tac x = 0 in allE, simp)
  2932  apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
  2933 (*TOO SLOW
  2934 apply (metis Suc_Suc_eq lessI less_trans_Suc nth_Cons_Suc)
  2935 *)
  2936 apply (erule_tac x = "Suc i" in allE, simp)
  2937 apply (erule_tac x = "Suc j" in allE, simp)
  2938 done
  2939 
  2940 lemma nth_eq_iff_index_eq:
  2941  "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
  2942 by(auto simp: distinct_conv_nth)
  2943 
  2944 lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
  2945 by (induct xs) auto
  2946 
  2947 lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
  2948 proof (induct xs)
  2949   case Nil thus ?case by simp
  2950 next
  2951   case (Cons x xs)
  2952   show ?case
  2953   proof (cases "x \<in> set xs")
  2954     case False with Cons show ?thesis by simp
  2955   next
  2956     case True with Cons.prems
  2957     have "card (set xs) = Suc (length xs)"
  2958       by (simp add: card_insert_if split: split_if_asm)
  2959     moreover have "card (set xs) \<le> length xs" by (rule card_length)
  2960     ultimately have False by simp
  2961     thus ?thesis ..
  2962   qed
  2963 qed
  2964 
  2965 lemma distinct_length_filter: "distinct xs \<Longrightarrow> length (filter P xs) = card ({x. P x} Int set xs)"
  2966 by (induct xs) (auto)
  2967 
  2968 lemma not_distinct_decomp: "~ distinct ws ==> EX xs ys zs y. ws = xs@[y]@ys@[y]@zs"
  2969 apply (induct n == "length ws" arbitrary:ws) apply simp
  2970 apply(case_tac ws) apply simp
  2971 apply (simp split:split_if_asm)
  2972 apply (metis Cons_eq_appendI eq_Nil_appendI split_list)
  2973 done
  2974 
  2975 lemma not_distinct_conv_prefix:
  2976   defines "dec as xs y ys \<equiv> y \<in> set xs \<and> distinct xs \<and> as = xs @ y # ys"
  2977   shows "\<not>distinct as \<longleftrightarrow> (\<exists>xs y ys. dec as xs y ys)" (is "?L = ?R")
  2978 proof
  2979   assume "?L" then show "?R"
  2980   proof (induct "length as" arbitrary: as rule: less_induct)
  2981     case less
  2982     obtain xs ys zs y where decomp: "as = (xs @ y # ys) @ y # zs"
  2983       using not_distinct_decomp[OF less.prems] by auto
  2984     show ?case
  2985     proof (cases "distinct (xs @ y # ys)")
  2986       case True
  2987       with decomp have "dec as (xs @ y # ys) y zs" by (simp add: dec_def)
  2988       then show ?thesis by blast
  2989     next
  2990       case False
  2991       with less decomp obtain xs' y' ys' where "dec (xs @ y # ys) xs' y' ys'"
  2992         by atomize_elim auto
  2993       with decomp have "dec as xs' y' (ys' @ y # zs)" by (simp add: dec_def)
  2994       then show ?thesis by blast
  2995     qed
  2996   qed
  2997 qed (auto simp: dec_def)
  2998 
  2999 lemma length_remdups_concat:
  3000   "length (remdups (concat xss)) = card (\<Union>xs\<in>set xss. set xs)"
  3001   by (simp add: distinct_card [symmetric])
  3002 
  3003 lemma length_remdups_card_conv: "length(remdups xs) = card(set xs)"
  3004 proof -
  3005   have xs: "concat[xs] = xs" by simp
  3006   from length_remdups_concat[of "[xs]"] show ?thesis unfolding xs by simp
  3007 qed
  3008 
  3009 lemma remdups_remdups:
  3010   "remdups (remdups xs) = remdups xs"
  3011   by (induct xs) simp_all
  3012 
  3013 lemma distinct_butlast:
  3014   assumes "distinct xs"
  3015   shows "distinct (butlast xs)"
  3016 proof (cases "xs = []")
  3017   case False
  3018     from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  3019     with `distinct xs` show ?thesis by simp
  3020 qed (auto)
  3021 
  3022 lemma remdups_map_remdups:
  3023   "remdups (map f (remdups xs)) = remdups (map f xs)"
  3024   by (induct xs) simp_all
  3025 
  3026 lemma distinct_zipI1:
  3027   assumes "distinct xs"
  3028   shows "distinct (zip xs ys)"
  3029 proof (rule zip_obtain_same_length)
  3030   fix xs' :: "'a list" and ys' :: "'b list" and n
  3031   assume "length xs' = length ys'"
  3032   assume "xs' = take n xs"
  3033   with assms have "distinct xs'" by simp
  3034   with `length xs' = length ys'` show "distinct (zip xs' ys')"
  3035     by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
  3036 qed
  3037 
  3038 lemma distinct_zipI2:
  3039   assumes "distinct ys"
  3040   shows "distinct (zip xs ys)"
  3041 proof (rule zip_obtain_same_length)
  3042   fix xs' :: "'b list" and ys' :: "'a list" and n
  3043   assume "length xs' = length ys'"
  3044   assume "ys' = take n ys"
  3045   with assms have "distinct ys'" by simp
  3046   with `length xs' = length ys'` show "distinct (zip xs' ys')"
  3047     by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
  3048 qed
  3049 
  3050 lemma set_take_disj_set_drop_if_distinct:
  3051   "distinct vs \<Longrightarrow> i \<le> j \<Longrightarrow> set (take i vs) \<inter> set (drop j vs) = {}"
  3052 by (auto simp: in_set_conv_nth distinct_conv_nth)
  3053 
  3054 (* The next two lemmas help Sledgehammer. *)
  3055 
  3056 lemma distinct_singleton: "distinct [x]" by simp
  3057 
  3058 lemma distinct_length_2_or_more:
  3059 "distinct (a # b # xs) \<longleftrightarrow> (a \<noteq> b \<and> distinct (a # xs) \<and> distinct (b # xs))"
  3060 by (metis distinct.simps(2) hd.simps hd_in_set list.simps(2) set_ConsD set_rev_mp set_subset_Cons)
  3061 
  3062 subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
  3063 
  3064 lemma (in monoid_add) listsum_simps [simp]:
  3065   "listsum [] = 0"
  3066   "listsum (x # xs) = x + listsum xs"
  3067   by (simp_all add: listsum_def)
  3068 
  3069 lemma (in monoid_add) listsum_append [simp]:
  3070   "listsum (xs @ ys) = listsum xs + listsum ys"
  3071   by (induct xs) (simp_all add: add.assoc)
  3072 
  3073 lemma (in comm_monoid_add) listsum_rev [simp]:
  3074   "listsum (rev xs) = listsum xs"
  3075   by (simp add: listsum_def foldr_fold fold_rev fun_eq_iff add_ac)
  3076 
  3077 lemma (in monoid_add) fold_plus_listsum_rev:
  3078   "fold plus xs = plus (listsum (rev xs))"
  3079 proof
  3080   fix x
  3081   have "fold plus xs x = fold plus xs (x + 0)" by simp
  3082   also have "\<dots> = fold plus (x # xs) 0" by simp
  3083   also have "\<dots> = foldr plus (rev xs @ [x]) 0" by (simp add: foldr_conv_fold)
  3084   also have "\<dots> = listsum (rev xs @ [x])" by (simp add: listsum_def)
  3085   also have "\<dots> = listsum (rev xs) + listsum [x]" by simp
  3086   finally show "fold plus xs x = listsum (rev xs) + x" by simp
  3087 qed
  3088 
  3089 text{* Some syntactic sugar for summing a function over a list: *}
  3090 
  3091 syntax
  3092   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
  3093 syntax (xsymbols)
  3094   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
  3095 syntax (HTML output)
  3096   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
  3097 
  3098 translations -- {* Beware of argument permutation! *}
  3099   "SUM x<-xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  3100   "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  3101 
  3102 lemma (in comm_monoid_add) listsum_map_remove1:
  3103   "x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))"
  3104   by (induct xs) (auto simp add: ac_simps)
  3105 
  3106 lemma (in monoid_add) list_size_conv_listsum:
  3107   "list_size f xs = listsum (map f xs) + size xs"
  3108   by (induct xs) auto
  3109 
  3110 lemma (in monoid_add) length_concat:
  3111   "length (concat xss) = listsum (map length xss)"
  3112   by (induct xss) simp_all
  3113 
  3114 lemma (in monoid_add) listsum_map_filter:
  3115   assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
  3116   shows "listsum (map f (filter P xs)) = listsum (map f xs)"
  3117   using assms by (induct xs) auto
  3118 
  3119 lemma (in monoid_add) distinct_listsum_conv_Setsum:
  3120   "distinct xs \<Longrightarrow> listsum xs = Setsum (set xs)"
  3121   by (induct xs) simp_all
  3122 
  3123 lemma listsum_eq_0_nat_iff_nat [simp]:
  3124   "listsum ns = (0::nat) \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
  3125   by (induct ns) simp_all
  3126 
  3127 lemma member_le_listsum_nat:
  3128   "(n :: nat) \<in> set ns \<Longrightarrow> n \<le> listsum ns"
  3129   by (induct ns) auto
  3130 
  3131 lemma elem_le_listsum_nat:
  3132   "k < size ns \<Longrightarrow> ns ! k \<le> listsum (ns::nat list)"
  3133   by (rule member_le_listsum_nat) simp
  3134 
  3135 lemma listsum_update_nat:
  3136   "k < size ns \<Longrightarrow> listsum (ns[k := (n::nat)]) = listsum ns + n - ns ! k"
  3137 apply(induct ns arbitrary:k)
  3138  apply (auto split:nat.split)
  3139 apply(drule elem_le_listsum_nat)
  3140 apply arith
  3141 done
  3142 
  3143 lemma (in monoid_add) listsum_triv:
  3144   "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
  3145   by (induct xs) (simp_all add: left_distrib)
  3146 
  3147 lemma (in monoid_add) listsum_0 [simp]:
  3148   "(\<Sum>x\<leftarrow>xs. 0) = 0"
  3149   by (induct xs) (simp_all add: left_distrib)
  3150 
  3151 text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
  3152 lemma (in ab_group_add) uminus_listsum_map:
  3153   "- listsum (map f xs) = listsum (map (uminus \<circ> f) xs)"
  3154   by (induct xs) simp_all
  3155 
  3156 lemma (in comm_monoid_add) listsum_addf:
  3157   "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
  3158   by (induct xs) (simp_all add: algebra_simps)
  3159 
  3160 lemma (in ab_group_add) listsum_subtractf:
  3161   "(\<Sum>x\<leftarrow>xs. f x - g x) = listsum (map f xs) - listsum (map g xs)"
  3162   by (induct xs) (simp_all add: algebra_simps)
  3163 
  3164 lemma (in semiring_0) listsum_const_mult:
  3165   "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
  3166   by (induct xs) (simp_all add: algebra_simps)
  3167 
  3168 lemma (in semiring_0) listsum_mult_const:
  3169   "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
  3170   by (induct xs) (simp_all add: algebra_simps)
  3171 
  3172 lemma (in ordered_ab_group_add_abs) listsum_abs:
  3173   "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
  3174   by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])
  3175 
  3176 lemma listsum_mono:
  3177   fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}"
  3178   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)"
  3179   by (induct xs) (simp, simp add: add_mono)
  3180 
  3181 lemma (in monoid_add) listsum_distinct_conv_setsum_set:
  3182   "distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)"
  3183   by (induct xs) simp_all
  3184 
  3185 lemma (in monoid_add) interv_listsum_conv_setsum_set_nat:
  3186   "listsum (map f [m..<n]) = setsum f (set [m..<n])"
  3187   by (simp add: listsum_distinct_conv_setsum_set)
  3188 
  3189 lemma (in monoid_add) interv_listsum_conv_setsum_set_int:
  3190   "listsum (map f [k..l]) = setsum f (set [k..l])"
  3191   by (simp add: listsum_distinct_conv_setsum_set)
  3192 
  3193 text {* General equivalence between @{const listsum} and @{const setsum} *}
  3194 lemma (in monoid_add) listsum_setsum_nth:
  3195   "listsum xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
  3196   using interv_listsum_conv_setsum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth)
  3197 
  3198 
  3199 subsubsection {* @{const insert} *}
  3200 
  3201 lemma in_set_insert [simp]:
  3202   "x \<in> set xs \<Longrightarrow> List.insert x xs = xs"
  3203   by (simp add: List.insert_def)
  3204 
  3205 lemma not_in_set_insert [simp]:
  3206   "x \<notin> set xs \<Longrightarrow> List.insert x xs = x # xs"
  3207   by (simp add: List.insert_def)
  3208 
  3209 lemma insert_Nil [simp]:
  3210   "List.insert x [] = [x]"
  3211   by simp
  3212 
  3213 lemma set_insert [simp]:
  3214   "set (List.insert x xs) = insert x (set xs)"
  3215   by (auto simp add: List.insert_def)
  3216 
  3217 lemma distinct_insert [simp]:
  3218   "distinct xs \<Longrightarrow> distinct (List.insert x xs)"
  3219   by (simp add: List.insert_def)
  3220 
  3221 lemma insert_remdups:
  3222   "List.insert x (remdups xs) = remdups (List.insert x xs)"
  3223   by (simp add: List.insert_def)
  3224 
  3225 
  3226 subsubsection {* @{const List.find} *}
  3227 
  3228 lemma find_None_iff: "List.find P xs = None \<longleftrightarrow> \<not> (\<exists>x. x \<in> set xs \<and> P x)"
  3229 proof (induction xs)
  3230   case Nil thus ?case by simp
  3231 next
  3232   case (Cons x xs) thus ?case by (fastforce split: if_splits)
  3233 qed
  3234 
  3235 lemma find_Some_iff:
  3236   "List.find P xs = Some x \<longleftrightarrow>
  3237   (\<exists>i<length xs. P (xs!i) \<and> x = xs!i \<and> (\<forall>j<i. \<not> P (xs!j)))"
  3238 proof (induction xs)
  3239   case Nil thus ?case by simp
  3240 next
  3241   case (Cons x xs) thus ?case
  3242     by(auto simp: nth_Cons' split: if_splits)
  3243       (metis One_nat_def diff_Suc_1 less_Suc_eq_0_disj)
  3244 qed
  3245 
  3246 lemma find_cong[fundef_cong]:
  3247   assumes "xs = ys" and "\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x" 
  3248   shows "List.find P xs = List.find Q ys"
  3249 proof (cases "List.find P xs")
  3250   case None thus ?thesis by (metis find_None_iff assms)
  3251 next
  3252   case (Some x)
  3253   hence "List.find Q ys = Some x" using assms
  3254     by (auto simp add: find_Some_iff)
  3255   thus ?thesis using Some by auto
  3256 qed
  3257 
  3258 
  3259 subsubsection {* @{const remove1} *}
  3260 
  3261 lemma remove1_append:
  3262   "remove1 x (xs @ ys) =
  3263   (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
  3264 by (induct xs) auto
  3265 
  3266 lemma remove1_commute: "remove1 x (remove1 y zs) = remove1 y (remove1 x zs)"
