src/HOL/List.thy
author haftmann
Fri Feb 24 22:46:44 2012 +0100 (2012-02-24)
changeset 46664 1f6c140f9c72
parent 46638 fc315796794e
child 46669 c1d2ab32174a
permissions -rw-r--r--
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
     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 -- {* canonical argument order *}
    89   fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where
    90     "fold f [] = id"
    91   | "fold f (x # xs) = fold f xs \<circ> f x"
    92 
    93 definition 
    94   foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where
    95   [code_abbrev]: "foldr f xs = fold f (rev xs)"
    96 
    97 definition
    98   foldl :: "('b \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b" where
    99   "foldl f s xs = fold (\<lambda>x s. f s x)  xs s"
   100 
   101 primrec
   102   concat:: "'a list list \<Rightarrow> 'a list" where
   103     "concat [] = []"
   104   | "concat (x # xs) = x @ concat xs"
   105 
   106 definition (in monoid_add)
   107   listsum :: "'a list \<Rightarrow> 'a" where
   108   "listsum xs = foldr plus xs 0"
   109 
   110 primrec
   111   drop:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   112     drop_Nil: "drop n [] = []"
   113   | drop_Cons: "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
   114   -- {*Warning: simpset does not contain this definition, but separate
   115        theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
   116 
   117 primrec
   118   take:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   119     take_Nil:"take n [] = []"
   120   | take_Cons: "take n (x # xs) = (case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # take m xs)"
   121   -- {*Warning: simpset does not contain this definition, but separate
   122        theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
   123 
   124 primrec
   125   nth :: "'a list => nat => 'a" (infixl "!" 100) where
   126   nth_Cons: "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
   127   -- {*Warning: simpset does not contain this definition, but separate
   128        theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
   129 
   130 primrec
   131   list_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
   132     "list_update [] i v = []"
   133   | "list_update (x # xs) i v = (case i of 0 \<Rightarrow> v # xs | Suc j \<Rightarrow> x # list_update xs j v)"
   134 
   135 nonterminal lupdbinds and lupdbind
   136 
   137 syntax
   138   "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
   139   "" :: "lupdbind => lupdbinds"    ("_")
   140   "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
   141   "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
   142 
   143 translations
   144   "_LUpdate xs (_lupdbinds b bs)" == "_LUpdate (_LUpdate xs b) bs"
   145   "xs[i:=x]" == "CONST list_update xs i x"
   146 
   147 primrec
   148   takeWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   149     "takeWhile P [] = []"
   150   | "takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])"
   151 
   152 primrec
   153   dropWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   154     "dropWhile P [] = []"
   155   | "dropWhile P (x # xs) = (if P x then dropWhile P xs else x # xs)"
   156 
   157 primrec
   158   zip :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
   159     "zip xs [] = []"
   160   | zip_Cons: "zip xs (y # ys) = (case xs of [] => [] | z # zs => (z, y) # zip zs ys)"
   161   -- {*Warning: simpset does not contain this definition, but separate
   162        theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
   163 
   164 primrec 
   165   upt :: "nat \<Rightarrow> nat \<Rightarrow> nat list" ("(1[_..</_'])") where
   166     upt_0: "[i..<0] = []"
   167   | upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
   168 
   169 definition
   170   insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   171   "insert x xs = (if x \<in> set xs then xs else x # xs)"
   172 
   173 hide_const (open) insert
   174 hide_fact (open) insert_def
   175 
   176 primrec
   177   remove1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   178     "remove1 x [] = []"
   179   | "remove1 x (y # xs) = (if x = y then xs else y # remove1 x xs)"
   180 
   181 primrec
   182   removeAll :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   183     "removeAll x [] = []"
   184   | "removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)"
   185 
   186 primrec
   187   distinct :: "'a list \<Rightarrow> bool" where
   188     "distinct [] \<longleftrightarrow> True"
   189   | "distinct (x # xs) \<longleftrightarrow> x \<notin> set xs \<and> distinct xs"
   190 
   191 primrec
   192   remdups :: "'a list \<Rightarrow> 'a list" where
   193     "remdups [] = []"
   194   | "remdups (x # xs) = (if x \<in> set xs then remdups xs else x # remdups xs)"
   195 
   196 primrec
   197   replicate :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
   198     replicate_0: "replicate 0 x = []"
   199   | replicate_Suc: "replicate (Suc n) x = x # replicate n x"
   200 
   201 text {*
   202   Function @{text size} is overloaded for all datatypes. Users may
   203   refer to the list version as @{text length}. *}
   204 
   205 abbreviation
   206   length :: "'a list \<Rightarrow> nat" where
   207   "length \<equiv> size"
   208 
   209 primrec rotate1 :: "'a list \<Rightarrow> 'a list" where
   210   "rotate1 [] = []" |
   211   "rotate1 (x # xs) = xs @ [x]"
   212 
   213 definition
   214   rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   215   "rotate n = rotate1 ^^ n"
   216 
   217 definition
   218   list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
   219   "list_all2 P xs ys =
   220     (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"
   221 
   222 definition
   223   sublist :: "'a list => nat set => 'a list" where
   224   "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"
   225 
   226 fun splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   227 "splice [] ys = ys" |
   228 "splice xs [] = xs" |
   229 "splice (x#xs) (y#ys) = x # y # splice xs ys"
   230 
   231 text{*
   232 \begin{figure}[htbp]
   233 \fbox{
   234 \begin{tabular}{l}
   235 @{lemma "[a,b]@[c,d] = [a,b,c,d]" by simp}\\
   236 @{lemma "length [a,b,c] = 3" by simp}\\
   237 @{lemma "set [a,b,c] = {a,b,c}" by simp}\\
   238 @{lemma "map f [a,b,c] = [f a, f b, f c]" by simp}\\
   239 @{lemma "rev [a,b,c] = [c,b,a]" by simp}\\
   240 @{lemma "hd [a,b,c,d] = a" by simp}\\
   241 @{lemma "tl [a,b,c,d] = [b,c,d]" by simp}\\
   242 @{lemma "last [a,b,c,d] = d" by simp}\\
   243 @{lemma "butlast [a,b,c,d] = [a,b,c]" by simp}\\
   244 @{lemma[source] "filter (\<lambda>n::nat. n<2) [0,2,1] = [0,1]" by simp}\\
   245 @{lemma "concat [[a,b],[c,d,e],[],[f]] = [a,b,c,d,e,f]" by simp}\\
   246 @{lemma "fold f [a,b,c] x = f c (f b (f a x))" by simp}\\
   247 @{lemma "foldr f [a,b,c] x = f a (f b (f c x))" by (simp add: foldr_def)}\\
   248 @{lemma "foldl f x [a,b,c] = f (f (f x a) b) c" by (simp add: foldl_def)}\\
   249 @{lemma "zip [a,b,c] [x,y,z] = [(a,x),(b,y),(c,z)]" by simp}\\
   250 @{lemma "zip [a,b] [x,y,z] = [(a,x),(b,y)]" by simp}\\
   251 @{lemma "splice [a,b,c] [x,y,z] = [a,x,b,y,c,z]" by simp}\\
   252 @{lemma "splice [a,b,c,d] [x,y] = [a,x,b,y,c,d]" by simp}\\
   253 @{lemma "take 2 [a,b,c,d] = [a,b]" by simp}\\
   254 @{lemma "take 6 [a,b,c,d] = [a,b,c,d]" by simp}\\
   255 @{lemma "drop 2 [a,b,c,d] = [c,d]" by simp}\\
   256 @{lemma "drop 6 [a,b,c,d] = []" by simp}\\
   257 @{lemma "takeWhile (%n::nat. n<3) [1,2,3,0] = [1,2]" by simp}\\
   258 @{lemma "dropWhile (%n::nat. n<3) [1,2,3,0] = [3,0]" by simp}\\
   259 @{lemma "distinct [2,0,1::nat]" by simp}\\
   260 @{lemma "remdups [2,0,2,1::nat,2] = [0,1,2]" by simp}\\
   261 @{lemma "List.insert 2 [0::nat,1,2] = [0,1,2]" by (simp add: List.insert_def)}\\
   262 @{lemma "List.insert 3 [0::nat,1,2] = [3,0,1,2]" by (simp add: List.insert_def)}\\
   263 @{lemma "remove1 2 [2,0,2,1::nat,2] = [0,2,1,2]" by simp}\\
   264 @{lemma "removeAll 2 [2,0,2,1::nat,2] = [0,1]" by simp}\\
   265 @{lemma "nth [a,b,c,d] 2 = c" by simp}\\
   266 @{lemma "[a,b,c,d][2 := x] = [a,b,x,d]" by simp}\\
   267 @{lemma "sublist [a,b,c,d,e] {0,2,3} = [a,c,d]" by (simp add:sublist_def)}\\
   268 @{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by simp}\\
   269 @{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate_def eval_nat_numeral)}\\
   270 @{lemma "replicate 4 a = [a,a,a,a]" by (simp add:eval_nat_numeral)}\\
   271 @{lemma "[2..<5] = [2,3,4]" by (simp add:eval_nat_numeral)}\\
   272 @{lemma "listsum [1,2,3::nat] = 6" by (simp add: listsum_def foldr_def)}
   273 \end{tabular}}
   274 \caption{Characteristic examples}
   275 \label{fig:Characteristic}
   276 \end{figure}
   277 Figure~\ref{fig:Characteristic} shows characteristic examples
   278 that should give an intuitive understanding of the above functions.
   279 *}
   280 
   281 text{* The following simple sort functions are intended for proofs,
   282 not for efficient implementations. *}
   283 
   284 context linorder
   285 begin
   286 
   287 inductive sorted :: "'a list \<Rightarrow> bool" where
   288   Nil [iff]: "sorted []"
   289 | Cons: "\<forall>y\<in>set xs. x \<le> y \<Longrightarrow> sorted xs \<Longrightarrow> sorted (x # xs)"
   290 
   291 lemma sorted_single [iff]:
   292   "sorted [x]"
   293   by (rule sorted.Cons) auto
   294 
   295 lemma sorted_many:
   296   "x \<le> y \<Longrightarrow> sorted (y # zs) \<Longrightarrow> sorted (x # y # zs)"
   297   by (rule sorted.Cons) (cases "y # zs" rule: sorted.cases, auto)
   298 
   299 lemma sorted_many_eq [simp, code]:
   300   "sorted (x # y # zs) \<longleftrightarrow> x \<le> y \<and> sorted (y # zs)"
   301   by (auto intro: sorted_many elim: sorted.cases)
   302 
   303 lemma [code]:
   304   "sorted [] \<longleftrightarrow> True"
   305   "sorted [x] \<longleftrightarrow> True"
   306   by simp_all
   307 
   308 primrec insort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
   309   "insort_key f x [] = [x]" |
   310   "insort_key f x (y#ys) = (if f x \<le> f y then (x#y#ys) else y#(insort_key f x ys))"
   311 
   312 definition sort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b list \<Rightarrow> 'b list" where
   313   "sort_key f xs = foldr (insort_key f) xs []"
   314 
   315 definition insort_insert_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
   316   "insort_insert_key f x xs = (if f x \<in> f ` set xs then xs else insort_key f x xs)"
   317 
   318 abbreviation "sort \<equiv> sort_key (\<lambda>x. x)"
   319 abbreviation "insort \<equiv> insort_key (\<lambda>x. x)"
   320 abbreviation "insort_insert \<equiv> insort_insert_key (\<lambda>x. x)"
   321 
   322 end
   323 
   324 
   325 subsubsection {* List comprehension *}
   326 
   327 text{* Input syntax for Haskell-like list comprehension notation.
   328 Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
   329 the list of all pairs of distinct elements from @{text xs} and @{text ys}.
   330 The syntax is as in Haskell, except that @{text"|"} becomes a dot
   331 (like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
   332 \verb![e| x <- xs, ...]!.
   333 
   334 The qualifiers after the dot are
   335 \begin{description}
   336 \item[generators] @{text"p \<leftarrow> xs"},
   337  where @{text p} is a pattern and @{text xs} an expression of list type, or
   338 \item[guards] @{text"b"}, where @{text b} is a boolean expression.
   339 %\item[local bindings] @ {text"let x = e"}.
   340 \end{description}
   341 
   342 Just like in Haskell, list comprehension is just a shorthand. To avoid
   343 misunderstandings, the translation into desugared form is not reversed
   344 upon output. Note that the translation of @{text"[e. x \<leftarrow> xs]"} is
   345 optmized to @{term"map (%x. e) xs"}.
   346 
   347 It is easy to write short list comprehensions which stand for complex
   348 expressions. During proofs, they may become unreadable (and
   349 mangled). In such cases it can be advisable to introduce separate
   350 definitions for the list comprehensions in question.  *}
   351 
   352 nonterminal lc_qual and lc_quals
   353 
   354 syntax
   355   "_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
   356   "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ <- _")
   357   "_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
   358   (*"_lc_let" :: "letbinds => lc_qual"  ("let _")*)
   359   "_lc_end" :: "lc_quals" ("]")
   360   "_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals"  (", __")
   361   "_lc_abs" :: "'a => 'b list => 'b list"
   362 
   363 (* These are easier than ML code but cannot express the optimized
   364    translation of [e. p<-xs]
   365 translations
   366   "[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
   367   "_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
   368    => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
   369   "[e. P]" => "if P then [e] else []"
   370   "_listcompr e (_lc_test P) (_lc_quals Q Qs)"
   371    => "if P then (_listcompr e Q Qs) else []"
   372   "_listcompr e (_lc_let b) (_lc_quals Q Qs)"
   373    => "_Let b (_listcompr e Q Qs)"
   374 *)
   375 
   376 syntax (xsymbols)
   377   "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ \<leftarrow> _")
   378 syntax (HTML output)
   379   "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ \<leftarrow> _")
   380 
   381 parse_translation (advanced) {*
   382   let
   383     val NilC = Syntax.const @{const_syntax Nil};
   384     val ConsC = Syntax.const @{const_syntax Cons};
   385     val mapC = Syntax.const @{const_syntax map};
   386     val concatC = Syntax.const @{const_syntax concat};
   387     val IfC = Syntax.const @{const_syntax If};
   388 
   389     fun single x = ConsC $ x $ NilC;
   390 
   391     fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
   392       let
   393         (* FIXME proper name context!? *)
   394         val x =
   395           Free (singleton (Name.variant_list (fold Term.add_free_names [p, e] [])) "x", dummyT);
   396         val e = if opti then single e else e;
   397         val case1 = Syntax.const @{syntax_const "_case1"} $ p $ e;
   398         val case2 =
   399           Syntax.const @{syntax_const "_case1"} $
   400             Syntax.const @{const_syntax dummy_pattern} $ NilC;
   401         val cs = Syntax.const @{syntax_const "_case2"} $ case1 $ case2;
   402       in Syntax_Trans.abs_tr [x, Datatype_Case.case_tr false ctxt [x, cs]] end;
   403 
   404     fun abs_tr ctxt p e opti =
   405       (case Term_Position.strip_positions p of
   406         Free (s, T) =>
   407           let
   408             val thy = Proof_Context.theory_of ctxt;
   409             val s' = Proof_Context.intern_const ctxt s;
   410           in
   411             if Sign.declared_const thy s'
   412             then (pat_tr ctxt p e opti, false)
   413             else (Syntax_Trans.abs_tr [p, e], true)
   414           end
   415       | _ => (pat_tr ctxt p e opti, false));
   416 
   417     fun lc_tr ctxt [e, Const (@{syntax_const "_lc_test"}, _) $ b, qs] =
   418           let
   419             val res =
   420               (case qs of
   421                 Const (@{syntax_const "_lc_end"}, _) => single e
   422               | Const (@{syntax_const "_lc_quals"}, _) $ q $ qs => lc_tr ctxt [e, q, qs]);
   423           in IfC $ b $ res $ NilC end
   424       | lc_tr ctxt
   425             [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
   426               Const(@{syntax_const "_lc_end"}, _)] =
   427           (case abs_tr ctxt p e true of
   428             (f, true) => mapC $ f $ es
   429           | (f, false) => concatC $ (mapC $ f $ es))
   430       | lc_tr ctxt
   431             [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
   432               Const (@{syntax_const "_lc_quals"}, _) $ q $ qs] =
   433           let val e' = lc_tr ctxt [e, q, qs];
   434           in concatC $ (mapC $ (fst (abs_tr ctxt p e' false)) $ es) end;
   435 
   436   in [(@{syntax_const "_listcompr"}, lc_tr)] end
   437 *}
   438 
   439 ML {*
   440   let
   441     val read = Syntax.read_term @{context};
   442     fun check s1 s2 = read s1 aconv read s2 orelse error ("Check failed: " ^ quote s1);
   443   in
   444     check "[(x,y,z). b]" "if b then [(x, y, z)] else []";
   445     check "[(x,y,z). x\<leftarrow>xs]" "map (\<lambda>x. (x, y, z)) xs";
   446     check "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]" "concat (map (\<lambda>x. map (\<lambda>y. e x y) ys) xs)";
   447     check "[(x,y,z). x<a, x>b]" "if x < a then if b < x then [(x, y, z)] else [] else []";
   448     check "[(x,y,z). x\<leftarrow>xs, x>b]" "concat (map (\<lambda>x. if b < x then [(x, y, z)] else []) xs)";
   449     check "[(x,y,z). x<a, x\<leftarrow>xs]" "if x < a then map (\<lambda>x. (x, y, z)) xs else []";
   450     check "[(x,y). Cons True x \<leftarrow> xs]"
   451       "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | True # x \<Rightarrow> [(x, y)] | False # x \<Rightarrow> []) xs)";
   452     check "[(x,y,z). Cons x [] \<leftarrow> xs]"
   453       "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | [x] \<Rightarrow> [(x, y, z)] | x # aa # lista \<Rightarrow> []) xs)";
   454     check "[(x,y,z). x<a, x>b, x=d]"
   455       "if x < a then if b < x then if x = d then [(x, y, z)] else [] else [] else []";
   456     check "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
   457       "if x < a then if b < x then map (\<lambda>y. (x, y, z)) ys else [] else []";
   458     check "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
   459       "if x < a then concat (map (\<lambda>x. if b < y then [(x, y, z)] else []) xs) else []";
   460     check "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
   461       "if x < a then concat (map (\<lambda>x. map (\<lambda>y. (x, y, z)) ys) xs) else []";
   462     check "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
   463       "concat (map (\<lambda>x. if b < x then if y < a then [(x, y, z)] else [] else []) xs)";
   464     check "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
   465       "concat (map (\<lambda>x. if b < x then map (\<lambda>y. (x, y, z)) ys else []) xs)";
   466     check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
   467       "concat (map (\<lambda>x. concat (map (\<lambda>y. if x < y then [(x, y, z)] else []) ys)) xs)";
   468     check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
   469       "concat (map (\<lambda>x. concat (map (\<lambda>y. map (\<lambda>z. (x, y, z)) zs) ys)) xs)"
   470   end;
   471 *}
   472 
   473 (*
   474 term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
   475 *)
   476 
   477 
   478 use "Tools/list_to_set_comprehension.ML"
   479 
   480 simproc_setup list_to_set_comprehension ("set xs") = {* K List_to_Set_Comprehension.simproc *}
   481 
   482 code_datatype set coset
   483 
   484 hide_const (open) coset
   485 
   486 subsubsection {* @{const Nil} and @{const Cons} *}
   487 
   488 lemma not_Cons_self [simp]:
   489   "xs \<noteq> x # xs"
   490 by (induct xs) auto
   491 
   492 lemma not_Cons_self2 [simp]:
   493   "x # xs \<noteq> xs"
   494 by (rule not_Cons_self [symmetric])
   495 
   496 lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
   497 by (induct xs) auto
   498 
   499 lemma length_induct:
   500   "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
   501 by (rule measure_induct [of length]) iprover
   502 
   503 lemma list_nonempty_induct [consumes 1, case_names single cons]:
   504   assumes "xs \<noteq> []"
   505   assumes single: "\<And>x. P [x]"
   506   assumes cons: "\<And>x xs. xs \<noteq> [] \<Longrightarrow> P xs \<Longrightarrow> P (x # xs)"
   507   shows "P xs"
   508 using `xs \<noteq> []` proof (induct xs)
   509   case Nil then show ?case by simp
   510 next
   511   case (Cons x xs) show ?case proof (cases xs)
   512     case Nil with single show ?thesis by simp
   513   next
   514     case Cons then have "xs \<noteq> []" by simp
   515     moreover with Cons.hyps have "P xs" .
