src/HOL/List.thy
author haftmann
Thu Aug 16 14:07:32 2012 +0200 (2012-08-16)
changeset 48828 441a4eed7823
parent 48619 558e4e77ce69
child 48891 c0eafbd55de3
permissions -rw-r--r--
prefer eta-expanded code equations for fold, to accomodate tail recursion optimisation in Scala
     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_simps [code]: -- {* eta-expanded variant for generated code -- enables tail-recursion optimisation in Scala *}
  2396   "fold f [] s = s"
  2397   "fold f (x # xs) s = fold f xs (f x s)" 
  2398   by simp_all
  2399 
  2400 lemma fold_remove1_split:
  2401   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"
  2402     and x: "x \<in> set xs"
  2403   shows "fold f xs = fold f (remove1 x xs) \<circ> f x"
  2404   using assms by (induct xs) (auto simp add: o_assoc [symmetric])
  2405 
  2406 lemma fold_cong [fundef_cong]:
  2407   "a = b \<Longrightarrow> xs = ys \<Longrightarrow> (\<And>x. x \<in> set xs \<Longrightarrow> f x = g x)
  2408     \<Longrightarrow> fold f xs a = fold g ys b"
  2409   by (induct ys arbitrary: a b xs) simp_all
  2410 
  2411 lemma fold_id:
  2412   assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x = id"
  2413   shows "fold f xs = id"
  2414   using assms by (induct xs) simp_all
  2415 
  2416 lemma fold_commute:
  2417   assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
  2418   shows "h \<circ> fold g xs = fold f xs \<circ> h"
  2419   using assms by (induct xs) (simp_all add: fun_eq_iff)
  2420 
  2421 lemma fold_commute_apply:
  2422   assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
  2423   shows "h (fold g xs s) = fold f xs (h s)"
  2424 proof -
  2425   from assms have "h \<circ> fold g xs = fold f xs \<circ> h" by (rule fold_commute)
  2426   then show ?thesis by (simp add: fun_eq_iff)
  2427 qed
  2428 
  2429 lemma fold_invariant: 
  2430   assumes "\<And>x. x \<in> set xs \<Longrightarrow> Q x" and "P s"
  2431     and "\<And>x s. Q x \<Longrightarrow> P s \<Longrightarrow> P (f x s)"
  2432   shows "P (fold f xs s)"
  2433   using assms by (induct xs arbitrary: s) simp_all
  2434 
  2435 lemma fold_append [simp]:
  2436   "fold f (xs @ ys) = fold f ys \<circ> fold f xs"
  2437   by (induct xs) simp_all
  2438 
  2439 lemma fold_map [code_unfold]:
  2440   "fold g (map f xs) = fold (g o f) xs"
  2441   by (induct xs) simp_all
  2442 
  2443 lemma fold_rev:
  2444   assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
  2445   shows "fold f (rev xs) = fold f xs"
  2446 using assms by (induct xs) (simp_all add: fold_commute_apply fun_eq_iff)
  2447 
  2448 lemma fold_Cons_rev:
  2449   "fold Cons xs = append (rev xs)"
  2450   by (induct xs) simp_all
  2451 
  2452 lemma rev_conv_fold [code]:
  2453   "rev xs = fold Cons xs []"
  2454   by (simp add: fold_Cons_rev)
  2455 
  2456 lemma fold_append_concat_rev:
  2457   "fold append xss = append (concat (rev xss))"
  2458   by (induct xss) simp_all
  2459 
  2460 text {* @{const Finite_Set.fold} and @{const fold} *}
  2461 
  2462 lemma (in comp_fun_commute) fold_set_fold_remdups:
  2463   "Finite_Set.fold f y (set xs) = fold f (remdups xs) y"
  2464   by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm insert_absorb)
  2465 
  2466 lemma (in ab_semigroup_mult) fold1_distinct_set_fold:
  2467   assumes "xs \<noteq> []"
  2468   assumes d: "distinct xs"
  2469   shows "Finite_Set.fold1 times (set xs) = List.fold times (tl xs) (hd xs)"
  2470 proof -
  2471   interpret comp_fun_commute times by (fact comp_fun_commute)
  2472   from assms obtain y ys where xs: "xs = y # ys"
  2473     by (cases xs) auto
  2474   then have *: "y \<notin> set ys" using assms by simp
  2475   from xs d have **: "remdups ys = ys"  by safe (induct ys, auto)
  2476   show ?thesis
  2477   proof (cases "set ys = {}")
  2478     case True with xs show ?thesis by simp
  2479   next
  2480     case False
  2481     then have "fold1 times (Set.insert y (set ys)) = Finite_Set.fold times y (set ys)"
  2482       by (simp_all add: fold1_eq_fold *)
  2483     with xs show ?thesis
  2484       by (simp add: fold_set_fold_remdups **)
  2485   qed
  2486 qed
  2487 
  2488 lemma (in comp_fun_idem) fold_set_fold:
  2489   "Finite_Set.fold f y (set xs) = fold f xs y"
  2490   by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm)
  2491 
  2492 lemma (in ab_semigroup_idem_mult) fold1_set_fold:
  2493   assumes "xs \<noteq> []"
  2494   shows "Finite_Set.fold1 times (set xs) = fold times (tl xs) (hd xs)"
  2495 proof -
  2496   interpret comp_fun_idem times by (fact comp_fun_idem)
  2497   from assms obtain y ys where xs: "xs = y # ys"
  2498     by (cases xs) auto
  2499   show ?thesis
  2500   proof (cases "set ys = {}")
  2501     case True with xs show ?thesis by simp
  2502   next
  2503     case False
  2504     then have "fold1 times (insert y (set ys)) = Finite_Set.fold times y (set ys)"
  2505       by (simp only: finite_set fold1_eq_fold_idem)
  2506     with xs show ?thesis by (simp add: fold_set_fold mult_commute)
  2507   qed
  2508 qed
  2509 
  2510 lemma union_set_fold [code]:
  2511   "set xs \<union> A = fold Set.insert xs A"
  2512 proof -
  2513   interpret comp_fun_idem Set.insert
  2514     by (fact comp_fun_idem_insert)
  2515   show ?thesis by (simp add: union_fold_insert fold_set_fold)
  2516 qed
  2517 
  2518 lemma union_coset_filter [code]:
  2519   "List.coset xs \<union> A = List.coset (List.filter (\<lambda>x. x \<notin> A) xs)"
  2520   by auto
  2521 
  2522 lemma minus_set_fold [code]:
  2523   "A - set xs = fold Set.remove xs A"
  2524 proof -
  2525   interpret comp_fun_idem Set.remove
  2526     by (fact comp_fun_idem_remove)
  2527   show ?thesis
  2528     by (simp add: minus_fold_remove [of _ A] fold_set_fold)
  2529 qed
  2530 
  2531 lemma minus_coset_filter [code]:
  2532   "A - List.coset xs = set (List.filter (\<lambda>x. x \<in> A) xs)"
  2533   by auto
  2534 
  2535 lemma inter_set_filter [code]:
  2536   "A \<inter> set xs = set (List.filter (\<lambda>x. x \<in> A) xs)"
  2537   by auto
  2538 
  2539 lemma inter_coset_fold [code]:
  2540   "A \<inter> List.coset xs = fold Set.remove xs A"
  2541   by (simp add: Diff_eq [symmetric] minus_set_fold)
  2542 
  2543 lemma (in lattice) Inf_fin_set_fold:
  2544   "Inf_fin (set (x # xs)) = fold inf xs x"
  2545 proof -
  2546   interpret ab_semigroup_idem_mult "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2547     by (fact ab_semigroup_idem_mult_inf)
  2548   show ?thesis
  2549     by (simp add: Inf_fin_def fold1_set_fold del: set.simps)
  2550 qed
  2551 
  2552 declare Inf_fin_set_fold [code]
  2553 
  2554 lemma (in lattice) Sup_fin_set_fold:
  2555   "Sup_fin (set (x # xs)) = fold sup xs x"
  2556 proof -
  2557   interpret ab_semigroup_idem_mult "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2558     by (fact ab_semigroup_idem_mult_sup)
  2559   show ?thesis
  2560     by (simp add: Sup_fin_def fold1_set_fold del: set.simps)
  2561 qed
  2562 
  2563 declare Sup_fin_set_fold [code]
  2564 
  2565 lemma (in linorder) Min_fin_set_fold:
  2566   "Min (set (x # xs)) = fold min xs x"
  2567 proof -
  2568   interpret ab_semigroup_idem_mult "min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2569     by (fact ab_semigroup_idem_mult_min)
  2570   show ?thesis
  2571     by (simp add: Min_def fold1_set_fold del: set.simps)
  2572 qed
  2573 
  2574 declare Min_fin_set_fold [code]
  2575 
  2576 lemma (in linorder) Max_fin_set_fold:
  2577   "Max (set (x # xs)) = fold max xs x"
  2578 proof -
  2579   interpret ab_semigroup_idem_mult "max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2580     by (fact ab_semigroup_idem_mult_max)
  2581   show ?thesis
  2582     by (simp add: Max_def fold1_set_fold del: set.simps)
  2583 qed
  2584 
  2585 declare Max_fin_set_fold [code]
  2586 
  2587 lemma (in complete_lattice) Inf_set_fold:
  2588   "Inf (set xs) = fold inf xs top"
  2589 proof -
  2590   interpret comp_fun_idem "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2591     by (fact comp_fun_idem_inf)
  2592   show ?thesis by (simp add: Inf_fold_inf fold_set_fold inf_commute)
  2593 qed
  2594 
  2595 declare Inf_set_fold [where 'a = "'a set", code]
  2596 
  2597 lemma (in complete_lattice) Sup_set_fold:
  2598   "Sup (set xs) = fold sup xs bot"
  2599 proof -
  2600   interpret comp_fun_idem "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2601     by (fact comp_fun_idem_sup)
  2602   show ?thesis by (simp add: Sup_fold_sup fold_set_fold sup_commute)
  2603 qed
  2604 
  2605 declare Sup_set_fold [where 'a = "'a set", code]
  2606 
  2607 lemma (in complete_lattice) INF_set_fold:
  2608   "INFI (set xs) f = fold (inf \<circ> f) xs top"
  2609   unfolding INF_def set_map [symmetric] Inf_set_fold fold_map ..
  2610 
  2611 declare INF_set_fold [code]
  2612 
  2613 lemma (in complete_lattice) SUP_set_fold:
  2614   "SUPR (set xs) f = fold (sup \<circ> f) xs bot"
  2615   unfolding SUP_def set_map [symmetric] Sup_set_fold fold_map ..
  2616 
  2617 declare SUP_set_fold [code]
  2618 
  2619 subsubsection {* Fold variants: @{const foldr} and @{const foldl} *}
  2620 
  2621 text {* Correspondence *}
  2622 
  2623 lemma foldr_conv_fold [code_abbrev]:
  2624   "foldr f xs = fold f (rev xs)"
  2625   by (induct xs) simp_all
  2626 
  2627 lemma foldl_conv_fold:
  2628   "foldl f s xs = fold (\<lambda>x s. f s x) xs s"
  2629   by (induct xs arbitrary: s) simp_all
  2630 
  2631 lemma foldr_conv_foldl: -- {* The ``Third Duality Theorem'' in Bird \& Wadler: *}
  2632   "foldr f xs a = foldl (\<lambda>x y. f y x) a (rev xs)"
  2633   by (simp add: foldr_conv_fold foldl_conv_fold)
  2634 
  2635 lemma foldl_conv_foldr:
  2636   "foldl f a xs = foldr (\<lambda>x y. f y x) (rev xs) a"
  2637   by (simp add: foldr_conv_fold foldl_conv_fold)
  2638 
  2639 lemma foldr_fold:
  2640   assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
  2641   shows "foldr f xs = fold f xs"
  2642   using assms unfolding foldr_conv_fold by (rule fold_rev)
  2643 
  2644 lemma foldr_cong [fundef_cong]:
  2645   "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"
  2646   by (auto simp add: foldr_conv_fold intro!: fold_cong)
  2647 
  2648 lemma foldl_cong [fundef_cong]:
  2649   "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"
  2650   by (auto simp add: foldl_conv_fold intro!: fold_cong)
  2651 
  2652 lemma foldr_append [simp]:
  2653   "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
  2654   by (simp add: foldr_conv_fold)
  2655 
  2656 lemma foldl_append [simp]:
  2657   "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
  2658   by (simp add: foldl_conv_fold)
  2659 
  2660 lemma foldr_map [code_unfold]:
  2661   "foldr g (map f xs) a = foldr (g o f) xs a"
  2662   by (simp add: foldr_conv_fold fold_map rev_map)
  2663 
  2664 lemma foldl_map [code_unfold]:
  2665   "foldl g a (map f xs) = foldl (\<lambda>a x. g a (f x)) a xs"
  2666   by (simp add: foldl_conv_fold fold_map comp_def)
  2667 
  2668 lemma concat_conv_foldr [code]:
  2669   "concat xss = foldr append xss []"
  2670   by (simp add: fold_append_concat_rev foldr_conv_fold)
  2671 
  2672 
  2673 subsubsection {* @{text upt} *}
  2674 
  2675 lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
  2676 -- {* simp does not terminate! *}
  2677 by (induct j) auto
  2678 
  2679 lemmas upt_rec_numeral[simp] = upt_rec[of "numeral m" "numeral n"] for m n
  2680 
  2681 lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
  2682 by (subst upt_rec) simp
  2683 
  2684 lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
  2685 by(induct j)simp_all
  2686 
  2687 lemma upt_eq_Cons_conv:
  2688  "([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
  2689 apply(induct j arbitrary: x xs)
  2690  apply simp
  2691 apply(clarsimp simp add: append_eq_Cons_conv)
  2692 apply arith
  2693 done
  2694 
  2695 lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
  2696 -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
  2697 by simp
  2698 
  2699 lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
  2700   by (simp add: upt_rec)
  2701 
  2702 lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
  2703 -- {* LOOPS as a simprule, since @{text "j <= j"}. *}
  2704 by (induct k) auto
  2705 
  2706 lemma length_upt [simp]: "length [i..<j] = j - i"
  2707 by (induct j) (auto simp add: Suc_diff_le)
  2708 
  2709 lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
  2710 apply (induct j)
  2711 apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
  2712 done
  2713 
  2714 
  2715 lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
  2716 by(simp add:upt_conv_Cons)
  2717 
  2718 lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
  2719 apply(cases j)
  2720  apply simp
  2721 by(simp add:upt_Suc_append)
  2722 
  2723 lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]"
  2724 apply (induct m arbitrary: i, simp)
  2725 apply (subst upt_rec)
  2726 apply (rule sym)
  2727 apply (subst upt_rec)
  2728 apply (simp del: upt.simps)
  2729 done
  2730 
  2731 lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
  2732 apply(induct j)
  2733 apply auto
  2734 done
  2735 
  2736 lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]"
  2737 by (induct n) auto
  2738 
  2739 lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
  2740 apply (induct n m  arbitrary: i rule: diff_induct)
  2741 prefer 3 apply (subst map_Suc_upt[symmetric])
  2742 apply (auto simp add: less_diff_conv)
  2743 done
  2744 
  2745 lemma nth_take_lemma:
  2746   "k <= length xs ==> k <= length ys ==>
  2747      (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
  2748 apply (atomize, induct k arbitrary: xs ys)
  2749 apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
  2750 txt {* Both lists must be non-empty *}
  2751 apply (case_tac xs, simp)
  2752 apply (case_tac ys, clarify)
  2753  apply (simp (no_asm_use))
  2754 apply clarify
  2755 txt {* prenexing's needed, not miniscoping *}
  2756 apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
  2757 apply blast
  2758 done
  2759 
  2760 lemma nth_equalityI:
  2761  "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
  2762   by (frule nth_take_lemma [OF le_refl eq_imp_le]) simp_all
  2763 
  2764 lemma map_nth:
  2765   "map (\<lambda>i. xs ! i) [0..<length xs] = xs"
  2766   by (rule nth_equalityI, auto)
  2767 
  2768 (* needs nth_equalityI *)
  2769 lemma list_all2_antisym:
  2770   "\<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> 
  2771   \<Longrightarrow> xs = ys"
  2772   apply (simp add: list_all2_conv_all_nth) 
  2773   apply (rule nth_equalityI, blast, simp)
  2774   done
  2775 
  2776 lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
  2777 -- {* The famous take-lemma. *}
  2778 apply (drule_tac x = "max (length xs) (length ys)" in spec)
  2779 apply (simp add: le_max_iff_disj)
  2780 done
  2781 
  2782 
  2783 lemma take_Cons':
  2784      "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
  2785 by (cases n) simp_all
  2786 
  2787 lemma drop_Cons':
  2788      "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
  2789 by (cases n) simp_all
  2790 
  2791 lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
  2792 by (cases n) simp_all
  2793 
  2794 lemma take_Cons_numeral [simp]:
  2795   "take (numeral v) (x # xs) = x # take (numeral v - 1) xs"
  2796 by (simp add: take_Cons')
  2797 
  2798 lemma drop_Cons_numeral [simp]:
  2799   "drop (numeral v) (x # xs) = drop (numeral v - 1) xs"
  2800 by (simp add: drop_Cons')
  2801 
  2802 lemma nth_Cons_numeral [simp]:
  2803   "(x # xs) ! numeral v = xs ! (numeral v - 1)"
  2804 by (simp add: nth_Cons')
  2805 
  2806 
  2807 subsubsection {* @{text upto}: interval-list on @{typ int} *}
  2808 
  2809 (* FIXME make upto tail recursive? *)
  2810 
  2811 function upto :: "int \<Rightarrow> int \<Rightarrow> int list" ("(1[_../_])") where
  2812 "upto i j = (if i \<le> j then i # [i+1..j] else [])"
  2813 by auto
  2814 termination
  2815 by(relation "measure(%(i::int,j). nat(j - i + 1))") auto
  2816 
  2817 declare upto.simps[code, simp del]
  2818 
  2819 lemmas upto_rec_numeral [simp] =
  2820   upto.simps[of "numeral m" "numeral n"]
  2821   upto.simps[of "numeral m" "neg_numeral n"]
  2822   upto.simps[of "neg_numeral m" "numeral n"]
  2823   upto.simps[of "neg_numeral m" "neg_numeral n"] for m n
  2824 
  2825 lemma upto_empty[simp]: "j < i \<Longrightarrow> [i..j] = []"
  2826 by(simp add: upto.simps)
  2827 
  2828 lemma set_upto[simp]: "set[i..j] = {i..j}"
  2829 proof(induct i j rule:upto.induct)
  2830   case (1 i j)
  2831   from this show ?case
  2832     unfolding upto.simps[of i j] simp_from_to[of i j] by auto
  2833 qed
  2834 
  2835 
  2836 subsubsection {* @{text "distinct"} and @{text remdups} *}
  2837 
  2838 lemma distinct_tl:
  2839   "distinct xs \<Longrightarrow> distinct (tl xs)"
  2840   by (cases xs) simp_all
  2841 
  2842 lemma distinct_append [simp]:
  2843 "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