  3267 by (induct zs) auto
  3268 
  3269 lemma in_set_remove1[simp]:
  3270   "a \<noteq> b \<Longrightarrow> a : set(remove1 b xs) = (a : set xs)"
  3271 apply (induct xs)
  3272 apply auto
  3273 done
  3274 
  3275 lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
  3276 apply(induct xs)
  3277  apply simp
  3278 apply simp
  3279 apply blast
  3280 done
  3281 
  3282 lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
  3283 apply(induct xs)
  3284  apply simp
  3285 apply simp
  3286 apply blast
  3287 done
  3288 
  3289 lemma length_remove1:
  3290   "length(remove1 x xs) = (if x : set xs then length xs - 1 else length xs)"
  3291 apply (induct xs)
  3292  apply (auto dest!:length_pos_if_in_set)
  3293 done
  3294 
  3295 lemma remove1_filter_not[simp]:
  3296   "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
  3297 by(induct xs) auto
  3298 
  3299 lemma filter_remove1:
  3300   "filter Q (remove1 x xs) = remove1 x (filter Q xs)"
  3301 by (induct xs) auto
  3302 
  3303 lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
  3304 apply(insert set_remove1_subset)
  3305 apply fast
  3306 done
  3307 
  3308 lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
  3309 by (induct xs) simp_all
  3310 
  3311 lemma remove1_remdups:
  3312   "distinct xs \<Longrightarrow> remove1 x (remdups xs) = remdups (remove1 x xs)"
  3313   by (induct xs) simp_all
  3314 
  3315 lemma remove1_idem:
  3316   assumes "x \<notin> set xs"
  3317   shows "remove1 x xs = xs"
  3318   using assms by (induct xs) simp_all
  3319 
  3320 
  3321 subsubsection {* @{text removeAll} *}
  3322 
  3323 lemma removeAll_filter_not_eq:
  3324   "removeAll x = filter (\<lambda>y. x \<noteq> y)"
  3325 proof
  3326   fix xs
  3327   show "removeAll x xs = filter (\<lambda>y. x \<noteq> y) xs"
  3328     by (induct xs) auto
  3329 qed
  3330 
  3331 lemma removeAll_append[simp]:
  3332   "removeAll x (xs @ ys) = removeAll x xs @ removeAll x ys"
  3333 by (induct xs) auto
  3334 
  3335 lemma set_removeAll[simp]: "set(removeAll x xs) = set xs - {x}"
  3336 by (induct xs) auto
  3337 
  3338 lemma removeAll_id[simp]: "x \<notin> set xs \<Longrightarrow> removeAll x xs = xs"
  3339 by (induct xs) auto
  3340 
  3341 (* Needs count:: 'a \<Rightarrow> 'a list \<Rightarrow> nat
  3342 lemma length_removeAll:
  3343   "length(removeAll x xs) = length xs - count x xs"
  3344 *)
  3345 
  3346 lemma removeAll_filter_not[simp]:
  3347   "\<not> P x \<Longrightarrow> removeAll x (filter P xs) = filter P xs"
  3348 by(induct xs) auto
  3349 
  3350 lemma distinct_removeAll:
  3351   "distinct xs \<Longrightarrow> distinct (removeAll x xs)"
  3352   by (simp add: removeAll_filter_not_eq)
  3353 
  3354 lemma distinct_remove1_removeAll:
  3355   "distinct xs ==> remove1 x xs = removeAll x xs"
  3356 by (induct xs) simp_all
  3357 
  3358 lemma map_removeAll_inj_on: "inj_on f (insert x (set xs)) \<Longrightarrow>
  3359   map f (removeAll x xs) = removeAll (f x) (map f xs)"
  3360 by (induct xs) (simp_all add:inj_on_def)
  3361 
  3362 lemma map_removeAll_inj: "inj f \<Longrightarrow>
  3363   map f (removeAll x xs) = removeAll (f x) (map f xs)"
  3364 by(metis map_removeAll_inj_on subset_inj_on subset_UNIV)
  3365 
  3366 
  3367 subsubsection {* @{text replicate} *}
  3368 
  3369 lemma length_replicate [simp]: "length (replicate n x) = n"
  3370 by (induct n) auto
  3371 
  3372 lemma Ex_list_of_length: "\<exists>xs. length xs = n"
  3373 by (rule exI[of _ "replicate n undefined"]) simp
  3374 
  3375 lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
  3376 by (induct n) auto
  3377 
  3378 lemma map_replicate_const:
  3379   "map (\<lambda> x. k) lst = replicate (length lst) k"
  3380   by (induct lst) auto
  3381 
  3382 lemma replicate_app_Cons_same:
  3383 "(replicate n x) @ (x # xs) = x # replicate n x @ xs"
  3384 by (induct n) auto
  3385 
  3386 lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
  3387 apply (induct n, simp)
  3388 apply (simp add: replicate_app_Cons_same)
  3389 done
  3390 
  3391 lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
  3392 by (induct n) auto
  3393 
  3394 text{* Courtesy of Matthias Daum: *}
  3395 lemma append_replicate_commute:
  3396   "replicate n x @ replicate k x = replicate k x @ replicate n x"
  3397 apply (simp add: replicate_add [THEN sym])
  3398 apply (simp add: add_commute)
  3399 done
  3400 
  3401 text{* Courtesy of Andreas Lochbihler: *}
  3402 lemma filter_replicate:
  3403   "filter P (replicate n x) = (if P x then replicate n x else [])"
  3404 by(induct n) auto
  3405 
  3406 lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
  3407 by (induct n) auto
  3408 
  3409 lemma tl_replicate [simp]: "tl (replicate n x) = replicate (n - 1) x"
  3410 by (induct n) auto
  3411 
  3412 lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
  3413 by (atomize (full), induct n) auto
  3414 
  3415 lemma nth_replicate[simp]: "i < n ==> (replicate n x)!i = x"
  3416 apply (induct n arbitrary: i, simp)
  3417 apply (simp add: nth_Cons split: nat.split)
  3418 done
  3419 
  3420 text{* Courtesy of Matthias Daum (2 lemmas): *}
  3421 lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
  3422 apply (case_tac "k \<le> i")
  3423  apply  (simp add: min_def)
  3424 apply (drule not_leE)
  3425 apply (simp add: min_def)
  3426 apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
  3427  apply  simp
  3428 apply (simp add: replicate_add [symmetric])
  3429 done
  3430 
  3431 lemma drop_replicate[simp]: "drop i (replicate k x) = replicate (k-i) x"
  3432 apply (induct k arbitrary: i)
  3433  apply simp
  3434 apply clarsimp
  3435 apply (case_tac i)
  3436  apply simp
  3437 apply clarsimp
  3438 done
  3439 
  3440 
  3441 lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
  3442 by (induct n) auto
  3443 
  3444 lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
  3445 by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
  3446 
  3447 lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
  3448 by auto
  3449 
  3450 lemma in_set_replicate[simp]: "(x : set (replicate n y)) = (x = y & n \<noteq> 0)"
  3451 by (simp add: set_replicate_conv_if)
  3452 
  3453 lemma Ball_set_replicate[simp]:
  3454   "(ALL x : set(replicate n a). P x) = (P a | n=0)"
  3455 by(simp add: set_replicate_conv_if)
  3456 
  3457 lemma Bex_set_replicate[simp]:
  3458   "(EX x : set(replicate n a). P x) = (P a & n\<noteq>0)"
  3459 by(simp add: set_replicate_conv_if)
  3460 
  3461 lemma replicate_append_same:
  3462   "replicate i x @ [x] = x # replicate i x"
  3463   by (induct i) simp_all
  3464 
  3465 lemma map_replicate_trivial:
  3466   "map (\<lambda>i. x) [0..<i] = replicate i x"
  3467   by (induct i) (simp_all add: replicate_append_same)
  3468 
  3469 lemma concat_replicate_trivial[simp]:
  3470   "concat (replicate i []) = []"
  3471   by (induct i) (auto simp add: map_replicate_const)
  3472 
  3473 lemma replicate_empty[simp]: "(replicate n x = []) \<longleftrightarrow> n=0"
  3474 by (induct n) auto
  3475 
  3476 lemma empty_replicate[simp]: "([] = replicate n x) \<longleftrightarrow> n=0"
  3477 by (induct n) auto
  3478 
  3479 lemma replicate_eq_replicate[simp]:
  3480   "(replicate m x = replicate n y) \<longleftrightarrow> (m=n & (m\<noteq>0 \<longrightarrow> x=y))"
  3481 apply(induct m arbitrary: n)
  3482  apply simp
  3483 apply(induct_tac n)
  3484 apply auto
  3485 done
  3486 
  3487 lemma replicate_length_filter:
  3488   "replicate (length (filter (\<lambda>y. x = y) xs)) x = filter (\<lambda>y. x = y) xs"
  3489   by (induct xs) auto
  3490 
  3491 lemma comm_append_are_replicate:
  3492   fixes xs ys :: "'a list"
  3493   assumes "xs \<noteq> []" "ys \<noteq> []"
  3494   assumes "xs @ ys = ys @ xs"
  3495   shows "\<exists>m n zs. concat (replicate m zs) = xs \<and> concat (replicate n zs) = ys"
  3496   using assms
  3497 proof (induct "length (xs @ ys)" arbitrary: xs ys rule: less_induct)
  3498   case less
  3499 
  3500   def xs' \<equiv> "if (length xs \<le> length ys) then xs else ys"
  3501     and ys' \<equiv> "if (length xs \<le> length ys) then ys else xs"
  3502   then have
  3503     prems': "length xs' \<le> length ys'"
  3504             "xs' @ ys' = ys' @ xs'"
  3505       and "xs' \<noteq> []"
  3506       and len: "length (xs @ ys) = length (xs' @ ys')"
  3507     using less by (auto intro: less.hyps)
  3508 
  3509   from prems'
  3510   obtain ws where "ys' = xs' @ ws"
  3511     by (auto simp: append_eq_append_conv2)
  3512 
  3513   have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ys'"
  3514   proof (cases "ws = []")
  3515     case True
  3516     then have "concat (replicate 1 xs') = xs'"
  3517       and "concat (replicate 1 xs') = ys'"
  3518       using `ys' = xs' @ ws` by auto
  3519     then show ?thesis by blast
  3520   next
  3521     case False
  3522     from `ys' = xs' @ ws` and `xs' @ ys' = ys' @ xs'`
  3523     have "xs' @ ws = ws @ xs'" by simp
  3524     then have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ws"
  3525       using False and `xs' \<noteq> []` and `ys' = xs' @ ws` and len
  3526       by (intro less.hyps) auto
  3527     then obtain m n zs where "concat (replicate m zs) = xs'"
  3528       and "concat (replicate n zs) = ws" by blast
  3529     moreover
  3530     then have "concat (replicate (m + n) zs) = ys'"
  3531       using `ys' = xs' @ ws`
  3532       by (simp add: replicate_add)
  3533     ultimately
  3534     show ?thesis by blast
  3535   qed
  3536   then show ?case
  3537     using xs'_def ys'_def by metis
  3538 qed
  3539 
  3540 lemma comm_append_is_replicate:
  3541   fixes xs ys :: "'a list"
  3542   assumes "xs \<noteq> []" "ys \<noteq> []"
  3543   assumes "xs @ ys = ys @ xs"
  3544   shows "\<exists>n zs. n > 1 \<and> concat (replicate n zs) = xs @ ys"
  3545 
  3546 proof -
  3547   obtain m n zs where "concat (replicate m zs) = xs"
  3548     and "concat (replicate n zs) = ys"
  3549     using assms by (metis comm_append_are_replicate)
  3550   then have "m + n > 1" and "concat (replicate (m+n) zs) = xs @ ys"
  3551     using `xs \<noteq> []` and `ys \<noteq> []`
  3552     by (auto simp: replicate_add)
  3553   then show ?thesis by blast
  3554 qed
  3555 
  3556 
  3557 subsubsection{*@{text rotate1} and @{text rotate}*}
  3558 
  3559 lemma rotate0[simp]: "rotate 0 = id"
  3560 by(simp add:rotate_def)
  3561 
  3562 lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
  3563 by(simp add:rotate_def)
  3564 
  3565 lemma rotate_add:
  3566   "rotate (m+n) = rotate m o rotate n"
  3567 by(simp add:rotate_def funpow_add)
  3568 
  3569 lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
  3570 by(simp add:rotate_add)
  3571 
  3572 lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
  3573 by(simp add:rotate_def funpow_swap1)
  3574 
  3575 lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
  3576 by(cases xs) simp_all
  3577 
  3578 lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
  3579 apply(induct n)
  3580  apply simp
  3581 apply (simp add:rotate_def)
  3582 done
  3583 
  3584 lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
  3585 by (cases xs) simp_all
  3586 
  3587 lemma rotate_drop_take:
  3588   "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
  3589 apply(induct n)
  3590  apply simp
  3591 apply(simp add:rotate_def)
  3592 apply(cases "xs = []")
  3593  apply (simp)
  3594 apply(case_tac "n mod length xs = 0")
  3595  apply(simp add:mod_Suc)
  3596  apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
  3597 apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
  3598                 take_hd_drop linorder_not_le)
  3599 done
  3600 
  3601 lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
  3602 by(simp add:rotate_drop_take)
  3603 
  3604 lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
  3605 by(simp add:rotate_drop_take)
  3606 
  3607 lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
  3608 by (cases xs) simp_all
  3609 
  3610 lemma length_rotate[simp]: "length(rotate n xs) = length xs"
  3611 by (induct n arbitrary: xs) (simp_all add:rotate_def)
  3612 
  3613 lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
  3614 by (cases xs) auto
  3615 
  3616 lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
  3617 by (induct n) (simp_all add:rotate_def)
  3618 
  3619 lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
  3620 by(simp add:rotate_drop_take take_map drop_map)
  3621 
  3622 lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
  3623 by (cases xs) auto
  3624 
  3625 lemma set_rotate[simp]: "set(rotate n xs) = set xs"
  3626 by (induct n) (simp_all add:rotate_def)
  3627 
  3628 lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
  3629 by (cases xs) auto
  3630 
  3631 lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
  3632 by (induct n) (simp_all add:rotate_def)
  3633 
  3634 lemma rotate_rev:
  3635   "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
  3636 apply(simp add:rotate_drop_take rev_drop rev_take)
  3637 apply(cases "length xs = 0")
  3638  apply simp
  3639 apply(cases "n mod length xs = 0")
  3640  apply simp
  3641 apply(simp add:rotate_drop_take rev_drop rev_take)
  3642 done
  3643 
  3644 lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
  3645 apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
  3646 apply(subgoal_tac "length xs \<noteq> 0")
  3647  prefer 2 apply simp
  3648 using mod_less_divisor[of "length xs" n] by arith