   516     ultimately show ?thesis by (rule cons)
   517   qed
   518 qed
   519 
   520 lemma inj_split_Cons: "inj_on (\<lambda>(xs, n). n#xs) X"
   521   by (auto intro!: inj_onI)
   522 
   523 subsubsection {* @{const length} *}
   524 
   525 text {*
   526   Needs to come before @{text "@"} because of theorem @{text
   527   append_eq_append_conv}.
   528 *}
   529 
   530 lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
   531 by (induct xs) auto
   532 
   533 lemma length_map [simp]: "length (map f xs) = length xs"
   534 by (induct xs) auto
   535 
   536 lemma length_rev [simp]: "length (rev xs) = length xs"
   537 by (induct xs) auto
   538 
   539 lemma length_tl [simp]: "length (tl xs) = length xs - 1"
   540 by (cases xs) auto
   541 
   542 lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
   543 by (induct xs) auto
   544 
   545 lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
   546 by (induct xs) auto
   547 
   548 lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
   549 by auto
   550 
   551 lemma length_Suc_conv:
   552 "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
   553 by (induct xs) auto
   554 
   555 lemma Suc_length_conv:
   556 "(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
   557 apply (induct xs, simp, simp)
   558 apply blast
   559 done
   560 
   561 lemma impossible_Cons: "length xs <= length ys ==> xs = x # ys = False"
   562   by (induct xs) auto
   563 
   564 lemma list_induct2 [consumes 1, case_names Nil Cons]:
   565   "length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
   566    (\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
   567    \<Longrightarrow> P xs ys"
   568 proof (induct xs arbitrary: ys)
   569   case Nil then show ?case by simp
   570 next
   571   case (Cons x xs ys) then show ?case by (cases ys) simp_all
   572 qed
   573 
   574 lemma list_induct3 [consumes 2, case_names Nil Cons]:
   575   "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
   576    (\<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))
   577    \<Longrightarrow> P xs ys zs"
   578 proof (induct xs arbitrary: ys zs)
   579   case Nil then show ?case by simp
   580 next
   581   case (Cons x xs ys zs) then show ?case by (cases ys, simp_all)
   582     (cases zs, simp_all)
   583 qed
   584 
   585 lemma list_induct4 [consumes 3, case_names Nil Cons]:
   586   "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow>
   587    P [] [] [] [] \<Longrightarrow> (\<And>x xs y ys z zs w ws. length xs = length ys \<Longrightarrow>
   588    length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow> P xs ys zs ws \<Longrightarrow>
   589    P (x#xs) (y#ys) (z#zs) (w#ws)) \<Longrightarrow> P xs ys zs ws"
   590 proof (induct xs arbitrary: ys zs ws)
   591   case Nil then show ?case by simp
   592 next
   593   case (Cons x xs ys zs ws) then show ?case by ((cases ys, simp_all), (cases zs,simp_all)) (cases ws, simp_all)
   594 qed
   595 
   596 lemma list_induct2': 
   597   "\<lbrakk> P [] [];
   598   \<And>x xs. P (x#xs) [];
   599   \<And>y ys. P [] (y#ys);
   600    \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
   601  \<Longrightarrow> P xs ys"
   602 by (induct xs arbitrary: ys) (case_tac x, auto)+
   603 
   604 lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
   605 by (rule Eq_FalseI) auto
   606 
   607 simproc_setup list_neq ("(xs::'a list) = ys") = {*
   608 (*
   609 Reduces xs=ys to False if xs and ys cannot be of the same length.
   610 This is the case if the atomic sublists of one are a submultiset
   611 of those of the other list and there are fewer Cons's in one than the other.
   612 *)
   613 
   614 let
   615 
   616 fun len (Const(@{const_name Nil},_)) acc = acc
   617   | len (Const(@{const_name Cons},_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
   618   | len (Const(@{const_name append},_) $ xs $ ys) acc = len xs (len ys acc)
   619   | len (Const(@{const_name rev},_) $ xs) acc = len xs acc
   620   | len (Const(@{const_name map},_) $ _ $ xs) acc = len xs acc
   621   | len t (ts,n) = (t::ts,n);
   622 
   623 fun list_neq _ ss ct =
   624   let
   625     val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
   626     val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
   627     fun prove_neq() =
   628       let
   629         val Type(_,listT::_) = eqT;
   630         val size = HOLogic.size_const listT;
   631         val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
   632         val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
   633         val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
   634           (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
   635       in SOME (thm RS @{thm neq_if_length_neq}) end
   636   in
   637     if m < n andalso submultiset (op aconv) (ls,rs) orelse
   638        n < m andalso submultiset (op aconv) (rs,ls)
   639     then prove_neq() else NONE
   640   end;
   641 in list_neq end;
   642 *}
   643 
   644 
   645 subsubsection {* @{text "@"} -- append *}
   646 
   647 lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
   648 by (induct xs) auto
   649 
   650 lemma append_Nil2 [simp]: "xs @ [] = xs"
   651 by (induct xs) auto
   652 
   653 lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
   654 by (induct xs) auto
   655 
   656 lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
   657 by (induct xs) auto
   658 
   659 lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
   660 by (induct xs) auto
   661 
   662 lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
   663 by (induct xs) auto
   664 
   665 lemma append_eq_append_conv [simp, no_atp]:
   666  "length xs = length ys \<or> length us = length vs
   667  ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
   668 apply (induct xs arbitrary: ys)
   669  apply (case_tac ys, simp, force)
   670 apply (case_tac ys, force, simp)
   671 done
   672 
   673 lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
   674   (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
   675 apply (induct xs arbitrary: ys zs ts)
   676  apply fastforce
   677 apply(case_tac zs)
   678  apply simp
   679 apply fastforce
   680 done
   681 
   682 lemma same_append_eq [iff, induct_simp]: "(xs @ ys = xs @ zs) = (ys = zs)"
   683 by simp
   684 
   685 lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
   686 by simp
   687 
   688 lemma append_same_eq [iff, induct_simp]: "(ys @ xs = zs @ xs) = (ys = zs)"
   689 by simp
   690 
   691 lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
   692 using append_same_eq [of _ _ "[]"] by auto
   693 
   694 lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
   695 using append_same_eq [of "[]"] by auto
   696 
   697 lemma hd_Cons_tl [simp,no_atp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
   698 by (induct xs) auto
   699 
   700 lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
   701 by (induct xs) auto
   702 
   703 lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
   704 by (simp add: hd_append split: list.split)
   705 
   706 lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
   707 by (simp split: list.split)
   708 
   709 lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
   710 by (simp add: tl_append split: list.split)
   711 
   712 
   713 lemma Cons_eq_append_conv: "x#xs = ys@zs =
   714  (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
   715 by(cases ys) auto
   716 
   717 lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
   718  (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
   719 by(cases ys) auto
   720 
   721 
   722 text {* Trivial rules for solving @{text "@"}-equations automatically. *}
   723 
   724 lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
   725 by simp
   726 
   727 lemma Cons_eq_appendI:
   728 "[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
   729 by (drule sym) simp
   730 
   731 lemma append_eq_appendI:
   732 "[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
   733 by (drule sym) simp
   734 
   735 
   736 text {*
   737 Simplification procedure for all list equalities.
   738 Currently only tries to rearrange @{text "@"} to see if
   739 - both lists end in a singleton list,
   740 - or both lists end in the same list.
   741 *}
   742 
   743 simproc_setup list_eq ("(xs::'a list) = ys")  = {*
   744   let
   745     fun last (cons as Const (@{const_name Cons}, _) $ _ $ xs) =
   746           (case xs of Const (@{const_name Nil}, _) => cons | _ => last xs)
   747       | last (Const(@{const_name append},_) $ _ $ ys) = last ys
   748       | last t = t;
   749     
   750     fun list1 (Const(@{const_name Cons},_) $ _ $ Const(@{const_name Nil},_)) = true
   751       | list1 _ = false;
   752     
   753     fun butlast ((cons as Const(@{const_name Cons},_) $ x) $ xs) =
   754           (case xs of Const (@{const_name Nil}, _) => xs | _ => cons $ butlast xs)
   755       | butlast ((app as Const (@{const_name append}, _) $ xs) $ ys) = app $ butlast ys
   756       | butlast xs = Const(@{const_name Nil}, fastype_of xs);
   757     
   758     val rearr_ss =
   759       HOL_basic_ss addsimps [@{thm append_assoc}, @{thm append_Nil}, @{thm append_Cons}];
   760     
   761     fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
   762       let
   763         val lastl = last lhs and lastr = last rhs;
   764         fun rearr conv =
   765           let
   766             val lhs1 = butlast lhs and rhs1 = butlast rhs;
   767             val Type(_,listT::_) = eqT
   768             val appT = [listT,listT] ---> listT
   769             val app = Const(@{const_name append},appT)
   770             val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
   771             val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
   772             val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
   773               (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
   774           in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
   775       in
   776         if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
   777         else if lastl aconv lastr then rearr @{thm append_same_eq}
   778         else NONE
   779       end;
   780   in fn _ => fn ss => fn ct => list_eq ss (term_of ct) end;
   781 *}
   782 
   783 
   784 subsubsection {* @{text map} *}
   785 
   786 lemma hd_map:
   787   "xs \<noteq> [] \<Longrightarrow> hd (map f xs) = f (hd xs)"
   788   by (cases xs) simp_all
   789 
   790 lemma map_tl:
   791   "map f (tl xs) = tl (map f xs)"
   792   by (cases xs) simp_all
   793 
   794 lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
   795 by (induct xs) simp_all
   796 
   797 lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
   798 by (rule ext, induct_tac xs) auto
   799 
   800 lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
   801 by (induct xs) auto
   802 
   803 lemma map_map [simp]: "map f (map g xs) = map (f \<circ> g) xs"
   804 by (induct xs) auto
   805 
   806 lemma map_comp_map[simp]: "((map f) o (map g)) = map(f o g)"
   807 apply(rule ext)
   808 apply(simp)
   809 done
   810 
   811 lemma rev_map: "rev (map f xs) = map f (rev xs)"
   812 by (induct xs) auto
   813 
   814 lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
   815 by (induct xs) auto
   816 
   817 lemma map_cong [fundef_cong]:
   818   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> map f xs = map g ys"
   819   by simp
   820 
   821 lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
   822 by (cases xs) auto
   823 
   824 lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
   825 by (cases xs) auto
   826 
   827 lemma map_eq_Cons_conv:
   828  "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
   829 by (cases xs) auto
   830 
   831 lemma Cons_eq_map_conv:
   832  "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
   833 by (cases ys) auto
   834 
   835 lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
   836 lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
   837 declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
   838 
   839 lemma ex_map_conv:
   840   "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
   841 by(induct ys, auto simp add: Cons_eq_map_conv)
   842 
   843 lemma map_eq_imp_length_eq:
   844   assumes "map f xs = map g ys"
   845   shows "length xs = length ys"
   846 using assms proof (induct ys arbitrary: xs)
   847   case Nil then show ?case by simp
   848 next
   849   case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto
   850   from Cons xs have "map f zs = map g ys" by simp
   851   moreover with Cons have "length zs = length ys" by blast
   852   with xs show ?case by simp
   853 qed
   854   
   855 lemma map_inj_on:
   856  "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
   857   ==> xs = ys"
   858 apply(frule map_eq_imp_length_eq)
   859 apply(rotate_tac -1)
   860 apply(induct rule:list_induct2)
   861  apply simp
   862 apply(simp)
   863 apply (blast intro:sym)
   864 done
   865 
   866 lemma inj_on_map_eq_map:
   867  "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
   868 by(blast dest:map_inj_on)
   869 
   870 lemma map_injective:
   871  "map f xs = map f ys ==> inj f ==> xs = ys"
   872 by (induct ys arbitrary: xs) (auto dest!:injD)
   873 
   874 lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
   875 by(blast dest:map_injective)
   876 
   877 lemma inj_mapI: "inj f ==> inj (map f)"
   878 by (iprover dest: map_injective injD intro: inj_onI)
   879 
   880 lemma inj_mapD: "inj (map f) ==> inj f"
   881 apply (unfold inj_on_def, clarify)
   882 apply (erule_tac x = "[x]" in ballE)
   883  apply (erule_tac x = "[y]" in ballE, simp, blast)
   884 apply blast
   885 done
   886 
   887 lemma inj_map[iff]: "inj (map f) = inj f"
   888 by (blast dest: inj_mapD intro: inj_mapI)
   889 
   890 lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
   891 apply(rule inj_onI)
   892 apply(erule map_inj_on)
   893 apply(blast intro:inj_onI dest:inj_onD)
   894 done
   895 
   896 lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
   897 by (induct xs, auto)
   898 
   899 lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
   900 by (induct xs) auto
   901 
   902 lemma map_fst_zip[simp]:
   903   "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
   904 by (induct rule:list_induct2, simp_all)
   905 
   906 lemma map_snd_zip[simp]:
   907   "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
   908 by (induct rule:list_induct2, simp_all)
   909 
   910 enriched_type map: map
   911   by (simp_all add: fun_eq_iff id_def)
   912 
   913 
   914 subsubsection {* @{text rev} *}
   915 
   916 lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
   917 by (induct xs) auto
   918 
   919 lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
   920 by (induct xs) auto
   921 
   922 lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
   923 by auto
   924 
   925 lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
   926 by (induct xs) auto
   927 
   928 lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
   929 by (induct xs) auto
   930 
   931 lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
   932 by (cases xs) auto
   933 
   934 lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
   935 by (cases xs) auto
   936 
   937 lemma rev_is_rev_conv [iff, no_atp]: "(rev xs = rev ys) = (xs = ys)"
   938 apply (induct xs arbitrary: ys, force)
   939 apply (case_tac ys, simp, force)
   940 done
   941 
   942 lemma inj_on_rev[iff]: "inj_on rev A"
   943 by(simp add:inj_on_def)
   944 
   945 lemma rev_induct [case_names Nil snoc]:
   946   "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
   947 apply(simplesubst rev_rev_ident[symmetric])
   948 apply(rule_tac list = "rev xs" in list.induct, simp_all)
   949 done
   950 
   951 lemma rev_exhaust [case_names Nil snoc]:
   952   "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
   953 by (induct xs rule: rev_induct) auto
   954 
   955 lemmas rev_cases = rev_exhaust
   956 
   957 lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
   958 by(rule rev_cases[of xs]) auto
   959 
   960 
   961 subsubsection {* @{text set} *}
   962 
   963 lemma finite_set [iff]: "finite (set xs)"
   964 by (induct xs) auto
   965 
   966 lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
   967 by (induct xs) auto
   968 
   969 lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