  2844 by (induct xs) auto
  2845 
  2846 lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
  2847 by(induct xs) auto
  2848 
  2849 lemma set_remdups [simp]: "set (remdups xs) = set xs"
  2850 by (induct xs) (auto simp add: insert_absorb)
  2851 
  2852 lemma distinct_remdups [iff]: "distinct (remdups xs)"
  2853 by (induct xs) auto
  2854 
  2855 lemma distinct_remdups_id: "distinct xs ==> remdups xs = xs"
  2856 by (induct xs, auto)
  2857 
  2858 lemma remdups_id_iff_distinct [simp]: "remdups xs = xs \<longleftrightarrow> distinct xs"
  2859 by (metis distinct_remdups distinct_remdups_id)
  2860 
  2861 lemma finite_distinct_list: "finite A \<Longrightarrow> EX xs. set xs = A & distinct xs"
  2862 by (metis distinct_remdups finite_list set_remdups)
  2863 
  2864 lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
  2865 by (induct x, auto)
  2866 
  2867 lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
  2868 by (induct x, auto)
  2869 
  2870 lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
  2871 by (induct xs) auto
  2872 
  2873 lemma length_remdups_eq[iff]:
  2874   "(length (remdups xs) = length xs) = (remdups xs = xs)"
  2875 apply(induct xs)
  2876  apply auto
  2877 apply(subgoal_tac "length (remdups xs) <= length xs")
  2878  apply arith
  2879 apply(rule length_remdups_leq)
  2880 done
  2881 
  2882 lemma remdups_filter: "remdups(filter P xs) = filter P (remdups xs)"
  2883 apply(induct xs)
  2884 apply auto
  2885 done
  2886 
  2887 lemma distinct_map:
  2888   "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
  2889 by (induct xs) auto
  2890 
  2891 
  2892 lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
  2893 by (induct xs) auto
  2894 
  2895 lemma distinct_upt[simp]: "distinct[i..<j]"
  2896 by (induct j) auto
  2897 
  2898 lemma distinct_upto[simp]: "distinct[i..j]"
  2899 apply(induct i j rule:upto.induct)
  2900 apply(subst upto.simps)
  2901 apply(simp)
  2902 done
  2903 
  2904 lemma distinct_take[simp]: "distinct xs \<Longrightarrow> distinct (take i xs)"
  2905 apply(induct xs arbitrary: i)
  2906  apply simp
  2907 apply (case_tac i)
  2908  apply simp_all
  2909 apply(blast dest:in_set_takeD)
  2910 done
  2911 
  2912 lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)"
  2913 apply(induct xs arbitrary: i)
  2914  apply simp
  2915 apply (case_tac i)
  2916  apply simp_all
  2917 done
  2918 
  2919 lemma distinct_list_update:
  2920 assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
  2921 shows "distinct (xs[i:=a])"
  2922 proof (cases "i < length xs")
  2923   case True
  2924   with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
  2925     apply (drule_tac id_take_nth_drop) by simp
  2926   with d True show ?thesis
  2927     apply (simp add: upd_conv_take_nth_drop)
  2928     apply (drule subst [OF id_take_nth_drop]) apply assumption
  2929     apply simp apply (cases "a = xs!i") apply simp by blast
  2930 next
  2931   case False with d show ?thesis by auto
  2932 qed
  2933 
  2934 lemma distinct_concat:
  2935   assumes "distinct xs"
  2936   and "\<And> ys. ys \<in> set xs \<Longrightarrow> distinct ys"
  2937   and "\<And> ys zs. \<lbrakk> ys \<in> set xs ; zs \<in> set xs ; ys \<noteq> zs \<rbrakk> \<Longrightarrow> set ys \<inter> set zs = {}"
  2938   shows "distinct (concat xs)"
  2939   using assms by (induct xs) auto
  2940 
  2941 text {* It is best to avoid this indexed version of distinct, but
  2942 sometimes it is useful. *}
  2943 
  2944 lemma distinct_conv_nth:
  2945 "distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
  2946 apply (induct xs, simp, simp)
  2947 apply (rule iffI, clarsimp)
  2948  apply (case_tac i)
  2949 apply (case_tac j, simp)
  2950 apply (simp add: set_conv_nth)
  2951  apply (case_tac j)
  2952 apply (clarsimp simp add: set_conv_nth, simp)
  2953 apply (rule conjI)
  2954 (*TOO SLOW
  2955 apply (metis Zero_neq_Suc gr0_conv_Suc in_set_conv_nth lessI less_trans_Suc nth_Cons' nth_Cons_Suc)
  2956 *)
  2957  apply (clarsimp simp add: set_conv_nth)
  2958  apply (erule_tac x = 0 in allE, simp)
  2959  apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
  2960 (*TOO SLOW
  2961 apply (metis Suc_Suc_eq lessI less_trans_Suc nth_Cons_Suc)
  2962 *)
  2963 apply (erule_tac x = "Suc i" in allE, simp)
  2964 apply (erule_tac x = "Suc j" in allE, simp)
  2965 done
  2966 
  2967 lemma nth_eq_iff_index_eq:
  2968  "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
  2969 by(auto simp: distinct_conv_nth)
  2970 
  2971 lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
  2972 by (induct xs) auto
  2973 
  2974 lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
  2975 proof (induct xs)
  2976   case Nil thus ?case by simp
  2977 next
  2978   case (Cons x xs)
  2979   show ?case
  2980   proof (cases "x \<in> set xs")
  2981     case False with Cons show ?thesis by simp
  2982   next
  2983     case True with Cons.prems
  2984     have "card (set xs) = Suc (length xs)"
  2985       by (simp add: card_insert_if split: split_if_asm)
  2986     moreover have "card (set xs) \<le> length xs" by (rule card_length)
  2987     ultimately have False by simp
  2988     thus ?thesis ..
  2989   qed
  2990 qed
  2991 
  2992 lemma distinct_length_filter: "distinct xs \<Longrightarrow> length (filter P xs) = card ({x. P x} Int set xs)"
  2993 by (induct xs) (auto)
  2994 
  2995 lemma not_distinct_decomp: "~ distinct ws ==> EX xs ys zs y. ws = xs@[y]@ys@[y]@zs"
  2996 apply (induct n == "length ws" arbitrary:ws) apply simp
  2997 apply(case_tac ws) apply simp
  2998 apply (simp split:split_if_asm)
  2999 apply (metis Cons_eq_appendI eq_Nil_appendI split_list)
  3000 done
  3001 
  3002 lemma not_distinct_conv_prefix:
  3003   defines "dec as xs y ys \<equiv> y \<in> set xs \<and> distinct xs \<and> as = xs @ y # ys"
  3004   shows "\<not>distinct as \<longleftrightarrow> (\<exists>xs y ys. dec as xs y ys)" (is "?L = ?R")
  3005 proof
  3006   assume "?L" then show "?R"
  3007   proof (induct "length as" arbitrary: as rule: less_induct)
  3008     case less
  3009     obtain xs ys zs y where decomp: "as = (xs @ y # ys) @ y # zs"
  3010       using not_distinct_decomp[OF less.prems] by auto
  3011     show ?case
  3012     proof (cases "distinct (xs @ y # ys)")
  3013       case True
  3014       with decomp have "dec as (xs @ y # ys) y zs" by (simp add: dec_def)
  3015       then show ?thesis by blast
  3016     next
  3017       case False
  3018       with less decomp obtain xs' y' ys' where "dec (xs @ y # ys) xs' y' ys'"
  3019         by atomize_elim auto
  3020       with decomp have "dec as xs' y' (ys' @ y # zs)" by (simp add: dec_def)
  3021       then show ?thesis by blast
  3022     qed
  3023   qed
  3024 qed (auto simp: dec_def)
  3025 
  3026 lemma length_remdups_concat:
  3027   "length (remdups (concat xss)) = card (\<Union>xs\<in>set xss. set xs)"
  3028   by (simp add: distinct_card [symmetric])
  3029 
  3030 lemma length_remdups_card_conv: "length(remdups xs) = card(set xs)"
  3031 proof -
  3032   have xs: "concat[xs] = xs" by simp
  3033   from length_remdups_concat[of "[xs]"] show ?thesis unfolding xs by simp
  3034 qed
  3035 
  3036 lemma remdups_remdups:
  3037   "remdups (remdups xs) = remdups xs"
  3038   by (induct xs) simp_all
  3039 
  3040 lemma distinct_butlast:
  3041   assumes "distinct xs"
  3042   shows "distinct (butlast xs)"
  3043 proof (cases "xs = []")
  3044   case False
  3045     from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  3046     with `distinct xs` show ?thesis by simp
  3047 qed (auto)
  3048 
  3049 lemma remdups_map_remdups:
  3050   "remdups (map f (remdups xs)) = remdups (map f xs)"
  3051   by (induct xs) simp_all
  3052 
  3053 lemma distinct_zipI1:
  3054   assumes "distinct xs"
  3055   shows "distinct (zip xs ys)"
  3056 proof (rule zip_obtain_same_length)
  3057   fix xs' :: "'a list" and ys' :: "'b list" and n
  3058   assume "length xs' = length ys'"
  3059   assume "xs' = take n xs"
  3060   with assms have "distinct xs'" by simp
  3061   with `length xs' = length ys'` show "distinct (zip xs' ys')"
  3062     by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
  3063 qed
  3064 
  3065 lemma distinct_zipI2:
  3066   assumes "distinct ys"
  3067   shows "distinct (zip xs ys)"
  3068 proof (rule zip_obtain_same_length)
  3069   fix xs' :: "'b list" and ys' :: "'a list" and n
  3070   assume "length xs' = length ys'"
  3071   assume "ys' = take n ys"
  3072   with assms have "distinct ys'" by simp
  3073   with `length xs' = length ys'` show "distinct (zip xs' ys')"
  3074     by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
  3075 qed
  3076 
  3077 lemma set_take_disj_set_drop_if_distinct:
  3078   "distinct vs \<Longrightarrow> i \<le> j \<Longrightarrow> set (take i vs) \<inter> set (drop j vs) = {}"
  3079 by (auto simp: in_set_conv_nth distinct_conv_nth)
  3080 
  3081 (* The next two lemmas help Sledgehammer. *)
  3082 
  3083 lemma distinct_singleton: "distinct [x]" by simp
  3084 
  3085 lemma distinct_length_2_or_more:
  3086 "distinct (a # b # xs) \<longleftrightarrow> (a \<noteq> b \<and> distinct (a # xs) \<and> distinct (b # xs))"
  3087 by (metis distinct.simps(2) hd.simps hd_in_set list.simps(2) set_ConsD set_rev_mp set_subset_Cons)
  3088 
  3089 subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
  3090 
  3091 lemma (in monoid_add) listsum_simps [simp]:
  3092   "listsum [] = 0"
  3093   "listsum (x # xs) = x + listsum xs"
  3094   by (simp_all add: listsum_def)
  3095 
  3096 lemma (in monoid_add) listsum_append [simp]:
  3097   "listsum (xs @ ys) = listsum xs + listsum ys"
  3098   by (induct xs) (simp_all add: add.assoc)
  3099 
  3100 lemma (in comm_monoid_add) listsum_rev [simp]:
  3101   "listsum (rev xs) = listsum xs"
  3102   by (simp add: listsum_def foldr_fold fold_rev fun_eq_iff add_ac)
  3103 
  3104 lemma (in monoid_add) fold_plus_listsum_rev:
  3105   "fold plus xs = plus (listsum (rev xs))"
  3106 proof
  3107   fix x
  3108   have "fold plus xs x = fold plus xs (x + 0)" by simp
  3109   also have "\<dots> = fold plus (x # xs) 0" by simp
  3110   also have "\<dots> = foldr plus (rev xs @ [x]) 0" by (simp add: foldr_conv_fold)
  3111   also have "\<dots> = listsum (rev xs @ [x])" by (simp add: listsum_def)
  3112   also have "\<dots> = listsum (rev xs) + listsum [x]" by simp
  3113   finally show "fold plus xs x = listsum (rev xs) + x" by simp
  3114 qed
  3115 
  3116 text{* Some syntactic sugar for summing a function over a list: *}
  3117 
  3118 syntax
  3119   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
  3120 syntax (xsymbols)
  3121   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
  3122 syntax (HTML output)
  3123   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
  3124 
  3125 translations -- {* Beware of argument permutation! *}
  3126   "SUM x<-xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  3127   "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  3128 
  3129 lemma (in comm_monoid_add) listsum_map_remove1:
  3130   "x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))"
  3131   by (induct xs) (auto simp add: ac_simps)
  3132 
  3133 lemma (in monoid_add) list_size_conv_listsum:
  3134   "list_size f xs = listsum (map f xs) + size xs"
  3135   by (induct xs) auto
  3136 
  3137 lemma (in monoid_add) length_concat:
  3138   "length (concat xss) = listsum (map length xss)"
  3139   by (induct xss) simp_all
  3140 
  3141 lemma (in monoid_add) listsum_map_filter:
  3142   assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
  3143   shows "listsum (map f (filter P xs)) = listsum (map f xs)"
  3144   using assms by (induct xs) auto
  3145 
  3146 lemma (in monoid_add) distinct_listsum_conv_Setsum:
  3147   "distinct xs \<Longrightarrow> listsum xs = Setsum (set xs)"
  3148   by (induct xs) simp_all
  3149 
  3150 lemma listsum_eq_0_nat_iff_nat [simp]:
  3151   "listsum ns = (0::nat) \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
  3152   by (induct ns) simp_all
  3153 
  3154 lemma member_le_listsum_nat:
  3155   "(n :: nat) \<in> set ns \<Longrightarrow> n \<le> listsum ns"
  3156   by (induct ns) auto
  3157 
  3158 lemma elem_le_listsum_nat:
  3159   "k < size ns \<Longrightarrow> ns ! k \<le> listsum (ns::nat list)"
  3160   by (rule member_le_listsum_nat) simp
  3161 
  3162 lemma listsum_update_nat:
  3163   "k < size ns \<Longrightarrow> listsum (ns[k := (n::nat)]) = listsum ns + n - ns ! k"
  3164 apply(induct ns arbitrary:k)
  3165  apply (auto split:nat.split)
  3166 apply(drule elem_le_listsum_nat)
  3167 apply arith
  3168 done
  3169 
  3170 lemma (in monoid_add) listsum_triv:
  3171   "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
  3172   by (induct xs) (simp_all add: left_distrib)
  3173 
  3174 lemma (in monoid_add) listsum_0 [simp]:
  3175   "(\<Sum>x\<leftarrow>xs. 0) = 0"
  3176   by (induct xs) (simp_all add: left_distrib)
  3177 
  3178 text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
  3179 lemma (in ab_group_add) uminus_listsum_map:
  3180   "- listsum (map f xs) = listsum (map (uminus \<circ> f) xs)"
  3181   by (induct xs) simp_all
  3182 
  3183 lemma (in comm_monoid_add) listsum_addf:
  3184   "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
  3185   by (induct xs) (simp_all add: algebra_simps)
  3186 
  3187 lemma (in ab_group_add) listsum_subtractf:
  3188   "(\<Sum>x\<leftarrow>xs. f x - g x) = listsum (map f xs) - listsum (map g xs)"
  3189   by (induct xs) (simp_all add: algebra_simps)
  3190 
  3191 lemma (in semiring_0) listsum_const_mult:
  3192   "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
  3193   by (induct xs) (simp_all add: algebra_simps)
  3194 
  3195 lemma (in semiring_0) listsum_mult_const:
  3196   "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
  3197   by (induct xs) (simp_all add: algebra_simps)
  3198 
  3199 lemma (in ordered_ab_group_add_abs) listsum_abs:
  3200   "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
  3201   by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])
  3202 
  3203 lemma listsum_mono:
  3204   fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}"
  3205   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)"
  3206   by (induct xs) (simp, simp add: add_mono)
  3207 
  3208 lemma (in monoid_add) listsum_distinct_conv_setsum_set:
  3209   "distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)"
  3210   by (induct xs) simp_all
  3211 
  3212 lemma (in monoid_add) interv_listsum_conv_setsum_set_nat:
  3213   "listsum (map f [m..<n]) = setsum f (set [m..<n])"
  3214   by (simp add: listsum_distinct_conv_setsum_set)
  3215 
  3216 lemma (in monoid_add) interv_listsum_conv_setsum_set_int:
  3217   "listsum (map f [k..l]) = setsum f (set [k..l])"
  3218   by (simp add: listsum_distinct_conv_setsum_set)
  3219 
  3220 text {* General equivalence between @{const listsum} and @{const setsum} *}
  3221 lemma (in monoid_add) listsum_setsum_nth:
  3222   "listsum xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
  3223   using interv_listsum_conv_setsum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth)
  3224 
  3225 
  3226 subsubsection {* @{const insert} *}
  3227 
  3228 lemma in_set_insert [simp]:
  3229   "x \<in> set xs \<Longrightarrow> List.insert x xs = xs"
  3230   by (simp add: List.insert_def)
  3231 
  3232 lemma not_in_set_insert [simp]:
  3233   "x \<notin> set xs \<Longrightarrow> List.insert x xs = x # xs"
  3234   by (simp add: List.insert_def)
  3235 
  3236 lemma insert_Nil [simp]:
  3237   "List.insert x [] = [x]"
  3238   by simp
  3239 
  3240 lemma set_insert [simp]:
  3241   "set (List.insert x xs) = insert x (set xs)"
  3242   by (auto simp add: List.insert_def)
  3243 
  3244 lemma distinct_insert [simp]:
  3245   "distinct xs \<Longrightarrow> distinct (List.insert x xs)"
  3246   by (simp add: List.insert_def)
  3247 
  3248 lemma insert_remdups:
  3249   "List.insert x (remdups xs) = remdups (List.insert x xs)"
  3250   by (simp add: List.insert_def)
  3251 
  3252 
  3253 subsubsection {* @{const List.find} *}
  3254 
  3255 lemma find_None_iff: "List.find P xs = None \<longleftrightarrow> \<not> (\<exists>x. x \<in> set xs \<and> P x)"
  3256 proof (induction xs)
  3257   case Nil thus ?case by simp
  3258 next
  3259   case (Cons x xs) thus ?case by (fastforce split: if_splits)
  3260 qed
  3261 
  3262 lemma find_Some_iff:
  3263   "List.find P xs = Some x \<longleftrightarrow>
  3264   (\<exists>i<length xs. P (xs!i) \<and> x = xs!i \<and> (\<forall>j<i. \<not> P (xs!j)))"
  3265 proof (induction xs)
  3266   case Nil thus ?case by simp
  3267 next
  3268   case (Cons x xs) thus ?case
  3269     by(auto simp: nth_Cons' split: if_splits)
  3270       (metis One_nat_def diff_Suc_1 less_Suc_eq_0_disj)
  3271 qed
  3272 
  3273 lemma find_cong[fundef_cong]:
  3274   assumes "xs = ys" and "\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x" 