  3649 
  3650 
  3651 subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
  3652 
  3653 lemma sublist_empty [simp]: "sublist xs {} = []"
  3654 by (auto simp add: sublist_def)
  3655 
  3656 lemma sublist_nil [simp]: "sublist [] A = []"
  3657 by (auto simp add: sublist_def)
  3658 
  3659 lemma length_sublist:
  3660   "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
  3661 by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
  3662 
  3663 lemma sublist_shift_lemma_Suc:
  3664   "map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
  3665    map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
  3666 apply(induct xs arbitrary: "is")
  3667  apply simp
  3668 apply (case_tac "is")
  3669  apply simp
  3670 apply simp
  3671 done
  3672 
  3673 lemma sublist_shift_lemma:
  3674      "map fst [p<-zip xs [i..<i + length xs] . snd p : A] =
  3675       map fst [p<-zip xs [0..<length xs] . snd p + i : A]"
  3676 by (induct xs rule: rev_induct) (simp_all add: add_commute)
  3677 
  3678 lemma sublist_append:
  3679      "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
  3680 apply (unfold sublist_def)
  3681 apply (induct l' rule: rev_induct, simp)
  3682 apply (simp add: upt_add_eq_append[of 0] sublist_shift_lemma)
  3683 apply (simp add: add_commute)
  3684 done
  3685 
  3686 lemma sublist_Cons:
  3687 "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
  3688 apply (induct l rule: rev_induct)
  3689  apply (simp add: sublist_def)
  3690 apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
  3691 done
  3692 
  3693 lemma set_sublist: "set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
  3694 apply(induct xs arbitrary: I)
  3695 apply(auto simp: sublist_Cons nth_Cons split:nat.split dest!: gr0_implies_Suc)
  3696 done
  3697 
  3698 lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
  3699 by(auto simp add:set_sublist)
  3700 
  3701 lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
  3702 by(auto simp add:set_sublist)
  3703 
  3704 lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
  3705 by(auto simp add:set_sublist)
  3706 
  3707 lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
  3708 by (simp add: sublist_Cons)
  3709 
  3710 
  3711 lemma distinct_sublistI[simp]: "distinct xs \<Longrightarrow> distinct(sublist xs I)"
  3712 apply(induct xs arbitrary: I)
  3713  apply simp
  3714 apply(auto simp add:sublist_Cons)
  3715 done
  3716 
  3717 
  3718 lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
  3719 apply (induct l rule: rev_induct, simp)
  3720 apply (simp split: nat_diff_split add: sublist_append)
  3721 done
  3722 
  3723 lemma filter_in_sublist:
  3724  "distinct xs \<Longrightarrow> filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
  3725 proof (induct xs arbitrary: s)
  3726   case Nil thus ?case by simp
  3727 next
  3728   case (Cons a xs)
  3729   moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
  3730   ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
  3731 qed
  3732 
  3733 
  3734 subsubsection {* @{const splice} *}
  3735 
  3736 lemma splice_Nil2 [simp, code]: "splice xs [] = xs"
  3737 by (cases xs) simp_all
  3738 
  3739 declare splice.simps(1,3)[code]
  3740 declare splice.simps(2)[simp del]
  3741 
  3742 lemma length_splice[simp]: "length(splice xs ys) = length xs + length ys"
  3743 by (induct xs ys rule: splice.induct) auto
  3744 
  3745 
  3746 subsubsection {* Transpose *}
  3747 
  3748 function transpose where
  3749 "transpose []             = []" |
  3750 "transpose ([]     # xss) = transpose xss" |
  3751 "transpose ((x#xs) # xss) =
  3752   (x # [h. (h#t) \<leftarrow> xss]) # transpose (xs # [t. (h#t) \<leftarrow> xss])"
  3753 by pat_completeness auto
  3754 
  3755 lemma transpose_aux_filter_head:
  3756   "concat (map (list_case [] (\<lambda>h t. [h])) xss) =
  3757   map (\<lambda>xs. hd xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  3758   by (induct xss) (auto split: list.split)
  3759 
  3760 lemma transpose_aux_filter_tail:
  3761   "concat (map (list_case [] (\<lambda>h t. [t])) xss) =
  3762   map (\<lambda>xs. tl xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  3763   by (induct xss) (auto split: list.split)
  3764 
  3765 lemma transpose_aux_max:
  3766   "max (Suc (length xs)) (foldr (\<lambda>xs. max (length xs)) xss 0) =
  3767   Suc (max (length xs) (foldr (\<lambda>x. max (length x - Suc 0)) [ys\<leftarrow>xss . ys\<noteq>[]] 0))"
  3768   (is "max _ ?foldB = Suc (max _ ?foldA)")
  3769 proof (cases "[ys\<leftarrow>xss . ys\<noteq>[]] = []")
  3770   case True
  3771   hence "foldr (\<lambda>xs. max (length xs)) xss 0 = 0"
  3772   proof (induct xss)
  3773     case (Cons x xs)
  3774     moreover hence "x = []" by (cases x) auto
  3775     ultimately show ?case by auto
  3776   qed simp
  3777   thus ?thesis using True by simp
  3778 next
  3779   case False
  3780 
  3781   have foldA: "?foldA = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0 - 1"
  3782     by (induct xss) auto
  3783   have foldB: "?foldB = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0"
  3784     by (induct xss) auto
  3785 
  3786   have "0 < ?foldB"
  3787   proof -
  3788     from False
  3789     obtain z zs where zs: "[ys\<leftarrow>xss . ys \<noteq> []] = z#zs" by (auto simp: neq_Nil_conv)
  3790     hence "z \<in> set ([ys\<leftarrow>xss . ys \<noteq> []])" by auto
  3791     hence "z \<noteq> []" by auto
  3792     thus ?thesis
  3793       unfolding foldB zs
  3794       by (auto simp: max_def intro: less_le_trans)
  3795   qed
  3796   thus ?thesis
  3797     unfolding foldA foldB max_Suc_Suc[symmetric]
  3798     by simp
  3799 qed
  3800 
  3801 termination transpose
  3802   by (relation "measure (\<lambda>xs. foldr (\<lambda>xs. max (length xs)) xs 0 + length xs)")
  3803      (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max less_Suc_eq_le)
  3804 
  3805 lemma transpose_empty: "(transpose xs = []) \<longleftrightarrow> (\<forall>x \<in> set xs. x = [])"
  3806   by (induct rule: transpose.induct) simp_all
  3807 
  3808 lemma length_transpose:
  3809   fixes xs :: "'a list list"
  3810   shows "length (transpose xs) = foldr (\<lambda>xs. max (length xs)) xs 0"
  3811   by (induct rule: transpose.induct)
  3812     (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max
  3813                 max_Suc_Suc[symmetric] simp del: max_Suc_Suc)
  3814 
  3815 lemma nth_transpose:
  3816   fixes xs :: "'a list list"
  3817   assumes "i < length (transpose xs)"
  3818   shows "transpose xs ! i = map (\<lambda>xs. xs ! i) [ys \<leftarrow> xs. i < length ys]"
  3819 using assms proof (induct arbitrary: i rule: transpose.induct)
  3820   case (3 x xs xss)
  3821   def XS == "(x # xs) # xss"
  3822   hence [simp]: "XS \<noteq> []" by auto
  3823   thus ?case
  3824   proof (cases i)
  3825     case 0
  3826     thus ?thesis by (simp add: transpose_aux_filter_head hd_conv_nth)
  3827   next
  3828     case (Suc j)
  3829     have *: "\<And>xss. xs # map tl xss = map tl ((x#xs)#xss)" by simp
  3830     have **: "\<And>xss. (x#xs) # filter (\<lambda>ys. ys \<noteq> []) xss = filter (\<lambda>ys. ys \<noteq> []) ((x#xs)#xss)" by simp
  3831     { fix x have "Suc j < length x \<longleftrightarrow> x \<noteq> [] \<and> j < length x - Suc 0"
  3832       by (cases x) simp_all
  3833     } note *** = this
  3834 
  3835     have j_less: "j < length (transpose (xs # concat (map (list_case [] (\<lambda>h t. [t])) xss)))"
  3836       using "3.prems" by (simp add: transpose_aux_filter_tail length_transpose Suc)
  3837 
  3838     show ?thesis
  3839       unfolding transpose.simps `i = Suc j` nth_Cons_Suc "3.hyps"[OF j_less]
  3840       apply (auto simp: transpose_aux_filter_tail filter_map comp_def length_transpose * ** *** XS_def[symmetric])
  3841       apply (rule_tac y=x in list.exhaust)
  3842       by auto
  3843   qed
  3844 qed simp_all
  3845 
  3846 lemma transpose_map_map:
  3847   "transpose (map (map f) xs) = map (map f) (transpose xs)"
  3848 proof (rule nth_equalityI, safe)
  3849   have [simp]: "length (transpose (map (map f) xs)) = length (transpose xs)"
  3850     by (simp add: length_transpose foldr_map comp_def)
  3851   show "length (transpose (map (map f) xs)) = length (map (map f) (transpose xs))" by simp
  3852 
  3853   fix i assume "i < length (transpose (map (map f) xs))"
  3854   thus "transpose (map (map f) xs) ! i = map (map f) (transpose xs) ! i"
  3855     by (simp add: nth_transpose filter_map comp_def)
  3856 qed
  3857 
  3858 
  3859 subsubsection {* (In)finiteness *}
  3860 
  3861 lemma finite_maxlen:
  3862   "finite (M::'a list set) ==> EX n. ALL s:M. size s < n"
  3863 proof (induct rule: finite.induct)
  3864   case emptyI show ?case by simp
  3865 next
  3866   case (insertI M xs)
  3867   then obtain n where "\<forall>s\<in>M. length s < n" by blast
  3868   hence "ALL s:insert xs M. size s < max n (size xs) + 1" by auto
  3869   thus ?case ..
  3870 qed
  3871 
  3872 lemma lists_length_Suc_eq:
  3873   "{xs. set xs \<subseteq> A \<and> length xs = Suc n} =
  3874     (\<lambda>(xs, n). n#xs) ` ({xs. set xs \<subseteq> A \<and> length xs = n} \<times> A)"
  3875   by (auto simp: length_Suc_conv)
  3876 
  3877 lemma
  3878   assumes "finite A"
  3879   shows finite_lists_length_eq: "finite {xs. set xs \<subseteq> A \<and> length xs = n}"
  3880   and card_lists_length_eq: "card {xs. set xs \<subseteq> A \<and> length xs = n} = (card A)^n"
  3881   using `finite A`
  3882   by (induct n)
  3883      (auto simp: card_image inj_split_Cons lists_length_Suc_eq cong: conj_cong)
  3884 
  3885 lemma finite_lists_length_le:
  3886   assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs \<le> n}"
  3887  (is "finite ?S")
  3888 proof-
  3889   have "?S = (\<Union>n\<in>{0..n}. {xs. set xs \<subseteq> A \<and> length xs = n})" by auto
  3890   thus ?thesis by (auto intro: finite_lists_length_eq[OF `finite A`])
  3891 qed
  3892 
  3893 lemma card_lists_length_le:
  3894   assumes "finite A" shows "card {xs. set xs \<subseteq> A \<and> length xs \<le> n} = (\<Sum>i\<le>n. card A^i)"
  3895 proof -
  3896   have "(\<Sum>i\<le>n. card A^i) = card (\<Union>i\<le>n. {xs. set xs \<subseteq> A \<and> length xs = i})"
  3897     using `finite A`
  3898     by (subst card_UN_disjoint)
  3899        (auto simp add: card_lists_length_eq finite_lists_length_eq)
  3900   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}"
  3901     by auto
  3902   finally show ?thesis by simp
  3903 qed
  3904 
  3905 lemma card_lists_distinct_length_eq:
  3906   assumes "k < card A"
  3907   shows "card {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A} = \<Prod>{card A - k + 1 .. card A}"
  3908 using assms
  3909 proof (induct k)
  3910   case 0
  3911   then have "{xs. length xs = 0 \<and> distinct xs \<and> set xs \<subseteq> A} = {[]}" by auto
  3912   then show ?case by simp
  3913 next
  3914   case (Suc k)
  3915   let "?k_list" = "\<lambda>k xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A"
  3916   have inj_Cons: "\<And>A. inj_on (\<lambda>(xs, n). n # xs) A"  by (rule inj_onI) auto
  3917 
  3918   from Suc have "k < card A" by simp
  3919   moreover have "finite A" using assms by (simp add: card_ge_0_finite)
  3920   moreover have "finite {xs. ?k_list k xs}"
  3921     using finite_lists_length_eq[OF `finite A`, of k]
  3922     by - (rule finite_subset, auto)
  3923   moreover have "\<And>i j. i \<noteq> j \<longrightarrow> {i} \<times> (A - set i) \<inter> {j} \<times> (A - set j) = {}"
  3924     by auto
  3925   moreover have "\<And>i. i \<in>Collect (?k_list k) \<Longrightarrow> card (A - set i) = card A - k"
  3926     by (simp add: card_Diff_subset distinct_card)
  3927   moreover have "{xs. ?k_list (Suc k) xs} =
  3928       (\<lambda>(xs, n). n#xs) ` \<Union>(\<lambda>xs. {xs} \<times> (A - set xs)) ` {xs. ?k_list k xs}"
  3929     by (auto simp: length_Suc_conv)
  3930   moreover
  3931   have "Suc (card A - Suc k) = card A - k" using Suc.prems by simp
  3932   then have "(card A - k) * \<Prod>{Suc (card A - k)..card A} = \<Prod>{Suc (card A - Suc k)..card A}"
  3933     by (subst setprod_insert[symmetric]) (simp add: atLeastAtMost_insertL)+
  3934   ultimately show ?case
  3935     by (simp add: card_image inj_Cons card_UN_disjoint Suc.hyps algebra_simps)
  3936 qed
  3937 
  3938 lemma infinite_UNIV_listI: "~ finite(UNIV::'a list set)"
  3939 apply(rule notI)
  3940 apply(drule finite_maxlen)
  3941 apply (metis UNIV_I length_replicate less_not_refl)
  3942 done
  3943 
  3944 
  3945 subsection {* Sorting *}
  3946 
  3947 text{* Currently it is not shown that @{const sort} returns a
  3948 permutation of its input because the nicest proof is via multisets,
  3949 which are not yet available. Alternatively one could define a function
  3950 that counts the number of occurrences of an element in a list and use
  3951 that instead of multisets to state the correctness property. *}
  3952 
  3953 context linorder
  3954 begin
  3955 
  3956 lemma length_insort [simp]:
  3957   "length (insort_key f x xs) = Suc (length xs)"
  3958   by (induct xs) simp_all
  3959 
  3960 lemma insort_key_left_comm:
  3961   assumes "f x \<noteq> f y"
  3962   shows "insort_key f y (insort_key f x xs) = insort_key f x (insort_key f y xs)"
  3963   by (induct xs) (auto simp add: assms dest: antisym)
  3964 
  3965 lemma insort_left_comm:
  3966   "insort x (insort y xs) = insort y (insort x xs)"
  3967   by (cases "x = y") (auto intro: insort_key_left_comm)