   970 by(cases xs) auto
   971 
   972 lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
   973 by auto
   974 
   975 lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
   976 by auto
   977 
   978 lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
   979 by (induct xs) auto
   980 
   981 lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
   982 by(induct xs) auto
   983 
   984 lemma set_rev [simp]: "set (rev xs) = set xs"
   985 by (induct xs) auto
   986 
   987 lemma set_map [simp]: "set (map f xs) = f`(set xs)"
   988 by (induct xs) auto
   989 
   990 lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
   991 by (induct xs) auto
   992 
   993 lemma set_upt [simp]: "set[i..<j] = {i..<j}"
   994 by (induct j) auto
   995 
   996 
   997 lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs"
   998 proof (induct xs)
   999   case Nil thus ?case by simp
  1000 next
  1001   case Cons thus ?case by (auto intro: Cons_eq_appendI)
  1002 qed
  1003 
  1004 lemma in_set_conv_decomp: "x \<in> set xs \<longleftrightarrow> (\<exists>ys zs. xs = ys @ x # zs)"
  1005   by (auto elim: split_list)
  1006 
  1007 lemma split_list_first: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
  1008 proof (induct xs)
  1009   case Nil thus ?case by simp
  1010 next
  1011   case (Cons a xs)
  1012   show ?case
  1013   proof cases
  1014     assume "x = a" thus ?case using Cons by fastforce
  1015   next
  1016     assume "x \<noteq> a" thus ?case using Cons by(fastforce intro!: Cons_eq_appendI)
  1017   qed
  1018 qed
  1019 
  1020 lemma in_set_conv_decomp_first:
  1021   "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
  1022   by (auto dest!: split_list_first)
  1023 
  1024 lemma split_list_last: "x \<in> set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
  1025 proof (induct xs rule: rev_induct)
  1026   case Nil thus ?case by simp
  1027 next
  1028   case (snoc a xs)
  1029   show ?case
  1030   proof cases
  1031     assume "x = a" thus ?case using snoc by (metis List.set.simps(1) emptyE)
  1032   next
  1033     assume "x \<noteq> a" thus ?case using snoc by fastforce
  1034   qed
  1035 qed
  1036 
  1037 lemma in_set_conv_decomp_last:
  1038   "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
  1039   by (auto dest!: split_list_last)
  1040 
  1041 lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs & P x"
  1042 proof (induct xs)
  1043   case Nil thus ?case by simp
  1044 next
  1045   case Cons thus ?case
  1046     by(simp add:Bex_def)(metis append_Cons append.simps(1))
  1047 qed
  1048 
  1049 lemma split_list_propE:
  1050   assumes "\<exists>x \<in> set xs. P x"
  1051   obtains ys x zs where "xs = ys @ x # zs" and "P x"
  1052 using split_list_prop [OF assms] by blast
  1053 
  1054 lemma split_list_first_prop:
  1055   "\<exists>x \<in> set xs. P x \<Longrightarrow>
  1056    \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)"
  1057 proof (induct xs)
  1058   case Nil thus ?case by simp
  1059 next
  1060   case (Cons x xs)
  1061   show ?case
  1062   proof cases
  1063     assume "P x"
  1064     thus ?thesis by simp (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
  1065   next
  1066     assume "\<not> P x"
  1067     hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
  1068     thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
  1069   qed
  1070 qed
  1071 
  1072 lemma split_list_first_propE:
  1073   assumes "\<exists>x \<in> set xs. P x"
  1074   obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y"
  1075 using split_list_first_prop [OF assms] by blast
  1076 
  1077 lemma split_list_first_prop_iff:
  1078   "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
  1079    (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))"
  1080 by (rule, erule split_list_first_prop) auto
  1081 
  1082 lemma split_list_last_prop:
  1083   "\<exists>x \<in> set xs. P x \<Longrightarrow>
  1084    \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)"
  1085 proof(induct xs rule:rev_induct)
  1086   case Nil thus ?case by simp
  1087 next
  1088   case (snoc x xs)
  1089   show ?case
  1090   proof cases
  1091     assume "P x" thus ?thesis by (metis emptyE set_empty)
  1092   next
  1093     assume "\<not> P x"
  1094     hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
  1095     thus ?thesis using `\<not> P x` snoc(1) by fastforce
  1096   qed
  1097 qed
  1098 
  1099 lemma split_list_last_propE:
  1100   assumes "\<exists>x \<in> set xs. P x"
  1101   obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z"
  1102 using split_list_last_prop [OF assms] by blast
  1103 
  1104 lemma split_list_last_prop_iff:
  1105   "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
  1106    (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
  1107 by (metis split_list_last_prop [where P=P] in_set_conv_decomp)
  1108 
  1109 lemma finite_list: "finite A ==> EX xs. set xs = A"
  1110   by (erule finite_induct)
  1111     (auto simp add: set.simps(2) [symmetric] simp del: set.simps(2))
  1112 
  1113 lemma card_length: "card (set xs) \<le> length xs"
  1114 by (induct xs) (auto simp add: card_insert_if)
  1115 
  1116 lemma set_minus_filter_out:
  1117   "set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)"
  1118   by (induct xs) auto
  1119 
  1120 
  1121 subsubsection {* @{text filter} *}
  1122 
  1123 lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
  1124 by (induct xs) auto
  1125 
  1126 lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
  1127 by (induct xs) simp_all
  1128 
  1129 lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
  1130 by (induct xs) auto
  1131 
  1132 lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
  1133 by (induct xs) (auto simp add: le_SucI)
  1134 
  1135 lemma sum_length_filter_compl:
  1136   "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
  1137 by(induct xs) simp_all
  1138 
  1139 lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
  1140 by (induct xs) auto
  1141 
  1142 lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
  1143 by (induct xs) auto
  1144 
  1145 lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
  1146 by (induct xs) simp_all
  1147 
  1148 lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
  1149 apply (induct xs)
  1150  apply auto
  1151 apply(cut_tac P=P and xs=xs in length_filter_le)
  1152 apply simp
  1153 done
  1154 
  1155 lemma filter_map:
  1156   "filter P (map f xs) = map f (filter (P o f) xs)"
  1157 by (induct xs) simp_all
  1158 
  1159 lemma length_filter_map[simp]:
  1160   "length (filter P (map f xs)) = length(filter (P o f) xs)"
  1161 by (simp add:filter_map)
  1162 
  1163 lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
  1164 by auto
  1165 
  1166 lemma length_filter_less:
  1167   "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
  1168 proof (induct xs)
  1169   case Nil thus ?case by simp
  1170 next
  1171   case (Cons x xs) thus ?case
  1172     apply (auto split:split_if_asm)
  1173     using length_filter_le[of P xs] apply arith
  1174   done
  1175 qed
  1176 
  1177 lemma length_filter_conv_card:
  1178  "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
  1179 proof (induct xs)
  1180   case Nil thus ?case by simp
  1181 next
  1182   case (Cons x xs)
  1183   let ?S = "{i. i < length xs & p(xs!i)}"
  1184   have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
  1185   show ?case (is "?l = card ?S'")
  1186   proof (cases)
  1187     assume "p x"
  1188     hence eq: "?S' = insert 0 (Suc ` ?S)"
  1189       by(auto simp: image_def split:nat.split dest:gr0_implies_Suc)
  1190     have "length (filter p (x # xs)) = Suc(card ?S)"
  1191       using Cons `p x` by simp
  1192     also have "\<dots> = Suc(card(Suc ` ?S))" using fin
  1193       by (simp add: card_image)
  1194     also have "\<dots> = card ?S'" using eq fin
  1195       by (simp add:card_insert_if) (simp add:image_def)
  1196     finally show ?thesis .
  1197   next
  1198     assume "\<not> p x"
  1199     hence eq: "?S' = Suc ` ?S"
  1200       by(auto simp add: image_def split:nat.split elim:lessE)
  1201     have "length (filter p (x # xs)) = card ?S"
  1202       using Cons `\<not> p x` by simp
  1203     also have "\<dots> = card(Suc ` ?S)" using fin
  1204       by (simp add: card_image)
  1205     also have "\<dots> = card ?S'" using eq fin
  1206       by (simp add:card_insert_if)
  1207     finally show ?thesis .
  1208   qed
  1209 qed
  1210 
  1211 lemma Cons_eq_filterD:
  1212  "x#xs = filter P ys \<Longrightarrow>
  1213   \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
  1214   (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
  1215 proof(induct ys)
  1216   case Nil thus ?case by simp
  1217 next
  1218   case (Cons y ys)
  1219   show ?case (is "\<exists>x. ?Q x")
  1220   proof cases
  1221     assume Py: "P y"
  1222     show ?thesis
  1223     proof cases
  1224       assume "x = y"
  1225       with Py Cons.prems have "?Q []" by simp
  1226       then show ?thesis ..
  1227     next
  1228       assume "x \<noteq> y"
  1229       with Py Cons.prems show ?thesis by simp
  1230     qed
  1231   next
  1232     assume "\<not> P y"
  1233     with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastforce
  1234     then have "?Q (y#us)" by simp
  1235     then show ?thesis ..
  1236   qed
  1237 qed
  1238 
  1239 lemma filter_eq_ConsD:
  1240  "filter P ys = x#xs \<Longrightarrow>
  1241   \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
  1242 by(rule Cons_eq_filterD) simp
  1243 
  1244 lemma filter_eq_Cons_iff:
  1245  "(filter P ys = x#xs) =
  1246   (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
  1247 by(auto dest:filter_eq_ConsD)
  1248 
  1249 lemma Cons_eq_filter_iff:
  1250  "(x#xs = filter P ys) =
  1251   (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
  1252 by(auto dest:Cons_eq_filterD)
  1253 
  1254 lemma filter_cong[fundef_cong]:
  1255  "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
  1256 apply simp
  1257 apply(erule thin_rl)
  1258 by (induct ys) simp_all
  1259 
  1260 
  1261 subsubsection {* List partitioning *}
  1262 
  1263 primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where
  1264   "partition P [] = ([], [])"
  1265   | "partition P (x # xs) = 
  1266       (let (yes, no) = partition P xs
  1267       in if P x then (x # yes, no) else (yes, x # no))"
  1268 
  1269 lemma partition_filter1:
  1270     "fst (partition P xs) = filter P xs"
  1271 by (induct xs) (auto simp add: Let_def split_def)
  1272 
  1273 lemma partition_filter2:
  1274     "snd (partition P xs) = filter (Not o P) xs"
  1275 by (induct xs) (auto simp add: Let_def split_def)
  1276 
  1277 lemma partition_P:
  1278   assumes "partition P xs = (yes, no)"
  1279   shows "(\<forall>p \<in> set yes.  P p) \<and> (\<forall>p  \<in> set no. \<not> P p)"
  1280 proof -
  1281   from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
  1282     by simp_all
  1283   then show ?thesis by (simp_all add: partition_filter1 partition_filter2)
  1284 qed
  1285 
  1286 lemma partition_set:
  1287   assumes "partition P xs = (yes, no)"
  1288   shows "set yes \<union> set no = set xs"
  1289 proof -
  1290   from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
  1291     by simp_all
  1292   then show ?thesis by (auto simp add: partition_filter1 partition_filter2) 
  1293 qed
  1294 
  1295 lemma partition_filter_conv[simp]:
  1296   "partition f xs = (filter f xs,filter (Not o f) xs)"
  1297 unfolding partition_filter2[symmetric]
  1298 unfolding partition_filter1[symmetric] by simp
  1299 
  1300 declare partition.simps[simp del]
  1301 
  1302 
  1303 subsubsection {* @{text concat} *}
  1304 
  1305 lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
  1306 by (induct xs) auto
  1307 
  1308 lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
  1309 by (induct xss) auto
  1310 
  1311 lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
  1312 by (induct xss) auto
  1313 
  1314 lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
  1315 by (induct xs) auto
  1316 
  1317 lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs"
  1318 by (induct xs) auto
  1319 
  1320 lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
  1321 by (induct xs) auto
  1322 
  1323 lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
  1324 by (induct xs) auto
  1325 
  1326 lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
  1327 by (induct xs) auto
  1328 
  1329 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)"
  1330 proof (induct xs arbitrary: ys)
  1331   case (Cons x xs ys)
  1332   thus ?case by (cases ys) auto
  1333 qed (auto)
  1334 
  1335 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"
  1336 by (simp add: concat_eq_concat_iff)
  1337 
  1338 
  1339 subsubsection {* @{text nth} *}
  1340 
  1341 lemma nth_Cons_0 [simp, code]: "(x # xs)!0 = x"
  1342 by auto
  1343 
  1344 lemma nth_Cons_Suc [simp, code]: "(x # xs)!(Suc n) = xs!n"
  1345 by auto
  1346 
  1347 declare nth.simps [simp del]
  1348 
  1349 lemma nth_Cons_pos[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
  1350 by(auto simp: Nat.gr0_conv_Suc)
  1351 
  1352 lemma nth_append:
  1353   "(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
  1354 apply (induct xs arbitrary: n, simp)
  1355 apply (case_tac n, auto)
  1356 done
  1357 
  1358 lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
  1359 by (induct xs) auto
  1360 
  1361 lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
  1362 by (induct xs) auto
  1363 
  1364 lemma nth_map [simp]: "n < length xs ==> (map f xs)!n = f(xs!n)"
  1365 apply (induct xs arbitrary: n, simp)
  1366 apply (case_tac n, auto)
  1367 done
  1368 
  1369 lemma nth_tl:
  1370   assumes "n < length (tl x)" shows "tl x ! n = x ! Suc n"
  1371 using assms by (induct x) auto
  1372 
  1373 lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
  1374 by(cases xs) simp_all
  1375 
  1376 
  1377 lemma list_eq_iff_nth_eq:
  1378  "(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
  1379 apply(induct xs arbitrary: ys)
  1380  apply force
  1381 apply(case_tac ys)
  1382  apply simp
  1383 apply(simp add:nth_Cons split:nat.split)apply blast
  1384 done
  1385 
  1386 lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
  1387 apply (induct xs, simp, simp)
  1388 apply safe
  1389 apply (metis nat_case_0 nth.simps zero_less_Suc)
  1390 apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
  1391 apply (case_tac i, simp)
  1392 apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff)
  1393 done
  1394 
  1395 lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
  1396 by(auto simp:set_conv_nth)
  1397 
  1398 lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
  1399 by (auto simp add: set_conv_nth)
  1400 
  1401 lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
  1402 by (auto simp add: set_conv_nth)
  1403 
  1404 lemma all_nth_imp_all_set:
  1405 "[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
  1406 by (auto simp add: set_conv_nth)
  1407 
  1408 lemma all_set_conv_all_nth:
  1409 "(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
  1410 by (auto simp add: set_conv_nth)
  1411 
  1412 lemma rev_nth:
  1413   "n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)"
  1414 proof (induct xs arbitrary: n)
  1415   case Nil thus ?case by simp
  1416 next
  1417   case (Cons x xs)
  1418   hence n: "n < Suc (length xs)" by simp
  1419   moreover
  1420   { assume "n < length xs"
  1421     with n obtain n' where "length xs - n = Suc n'"
  1422       by (cases "length xs - n", auto)
  1423     moreover
  1424     then have "length xs - Suc n = n'" by simp
  1425     ultimately
  1426     have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp
  1427   }
  1428   ultimately
  1429   show ?case by (clarsimp simp add: Cons nth_append)
  1430 qed
  1431 
  1432 lemma Skolem_list_nth:
  1433   "(ALL i<k. EX x. P i x) = (EX xs. size xs = k & (ALL i<k. P i (xs!i)))"
  1434   (is "_ = (EX xs. ?P k xs)")
  1435 proof(induct k)
  1436   case 0 show ?case by simp
  1437 next
  1438   case (Suc k)
  1439   show ?case (is "?L = ?R" is "_ = (EX xs. ?P' xs)")
  1440   proof
  1441     assume "?R" thus "?L" using Suc by auto
  1442   next
  1443     assume "?L"
  1444     with Suc obtain x xs where "?P k xs & P k x" by (metis less_Suc_eq)
  1445     hence "?P'(xs@[x])" by(simp add:nth_append less_Suc_eq)
  1446     thus "?R" ..