  3275   shows "List.find P xs = List.find Q ys"
  3276 proof (cases "List.find P xs")
  3277   case None thus ?thesis by (metis find_None_iff assms)
  3278 next
  3279   case (Some x)
  3280   hence "List.find Q ys = Some x" using assms
  3281     by (auto simp add: find_Some_iff)
  3282   thus ?thesis using Some by auto
  3283 qed
  3284 
  3285 
  3286 subsubsection {* @{const remove1} *}
  3287 
  3288 lemma remove1_append:
  3289   "remove1 x (xs @ ys) =
  3290   (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
  3291 by (induct xs) auto
  3292 
  3293 lemma remove1_commute: "remove1 x (remove1 y zs) = remove1 y (remove1 x zs)"
  3294 by (induct zs) auto
  3295 
  3296 lemma in_set_remove1[simp]:
  3297   "a \<noteq> b \<Longrightarrow> a : set(remove1 b xs) = (a : set xs)"
  3298 apply (induct xs)
  3299 apply auto
  3300 done
  3301 
  3302 lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
  3303 apply(induct xs)
  3304  apply simp
  3305 apply simp
  3306 apply blast
  3307 done
  3308 
  3309 lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
  3310 apply(induct xs)
  3311  apply simp
  3312 apply simp
  3313 apply blast
  3314 done
  3315 
  3316 lemma length_remove1:
  3317   "length(remove1 x xs) = (if x : set xs then length xs - 1 else length xs)"
  3318 apply (induct xs)
  3319  apply (auto dest!:length_pos_if_in_set)
  3320 done
  3321 
  3322 lemma remove1_filter_not[simp]:
  3323   "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
  3324 by(induct xs) auto
  3325 
  3326 lemma filter_remove1:
  3327   "filter Q (remove1 x xs) = remove1 x (filter Q xs)"
  3328 by (induct xs) auto
  3329 
  3330 lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
  3331 apply(insert set_remove1_subset)
  3332 apply fast
  3333 done
  3334 
  3335 lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
  3336 by (induct xs) simp_all
  3337 
  3338 lemma remove1_remdups:
  3339   "distinct xs \<Longrightarrow> remove1 x (remdups xs) = remdups (remove1 x xs)"
  3340   by (induct xs) simp_all
  3341 
  3342 lemma remove1_idem:
  3343   assumes "x \<notin> set xs"
  3344   shows "remove1 x xs = xs"
  3345   using assms by (induct xs) simp_all
  3346 
  3347 
  3348 subsubsection {* @{text removeAll} *}
  3349 
  3350 lemma removeAll_filter_not_eq:
  3351   "removeAll x = filter (\<lambda>y. x \<noteq> y)"
  3352 proof
  3353   fix xs
  3354   show "removeAll x xs = filter (\<lambda>y. x \<noteq> y) xs"
  3355     by (induct xs) auto
  3356 qed
  3357 
  3358 lemma removeAll_append[simp]:
  3359   "removeAll x (xs @ ys) = removeAll x xs @ removeAll x ys"
  3360 by (induct xs) auto
  3361 
  3362 lemma set_removeAll[simp]: "set(removeAll x xs) = set xs - {x}"
  3363 by (induct xs) auto
  3364 
  3365 lemma removeAll_id[simp]: "x \<notin> set xs \<Longrightarrow> removeAll x xs = xs"
  3366 by (induct xs) auto
  3367 
  3368 (* Needs count:: 'a \<Rightarrow> 'a list \<Rightarrow> nat
  3369 lemma length_removeAll:
  3370   "length(removeAll x xs) = length xs - count x xs"
  3371 *)
  3372 
  3373 lemma removeAll_filter_not[simp]:
  3374   "\<not> P x \<Longrightarrow> removeAll x (filter P xs) = filter P xs"
  3375 by(induct xs) auto
  3376 
  3377 lemma distinct_removeAll:
  3378   "distinct xs \<Longrightarrow> distinct (removeAll x xs)"
  3379   by (simp add: removeAll_filter_not_eq)
  3380 
  3381 lemma distinct_remove1_removeAll:
  3382   "distinct xs ==> remove1 x xs = removeAll x xs"
  3383 by (induct xs) simp_all
  3384 
  3385 lemma map_removeAll_inj_on: "inj_on f (insert x (set xs)) \<Longrightarrow>
  3386   map f (removeAll x xs) = removeAll (f x) (map f xs)"
  3387 by (induct xs) (simp_all add:inj_on_def)
  3388 
  3389 lemma map_removeAll_inj: "inj f \<Longrightarrow>
  3390   map f (removeAll x xs) = removeAll (f x) (map f xs)"
  3391 by(metis map_removeAll_inj_on subset_inj_on subset_UNIV)
  3392 
  3393 
  3394 subsubsection {* @{text replicate} *}
  3395 
  3396 lemma length_replicate [simp]: "length (replicate n x) = n"
  3397 by (induct n) auto
  3398 
  3399 lemma Ex_list_of_length: "\<exists>xs. length xs = n"
  3400 by (rule exI[of _ "replicate n undefined"]) simp
  3401 
  3402 lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
  3403 by (induct n) auto
  3404 
  3405 lemma map_replicate_const:
  3406   "map (\<lambda> x. k) lst = replicate (length lst) k"
  3407   by (induct lst) auto
  3408 
  3409 lemma replicate_app_Cons_same:
  3410 "(replicate n x) @ (x # xs) = x # replicate n x @ xs"
  3411 by (induct n) auto
  3412 
  3413 lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
  3414 apply (induct n, simp)
  3415 apply (simp add: replicate_app_Cons_same)
  3416 done
  3417 
  3418 lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
  3419 by (induct n) auto
  3420 
  3421 text{* Courtesy of Matthias Daum: *}
  3422 lemma append_replicate_commute:
  3423   "replicate n x @ replicate k x = replicate k x @ replicate n x"
  3424 apply (simp add: replicate_add [THEN sym])
  3425 apply (simp add: add_commute)
  3426 done
  3427 
  3428 text{* Courtesy of Andreas Lochbihler: *}
  3429 lemma filter_replicate:
  3430   "filter P (replicate n x) = (if P x then replicate n x else [])"
  3431 by(induct n) auto
  3432 
  3433 lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
  3434 by (induct n) auto
  3435 
  3436 lemma tl_replicate [simp]: "tl (replicate n x) = replicate (n - 1) x"
  3437 by (induct n) auto
  3438 
  3439 lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
  3440 by (atomize (full), induct n) auto
  3441 
  3442 lemma nth_replicate[simp]: "i < n ==> (replicate n x)!i = x"
  3443 apply (induct n arbitrary: i, simp)
  3444 apply (simp add: nth_Cons split: nat.split)
  3445 done
  3446 
  3447 text{* Courtesy of Matthias Daum (2 lemmas): *}
  3448 lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
  3449 apply (case_tac "k \<le> i")
  3450  apply  (simp add: min_def)
  3451 apply (drule not_leE)
  3452 apply (simp add: min_def)
  3453 apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
  3454  apply  simp
  3455 apply (simp add: replicate_add [symmetric])
  3456 done
  3457 
  3458 lemma drop_replicate[simp]: "drop i (replicate k x) = replicate (k-i) x"
  3459 apply (induct k arbitrary: i)
  3460  apply simp
  3461 apply clarsimp
  3462 apply (case_tac i)
  3463  apply simp
  3464 apply clarsimp
  3465 done
  3466 
  3467 
  3468 lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
  3469 by (induct n) auto
  3470 
  3471 lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
  3472 by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
  3473 
  3474 lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
  3475 by auto
  3476 
  3477 lemma in_set_replicate[simp]: "(x : set (replicate n y)) = (x = y & n \<noteq> 0)"
  3478 by (simp add: set_replicate_conv_if)
  3479 
  3480 lemma Ball_set_replicate[simp]:
  3481   "(ALL x : set(replicate n a). P x) = (P a | n=0)"
  3482 by(simp add: set_replicate_conv_if)
  3483 
  3484 lemma Bex_set_replicate[simp]:
  3485   "(EX x : set(replicate n a). P x) = (P a & n\<noteq>0)"
  3486 by(simp add: set_replicate_conv_if)
  3487 
  3488 lemma replicate_append_same:
  3489   "replicate i x @ [x] = x # replicate i x"
  3490   by (induct i) simp_all
  3491 
  3492 lemma map_replicate_trivial:
  3493   "map (\<lambda>i. x) [0..<i] = replicate i x"
  3494   by (induct i) (simp_all add: replicate_append_same)
  3495 
  3496 lemma concat_replicate_trivial[simp]:
  3497   "concat (replicate i []) = []"
  3498   by (induct i) (auto simp add: map_replicate_const)
  3499 
  3500 lemma replicate_empty[simp]: "(replicate n x = []) \<longleftrightarrow> n=0"
  3501 by (induct n) auto
  3502 
  3503 lemma empty_replicate[simp]: "([] = replicate n x) \<longleftrightarrow> n=0"
  3504 by (induct n) auto
  3505 
  3506 lemma replicate_eq_replicate[simp]:
  3507   "(replicate m x = replicate n y) \<longleftrightarrow> (m=n & (m\<noteq>0 \<longrightarrow> x=y))"
  3508 apply(induct m arbitrary: n)
  3509  apply simp
  3510 apply(induct_tac n)
  3511 apply auto
  3512 done
  3513 
  3514 lemma replicate_length_filter:
  3515   "replicate (length (filter (\<lambda>y. x = y) xs)) x = filter (\<lambda>y. x = y) xs"
  3516   by (induct xs) auto
  3517 
  3518 lemma comm_append_are_replicate:
  3519   fixes xs ys :: "'a list"
  3520   assumes "xs \<noteq> []" "ys \<noteq> []"
  3521   assumes "xs @ ys = ys @ xs"
  3522   shows "\<exists>m n zs. concat (replicate m zs) = xs \<and> concat (replicate n zs) = ys"
  3523   using assms
  3524 proof (induct "length (xs @ ys)" arbitrary: xs ys rule: less_induct)
  3525   case less
  3526 
  3527   def xs' \<equiv> "if (length xs \<le> length ys) then xs else ys"
  3528     and ys' \<equiv> "if (length xs \<le> length ys) then ys else xs"
  3529   then have
  3530     prems': "length xs' \<le> length ys'"
  3531             "xs' @ ys' = ys' @ xs'"
  3532       and "xs' \<noteq> []"
  3533       and len: "length (xs @ ys) = length (xs' @ ys')"
  3534     using less by (auto intro: less.hyps)
  3535 
  3536   from prems'
  3537   obtain ws where "ys' = xs' @ ws"
  3538     by (auto simp: append_eq_append_conv2)
  3539 
  3540   have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ys'"
  3541   proof (cases "ws = []")
  3542     case True
  3543     then have "concat (replicate 1 xs') = xs'"
  3544       and "concat (replicate 1 xs') = ys'"
  3545       using `ys' = xs' @ ws` by auto
  3546     then show ?thesis by blast
  3547   next
  3548     case False
  3549     from `ys' = xs' @ ws` and `xs' @ ys' = ys' @ xs'`
  3550     have "xs' @ ws = ws @ xs'" by simp
  3551     then have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ws"
  3552       using False and `xs' \<noteq> []` and `ys' = xs' @ ws` and len
  3553       by (intro less.hyps) auto
  3554     then obtain m n zs where "concat (replicate m zs) = xs'"
  3555       and "concat (replicate n zs) = ws" by blast
  3556     moreover
  3557     then have "concat (replicate (m + n) zs) = ys'"
  3558       using `ys' = xs' @ ws`
  3559       by (simp add: replicate_add)
  3560     ultimately
  3561     show ?thesis by blast
  3562   qed
  3563   then show ?case
  3564     using xs'_def ys'_def by metis
  3565 qed
  3566 
  3567 lemma comm_append_is_replicate:
  3568   fixes xs ys :: "'a list"
  3569   assumes "xs \<noteq> []" "ys \<noteq> []"
  3570   assumes "xs @ ys = ys @ xs"
  3571   shows "\<exists>n zs. n > 1 \<and> concat (replicate n zs) = xs @ ys"
  3572 
  3573 proof -
  3574   obtain m n zs where "concat (replicate m zs) = xs"
  3575     and "concat (replicate n zs) = ys"
  3576     using assms by (metis comm_append_are_replicate)
  3577   then have "m + n > 1" and "concat (replicate (m+n) zs) = xs @ ys"
  3578     using `xs \<noteq> []` and `ys \<noteq> []`
  3579     by (auto simp: replicate_add)
  3580   then show ?thesis by blast
  3581 qed
  3582 
  3583 
  3584 subsubsection{*@{text rotate1} and @{text rotate}*}
  3585 
  3586 lemma rotate0[simp]: "rotate 0 = id"
  3587 by(simp add:rotate_def)
  3588 
  3589 lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
  3590 by(simp add:rotate_def)
  3591 
  3592 lemma rotate_add:
  3593   "rotate (m+n) = rotate m o rotate n"
  3594 by(simp add:rotate_def funpow_add)
  3595 
  3596 lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
  3597 by(simp add:rotate_add)
  3598 
  3599 lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
  3600 by(simp add:rotate_def funpow_swap1)
  3601 
  3602 lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
  3603 by(cases xs) simp_all
  3604 
  3605 lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
  3606 apply(induct n)
  3607  apply simp
  3608 apply (simp add:rotate_def)
  3609 done
  3610 
  3611 lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
  3612 by (cases xs) simp_all
  3613 
  3614 lemma rotate_drop_take:
  3615   "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
  3616 apply(induct n)
  3617  apply simp
  3618 apply(simp add:rotate_def)
  3619 apply(cases "xs = []")
  3620  apply (simp)
  3621 apply(case_tac "n mod length xs = 0")
  3622  apply(simp add:mod_Suc)
  3623  apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
  3624 apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
  3625                 take_hd_drop linorder_not_le)
  3626 done
  3627 
  3628 lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
  3629 by(simp add:rotate_drop_take)
  3630 
  3631 lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
  3632 by(simp add:rotate_drop_take)
  3633 
  3634 lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
  3635 by (cases xs) simp_all
  3636 
  3637 lemma length_rotate[simp]: "length(rotate n xs) = length xs"
  3638 by (induct n arbitrary: xs) (simp_all add:rotate_def)
  3639 
  3640 lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
  3641 by (cases xs) auto
  3642 
  3643 lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
  3644 by (induct n) (simp_all add:rotate_def)
  3645 
  3646 lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
  3647 by(simp add:rotate_drop_take take_map drop_map)
  3648 
  3649 lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
  3650 by (cases xs) auto
  3651 
  3652 lemma set_rotate[simp]: "set(rotate n xs) = set xs"
  3653 by (induct n) (simp_all add:rotate_def)
  3654 
  3655 lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
  3656 by (cases xs) auto
  3657 
  3658 lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
  3659 by (induct n) (simp_all add:rotate_def)
  3660 
  3661 lemma rotate_rev:
  3662   "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
  3663 apply(simp add:rotate_drop_take rev_drop rev_take)
  3664 apply(cases "length xs = 0")
  3665  apply simp
  3666 apply(cases "n mod length xs = 0")
  3667  apply simp
  3668 apply(simp add:rotate_drop_take rev_drop rev_take)
  3669 done
  3670 
  3671 lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
  3672 apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
  3673 apply(subgoal_tac "length xs \<noteq> 0")
  3674  prefer 2 apply simp
  3675 using mod_less_divisor[of "length xs" n] by arith
  3676 
  3677 
  3678 subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
  3679 
  3680 lemma sublist_empty [simp]: "sublist xs {} = []"
  3681 by (auto simp add: sublist_def)
  3682 
  3683 lemma sublist_nil [simp]: "sublist [] A = []"
  3684 by (auto simp add: sublist_def)
  3685 
  3686 lemma length_sublist:
  3687   "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
  3688 by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
  3689 
  3690 lemma sublist_shift_lemma_Suc:
  3691   "map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
  3692    map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
  3693 apply(induct xs arbitrary: "is")
  3694  apply simp
  3695 apply (case_tac "is")
  3696  apply simp
  3697 apply simp
  3698 done
  3699 
  3700 lemma sublist_shift_lemma:
  3701      "map fst [p<-zip xs [i..<i + length xs] . snd p : A] =
  3702       map fst [p<-zip xs [0..<length xs] . snd p + i : A]"
  3703 by (induct xs rule: rev_induct) (simp_all add: add_commute)
  3704 
  3705 lemma sublist_append:
  3706      "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
  3707 apply (unfold sublist_def)
  3708 apply (induct l' rule: rev_induct, simp)
  3709 apply (simp add: upt_add_eq_append[of 0] sublist_shift_lemma)
  3710 apply (simp add: add_commute)
  3711 done
  3712 
  3713 lemma sublist_Cons:
  3714 "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
  3715 apply (induct l rule: rev_induct)
  3716  apply (simp add: sublist_def)
  3717 apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
  3718 done
  3719 
  3720 lemma set_sublist: "set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
  3721 apply(induct xs arbitrary: I)
  3722 apply(auto simp: sublist_Cons nth_Cons split:nat.split dest!: gr0_implies_Suc)
  3723 done
  3724 
  3725 lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
  3726 by(auto simp add:set_sublist)
  3727 
  3728 lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
  3729 by(auto simp add:set_sublist)
  3730 
  3731 lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
  3732 by(auto simp add:set_sublist)
  3733 
  3734 lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
  3735 by (simp add: sublist_Cons)
  3736 
  3737 
  3738 lemma distinct_sublistI[simp]: "distinct xs \<Longrightarrow> distinct(sublist xs I)"
  3739 apply(induct xs arbitrary: I)
  3740  apply simp
  3741 apply(auto simp add:sublist_Cons)
  3742 done
  3743 
  3744 
  3745 lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
  3746 apply (induct l rule: rev_induct, simp)
  3747 apply (simp split: nat_diff_split add: sublist_append)
  3748 done
  3749 
  3750 lemma filter_in_sublist:
  3751  "distinct xs \<Longrightarrow> filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
  3752 proof (induct xs arbitrary: s)
  3753   case Nil thus ?case by simp
  3754 next
  3755   case (Cons a xs)