  3968 
  3969 lemma comp_fun_commute_insort:
  3970   "comp_fun_commute insort"
  3971 proof
  3972 qed (simp add: insort_left_comm fun_eq_iff)
  3973 
  3974 lemma sort_key_simps [simp]:
  3975   "sort_key f [] = []"
  3976   "sort_key f (x#xs) = insort_key f x (sort_key f xs)"
  3977   by (simp_all add: sort_key_def)
  3978 
  3979 lemma (in linorder) sort_key_conv_fold:
  3980   assumes "inj_on f (set xs)"
  3981   shows "sort_key f xs = fold (insort_key f) xs []"
  3982 proof -
  3983   have "fold (insort_key f) (rev xs) = fold (insort_key f) xs"
  3984   proof (rule fold_rev, rule ext)
  3985     fix zs
  3986     fix x y
  3987     assume "x \<in> set xs" "y \<in> set xs"
  3988     with assms have *: "f y = f x \<Longrightarrow> y = x" by (auto dest: inj_onD)
  3989     have **: "x = y \<longleftrightarrow> y = x" by auto
  3990     show "(insort_key f y \<circ> insort_key f x) zs = (insort_key f x \<circ> insort_key f y) zs"
  3991       by (induct zs) (auto intro: * simp add: **)
  3992   qed
  3993   then show ?thesis by (simp add: sort_key_def foldr_conv_fold)
  3994 qed
  3995 
  3996 lemma (in linorder) sort_conv_fold:
  3997   "sort xs = fold insort xs []"
  3998   by (rule sort_key_conv_fold) simp
  3999 
  4000 lemma length_sort[simp]: "length (sort_key f xs) = length xs"
  4001 by (induct xs, auto)
  4002 
  4003 lemma sorted_Cons: "sorted (x#xs) = (sorted xs & (ALL y:set xs. x <= y))"
  4004 apply(induct xs arbitrary: x) apply simp
  4005 by simp (blast intro: order_trans)
  4006 
  4007 lemma sorted_tl:
  4008   "sorted xs \<Longrightarrow> sorted (tl xs)"
  4009   by (cases xs) (simp_all add: sorted_Cons)
  4010 
  4011 lemma sorted_append:
  4012   "sorted (xs@ys) = (sorted xs & sorted ys & (\<forall>x \<in> set xs. \<forall>y \<in> set ys. x\<le>y))"
  4013 by (induct xs) (auto simp add:sorted_Cons)
  4014 
  4015 lemma sorted_nth_mono:
  4016   "sorted xs \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!i \<le> xs!j"
  4017 by (induct xs arbitrary: i j) (auto simp:nth_Cons' sorted_Cons)
  4018 
  4019 lemma sorted_rev_nth_mono:
  4020   "sorted (rev xs) \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!j \<le> xs!i"
  4021 using sorted_nth_mono[ of "rev xs" "length xs - j - 1" "length xs - i - 1"]
  4022       rev_nth[of "length xs - i - 1" "xs"] rev_nth[of "length xs - j - 1" "xs"]
  4023 by auto
  4024 
  4025 lemma sorted_nth_monoI:
  4026   "(\<And> i j. \<lbrakk> i \<le> j ; j < length xs \<rbrakk> \<Longrightarrow> xs ! i \<le> xs ! j) \<Longrightarrow> sorted xs"
  4027 proof (induct xs)
  4028   case (Cons x xs)
  4029   have "sorted xs"
  4030   proof (rule Cons.hyps)
  4031     fix i j assume "i \<le> j" and "j < length xs"
  4032     with Cons.prems[of "Suc i" "Suc j"]
  4033     show "xs ! i \<le> xs ! j" by auto
  4034   qed
  4035   moreover
  4036   {
  4037     fix y assume "y \<in> set xs"
  4038     then obtain j where "j < length xs" and "xs ! j = y"
  4039       unfolding in_set_conv_nth by blast
  4040     with Cons.prems[of 0 "Suc j"]
  4041     have "x \<le> y"
  4042       by auto
  4043   }
  4044   ultimately
  4045   show ?case
  4046     unfolding sorted_Cons by auto
  4047 qed simp
  4048 
  4049 lemma sorted_equals_nth_mono:
  4050   "sorted xs = (\<forall>j < length xs. \<forall>i \<le> j. xs ! i \<le> xs ! j)"
  4051 by (auto intro: sorted_nth_monoI sorted_nth_mono)
  4052 
  4053 lemma set_insort: "set(insort_key f x xs) = insert x (set xs)"
  4054 by (induct xs) auto
  4055 
  4056 lemma set_sort[simp]: "set(sort_key f xs) = set xs"
  4057 by (induct xs) (simp_all add:set_insort)
  4058 
  4059 lemma distinct_insort: "distinct (insort_key f x xs) = (x \<notin> set xs \<and> distinct xs)"
  4060 by(induct xs)(auto simp:set_insort)
  4061 
  4062 lemma distinct_sort[simp]: "distinct (sort_key f xs) = distinct xs"
  4063   by (induct xs) (simp_all add: distinct_insort)
  4064 
  4065 lemma sorted_insort_key: "sorted (map f (insort_key f x xs)) = sorted (map f xs)"
  4066   by (induct xs) (auto simp:sorted_Cons set_insort)
  4067 
  4068 lemma sorted_insort: "sorted (insort x xs) = sorted xs"
  4069   using sorted_insort_key [where f="\<lambda>x. x"] by simp
  4070 
  4071 theorem sorted_sort_key [simp]: "sorted (map f (sort_key f xs))"
  4072   by (induct xs) (auto simp:sorted_insort_key)
  4073 
  4074 theorem sorted_sort [simp]: "sorted (sort xs)"
  4075   using sorted_sort_key [where f="\<lambda>x. x"] by simp
  4076 
  4077 lemma sorted_butlast:
  4078   assumes "xs \<noteq> []" and "sorted xs"
  4079   shows "sorted (butlast xs)"
  4080 proof -
  4081   from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  4082   with `sorted xs` show ?thesis by (simp add: sorted_append)
  4083 qed
  4084   
  4085 lemma insort_not_Nil [simp]:
  4086   "insort_key f a xs \<noteq> []"
  4087   by (induct xs) simp_all
  4088 
  4089 lemma insort_is_Cons: "\<forall>x\<in>set xs. f a \<le> f x \<Longrightarrow> insort_key f a xs = a # xs"
  4090 by (cases xs) auto
  4091 
  4092 lemma sorted_sort_id: "sorted xs \<Longrightarrow> sort xs = xs"
  4093   by (induct xs) (auto simp add: sorted_Cons insort_is_Cons)
  4094 
  4095 lemma sorted_map_remove1:
  4096   "sorted (map f xs) \<Longrightarrow> sorted (map f (remove1 x xs))"
  4097   by (induct xs) (auto simp add: sorted_Cons)
  4098 
  4099 lemma sorted_remove1: "sorted xs \<Longrightarrow> sorted (remove1 a xs)"
  4100   using sorted_map_remove1 [of "\<lambda>x. x"] by simp
  4101 
  4102 lemma insort_key_remove1:
  4103   assumes "a \<in> set xs" and "sorted (map f xs)" and "hd (filter (\<lambda>x. f a = f x) xs) = a"
  4104   shows "insort_key f a (remove1 a xs) = xs"
  4105 using assms proof (induct xs)
  4106   case (Cons x xs)
  4107   then show ?case
  4108   proof (cases "x = a")
  4109     case False
  4110     then have "f x \<noteq> f a" using Cons.prems by auto
  4111     then have "f x < f a" using Cons.prems by (auto simp: sorted_Cons)
  4112     with `f x \<noteq> f a` show ?thesis using Cons by (auto simp: sorted_Cons insort_is_Cons)
  4113   qed (auto simp: sorted_Cons insort_is_Cons)
  4114 qed simp
  4115 
  4116 lemma insort_remove1:
  4117   assumes "a \<in> set xs" and "sorted xs"
  4118   shows "insort a (remove1 a xs) = xs"
  4119 proof (rule insort_key_remove1)
  4120   from `a \<in> set xs` show "a \<in> set xs" .
  4121   from `sorted xs` show "sorted (map (\<lambda>x. x) xs)" by simp
  4122   from `a \<in> set xs` have "a \<in> set (filter (op = a) xs)" by auto
  4123   then have "set (filter (op = a) xs) \<noteq> {}" by auto
  4124   then have "filter (op = a) xs \<noteq> []" by (auto simp only: set_empty)
  4125   then have "length (filter (op = a) xs) > 0" by simp
  4126   then obtain n where n: "Suc n = length (filter (op = a) xs)"
  4127     by (cases "length (filter (op = a) xs)") simp_all
  4128   moreover have "replicate (Suc n) a = a # replicate n a"
  4129     by simp
  4130   ultimately show "hd (filter (op = a) xs) = a" by (simp add: replicate_length_filter)
  4131 qed
  4132 
  4133 lemma sorted_remdups[simp]:
  4134   "sorted l \<Longrightarrow> sorted (remdups l)"
  4135 by (induct l) (auto simp: sorted_Cons)
  4136 
  4137 lemma sorted_distinct_set_unique:
  4138 assumes "sorted xs" "distinct xs" "sorted ys" "distinct ys" "set xs = set ys"
  4139 shows "xs = ys"
  4140 proof -
  4141   from assms have 1: "length xs = length ys" by (auto dest!: distinct_card)
  4142   from assms show ?thesis
  4143   proof(induct rule:list_induct2[OF 1])
  4144     case 1 show ?case by simp
  4145   next
  4146     case 2 thus ?case by (simp add:sorted_Cons)
  4147        (metis Diff_insert_absorb antisym insertE insert_iff)
  4148   qed
  4149 qed
  4150 
  4151 lemma map_sorted_distinct_set_unique:
  4152   assumes "inj_on f (set xs \<union> set ys)"
  4153   assumes "sorted (map f xs)" "distinct (map f xs)"
  4154     "sorted (map f ys)" "distinct (map f ys)"
  4155   assumes "set xs = set ys"
  4156   shows "xs = ys"
  4157 proof -
  4158   from assms have "map f xs = map f ys"
  4159     by (simp add: sorted_distinct_set_unique)
  4160   moreover with `inj_on f (set xs \<union> set ys)` show "xs = ys"
  4161     by (blast intro: map_inj_on)
  4162 qed
  4163 
  4164 lemma finite_sorted_distinct_unique:
  4165 shows "finite A \<Longrightarrow> EX! xs. set xs = A & sorted xs & distinct xs"
  4166 apply(drule finite_distinct_list)
  4167 apply clarify
  4168 apply(rule_tac a="sort xs" in ex1I)
  4169 apply (auto simp: sorted_distinct_set_unique)
  4170 done
  4171 
  4172 lemma
  4173   assumes "sorted xs"
  4174   shows sorted_take: "sorted (take n xs)"
  4175   and sorted_drop: "sorted (drop n xs)"
  4176 proof -
  4177   from assms have "sorted (take n xs @ drop n xs)" by simp
  4178   then show "sorted (take n xs)" and "sorted (drop n xs)"
  4179     unfolding sorted_append by simp_all
  4180 qed
  4181 
  4182 lemma sorted_dropWhile: "sorted xs \<Longrightarrow> sorted (dropWhile P xs)"
  4183   by (auto dest: sorted_drop simp add: dropWhile_eq_drop)
  4184 
  4185 lemma sorted_takeWhile: "sorted xs \<Longrightarrow> sorted (takeWhile P xs)"
  4186   by (subst takeWhile_eq_take) (auto dest: sorted_take)
  4187 
  4188 lemma sorted_filter:
  4189   "sorted (map f xs) \<Longrightarrow> sorted (map f (filter P xs))"
  4190   by (induct xs) (simp_all add: sorted_Cons)
  4191 
  4192 lemma foldr_max_sorted:
  4193   assumes "sorted (rev xs)"
  4194   shows "foldr max xs y = (if xs = [] then y else max (xs ! 0) y)"
  4195 using assms proof (induct xs)
  4196   case (Cons x xs)
  4197   moreover hence "sorted (rev xs)" using sorted_append by auto
  4198   ultimately show ?case
  4199     by (cases xs, auto simp add: sorted_append max_def)
  4200 qed simp
  4201 
  4202 lemma filter_equals_takeWhile_sorted_rev:
  4203   assumes sorted: "sorted (rev (map f xs))"
  4204   shows "[x \<leftarrow> xs. t < f x] = takeWhile (\<lambda> x. t < f x) xs"
  4205     (is "filter ?P xs = ?tW")
  4206 proof (rule takeWhile_eq_filter[symmetric])
  4207   let "?dW" = "dropWhile ?P xs"
  4208   fix x assume "x \<in> set ?dW"
  4209   then obtain i where i: "i < length ?dW" and nth_i: "x = ?dW ! i"
  4210     unfolding in_set_conv_nth by auto
  4211   hence "length ?tW + i < length (?tW @ ?dW)"
  4212     unfolding length_append by simp
  4213   hence i': "length (map f ?tW) + i < length (map f xs)" by simp
  4214   have "(map f ?tW @ map f ?dW) ! (length (map f ?tW) + i) \<le>
  4215         (map f ?tW @ map f ?dW) ! (length (map f ?tW) + 0)"
  4216     using sorted_rev_nth_mono[OF sorted _ i', of "length ?tW"]
  4217     unfolding map_append[symmetric] by simp
  4218   hence "f x \<le> f (?dW ! 0)"
  4219     unfolding nth_append_length_plus nth_i
  4220     using i preorder_class.le_less_trans[OF le0 i] by simp
  4221   also have "... \<le> t"
  4222     using hd_dropWhile[of "?P" xs] le0[THEN preorder_class.le_less_trans, OF i]
  4223     using hd_conv_nth[of "?dW"] by simp
  4224   finally show "\<not> t < f x" by simp
  4225 qed
  4226 
  4227 lemma insort_insert_key_triv:
  4228   "f x \<in> f ` set xs \<Longrightarrow> insort_insert_key f x xs = xs"
  4229   by (simp add: insort_insert_key_def)
  4230 
  4231 lemma insort_insert_triv:
  4232   "x \<in> set xs \<Longrightarrow> insort_insert x xs = xs"
  4233   using insort_insert_key_triv [of "\<lambda>x. x"] by simp
  4234 
  4235 lemma insort_insert_insort_key:
  4236   "f x \<notin> f ` set xs \<Longrightarrow> insort_insert_key f x xs = insort_key f x xs"
  4237   by (simp add: insort_insert_key_def)
  4238 
  4239 lemma insort_insert_insort:
  4240   "x \<notin> set xs \<Longrightarrow> insort_insert x xs = insort x xs"
  4241   using insort_insert_insort_key [of "\<lambda>x. x"] by simp
  4242 
  4243 lemma set_insort_insert:
  4244   "set (insort_insert x xs) = insert x (set xs)"
  4245   by (auto simp add: insort_insert_key_def set_insort)
  4246 
  4247 lemma distinct_insort_insert:
  4248   assumes "distinct xs"
  4249   shows "distinct (insort_insert_key f x xs)"
  4250   using assms by (induct xs) (auto simp add: insort_insert_key_def set_insort)
  4251 
  4252 lemma sorted_insort_insert_key:
  4253   assumes "sorted (map f xs)"
  4254   shows "sorted (map f (insort_insert_key f x xs))"
  4255   using assms by (simp add: insort_insert_key_def sorted_insort_key)
  4256 
  4257 lemma sorted_insort_insert:
  4258   assumes "sorted xs"
  4259   shows "sorted (insort_insert x xs)"
  4260   using assms sorted_insort_insert_key [of "\<lambda>x. x"] by simp
  4261 
  4262 lemma filter_insort_triv:
  4263   "\<not> P x \<Longrightarrow> filter P (insort_key f x xs) = filter P xs"
  4264   by (induct xs) simp_all
  4265 
  4266 lemma filter_insort:
  4267   "sorted (map f xs) \<Longrightarrow> P x \<Longrightarrow> filter P (insort_key f x xs) = insort_key f x (filter P xs)"
  4268   using assms by (induct xs)
  4269     (auto simp add: sorted_Cons, subst insort_is_Cons, auto)
  4270 
  4271 lemma filter_sort:
  4272   "filter P (sort_key f xs) = sort_key f (filter P xs)"
  4273   by (induct xs) (simp_all add: filter_insort_triv filter_insort)
  4274 
  4275 lemma sorted_map_same:
  4276   "sorted (map f [x\<leftarrow>xs. f x = g xs])"
  4277 proof (induct xs arbitrary: g)
  4278   case Nil then show ?case by simp
  4279 next
  4280   case (Cons x xs)
  4281   then have "sorted (map f [y\<leftarrow>xs . f y = (\<lambda>xs. f x) xs])" .
  4282   moreover from Cons have "sorted (map f [y\<leftarrow>xs . f y = (g \<circ> Cons x) xs])" .