  1447   qed
  1448 qed
  1449 
  1450 
  1451 subsubsection {* @{text list_update} *}
  1452 
  1453 lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
  1454 by (induct xs arbitrary: i) (auto split: nat.split)
  1455 
  1456 lemma nth_list_update:
  1457 "i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
  1458 by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
  1459 
  1460 lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
  1461 by (simp add: nth_list_update)
  1462 
  1463 lemma nth_list_update_neq [simp]: "i \<noteq> j ==> xs[i:=x]!j = xs!j"
  1464 by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
  1465 
  1466 lemma list_update_id[simp]: "xs[i := xs!i] = xs"
  1467 by (induct xs arbitrary: i) (simp_all split:nat.splits)
  1468 
  1469 lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
  1470 apply (induct xs arbitrary: i)
  1471  apply simp
  1472 apply (case_tac i)
  1473 apply simp_all
  1474 done
  1475 
  1476 lemma list_update_nonempty[simp]: "xs[k:=x] = [] \<longleftrightarrow> xs=[]"
  1477 by(metis length_0_conv length_list_update)
  1478 
  1479 lemma list_update_same_conv:
  1480 "i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
  1481 by (induct xs arbitrary: i) (auto split: nat.split)
  1482 
  1483 lemma list_update_append1:
  1484  "i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
  1485 apply (induct xs arbitrary: i, simp)
  1486 apply(simp split:nat.split)
  1487 done
  1488 
  1489 lemma list_update_append:
  1490   "(xs @ ys) [n:= x] = 
  1491   (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
  1492 by (induct xs arbitrary: n) (auto split:nat.splits)
  1493 
  1494 lemma list_update_length [simp]:
  1495  "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
  1496 by (induct xs, auto)
  1497 
  1498 lemma map_update: "map f (xs[k:= y]) = (map f xs)[k := f y]"
  1499 by(induct xs arbitrary: k)(auto split:nat.splits)
  1500 
  1501 lemma rev_update:
  1502   "k < length xs \<Longrightarrow> rev (xs[k:= y]) = (rev xs)[length xs - k - 1 := y]"
  1503 by (induct xs arbitrary: k) (auto simp: list_update_append split:nat.splits)
  1504 
  1505 lemma update_zip:
  1506   "(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
  1507 by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split)
  1508 
  1509 lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)"
  1510 by (induct xs arbitrary: i) (auto split: nat.split)
  1511 
  1512 lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
  1513 by (blast dest!: set_update_subset_insert [THEN subsetD])
  1514 
  1515 lemma set_update_memI: "n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
  1516 by (induct xs arbitrary: n) (auto split:nat.splits)
  1517 
  1518 lemma list_update_overwrite[simp]:
  1519   "xs [i := x, i := y] = xs [i := y]"
  1520 apply (induct xs arbitrary: i) apply simp
  1521 apply (case_tac i, simp_all)
  1522 done
  1523 
  1524 lemma list_update_swap:
  1525   "i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]"
  1526 apply (induct xs arbitrary: i i')
  1527 apply simp
  1528 apply (case_tac i, case_tac i')
  1529 apply auto
  1530 apply (case_tac i')
  1531 apply auto
  1532 done
  1533 
  1534 lemma list_update_code [code]:
  1535   "[][i := y] = []"
  1536   "(x # xs)[0 := y] = y # xs"
  1537   "(x # xs)[Suc i := y] = x # xs[i := y]"
  1538   by simp_all
  1539 
  1540 
  1541 subsubsection {* @{text last} and @{text butlast} *}
  1542 
  1543 lemma last_snoc [simp]: "last (xs @ [x]) = x"
  1544 by (induct xs) auto
  1545 
  1546 lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
  1547 by (induct xs) auto
  1548 
  1549 lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
  1550   by simp
  1551 
  1552 lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
  1553   by simp
  1554 
  1555 lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
  1556 by (induct xs) (auto)
  1557 
  1558 lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
  1559 by(simp add:last_append)
  1560 
  1561 lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
  1562 by(simp add:last_append)
  1563 
  1564 lemma last_tl: "xs = [] \<or> tl xs \<noteq> [] \<Longrightarrow>last (tl xs) = last xs"
  1565 by (induct xs) simp_all
  1566 
  1567 lemma butlast_tl: "butlast (tl xs) = tl (butlast xs)"
  1568 by (induct xs) simp_all
  1569 
  1570 lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
  1571 by(rule rev_exhaust[of xs]) simp_all
  1572 
  1573 lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
  1574 by(cases xs) simp_all
  1575 
  1576 lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
  1577 by (induct as) auto
  1578 
  1579 lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
  1580 by (induct xs rule: rev_induct) auto
  1581 
  1582 lemma butlast_append:
  1583   "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
  1584 by (induct xs arbitrary: ys) auto
  1585 
  1586 lemma append_butlast_last_id [simp]:
  1587 "xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
  1588 by (induct xs) auto
  1589 
  1590 lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
  1591 by (induct xs) (auto split: split_if_asm)
  1592 
  1593 lemma in_set_butlast_appendI:
  1594 "x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
  1595 by (auto dest: in_set_butlastD simp add: butlast_append)
  1596 
  1597 lemma last_drop[simp]: "n < length xs \<Longrightarrow> last (drop n xs) = last xs"
  1598 apply (induct xs arbitrary: n)
  1599  apply simp
  1600 apply (auto split:nat.split)
  1601 done
  1602 
  1603 lemma nth_butlast:
  1604   assumes "n < length (butlast xs)" shows "butlast xs ! n = xs ! n"
  1605 proof (cases xs)
  1606   case (Cons y ys)
  1607   moreover from assms have "butlast xs ! n = (butlast xs @ [last xs]) ! n"
  1608     by (simp add: nth_append)
  1609   ultimately show ?thesis using append_butlast_last_id by simp
  1610 qed simp
  1611 
  1612 lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
  1613 by(induct xs)(auto simp:neq_Nil_conv)
  1614 
  1615 lemma butlast_conv_take: "butlast xs = take (length xs - 1) xs"
  1616 by (induct xs, simp, case_tac xs, simp_all)
  1617 
  1618 lemma last_list_update:
  1619   "xs \<noteq> [] \<Longrightarrow> last(xs[k:=x]) = (if k = size xs - 1 then x else last xs)"
  1620 by (auto simp: last_conv_nth)
  1621 
  1622 lemma butlast_list_update:
  1623   "butlast(xs[k:=x]) =
  1624  (if k = size xs - 1 then butlast xs else (butlast xs)[k:=x])"
  1625 apply(cases xs rule:rev_cases)
  1626 apply simp
  1627 apply(simp add:list_update_append split:nat.splits)
  1628 done
  1629 
  1630 lemma last_map:
  1631   "xs \<noteq> [] \<Longrightarrow> last (map f xs) = f (last xs)"
  1632   by (cases xs rule: rev_cases) simp_all
  1633 
  1634 lemma map_butlast:
  1635   "map f (butlast xs) = butlast (map f xs)"
  1636   by (induct xs) simp_all
  1637 
  1638 lemma snoc_eq_iff_butlast:
  1639   "xs @ [x] = ys \<longleftrightarrow> (ys \<noteq> [] & butlast ys = xs & last ys = x)"
  1640 by (metis append_butlast_last_id append_is_Nil_conv butlast_snoc last_snoc not_Cons_self)
  1641 
  1642 
  1643 subsubsection {* @{text take} and @{text drop} *}
  1644 
  1645 lemma take_0 [simp]: "take 0 xs = []"
  1646 by (induct xs) auto
  1647 
  1648 lemma drop_0 [simp]: "drop 0 xs = xs"
  1649 by (induct xs) auto
  1650 
  1651 lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
  1652 by simp
  1653 
  1654 lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
  1655 by simp
  1656 
  1657 declare take_Cons [simp del] and drop_Cons [simp del]
  1658 
  1659 lemma take_1_Cons [simp]: "take 1 (x # xs) = [x]"
  1660   unfolding One_nat_def by simp
  1661 
  1662 lemma drop_1_Cons [simp]: "drop 1 (x # xs) = xs"
  1663   unfolding One_nat_def by simp
  1664 
  1665 lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
  1666 by(clarsimp simp add:neq_Nil_conv)
  1667 
  1668 lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
  1669 by(cases xs, simp_all)
  1670 
  1671 lemma take_tl: "take n (tl xs) = tl (take (Suc n) xs)"
  1672 by (induct xs arbitrary: n) simp_all
  1673 
  1674 lemma drop_tl: "drop n (tl xs) = tl(drop n xs)"
  1675 by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split)
  1676 
  1677 lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)"
  1678 by (cases n, simp, cases xs, auto)
  1679 
  1680 lemma tl_drop: "tl (drop n xs) = drop n (tl xs)"
  1681 by (simp only: drop_tl)
  1682 
  1683 lemma nth_via_drop: "drop n xs = y#ys \<Longrightarrow> xs!n = y"
  1684 apply (induct xs arbitrary: n, simp)
  1685 apply(simp add:drop_Cons nth_Cons split:nat.splits)
  1686 done
  1687 
  1688 lemma take_Suc_conv_app_nth:
  1689   "i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
  1690 apply (induct xs arbitrary: i, simp)
  1691 apply (case_tac i, auto)
  1692 done
  1693 
  1694 lemma drop_Suc_conv_tl:
  1695   "i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
  1696 apply (induct xs arbitrary: i, simp)
  1697 apply (case_tac i, auto)
  1698 done
  1699 
  1700 lemma length_take [simp]: "length (take n xs) = min (length xs) n"
  1701 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1702 
  1703 lemma length_drop [simp]: "length (drop n xs) = (length xs - n)"
  1704 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1705 
  1706 lemma take_all [simp]: "length xs <= n ==> take n xs = xs"
  1707 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1708 
  1709 lemma drop_all [simp]: "length xs <= n ==> drop n xs = []"
  1710 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1711 
  1712 lemma take_append [simp]:
  1713   "take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
  1714 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1715 
  1716 lemma drop_append [simp]:
  1717   "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
  1718 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1719 
  1720 lemma take_take [simp]: "take n (take m xs) = take (min n m) xs"
  1721 apply (induct m arbitrary: xs n, auto)
  1722 apply (case_tac xs, auto)
  1723 apply (case_tac n, auto)
  1724 done
  1725 
  1726 lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs"
  1727 apply (induct m arbitrary: xs, auto)
  1728 apply (case_tac xs, auto)
  1729 done
  1730 
  1731 lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)"
  1732 apply (induct m arbitrary: xs n, auto)
  1733 apply (case_tac xs, auto)
  1734 done
  1735 
  1736 lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)"
  1737 apply(induct xs arbitrary: m n)
  1738  apply simp
  1739 apply(simp add: take_Cons drop_Cons split:nat.split)
  1740 done
  1741 
  1742 lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs"
  1743 apply (induct n arbitrary: xs, auto)
  1744 apply (case_tac xs, auto)
  1745 done
  1746 
  1747 lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])"
  1748 apply(induct xs arbitrary: n)
  1749  apply simp
  1750 apply(simp add:take_Cons split:nat.split)
  1751 done
  1752 
  1753 lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)"
  1754 apply(induct xs arbitrary: n)
  1755 apply simp
  1756 apply(simp add:drop_Cons split:nat.split)
  1757 done
  1758 
  1759 lemma take_map: "take n (map f xs) = map f (take n xs)"
  1760 apply (induct n arbitrary: xs, auto)
  1761 apply (case_tac xs, auto)
  1762 done
  1763 
  1764 lemma drop_map: "drop n (map f xs) = map f (drop n xs)"
  1765 apply (induct n arbitrary: xs, auto)
  1766 apply (case_tac xs, auto)
  1767 done
  1768 
  1769 lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)"
  1770 apply (induct xs arbitrary: i, auto)
  1771 apply (case_tac i, auto)
  1772 done
  1773 
  1774 lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)"
  1775 apply (induct xs arbitrary: i, auto)
  1776 apply (case_tac i, auto)
  1777 done
  1778 
  1779 lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i"
  1780 apply (induct xs arbitrary: i n, auto)
  1781 apply (case_tac n, blast)
  1782 apply (case_tac i, auto)
  1783 done
  1784 
  1785 lemma nth_drop [simp]:
  1786   "n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
  1787 apply (induct n arbitrary: xs i, auto)
  1788 apply (case_tac xs, auto)
  1789 done
  1790 
  1791 lemma butlast_take:
  1792   "n <= length xs ==> butlast (take n xs) = take (n - 1) xs"
  1793 by (simp add: butlast_conv_take min_max.inf_absorb1 min_max.inf_absorb2)
  1794 
  1795 lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)"
  1796 by (simp add: butlast_conv_take drop_take add_ac)
  1797 
  1798 lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs"
  1799 by (simp add: butlast_conv_take min_max.inf_absorb1)
  1800 
  1801 lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)"
  1802 by (simp add: butlast_conv_take drop_take add_ac)
  1803 
  1804 lemma hd_drop_conv_nth: "n < length xs \<Longrightarrow> hd(drop n xs) = xs!n"
  1805 by(simp add: hd_conv_nth)
  1806 
  1807 lemma set_take_subset_set_take:
  1808   "m <= n \<Longrightarrow> set(take m xs) <= set(take n xs)"
  1809 apply (induct xs arbitrary: m n)
  1810 apply simp
  1811 apply (case_tac n)
  1812 apply (auto simp: take_Cons)
  1813 done
  1814 
  1815 lemma set_take_subset: "set(take n xs) \<subseteq> set xs"
  1816 by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split)
  1817 
  1818 lemma set_drop_subset: "set(drop n xs) \<subseteq> set xs"
  1819 by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split)
  1820 
  1821 lemma set_drop_subset_set_drop:
  1822   "m >= n \<Longrightarrow> set(drop m xs) <= set(drop n xs)"
  1823 apply(induct xs arbitrary: m n)
  1824 apply(auto simp:drop_Cons split:nat.split)
  1825 apply (metis set_drop_subset subset_iff)
  1826 done
  1827 
  1828 lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
  1829 using set_take_subset by fast
  1830 
  1831 lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
  1832 using set_drop_subset by fast
  1833 
  1834 lemma append_eq_conv_conj:
  1835   "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
  1836 apply (induct xs arbitrary: zs, simp, clarsimp)
  1837 apply (case_tac zs, auto)
  1838 done
  1839 
  1840 lemma take_add: 
  1841   "take (i+j) xs = take i xs @ take j (drop i xs)"
  1842 apply (induct xs arbitrary: i, auto) 
  1843 apply (case_tac i, simp_all)
  1844 done
  1845 
  1846 lemma append_eq_append_conv_if:
  1847  "(xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
  1848   (if size xs\<^isub>1 \<le> size ys\<^isub>1
  1849    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
  1850    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)"
  1851 apply(induct xs\<^isub>1 arbitrary: ys\<^isub>1)
  1852  apply simp
  1853 apply(case_tac ys\<^isub>1)
  1854 apply simp_all
  1855 done
  1856 
  1857 lemma take_hd_drop:
  1858   "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (Suc n) xs"
  1859 apply(induct xs arbitrary: n)
  1860 apply simp
  1861 apply(simp add:drop_Cons split:nat.split)
  1862 done
  1863 
  1864 lemma id_take_nth_drop:
  1865  "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
  1866 proof -
  1867   assume si: "i < length xs"
  1868   hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
  1869   moreover
  1870   from si have "take (Suc i) xs = take i xs @ [xs!i]"
  1871     apply (rule_tac take_Suc_conv_app_nth) by arith
  1872   ultimately show ?thesis by auto
  1873 qed
  1874   
  1875 lemma upd_conv_take_nth_drop:
  1876  "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
  1877 proof -
  1878   assume i: "i < length xs"
  1879   have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
  1880     by(rule arg_cong[OF id_take_nth_drop[OF i]])
  1881   also have "\<dots> = take i xs @ a # drop (Suc i) xs"
  1882     using i by (simp add: list_update_append)
  1883   finally show ?thesis .
  1884 qed
  1885 
  1886 lemma nth_drop':
  1887   "i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs"
  1888 apply (induct i arbitrary: xs)
  1889 apply (simp add: neq_Nil_conv)
  1890 apply (erule exE)+
  1891 apply simp
  1892 apply (case_tac xs)
  1893 apply simp_all
  1894 done
  1895 
  1896 
  1897 subsubsection {* @{text takeWhile} and @{text dropWhile} *}
  1898 
  1899 lemma length_takeWhile_le: "length (takeWhile P xs) \<le> length xs"
  1900   by (induct xs) auto
  1901 
  1902 lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
  1903 by (induct xs) auto
  1904 
  1905 lemma takeWhile_append1 [simp]:
  1906 "[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
  1907 by (induct xs) auto
  1908 
  1909 lemma takeWhile_append2 [simp]:
  1910 "(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
  1911 by (induct xs) auto
  1912 
  1913 lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
  1914 by (induct xs) auto
  1915 
  1916 lemma takeWhile_nth: "j < length (takeWhile P xs) \<Longrightarrow> takeWhile P xs ! j = xs ! j"
  1917 apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
  1918 
  1919 lemma dropWhile_nth: "j < length (dropWhile P xs) \<Longrightarrow> dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))"
  1920 apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
  1921 
  1922 lemma length_dropWhile_le: "length (dropWhile P xs) \<le> length xs"
  1923 by (induct xs) auto
  1924 
  1925 lemma dropWhile_append1 [simp]:
  1926 "[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
  1927 by (induct xs) auto
  1928 
  1929 lemma dropWhile_append2 [simp]:
  1930 "(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
  1931 by (induct xs) auto
  1932 
  1933 lemma dropWhile_append3:
  1934   "\<not> P y \<Longrightarrow>dropWhile P (xs @ y # ys) = dropWhile P xs @ y # ys"
  1935 by (induct xs) auto
  1936 
  1937 lemma dropWhile_last:
  1938   "x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> last (dropWhile P xs) = last xs"
  1939 by (auto simp add: dropWhile_append3 in_set_conv_decomp)
  1940 
  1941 lemma set_dropWhileD: "x \<in> set (dropWhile P xs) \<Longrightarrow> x \<in> set xs"
  1942 by (induct xs) (auto split: split_if_asm)
  1943 
  1944 lemma set_takeWhileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
  1945 by (induct xs) (auto split: split_if_asm)
  1946 
  1947 lemma takeWhile_eq_all_conv[simp]:
  1948  "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
  1949 by(induct xs, auto)
  1950 
  1951 lemma dropWhile_eq_Nil_conv[simp]:
  1952  "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
  1953 by(induct xs, auto)
  1954 
  1955 lemma dropWhile_eq_Cons_conv:
  1956  "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
  1957 by(induct xs, auto)
  1958 
  1959 lemma distinct_takeWhile[simp]: "distinct xs ==> distinct (takeWhile P xs)"
  1960 by (induct xs) (auto dest: set_takeWhileD)
  1961 
  1962 lemma distinct_dropWhile[simp]: "distinct xs ==> distinct (dropWhile P xs)"
  1963 by (induct xs) auto
  1964 
  1965 lemma takeWhile_map: "takeWhile P (map f xs) = map f (takeWhile (P \<circ> f) xs)"
  1966 by (induct xs) auto
  1967 
  1968 lemma dropWhile_map: "dropWhile P (map f xs) = map f (dropWhile (P \<circ> f) xs)"
  1969 by (induct xs) auto
  1970 
  1971 lemma takeWhile_eq_take: "takeWhile P xs = take (length (takeWhile P xs)) xs"
  1972 by (induct xs) auto
  1973 
  1974 lemma dropWhile_eq_drop: "dropWhile P xs = drop (length (takeWhile P xs)) xs"
  1975 by (induct xs) auto
  1976 
  1977 lemma hd_dropWhile:
  1978   "dropWhile P xs \<noteq> [] \<Longrightarrow> \<not> P (hd (dropWhile P xs))"
  1979 using assms by (induct xs) auto
  1980 
  1981 lemma takeWhile_eq_filter:
  1982   assumes "\<And> x. x \<in> set (dropWhile P xs) \<Longrightarrow> \<not> P x"
  1983   shows "takeWhile P xs = filter P xs"
  1984 proof -
  1985   have A: "filter P xs = filter P (takeWhile P xs @ dropWhile P xs)"
  1986     by simp
  1987   have B: "filter P (dropWhile P xs) = []"
  1988     unfolding filter_empty_conv using assms by blast
  1989   have "filter P xs = takeWhile P xs"
  1990     unfolding A filter_append B
  1991     by (auto simp add: filter_id_conv dest: set_takeWhileD)
  1992   thus ?thesis ..