  3756   moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
  3757   ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
  3758 qed
  3759 
  3760 
  3761 subsubsection {* @{const splice} *}
  3762 
  3763 lemma splice_Nil2 [simp, code]: "splice xs [] = xs"
  3764 by (cases xs) simp_all
  3765 
  3766 declare splice.simps(1,3)[code]
  3767 declare splice.simps(2)[simp del]
  3768 
  3769 lemma length_splice[simp]: "length(splice xs ys) = length xs + length ys"
  3770 by (induct xs ys rule: splice.induct) auto
  3771 
  3772 
  3773 subsubsection {* Transpose *}
  3774 
  3775 function transpose where
  3776 "transpose []             = []" |
  3777 "transpose ([]     # xss) = transpose xss" |
  3778 "transpose ((x#xs) # xss) =
  3779   (x # [h. (h#t) \<leftarrow> xss]) # transpose (xs # [t. (h#t) \<leftarrow> xss])"
  3780 by pat_completeness auto
  3781 
  3782 lemma transpose_aux_filter_head:
  3783   "concat (map (list_case [] (\<lambda>h t. [h])) xss) =
  3784   map (\<lambda>xs. hd xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  3785   by (induct xss) (auto split: list.split)
  3786 
  3787 lemma transpose_aux_filter_tail:
  3788   "concat (map (list_case [] (\<lambda>h t. [t])) xss) =
  3789   map (\<lambda>xs. tl xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  3790   by (induct xss) (auto split: list.split)
  3791 
  3792 lemma transpose_aux_max:
  3793   "max (Suc (length xs)) (foldr (\<lambda>xs. max (length xs)) xss 0) =
  3794   Suc (max (length xs) (foldr (\<lambda>x. max (length x - Suc 0)) [ys\<leftarrow>xss . ys\<noteq>[]] 0))"
  3795   (is "max _ ?foldB = Suc (max _ ?foldA)")
  3796 proof (cases "[ys\<leftarrow>xss . ys\<noteq>[]] = []")
  3797   case True
  3798   hence "foldr (\<lambda>xs. max (length xs)) xss 0 = 0"
  3799   proof (induct xss)
  3800     case (Cons x xs)
  3801     moreover hence "x = []" by (cases x) auto
  3802     ultimately show ?case by auto
  3803   qed simp
  3804   thus ?thesis using True by simp
  3805 next
  3806   case False
  3807 
  3808   have foldA: "?foldA = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0 - 1"
  3809     by (induct xss) auto
  3810   have foldB: "?foldB = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0"
  3811     by (induct xss) auto
  3812 
  3813   have "0 < ?foldB"
  3814   proof -
  3815     from False
  3816     obtain z zs where zs: "[ys\<leftarrow>xss . ys \<noteq> []] = z#zs" by (auto simp: neq_Nil_conv)
  3817     hence "z \<in> set ([ys\<leftarrow>xss . ys \<noteq> []])" by auto
  3818     hence "z \<noteq> []" by auto
  3819     thus ?thesis
  3820       unfolding foldB zs
  3821       by (auto simp: max_def intro: less_le_trans)
  3822   qed
  3823   thus ?thesis
  3824     unfolding foldA foldB max_Suc_Suc[symmetric]
  3825     by simp
  3826 qed
  3827 
  3828 termination transpose
  3829   by (relation "measure (\<lambda>xs. foldr (\<lambda>xs. max (length xs)) xs 0 + length xs)")
  3830      (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max less_Suc_eq_le)
  3831 
  3832 lemma transpose_empty: "(transpose xs = []) \<longleftrightarrow> (\<forall>x \<in> set xs. x = [])"
  3833   by (induct rule: transpose.induct) simp_all
  3834 
  3835 lemma length_transpose:
  3836   fixes xs :: "'a list list"
  3837   shows "length (transpose xs) = foldr (\<lambda>xs. max (length xs)) xs 0"
  3838   by (induct rule: transpose.induct)
  3839     (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max
  3840                 max_Suc_Suc[symmetric] simp del: max_Suc_Suc)
  3841 
  3842 lemma nth_transpose:
  3843   fixes xs :: "'a list list"
  3844   assumes "i < length (transpose xs)"
  3845   shows "transpose xs ! i = map (\<lambda>xs. xs ! i) [ys \<leftarrow> xs. i < length ys]"
  3846 using assms proof (induct arbitrary: i rule: transpose.induct)
  3847   case (3 x xs xss)
  3848   def XS == "(x # xs) # xss"
  3849   hence [simp]: "XS \<noteq> []" by auto
  3850   thus ?case
  3851   proof (cases i)
  3852     case 0
  3853     thus ?thesis by (simp add: transpose_aux_filter_head hd_conv_nth)
  3854   next
  3855     case (Suc j)
  3856     have *: "\<And>xss. xs # map tl xss = map tl ((x#xs)#xss)" by simp
  3857     have **: "\<And>xss. (x#xs) # filter (\<lambda>ys. ys \<noteq> []) xss = filter (\<lambda>ys. ys \<noteq> []) ((x#xs)#xss)" by simp
  3858     { fix x have "Suc j < length x \<longleftrightarrow> x \<noteq> [] \<and> j < length x - Suc 0"
  3859       by (cases x) simp_all
  3860     } note *** = this
  3861 
  3862     have j_less: "j < length (transpose (xs # concat (map (list_case [] (\<lambda>h t. [t])) xss)))"
  3863       using "3.prems" by (simp add: transpose_aux_filter_tail length_transpose Suc)
  3864 
  3865     show ?thesis
  3866       unfolding transpose.simps `i = Suc j` nth_Cons_Suc "3.hyps"[OF j_less]
  3867       apply (auto simp: transpose_aux_filter_tail filter_map comp_def length_transpose * ** *** XS_def[symmetric])
  3868       apply (rule_tac y=x in list.exhaust)
  3869       by auto
  3870   qed
  3871 qed simp_all
  3872 
  3873 lemma transpose_map_map:
  3874   "transpose (map (map f) xs) = map (map f) (transpose xs)"
  3875 proof (rule nth_equalityI, safe)
  3876   have [simp]: "length (transpose (map (map f) xs)) = length (transpose xs)"
  3877     by (simp add: length_transpose foldr_map comp_def)
  3878   show "length (transpose (map (map f) xs)) = length (map (map f) (transpose xs))" by simp
  3879 
  3880   fix i assume "i < length (transpose (map (map f) xs))"
  3881   thus "transpose (map (map f) xs) ! i = map (map f) (transpose xs) ! i"
  3882     by (simp add: nth_transpose filter_map comp_def)
  3883 qed
  3884 
  3885 
  3886 subsubsection {* (In)finiteness *}
  3887 
  3888 lemma finite_maxlen:
  3889   "finite (M::'a list set) ==> EX n. ALL s:M. size s < n"
  3890 proof (induct rule: finite.induct)
  3891   case emptyI show ?case by simp
  3892 next
  3893   case (insertI M xs)
  3894   then obtain n where "\<forall>s\<in>M. length s < n" by blast
  3895   hence "ALL s:insert xs M. size s < max n (size xs) + 1" by auto
  3896   thus ?case ..
  3897 qed
  3898 
  3899 lemma lists_length_Suc_eq:
  3900   "{xs. set xs \<subseteq> A \<and> length xs = Suc n} =
  3901     (\<lambda>(xs, n). n#xs) ` ({xs. set xs \<subseteq> A \<and> length xs = n} \<times> A)"
  3902   by (auto simp: length_Suc_conv)
  3903 
  3904 lemma
  3905   assumes "finite A"
  3906   shows finite_lists_length_eq: "finite {xs. set xs \<subseteq> A \<and> length xs = n}"
  3907   and card_lists_length_eq: "card {xs. set xs \<subseteq> A \<and> length xs = n} = (card A)^n"
  3908   using `finite A`
  3909   by (induct n)
  3910      (auto simp: card_image inj_split_Cons lists_length_Suc_eq cong: conj_cong)
  3911 
  3912 lemma finite_lists_length_le:
  3913   assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs \<le> n}"
  3914  (is "finite ?S")
  3915 proof-
  3916   have "?S = (\<Union>n\<in>{0..n}. {xs. set xs \<subseteq> A \<and> length xs = n})" by auto
  3917   thus ?thesis by (auto intro: finite_lists_length_eq[OF `finite A`])
  3918 qed
  3919 
  3920 lemma card_lists_length_le:
  3921   assumes "finite A" shows "card {xs. set xs \<subseteq> A \<and> length xs \<le> n} = (\<Sum>i\<le>n. card A^i)"
  3922 proof -
  3923   have "(\<Sum>i\<le>n. card A^i) = card (\<Union>i\<le>n. {xs. set xs \<subseteq> A \<and> length xs = i})"
  3924     using `finite A`
  3925     by (subst card_UN_disjoint)
  3926        (auto simp add: card_lists_length_eq finite_lists_length_eq)
  3927   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}"
  3928     by auto
  3929   finally show ?thesis by simp
  3930 qed
  3931 
  3932 lemma card_lists_distinct_length_eq:
  3933   assumes "k < card A"
  3934   shows "card {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A} = \<Prod>{card A - k + 1 .. card A}"
  3935 using assms
  3936 proof (induct k)
  3937   case 0
  3938   then have "{xs. length xs = 0 \<and> distinct xs \<and> set xs \<subseteq> A} = {[]}" by auto
  3939   then show ?case by simp
  3940 next
  3941   case (Suc k)
  3942   let "?k_list" = "\<lambda>k xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A"
  3943   have inj_Cons: "\<And>A. inj_on (\<lambda>(xs, n). n # xs) A"  by (rule inj_onI) auto
  3944 
  3945   from Suc have "k < card A" by simp
  3946   moreover have "finite A" using assms by (simp add: card_ge_0_finite)
  3947   moreover have "finite {xs. ?k_list k xs}"
  3948     using finite_lists_length_eq[OF `finite A`, of k]
  3949     by - (rule finite_subset, auto)
  3950   moreover have "\<And>i j. i \<noteq> j \<longrightarrow> {i} \<times> (A - set i) \<inter> {j} \<times> (A - set j) = {}"
  3951     by auto
  3952   moreover have "\<And>i. i \<in>Collect (?k_list k) \<Longrightarrow> card (A - set i) = card A - k"
  3953     by (simp add: card_Diff_subset distinct_card)
  3954   moreover have "{xs. ?k_list (Suc k) xs} =
  3955       (\<lambda>(xs, n). n#xs) ` \<Union>(\<lambda>xs. {xs} \<times> (A - set xs)) ` {xs. ?k_list k xs}"
  3956     by (auto simp: length_Suc_conv)
  3957   moreover
  3958   have "Suc (card A - Suc k) = card A - k" using Suc.prems by simp
  3959   then have "(card A - k) * \<Prod>{Suc (card A - k)..card A} = \<Prod>{Suc (card A - Suc k)..card A}"
  3960     by (subst setprod_insert[symmetric]) (simp add: atLeastAtMost_insertL)+
  3961   ultimately show ?case
  3962     by (simp add: card_image inj_Cons card_UN_disjoint Suc.hyps algebra_simps)
  3963 qed
  3964 
  3965 lemma infinite_UNIV_listI: "~ finite(UNIV::'a list set)"
  3966 apply(rule notI)
  3967 apply(drule finite_maxlen)
  3968 apply (metis UNIV_I length_replicate less_not_refl)
  3969 done
  3970 
  3971 
  3972 subsection {* Sorting *}
  3973 
  3974 text{* Currently it is not shown that @{const sort} returns a
  3975 permutation of its input because the nicest proof is via multisets,
  3976 which are not yet available. Alternatively one could define a function
  3977 that counts the number of occurrences of an element in a list and use
  3978 that instead of multisets to state the correctness property. *}
  3979 
  3980 context linorder
  3981 begin
  3982 
  3983 lemma length_insort [simp]:
  3984   "length (insort_key f x xs) = Suc (length xs)"
  3985   by (induct xs) simp_all
  3986 
  3987 lemma insort_key_left_comm:
  3988   assumes "f x \<noteq> f y"
  3989   shows "insort_key f y (insort_key f x xs) = insort_key f x (insort_key f y xs)"
  3990   by (induct xs) (auto simp add: assms dest: antisym)
  3991 
  3992 lemma insort_left_comm:
  3993   "insort x (insort y xs) = insort y (insort x xs)"
  3994   by (cases "x = y") (auto intro: insort_key_left_comm)
  3995 
  3996 lemma comp_fun_commute_insort:
  3997   "comp_fun_commute insort"
  3998 proof
  3999 qed (simp add: insort_left_comm fun_eq_iff)
  4000 
  4001 lemma sort_key_simps [simp]:
  4002   "sort_key f [] = []"
  4003   "sort_key f (x#xs) = insort_key f x (sort_key f xs)"
  4004   by (simp_all add: sort_key_def)
  4005 
  4006 lemma (in linorder) sort_key_conv_fold:
  4007   assumes "inj_on f (set xs)"
  4008   shows "sort_key f xs = fold (insort_key f) xs []"
  4009 proof -
  4010   have "fold (insort_key f) (rev xs) = fold (insort_key f) xs"
  4011   proof (rule fold_rev, rule ext)
  4012     fix zs
  4013     fix x y
  4014     assume "x \<in> set xs" "y \<in> set xs"
  4015     with assms have *: "f y = f x \<Longrightarrow> y = x" by (auto dest: inj_onD)
  4016     have **: "x = y \<longleftrightarrow> y = x" by auto
  4017     show "(insort_key f y \<circ> insort_key f x) zs = (insort_key f x \<circ> insort_key f y) zs"
  4018       by (induct zs) (auto intro: * simp add: **)
  4019   qed
  4020   then show ?thesis by (simp add: sort_key_def foldr_conv_fold)
  4021 qed
  4022 
  4023 lemma (in linorder) sort_conv_fold:
  4024   "sort xs = fold insort xs []"
  4025   by (rule sort_key_conv_fold) simp
  4026 
  4027 lemma length_sort[simp]: "length (sort_key f xs) = length xs"
  4028 by (induct xs, auto)
  4029 
  4030 lemma sorted_Cons: "sorted (x#xs) = (sorted xs & (ALL y:set xs. x <= y))"
  4031 apply(induct xs arbitrary: x) apply simp
  4032 by simp (blast intro: order_trans)
  4033 
  4034 lemma sorted_tl:
  4035   "sorted xs \<Longrightarrow> sorted (tl xs)"
  4036   by (cases xs) (simp_all add: sorted_Cons)
  4037 
  4038 lemma sorted_append:
  4039   "sorted (xs@ys) = (sorted xs & sorted ys & (\<forall>x \<in> set xs. \<forall>y \<in> set ys. x\<le>y))"
  4040 by (induct xs) (auto simp add:sorted_Cons)
  4041 
  4042 lemma sorted_nth_mono:
  4043   "sorted xs \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!i \<le> xs!j"
  4044 by (induct xs arbitrary: i j) (auto simp:nth_Cons' sorted_Cons)
  4045 
  4046 lemma sorted_rev_nth_mono:
  4047   "sorted (rev xs) \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!j \<le> xs!i"
  4048 using sorted_nth_mono[ of "rev xs" "length xs - j - 1" "length xs - i - 1"]
  4049       rev_nth[of "length xs - i - 1" "xs"] rev_nth[of "length xs - j - 1" "xs"]
  4050 by auto
  4051 
  4052 lemma sorted_nth_monoI:
  4053   "(\<And> i j. \<lbrakk> i \<le> j ; j < length xs \<rbrakk> \<Longrightarrow> xs ! i \<le> xs ! j) \<Longrightarrow> sorted xs"
  4054 proof (induct xs)
  4055   case (Cons x xs)
  4056   have "sorted xs"
  4057   proof (rule Cons.hyps)
  4058     fix i j assume "i \<le> j" and "j < length xs"
  4059     with Cons.prems[of "Suc i" "Suc j"]
  4060     show "xs ! i \<le> xs ! j" by auto
  4061   qed
  4062   moreover
  4063   {
  4064     fix y assume "y \<in> set xs"
  4065     then obtain j where "j < length xs" and "xs ! j = y"
  4066       unfolding in_set_conv_nth by blast
  4067     with Cons.prems[of 0 "Suc j"]
  4068     have "x \<le> y"
  4069       by auto
  4070   }
  4071   ultimately
  4072   show ?case
  4073     unfolding sorted_Cons by auto
  4074 qed simp
  4075 
  4076 lemma sorted_equals_nth_mono:
  4077   "sorted xs = (\<forall>j < length xs. \<forall>i \<le> j. xs ! i \<le> xs ! j)"
  4078 by (auto intro: sorted_nth_monoI sorted_nth_mono)
  4079 
  4080 lemma set_insort: "set(insort_key f x xs) = insert x (set xs)"
  4081 by (induct xs) auto
  4082 
  4083 lemma set_sort[simp]: "set(sort_key f xs) = set xs"
  4084 by (induct xs) (simp_all add:set_insort)
  4085 
  4086 lemma distinct_insort: "distinct (insort_key f x xs) = (x \<notin> set xs \<and> distinct xs)"
  4087 by(induct xs)(auto simp:set_insort)
  4088 
  4089 lemma distinct_sort[simp]: "distinct (sort_key f xs) = distinct xs"
  4090   by (induct xs) (simp_all add: distinct_insort)
  4091 
  4092 lemma sorted_insort_key: "sorted (map f (insort_key f x xs)) = sorted (map f xs)"
  4093   by (induct xs) (auto simp:sorted_Cons set_insort)
  4094 
  4095 lemma sorted_insort: "sorted (insort x xs) = sorted xs"
  4096   using sorted_insort_key [where f="\<lambda>x. x"] by simp
  4097 
  4098 theorem sorted_sort_key [simp]: "sorted (map f (sort_key f xs))"
  4099   by (induct xs) (auto simp:sorted_insort_key)
  4100 
  4101 theorem sorted_sort [simp]: "sorted (sort xs)"
  4102   using sorted_sort_key [where f="\<lambda>x. x"] by simp
  4103 
  4104 lemma sorted_butlast:
  4105   assumes "xs \<noteq> []" and "sorted xs"
  4106   shows "sorted (butlast xs)"
  4107 proof -
  4108   from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  4109   with `sorted xs` show ?thesis by (simp add: sorted_append)
  4110 qed
  4111   
  4112 lemma insort_not_Nil [simp]:
  4113   "insort_key f a xs \<noteq> []"
  4114   by (induct xs) simp_all
  4115 
  4116 lemma insort_is_Cons: "\<forall>x\<in>set xs. f a \<le> f x \<Longrightarrow> insort_key f a xs = a # xs"
  4117 by (cases xs) auto
  4118 
  4119 lemma sorted_sort_id: "sorted xs \<Longrightarrow> sort xs = xs"
  4120   by (induct xs) (auto simp add: sorted_Cons insort_is_Cons)
  4121 
  4122 lemma sorted_map_remove1:
  4123   "sorted (map f xs) \<Longrightarrow> sorted (map f (remove1 x xs))"
  4124   by (induct xs) (auto simp add: sorted_Cons)
  4125 
  4126 lemma sorted_remove1: "sorted xs \<Longrightarrow> sorted (remove1 a xs)"
  4127   using sorted_map_remove1 [of "\<lambda>x. x"] by simp
  4128 
  4129 lemma insort_key_remove1:
  4130   assumes "a \<in> set xs" and "sorted (map f xs)" and "hd (filter (\<lambda>x. f a = f x) xs) = a"
  4131   shows "insort_key f a (remove1 a xs) = xs"
  4132 using assms proof (induct xs)
  4133   case (Cons x xs)
  4134   then show ?case
  4135   proof (cases "x = a")
  4136     case False
  4137     then have "f x \<noteq> f a" using Cons.prems by auto
  4138     then have "f x < f a" using Cons.prems by (auto simp: sorted_Cons)
  4139     with `f x \<noteq> f a` show ?thesis using Cons by (auto simp: sorted_Cons insort_is_Cons)
  4140   qed (auto simp: sorted_Cons insort_is_Cons)
  4141 qed simp
  4142 
  4143 lemma insort_remove1:
  4144   assumes "a \<in> set xs" and "sorted xs"
  4145   shows "insort a (remove1 a xs) = xs"
  4146 proof (rule insort_key_remove1)
  4147   from `a \<in> set xs` show "a \<in> set xs" .