  4283   ultimately show ?case by (simp_all add: sorted_Cons)
  4284 qed
  4285 
  4286 lemma sorted_same:
  4287   "sorted [x\<leftarrow>xs. x = g xs]"
  4288   using sorted_map_same [of "\<lambda>x. x"] by simp
  4289 
  4290 lemma remove1_insort [simp]:
  4291   "remove1 x (insort x xs) = xs"
  4292   by (induct xs) simp_all
  4293 
  4294 end
  4295 
  4296 lemma sorted_upt[simp]: "sorted[i..<j]"
  4297 by (induct j) (simp_all add:sorted_append)
  4298 
  4299 lemma sorted_upto[simp]: "sorted[i..j]"
  4300 apply(induct i j rule:upto.induct)
  4301 apply(subst upto.simps)
  4302 apply(simp add:sorted_Cons)
  4303 done
  4304 
  4305 
  4306 subsubsection {* @{const transpose} on sorted lists *}
  4307 
  4308 lemma sorted_transpose[simp]:
  4309   shows "sorted (rev (map length (transpose xs)))"
  4310   by (auto simp: sorted_equals_nth_mono rev_nth nth_transpose
  4311     length_filter_conv_card intro: card_mono)
  4312 
  4313 lemma transpose_max_length:
  4314   "foldr (\<lambda>xs. max (length xs)) (transpose xs) 0 = length [x \<leftarrow> xs. x \<noteq> []]"
  4315   (is "?L = ?R")
  4316 proof (cases "transpose xs = []")
  4317   case False
  4318   have "?L = foldr max (map length (transpose xs)) 0"
  4319     by (simp add: foldr_map comp_def)
  4320   also have "... = length (transpose xs ! 0)"
  4321     using False sorted_transpose by (simp add: foldr_max_sorted)
  4322   finally show ?thesis
  4323     using False by (simp add: nth_transpose)
  4324 next
  4325   case True
  4326   hence "[x \<leftarrow> xs. x \<noteq> []] = []"
  4327     by (auto intro!: filter_False simp: transpose_empty)
  4328   thus ?thesis by (simp add: transpose_empty True)
  4329 qed
  4330 
  4331 lemma length_transpose_sorted:
  4332   fixes xs :: "'a list list"
  4333   assumes sorted: "sorted (rev (map length xs))"
  4334   shows "length (transpose xs) = (if xs = [] then 0 else length (xs ! 0))"
  4335 proof (cases "xs = []")
  4336   case False
  4337   thus ?thesis
  4338     using foldr_max_sorted[OF sorted] False
  4339     unfolding length_transpose foldr_map comp_def
  4340     by simp
  4341 qed simp
  4342 
  4343 lemma nth_nth_transpose_sorted[simp]:
  4344   fixes xs :: "'a list list"
  4345   assumes sorted: "sorted (rev (map length xs))"
  4346   and i: "i < length (transpose xs)"
  4347   and j: "j < length [ys \<leftarrow> xs. i < length ys]"
  4348   shows "transpose xs ! i ! j = xs ! j  ! i"
  4349   using j filter_equals_takeWhile_sorted_rev[OF sorted, of i]
  4350     nth_transpose[OF i] nth_map[OF j]
  4351   by (simp add: takeWhile_nth)
  4352 
  4353 lemma transpose_column_length:
  4354   fixes xs :: "'a list list"
  4355   assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  4356   shows "length (filter (\<lambda>ys. i < length ys) (transpose xs)) = length (xs ! i)"
  4357 proof -
  4358   have "xs \<noteq> []" using `i < length xs` by auto
  4359   note filter_equals_takeWhile_sorted_rev[OF sorted, simp]
  4360   { fix j assume "j \<le> i"
  4361     note sorted_rev_nth_mono[OF sorted, of j i, simplified, OF this `i < length xs`]
  4362   } note sortedE = this[consumes 1]
  4363 
  4364   have "{j. j < length (transpose xs) \<and> i < length (transpose xs ! j)}
  4365     = {..< length (xs ! i)}"
  4366   proof safe
  4367     fix j
  4368     assume "j < length (transpose xs)" and "i < length (transpose xs ! j)"
  4369     with this(2) nth_transpose[OF this(1)]
  4370     have "i < length (takeWhile (\<lambda>ys. j < length ys) xs)" by simp
  4371     from nth_mem[OF this] takeWhile_nth[OF this]
  4372     show "j < length (xs ! i)" by (auto dest: set_takeWhileD)
  4373   next
  4374     fix j assume "j < length (xs ! i)"
  4375     thus "j < length (transpose xs)"
  4376       using foldr_max_sorted[OF sorted] `xs \<noteq> []` sortedE[OF le0]
  4377       by (auto simp: length_transpose comp_def foldr_map)
  4378 
  4379     have "Suc i \<le> length (takeWhile (\<lambda>ys. j < length ys) xs)"
  4380       using `i < length xs` `j < length (xs ! i)` less_Suc_eq_le
  4381       by (auto intro!: length_takeWhile_less_P_nth dest!: sortedE)
  4382     with nth_transpose[OF `j < length (transpose xs)`]
  4383     show "i < length (transpose xs ! j)" by simp
  4384   qed
  4385   thus ?thesis by (simp add: length_filter_conv_card)
  4386 qed
  4387 
  4388 lemma transpose_column:
  4389   fixes xs :: "'a list list"
  4390   assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  4391   shows "map (\<lambda>ys. ys ! i) (filter (\<lambda>ys. i < length ys) (transpose xs))
  4392     = xs ! i" (is "?R = _")
  4393 proof (rule nth_equalityI, safe)
  4394   show length: "length ?R = length (xs ! i)"
  4395     using transpose_column_length[OF assms] by simp
  4396 
  4397   fix j assume j: "j < length ?R"
  4398   note * = less_le_trans[OF this, unfolded length_map, OF length_filter_le]
  4399   from j have j_less: "j < length (xs ! i)" using length by simp
  4400   have i_less_tW: "Suc i \<le> length (takeWhile (\<lambda>ys. Suc j \<le> length ys) xs)"
  4401   proof (rule length_takeWhile_less_P_nth)
  4402     show "Suc i \<le> length xs" using `i < length xs` by simp
  4403     fix k assume "k < Suc i"
  4404     hence "k \<le> i" by auto
  4405     with sorted_rev_nth_mono[OF sorted this] `i < length xs`
  4406     have "length (xs ! i) \<le> length (xs ! k)" by simp
  4407     thus "Suc j \<le> length (xs ! k)" using j_less by simp
  4408   qed
  4409   have i_less_filter: "i < length [ys\<leftarrow>xs . j < length ys]"
  4410     unfolding filter_equals_takeWhile_sorted_rev[OF sorted, of j]
  4411     using i_less_tW by (simp_all add: Suc_le_eq)
  4412   from j show "?R ! j = xs ! i ! j"
  4413     unfolding filter_equals_takeWhile_sorted_rev[OF sorted_transpose, of i]
  4414     by (simp add: takeWhile_nth nth_nth_transpose_sorted[OF sorted * i_less_filter])
  4415 qed
  4416 
  4417 lemma transpose_transpose:
  4418   fixes xs :: "'a list list"
  4419   assumes sorted: "sorted (rev (map length xs))"
  4420   shows "transpose (transpose xs) = takeWhile (\<lambda>x. x \<noteq> []) xs" (is "?L = ?R")
  4421 proof -
  4422   have len: "length ?L = length ?R"
  4423     unfolding length_transpose transpose_max_length
  4424     using filter_equals_takeWhile_sorted_rev[OF sorted, of 0]
  4425     by simp
  4426 
  4427   { fix i assume "i < length ?R"
  4428     with less_le_trans[OF _ length_takeWhile_le[of _ xs]]
  4429     have "i < length xs" by simp
  4430   } note * = this
  4431   show ?thesis
  4432     by (rule nth_equalityI)
  4433        (simp_all add: len nth_transpose transpose_column[OF sorted] * takeWhile_nth)
  4434 qed
  4435 
  4436 theorem transpose_rectangle:
  4437   assumes "xs = [] \<Longrightarrow> n = 0"
  4438   assumes rect: "\<And> i. i < length xs \<Longrightarrow> length (xs ! i) = n"
  4439   shows "transpose xs = map (\<lambda> i. map (\<lambda> j. xs ! j ! i) [0..<length xs]) [0..<n]"
  4440     (is "?trans = ?map")
  4441 proof (rule nth_equalityI)
  4442   have "sorted (rev (map length xs))"
  4443     by (auto simp: rev_nth rect intro!: sorted_nth_monoI)
  4444   from foldr_max_sorted[OF this] assms
  4445   show len: "length ?trans = length ?map"
  4446     by (simp_all add: length_transpose foldr_map comp_def)
  4447   moreover
  4448   { fix i assume "i < n" hence "[ys\<leftarrow>xs . i < length ys] = xs"
  4449       using rect by (auto simp: in_set_conv_nth intro!: filter_True) }
  4450   ultimately show "\<forall>i < length ?trans. ?trans ! i = ?map ! i"
  4451     by (auto simp: nth_transpose intro: nth_equalityI)
  4452 qed
  4453 
  4454 
  4455 subsubsection {* @{text sorted_list_of_set} *}
  4456 
  4457 text{* This function maps (finite) linearly ordered sets to sorted
  4458 lists. Warning: in most cases it is not a good idea to convert from
  4459 sets to lists but one should convert in the other direction (via
  4460 @{const set}). *}
  4461 
  4462 context linorder
  4463 begin
  4464 
  4465 definition sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where
  4466   "sorted_list_of_set = Finite_Set.fold insort []"
  4467 
  4468 lemma sorted_list_of_set_empty [simp]:
  4469   "sorted_list_of_set {} = []"
  4470   by (simp add: sorted_list_of_set_def)
  4471 
  4472 lemma sorted_list_of_set_insert [simp]:
  4473   assumes "finite A"
  4474   shows "sorted_list_of_set (insert x A) = insort x (sorted_list_of_set (A - {x}))"
  4475 proof -
  4476   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  4477   from assms show ?thesis
  4478     by (simp add: sorted_list_of_set_def fold_insert_remove)
  4479 qed
  4480 
  4481 lemma sorted_list_of_set [simp]:
  4482   "finite A \<Longrightarrow> set (sorted_list_of_set A) = A \<and> sorted (sorted_list_of_set A) 
  4483     \<and> distinct (sorted_list_of_set A)"
  4484   by (induct A rule: finite_induct) (simp_all add: set_insort sorted_insort distinct_insort)
  4485 
  4486 lemma sorted_list_of_set_sort_remdups [code]:
  4487   "sorted_list_of_set (set xs) = sort (remdups xs)"
  4488 proof -
  4489   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  4490   show ?thesis by (simp add: sorted_list_of_set_def sort_conv_fold fold_set_fold_remdups)
  4491 qed
  4492 
  4493 lemma sorted_list_of_set_remove:
  4494   assumes "finite A"
  4495   shows "sorted_list_of_set (A - {x}) = remove1 x (sorted_list_of_set A)"
  4496 proof (cases "x \<in> A")
  4497   case False with assms have "x \<notin> set (sorted_list_of_set A)" by simp
  4498   with False show ?thesis by (simp add: remove1_idem)
  4499 next
  4500   case True then obtain B where A: "A = insert x B" by (rule Set.set_insert)
  4501   with assms show ?thesis by simp
  4502 qed
  4503 
  4504 end
  4505 
  4506 lemma sorted_list_of_set_range [simp]:
  4507   "sorted_list_of_set {m..<n} = [m..<n]"
  4508   by (rule sorted_distinct_set_unique) simp_all
  4509 
  4510 
  4511 subsubsection {* @{text lists}: the list-forming operator over sets *}
  4512 
  4513 inductive_set
  4514   lists :: "'a set => 'a list set"
  4515   for A :: "'a set"
  4516 where
  4517     Nil [intro!, simp]: "[]: lists A"
  4518   | Cons [intro!, simp, no_atp]: "[| a: A; l: lists A|] ==> a#l : lists A"
  4519 
  4520 inductive_cases listsE [elim!,no_atp]: "x#l : lists A"
  4521 inductive_cases listspE [elim!,no_atp]: "listsp A (x # l)"
  4522 
  4523 inductive_simps listsp_simps[code]:
  4524   "listsp A []"
  4525   "listsp A (x # xs)"
  4526 
  4527 lemma listsp_mono [mono]: "A \<le> B ==> listsp A \<le> listsp B"
  4528 by (rule predicate1I, erule listsp.induct, blast+)
  4529 
  4530 lemmas lists_mono = listsp_mono [to_set]
  4531 
  4532 lemma listsp_infI:
  4533   assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
  4534 by induct blast+
  4535 
  4536 lemmas lists_IntI = listsp_infI [to_set]
  4537 
  4538 lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
  4539 proof (rule mono_inf [where f=listsp, THEN order_antisym])
  4540   show "mono listsp" by (simp add: mono_def listsp_mono)
  4541   show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro!: listsp_infI)
  4542 qed
  4543 
  4544 lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_def inf_bool_def]
  4545 
  4546 lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set]
  4547 
  4548 lemma Cons_in_lists_iff[simp]: "x#xs : lists A \<longleftrightarrow> x:A \<and> xs : lists A"
  4549 by auto
  4550 
  4551 lemma append_in_listsp_conv [iff]:
  4552      "(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
  4553 by (induct xs) auto
  4554 
  4555 lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]
  4556 
  4557 lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
  4558 -- {* eliminate @{text listsp} in favour of @{text set} *}
  4559 by (induct xs) auto
  4560 
  4561 lemmas in_lists_conv_set [code_unfold] = in_listsp_conv_set [to_set]
  4562 
  4563 lemma in_listspD [dest!,no_atp]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
  4564 by (rule in_listsp_conv_set [THEN iffD1])
  4565 
  4566 lemmas in_listsD [dest!,no_atp] = in_listspD [to_set]
  4567 
  4568 lemma in_listspI [intro!,no_atp]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
  4569 by (rule in_listsp_conv_set [THEN iffD2])
  4570 
  4571 lemmas in_listsI [intro!,no_atp] = in_listspI [to_set]
  4572 
  4573 lemma lists_eq_set: "lists A = {xs. set xs <= A}"
  4574 by auto
  4575 
  4576 lemma lists_empty [simp]: "lists {} = {[]}"
  4577 by auto
  4578 
  4579 lemma lists_UNIV [simp]: "lists UNIV = UNIV"
  4580 by auto
  4581 
  4582 
  4583 subsubsection {* Inductive definition for membership *}
  4584 
  4585 inductive ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
  4586 where
  4587     elem:  "ListMem x (x # xs)"
  4588   | insert:  "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"
  4589 
  4590 lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
  4591 apply (rule iffI)
  4592  apply (induct set: ListMem)
  4593   apply auto
  4594 apply (induct xs)
  4595  apply (auto intro: ListMem.intros)
  4596 done
  4597 
  4598 
  4599 subsubsection {* Lists as Cartesian products *}
  4600 
  4601 text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
  4602 @{term A} and tail drawn from @{term Xs}.*}
  4603 
  4604 definition
  4605   set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set" where
  4606   "set_Cons A XS = {z. \<exists>x xs. z = x # xs \<and> x \<in> A \<and> xs \<in> XS}"
  4607 
  4608 lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
  4609 by (auto simp add: set_Cons_def)
  4610 
  4611 text{*Yields the set of lists, all of the same length as the argument and
  4612 with elements drawn from the corresponding element of the argument.*}
  4613 
  4614 primrec
  4615   listset :: "'a set list \<Rightarrow> 'a list set" where
  4616      "listset [] = {[]}"
  4617   |  "listset (A # As) = set_Cons A (listset As)"
  4618 
  4619 
  4620 subsection {* Relations on Lists *}
  4621 
  4622 subsubsection {* Length Lexicographic Ordering *}
  4623 
  4624 text{*These orderings preserve well-foundedness: shorter lists 
  4625   precede longer lists. These ordering are not used in dictionaries.*}
  4626         
  4627 primrec -- {*The lexicographic ordering for lists of the specified length*}
  4628   lexn :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> ('a list \<times> 'a list) set" where
  4629     "lexn r 0 = {}"
  4630   | "lexn r (Suc n) = (map_pair (%(x, xs). x#xs) (%(x, xs). x#xs) ` (r <*lex*> lexn r n)) Int
  4631       {(xs, ys). length xs = Suc n \<and> length ys = Suc n}"
  4632 
  4633 definition
  4634   lex :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4635   "lex r = (\<Union>n. lexn r n)" -- {*Holds only between lists of the same length*}
  4636 
  4637 definition
  4638   lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" where
  4639   "lenlex r = inv_image (less_than <*lex*> lex r) (\<lambda>xs. (length xs, xs))"
  4640         -- {*Compares lists by their length and then lexicographically*}
  4641 
  4642 lemma wf_lexn: "wf r ==> wf (lexn r n)"
  4643 apply (induct n, simp, simp)
  4644 apply(rule wf_subset)
  4645  prefer 2 apply (rule Int_lower1)
  4646 apply(rule wf_map_pair_image)
  4647  prefer 2 apply (rule inj_onI, auto)
  4648 done
  4649 
  4650 lemma lexn_length:
  4651   "(xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
  4652 by (induct n arbitrary: xs ys) auto
  4653 
  4654 lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
  4655 apply (unfold lex_def)
  4656 apply (rule wf_UN)
  4657 apply (blast intro: wf_lexn, clarify)
  4658 apply (rename_tac m n)
  4659 apply (subgoal_tac "m \<noteq> n")
  4660  prefer 2 apply blast
  4661 apply (blast dest: lexn_length not_sym)
  4662 done
  4663 
  4664 lemma lexn_conv:
  4665   "lexn r n =
  4666     {(xs,ys). length xs = n \<and> length ys = n \<and>
  4667     (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
  4668 apply (induct n, simp)
  4669 apply (simp add: image_Collect lex_prod_def, safe, blast)
  4670  apply (rule_tac x = "ab # xys" in exI, simp)
  4671 apply (case_tac xys, simp_all, blast)
  4672 done
  4673 
  4674 lemma lex_conv:
  4675   "lex r =
  4676     {(xs,ys). length xs = length ys \<and>
  4677     (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
  4678 by (force simp add: lex_def lexn_conv)
  4679 
  4680 lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
  4681 by (unfold lenlex_def) blast
  4682 
  4683 lemma lenlex_conv:
  4684     "lenlex r = {(xs,ys). length xs < length ys |
  4685                  length xs = length ys \<and> (xs, ys) : lex r}"
  4686 by (simp add: lenlex_def Id_on_def lex_prod_def inv_image_def)
  4687 
  4688 lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
  4689 by (simp add: lex_conv)
  4690 
  4691 lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
  4692 by (simp add:lex_conv)
  4693 
  4694 lemma Cons_in_lex [simp]:
  4695     "((x # xs, y # ys) : lex r) =
  4696       ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
  4697 apply (simp add: lex_conv)
  4698 apply (rule iffI)
  4699  prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
  4700 apply (case_tac xys, simp, simp)