  1993 qed
  1994 
  1995 lemma takeWhile_eq_take_P_nth:
  1996   "\<lbrakk> \<And> i. \<lbrakk> i < n ; i < length xs \<rbrakk> \<Longrightarrow> P (xs ! i) ; n < length xs \<Longrightarrow> \<not> P (xs ! n) \<rbrakk> \<Longrightarrow>
  1997   takeWhile P xs = take n xs"
  1998 proof (induct xs arbitrary: n)
  1999   case (Cons x xs)
  2000   thus ?case
  2001   proof (cases n)
  2002     case (Suc n') note this[simp]
  2003     have "P x" using Cons.prems(1)[of 0] by simp
  2004     moreover have "takeWhile P xs = take n' xs"
  2005     proof (rule Cons.hyps)
  2006       case goal1 thus "P (xs ! i)" using Cons.prems(1)[of "Suc i"] by simp
  2007     next case goal2 thus ?case using Cons by auto
  2008     qed
  2009     ultimately show ?thesis by simp
  2010    qed simp
  2011 qed simp
  2012 
  2013 lemma nth_length_takeWhile:
  2014   "length (takeWhile P xs) < length xs \<Longrightarrow> \<not> P (xs ! length (takeWhile P xs))"
  2015 by (induct xs) auto
  2016 
  2017 lemma length_takeWhile_less_P_nth:
  2018   assumes all: "\<And> i. i < j \<Longrightarrow> P (xs ! i)" and "j \<le> length xs"
  2019   shows "j \<le> length (takeWhile P xs)"
  2020 proof (rule classical)
  2021   assume "\<not> ?thesis"
  2022   hence "length (takeWhile P xs) < length xs" using assms by simp
  2023   thus ?thesis using all `\<not> ?thesis` nth_length_takeWhile[of P xs] by auto
  2024 qed
  2025 
  2026 text{* The following two lemmmas could be generalized to an arbitrary
  2027 property. *}
  2028 
  2029 lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
  2030  takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
  2031 by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
  2032 
  2033 lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
  2034   dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
  2035 apply(induct xs)
  2036  apply simp
  2037 apply auto
  2038 apply(subst dropWhile_append2)
  2039 apply auto
  2040 done
  2041 
  2042 lemma takeWhile_not_last:
  2043  "distinct xs \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
  2044 apply(induct xs)
  2045  apply simp
  2046 apply(case_tac xs)
  2047 apply(auto)
  2048 done
  2049 
  2050 lemma takeWhile_cong [fundef_cong]:
  2051   "[| l = k; !!x. x : set l ==> P x = Q x |] 
  2052   ==> takeWhile P l = takeWhile Q k"
  2053 by (induct k arbitrary: l) (simp_all)
  2054 
  2055 lemma dropWhile_cong [fundef_cong]:
  2056   "[| l = k; !!x. x : set l ==> P x = Q x |] 
  2057   ==> dropWhile P l = dropWhile Q k"
  2058 by (induct k arbitrary: l, simp_all)
  2059 
  2060 
  2061 subsubsection {* @{text zip} *}
  2062 
  2063 lemma zip_Nil [simp]: "zip [] ys = []"
  2064 by (induct ys) auto
  2065 
  2066 lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
  2067 by simp
  2068 
  2069 declare zip_Cons [simp del]
  2070 
  2071 lemma [code]:
  2072   "zip [] ys = []"
  2073   "zip xs [] = []"
  2074   "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
  2075   by (fact zip_Nil zip.simps(1) zip_Cons_Cons)+
  2076 
  2077 lemma zip_Cons1:
  2078  "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
  2079 by(auto split:list.split)
  2080 
  2081 lemma length_zip [simp]:
  2082 "length (zip xs ys) = min (length xs) (length ys)"
  2083 by (induct xs ys rule:list_induct2') auto
  2084 
  2085 lemma zip_obtain_same_length:
  2086   assumes "\<And>zs ws n. length zs = length ws \<Longrightarrow> n = min (length xs) (length ys)
  2087     \<Longrightarrow> zs = take n xs \<Longrightarrow> ws = take n ys \<Longrightarrow> P (zip zs ws)"
  2088   shows "P (zip xs ys)"
  2089 proof -
  2090   let ?n = "min (length xs) (length ys)"
  2091   have "P (zip (take ?n xs) (take ?n ys))"
  2092     by (rule assms) simp_all
  2093   moreover have "zip xs ys = zip (take ?n xs) (take ?n ys)"
  2094   proof (induct xs arbitrary: ys)
  2095     case Nil then show ?case by simp
  2096   next
  2097     case (Cons x xs) then show ?case by (cases ys) simp_all
  2098   qed
  2099   ultimately show ?thesis by simp
  2100 qed
  2101 
  2102 lemma zip_append1:
  2103 "zip (xs @ ys) zs =
  2104 zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
  2105 by (induct xs zs rule:list_induct2') auto
  2106 
  2107 lemma zip_append2:
  2108 "zip xs (ys @ zs) =
  2109 zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
  2110 by (induct xs ys rule:list_induct2') auto
  2111 
  2112 lemma zip_append [simp]:
  2113  "[| length xs = length us |] ==>
  2114 zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
  2115 by (simp add: zip_append1)
  2116 
  2117 lemma zip_rev:
  2118 "length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
  2119 by (induct rule:list_induct2, simp_all)
  2120 
  2121 lemma zip_map_map:
  2122   "zip (map f xs) (map g ys) = map (\<lambda> (x, y). (f x, g y)) (zip xs ys)"
  2123 proof (induct xs arbitrary: ys)
  2124   case (Cons x xs) note Cons_x_xs = Cons.hyps
  2125   show ?case
  2126   proof (cases ys)
  2127     case (Cons y ys')
  2128     show ?thesis unfolding Cons using Cons_x_xs by simp
  2129   qed simp
  2130 qed simp
  2131 
  2132 lemma zip_map1:
  2133   "zip (map f xs) ys = map (\<lambda>(x, y). (f x, y)) (zip xs ys)"
  2134 using zip_map_map[of f xs "\<lambda>x. x" ys] by simp
  2135 
  2136 lemma zip_map2:
  2137   "zip xs (map f ys) = map (\<lambda>(x, y). (x, f y)) (zip xs ys)"
  2138 using zip_map_map[of "\<lambda>x. x" xs f ys] by simp
  2139 
  2140 lemma map_zip_map:
  2141   "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
  2142 unfolding zip_map1 by auto
  2143 
  2144 lemma map_zip_map2:
  2145   "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
  2146 unfolding zip_map2 by auto
  2147 
  2148 text{* Courtesy of Andreas Lochbihler: *}
  2149 lemma zip_same_conv_map: "zip xs xs = map (\<lambda>x. (x, x)) xs"
  2150 by(induct xs) auto
  2151 
  2152 lemma nth_zip [simp]:
  2153 "[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
  2154 apply (induct ys arbitrary: i xs, simp)
  2155 apply (case_tac xs)
  2156  apply (simp_all add: nth.simps split: nat.split)
  2157 done
  2158 
  2159 lemma set_zip:
  2160 "set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
  2161 by(simp add: set_conv_nth cong: rev_conj_cong)
  2162 
  2163 lemma zip_same: "((a,b) \<in> set (zip xs xs)) = (a \<in> set xs \<and> a = b)"
  2164 by(induct xs) auto
  2165 
  2166 lemma zip_update:
  2167   "zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
  2168 by(rule sym, simp add: update_zip)
  2169 
  2170 lemma zip_replicate [simp]:
  2171   "zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
  2172 apply (induct i arbitrary: j, auto)
  2173 apply (case_tac j, auto)
  2174 done
  2175 
  2176 lemma take_zip:
  2177   "take n (zip xs ys) = zip (take n xs) (take n ys)"
  2178 apply (induct n arbitrary: xs ys)
  2179  apply simp
  2180 apply (case_tac xs, simp)
  2181 apply (case_tac ys, simp_all)
  2182 done
  2183 
  2184 lemma drop_zip:
  2185   "drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
  2186 apply (induct n arbitrary: xs ys)
  2187  apply simp
  2188 apply (case_tac xs, simp)
  2189 apply (case_tac ys, simp_all)
  2190 done
  2191 
  2192 lemma zip_takeWhile_fst: "zip (takeWhile P xs) ys = takeWhile (P \<circ> fst) (zip xs ys)"
  2193 proof (induct xs arbitrary: ys)
  2194   case (Cons x xs) thus ?case by (cases ys) auto
  2195 qed simp
  2196 
  2197 lemma zip_takeWhile_snd: "zip xs (takeWhile P ys) = takeWhile (P \<circ> snd) (zip xs ys)"
  2198 proof (induct xs arbitrary: ys)
  2199   case (Cons x xs) thus ?case by (cases ys) auto
  2200 qed simp
  2201 
  2202 lemma set_zip_leftD:
  2203   "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
  2204 by (induct xs ys rule:list_induct2') auto
  2205 
  2206 lemma set_zip_rightD:
  2207   "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
  2208 by (induct xs ys rule:list_induct2') auto
  2209 
  2210 lemma in_set_zipE:
  2211   "(x,y) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
  2212 by(blast dest: set_zip_leftD set_zip_rightD)
  2213 
  2214 lemma zip_map_fst_snd:
  2215   "zip (map fst zs) (map snd zs) = zs"
  2216   by (induct zs) simp_all
  2217 
  2218 lemma zip_eq_conv:
  2219   "length xs = length ys \<Longrightarrow> zip xs ys = zs \<longleftrightarrow> map fst zs = xs \<and> map snd zs = ys"
  2220   by (auto simp add: zip_map_fst_snd)
  2221 
  2222 
  2223 subsubsection {* @{text list_all2} *}
  2224 
  2225 lemma list_all2_lengthD [intro?]: 
  2226   "list_all2 P xs ys ==> length xs = length ys"
  2227 by (simp add: list_all2_def)
  2228 
  2229 lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
  2230 by (simp add: list_all2_def)
  2231 
  2232 lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
  2233 by (simp add: list_all2_def)
  2234 
  2235 lemma list_all2_Cons [iff, code]:
  2236   "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
  2237 by (auto simp add: list_all2_def)
  2238 
  2239 lemma list_all2_Cons1:
  2240 "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
  2241 by (cases ys) auto
  2242 
  2243 lemma list_all2_Cons2:
  2244 "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
  2245 by (cases xs) auto
  2246 
  2247 lemma list_all2_induct
  2248   [consumes 1, case_names Nil Cons, induct set: list_all2]:
  2249   assumes P: "list_all2 P xs ys"
  2250   assumes Nil: "R [] []"
  2251   assumes Cons: "\<And>x xs y ys. \<lbrakk>P x y; R xs ys\<rbrakk> \<Longrightarrow> R (x # xs) (y # ys)"
  2252   shows "R xs ys"
  2253 using P
  2254 by (induct xs arbitrary: ys) (auto simp add: list_all2_Cons1 Nil Cons)
  2255 
  2256 lemma list_all2_rev [iff]:
  2257 "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
  2258 by (simp add: list_all2_def zip_rev cong: conj_cong)
  2259 
  2260 lemma list_all2_rev1:
  2261 "list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
  2262 by (subst list_all2_rev [symmetric]) simp
  2263 
  2264 lemma list_all2_append1:
  2265 "list_all2 P (xs @ ys) zs =
  2266 (EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
  2267 list_all2 P xs us \<and> list_all2 P ys vs)"
  2268 apply (simp add: list_all2_def zip_append1)
  2269 apply (rule iffI)
  2270  apply (rule_tac x = "take (length xs) zs" in exI)
  2271  apply (rule_tac x = "drop (length xs) zs" in exI)
  2272  apply (force split: nat_diff_split simp add: min_def, clarify)
  2273 apply (simp add: ball_Un)
  2274 done
  2275 
  2276 lemma list_all2_append2:
  2277 "list_all2 P xs (ys @ zs) =
  2278 (EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
  2279 list_all2 P us ys \<and> list_all2 P vs zs)"
  2280 apply (simp add: list_all2_def zip_append2)
  2281 apply (rule iffI)
  2282  apply (rule_tac x = "take (length ys) xs" in exI)
  2283  apply (rule_tac x = "drop (length ys) xs" in exI)
  2284  apply (force split: nat_diff_split simp add: min_def, clarify)
  2285 apply (simp add: ball_Un)
  2286 done
  2287 
  2288 lemma list_all2_append:
  2289   "length xs = length ys \<Longrightarrow>
  2290   list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
  2291 by (induct rule:list_induct2, simp_all)
  2292 
  2293 lemma list_all2_appendI [intro?, trans]:
  2294   "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
  2295 by (simp add: list_all2_append list_all2_lengthD)
  2296 
  2297 lemma list_all2_conv_all_nth:
  2298 "list_all2 P xs ys =
  2299 (length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
  2300 by (force simp add: list_all2_def set_zip)
  2301 
  2302 lemma list_all2_trans:
  2303   assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
  2304   shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
  2305         (is "!!bs cs. PROP ?Q as bs cs")
  2306 proof (induct as)
  2307   fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
  2308   show "!!cs. PROP ?Q (x # xs) bs cs"
  2309   proof (induct bs)
  2310     fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
  2311     show "PROP ?Q (x # xs) (y # ys) cs"
  2312       by (induct cs) (auto intro: tr I1 I2)
  2313   qed simp
  2314 qed simp
  2315 
  2316 lemma list_all2_all_nthI [intro?]:
  2317   "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
  2318 by (simp add: list_all2_conv_all_nth)
  2319 
  2320 lemma list_all2I:
  2321   "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
  2322 by (simp add: list_all2_def)
  2323 
  2324 lemma list_all2_nthD:
  2325   "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
  2326 by (simp add: list_all2_conv_all_nth)
  2327 
  2328 lemma list_all2_nthD2:
  2329   "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
  2330 by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
  2331 
  2332 lemma list_all2_map1: 
  2333   "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
  2334 by (simp add: list_all2_conv_all_nth)
  2335 
  2336 lemma list_all2_map2: 
  2337   "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
  2338 by (auto simp add: list_all2_conv_all_nth)
  2339 
  2340 lemma list_all2_refl [intro?]:
  2341   "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
  2342 by (simp add: list_all2_conv_all_nth)
  2343 
  2344 lemma list_all2_update_cong:
  2345   "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
  2346 by (simp add: list_all2_conv_all_nth nth_list_update)
  2347 
  2348 lemma list_all2_update_cong2:
  2349   "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
  2350 by (simp add: list_all2_lengthD list_all2_update_cong)
  2351 
  2352 lemma list_all2_takeI [simp,intro?]:
  2353   "list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
  2354 apply (induct xs arbitrary: n ys)
  2355  apply simp
  2356 apply (clarsimp simp add: list_all2_Cons1)
  2357 apply (case_tac n)
  2358 apply auto
  2359 done
  2360 
  2361 lemma list_all2_dropI [simp,intro?]:
  2362   "list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
  2363 apply (induct as arbitrary: n bs, simp)
  2364 apply (clarsimp simp add: list_all2_Cons1)
  2365 apply (case_tac n, simp, simp)
  2366 done
  2367 
  2368 lemma list_all2_mono [intro?]:
  2369   "list_all2 P xs ys \<Longrightarrow> (\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys) \<Longrightarrow> list_all2 Q xs ys"
  2370 apply (induct xs arbitrary: ys, simp)
  2371 apply (case_tac ys, auto)
  2372 done
  2373 
  2374 lemma list_all2_eq:
  2375   "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
  2376 by (induct xs ys rule: list_induct2') auto
  2377 
  2378 lemma list_eq_iff_zip_eq:
  2379   "xs = ys \<longleftrightarrow> length xs = length ys \<and> (\<forall>(x,y) \<in> set (zip xs ys). x = y)"
  2380 by(auto simp add: set_zip list_all2_eq list_all2_conv_all_nth cong: conj_cong)
  2381 
  2382 
  2383 subsubsection {* @{const fold} with canonical argument order *}
  2384 
  2385 lemma fold_remove1_split:
  2386   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"
  2387     and x: "x \<in> set xs"
  2388   shows "fold f xs = fold f (remove1 x xs) \<circ> f x"
  2389   using assms by (induct xs) (auto simp add: o_assoc [symmetric])
  2390 
  2391 lemma fold_cong [fundef_cong]:
  2392   "a = b \<Longrightarrow> xs = ys \<Longrightarrow> (\<And>x. x \<in> set xs \<Longrightarrow> f x = g x)
  2393     \<Longrightarrow> fold f xs a = fold g ys b"
  2394   by (induct ys arbitrary: a b xs) simp_all
  2395 
  2396 lemma fold_id:
  2397   assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x = id"
  2398   shows "fold f xs = id"
  2399   using assms by (induct xs) simp_all
  2400 
  2401 lemma fold_commute:
  2402   assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
  2403   shows "h \<circ> fold g xs = fold f xs \<circ> h"
  2404   using assms by (induct xs) (simp_all add: fun_eq_iff)
  2405 
  2406 lemma fold_commute_apply:
  2407   assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
  2408   shows "h (fold g xs s) = fold f xs (h s)"
  2409 proof -
  2410   from assms have "h \<circ> fold g xs = fold f xs \<circ> h" by (rule fold_commute)
  2411   then show ?thesis by (simp add: fun_eq_iff)
  2412 qed
  2413 
  2414 lemma fold_invariant: 
  2415   assumes "\<And>x. x \<in> set xs \<Longrightarrow> Q x" and "P s"
  2416     and "\<And>x s. Q x \<Longrightarrow> P s \<Longrightarrow> P (f x s)"
  2417   shows "P (fold f xs s)"
  2418   using assms by (induct xs arbitrary: s) simp_all
  2419 
  2420 lemma fold_append [simp]:
  2421   "fold f (xs @ ys) = fold f ys \<circ> fold f xs"
  2422   by (induct xs) simp_all
  2423 
  2424 lemma fold_map [code_unfold]:
  2425   "fold g (map f xs) = fold (g o f) xs"
  2426   by (induct xs) simp_all
  2427 
  2428 lemma fold_rev:
  2429   assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
  2430   shows "fold f (rev xs) = fold f xs"
  2431 using assms by (induct xs) (simp_all add: fold_commute_apply fun_eq_iff)
  2432 
  2433 lemma fold_Cons_rev:
  2434   "fold Cons xs = append (rev xs)"
  2435   by (induct xs) simp_all
  2436 
  2437 lemma rev_conv_fold [code]:
  2438   "rev xs = fold Cons xs []"
  2439   by (simp add: fold_Cons_rev)
  2440 
  2441 lemma fold_append_concat_rev:
  2442   "fold append xss = append (concat (rev xss))"
  2443   by (induct xss) simp_all
  2444 
  2445 text {* @{const Finite_Set.fold} and @{const fold} *}
  2446 
  2447 lemma (in comp_fun_commute) fold_set_fold_remdups:
  2448   "Finite_Set.fold f y (set xs) = fold f (remdups xs) y"
  2449   by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm insert_absorb)
  2450 
  2451 lemma (in comp_fun_idem) fold_set_fold:
  2452   "Finite_Set.fold f y (set xs) = fold f xs y"
  2453   by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm)
  2454 
  2455 lemma (in ab_semigroup_idem_mult) fold1_set_fold:
  2456   assumes "xs \<noteq> []"
  2457   shows "Finite_Set.fold1 times (set xs) = fold times (tl xs) (hd xs)"
  2458 proof -
  2459   interpret comp_fun_idem times by (fact comp_fun_idem)
  2460   from assms obtain y ys where xs: "xs = y # ys"
  2461     by (cases xs) auto
  2462   show ?thesis
  2463   proof (cases "set ys = {}")
  2464     case True with xs show ?thesis by simp
  2465   next
  2466     case False
  2467     then have "fold1 times (insert y (set ys)) = Finite_Set.fold times y (set ys)"
  2468       by (simp only: finite_set fold1_eq_fold_idem)
  2469     with xs show ?thesis by (simp add: fold_set_fold mult_commute)
  2470   qed
  2471 qed
  2472 
  2473 lemma union_set_fold:
  2474   "set xs \<union> A = fold Set.insert xs A"
  2475 proof -
  2476   interpret comp_fun_idem Set.insert
  2477     by (fact comp_fun_idem_insert)
  2478   show ?thesis by (simp add: union_fold_insert fold_set_fold)
  2479 qed
  2480 
  2481 lemma minus_set_fold:
  2482   "A - set xs = fold Set.remove xs A"
  2483 proof -
  2484   interpret comp_fun_idem Set.remove
  2485     by (fact comp_fun_idem_remove)
  2486   show ?thesis
  2487     by (simp add: minus_fold_remove [of _ A] fold_set_fold)
  2488 qed
  2489 
  2490 lemma (in lattice) Inf_fin_set_fold:
  2491   "Inf_fin (set (x # xs)) = fold inf xs x"
  2492 proof -
  2493   interpret ab_semigroup_idem_mult "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2494     by (fact ab_semigroup_idem_mult_inf)
  2495   show ?thesis
  2496     by (simp add: Inf_fin_def fold1_set_fold del: set.simps)
  2497 qed
  2498 
  2499 lemma (in lattice) Sup_fin_set_fold:
  2500   "Sup_fin (set (x # xs)) = fold sup xs x"
  2501 proof -
  2502   interpret ab_semigroup_idem_mult "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2503     by (fact ab_semigroup_idem_mult_sup)
  2504   show ?thesis
  2505     by (simp add: Sup_fin_def fold1_set_fold del: set.simps)
  2506 qed
  2507 
  2508 lemma (in linorder) Min_fin_set_fold:
  2509   "Min (set (x # xs)) = fold min xs x"
  2510 proof -
  2511   interpret ab_semigroup_idem_mult "min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2512     by (fact ab_semigroup_idem_mult_min)
  2513   show ?thesis
  2514     by (simp add: Min_def fold1_set_fold del: set.simps)
  2515 qed
  2516 
  2517 lemma (in linorder) Max_fin_set_fold:
  2518   "Max (set (x # xs)) = fold max xs x"
  2519 proof -
  2520   interpret ab_semigroup_idem_mult "max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2521     by (fact ab_semigroup_idem_mult_max)
  2522   show ?thesis
  2523     by (simp add: Max_def fold1_set_fold del: set.simps)
  2524 qed
  2525 
  2526 lemma (in complete_lattice) Inf_set_fold:
  2527   "Inf (set xs) = fold inf xs top"
  2528 proof -
  2529   interpret comp_fun_idem "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2530     by (fact comp_fun_idem_inf)
  2531   show ?thesis by (simp add: Inf_fold_inf fold_set_fold inf_commute)
  2532 qed
  2533 
  2534 lemma (in complete_lattice) Sup_set_fold:
  2535   "Sup (set xs) = fold sup xs bot"
  2536 proof -
  2537   interpret comp_fun_idem "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2538     by (fact comp_fun_idem_sup)
  2539   show ?thesis by (simp add: Sup_fold_sup fold_set_fold sup_commute)
  2540 qed
  2541 
  2542 lemma (in complete_lattice) INF_set_fold:
  2543   "INFI (set xs) f = fold (inf \<circ> f) xs top"
  2544   unfolding INF_def set_map [symmetric] Inf_set_fold fold_map ..