  4148   from `sorted xs` show "sorted (map (\<lambda>x. x) xs)" by simp
  4149   from `a \<in> set xs` have "a \<in> set (filter (op = a) xs)" by auto
  4150   then have "set (filter (op = a) xs) \<noteq> {}" by auto
  4151   then have "filter (op = a) xs \<noteq> []" by (auto simp only: set_empty)
  4152   then have "length (filter (op = a) xs) > 0" by simp
  4153   then obtain n where n: "Suc n = length (filter (op = a) xs)"
  4154     by (cases "length (filter (op = a) xs)") simp_all
  4155   moreover have "replicate (Suc n) a = a # replicate n a"
  4156     by simp
  4157   ultimately show "hd (filter (op = a) xs) = a" by (simp add: replicate_length_filter)
  4158 qed
  4159 
  4160 lemma sorted_remdups[simp]:
  4161   "sorted l \<Longrightarrow> sorted (remdups l)"
  4162 by (induct l) (auto simp: sorted_Cons)
  4163 
  4164 lemma sorted_distinct_set_unique:
  4165 assumes "sorted xs" "distinct xs" "sorted ys" "distinct ys" "set xs = set ys"
  4166 shows "xs = ys"
  4167 proof -
  4168   from assms have 1: "length xs = length ys" by (auto dest!: distinct_card)
  4169   from assms show ?thesis
  4170   proof(induct rule:list_induct2[OF 1])
  4171     case 1 show ?case by simp
  4172   next
  4173     case 2 thus ?case by (simp add:sorted_Cons)
  4174        (metis Diff_insert_absorb antisym insertE insert_iff)
  4175   qed
  4176 qed
  4177 
  4178 lemma map_sorted_distinct_set_unique:
  4179   assumes "inj_on f (set xs \<union> set ys)"
  4180   assumes "sorted (map f xs)" "distinct (map f xs)"
  4181     "sorted (map f ys)" "distinct (map f ys)"
  4182   assumes "set xs = set ys"
  4183   shows "xs = ys"
  4184 proof -
  4185   from assms have "map f xs = map f ys"
  4186     by (simp add: sorted_distinct_set_unique)
  4187   moreover with `inj_on f (set xs \<union> set ys)` show "xs = ys"
  4188     by (blast intro: map_inj_on)
  4189 qed
  4190 
  4191 lemma finite_sorted_distinct_unique:
  4192 shows "finite A \<Longrightarrow> EX! xs. set xs = A & sorted xs & distinct xs"
  4193 apply(drule finite_distinct_list)
  4194 apply clarify
  4195 apply(rule_tac a="sort xs" in ex1I)
  4196 apply (auto simp: sorted_distinct_set_unique)
  4197 done
  4198 
  4199 lemma
  4200   assumes "sorted xs"
  4201   shows sorted_take: "sorted (take n xs)"
  4202   and sorted_drop: "sorted (drop n xs)"
  4203 proof -
  4204   from assms have "sorted (take n xs @ drop n xs)" by simp
  4205   then show "sorted (take n xs)" and "sorted (drop n xs)"
  4206     unfolding sorted_append by simp_all
  4207 qed
  4208 
  4209 lemma sorted_dropWhile: "sorted xs \<Longrightarrow> sorted (dropWhile P xs)"
  4210   by (auto dest: sorted_drop simp add: dropWhile_eq_drop)
  4211 
  4212 lemma sorted_takeWhile: "sorted xs \<Longrightarrow> sorted (takeWhile P xs)"
  4213   by (subst takeWhile_eq_take) (auto dest: sorted_take)
  4214 
  4215 lemma sorted_filter:
  4216   "sorted (map f xs) \<Longrightarrow> sorted (map f (filter P xs))"
  4217   by (induct xs) (simp_all add: sorted_Cons)
  4218 
  4219 lemma foldr_max_sorted:
  4220   assumes "sorted (rev xs)"
  4221   shows "foldr max xs y = (if xs = [] then y else max (xs ! 0) y)"
  4222 using assms proof (induct xs)
  4223   case (Cons x xs)
  4224   moreover hence "sorted (rev xs)" using sorted_append by auto
  4225   ultimately show ?case
  4226     by (cases xs, auto simp add: sorted_append max_def)
  4227 qed simp
  4228 
  4229 lemma filter_equals_takeWhile_sorted_rev:
  4230   assumes sorted: "sorted (rev (map f xs))"
  4231   shows "[x \<leftarrow> xs. t < f x] = takeWhile (\<lambda> x. t < f x) xs"
  4232     (is "filter ?P xs = ?tW")
  4233 proof (rule takeWhile_eq_filter[symmetric])
  4234   let "?dW" = "dropWhile ?P xs"
  4235   fix x assume "x \<in> set ?dW"
  4236   then obtain i where i: "i < length ?dW" and nth_i: "x = ?dW ! i"
  4237     unfolding in_set_conv_nth by auto
  4238   hence "length ?tW + i < length (?tW @ ?dW)"
  4239     unfolding length_append by simp
  4240   hence i': "length (map f ?tW) + i < length (map f xs)" by simp
  4241   have "(map f ?tW @ map f ?dW) ! (length (map f ?tW) + i) \<le>
  4242         (map f ?tW @ map f ?dW) ! (length (map f ?tW) + 0)"
  4243     using sorted_rev_nth_mono[OF sorted _ i', of "length ?tW"]
  4244     unfolding map_append[symmetric] by simp
  4245   hence "f x \<le> f (?dW ! 0)"
  4246     unfolding nth_append_length_plus nth_i
  4247     using i preorder_class.le_less_trans[OF le0 i] by simp
  4248   also have "... \<le> t"
  4249     using hd_dropWhile[of "?P" xs] le0[THEN preorder_class.le_less_trans, OF i]
  4250     using hd_conv_nth[of "?dW"] by simp
  4251   finally show "\<not> t < f x" by simp
  4252 qed
  4253 
  4254 lemma insort_insert_key_triv:
  4255   "f x \<in> f ` set xs \<Longrightarrow> insort_insert_key f x xs = xs"
  4256   by (simp add: insort_insert_key_def)
  4257 
  4258 lemma insort_insert_triv:
  4259   "x \<in> set xs \<Longrightarrow> insort_insert x xs = xs"
  4260   using insort_insert_key_triv [of "\<lambda>x. x"] by simp
  4261 
  4262 lemma insort_insert_insort_key:
  4263   "f x \<notin> f ` set xs \<Longrightarrow> insort_insert_key f x xs = insort_key f x xs"
  4264   by (simp add: insort_insert_key_def)
  4265 
  4266 lemma insort_insert_insort:
  4267   "x \<notin> set xs \<Longrightarrow> insort_insert x xs = insort x xs"
  4268   using insort_insert_insort_key [of "\<lambda>x. x"] by simp
  4269 
  4270 lemma set_insort_insert:
  4271   "set (insort_insert x xs) = insert x (set xs)"
  4272   by (auto simp add: insort_insert_key_def set_insort)
  4273 
  4274 lemma distinct_insort_insert:
  4275   assumes "distinct xs"
  4276   shows "distinct (insort_insert_key f x xs)"
  4277   using assms by (induct xs) (auto simp add: insort_insert_key_def set_insort)
  4278 
  4279 lemma sorted_insort_insert_key:
  4280   assumes "sorted (map f xs)"
  4281   shows "sorted (map f (insort_insert_key f x xs))"
  4282   using assms by (simp add: insort_insert_key_def sorted_insort_key)
  4283 
  4284 lemma sorted_insort_insert:
  4285   assumes "sorted xs"
  4286   shows "sorted (insort_insert x xs)"
  4287   using assms sorted_insort_insert_key [of "\<lambda>x. x"] by simp
  4288 
  4289 lemma filter_insort_triv:
  4290   "\<not> P x \<Longrightarrow> filter P (insort_key f x xs) = filter P xs"
  4291   by (induct xs) simp_all
  4292 
  4293 lemma filter_insort:
  4294   "sorted (map f xs) \<Longrightarrow> P x \<Longrightarrow> filter P (insort_key f x xs) = insort_key f x (filter P xs)"
  4295   using assms by (induct xs)
  4296     (auto simp add: sorted_Cons, subst insort_is_Cons, auto)
  4297 
  4298 lemma filter_sort:
  4299   "filter P (sort_key f xs) = sort_key f (filter P xs)"
  4300   by (induct xs) (simp_all add: filter_insort_triv filter_insort)
  4301 
  4302 lemma sorted_map_same:
  4303   "sorted (map f [x\<leftarrow>xs. f x = g xs])"
  4304 proof (induct xs arbitrary: g)
  4305   case Nil then show ?case by simp
  4306 next
  4307   case (Cons x xs)
  4308   then have "sorted (map f [y\<leftarrow>xs . f y = (\<lambda>xs. f x) xs])" .
  4309   moreover from Cons have "sorted (map f [y\<leftarrow>xs . f y = (g \<circ> Cons x) xs])" .
  4310   ultimately show ?case by (simp_all add: sorted_Cons)
  4311 qed
  4312 
  4313 lemma sorted_same:
  4314   "sorted [x\<leftarrow>xs. x = g xs]"
  4315   using sorted_map_same [of "\<lambda>x. x"] by simp
  4316 
  4317 lemma remove1_insort [simp]:
  4318   "remove1 x (insort x xs) = xs"
  4319   by (induct xs) simp_all
  4320 
  4321 end
  4322 
  4323 lemma sorted_upt[simp]: "sorted[i..<j]"
  4324 by (induct j) (simp_all add:sorted_append)
  4325 
  4326 lemma sorted_upto[simp]: "sorted[i..j]"
  4327 apply(induct i j rule:upto.induct)
  4328 apply(subst upto.simps)
  4329 apply(simp add:sorted_Cons)
  4330 done
  4331 
  4332 
  4333 subsubsection {* @{const transpose} on sorted lists *}
  4334 
  4335 lemma sorted_transpose[simp]:
  4336   shows "sorted (rev (map length (transpose xs)))"
  4337   by (auto simp: sorted_equals_nth_mono rev_nth nth_transpose
  4338     length_filter_conv_card intro: card_mono)
  4339 
  4340 lemma transpose_max_length:
  4341   "foldr (\<lambda>xs. max (length xs)) (transpose xs) 0 = length [x \<leftarrow> xs. x \<noteq> []]"
  4342   (is "?L = ?R")
  4343 proof (cases "transpose xs = []")
  4344   case False
  4345   have "?L = foldr max (map length (transpose xs)) 0"
  4346     by (simp add: foldr_map comp_def)
  4347   also have "... = length (transpose xs ! 0)"
  4348     using False sorted_transpose by (simp add: foldr_max_sorted)
  4349   finally show ?thesis
  4350     using False by (simp add: nth_transpose)
  4351 next
  4352   case True
  4353   hence "[x \<leftarrow> xs. x \<noteq> []] = []"
  4354     by (auto intro!: filter_False simp: transpose_empty)
  4355   thus ?thesis by (simp add: transpose_empty True)
  4356 qed
  4357 
  4358 lemma length_transpose_sorted:
  4359   fixes xs :: "'a list list"
  4360   assumes sorted: "sorted (rev (map length xs))"
  4361   shows "length (transpose xs) = (if xs = [] then 0 else length (xs ! 0))"
  4362 proof (cases "xs = []")
  4363   case False
  4364   thus ?thesis
  4365     using foldr_max_sorted[OF sorted] False
  4366     unfolding length_transpose foldr_map comp_def
  4367     by simp
  4368 qed simp
  4369 
  4370 lemma nth_nth_transpose_sorted[simp]:
  4371   fixes xs :: "'a list list"
  4372   assumes sorted: "sorted (rev (map length xs))"
  4373   and i: "i < length (transpose xs)"
  4374   and j: "j < length [ys \<leftarrow> xs. i < length ys]"
  4375   shows "transpose xs ! i ! j = xs ! j  ! i"
  4376   using j filter_equals_takeWhile_sorted_rev[OF sorted, of i]
  4377     nth_transpose[OF i] nth_map[OF j]
  4378   by (simp add: takeWhile_nth)
  4379 
  4380 lemma transpose_column_length:
  4381   fixes xs :: "'a list list"
  4382   assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  4383   shows "length (filter (\<lambda>ys. i < length ys) (transpose xs)) = length (xs ! i)"
  4384 proof -
  4385   have "xs \<noteq> []" using `i < length xs` by auto
  4386   note filter_equals_takeWhile_sorted_rev[OF sorted, simp]
  4387   { fix j assume "j \<le> i"
  4388     note sorted_rev_nth_mono[OF sorted, of j i, simplified, OF this `i < length xs`]
  4389   } note sortedE = this[consumes 1]
  4390 
  4391   have "{j. j < length (transpose xs) \<and> i < length (transpose xs ! j)}
  4392     = {..< length (xs ! i)}"
  4393   proof safe
  4394     fix j
  4395     assume "j < length (transpose xs)" and "i < length (transpose xs ! j)"
  4396     with this(2) nth_transpose[OF this(1)]
  4397     have "i < length (takeWhile (\<lambda>ys. j < length ys) xs)" by simp
  4398     from nth_mem[OF this] takeWhile_nth[OF this]
  4399     show "j < length (xs ! i)" by (auto dest: set_takeWhileD)
  4400   next
  4401     fix j assume "j < length (xs ! i)"
  4402     thus "j < length (transpose xs)"
  4403       using foldr_max_sorted[OF sorted] `xs \<noteq> []` sortedE[OF le0]
  4404       by (auto simp: length_transpose comp_def foldr_map)
  4405 
  4406     have "Suc i \<le> length (takeWhile (\<lambda>ys. j < length ys) xs)"
  4407       using `i < length xs` `j < length (xs ! i)` less_Suc_eq_le
  4408       by (auto intro!: length_takeWhile_less_P_nth dest!: sortedE)
  4409     with nth_transpose[OF `j < length (transpose xs)`]
  4410     show "i < length (transpose xs ! j)" by simp
  4411   qed
  4412   thus ?thesis by (simp add: length_filter_conv_card)
  4413 qed
  4414 
  4415 lemma transpose_column:
  4416   fixes xs :: "'a list list"
  4417   assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  4418   shows "map (\<lambda>ys. ys ! i) (filter (\<lambda>ys. i < length ys) (transpose xs))
  4419     = xs ! i" (is "?R = _")
  4420 proof (rule nth_equalityI, safe)
  4421   show length: "length ?R = length (xs ! i)"
  4422     using transpose_column_length[OF assms] by simp
  4423 
  4424   fix j assume j: "j < length ?R"
  4425   note * = less_le_trans[OF this, unfolded length_map, OF length_filter_le]
  4426   from j have j_less: "j < length (xs ! i)" using length by simp
  4427   have i_less_tW: "Suc i \<le> length (takeWhile (\<lambda>ys. Suc j \<le> length ys) xs)"
  4428   proof (rule length_takeWhile_less_P_nth)
  4429     show "Suc i \<le> length xs" using `i < length xs` by simp
  4430     fix k assume "k < Suc i"
  4431     hence "k \<le> i" by auto
  4432     with sorted_rev_nth_mono[OF sorted this] `i < length xs`
  4433     have "length (xs ! i) \<le> length (xs ! k)" by simp
  4434     thus "Suc j \<le> length (xs ! k)" using j_less by simp
  4435   qed
  4436   have i_less_filter: "i < length [ys\<leftarrow>xs . j < length ys]"
  4437     unfolding filter_equals_takeWhile_sorted_rev[OF sorted, of j]
  4438     using i_less_tW by (simp_all add: Suc_le_eq)
  4439   from j show "?R ! j = xs ! i ! j"
  4440     unfolding filter_equals_takeWhile_sorted_rev[OF sorted_transpose, of i]
  4441     by (simp add: takeWhile_nth nth_nth_transpose_sorted[OF sorted * i_less_filter])
  4442 qed
  4443 
  4444 lemma transpose_transpose:
  4445   fixes xs :: "'a list list"
  4446   assumes sorted: "sorted (rev (map length xs))"
  4447   shows "transpose (transpose xs) = takeWhile (\<lambda>x. x \<noteq> []) xs" (is "?L = ?R")
  4448 proof -
  4449   have len: "length ?L = length ?R"
  4450     unfolding length_transpose transpose_max_length
  4451     using filter_equals_takeWhile_sorted_rev[OF sorted, of 0]
  4452     by simp
  4453 
  4454   { fix i assume "i < length ?R"
  4455     with less_le_trans[OF _ length_takeWhile_le[of _ xs]]
  4456     have "i < length xs" by simp
  4457   } note * = this
  4458   show ?thesis
  4459     by (rule nth_equalityI)
  4460        (simp_all add: len nth_transpose transpose_column[OF sorted] * takeWhile_nth)
  4461 qed
  4462 
  4463 theorem transpose_rectangle:
  4464   assumes "xs = [] \<Longrightarrow> n = 0"
  4465   assumes rect: "\<And> i. i < length xs \<Longrightarrow> length (xs ! i) = n"
  4466   shows "transpose xs = map (\<lambda> i. map (\<lambda> j. xs ! j ! i) [0..<length xs]) [0..<n]"
  4467     (is "?trans = ?map")
  4468 proof (rule nth_equalityI)
  4469   have "sorted (rev (map length xs))"
  4470     by (auto simp: rev_nth rect intro!: sorted_nth_monoI)
  4471   from foldr_max_sorted[OF this] assms
  4472   show len: "length ?trans = length ?map"
  4473     by (simp_all add: length_transpose foldr_map comp_def)
  4474   moreover
  4475   { fix i assume "i < n" hence "[ys\<leftarrow>xs . i < length ys] = xs"
  4476       using rect by (auto simp: in_set_conv_nth intro!: filter_True) }
  4477   ultimately show "\<forall>i < length ?trans. ?trans ! i = ?map ! i"
  4478     by (auto simp: nth_transpose intro: nth_equalityI)
  4479 qed
  4480 
  4481 
  4482 subsubsection {* @{text sorted_list_of_set} *}
  4483 
  4484 text{* This function maps (finite) linearly ordered sets to sorted
  4485 lists. Warning: in most cases it is not a good idea to convert from
  4486 sets to lists but one should convert in the other direction (via
  4487 @{const set}). *}
  4488 
  4489 context linorder
  4490 begin
  4491 
  4492 definition sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where