  4701 apply blast
  4702 done
  4703 
  4704 
  4705 subsubsection {* Lexicographic Ordering *}
  4706 
  4707 text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
  4708     This ordering does \emph{not} preserve well-foundedness.
  4709      Author: N. Voelker, March 2005. *} 
  4710 
  4711 definition
  4712   lexord :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4713   "lexord r = {(x,y ). \<exists> a v. y = x @ a # v \<or>
  4714             (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
  4715 
  4716 lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
  4717 by (unfold lexord_def, induct_tac y, auto) 
  4718 
  4719 lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
  4720 by (unfold lexord_def, induct_tac x, auto)
  4721 
  4722 lemma lexord_cons_cons[simp]:
  4723      "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
  4724   apply (unfold lexord_def, safe, simp_all)
  4725   apply (case_tac u, simp, simp)
  4726   apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
  4727   apply (erule_tac x="b # u" in allE)
  4728   by force
  4729 
  4730 lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
  4731 
  4732 lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
  4733 by (induct_tac x, auto)  
  4734 
  4735 lemma lexord_append_left_rightI:
  4736      "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
  4737 by (induct_tac u, auto)
  4738 
  4739 lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
  4740 by (induct x, auto)
  4741 
  4742 lemma lexord_append_leftD:
  4743      "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
  4744 by (erule rev_mp, induct_tac x, auto)
  4745 
  4746 lemma lexord_take_index_conv: 
  4747    "((x,y) : lexord r) = 
  4748     ((length x < length y \<and> take (length x) y = x) \<or> 
  4749      (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
  4750   apply (unfold lexord_def Let_def, clarsimp) 
  4751   apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
  4752   apply auto 
  4753   apply (rule_tac x="hd (drop (length x) y)" in exI)
  4754   apply (rule_tac x="tl (drop (length x) y)" in exI)
  4755   apply (erule subst, simp add: min_def) 
  4756   apply (rule_tac x ="length u" in exI, simp) 
  4757   apply (rule_tac x ="take i x" in exI) 
  4758   apply (rule_tac x ="x ! i" in exI) 
  4759   apply (rule_tac x ="y ! i" in exI, safe) 
  4760   apply (rule_tac x="drop (Suc i) x" in exI)
  4761   apply (drule sym, simp add: drop_Suc_conv_tl) 
  4762   apply (rule_tac x="drop (Suc i) y" in exI)
  4763   by (simp add: drop_Suc_conv_tl) 
  4764 
  4765 -- {* lexord is extension of partial ordering List.lex *} 
  4766 lemma lexord_lex: "(x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
  4767   apply (rule_tac x = y in spec) 
  4768   apply (induct_tac x, clarsimp) 
  4769   by (clarify, case_tac x, simp, force)
  4770 
  4771 lemma lexord_irreflexive: "ALL x. (x,x) \<notin> r \<Longrightarrow> (xs,xs) \<notin> lexord r"
  4772 by (induct xs) auto
  4773 
  4774 text{* By Ren\'e Thiemann: *}
  4775 lemma lexord_partial_trans: 
  4776   "(\<And>x y z. x \<in> set xs \<Longrightarrow> (x,y) \<in> r \<Longrightarrow> (y,z) \<in> r \<Longrightarrow> (x,z) \<in> r)
  4777    \<Longrightarrow>  (xs,ys) \<in> lexord r  \<Longrightarrow>  (ys,zs) \<in> lexord r \<Longrightarrow>  (xs,zs) \<in> lexord r"
  4778 proof (induct xs arbitrary: ys zs)
  4779   case Nil
  4780   from Nil(3) show ?case unfolding lexord_def by (cases zs, auto)
  4781 next
  4782   case (Cons x xs yys zzs)
  4783   from Cons(3) obtain y ys where yys: "yys = y # ys" unfolding lexord_def
  4784     by (cases yys, auto)
  4785   note Cons = Cons[unfolded yys]
  4786   from Cons(3) have one: "(x,y) \<in> r \<or> x = y \<and> (xs,ys) \<in> lexord r" by auto
  4787   from Cons(4) obtain z zs where zzs: "zzs = z # zs" unfolding lexord_def
  4788     by (cases zzs, auto)
  4789   note Cons = Cons[unfolded zzs]
  4790   from Cons(4) have two: "(y,z) \<in> r \<or> y = z \<and> (ys,zs) \<in> lexord r" by auto
  4791   {
  4792     assume "(xs,ys) \<in> lexord r" and "(ys,zs) \<in> lexord r"
  4793     from Cons(1)[OF _ this] Cons(2)
  4794     have "(xs,zs) \<in> lexord r" by auto
  4795   } note ind1 = this
  4796   {
  4797     assume "(x,y) \<in> r" and "(y,z) \<in> r"
  4798     from Cons(2)[OF _ this] have "(x,z) \<in> r" by auto
  4799   } note ind2 = this
  4800   from one two ind1 ind2
  4801   have "(x,z) \<in> r \<or> x = z \<and> (xs,zs) \<in> lexord r" by blast
  4802   thus ?case unfolding zzs by auto
  4803 qed
  4804 
  4805 lemma lexord_trans: 
  4806     "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
  4807 by(auto simp: trans_def intro:lexord_partial_trans)
  4808 
  4809 lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
  4810 by (rule transI, drule lexord_trans, blast) 
  4811 
  4812 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"
  4813   apply (rule_tac x = y in spec) 
  4814   apply (induct_tac x, rule allI) 
  4815   apply (case_tac x, simp, simp) 
  4816   apply (rule allI, case_tac x, simp, simp) 
  4817   by blast
  4818 
  4819 
  4820 subsubsection {* Lexicographic combination of measure functions *}
  4821 
  4822 text {* These are useful for termination proofs *}
  4823 
  4824 definition
  4825   "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"
  4826 
  4827 lemma wf_measures[simp]: "wf (measures fs)"
  4828 unfolding measures_def
  4829 by blast
  4830 
  4831 lemma in_measures[simp]: 
  4832   "(x, y) \<in> measures [] = False"
  4833   "(x, y) \<in> measures (f # fs)
  4834          = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"  
  4835 unfolding measures_def
  4836 by auto
  4837 
  4838 lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
  4839 by simp
  4840 
  4841 lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
  4842 by auto
  4843 
  4844 
  4845 subsubsection {* Lifting Relations to Lists: one element *}
  4846 
  4847 definition listrel1 :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4848 "listrel1 r = {(xs,ys).
  4849    \<exists>us z z' vs. xs = us @ z # vs \<and> (z,z') \<in> r \<and> ys = us @ z' # vs}"
  4850 
  4851 lemma listrel1I:
  4852   "\<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow>
  4853   (xs, ys) \<in> listrel1 r"
  4854 unfolding listrel1_def by auto
  4855 
  4856 lemma listrel1E:
  4857   "\<lbrakk> (xs, ys) \<in> listrel1 r;
  4858      !!x y us vs. \<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow> P
  4859    \<rbrakk> \<Longrightarrow> P"
  4860 unfolding listrel1_def by auto
  4861 
  4862 lemma not_Nil_listrel1 [iff]: "([], xs) \<notin> listrel1 r"
  4863 unfolding listrel1_def by blast
  4864 
  4865 lemma not_listrel1_Nil [iff]: "(xs, []) \<notin> listrel1 r"
  4866 unfolding listrel1_def by blast
  4867 
  4868 lemma Cons_listrel1_Cons [iff]:
  4869   "(x # xs, y # ys) \<in> listrel1 r \<longleftrightarrow>
  4870    (x,y) \<in> r \<and> xs = ys \<or> x = y \<and> (xs, ys) \<in> listrel1 r"
  4871 by (simp add: listrel1_def Cons_eq_append_conv) (blast)
  4872 
  4873 lemma listrel1I1: "(x,y) \<in> r \<Longrightarrow> (x # xs, y # xs) \<in> listrel1 r"
  4874 by (metis Cons_listrel1_Cons)
  4875 
  4876 lemma listrel1I2: "(xs, ys) \<in> listrel1 r \<Longrightarrow> (x # xs, x # ys) \<in> listrel1 r"
  4877 by (metis Cons_listrel1_Cons)
  4878 
  4879 lemma append_listrel1I:
  4880   "(xs, ys) \<in> listrel1 r \<and> us = vs \<or> xs = ys \<and> (us, vs) \<in> listrel1 r
  4881     \<Longrightarrow> (xs @ us, ys @ vs) \<in> listrel1 r"
  4882 unfolding listrel1_def
  4883 by auto (blast intro: append_eq_appendI)+
  4884 
  4885 lemma Cons_listrel1E1[elim!]:
  4886   assumes "(x # xs, ys) \<in> listrel1 r"
  4887     and "\<And>y. ys = y # xs \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
  4888     and "\<And>zs. ys = x # zs \<Longrightarrow> (xs, zs) \<in> listrel1 r \<Longrightarrow> R"
  4889   shows R
  4890 using assms by (cases ys) blast+
  4891 
  4892 lemma Cons_listrel1E2[elim!]:
  4893   assumes "(xs, y # ys) \<in> listrel1 r"
  4894     and "\<And>x. xs = x # ys \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
  4895     and "\<And>zs. xs = y # zs \<Longrightarrow> (zs, ys) \<in> listrel1 r \<Longrightarrow> R"
  4896   shows R
  4897 using assms by (cases xs) blast+
  4898 
  4899 lemma snoc_listrel1_snoc_iff:
  4900   "(xs @ [x], ys @ [y]) \<in> listrel1 r
  4901     \<longleftrightarrow> (xs, ys) \<in> listrel1 r \<and> x = y \<or> xs = ys \<and> (x,y) \<in> r" (is "?L \<longleftrightarrow> ?R")
  4902 proof
  4903   assume ?L thus ?R
  4904     by (fastforce simp: listrel1_def snoc_eq_iff_butlast butlast_append)
  4905 next
  4906   assume ?R then show ?L unfolding listrel1_def by force
  4907 qed
  4908 
  4909 lemma listrel1_eq_len: "(xs,ys) \<in> listrel1 r \<Longrightarrow> length xs = length ys"
  4910 unfolding listrel1_def by auto
  4911 
  4912 lemma listrel1_mono:
  4913   "r \<subseteq> s \<Longrightarrow> listrel1 r \<subseteq> listrel1 s"
  4914 unfolding listrel1_def by blast
  4915 
  4916 
  4917 lemma listrel1_converse: "listrel1 (r^-1) = (listrel1 r)^-1"
  4918 unfolding listrel1_def by blast
  4919 
  4920 lemma in_listrel1_converse:
  4921   "(x,y) : listrel1 (r^-1) \<longleftrightarrow> (x,y) : (listrel1 r)^-1"
  4922 unfolding listrel1_def by blast
  4923 
  4924 lemma listrel1_iff_update:
  4925   "(xs,ys) \<in> (listrel1 r)
  4926    \<longleftrightarrow> (\<exists>y n. (xs ! n, y) \<in> r \<and> n < length xs \<and> ys = xs[n:=y])" (is "?L \<longleftrightarrow> ?R")
  4927 proof
  4928   assume "?L"
  4929   then obtain x y u v where "xs = u @ x # v"  "ys = u @ y # v"  "(x,y) \<in> r"
  4930     unfolding listrel1_def by auto
  4931   then have "ys = xs[length u := y]" and "length u < length xs"
  4932     and "(xs ! length u, y) \<in> r" by auto
  4933   then show "?R" by auto
  4934 next
  4935   assume "?R"
  4936   then obtain x y n where "(xs!n, y) \<in> r" "n < size xs" "ys = xs[n:=y]" "x = xs!n"
  4937     by auto
  4938   then obtain u v where "xs = u @ x # v" and "ys = u @ y # v" and "(x, y) \<in> r"
  4939     by (auto intro: upd_conv_take_nth_drop id_take_nth_drop)
  4940   then show "?L" by (auto simp: listrel1_def)
  4941 qed
  4942 
  4943 
  4944 text{* Accessible part and wellfoundedness: *}
  4945 
  4946 lemma Cons_acc_listrel1I [intro!]:
  4947   "x \<in> acc r \<Longrightarrow> xs \<in> acc (listrel1 r) \<Longrightarrow> (x # xs) \<in> acc (listrel1 r)"
  4948 apply (induct arbitrary: xs set: acc)
  4949 apply (erule thin_rl)
  4950 apply (erule acc_induct)
  4951 apply (rule accI)
  4952 apply (blast)
  4953 done
  4954 
  4955 lemma lists_accD: "xs \<in> lists (acc r) \<Longrightarrow> xs \<in> acc (listrel1 r)"
  4956 apply (induct set: lists)
  4957  apply (rule accI)
  4958  apply simp
  4959 apply (rule accI)
  4960 apply (fast dest: acc_downward)
  4961 done
  4962 
  4963 lemma lists_accI: "xs \<in> acc (listrel1 r) \<Longrightarrow> xs \<in> lists (acc r)"
  4964 apply (induct set: acc)
  4965 apply clarify
  4966 apply (rule accI)
  4967 apply (fastforce dest!: in_set_conv_decomp[THEN iffD1] simp: listrel1_def)
  4968 done
  4969 
  4970 lemma wf_listrel1_iff[simp]: "wf(listrel1 r) = wf r"
  4971 by(metis wf_acc_iff in_lists_conv_set lists_accI lists_accD Cons_in_lists_iff)
  4972 
  4973 
  4974 subsubsection {* Lifting Relations to Lists: all elements *}
  4975 
  4976 inductive_set
  4977   listrel :: "('a \<times> 'b) set \<Rightarrow> ('a list \<times> 'b list) set"
  4978   for r :: "('a \<times> 'b) set"
  4979 where
  4980     Nil:  "([],[]) \<in> listrel r"
  4981   | Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
  4982 
  4983 inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
  4984 inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
  4985 inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
  4986 inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
  4987 
  4988 
  4989 lemma listrel_eq_len:  "(xs, ys) \<in> listrel r \<Longrightarrow> length xs = length ys"
  4990 by(induct rule: listrel.induct) auto
  4991 
  4992 lemma listrel_iff_zip [code_unfold]: "(xs,ys) : listrel r \<longleftrightarrow>
  4993   length xs = length ys & (\<forall>(x,y) \<in> set(zip xs ys). (x,y) \<in> r)" (is "?L \<longleftrightarrow> ?R")
  4994 proof
  4995   assume ?L thus ?R by induct (auto intro: listrel_eq_len)
  4996 next
  4997   assume ?R thus ?L
  4998     apply (clarify)
  4999     by (induct rule: list_induct2) (auto intro: listrel.intros)
  5000 qed
  5001 
  5002 lemma listrel_iff_nth: "(xs,ys) : listrel r \<longleftrightarrow>
  5003   length xs = length ys & (\<forall>n < length xs. (xs!n, ys!n) \<in> r)" (is "?L \<longleftrightarrow> ?R")
  5004 by (auto simp add: all_set_conv_all_nth listrel_iff_zip)
  5005 
  5006 
  5007 lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
  5008 apply clarify  
  5009 apply (erule listrel.induct)
  5010 apply (blast intro: listrel.intros)+
  5011 done
  5012 
  5013 lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
  5014 apply clarify 
  5015 apply (erule listrel.induct, auto) 
  5016 done
  5017 
  5018 lemma listrel_refl_on: "refl_on A r \<Longrightarrow> refl_on (lists A) (listrel r)" 
  5019 apply (simp add: refl_on_def listrel_subset Ball_def)
  5020 apply (rule allI) 
  5021 apply (induct_tac x) 
  5022 apply (auto intro: listrel.intros)
  5023 done
  5024 
  5025 lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
  5026 apply (auto simp add: sym_def)
  5027 apply (erule listrel.induct) 
  5028 apply (blast intro: listrel.intros)+
  5029 done
  5030 
  5031 lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
  5032 apply (simp add: trans_def)
  5033 apply (intro allI) 
  5034 apply (rule impI) 
  5035 apply (erule listrel.induct) 
  5036 apply (blast intro: listrel.intros)+
  5037 done
  5038 
  5039 theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