  2545 
  2546 lemma (in complete_lattice) SUP_set_fold:
  2547   "SUPR (set xs) f = fold (sup \<circ> f) xs bot"
  2548   unfolding SUP_def set_map [symmetric] Sup_set_fold fold_map ..
  2549 
  2550 
  2551 subsubsection {* Fold variants: @{const foldr} and @{const foldl} *}
  2552 
  2553 text {* Correspondence *}
  2554 
  2555 lemma foldr_foldl: -- {* The ``Third Duality Theorem'' in Bird \& Wadler: *}
  2556   "foldr f xs a = foldl (\<lambda>x y. f y x) a (rev xs)"
  2557   by (simp add: foldr_def foldl_def)
  2558 
  2559 lemma foldl_foldr:
  2560   "foldl f a xs = foldr (\<lambda>x y. f y x) (rev xs) a"
  2561   by (simp add: foldr_def foldl_def)
  2562 
  2563 lemma foldr_fold:
  2564   assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
  2565   shows "foldr f xs = fold f xs"
  2566   using assms unfolding foldr_def by (rule fold_rev)
  2567 
  2568 lemma
  2569   foldr_Nil [code, simp]: "foldr f [] = id"
  2570   and foldr_Cons [code, simp]: "foldr f (x # xs) = f x \<circ> foldr f xs"
  2571   by (simp_all add: foldr_def)
  2572 
  2573 lemma
  2574   foldl_Nil [simp]: "foldl f a [] = a"
  2575   and foldl_Cons [simp]: "foldl f a (x # xs) = foldl f (f a x) xs"
  2576   by (simp_all add: foldl_def)
  2577 
  2578 lemma foldr_cong [fundef_cong]:
  2579   "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"
  2580   by (auto simp add: foldr_def intro!: fold_cong)
  2581 
  2582 lemma foldl_cong [fundef_cong]:
  2583   "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"
  2584   by (auto simp add: foldl_def intro!: fold_cong)
  2585 
  2586 lemma foldr_append [simp]:
  2587   "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
  2588   by (simp add: foldr_def)
  2589 
  2590 lemma foldl_append [simp]:
  2591   "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
  2592   by (simp add: foldl_def)
  2593 
  2594 lemma foldr_map [code_unfold]:
  2595   "foldr g (map f xs) a = foldr (g o f) xs a"
  2596   by (simp add: foldr_def fold_map rev_map)
  2597 
  2598 lemma foldl_map [code_unfold]:
  2599   "foldl g a (map f xs) = foldl (\<lambda>a x. g a (f x)) a xs"
  2600   by (simp add: foldl_def fold_map comp_def)
  2601 
  2602 text {* Executing operations in terms of @{const foldr} -- tail-recursive! *}
  2603 
  2604 lemma concat_conv_foldr [code]:
  2605   "concat xss = foldr append xss []"
  2606   by (simp add: fold_append_concat_rev foldr_def)
  2607 
  2608 lemma minus_set_foldr [code]:
  2609   "A - set xs = foldr Set.remove xs A"
  2610 proof -
  2611   have "\<And>x y :: 'a. Set.remove y \<circ> Set.remove x = Set.remove x \<circ> Set.remove y"
  2612     by (auto simp add: remove_def)
  2613   then show ?thesis by (simp add: minus_set_fold foldr_fold)
  2614 qed
  2615 
  2616 lemma subtract_coset_filter [code]:
  2617   "A - List.coset xs = set (List.filter (\<lambda>x. x \<in> A) xs)"
  2618   by auto
  2619 
  2620 lemma union_set_foldr [code]:
  2621   "set xs \<union> A = foldr Set.insert xs A"
  2622 proof -
  2623   have "\<And>x y :: 'a. insert y \<circ> insert x = insert x \<circ> insert y"
  2624     by auto
  2625   then show ?thesis by (simp add: union_set_fold foldr_fold)
  2626 qed
  2627 
  2628 lemma union_coset_foldr [code]:
  2629   "List.coset xs \<union> A = List.coset (List.filter (\<lambda>x. x \<notin> A) xs)"
  2630   by auto
  2631 
  2632 lemma inter_set_filer [code]:
  2633   "A \<inter> set xs = set (List.filter (\<lambda>x. x \<in> A) xs)"
  2634   by auto
  2635 
  2636 lemma inter_coset_foldr [code]:
  2637   "A \<inter> List.coset xs = foldr Set.remove xs A"
  2638   by (simp add: Diff_eq [symmetric] minus_set_foldr)
  2639 
  2640 lemma (in lattice) Inf_fin_set_foldr [code]:
  2641   "Inf_fin (set (x # xs)) = foldr inf xs x"
  2642   by (simp add: Inf_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
  2643 
  2644 lemma (in lattice) Sup_fin_set_foldr [code]:
  2645   "Sup_fin (set (x # xs)) = foldr sup xs x"
  2646   by (simp add: Sup_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
  2647 
  2648 lemma (in linorder) Min_fin_set_foldr [code]:
  2649   "Min (set (x # xs)) = foldr min xs x"
  2650   by (simp add: Min_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
  2651 
  2652 lemma (in linorder) Max_fin_set_foldr [code]:
  2653   "Max (set (x # xs)) = foldr max xs x"
  2654   by (simp add: Max_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
  2655 
  2656 lemma (in complete_lattice) Inf_set_foldr:
  2657   "Inf (set xs) = foldr inf xs top"
  2658   by (simp add: Inf_set_fold ac_simps foldr_fold fun_eq_iff)
  2659 
  2660 lemma (in complete_lattice) Sup_set_foldr:
  2661   "Sup (set xs) = foldr sup xs bot"
  2662   by (simp add: Sup_set_fold ac_simps foldr_fold fun_eq_iff)
  2663 
  2664 declare Inf_set_foldr [where 'a = "'a set", code] Sup_set_foldr [where 'a = "'a set", code]
  2665 
  2666 lemma (in complete_lattice) INF_set_foldr [code]:
  2667   "INFI (set xs) f = foldr (inf \<circ> f) xs top"
  2668   by (simp only: INF_def Inf_set_foldr foldr_map set_map [symmetric])
  2669 
  2670 lemma (in complete_lattice) SUP_set_foldr [code]:
  2671   "SUPR (set xs) f = foldr (sup \<circ> f) xs bot"
  2672   by (simp only: SUP_def Sup_set_foldr foldr_map set_map [symmetric])
  2673 
  2674 
  2675 subsubsection {* @{text upt} *}
  2676 
  2677 lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
  2678 -- {* simp does not terminate! *}
  2679 by (induct j) auto
  2680 
  2681 lemmas upt_rec_number_of[simp] = upt_rec[of "number_of m" "number_of n"] for m n
  2682 
  2683 lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
  2684 by (subst upt_rec) simp
  2685 
  2686 lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
  2687 by(induct j)simp_all
  2688 
  2689 lemma upt_eq_Cons_conv:
  2690  "([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
  2691 apply(induct j arbitrary: x xs)
  2692  apply simp
  2693 apply(clarsimp simp add: append_eq_Cons_conv)
  2694 apply arith
  2695 done
  2696 
  2697 lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
  2698 -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
  2699 by simp
  2700 
  2701 lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
  2702   by (simp add: upt_rec)
  2703 
  2704 lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
  2705 -- {* LOOPS as a simprule, since @{text "j <= j"}. *}
  2706 by (induct k) auto
  2707 
  2708 lemma length_upt [simp]: "length [i..<j] = j - i"
  2709 by (induct j) (auto simp add: Suc_diff_le)
  2710 
  2711 lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
  2712 apply (induct j)
  2713 apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
  2714 done
  2715 
  2716 
  2717 lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
  2718 by(simp add:upt_conv_Cons)
  2719 
  2720 lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
  2721 apply(cases j)
  2722  apply simp
  2723 by(simp add:upt_Suc_append)
  2724 
  2725 lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]"
  2726 apply (induct m arbitrary: i, simp)
  2727 apply (subst upt_rec)
  2728 apply (rule sym)
  2729 apply (subst upt_rec)
  2730 apply (simp del: upt.simps)
  2731 done
  2732 
  2733 lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
  2734 apply(induct j)
  2735 apply auto
  2736 done
  2737 
  2738 lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]"
  2739 by (induct n) auto
  2740 
  2741 lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
  2742 apply (induct n m  arbitrary: i rule: diff_induct)
  2743 prefer 3 apply (subst map_Suc_upt[symmetric])
  2744 apply (auto simp add: less_diff_conv)
  2745 done
  2746 
  2747 lemma nth_take_lemma:
  2748   "k <= length xs ==> k <= length ys ==>
  2749      (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
  2750 apply (atomize, induct k arbitrary: xs ys)
  2751 apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
  2752 txt {* Both lists must be non-empty *}
  2753 apply (case_tac xs, simp)
  2754 apply (case_tac ys, clarify)
  2755  apply (simp (no_asm_use))
  2756 apply clarify
  2757 txt {* prenexing's needed, not miniscoping *}
  2758 apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
  2759 apply blast
  2760 done
  2761 
  2762 lemma nth_equalityI:
  2763  "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
  2764   by (frule nth_take_lemma [OF le_refl eq_imp_le]) simp_all
  2765 
  2766 lemma map_nth:
  2767   "map (\<lambda>i. xs ! i) [0..<length xs] = xs"
  2768   by (rule nth_equalityI, auto)
  2769 
  2770 (* needs nth_equalityI *)
  2771 lemma list_all2_antisym:
  2772   "\<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> 
  2773   \<Longrightarrow> xs = ys"
  2774   apply (simp add: list_all2_conv_all_nth) 
  2775   apply (rule nth_equalityI, blast, simp)
  2776   done
  2777 
  2778 lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
  2779 -- {* The famous take-lemma. *}
  2780 apply (drule_tac x = "max (length xs) (length ys)" in spec)
  2781 apply (simp add: le_max_iff_disj)
  2782 done
  2783 
  2784 
  2785 lemma take_Cons':
  2786      "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
  2787 by (cases n) simp_all
  2788 
  2789 lemma drop_Cons':
  2790      "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
  2791 by (cases n) simp_all
  2792 
  2793 lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
  2794 by (cases n) simp_all
  2795 
  2796 lemmas take_Cons_number_of = take_Cons'[of "number_of v"] for v
  2797 lemmas drop_Cons_number_of = drop_Cons'[of "number_of v"] for v
  2798 lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v"] for v
  2799 
  2800 declare take_Cons_number_of [simp] 
  2801         drop_Cons_number_of [simp] 
  2802         nth_Cons_number_of [simp] 
  2803 
  2804 
  2805 subsubsection {* @{text upto}: interval-list on @{typ int} *}
  2806 
  2807 (* FIXME make upto tail recursive? *)
  2808 
  2809 function upto :: "int \<Rightarrow> int \<Rightarrow> int list" ("(1[_../_])") where
  2810 "upto i j = (if i \<le> j then i # [i+1..j] else [])"
  2811 by auto
  2812 termination
  2813 by(relation "measure(%(i::int,j). nat(j - i + 1))") auto
  2814 
  2815 declare upto.simps[code, simp del]
  2816 
  2817 lemmas upto_rec_number_of[simp] = upto.simps[of "number_of m" "number_of n"] for m n
  2818 
  2819 lemma upto_empty[simp]: "j < i \<Longrightarrow> [i..j] = []"
  2820 by(simp add: upto.simps)
  2821 
  2822 lemma set_upto[simp]: "set[i..j] = {i..j}"
  2823 proof(induct i j rule:upto.induct)
  2824   case (1 i j)
  2825   from this show ?case
  2826     unfolding upto.simps[of i j] simp_from_to[of i j] by auto
  2827 qed
  2828 
  2829 
  2830 subsubsection {* @{text "distinct"} and @{text remdups} *}
  2831 
  2832 lemma distinct_tl:
  2833   "distinct xs \<Longrightarrow> distinct (tl xs)"
  2834   by (cases xs) simp_all
  2835 
  2836 lemma distinct_append [simp]:
  2837 "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
  2838 by (induct xs) auto
  2839 
  2840 lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
  2841 by(induct xs) auto
  2842 
  2843 lemma set_remdups [simp]: "set (remdups xs) = set xs"
  2844 by (induct xs) (auto simp add: insert_absorb)
  2845 
  2846 lemma distinct_remdups [iff]: "distinct (remdups xs)"
  2847 by (induct xs) auto
  2848 
  2849 lemma distinct_remdups_id: "distinct xs ==> remdups xs = xs"
  2850 by (induct xs, auto)
  2851 
  2852 lemma remdups_id_iff_distinct [simp]: "remdups xs = xs \<longleftrightarrow> distinct xs"
  2853 by (metis distinct_remdups distinct_remdups_id)
  2854 
  2855 lemma finite_distinct_list: "finite A \<Longrightarrow> EX xs. set xs = A & distinct xs"
  2856 by (metis distinct_remdups finite_list set_remdups)
  2857 
  2858 lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
  2859 by (induct x, auto)
  2860 
  2861 lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
  2862 by (induct x, auto)
  2863 
  2864 lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
  2865 by (induct xs) auto
  2866 
  2867 lemma length_remdups_eq[iff]:
  2868   "(length (remdups xs) = length xs) = (remdups xs = xs)"
  2869 apply(induct xs)
  2870  apply auto
  2871 apply(subgoal_tac "length (remdups xs) <= length xs")
  2872  apply arith
  2873 apply(rule length_remdups_leq)
  2874 done
  2875 
  2876 lemma remdups_filter: "remdups(filter P xs) = filter P (remdups xs)"
  2877 apply(induct xs)
  2878 apply auto
  2879 done
  2880 
  2881 lemma distinct_map:
  2882   "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
  2883 by (induct xs) auto
  2884 
  2885 
  2886 lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
  2887 by (induct xs) auto
  2888 
  2889 lemma distinct_upt[simp]: "distinct[i..<j]"
  2890 by (induct j) auto
  2891 
  2892 lemma distinct_upto[simp]: "distinct[i..j]"
  2893 apply(induct i j rule:upto.induct)
  2894 apply(subst upto.simps)
  2895 apply(simp)
  2896 done
  2897 
  2898 lemma distinct_take[simp]: "distinct xs \<Longrightarrow> distinct (take i xs)"
  2899 apply(induct xs arbitrary: i)
  2900  apply simp
  2901 apply (case_tac i)
  2902  apply simp_all
  2903 apply(blast dest:in_set_takeD)
  2904 done
  2905 
  2906 lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)"
  2907 apply(induct xs arbitrary: i)
  2908  apply simp
  2909 apply (case_tac i)
  2910  apply simp_all
  2911 done
  2912 
  2913 lemma distinct_list_update:
  2914 assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
  2915 shows "distinct (xs[i:=a])"
  2916 proof (cases "i < length xs")
  2917   case True
  2918   with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
  2919     apply (drule_tac id_take_nth_drop) by simp
  2920   with d True show ?thesis
  2921     apply (simp add: upd_conv_take_nth_drop)
  2922     apply (drule subst [OF id_take_nth_drop]) apply assumption
  2923     apply simp apply (cases "a = xs!i") apply simp by blast
  2924 next
  2925   case False with d show ?thesis by auto
  2926 qed
  2927 
  2928 lemma distinct_concat:
  2929   assumes "distinct xs"
  2930   and "\<And> ys. ys \<in> set xs \<Longrightarrow> distinct ys"
  2931   and "\<And> ys zs. \<lbrakk> ys \<in> set xs ; zs \<in> set xs ; ys \<noteq> zs \<rbrakk> \<Longrightarrow> set ys \<inter> set zs = {}"
  2932   shows "distinct (concat xs)"
  2933   using assms by (induct xs) auto
  2934 
  2935 text {* It is best to avoid this indexed version of distinct, but
  2936 sometimes it is useful. *}
  2937 
  2938 lemma distinct_conv_nth:
  2939 "distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
  2940 apply (induct xs, simp, simp)
  2941 apply (rule iffI, clarsimp)
  2942  apply (case_tac i)
  2943 apply (case_tac j, simp)
  2944 apply (simp add: set_conv_nth)
  2945  apply (case_tac j)
  2946 apply (clarsimp simp add: set_conv_nth, simp)
  2947 apply (rule conjI)
  2948 (*TOO SLOW
  2949 apply (metis Zero_neq_Suc gr0_conv_Suc in_set_conv_nth lessI less_trans_Suc nth_Cons' nth_Cons_Suc)
  2950 *)
  2951  apply (clarsimp simp add: set_conv_nth)
  2952  apply (erule_tac x = 0 in allE, simp)
  2953  apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
  2954 (*TOO SLOW
  2955 apply (metis Suc_Suc_eq lessI less_trans_Suc nth_Cons_Suc)
  2956 *)
  2957 apply (erule_tac x = "Suc i" in allE, simp)
  2958 apply (erule_tac x = "Suc j" in allE, simp)
  2959 done
  2960 
  2961 lemma nth_eq_iff_index_eq:
  2962  "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
  2963 by(auto simp: distinct_conv_nth)
  2964 
  2965 lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
  2966 by (induct xs) auto
  2967 
  2968 lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
  2969 proof (induct xs)
  2970   case Nil thus ?case by simp
  2971 next
  2972   case (Cons x xs)
  2973   show ?case
  2974   proof (cases "x \<in> set xs")
  2975     case False with Cons show ?thesis by simp
  2976   next
  2977     case True with Cons.prems
  2978     have "card (set xs) = Suc (length xs)"
  2979       by (simp add: card_insert_if split: split_if_asm)
  2980     moreover have "card (set xs) \<le> length xs" by (rule card_length)
  2981     ultimately have False by simp
  2982     thus ?thesis ..