  4493   "sorted_list_of_set = Finite_Set.fold insort []"
  4494 
  4495 lemma sorted_list_of_set_empty [simp]:
  4496   "sorted_list_of_set {} = []"
  4497   by (simp add: sorted_list_of_set_def)
  4498 
  4499 lemma sorted_list_of_set_insert [simp]:
  4500   assumes "finite A"
  4501   shows "sorted_list_of_set (insert x A) = insort x (sorted_list_of_set (A - {x}))"
  4502 proof -
  4503   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  4504   from assms show ?thesis
  4505     by (simp add: sorted_list_of_set_def fold_insert_remove)
  4506 qed
  4507 
  4508 lemma sorted_list_of_set [simp]:
  4509   "finite A \<Longrightarrow> set (sorted_list_of_set A) = A \<and> sorted (sorted_list_of_set A) 
  4510     \<and> distinct (sorted_list_of_set A)"
  4511   by (induct A rule: finite_induct) (simp_all add: set_insort sorted_insort distinct_insort)
  4512 
  4513 lemma sorted_list_of_set_sort_remdups [code]:
  4514   "sorted_list_of_set (set xs) = sort (remdups xs)"
  4515 proof -
  4516   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  4517   show ?thesis by (simp add: sorted_list_of_set_def sort_conv_fold fold_set_fold_remdups)
  4518 qed
  4519 
  4520 lemma sorted_list_of_set_remove:
  4521   assumes "finite A"
  4522   shows "sorted_list_of_set (A - {x}) = remove1 x (sorted_list_of_set A)"
  4523 proof (cases "x \<in> A")
  4524   case False with assms have "x \<notin> set (sorted_list_of_set A)" by simp
  4525   with False show ?thesis by (simp add: remove1_idem)
  4526 next
  4527   case True then obtain B where A: "A = insert x B" by (rule Set.set_insert)
  4528   with assms show ?thesis by simp
  4529 qed
  4530 
  4531 end
  4532 
  4533 lemma sorted_list_of_set_range [simp]:
  4534   "sorted_list_of_set {m..<n} = [m..<n]"
  4535   by (rule sorted_distinct_set_unique) simp_all
  4536 
  4537 
  4538 subsubsection {* @{text lists}: the list-forming operator over sets *}
  4539 
  4540 inductive_set
  4541   lists :: "'a set => 'a list set"
  4542   for A :: "'a set"
  4543 where
  4544     Nil [intro!, simp]: "[]: lists A"
  4545   | Cons [intro!, simp, no_atp]: "[| a: A; l: lists A|] ==> a#l : lists A"
  4546 
  4547 inductive_cases listsE [elim!,no_atp]: "x#l : lists A"
  4548 inductive_cases listspE [elim!,no_atp]: "listsp A (x # l)"
  4549 
  4550 inductive_simps listsp_simps[code]:
  4551   "listsp A []"
  4552   "listsp A (x # xs)"
  4553 
  4554 lemma listsp_mono [mono]: "A \<le> B ==> listsp A \<le> listsp B"
  4555 by (rule predicate1I, erule listsp.induct, blast+)
  4556 
  4557 lemmas lists_mono = listsp_mono [to_set]
  4558 
  4559 lemma listsp_infI:
  4560   assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
  4561 by induct blast+
  4562 
  4563 lemmas lists_IntI = listsp_infI [to_set]
  4564 
  4565 lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
  4566 proof (rule mono_inf [where f=listsp, THEN order_antisym])
  4567   show "mono listsp" by (simp add: mono_def listsp_mono)
  4568   show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro!: listsp_infI)
  4569 qed
  4570 
  4571 lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_def inf_bool_def]
  4572 
  4573 lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set]
  4574 
  4575 lemma Cons_in_lists_iff[simp]: "x#xs : lists A \<longleftrightarrow> x:A \<and> xs : lists A"
  4576 by auto
  4577 
  4578 lemma append_in_listsp_conv [iff]:
  4579      "(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
  4580 by (induct xs) auto
  4581 
  4582 lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]
  4583 
  4584 lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
  4585 -- {* eliminate @{text listsp} in favour of @{text set} *}
  4586 by (induct xs) auto
  4587 
  4588 lemmas in_lists_conv_set [code_unfold] = in_listsp_conv_set [to_set]
  4589 
  4590 lemma in_listspD [dest!,no_atp]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
  4591 by (rule in_listsp_conv_set [THEN iffD1])
  4592 
  4593 lemmas in_listsD [dest!,no_atp] = in_listspD [to_set]
  4594 
  4595 lemma in_listspI [intro!,no_atp]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
  4596 by (rule in_listsp_conv_set [THEN iffD2])
  4597 
  4598 lemmas in_listsI [intro!,no_atp] = in_listspI [to_set]
  4599 
  4600 lemma lists_eq_set: "lists A = {xs. set xs <= A}"
  4601 by auto
  4602 
  4603 lemma lists_empty [simp]: "lists {} = {[]}"
  4604 by auto
  4605 
  4606 lemma lists_UNIV [simp]: "lists UNIV = UNIV"
  4607 by auto
  4608 
  4609 
  4610 subsubsection {* Inductive definition for membership *}
  4611 
  4612 inductive ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
  4613 where
  4614     elem:  "ListMem x (x # xs)"
  4615   | insert:  "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"
  4616 
  4617 lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
  4618 apply (rule iffI)
  4619  apply (induct set: ListMem)
  4620   apply auto
  4621 apply (induct xs)
  4622  apply (auto intro: ListMem.intros)
  4623 done
  4624 
  4625 
  4626 subsubsection {* Lists as Cartesian products *}
  4627 
  4628 text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
  4629 @{term A} and tail drawn from @{term Xs}.*}
  4630 
  4631 definition
  4632   set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set" where
  4633   "set_Cons A XS = {z. \<exists>x xs. z = x # xs \<and> x \<in> A \<and> xs \<in> XS}"
  4634 
  4635 lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
  4636 by (auto simp add: set_Cons_def)
  4637 
  4638 text{*Yields the set of lists, all of the same length as the argument and
  4639 with elements drawn from the corresponding element of the argument.*}
  4640 
  4641 primrec
  4642   listset :: "'a set list \<Rightarrow> 'a list set" where
  4643      "listset [] = {[]}"
  4644   |  "listset (A # As) = set_Cons A (listset As)"
  4645 
  4646 
  4647 subsection {* Relations on Lists *}
  4648 
  4649 subsubsection {* Length Lexicographic Ordering *}
  4650 
  4651 text{*These orderings preserve well-foundedness: shorter lists 
  4652   precede longer lists. These ordering are not used in dictionaries.*}
  4653         
  4654 primrec -- {*The lexicographic ordering for lists of the specified length*}
  4655   lexn :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> ('a list \<times> 'a list) set" where
  4656     "lexn r 0 = {}"
  4657   | "lexn r (Suc n) = (map_pair (%(x, xs). x#xs) (%(x, xs). x#xs) ` (r <*lex*> lexn r n)) Int
  4658       {(xs, ys). length xs = Suc n \<and> length ys = Suc n}"
  4659 
  4660 definition
  4661   lex :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4662   "lex r = (\<Union>n. lexn r n)" -- {*Holds only between lists of the same length*}
  4663 
  4664 definition
  4665   lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" where
  4666   "lenlex r = inv_image (less_than <*lex*> lex r) (\<lambda>xs. (length xs, xs))"
  4667         -- {*Compares lists by their length and then lexicographically*}
  4668 
  4669 lemma wf_lexn: "wf r ==> wf (lexn r n)"
  4670 apply (induct n, simp, simp)
  4671 apply(rule wf_subset)
  4672  prefer 2 apply (rule Int_lower1)
  4673 apply(rule wf_map_pair_image)
  4674  prefer 2 apply (rule inj_onI, auto)
  4675 done
  4676 
  4677 lemma lexn_length:
  4678   "(xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
  4679 by (induct n arbitrary: xs ys) auto
  4680 
  4681 lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
  4682 apply (unfold lex_def)
  4683 apply (rule wf_UN)
  4684 apply (blast intro: wf_lexn, clarify)
  4685 apply (rename_tac m n)
  4686 apply (subgoal_tac "m \<noteq> n")
  4687  prefer 2 apply blast
  4688 apply (blast dest: lexn_length not_sym)
  4689 done
  4690 
  4691 lemma lexn_conv:
  4692   "lexn r n =
  4693     {(xs,ys). length xs = n \<and> length ys = n \<and>
  4694     (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
  4695 apply (induct n, simp)
  4696 apply (simp add: image_Collect lex_prod_def, safe, blast)
  4697  apply (rule_tac x = "ab # xys" in exI, simp)
  4698 apply (case_tac xys, simp_all, blast)
  4699 done
  4700 
  4701 lemma lex_conv:
  4702   "lex r =
  4703     {(xs,ys). length xs = length ys \<and>
  4704     (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
  4705 by (force simp add: lex_def lexn_conv)
  4706 
  4707 lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
  4708 by (unfold lenlex_def) blast
  4709 
  4710 lemma lenlex_conv:
  4711     "lenlex r = {(xs,ys). length xs < length ys |
  4712                  length xs = length ys \<and> (xs, ys) : lex r}"
  4713 by (simp add: lenlex_def Id_on_def lex_prod_def inv_image_def)
  4714 
  4715 lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
  4716 by (simp add: lex_conv)
  4717 
  4718 lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
  4719 by (simp add:lex_conv)
  4720 
  4721 lemma Cons_in_lex [simp]:
  4722     "((x # xs, y # ys) : lex r) =
  4723       ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
  4724 apply (simp add: lex_conv)
  4725 apply (rule iffI)
  4726  prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
  4727 apply (case_tac xys, simp, simp)
  4728 apply blast
  4729 done
  4730 
  4731 
  4732 subsubsection {* Lexicographic Ordering *}
  4733 
  4734 text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
  4735     This ordering does \emph{not} preserve well-foundedness.
  4736      Author: N. Voelker, March 2005. *} 
  4737 
  4738 definition
  4739   lexord :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4740   "lexord r = {(x,y ). \<exists> a v. y = x @ a # v \<or>
  4741             (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
  4742 
  4743 lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
  4744 by (unfold lexord_def, induct_tac y, auto) 
  4745 
  4746 lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
  4747 by (unfold lexord_def, induct_tac x, auto)
  4748 
  4749 lemma lexord_cons_cons[simp]:
  4750      "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
  4751   apply (unfold lexord_def, safe, simp_all)
  4752   apply (case_tac u, simp, simp)
  4753   apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
  4754   apply (erule_tac x="b # u" in allE)
  4755   by force
  4756 
  4757 lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
  4758 
  4759 lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
  4760 by (induct_tac x, auto)  
  4761 
  4762 lemma lexord_append_left_rightI:
  4763      "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
  4764 by (induct_tac u, auto)
  4765 
  4766 lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
  4767 by (induct x, auto)
  4768 
  4769 lemma lexord_append_leftD:
  4770      "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
  4771 by (erule rev_mp, induct_tac x, auto)
  4772 
  4773 lemma lexord_take_index_conv: 
  4774    "((x,y) : lexord r) = 
  4775     ((length x < length y \<and> take (length x) y = x) \<or> 
  4776      (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
  4777   apply (unfold lexord_def Let_def, clarsimp) 
  4778   apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
  4779   apply auto 
  4780   apply (rule_tac x="hd (drop (length x) y)" in exI)
  4781   apply (rule_tac x="tl (drop (length x) y)" in exI)
  4782   apply (erule subst, simp add: min_def) 
  4783   apply (rule_tac x ="length u" in exI, simp) 
  4784   apply (rule_tac x ="take i x" in exI) 
  4785   apply (rule_tac x ="x ! i" in exI) 
  4786   apply (rule_tac x ="y ! i" in exI, safe) 
  4787   apply (rule_tac x="drop (Suc i) x" in exI)
  4788   apply (drule sym, simp add: drop_Suc_conv_tl) 
  4789   apply (rule_tac x="drop (Suc i) y" in exI)
  4790   by (simp add: drop_Suc_conv_tl) 
  4791 
  4792 -- {* lexord is extension of partial ordering List.lex *} 
  4793 lemma lexord_lex: "(x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
  4794   apply (rule_tac x = y in spec) 
  4795   apply (induct_tac x, clarsimp) 
  4796   by (clarify, case_tac x, simp, force)
  4797 
  4798 lemma lexord_irreflexive: "ALL x. (x,x) \<notin> r \<Longrightarrow> (xs,xs) \<notin> lexord r"
  4799 by (induct xs) auto
  4800 
  4801 text{* By Ren\'e Thiemann: *}
  4802 lemma lexord_partial_trans: 
  4803   "(\<And>x y z. x \<in> set xs \<Longrightarrow> (x,y) \<in> r \<Longrightarrow> (y,z) \<in> r \<Longrightarrow> (x,z) \<in> r)
  4804    \<Longrightarrow>  (xs,ys) \<in> lexord r  \<Longrightarrow>  (ys,zs) \<in> lexord r \<Longrightarrow>  (xs,zs) \<in> lexord r"
  4805 proof (induct xs arbitrary: ys zs)
  4806   case Nil
  4807   from Nil(3) show ?case unfolding lexord_def by (cases zs, auto)
  4808 next
  4809   case (Cons x xs yys zzs)
  4810   from Cons(3) obtain y ys where yys: "yys = y # ys" unfolding lexord_def
  4811     by (cases yys, auto)
  4812   note Cons = Cons[unfolded yys]
  4813   from Cons(3) have one: "(x,y) \<in> r \<or> x = y \<and> (xs,ys) \<in> lexord r" by auto
  4814   from Cons(4) obtain z zs where zzs: "zzs = z # zs" unfolding lexord_def
  4815     by (cases zzs, auto)
  4816   note Cons = Cons[unfolded zzs]
  4817   from Cons(4) have two: "(y,z) \<in> r \<or> y = z \<and> (ys,zs) \<in> lexord r" by auto
  4818   {
  4819     assume "(xs,ys) \<in> lexord r" and "(ys,zs) \<in> lexord r"
  4820     from Cons(1)[OF _ this] Cons(2)
  4821     have "(xs,zs) \<in> lexord r" by auto
  4822   } note ind1 = this
  4823   {
  4824     assume "(x,y) \<in> r" and "(y,z) \<in> r"
  4825     from Cons(2)[OF _ this] have "(x,z) \<in> r" by auto
  4826   } note ind2 = this
  4827   from one two ind1 ind2
  4828   have "(x,z) \<in> r \<or> x = z \<and> (xs,zs) \<in> lexord r" by blast
  4829   thus ?case unfolding zzs by auto
  4830 qed
  4831 
  4832 lemma lexord_trans: 
  4833     "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
  4834 by(auto simp: trans_def intro:lexord_partial_trans)
  4835 
  4836 lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
  4837 by (rule transI, drule lexord_trans, blast) 
  4838 
  4839 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"
  4840   apply (rule_tac x = y in spec) 
  4841   apply (induct_tac x, rule allI) 
  4842   apply (case_tac x, simp, simp) 
  4843   apply (rule allI, case_tac x, simp, simp) 
  4844   by blast
  4845 
  4846 
  4847 subsubsection {* Lexicographic combination of measure functions *}
  4848 
  4849 text {* These are useful for termination proofs *}
  4850 
  4851 definition
  4852   "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"
  4853 
  4854 lemma wf_measures[simp]: "wf (measures fs)"
  4855 unfolding measures_def
  4856 by blast
  4857 
  4858 lemma in_measures[simp]: 
  4859   "(x, y) \<in> measures [] = False"
  4860   "(x, y) \<in> measures (f # fs)
  4861          = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"  
  4862 unfolding measures_def
  4863 by auto
  4864 
  4865 lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
  4866 by simp
  4867 
  4868 lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
  4869 by auto
  4870 
  4871 
  4872 subsubsection {* Lifting Relations to Lists: one element *}
  4873 
  4874 definition listrel1 :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4875 "listrel1 r = {(xs,ys).