  5040 by (simp add: equiv_def listrel_refl_on listrel_sym listrel_trans) 
  5041 
  5042 lemma listrel_rtrancl_refl[iff]: "(xs,xs) : listrel(r^*)"
  5043 using listrel_refl_on[of UNIV, OF refl_rtrancl]
  5044 by(auto simp: refl_on_def)
  5045 
  5046 lemma listrel_rtrancl_trans:
  5047   "\<lbrakk> (xs,ys) : listrel(r^*);  (ys,zs) : listrel(r^*) \<rbrakk>
  5048   \<Longrightarrow> (xs,zs) : listrel(r^*)"
  5049 by (metis listrel_trans trans_def trans_rtrancl)
  5050 
  5051 
  5052 lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
  5053 by (blast intro: listrel.intros)
  5054 
  5055 lemma listrel_Cons:
  5056      "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})"
  5057 by (auto simp add: set_Cons_def intro: listrel.intros)
  5058 
  5059 text {* Relating @{term listrel1}, @{term listrel} and closures: *}
  5060 
  5061 lemma listrel1_rtrancl_subset_rtrancl_listrel1:
  5062   "listrel1 (r^*) \<subseteq> (listrel1 r)^*"
  5063 proof (rule subrelI)
  5064   fix xs ys assume 1: "(xs,ys) \<in> listrel1 (r^*)"
  5065   { fix x y us vs
  5066     have "(x,y) : r^* \<Longrightarrow> (us @ x # vs, us @ y # vs) : (listrel1 r)^*"
  5067     proof(induct rule: rtrancl.induct)
  5068       case rtrancl_refl show ?case by simp
  5069     next
  5070       case rtrancl_into_rtrancl thus ?case
  5071         by (metis listrel1I rtrancl.rtrancl_into_rtrancl)
  5072     qed }
  5073   thus "(xs,ys) \<in> (listrel1 r)^*" using 1 by(blast elim: listrel1E)
  5074 qed
  5075 
  5076 lemma rtrancl_listrel1_eq_len: "(x,y) \<in> (listrel1 r)^* \<Longrightarrow> length x = length y"
  5077 by (induct rule: rtrancl.induct) (auto intro: listrel1_eq_len)
  5078 
  5079 lemma rtrancl_listrel1_ConsI1:
  5080   "(xs,ys) : (listrel1 r)^* \<Longrightarrow> (x#xs,x#ys) : (listrel1 r)^*"
  5081 apply(induct rule: rtrancl.induct)
  5082  apply simp
  5083 by (metis listrel1I2 rtrancl.rtrancl_into_rtrancl)
  5084 
  5085 lemma rtrancl_listrel1_ConsI2:
  5086   "(x,y) \<in> r^* \<Longrightarrow> (xs, ys) \<in> (listrel1 r)^*
  5087   \<Longrightarrow> (x # xs, y # ys) \<in> (listrel1 r)^*"
  5088   by (blast intro: rtrancl_trans rtrancl_listrel1_ConsI1 
  5089     subsetD[OF listrel1_rtrancl_subset_rtrancl_listrel1 listrel1I1])
  5090 
  5091 lemma listrel1_subset_listrel:
  5092   "r \<subseteq> r' \<Longrightarrow> refl r' \<Longrightarrow> listrel1 r \<subseteq> listrel(r')"
  5093 by(auto elim!: listrel1E simp add: listrel_iff_zip set_zip refl_on_def)
  5094 
  5095 lemma listrel_reflcl_if_listrel1:
  5096   "(xs,ys) : listrel1 r \<Longrightarrow> (xs,ys) : listrel(r^*)"
  5097 by(erule listrel1E)(auto simp add: listrel_iff_zip set_zip)
  5098 
  5099 lemma listrel_rtrancl_eq_rtrancl_listrel1: "listrel (r^*) = (listrel1 r)^*"
  5100 proof
  5101   { fix x y assume "(x,y) \<in> listrel (r^*)"
  5102     then have "(x,y) \<in> (listrel1 r)^*"
  5103     by induct (auto intro: rtrancl_listrel1_ConsI2) }
  5104   then show "listrel (r^*) \<subseteq> (listrel1 r)^*"
  5105     by (rule subrelI)
  5106 next
  5107   show "listrel (r^*) \<supseteq> (listrel1 r)^*"
  5108   proof(rule subrelI)
  5109     fix xs ys assume "(xs,ys) \<in> (listrel1 r)^*"
  5110     then show "(xs,ys) \<in> listrel (r^*)"
  5111     proof induct
  5112       case base show ?case by(auto simp add: listrel_iff_zip set_zip)
  5113     next
  5114       case (step ys zs)
  5115       thus ?case  by (metis listrel_reflcl_if_listrel1 listrel_rtrancl_trans)
  5116     qed
  5117   qed
  5118 qed
  5119 
  5120 lemma rtrancl_listrel1_if_listrel:
  5121   "(xs,ys) : listrel r \<Longrightarrow> (xs,ys) : (listrel1 r)^*"
  5122 by(metis listrel_rtrancl_eq_rtrancl_listrel1 subsetD[OF listrel_mono] r_into_rtrancl subsetI)
  5123 
  5124 lemma listrel_subset_rtrancl_listrel1: "listrel r \<subseteq> (listrel1 r)^*"
  5125 by(fast intro:rtrancl_listrel1_if_listrel)
  5126 
  5127 
  5128 subsection {* Size function *}
  5129 
  5130 lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (list_size f)"
  5131 by (rule is_measure_trivial)
  5132 
  5133 lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (option_size f)"
  5134 by (rule is_measure_trivial)
  5135 
  5136 lemma list_size_estimation[termination_simp]: 
  5137   "x \<in> set xs \<Longrightarrow> y < f x \<Longrightarrow> y < list_size f xs"
  5138 by (induct xs) auto
  5139 
  5140 lemma list_size_estimation'[termination_simp]: 
  5141   "x \<in> set xs \<Longrightarrow> y \<le> f x \<Longrightarrow> y \<le> list_size f xs"
  5142 by (induct xs) auto
  5143 
  5144 lemma list_size_map[simp]: "list_size f (map g xs) = list_size (f o g) xs"
  5145 by (induct xs) auto
  5146 
  5147 lemma list_size_append[simp]: "list_size f (xs @ ys) = list_size f xs + list_size f ys"
  5148 by (induct xs, auto)
  5149 
  5150 lemma list_size_pointwise[termination_simp]: 
  5151   "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> list_size f xs \<le> list_size g xs"
  5152 by (induct xs) force+
  5153 
  5154 
  5155 subsection {* Monad operation *}
  5156 
  5157 definition bind :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b list) \<Rightarrow> 'b list" where
  5158   "bind xs f = concat (map f xs)"
  5159 
  5160 hide_const (open) bind
  5161 
  5162 lemma bind_simps [simp]:
  5163   "List.bind [] f = []"
  5164   "List.bind (x # xs) f = f x @ List.bind xs f"
  5165   by (simp_all add: bind_def)
  5166 
  5167 
  5168 subsection {* Transfer *}
  5169 
  5170 definition
  5171   embed_list :: "nat list \<Rightarrow> int list"
  5172 where
  5173   "embed_list l = map int l"
  5174 
  5175 definition
  5176   nat_list :: "int list \<Rightarrow> bool"
  5177 where
  5178   "nat_list l = nat_set (set l)"
  5179 
  5180 definition
  5181   return_list :: "int list \<Rightarrow> nat list"
  5182 where
  5183   "return_list l = map nat l"
  5184 
  5185 lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow>
  5186     embed_list (return_list l) = l"
  5187   unfolding embed_list_def return_list_def nat_list_def nat_set_def
  5188   apply (induct l)
  5189   apply auto
  5190 done
  5191 
  5192 lemma transfer_nat_int_list_functions:
  5193   "l @ m = return_list (embed_list l @ embed_list m)"
  5194   "[] = return_list []"
  5195   unfolding return_list_def embed_list_def
  5196   apply auto
  5197   apply (induct l, auto)
  5198   apply (induct m, auto)
  5199 done
  5200 
  5201 (*
  5202 lemma transfer_nat_int_fold1: "fold f l x =
  5203     fold (%x. f (nat x)) (embed_list l) x";
  5204 *)
  5205 
  5206 
  5207 subsection {* Code generation *}
  5208 
  5209 subsubsection {* Counterparts for set-related operations *}
  5210 
  5211 definition member :: "'a list \<Rightarrow> 'a \<Rightarrow> bool" where
  5212   [code_abbrev]: "member xs x \<longleftrightarrow> x \<in> set xs"
  5213 
  5214 text {*
  5215   Use @{text member} only for generating executable code.  Otherwise use
  5216   @{prop "x \<in> set xs"} instead --- it is much easier to reason about.
  5217 *}
  5218 
  5219 lemma member_rec [code]:
  5220   "member (x # xs) y \<longleftrightarrow> x = y \<or> member xs y"
  5221   "member [] y \<longleftrightarrow> False"
  5222   by (auto simp add: member_def)
  5223 
  5224 lemma in_set_member (* FIXME delete candidate *):
  5225   "x \<in> set xs \<longleftrightarrow> member xs x"
  5226   by (simp add: member_def)
  5227 
  5228 definition list_all :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5229   list_all_iff [code_abbrev]: "list_all P xs \<longleftrightarrow> Ball (set xs) P"
  5230 
  5231 definition list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5232   list_ex_iff [code_abbrev]: "list_ex P xs \<longleftrightarrow> Bex (set xs) P"
  5233 
  5234 definition list_ex1 :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5235   list_ex1_iff [code_abbrev]: "list_ex1 P xs \<longleftrightarrow> (\<exists>! x. x \<in> set xs \<and> P x)"
  5236 
  5237 text {*
  5238   Usually you should prefer @{text "\<forall>x\<in>set xs"}, @{text "\<exists>x\<in>set xs"}
  5239   and @{text "\<exists>!x. x\<in>set xs \<and> _"} over @{const list_all}, @{const list_ex}
  5240   and @{const list_ex1} in specifications.
  5241 *}
  5242 
  5243 lemma list_all_simps [simp, code]:
  5244   "list_all P (x # xs) \<longleftrightarrow> P x \<and> list_all P xs"
  5245   "list_all P [] \<longleftrightarrow> True"
  5246   by (simp_all add: list_all_iff)
  5247 
  5248 lemma list_ex_simps [simp, code]:
  5249   "list_ex P (x # xs) \<longleftrightarrow> P x \<or> list_ex P xs"
  5250   "list_ex P [] \<longleftrightarrow> False"
  5251   by (simp_all add: list_ex_iff)
  5252 
  5253 lemma list_ex1_simps [simp, code]:
  5254   "list_ex1 P [] = False"
  5255   "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)"
  5256   by (auto simp add: list_ex1_iff list_all_iff)
  5257 
  5258 lemma Ball_set_list_all: (* FIXME delete candidate *)
  5259   "Ball (set xs) P \<longleftrightarrow> list_all P xs"
  5260   by (simp add: list_all_iff)
  5261 
  5262 lemma Bex_set_list_ex: (* FIXME delete candidate *)
  5263   "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
  5264   by (simp add: list_ex_iff)
  5265 
  5266 lemma list_all_append [simp]:
  5267   "list_all P (xs @ ys) \<longleftrightarrow> list_all P xs \<and> list_all P ys"
  5268   by (auto simp add: list_all_iff)
  5269 
  5270 lemma list_ex_append [simp]:
  5271   "list_ex P (xs @ ys) \<longleftrightarrow> list_ex P xs \<or> list_ex P ys"
  5272   by (auto simp add: list_ex_iff)
  5273 
  5274 lemma list_all_rev [simp]:
  5275   "list_all P (rev xs) \<longleftrightarrow> list_all P xs"
  5276   by (simp add: list_all_iff)
  5277 
  5278 lemma list_ex_rev [simp]:
  5279   "list_ex P (rev xs) \<longleftrightarrow> list_ex P xs"
  5280   by (simp add: list_ex_iff)
  5281 
  5282 lemma list_all_length:
  5283   "list_all P xs \<longleftrightarrow> (\<forall>n < length xs. P (xs ! n))"
  5284   by (auto simp add: list_all_iff set_conv_nth)
  5285 
  5286 lemma list_ex_length:
  5287   "list_ex P xs \<longleftrightarrow> (\<exists>n < length xs. P (xs ! n))"
  5288   by (auto simp add: list_ex_iff set_conv_nth)
  5289 
  5290 lemma list_all_cong [fundef_cong]:
  5291   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_all f xs = list_all g ys"
  5292   by (simp add: list_all_iff)
  5293 
  5294 lemma list_ex_cong [fundef_cong]:
  5295   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_ex f xs = list_ex g ys"
  5296 by (simp add: list_ex_iff)
  5297 
  5298 text {* Executable checks for relations on sets *}
  5299 
  5300 definition listrel1p :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
  5301 "listrel1p r xs ys = ((xs, ys) \<in> listrel1 {(x, y). r x y})"
  5302 
  5303 lemma [code_unfold]:
  5304   "(xs, ys) \<in> listrel1 r = listrel1p (\<lambda>x y. (x, y) \<in> r) xs ys"
  5305 unfolding listrel1p_def by auto
  5306 
  5307 lemma [code]:
  5308   "listrel1p r [] xs = False"
  5309   "listrel1p r xs [] =  False"
  5310   "listrel1p r (x # xs) (y # ys) \<longleftrightarrow>
  5311      r x y \<and> xs = ys \<or> x = y \<and> listrel1p r xs ys"
  5312 by (simp add: listrel1p_def)+
  5313 
  5314 definition
  5315   lexordp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
  5316   "lexordp r xs ys = ((xs, ys) \<in> lexord {(x, y). r x y})"
  5317 
  5318 lemma [code_unfold]:
  5319   "(xs, ys) \<in> lexord r = lexordp (\<lambda>x y. (x, y) \<in> r) xs ys"
  5320 unfolding lexordp_def by auto
  5321 
  5322 lemma [code]:
  5323   "lexordp r xs [] = False"
  5324   "lexordp r [] (y#ys) = True"
  5325   "lexordp r (x # xs) (y # ys) = (r x y | (x = y & lexordp r xs ys))"
  5326 unfolding lexordp_def by auto
  5327 
  5328 text {* Bounded quantification and summation over nats. *}
  5329 
  5330 lemma atMost_upto [code_unfold]:
  5331   "{..n} = set [0..<Suc n]"
  5332   by auto
  5333 
  5334 lemma atLeast_upt [code_unfold]:
  5335   "{..<n} = set [0..<n]"
  5336   by auto
  5337 
  5338 lemma greaterThanLessThan_upt [code_unfold]:
  5339   "{n<..<m} = set [Suc n..<m]"
  5340   by auto
  5341 
  5342 lemmas atLeastLessThan_upt [code_unfold] = set_upt [symmetric]
  5343 
  5344 lemma greaterThanAtMost_upt [code_unfold]:
  5345   "{n<..m} = set [Suc n..<Suc m]"
  5346   by auto
  5347 
  5348 lemma atLeastAtMost_upt [code_unfold]:
  5349   "{n..m} = set [n..<Suc m]"
  5350   by auto
  5351 
  5352 lemma all_nat_less_eq [code_unfold]:
  5353   "(\<forall>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..<n}. P m)"
  5354   by auto
  5355 
  5356 lemma ex_nat_less_eq [code_unfold]:
  5357   "(\<exists>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..<n}. P m)"
  5358   by auto
  5359 
  5360 lemma all_nat_less [code_unfold]:
  5361   "(\<forall>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..n}. P m)"
  5362   by auto
  5363 
  5364 lemma ex_nat_less [code_unfold]:
  5365   "(\<exists>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..n}. P m)"
  5366   by auto
  5367 
  5368 lemma setsum_set_upt_conv_listsum_nat [code_unfold]:
  5369   "setsum f (set [m..<n]) = listsum (map f [m..<n])"
  5370   by (simp add: interv_listsum_conv_setsum_set_nat)
  5371 
  5372 text {* Summation over ints. *}
  5373 
  5374 lemma greaterThanLessThan_upto [code_unfold]:
  5375   "{i<..<j::int} = set [i+1..j - 1]"
  5376 by auto
  5377 
  5378 lemma atLeastLessThan_upto [code_unfold]:
  5379   "{i..<j::int} = set [i..j - 1]"
  5380 by auto
  5381 
  5382 lemma greaterThanAtMost_upto [code_unfold]:
  5383   "{i<..j::int} = set [i+1..j]"
  5384 by auto
  5385 
  5386 lemmas atLeastAtMost_upto [code_unfold] = set_upto [symmetric]
  5387 
  5388 lemma setsum_set_upto_conv_listsum_int [code_unfold]:
  5389   "setsum f (set [i..j::int]) = listsum (map f [i..j])"
  5390   by (simp add: interv_listsum_conv_setsum_set_int)
  5391 
  5392 
  5393 subsubsection {* Optimizing by rewriting *}
  5394 
  5395 definition null :: "'a list \<Rightarrow> bool" where
  5396   [code_abbrev]: "null xs \<longleftrightarrow> xs = []"