  2983   qed
  2984 qed
  2985 
  2986 lemma distinct_length_filter: "distinct xs \<Longrightarrow> length (filter P xs) = card ({x. P x} Int set xs)"
  2987 by (induct xs) (auto)
  2988 
  2989 lemma not_distinct_decomp: "~ distinct ws ==> EX xs ys zs y. ws = xs@[y]@ys@[y]@zs"
  2990 apply (induct n == "length ws" arbitrary:ws) apply simp
  2991 apply(case_tac ws) apply simp
  2992 apply (simp split:split_if_asm)
  2993 apply (metis Cons_eq_appendI eq_Nil_appendI split_list)
  2994 done
  2995 
  2996 lemma not_distinct_conv_prefix:
  2997   defines "dec as xs y ys \<equiv> y \<in> set xs \<and> distinct xs \<and> as = xs @ y # ys"
  2998   shows "\<not>distinct as \<longleftrightarrow> (\<exists>xs y ys. dec as xs y ys)" (is "?L = ?R")
  2999 proof
  3000   assume "?L" then show "?R"
  3001   proof (induct "length as" arbitrary: as rule: less_induct)
  3002     case less
  3003     obtain xs ys zs y where decomp: "as = (xs @ y # ys) @ y # zs"
  3004       using not_distinct_decomp[OF less.prems] by auto
  3005     show ?case
  3006     proof (cases "distinct (xs @ y # ys)")
  3007       case True
  3008       with decomp have "dec as (xs @ y # ys) y zs" by (simp add: dec_def)
  3009       then show ?thesis by blast
  3010     next
  3011       case False
  3012       with less decomp obtain xs' y' ys' where "dec (xs @ y # ys) xs' y' ys'"
  3013         by atomize_elim auto
  3014       with decomp have "dec as xs' y' (ys' @ y # zs)" by (simp add: dec_def)
  3015       then show ?thesis by blast
  3016     qed
  3017   qed
  3018 qed (auto simp: dec_def)
  3019 
  3020 lemma length_remdups_concat:
  3021   "length (remdups (concat xss)) = card (\<Union>xs\<in>set xss. set xs)"
  3022   by (simp add: distinct_card [symmetric])
  3023 
  3024 lemma length_remdups_card_conv: "length(remdups xs) = card(set xs)"
  3025 proof -
  3026   have xs: "concat[xs] = xs" by simp
  3027   from length_remdups_concat[of "[xs]"] show ?thesis unfolding xs by simp
  3028 qed
  3029 
  3030 lemma remdups_remdups:
  3031   "remdups (remdups xs) = remdups xs"
  3032   by (induct xs) simp_all
  3033 
  3034 lemma distinct_butlast:
  3035   assumes "distinct xs"
  3036   shows "distinct (butlast xs)"
  3037 proof (cases "xs = []")
  3038   case False
  3039     from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  3040     with `distinct xs` show ?thesis by simp
  3041 qed (auto)
  3042 
  3043 lemma remdups_map_remdups:
  3044   "remdups (map f (remdups xs)) = remdups (map f xs)"
  3045   by (induct xs) simp_all
  3046 
  3047 lemma distinct_zipI1:
  3048   assumes "distinct xs"
  3049   shows "distinct (zip xs ys)"
  3050 proof (rule zip_obtain_same_length)
  3051   fix xs' :: "'a list" and ys' :: "'b list" and n
  3052   assume "length xs' = length ys'"
  3053   assume "xs' = take n xs"
  3054   with assms have "distinct xs'" by simp
  3055   with `length xs' = length ys'` show "distinct (zip xs' ys')"
  3056     by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
  3057 qed
  3058 
  3059 lemma distinct_zipI2:
  3060   assumes "distinct ys"
  3061   shows "distinct (zip xs ys)"
  3062 proof (rule zip_obtain_same_length)
  3063   fix xs' :: "'b list" and ys' :: "'a list" and n
  3064   assume "length xs' = length ys'"
  3065   assume "ys' = take n ys"
  3066   with assms have "distinct ys'" by simp
  3067   with `length xs' = length ys'` show "distinct (zip xs' ys')"
  3068     by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
  3069 qed
  3070 
  3071 (* The next two lemmas help Sledgehammer. *)
  3072 
  3073 lemma distinct_singleton: "distinct [x]" by simp
  3074 
  3075 lemma distinct_length_2_or_more:
  3076 "distinct (a # b # xs) \<longleftrightarrow> (a \<noteq> b \<and> distinct (a # xs) \<and> distinct (b # xs))"
  3077 by (metis distinct.simps(2) hd.simps hd_in_set list.simps(2) set_ConsD set_rev_mp set_subset_Cons)
  3078 
  3079 subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
  3080 
  3081 lemma (in monoid_add) listsum_simps [simp]:
  3082   "listsum [] = 0"
  3083   "listsum (x # xs) = x + listsum xs"
  3084   by (simp_all add: listsum_def)
  3085 
  3086 lemma (in monoid_add) listsum_append [simp]:
  3087   "listsum (xs @ ys) = listsum xs + listsum ys"
  3088   by (induct xs) (simp_all add: add.assoc)
  3089 
  3090 lemma (in comm_monoid_add) listsum_rev [simp]:
  3091   "listsum (rev xs) = listsum xs"
  3092   by (simp add: listsum_def foldr_def fold_rev fun_eq_iff add_ac)
  3093 
  3094 lemma (in monoid_add) fold_plus_listsum_rev:
  3095   "fold plus xs = plus (listsum (rev xs))"
  3096 proof
  3097   fix x
  3098   have "fold plus xs x = fold plus xs (x + 0)" by simp
  3099   also have "\<dots> = fold plus (x # xs) 0" by simp
  3100   also have "\<dots> = foldr plus (rev xs @ [x]) 0" by (simp add: foldr_def)
  3101   also have "\<dots> = listsum (rev xs @ [x])" by (simp add: listsum_def)
  3102   also have "\<dots> = listsum (rev xs) + listsum [x]" by simp
  3103   finally show "fold plus xs x = listsum (rev xs) + x" by simp
  3104 qed
  3105 
  3106 lemma (in semigroup_add) foldl_assoc:
  3107   "foldl plus (x + y) zs = x + foldl plus y zs"
  3108   by (simp add: foldl_def fold_commute_apply [symmetric] fun_eq_iff add_assoc)
  3109 
  3110 lemma (in ab_semigroup_add) foldr_conv_foldl:
  3111   "foldr plus xs a = foldl plus a xs"
  3112   by (simp add: foldl_def foldr_fold fun_eq_iff add_ac)
  3113 
  3114 text {*
  3115   Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
  3116   difficult to use because it requires an additional transitivity step.
  3117 *}
  3118 
  3119 lemma start_le_sum:
  3120   fixes m n :: nat
  3121   shows "m \<le> n \<Longrightarrow> m \<le> foldl plus n ns"
  3122   by (simp add: foldl_def add_commute fold_plus_listsum_rev)
  3123 
  3124 lemma elem_le_sum:
  3125   fixes m n :: nat 
  3126   shows "n \<in> set ns \<Longrightarrow> n \<le> foldl plus 0 ns"
  3127   by (force intro: start_le_sum simp add: in_set_conv_decomp)
  3128 
  3129 lemma sum_eq_0_conv [iff]:
  3130   fixes m :: nat
  3131   shows "foldl plus m ns = 0 \<longleftrightarrow> m = 0 \<and> (\<forall>n \<in> set ns. n = 0)"
  3132   by (induct ns arbitrary: m) auto
  3133 
  3134 text{* Some syntactic sugar for summing a function over a list: *}
  3135 
  3136 syntax
  3137   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
  3138 syntax (xsymbols)
  3139   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
  3140 syntax (HTML output)
  3141   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
  3142 
  3143 translations -- {* Beware of argument permutation! *}
  3144   "SUM x<-xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  3145   "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  3146 
  3147 lemma (in comm_monoid_add) listsum_map_remove1:
  3148   "x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))"
  3149   by (induct xs) (auto simp add: ac_simps)
  3150 
  3151 lemma (in monoid_add) list_size_conv_listsum:
  3152   "list_size f xs = listsum (map f xs) + size xs"
  3153   by (induct xs) auto
  3154 
  3155 lemma (in monoid_add) length_concat:
  3156   "length (concat xss) = listsum (map length xss)"
  3157   by (induct xss) simp_all
  3158 
  3159 lemma (in monoid_add) listsum_map_filter:
  3160   assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
  3161   shows "listsum (map f (filter P xs)) = listsum (map f xs)"
  3162   using assms by (induct xs) auto
  3163 
  3164 lemma (in monoid_add) distinct_listsum_conv_Setsum:
  3165   "distinct xs \<Longrightarrow> listsum xs = Setsum (set xs)"
  3166   by (induct xs) simp_all
  3167 
  3168 lemma listsum_eq_0_nat_iff_nat [simp]:
  3169   "listsum ns = (0::nat) \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
  3170   by (simp add: listsum_def foldr_conv_foldl)
  3171 
  3172 lemma elem_le_listsum_nat:
  3173   "k < size ns \<Longrightarrow> ns ! k \<le> listsum (ns::nat list)"
  3174 apply(induct ns arbitrary: k)
  3175  apply simp
  3176 apply(fastforce simp add:nth_Cons split: nat.split)
  3177 done
  3178 
  3179 lemma listsum_update_nat:
  3180   "k<size ns \<Longrightarrow> listsum (ns[k := (n::nat)]) = listsum ns + n - ns ! k"
  3181 apply(induct ns arbitrary:k)
  3182  apply (auto split:nat.split)
  3183 apply(drule elem_le_listsum_nat)
  3184 apply arith
  3185 done
  3186 
  3187 lemma (in monoid_add) listsum_triv:
  3188   "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
  3189   by (induct xs) (simp_all add: left_distrib)
  3190 
  3191 lemma (in monoid_add) listsum_0 [simp]:
  3192   "(\<Sum>x\<leftarrow>xs. 0) = 0"
  3193   by (induct xs) (simp_all add: left_distrib)
  3194 
  3195 text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
  3196 lemma (in ab_group_add) uminus_listsum_map:
  3197   "- listsum (map f xs) = listsum (map (uminus \<circ> f) xs)"
  3198   by (induct xs) simp_all
  3199 
  3200 lemma (in comm_monoid_add) listsum_addf:
  3201   "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
  3202   by (induct xs) (simp_all add: algebra_simps)
  3203 
  3204 lemma (in ab_group_add) listsum_subtractf:
  3205   "(\<Sum>x\<leftarrow>xs. f x - g x) = listsum (map f xs) - listsum (map g xs)"
  3206   by (induct xs) (simp_all add: algebra_simps)
  3207 
  3208 lemma (in semiring_0) listsum_const_mult:
  3209   "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
  3210   by (induct xs) (simp_all add: algebra_simps)
  3211 
  3212 lemma (in semiring_0) listsum_mult_const:
  3213   "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
  3214   by (induct xs) (simp_all add: algebra_simps)
  3215 
  3216 lemma (in ordered_ab_group_add_abs) listsum_abs:
  3217   "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
  3218   by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])
  3219 
  3220 lemma listsum_mono:
  3221   fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}"
  3222   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)"
  3223   by (induct xs) (simp, simp add: add_mono)
  3224 
  3225 lemma (in monoid_add) listsum_distinct_conv_setsum_set:
  3226   "distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)"
  3227   by (induct xs) simp_all
  3228 
  3229 lemma (in monoid_add) interv_listsum_conv_setsum_set_nat:
  3230   "listsum (map f [m..<n]) = setsum f (set [m..<n])"
  3231   by (simp add: listsum_distinct_conv_setsum_set)
  3232 
  3233 lemma (in monoid_add) interv_listsum_conv_setsum_set_int:
  3234   "listsum (map f [k..l]) = setsum f (set [k..l])"
  3235   by (simp add: listsum_distinct_conv_setsum_set)
  3236 
  3237 text {* General equivalence between @{const listsum} and @{const setsum} *}
  3238 lemma (in monoid_add) listsum_setsum_nth:
  3239   "listsum xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
  3240   using interv_listsum_conv_setsum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth)
  3241 
  3242 
  3243 subsubsection {* @{const insert} *}
  3244 
  3245 lemma in_set_insert [simp]:
  3246   "x \<in> set xs \<Longrightarrow> List.insert x xs = xs"
  3247   by (simp add: List.insert_def)
  3248 
  3249 lemma not_in_set_insert [simp]:
  3250   "x \<notin> set xs \<Longrightarrow> List.insert x xs = x # xs"
  3251   by (simp add: List.insert_def)
  3252 
  3253 lemma insert_Nil [simp]:
  3254   "List.insert x [] = [x]"
  3255   by simp
  3256 
  3257 lemma set_insert [simp]:
  3258   "set (List.insert x xs) = insert x (set xs)"
  3259   by (auto simp add: List.insert_def)
  3260 
  3261 lemma distinct_insert [simp]:
  3262   "distinct xs \<Longrightarrow> distinct (List.insert x xs)"
  3263   by (simp add: List.insert_def)
  3264 
  3265 lemma insert_remdups:
  3266   "List.insert x (remdups xs) = remdups (List.insert x xs)"
  3267   by (simp add: List.insert_def)
  3268 
  3269 
  3270 subsubsection {* @{text remove1} *}
  3271 
  3272 lemma remove1_append:
  3273   "remove1 x (xs @ ys) =
  3274   (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
  3275 by (induct xs) auto
  3276 
  3277 lemma remove1_commute: "remove1 x (remove1 y zs) = remove1 y (remove1 x zs)"
  3278 by (induct zs) auto
  3279 
  3280 lemma in_set_remove1[simp]:
  3281   "a \<noteq> b \<Longrightarrow> a : set(remove1 b xs) = (a : set xs)"
  3282 apply (induct xs)
  3283 apply auto
  3284 done
  3285 
  3286 lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
  3287 apply(induct xs)
  3288  apply simp
  3289 apply simp
  3290 apply blast
  3291 done
  3292 
  3293 lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
  3294 apply(induct xs)
  3295  apply simp
  3296 apply simp
  3297 apply blast
  3298 done
  3299 
  3300 lemma length_remove1:
  3301   "length(remove1 x xs) = (if x : set xs then length xs - 1 else length xs)"
  3302 apply (induct xs)
  3303  apply (auto dest!:length_pos_if_in_set)
  3304 done
  3305 
  3306 lemma remove1_filter_not[simp]:
  3307   "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
  3308 by(induct xs) auto
  3309 
  3310 lemma filter_remove1:
  3311   "filter Q (remove1 x xs) = remove1 x (filter Q xs)"
  3312 by (induct xs) auto
  3313 
  3314 lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
  3315 apply(insert set_remove1_subset)
  3316 apply fast
  3317 done
  3318 
  3319 lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
  3320 by (induct xs) simp_all
  3321 
  3322 lemma remove1_remdups:
  3323   "distinct xs \<Longrightarrow> remove1 x (remdups xs) = remdups (remove1 x xs)"
  3324   by (induct xs) simp_all
  3325 
  3326 lemma remove1_idem:
  3327   assumes "x \<notin> set xs"
  3328   shows "remove1 x xs = xs"
  3329   using assms by (induct xs) simp_all
  3330 
  3331 
  3332 subsubsection {* @{text removeAll} *}
  3333 
  3334 lemma removeAll_filter_not_eq:
  3335   "removeAll x = filter (\<lambda>y. x \<noteq> y)"
  3336 proof
  3337   fix xs
  3338   show "removeAll x xs = filter (\<lambda>y. x \<noteq> y) xs"
  3339     by (induct xs) auto
  3340 qed
  3341 
  3342 lemma removeAll_append[simp]:
  3343   "removeAll x (xs @ ys) = removeAll x xs @ removeAll x ys"
  3344 by (induct xs) auto
  3345 
  3346 lemma set_removeAll[simp]: "set(removeAll x xs) = set xs - {x}"
  3347 by (induct xs) auto
  3348 
  3349 lemma removeAll_id[simp]: "x \<notin> set xs \<Longrightarrow> removeAll x xs = xs"
  3350 by (induct xs) auto
  3351 
  3352 (* Needs count:: 'a \<Rightarrow> 'a list \<Rightarrow> nat
  3353 lemma length_removeAll:
  3354   "length(removeAll x xs) = length xs - count x xs"
  3355 *)
  3356 