  4876    \<exists>us z z' vs. xs = us @ z # vs \<and> (z,z') \<in> r \<and> ys = us @ z' # vs}"
  4877 
  4878 lemma listrel1I:
  4879   "\<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow>
  4880   (xs, ys) \<in> listrel1 r"
  4881 unfolding listrel1_def by auto
  4882 
  4883 lemma listrel1E:
  4884   "\<lbrakk> (xs, ys) \<in> listrel1 r;
  4885      !!x y us vs. \<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow> P
  4886    \<rbrakk> \<Longrightarrow> P"
  4887 unfolding listrel1_def by auto
  4888 
  4889 lemma not_Nil_listrel1 [iff]: "([], xs) \<notin> listrel1 r"
  4890 unfolding listrel1_def by blast
  4891 
  4892 lemma not_listrel1_Nil [iff]: "(xs, []) \<notin> listrel1 r"
  4893 unfolding listrel1_def by blast
  4894 
  4895 lemma Cons_listrel1_Cons [iff]:
  4896   "(x # xs, y # ys) \<in> listrel1 r \<longleftrightarrow>
  4897    (x,y) \<in> r \<and> xs = ys \<or> x = y \<and> (xs, ys) \<in> listrel1 r"
  4898 by (simp add: listrel1_def Cons_eq_append_conv) (blast)
  4899 
  4900 lemma listrel1I1: "(x,y) \<in> r \<Longrightarrow> (x # xs, y # xs) \<in> listrel1 r"
  4901 by (metis Cons_listrel1_Cons)
  4902 
  4903 lemma listrel1I2: "(xs, ys) \<in> listrel1 r \<Longrightarrow> (x # xs, x # ys) \<in> listrel1 r"
  4904 by (metis Cons_listrel1_Cons)
  4905 
  4906 lemma append_listrel1I:
  4907   "(xs, ys) \<in> listrel1 r \<and> us = vs \<or> xs = ys \<and> (us, vs) \<in> listrel1 r
  4908     \<Longrightarrow> (xs @ us, ys @ vs) \<in> listrel1 r"
  4909 unfolding listrel1_def
  4910 by auto (blast intro: append_eq_appendI)+
  4911 
  4912 lemma Cons_listrel1E1[elim!]:
  4913   assumes "(x # xs, ys) \<in> listrel1 r"
  4914     and "\<And>y. ys = y # xs \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
  4915     and "\<And>zs. ys = x # zs \<Longrightarrow> (xs, zs) \<in> listrel1 r \<Longrightarrow> R"
  4916   shows R
  4917 using assms by (cases ys) blast+
  4918 
  4919 lemma Cons_listrel1E2[elim!]:
  4920   assumes "(xs, y # ys) \<in> listrel1 r"
  4921     and "\<And>x. xs = x # ys \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
  4922     and "\<And>zs. xs = y # zs \<Longrightarrow> (zs, ys) \<in> listrel1 r \<Longrightarrow> R"
  4923   shows R
  4924 using assms by (cases xs) blast+
  4925 
  4926 lemma snoc_listrel1_snoc_iff:
  4927   "(xs @ [x], ys @ [y]) \<in> listrel1 r
  4928     \<longleftrightarrow> (xs, ys) \<in> listrel1 r \<and> x = y \<or> xs = ys \<and> (x,y) \<in> r" (is "?L \<longleftrightarrow> ?R")
  4929 proof
  4930   assume ?L thus ?R
  4931     by (fastforce simp: listrel1_def snoc_eq_iff_butlast butlast_append)
  4932 next
  4933   assume ?R then show ?L unfolding listrel1_def by force
  4934 qed
  4935 
  4936 lemma listrel1_eq_len: "(xs,ys) \<in> listrel1 r \<Longrightarrow> length xs = length ys"
  4937 unfolding listrel1_def by auto
  4938 
  4939 lemma listrel1_mono:
  4940   "r \<subseteq> s \<Longrightarrow> listrel1 r \<subseteq> listrel1 s"
  4941 unfolding listrel1_def by blast
  4942 
  4943 
  4944 lemma listrel1_converse: "listrel1 (r^-1) = (listrel1 r)^-1"
  4945 unfolding listrel1_def by blast
  4946 
  4947 lemma in_listrel1_converse:
  4948   "(x,y) : listrel1 (r^-1) \<longleftrightarrow> (x,y) : (listrel1 r)^-1"
  4949 unfolding listrel1_def by blast
  4950 
  4951 lemma listrel1_iff_update:
  4952   "(xs,ys) \<in> (listrel1 r)
  4953    \<longleftrightarrow> (\<exists>y n. (xs ! n, y) \<in> r \<and> n < length xs \<and> ys = xs[n:=y])" (is "?L \<longleftrightarrow> ?R")
  4954 proof
  4955   assume "?L"
  4956   then obtain x y u v where "xs = u @ x # v"  "ys = u @ y # v"  "(x,y) \<in> r"
  4957     unfolding listrel1_def by auto
  4958   then have "ys = xs[length u := y]" and "length u < length xs"
  4959     and "(xs ! length u, y) \<in> r" by auto
  4960   then show "?R" by auto
  4961 next
  4962   assume "?R"
  4963   then obtain x y n where "(xs!n, y) \<in> r" "n < size xs" "ys = xs[n:=y]" "x = xs!n"
  4964     by auto
  4965   then obtain u v where "xs = u @ x # v" and "ys = u @ y # v" and "(x, y) \<in> r"
  4966     by (auto intro: upd_conv_take_nth_drop id_take_nth_drop)
  4967   then show "?L" by (auto simp: listrel1_def)
  4968 qed
  4969 
  4970 
  4971 text{* Accessible part and wellfoundedness: *}
  4972 
  4973 lemma Cons_acc_listrel1I [intro!]:
  4974   "x \<in> acc r \<Longrightarrow> xs \<in> acc (listrel1 r) \<Longrightarrow> (x # xs) \<in> acc (listrel1 r)"
  4975 apply (induct arbitrary: xs set: acc)
  4976 apply (erule thin_rl)
  4977 apply (erule acc_induct)
  4978 apply (rule accI)
  4979 apply (blast)
  4980 done
  4981 
  4982 lemma lists_accD: "xs \<in> lists (acc r) \<Longrightarrow> xs \<in> acc (listrel1 r)"
  4983 apply (induct set: lists)
  4984  apply (rule accI)
  4985  apply simp
  4986 apply (rule accI)
  4987 apply (fast dest: acc_downward)
  4988 done
  4989 
  4990 lemma lists_accI: "xs \<in> acc (listrel1 r) \<Longrightarrow> xs \<in> lists (acc r)"
  4991 apply (induct set: acc)
  4992 apply clarify
  4993 apply (rule accI)
  4994 apply (fastforce dest!: in_set_conv_decomp[THEN iffD1] simp: listrel1_def)
  4995 done
  4996 
  4997 lemma wf_listrel1_iff[simp]: "wf(listrel1 r) = wf r"
  4998 by(metis wf_acc_iff in_lists_conv_set lists_accI lists_accD Cons_in_lists_iff)
  4999 
  5000 
  5001 subsubsection {* Lifting Relations to Lists: all elements *}
  5002 
  5003 inductive_set
  5004   listrel :: "('a \<times> 'b) set \<Rightarrow> ('a list \<times> 'b list) set"
  5005   for r :: "('a \<times> 'b) set"
  5006 where
  5007     Nil:  "([],[]) \<in> listrel r"
  5008   | Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
  5009 
  5010 inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
  5011 inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
  5012 inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
  5013 inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
  5014 
  5015 
  5016 lemma listrel_eq_len:  "(xs, ys) \<in> listrel r \<Longrightarrow> length xs = length ys"
  5017 by(induct rule: listrel.induct) auto
  5018 
  5019 lemma listrel_iff_zip [code_unfold]: "(xs,ys) : listrel r \<longleftrightarrow>
  5020   length xs = length ys & (\<forall>(x,y) \<in> set(zip xs ys). (x,y) \<in> r)" (is "?L \<longleftrightarrow> ?R")
  5021 proof
  5022   assume ?L thus ?R by induct (auto intro: listrel_eq_len)
  5023 next
  5024   assume ?R thus ?L
  5025     apply (clarify)
  5026     by (induct rule: list_induct2) (auto intro: listrel.intros)
  5027 qed
  5028 
  5029 lemma listrel_iff_nth: "(xs,ys) : listrel r \<longleftrightarrow>
  5030   length xs = length ys & (\<forall>n < length xs. (xs!n, ys!n) \<in> r)" (is "?L \<longleftrightarrow> ?R")
  5031 by (auto simp add: all_set_conv_all_nth listrel_iff_zip)
  5032 
  5033 
  5034 lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
  5035 apply clarify  
  5036 apply (erule listrel.induct)
  5037 apply (blast intro: listrel.intros)+
  5038 done
  5039 
  5040 lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
  5041 apply clarify 
  5042 apply (erule listrel.induct, auto) 
  5043 done
  5044 
  5045 lemma listrel_refl_on: "refl_on A r \<Longrightarrow> refl_on (lists A) (listrel r)" 
  5046 apply (simp add: refl_on_def listrel_subset Ball_def)
  5047 apply (rule allI) 
  5048 apply (induct_tac x) 
  5049 apply (auto intro: listrel.intros)
  5050 done
  5051 
  5052 lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
  5053 apply (auto simp add: sym_def)
  5054 apply (erule listrel.induct) 
  5055 apply (blast intro: listrel.intros)+
  5056 done
  5057 
  5058 lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
  5059 apply (simp add: trans_def)
  5060 apply (intro allI) 
  5061 apply (rule impI) 
  5062 apply (erule listrel.induct) 
  5063 apply (blast intro: listrel.intros)+
  5064 done
  5065 
  5066 theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
  5067 by (simp add: equiv_def listrel_refl_on listrel_sym listrel_trans) 
  5068 
  5069 lemma listrel_rtrancl_refl[iff]: "(xs,xs) : listrel(r^*)"
  5070 using listrel_refl_on[of UNIV, OF refl_rtrancl]
  5071 by(auto simp: refl_on_def)
  5072 
  5073 lemma listrel_rtrancl_trans:
  5074   "\<lbrakk> (xs,ys) : listrel(r^*);  (ys,zs) : listrel(r^*) \<rbrakk>
  5075   \<Longrightarrow> (xs,zs) : listrel(r^*)"
  5076 by (metis listrel_trans trans_def trans_rtrancl)
  5077 
  5078 
  5079 lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
  5080 by (blast intro: listrel.intros)
  5081 
  5082 lemma listrel_Cons:
  5083      "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})"
  5084 by (auto simp add: set_Cons_def intro: listrel.intros)
  5085 
  5086 text {* Relating @{term listrel1}, @{term listrel} and closures: *}
  5087 
  5088 lemma listrel1_rtrancl_subset_rtrancl_listrel1:
  5089   "listrel1 (r^*) \<subseteq> (listrel1 r)^*"
  5090 proof (rule subrelI)
  5091   fix xs ys assume 1: "(xs,ys) \<in> listrel1 (r^*)"
  5092   { fix x y us vs
  5093     have "(x,y) : r^* \<Longrightarrow> (us @ x # vs, us @ y # vs) : (listrel1 r)^*"
  5094     proof(induct rule: rtrancl.induct)
  5095       case rtrancl_refl show ?case by simp
  5096     next
  5097       case rtrancl_into_rtrancl thus ?case
  5098         by (metis listrel1I rtrancl.rtrancl_into_rtrancl)
  5099     qed }
  5100   thus "(xs,ys) \<in> (listrel1 r)^*" using 1 by(blast elim: listrel1E)
  5101 qed
  5102 
  5103 lemma rtrancl_listrel1_eq_len: "(x,y) \<in> (listrel1 r)^* \<Longrightarrow> length x = length y"
  5104 by (induct rule: rtrancl.induct) (auto intro: listrel1_eq_len)
  5105 
  5106 lemma rtrancl_listrel1_ConsI1:
  5107   "(xs,ys) : (listrel1 r)^* \<Longrightarrow> (x#xs,x#ys) : (listrel1 r)^*"
  5108 apply(induct rule: rtrancl.induct)
  5109  apply simp
  5110 by (metis listrel1I2 rtrancl.rtrancl_into_rtrancl)
  5111 
  5112 lemma rtrancl_listrel1_ConsI2:
  5113   "(x,y) \<in> r^* \<Longrightarrow> (xs, ys) \<in> (listrel1 r)^*
  5114   \<Longrightarrow> (x # xs, y # ys) \<in> (listrel1 r)^*"
  5115   by (blast intro: rtrancl_trans rtrancl_listrel1_ConsI1 
  5116     subsetD[OF listrel1_rtrancl_subset_rtrancl_listrel1 listrel1I1])
  5117 
  5118 lemma listrel1_subset_listrel:
  5119   "r \<subseteq> r' \<Longrightarrow> refl r' \<Longrightarrow> listrel1 r \<subseteq> listrel(r')"
  5120 by(auto elim!: listrel1E simp add: listrel_iff_zip set_zip refl_on_def)
  5121 
  5122 lemma listrel_reflcl_if_listrel1:
  5123   "(xs,ys) : listrel1 r \<Longrightarrow> (xs,ys) : listrel(r^*)"
  5124 by(erule listrel1E)(auto simp add: listrel_iff_zip set_zip)
  5125 
  5126 lemma listrel_rtrancl_eq_rtrancl_listrel1: "listrel (r^*) = (listrel1 r)^*"
  5127 proof
  5128   { fix x y assume "(x,y) \<in> listrel (r^*)"
  5129     then have "(x,y) \<in> (listrel1 r)^*"
  5130     by induct (auto intro: rtrancl_listrel1_ConsI2) }
  5131   then show "listrel (r^*) \<subseteq> (listrel1 r)^*"
  5132     by (rule subrelI)
  5133 next
  5134   show "listrel (r^*) \<supseteq> (listrel1 r)^*"
  5135   proof(rule subrelI)
  5136     fix xs ys assume "(xs,ys) \<in> (listrel1 r)^*"
  5137     then show "(xs,ys) \<in> listrel (r^*)"
  5138     proof induct
  5139       case base show ?case by(auto simp add: listrel_iff_zip set_zip)
  5140     next
  5141       case (step ys zs)
  5142       thus ?case  by (metis listrel_reflcl_if_listrel1 listrel_rtrancl_trans)
  5143     qed
  5144   qed
  5145 qed
  5146 
  5147 lemma rtrancl_listrel1_if_listrel:
  5148   "(xs,ys) : listrel r \<Longrightarrow> (xs,ys) : (listrel1 r)^*"
  5149 by(metis listrel_rtrancl_eq_rtrancl_listrel1 subsetD[OF listrel_mono] r_into_rtrancl subsetI)
  5150 
  5151 lemma listrel_subset_rtrancl_listrel1: "listrel r \<subseteq> (listrel1 r)^*"
  5152 by(fast intro:rtrancl_listrel1_if_listrel)
  5153 
  5154 
  5155 subsection {* Size function *}
  5156 
  5157 lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (list_size f)"
  5158 by (rule is_measure_trivial)
  5159 
  5160 lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (option_size f)"
  5161 by (rule is_measure_trivial)
  5162 
  5163 lemma list_size_estimation[termination_simp]: 
  5164   "x \<in> set xs \<Longrightarrow> y < f x \<Longrightarrow> y < list_size f xs"
  5165 by (induct xs) auto
  5166 
  5167 lemma list_size_estimation'[termination_simp]: 
  5168   "x \<in> set xs \<Longrightarrow> y \<le> f x \<Longrightarrow> y \<le> list_size f xs"
  5169 by (induct xs) auto
  5170 
  5171 lemma list_size_map[simp]: "list_size f (map g xs) = list_size (f o g) xs"
  5172 by (induct xs) auto
  5173 
  5174 lemma list_size_append[simp]: "list_size f (xs @ ys) = list_size f xs + list_size f ys"
  5175 by (induct xs, auto)
  5176 
  5177 lemma list_size_pointwise[termination_simp]: 
  5178   "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> list_size f xs \<le> list_size g xs"
  5179 by (induct xs) force+
  5180 
  5181 
  5182 subsection {* Monad operation *}
  5183 
  5184 definition bind :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b list) \<Rightarrow> 'b list" where
  5185   "bind xs f = concat (map f xs)"
  5186 
  5187 hide_const (open) bind
  5188 
  5189 lemma bind_simps [simp]:
  5190   "List.bind [] f = []"
  5191   "List.bind (x # xs) f = f x @ List.bind xs f"
  5192   by (simp_all add: bind_def)
  5193 
  5194 
  5195 subsection {* Transfer *}
  5196 
  5197 definition
  5198   embed_list :: "nat list \<Rightarrow> int list"
  5199 where
  5200   "embed_list l = map int l"
  5201 
  5202 definition
  5203   nat_list :: "int list \<Rightarrow> bool"
  5204 where
  5205   "nat_list l = nat_set (set l)"
  5206 
  5207 definition
  5208   return_list :: "int list \<Rightarrow> nat list"
  5209 where
  5210   "return_list l = map nat l"
  5211 
  5212 lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow>
  5213     embed_list (return_list l) = l"
  5214   unfolding embed_list_def return_list_def nat_list_def nat_set_def
  5215   apply (induct l)
  5216   apply auto
  5217 done
  5218 
  5219 lemma transfer_nat_int_list_functions:
  5220   "l @ m = return_list (embed_list l @ embed_list m)"
  5221   "[] = return_list []"
  5222   unfolding return_list_def embed_list_def
  5223   apply auto
  5224   apply (induct l, auto)
  5225   apply (induct m, auto)
  5226 done
  5227 
  5228 (*
  5229 lemma transfer_nat_int_fold1: "fold f l x =
  5230     fold (%x. f (nat x)) (embed_list l) x";
  5231 *)
  5232 
  5233 
  5234 subsection {* Code generation *}
  5235 
  5236 subsubsection {* Counterparts for set-related operations *}
  5237 
  5238 definition member :: "'a list \<Rightarrow> 'a \<Rightarrow> bool" where
  5239   [code_abbrev]: "member xs x \<longleftrightarrow> x \<in> set xs"
  5240 
  5241 text {*
  5242   Use @{text member} only for generating executable code.  Otherwise use
  5243   @{prop "x \<in> set xs"} instead --- it is much easier to reason about.
  5244 *}
  5245 
  5246 lemma member_rec [code]:
  5247   "member (x # xs) y \<longleftrightarrow> x = y \<or> member xs y"
  5248   "member [] y \<longleftrightarrow> False"
  5249   by (auto simp add: member_def)
  5250 
  5251 lemma in_set_member (* FIXME delete candidate *):
  5252   "x \<in> set xs \<longleftrightarrow> member xs x"
  5253   by (simp add: member_def)
  5254 
  5255 definition list_all :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5256   list_all_iff [code_abbrev]: "list_all P xs \<longleftrightarrow> Ball (set xs) P"
  5257 
  5258 definition list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5259   list_ex_iff [code_abbrev]: "list_ex P xs \<longleftrightarrow> Bex (set xs) P"
  5260 
  5261 definition list_ex1 :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5262   list_ex1_iff [code_abbrev]: "list_ex1 P xs \<longleftrightarrow> (\<exists>! x. x \<in> set xs \<and> P x)"
  5263 
  5264 text {*
  5265   Usually you should prefer @{text "\<forall>x\<in>set xs"}, @{text "\<exists>x\<in>set xs"}
  5266   and @{text "\<exists>!x. x\<in>set xs \<and> _"} over @{const list_all}, @{const list_ex}
  5267   and @{const list_ex1} in specifications.