  5397 
  5398 text {*
  5399   Efficient emptyness check is implemented by @{const null}.
  5400 *}
  5401 
  5402 lemma null_rec [code]:
  5403   "null (x # xs) \<longleftrightarrow> False"
  5404   "null [] \<longleftrightarrow> True"
  5405   by (simp_all add: null_def)
  5406 
  5407 lemma eq_Nil_null: (* FIXME delete candidate *)
  5408   "xs = [] \<longleftrightarrow> null xs"
  5409   by (simp add: null_def)
  5410 
  5411 lemma equal_Nil_null [code_unfold]:
  5412   "HOL.equal xs [] \<longleftrightarrow> null xs"
  5413   by (simp add: equal eq_Nil_null)
  5414 
  5415 definition maps :: "('a \<Rightarrow> 'b list) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  5416   [code_abbrev]: "maps f xs = concat (map f xs)"
  5417 
  5418 definition map_filter :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  5419   [code_post]: "map_filter f xs = map (the \<circ> f) (filter (\<lambda>x. f x \<noteq> None) xs)"
  5420 
  5421 text {*
  5422   Operations @{const maps} and @{const map_filter} avoid
  5423   intermediate lists on execution -- do not use for proving.
  5424 *}
  5425 
  5426 lemma maps_simps [code]:
  5427   "maps f (x # xs) = f x @ maps f xs"
  5428   "maps f [] = []"
  5429   by (simp_all add: maps_def)
  5430 
  5431 lemma map_filter_simps [code]:
  5432   "map_filter f (x # xs) = (case f x of None \<Rightarrow> map_filter f xs | Some y \<Rightarrow> y # map_filter f xs)"
  5433   "map_filter f [] = []"
  5434   by (simp_all add: map_filter_def split: option.split)
  5435 
  5436 lemma concat_map_maps: (* FIXME delete candidate *)
  5437   "concat (map f xs) = maps f xs"
  5438   by (simp add: maps_def)
  5439 
  5440 lemma map_filter_map_filter [code_unfold]:
  5441   "map f (filter P xs) = map_filter (\<lambda>x. if P x then Some (f x) else None) xs"
  5442   by (simp add: map_filter_def)
  5443 
  5444 text {* Optimized code for @{text"\<forall>i\<in>{a..b::int}"} and @{text"\<forall>n:{a..<b::nat}"}
  5445 and similiarly for @{text"\<exists>"}. *}
  5446 
  5447 definition all_interval_nat :: "(nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
  5448   "all_interval_nat P i j \<longleftrightarrow> (\<forall>n \<in> {i..<j}. P n)"
  5449 
  5450 lemma [code]:
  5451   "all_interval_nat P i j \<longleftrightarrow> i \<ge> j \<or> P i \<and> all_interval_nat P (Suc i) j"
  5452 proof -
  5453   have *: "\<And>n. P i \<Longrightarrow> \<forall>n\<in>{Suc i..<j}. P n \<Longrightarrow> i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n"
  5454   proof -
  5455     fix n
  5456     assume "P i" "\<forall>n\<in>{Suc i..<j}. P n" "i \<le> n" "n < j"
  5457     then show "P n" by (cases "n = i") simp_all
  5458   qed
  5459   show ?thesis by (auto simp add: all_interval_nat_def intro: *)
  5460 qed
  5461 
  5462 lemma list_all_iff_all_interval_nat [code_unfold]:
  5463   "list_all P [i..<j] \<longleftrightarrow> all_interval_nat P i j"
  5464   by (simp add: list_all_iff all_interval_nat_def)
  5465 
  5466 lemma list_ex_iff_not_all_inverval_nat [code_unfold]:
  5467   "list_ex P [i..<j] \<longleftrightarrow> \<not> (all_interval_nat (Not \<circ> P) i j)"
  5468   by (simp add: list_ex_iff all_interval_nat_def)
  5469 
  5470 definition all_interval_int :: "(int \<Rightarrow> bool) \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool" where
  5471   "all_interval_int P i j \<longleftrightarrow> (\<forall>k \<in> {i..j}. P k)"
  5472 
  5473 lemma [code]:
  5474   "all_interval_int P i j \<longleftrightarrow> i > j \<or> P i \<and> all_interval_int P (i + 1) j"
  5475 proof -
  5476   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"
  5477   proof -
  5478     fix k
  5479     assume "P i" "\<forall>k\<in>{i+1..j}. P k" "i \<le> k" "k \<le> j"
  5480     then show "P k" by (cases "k = i") simp_all
  5481   qed
  5482   show ?thesis by (auto simp add: all_interval_int_def intro: *)
  5483 qed
  5484 
  5485 lemma list_all_iff_all_interval_int [code_unfold]:
  5486   "list_all P [i..j] \<longleftrightarrow> all_interval_int P i j"
  5487   by (simp add: list_all_iff all_interval_int_def)
  5488 
  5489 lemma list_ex_iff_not_all_inverval_int [code_unfold]:
  5490   "list_ex P [i..j] \<longleftrightarrow> \<not> (all_interval_int (Not \<circ> P) i j)"
  5491   by (simp add: list_ex_iff all_interval_int_def)
  5492 
  5493 hide_const (open) member null maps map_filter all_interval_nat all_interval_int
  5494 
  5495 
  5496 subsubsection {* Pretty lists *}
  5497 
  5498 use "Tools/list_code.ML"
  5499 
  5500 code_type list
  5501   (SML "_ list")
  5502   (OCaml "_ list")
  5503   (Haskell "![(_)]")
  5504   (Scala "List[(_)]")
  5505 
  5506 code_const Nil
  5507   (SML "[]")
  5508   (OCaml "[]")
  5509   (Haskell "[]")
  5510   (Scala "!Nil")
  5511 
  5512 code_instance list :: equal
  5513   (Haskell -)
  5514 
  5515 code_const "HOL.equal \<Colon> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
  5516   (Haskell infix 4 "==")
  5517 
  5518 code_reserved SML
  5519   list
  5520 
  5521 code_reserved OCaml
  5522   list
  5523 
  5524 setup {* fold (List_Code.add_literal_list) ["SML", "OCaml", "Haskell", "Scala"] *}
  5525 
  5526 
  5527 subsubsection {* Use convenient predefined operations *}
  5528 
  5529 code_const "op @"
  5530   (SML infixr 7 "@")
  5531   (OCaml infixr 6 "@")
  5532   (Haskell infixr 5 "++")
  5533   (Scala infixl 7 "++")
  5534 
  5535 code_const map
  5536   (Haskell "map")
  5537 
  5538 code_const filter
  5539   (Haskell "filter")
  5540 
  5541 code_const concat
  5542   (Haskell "concat")
  5543 
  5544 code_const List.maps
  5545   (Haskell "concatMap")
  5546 
  5547 code_const rev
  5548   (Haskell "reverse")
  5549 
  5550 code_const zip
  5551   (Haskell "zip")
  5552 
  5553 code_const List.null
  5554   (Haskell "null")
  5555 
  5556 code_const takeWhile
  5557   (Haskell "takeWhile")
  5558 
  5559 code_const dropWhile
  5560   (Haskell "dropWhile")
  5561 
  5562 code_const list_all
  5563   (Haskell "all")
  5564 
  5565 code_const list_ex
  5566   (Haskell "any")
  5567 
  5568 
  5569 subsubsection {* Implementation of sets by lists *}
  5570 
  5571 lemma is_empty_set [code]:
  5572   "Set.is_empty (set xs) \<longleftrightarrow> List.null xs"
  5573   by (simp add: Set.is_empty_def null_def)
  5574 
  5575 lemma empty_set [code]:
  5576   "{} = set []"
  5577   by simp
  5578 
  5579 lemma UNIV_coset [code]:
  5580   "UNIV = List.coset []"
  5581   by simp
  5582 
  5583 lemma compl_set [code]:
  5584   "- set xs = List.coset xs"
  5585   by simp
  5586 
  5587 lemma compl_coset [code]:
  5588   "- List.coset xs = set xs"
  5589   by simp
  5590 
  5591 lemma [code]:
  5592   "x \<in> set xs \<longleftrightarrow> List.member xs x"
  5593   "x \<in> List.coset xs \<longleftrightarrow> \<not> List.member xs x"
  5594   by (simp_all add: member_def)
  5595 
  5596 lemma insert_code [code]:
  5597   "insert x (set xs) = set (List.insert x xs)"
  5598   "insert x (List.coset xs) = List.coset (removeAll x xs)"
  5599   by simp_all
  5600 
  5601 lemma remove_code [code]:
  5602   "Set.remove x (set xs) = set (removeAll x xs)"
  5603   "Set.remove x (List.coset xs) = List.coset (List.insert x xs)"
  5604   by (simp_all add: remove_def Compl_insert)
  5605 
  5606 lemma project_set [code]:
  5607   "Set.project P (set xs) = set (filter P xs)"
  5608   by auto
  5609 
  5610 lemma image_set [code]:
  5611   "image f (set xs) = set (map f xs)"
  5612   by simp
  5613 
  5614 lemma subset_code [code]:
  5615   "set xs \<le> B \<longleftrightarrow> (\<forall>x\<in>set xs. x \<in> B)"
  5616   "A \<le> List.coset ys \<longleftrightarrow> (\<forall>y\<in>set ys. y \<notin> A)"
  5617   "List.coset [] \<le> set [] \<longleftrightarrow> False"
  5618   by auto
  5619 
  5620 text {* A frequent case – avoid intermediate sets *}
  5621 lemma [code_unfold]:
  5622   "set xs \<subseteq> set ys \<longleftrightarrow> list_all (\<lambda>x. x \<in> set ys) xs"
  5623   by (auto simp: list_all_iff)
  5624 
  5625 lemma Ball_set [code]:
  5626   "Ball (set xs) P \<longleftrightarrow> list_all P xs"
  5627   by (simp add: list_all_iff)
  5628 
  5629 lemma Bex_set [code]:
  5630   "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
  5631   by (simp add: list_ex_iff)
  5632 
  5633 lemma card_set [code]:
  5634   "card (set xs) = length (remdups xs)"
  5635 proof -
  5636   have "card (set (remdups xs)) = length (remdups xs)"
  5637     by (rule distinct_card) simp
  5638   then show ?thesis by simp
  5639 qed
  5640 
  5641 lemma the_elem_set [code]:
  5642   "the_elem (set [x]) = x"
  5643   by simp
  5644 
  5645 lemma Pow_set [code]:
  5646   "Pow (set []) = {{}}"
  5647   "Pow (set (x # xs)) = (let A = Pow (set xs) in A \<union> insert x ` A)"
  5648   by (simp_all add: Pow_insert Let_def)
  5649 
  5650 lemma setsum_code [code]:
  5651   "setsum f (set xs) = listsum (map f (remdups xs))"
  5652 by (simp add: listsum_distinct_conv_setsum_set)
  5653 
  5654 definition map_project :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a set \<Rightarrow> 'b set" where
  5655   "map_project f A = {b. \<exists> a \<in> A. f a = Some b}"
  5656 
  5657 lemma [code]:
  5658   "map_project f (set xs) = set (List.map_filter f xs)"
  5659   by (auto simp add: map_project_def map_filter_def image_def)
  5660 
  5661 hide_const (open) map_project
  5662 
  5663 text {* Operations on relations *}
  5664 
  5665 lemma product_code [code]:
  5666   "Product_Type.product (set xs) (set ys) = set [(x, y). x \<leftarrow> xs, y \<leftarrow> ys]"
  5667   by (auto simp add: Product_Type.product_def)
  5668 
  5669 lemma Id_on_set [code]:
  5670   "Id_on (set xs) = set [(x, x). x \<leftarrow> xs]"
  5671   by (auto simp add: Id_on_def)
  5672 
  5673 lemma [code]:
  5674   "R `` S = List.map_project (%(x, y). if x : S then Some y else None) R"
  5675 unfolding map_project_def by (auto split: prod.split split_if_asm)
  5676 
  5677 lemma trancl_set_ntrancl [code]:
  5678   "trancl (set xs) = ntrancl (card (set xs) - 1) (set xs)"
  5679   by (simp add: finite_trancl_ntranl)
  5680 
  5681 lemma set_relcomp [code]:
  5682   "set xys O set yzs = set ([(fst xy, snd yz). xy \<leftarrow> xys, yz \<leftarrow> yzs, snd xy = fst yz])"
  5683   by (auto simp add: Bex_def)
  5684 
  5685 lemma wf_set [code]:
  5686   "wf (set xs) = acyclic (set xs)"
  5687   by (simp add: wf_iff_acyclic_if_finite)
  5688 
  5689 end
  5690