  5268 *}
  5269 
  5270 lemma list_all_simps [simp, code]:
  5271   "list_all P (x # xs) \<longleftrightarrow> P x \<and> list_all P xs"
  5272   "list_all P [] \<longleftrightarrow> True"
  5273   by (simp_all add: list_all_iff)
  5274 
  5275 lemma list_ex_simps [simp, code]:
  5276   "list_ex P (x # xs) \<longleftrightarrow> P x \<or> list_ex P xs"
  5277   "list_ex P [] \<longleftrightarrow> False"
  5278   by (simp_all add: list_ex_iff)
  5279 
  5280 lemma list_ex1_simps [simp, code]:
  5281   "list_ex1 P [] = False"
  5282   "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)"
  5283   by (auto simp add: list_ex1_iff list_all_iff)
  5284 
  5285 lemma Ball_set_list_all: (* FIXME delete candidate *)
  5286   "Ball (set xs) P \<longleftrightarrow> list_all P xs"
  5287   by (simp add: list_all_iff)
  5288 
  5289 lemma Bex_set_list_ex: (* FIXME delete candidate *)
  5290   "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
  5291   by (simp add: list_ex_iff)
  5292 
  5293 lemma list_all_append [simp]:
  5294   "list_all P (xs @ ys) \<longleftrightarrow> list_all P xs \<and> list_all P ys"
  5295   by (auto simp add: list_all_iff)
  5296 
  5297 lemma list_ex_append [simp]:
  5298   "list_ex P (xs @ ys) \<longleftrightarrow> list_ex P xs \<or> list_ex P ys"
  5299   by (auto simp add: list_ex_iff)
  5300 
  5301 lemma list_all_rev [simp]:
  5302   "list_all P (rev xs) \<longleftrightarrow> list_all P xs"
  5303   by (simp add: list_all_iff)
  5304 
  5305 lemma list_ex_rev [simp]:
  5306   "list_ex P (rev xs) \<longleftrightarrow> list_ex P xs"
  5307   by (simp add: list_ex_iff)
  5308 
  5309 lemma list_all_length:
  5310   "list_all P xs \<longleftrightarrow> (\<forall>n < length xs. P (xs ! n))"
  5311   by (auto simp add: list_all_iff set_conv_nth)
  5312 
  5313 lemma list_ex_length:
  5314   "list_ex P xs \<longleftrightarrow> (\<exists>n < length xs. P (xs ! n))"
  5315   by (auto simp add: list_ex_iff set_conv_nth)
  5316 
  5317 lemma list_all_cong [fundef_cong]:
  5318   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_all f xs = list_all g ys"
  5319   by (simp add: list_all_iff)
  5320 
  5321 lemma list_ex_cong [fundef_cong]:
  5322   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_ex f xs = list_ex g ys"
  5323 by (simp add: list_ex_iff)
  5324 
  5325 text {* Executable checks for relations on sets *}
  5326 
  5327 definition listrel1p :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
  5328 "listrel1p r xs ys = ((xs, ys) \<in> listrel1 {(x, y). r x y})"
  5329 
  5330 lemma [code_unfold]:
  5331   "(xs, ys) \<in> listrel1 r = listrel1p (\<lambda>x y. (x, y) \<in> r) xs ys"
  5332 unfolding listrel1p_def by auto
  5333 
  5334 lemma [code]:
  5335   "listrel1p r [] xs = False"
  5336   "listrel1p r xs [] =  False"
  5337   "listrel1p r (x # xs) (y # ys) \<longleftrightarrow>
  5338      r x y \<and> xs = ys \<or> x = y \<and> listrel1p r xs ys"
  5339 by (simp add: listrel1p_def)+
  5340 
  5341 definition
  5342   lexordp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
  5343   "lexordp r xs ys = ((xs, ys) \<in> lexord {(x, y). r x y})"
  5344 
  5345 lemma [code_unfold]:
  5346   "(xs, ys) \<in> lexord r = lexordp (\<lambda>x y. (x, y) \<in> r) xs ys"
  5347 unfolding lexordp_def by auto
  5348 
  5349 lemma [code]:
  5350   "lexordp r xs [] = False"
  5351   "lexordp r [] (y#ys) = True"
  5352   "lexordp r (x # xs) (y # ys) = (r x y | (x = y & lexordp r xs ys))"
  5353 unfolding lexordp_def by auto
  5354 
  5355 text {* Bounded quantification and summation over nats. *}
  5356 
  5357 lemma atMost_upto [code_unfold]:
  5358   "{..n} = set [0..<Suc n]"
  5359   by auto
  5360 
  5361 lemma atLeast_upt [code_unfold]:
  5362   "{..<n} = set [0..<n]"
  5363   by auto
  5364 
  5365 lemma greaterThanLessThan_upt [code_unfold]:
  5366   "{n<..<m} = set [Suc n..<m]"
  5367   by auto
  5368 
  5369 lemmas atLeastLessThan_upt [code_unfold] = set_upt [symmetric]
  5370 
  5371 lemma greaterThanAtMost_upt [code_unfold]:
  5372   "{n<..m} = set [Suc n..<Suc m]"
  5373   by auto
  5374 
  5375 lemma atLeastAtMost_upt [code_unfold]:
  5376   "{n..m} = set [n..<Suc m]"
  5377   by auto
  5378 
  5379 lemma all_nat_less_eq [code_unfold]:
  5380   "(\<forall>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..<n}. P m)"
  5381   by auto
  5382 
  5383 lemma ex_nat_less_eq [code_unfold]:
  5384   "(\<exists>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..<n}. P m)"
  5385   by auto
  5386 
  5387 lemma all_nat_less [code_unfold]:
  5388   "(\<forall>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..n}. P m)"
  5389   by auto
  5390 
  5391 lemma ex_nat_less [code_unfold]:
  5392   "(\<exists>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..n}. P m)"
  5393   by auto
  5394 
  5395 lemma setsum_set_upt_conv_listsum_nat [code_unfold]:
  5396   "setsum f (set [m..<n]) = listsum (map f [m..<n])"
  5397   by (simp add: interv_listsum_conv_setsum_set_nat)
  5398 
  5399 text {* Summation over ints. *}
  5400 
  5401 lemma greaterThanLessThan_upto [code_unfold]:
  5402   "{i<..<j::int} = set [i+1..j - 1]"
  5403 by auto
  5404 
  5405 lemma atLeastLessThan_upto [code_unfold]:
  5406   "{i..<j::int} = set [i..j - 1]"
  5407 by auto
  5408 
  5409 lemma greaterThanAtMost_upto [code_unfold]:
  5410   "{i<..j::int} = set [i+1..j]"
  5411 by auto
  5412 
  5413 lemmas atLeastAtMost_upto [code_unfold] = set_upto [symmetric]
  5414 
  5415 lemma setsum_set_upto_conv_listsum_int [code_unfold]:
  5416   "setsum f (set [i..j::int]) = listsum (map f [i..j])"
  5417   by (simp add: interv_listsum_conv_setsum_set_int)
  5418 
  5419 
  5420 subsubsection {* Optimizing by rewriting *}
  5421 
  5422 definition null :: "'a list \<Rightarrow> bool" where
  5423   [code_abbrev]: "null xs \<longleftrightarrow> xs = []"
  5424 
  5425 text {*
  5426   Efficient emptyness check is implemented by @{const null}.
  5427 *}
  5428 
  5429 lemma null_rec [code]:
  5430   "null (x # xs) \<longleftrightarrow> False"
  5431   "null [] \<longleftrightarrow> True"
  5432   by (simp_all add: null_def)
  5433 
  5434 lemma eq_Nil_null: (* FIXME delete candidate *)
  5435   "xs = [] \<longleftrightarrow> null xs"
  5436   by (simp add: null_def)
  5437 
  5438 lemma equal_Nil_null [code_unfold]:
  5439   "HOL.equal xs [] \<longleftrightarrow> null xs"
  5440   by (simp add: equal eq_Nil_null)
  5441 
  5442 definition maps :: "('a \<Rightarrow> 'b list) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  5443   [code_abbrev]: "maps f xs = concat (map f xs)"
  5444 
  5445 definition map_filter :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  5446   [code_post]: "map_filter f xs = map (the \<circ> f) (filter (\<lambda>x. f x \<noteq> None) xs)"
  5447 
  5448 text {*
  5449   Operations @{const maps} and @{const map_filter} avoid
  5450   intermediate lists on execution -- do not use for proving.
  5451 *}
  5452 
  5453 lemma maps_simps [code]:
  5454   "maps f (x # xs) = f x @ maps f xs"
  5455   "maps f [] = []"
  5456   by (simp_all add: maps_def)
  5457 
  5458 lemma map_filter_simps [code]:
  5459   "map_filter f (x # xs) = (case f x of None \<Rightarrow> map_filter f xs | Some y \<Rightarrow> y # map_filter f xs)"
  5460   "map_filter f [] = []"
  5461   by (simp_all add: map_filter_def split: option.split)
  5462 
  5463 lemma concat_map_maps: (* FIXME delete candidate *)
  5464   "concat (map f xs) = maps f xs"
  5465   by (simp add: maps_def)
  5466 
  5467 lemma map_filter_map_filter [code_unfold]:
  5468   "map f (filter P xs) = map_filter (\<lambda>x. if P x then Some (f x) else None) xs"
  5469   by (simp add: map_filter_def)
  5470 
  5471 text {* Optimized code for @{text"\<forall>i\<in>{a..b::int}"} and @{text"\<forall>n:{a..<b::nat}"}
  5472 and similiarly for @{text"\<exists>"}. *}
  5473 
  5474 definition all_interval_nat :: "(nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
  5475   "all_interval_nat P i j \<longleftrightarrow> (\<forall>n \<in> {i..<j}. P n)"
  5476 
  5477 lemma [code]:
  5478   "all_interval_nat P i j \<longleftrightarrow> i \<ge> j \<or> P i \<and> all_interval_nat P (Suc i) j"
  5479 proof -
  5480   have *: "\<And>n. P i \<Longrightarrow> \<forall>n\<in>{Suc i..<j}. P n \<Longrightarrow> i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n"
  5481   proof -
  5482     fix n
  5483     assume "P i" "\<forall>n\<in>{Suc i..<j}. P n" "i \<le> n" "n < j"
  5484     then show "P n" by (cases "n = i") simp_all
  5485   qed
  5486   show ?thesis by (auto simp add: all_interval_nat_def intro: *)
  5487 qed
  5488 
  5489 lemma list_all_iff_all_interval_nat [code_unfold]:
  5490   "list_all P [i..<j] \<longleftrightarrow> all_interval_nat P i j"
  5491   by (simp add: list_all_iff all_interval_nat_def)
  5492 
  5493 lemma list_ex_iff_not_all_inverval_nat [code_unfold]:
  5494   "list_ex P [i..<j] \<longleftrightarrow> \<not> (all_interval_nat (Not \<circ> P) i j)"
  5495   by (simp add: list_ex_iff all_interval_nat_def)
  5496 
  5497 definition all_interval_int :: "(int \<Rightarrow> bool) \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool" where
  5498   "all_interval_int P i j \<longleftrightarrow> (\<forall>k \<in> {i..j}. P k)"
  5499 
  5500 lemma [code]:
  5501   "all_interval_int P i j \<longleftrightarrow> i > j \<or> P i \<and> all_interval_int P (i + 1) j"
  5502 proof -
  5503   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"
  5504   proof -
  5505     fix k
  5506     assume "P i" "\<forall>k\<in>{i+1..j}. P k" "i \<le> k" "k \<le> j"
  5507     then show "P k" by (cases "k = i") simp_all
  5508   qed
  5509   show ?thesis by (auto simp add: all_interval_int_def intro: *)
  5510 qed
  5511 
  5512 lemma list_all_iff_all_interval_int [code_unfold]:
  5513   "list_all P [i..j] \<longleftrightarrow> all_interval_int P i j"
  5514   by (simp add: list_all_iff all_interval_int_def)
  5515 
  5516 lemma list_ex_iff_not_all_inverval_int [code_unfold]:
  5517   "list_ex P [i..j] \<longleftrightarrow> \<not> (all_interval_int (Not \<circ> P) i j)"
  5518   by (simp add: list_ex_iff all_interval_int_def)
  5519 
  5520 hide_const (open) member null maps map_filter all_interval_nat all_interval_int
  5521 
  5522 
  5523 subsubsection {* Pretty lists *}
  5524 
  5525 use "Tools/list_code.ML"
  5526 
  5527 code_type list
  5528   (SML "_ list")
  5529   (OCaml "_ list")
  5530   (Haskell "![(_)]")
  5531   (Scala "List[(_)]")
  5532 
  5533 code_const Nil
  5534   (SML "[]")
  5535   (OCaml "[]")
  5536   (Haskell "[]")
  5537   (Scala "!Nil")
  5538 
  5539 code_instance list :: equal
  5540   (Haskell -)
  5541 
  5542 code_const "HOL.equal \<Colon> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
  5543   (Haskell infix 4 "==")
  5544 
  5545 code_reserved SML
  5546   list
  5547 
  5548 code_reserved OCaml
  5549   list
  5550 
  5551 setup {* fold (List_Code.add_literal_list) ["SML", "OCaml", "Haskell", "Scala"] *}
  5552 
  5553 
  5554 subsubsection {* Use convenient predefined operations *}
  5555 
  5556 code_const "op @"
  5557   (SML infixr 7 "@")
  5558   (OCaml infixr 6 "@")
  5559   (Haskell infixr 5 "++")
  5560   (Scala infixl 7 "++")
  5561 
  5562 code_const map
  5563   (Haskell "map")
  5564 
  5565 code_const filter
  5566   (Haskell "filter")
  5567 
  5568 code_const concat
  5569   (Haskell "concat")
  5570 
  5571 code_const List.maps
  5572   (Haskell "concatMap")
  5573 
  5574 code_const rev
  5575   (Haskell "reverse")
  5576 
  5577 code_const zip
  5578   (Haskell "zip")
  5579 
  5580 code_const List.null
  5581   (Haskell "null")
  5582 
  5583 code_const takeWhile
  5584   (Haskell "takeWhile")
  5585 
  5586 code_const dropWhile
  5587   (Haskell "dropWhile")
  5588 
  5589 code_const list_all
  5590   (Haskell "all")
  5591 
  5592 code_const list_ex
  5593   (Haskell "any")
  5594 
  5595 
  5596 subsubsection {* Implementation of sets by lists *}
  5597 
  5598 lemma is_empty_set [code]:
  5599   "Set.is_empty (set xs) \<longleftrightarrow> List.null xs"
  5600   by (simp add: Set.is_empty_def null_def)
  5601 
  5602 lemma empty_set [code]:
  5603   "{} = set []"
  5604   by simp
  5605 
  5606 lemma UNIV_coset [code]:
  5607   "UNIV = List.coset []"
  5608   by simp
  5609 
  5610 lemma compl_set [code]:
  5611   "- set xs = List.coset xs"
  5612   by simp
  5613 
  5614 lemma compl_coset [code]:
  5615   "- List.coset xs = set xs"
  5616   by simp
  5617 
  5618 lemma [code]:
  5619   "x \<in> set xs \<longleftrightarrow> List.member xs x"
  5620   "x \<in> List.coset xs \<longleftrightarrow> \<not> List.member xs x"
  5621   by (simp_all add: member_def)
  5622 
  5623 lemma insert_code [code]:
  5624   "insert x (set xs) = set (List.insert x xs)"
  5625   "insert x (List.coset xs) = List.coset (removeAll x xs)"
  5626   by simp_all
  5627 
  5628 lemma remove_code [code]:
  5629   "Set.remove x (set xs) = set (removeAll x xs)"
  5630   "Set.remove x (List.coset xs) = List.coset (List.insert x xs)"
  5631   by (simp_all add: remove_def Compl_insert)
  5632 
  5633 lemma project_set [code]:
  5634   "Set.project P (set xs) = set (filter P xs)"
  5635   by auto
  5636 
  5637 lemma image_set [code]:
  5638   "image f (set xs) = set (map f xs)"
  5639   by simp
  5640 
  5641 lemma subset_code [code]:
  5642   "set xs \<le> B \<longleftrightarrow> (\<forall>x\<in>set xs. x \<in> B)"
  5643   "A \<le> List.coset ys \<longleftrightarrow> (\<forall>y\<in>set ys. y \<notin> A)"
  5644   "List.coset [] \<le> set [] \<longleftrightarrow> False"
  5645   by auto
  5646 
  5647 text {* A frequent case – avoid intermediate sets *}
  5648 lemma [code_unfold]:
  5649   "set xs \<subseteq> set ys \<longleftrightarrow> list_all (\<lambda>x. x \<in> set ys) xs"
  5650   by (auto simp: list_all_iff)
  5651 
  5652 lemma Ball_set [code]:
  5653   "Ball (set xs) P \<longleftrightarrow> list_all P xs"
  5654   by (simp add: list_all_iff)
  5655 
  5656 lemma Bex_set [code]:
  5657   "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
  5658   by (simp add: list_ex_iff)
  5659 
  5660 lemma card_set [code]:
  5661   "card (set xs) = length (remdups xs)"
  5662 proof -
  5663   have "card (set (remdups xs)) = length (remdups xs)"
  5664     by (rule distinct_card) simp
  5665   then show ?thesis by simp
  5666 qed
  5667 
  5668 lemma the_elem_set [code]:
  5669   "the_elem (set [x]) = x"
  5670   by simp
  5671 
  5672 lemma Pow_set [code]:
  5673   "Pow (set []) = {{}}"
  5674   "Pow (set (x # xs)) = (let A = Pow (set xs) in A \<union> insert x ` A)"
  5675   by (simp_all add: Pow_insert Let_def)
  5676 
  5677 lemma setsum_code [code]:
  5678   "setsum f (set xs) = listsum (map f (remdups xs))"
  5679 by (simp add: listsum_distinct_conv_setsum_set)
  5680 
  5681 definition map_project :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a set \<Rightarrow> 'b set" where
  5682   "map_project f A = {b. \<exists> a \<in> A. f a = Some b}"
  5683 
  5684 lemma [code]:
  5685   "map_project f (set xs) = set (List.map_filter f xs)"
  5686   by (auto simp add: map_project_def map_filter_def image_def)
  5687 
  5688 hide_const (open) map_project
  5689 
  5690 text {* Operations on relations *}
  5691 
  5692 lemma product_code [code]:
  5693   "Product_Type.product (set xs) (set ys) = set [(x, y). x \<leftarrow> xs, y \<leftarrow> ys]"
  5694   by (auto simp add: Product_Type.product_def)
  5695 
  5696 lemma Id_on_set [code]:
  5697   "Id_on (set xs) = set [(x, x). x \<leftarrow> xs]"
  5698   by (auto simp add: Id_on_def)
  5699 
  5700 lemma [code]:
  5701   "R `` S = List.map_project (%(x, y). if x : S then Some y else None) R"
  5702 unfolding map_project_def by (auto split: prod.split split_if_asm)
  5703 
  5704 lemma trancl_set_ntrancl [code]:
  5705   "trancl (set xs) = ntrancl (card (set xs) - 1) (set xs)"
  5706   by (simp add: finite_trancl_ntranl)
  5707 
  5708 lemma set_relcomp [code]:
  5709   "set xys O set yzs = set ([(fst xy, snd yz). xy \<leftarrow> xys, yz \<leftarrow> yzs, snd xy = fst yz])"
  5710   by (auto simp add: Bex_def)
  5711 
  5712 lemma wf_set [code]:
  5713   "wf (set xs) = acyclic (set xs)"
  5714   by (simp add: wf_iff_acyclic_if_finite)
  5715 
  5716 end
  5717