  3357 lemma removeAll_filter_not[simp]:
  3358   "\<not> P x \<Longrightarrow> removeAll x (filter P xs) = filter P xs"
  3359 by(induct xs) auto
  3360 
  3361 lemma distinct_removeAll:
  3362   "distinct xs \<Longrightarrow> distinct (removeAll x xs)"
  3363   by (simp add: removeAll_filter_not_eq)
  3364 
  3365 lemma distinct_remove1_removeAll:
  3366   "distinct xs ==> remove1 x xs = removeAll x xs"
  3367 by (induct xs) simp_all
  3368 
  3369 lemma map_removeAll_inj_on: "inj_on f (insert x (set xs)) \<Longrightarrow>
  3370   map f (removeAll x xs) = removeAll (f x) (map f xs)"
  3371 by (induct xs) (simp_all add:inj_on_def)
  3372 
  3373 lemma map_removeAll_inj: "inj f \<Longrightarrow>
  3374   map f (removeAll x xs) = removeAll (f x) (map f xs)"
  3375 by(metis map_removeAll_inj_on subset_inj_on subset_UNIV)
  3376 
  3377 
  3378 subsubsection {* @{text replicate} *}
  3379 
  3380 lemma length_replicate [simp]: "length (replicate n x) = n"
  3381 by (induct n) auto
  3382 
  3383 lemma Ex_list_of_length: "\<exists>xs. length xs = n"
  3384 by (rule exI[of _ "replicate n undefined"]) simp
  3385 
  3386 lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
  3387 by (induct n) auto
  3388 
  3389 lemma map_replicate_const:
  3390   "map (\<lambda> x. k) lst = replicate (length lst) k"
  3391   by (induct lst) auto
  3392 
  3393 lemma replicate_app_Cons_same:
  3394 "(replicate n x) @ (x # xs) = x # replicate n x @ xs"
  3395 by (induct n) auto
  3396 
  3397 lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
  3398 apply (induct n, simp)
  3399 apply (simp add: replicate_app_Cons_same)
  3400 done
  3401 
  3402 lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
  3403 by (induct n) auto
  3404 
  3405 text{* Courtesy of Matthias Daum: *}
  3406 lemma append_replicate_commute:
  3407   "replicate n x @ replicate k x = replicate k x @ replicate n x"
  3408 apply (simp add: replicate_add [THEN sym])
  3409 apply (simp add: add_commute)
  3410 done
  3411 
  3412 text{* Courtesy of Andreas Lochbihler: *}
  3413 lemma filter_replicate:
  3414   "filter P (replicate n x) = (if P x then replicate n x else [])"
  3415 by(induct n) auto
  3416 
  3417 lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
  3418 by (induct n) auto
  3419 
  3420 lemma tl_replicate [simp]: "tl (replicate n x) = replicate (n - 1) x"
  3421 by (induct n) auto
  3422 
  3423 lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
  3424 by (atomize (full), induct n) auto
  3425 
  3426 lemma nth_replicate[simp]: "i < n ==> (replicate n x)!i = x"
  3427 apply (induct n arbitrary: i, simp)
  3428 apply (simp add: nth_Cons split: nat.split)
  3429 done
  3430 
  3431 text{* Courtesy of Matthias Daum (2 lemmas): *}
  3432 lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
  3433 apply (case_tac "k \<le> i")
  3434  apply  (simp add: min_def)
  3435 apply (drule not_leE)
  3436 apply (simp add: min_def)
  3437 apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
  3438  apply  simp
  3439 apply (simp add: replicate_add [symmetric])
  3440 done
  3441 
  3442 lemma drop_replicate[simp]: "drop i (replicate k x) = replicate (k-i) x"
  3443 apply (induct k arbitrary: i)
  3444  apply simp
  3445 apply clarsimp
  3446 apply (case_tac i)
  3447  apply simp
  3448 apply clarsimp
  3449 done
  3450 
  3451 
  3452 lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
  3453 by (induct n) auto
  3454 
  3455 lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
  3456 by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
  3457 
  3458 lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
  3459 by auto
  3460 
  3461 lemma in_set_replicate[simp]: "(x : set (replicate n y)) = (x = y & n \<noteq> 0)"
  3462 by (simp add: set_replicate_conv_if)
  3463 
  3464 lemma Ball_set_replicate[simp]:
  3465   "(ALL x : set(replicate n a). P x) = (P a | n=0)"
  3466 by(simp add: set_replicate_conv_if)
  3467 
  3468 lemma Bex_set_replicate[simp]:
  3469   "(EX x : set(replicate n a). P x) = (P a & n\<noteq>0)"
  3470 by(simp add: set_replicate_conv_if)
  3471 
  3472 lemma replicate_append_same:
  3473   "replicate i x @ [x] = x # replicate i x"
  3474   by (induct i) simp_all
  3475 
  3476 lemma map_replicate_trivial:
  3477   "map (\<lambda>i. x) [0..<i] = replicate i x"
  3478   by (induct i) (simp_all add: replicate_append_same)
  3479 
  3480 lemma concat_replicate_trivial[simp]:
  3481   "concat (replicate i []) = []"
  3482   by (induct i) (auto simp add: map_replicate_const)
  3483 
  3484 lemma replicate_empty[simp]: "(replicate n x = []) \<longleftrightarrow> n=0"
  3485 by (induct n) auto
  3486 
  3487 lemma empty_replicate[simp]: "([] = replicate n x) \<longleftrightarrow> n=0"
  3488 by (induct n) auto
  3489 
  3490 lemma replicate_eq_replicate[simp]:
  3491   "(replicate m x = replicate n y) \<longleftrightarrow> (m=n & (m\<noteq>0 \<longrightarrow> x=y))"
  3492 apply(induct m arbitrary: n)
  3493  apply simp
  3494 apply(induct_tac n)
  3495 apply auto
  3496 done
  3497 
  3498 lemma replicate_length_filter:
  3499   "replicate (length (filter (\<lambda>y. x = y) xs)) x = filter (\<lambda>y. x = y) xs"
  3500   by (induct xs) auto
  3501 
  3502 lemma comm_append_are_replicate:
  3503   fixes xs ys :: "'a list"
  3504   assumes "xs \<noteq> []" "ys \<noteq> []"
  3505   assumes "xs @ ys = ys @ xs"
  3506   shows "\<exists>m n zs. concat (replicate m zs) = xs \<and> concat (replicate n zs) = ys"
  3507   using assms
  3508 proof (induct "length (xs @ ys)" arbitrary: xs ys rule: less_induct)
  3509   case less
  3510 
  3511   def xs' \<equiv> "if (length xs \<le> length ys) then xs else ys"
  3512     and ys' \<equiv> "if (length xs \<le> length ys) then ys else xs"
  3513   then have
  3514     prems': "length xs' \<le> length ys'"
  3515             "xs' @ ys' = ys' @ xs'"
  3516       and "xs' \<noteq> []"
  3517       and len: "length (xs @ ys) = length (xs' @ ys')"
  3518     using less by (auto intro: less.hyps)
  3519 
  3520   from prems'
  3521   obtain ws where "ys' = xs' @ ws"
  3522     by (auto simp: append_eq_append_conv2)
  3523 
  3524   have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ys'"
  3525   proof (cases "ws = []")
  3526     case True
  3527     then have "concat (replicate 1 xs') = xs'"
  3528       and "concat (replicate 1 xs') = ys'"
  3529       using `ys' = xs' @ ws` by auto
  3530     then show ?thesis by blast
  3531   next
  3532     case False
  3533     from `ys' = xs' @ ws` and `xs' @ ys' = ys' @ xs'`
  3534     have "xs' @ ws = ws @ xs'" by simp
  3535     then have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ws"
  3536       using False and `xs' \<noteq> []` and `ys' = xs' @ ws` and len
  3537       by (intro less.hyps) auto
  3538     then obtain m n zs where "concat (replicate m zs) = xs'"
  3539       and "concat (replicate n zs) = ws" by blast
  3540     moreover
  3541     then have "concat (replicate (m + n) zs) = ys'"
  3542       using `ys' = xs' @ ws`
  3543       by (simp add: replicate_add)
  3544     ultimately
  3545     show ?thesis by blast
  3546   qed
  3547   then show ?case
  3548     using xs'_def ys'_def by metis
  3549 qed
  3550 
  3551 lemma comm_append_is_replicate:
  3552   fixes xs ys :: "'a list"
  3553   assumes "xs \<noteq> []" "ys \<noteq> []"
  3554   assumes "xs @ ys = ys @ xs"
  3555   shows "\<exists>n zs. n > 1 \<and> concat (replicate n zs) = xs @ ys"
  3556 
  3557 proof -
  3558   obtain m n zs where "concat (replicate m zs) = xs"
  3559     and "concat (replicate n zs) = ys"
  3560     using assms by (metis comm_append_are_replicate)
  3561   then have "m + n > 1" and "concat (replicate (m+n) zs) = xs @ ys"
  3562     using `xs \<noteq> []` and `ys \<noteq> []`
  3563     by (auto simp: replicate_add)
  3564   then show ?thesis by blast
  3565 qed
  3566 
  3567 
  3568 subsubsection{*@{text rotate1} and @{text rotate}*}
  3569 
  3570 lemma rotate0[simp]: "rotate 0 = id"
  3571 by(simp add:rotate_def)
  3572 
  3573 lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
  3574 by(simp add:rotate_def)
  3575 
  3576 lemma rotate_add:
  3577   "rotate (m+n) = rotate m o rotate n"
  3578 by(simp add:rotate_def funpow_add)
  3579 
  3580 lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
  3581 by(simp add:rotate_add)
  3582 
  3583 lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
  3584 by(simp add:rotate_def funpow_swap1)
  3585 
  3586 lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
  3587 by(cases xs) simp_all
  3588 
  3589 lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
  3590 apply(induct n)
  3591  apply simp
  3592 apply (simp add:rotate_def)
  3593 done
  3594 
  3595 lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
  3596 by (cases xs) simp_all
  3597 
  3598 lemma rotate_drop_take:
  3599   "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
  3600 apply(induct n)
  3601  apply simp
  3602 apply(simp add:rotate_def)
  3603 apply(cases "xs = []")
  3604  apply (simp)
  3605 apply(case_tac "n mod length xs = 0")
  3606  apply(simp add:mod_Suc)
  3607  apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
  3608 apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
  3609                 take_hd_drop linorder_not_le)
  3610 done
  3611 
  3612 lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
  3613 by(simp add:rotate_drop_take)
  3614 
  3615 lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
  3616 by(simp add:rotate_drop_take)
  3617 
  3618 lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
  3619 by (cases xs) simp_all
  3620 
  3621 lemma length_rotate[simp]: "length(rotate n xs) = length xs"
  3622 by (induct n arbitrary: xs) (simp_all add:rotate_def)
  3623 
  3624 lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
  3625 by (cases xs) auto
  3626 
  3627 lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
  3628 by (induct n) (simp_all add:rotate_def)
  3629 
  3630 lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
  3631 by(simp add:rotate_drop_take take_map drop_map)
  3632 
  3633 lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
  3634 by (cases xs) auto
  3635 
  3636 lemma set_rotate[simp]: "set(rotate n xs) = set xs"
  3637 by (induct n) (simp_all add:rotate_def)
  3638 
  3639 lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
  3640 by (cases xs) auto
  3641 
  3642 lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
  3643 by (induct n) (simp_all add:rotate_def)
  3644 
  3645 lemma rotate_rev:
  3646   "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
  3647 apply(simp add:rotate_drop_take rev_drop rev_take)
  3648 apply(cases "length xs = 0")
  3649  apply simp
  3650 apply(cases "n mod length xs = 0")
  3651  apply simp
  3652 apply(simp add:rotate_drop_take rev_drop rev_take)
  3653 done
  3654 
  3655 lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
  3656 apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
  3657 apply(subgoal_tac "length xs \<noteq> 0")
  3658  prefer 2 apply simp
  3659 using mod_less_divisor[of "length xs" n] by arith
  3660 
  3661 
  3662 subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
  3663 
  3664 lemma sublist_empty [simp]: "sublist xs {} = []"
  3665 by (auto simp add: sublist_def)
  3666 
  3667 lemma sublist_nil [simp]: "sublist [] A = []"
  3668 by (auto simp add: sublist_def)
  3669 
  3670 lemma length_sublist:
  3671   "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
  3672 by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
  3673 
  3674 lemma sublist_shift_lemma_Suc:
  3675   "map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
  3676    map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
  3677 apply(induct xs arbitrary: "is")
  3678  apply simp
  3679 apply (case_tac "is")
  3680  apply simp
  3681 apply simp
  3682 done
  3683 
  3684 lemma sublist_shift_lemma:
  3685      "map fst [p<-zip xs [i..<i + length xs] . snd p : A] =
  3686       map fst [p<-zip xs [0..<length xs] . snd p + i : A]"
  3687 by (induct xs rule: rev_induct) (simp_all add: add_commute)
  3688 
  3689 lemma sublist_append:
  3690      "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
  3691 apply (unfold sublist_def)
  3692 apply (induct l' rule: rev_induct, simp)
  3693 apply (simp add: upt_add_eq_append[of 0] sublist_shift_lemma)
  3694 apply (simp add: add_commute)
  3695 done
  3696 
  3697 lemma sublist_Cons:
  3698 "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
  3699 apply (induct l rule: rev_induct)
  3700  apply (simp add: sublist_def)
  3701 apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
  3702 done
  3703 
  3704 lemma set_sublist: "set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
  3705 apply(induct xs arbitrary: I)
  3706 apply(auto simp: sublist_Cons nth_Cons split:nat.split dest!: gr0_implies_Suc)
  3707 done
  3708 
  3709 lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
  3710 by(auto simp add:set_sublist)
  3711 
  3712 lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
  3713 by(auto simp add:set_sublist)
  3714 
  3715 lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
  3716 by(auto simp add:set_sublist)
  3717 
  3718 lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
  3719 by (simp add: sublist_Cons)
  3720 
  3721 
  3722 lemma distinct_sublistI[simp]: "distinct xs \<Longrightarrow> distinct(sublist xs I)"
  3723 apply(induct xs arbitrary: I)
  3724  apply simp
  3725 apply(auto simp add:sublist_Cons)
  3726 done
  3727 
  3728 
  3729 lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
  3730 apply (induct l rule: rev_induct, simp)
  3731 apply (simp split: nat_diff_split add: sublist_append)
  3732 done
  3733 
  3734 lemma filter_in_sublist:
  3735  "distinct xs \<Longrightarrow> filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
  3736 proof (induct xs arbitrary: s)
  3737   case Nil thus ?case by simp
  3738 next
  3739   case (Cons a xs)
  3740   moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
  3741   ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
  3742 qed
  3743 
  3744 
  3745 subsubsection {* @{const splice} *}
  3746 
  3747 lemma splice_Nil2 [simp, code]: "splice xs [] = xs"
  3748 by (cases xs) simp_all
  3749 
  3750 declare splice.simps(1,3)[code]
  3751 declare splice.simps(2)[simp del]
  3752 
  3753 lemma length_splice[simp]: "length(splice xs ys) = length xs + length ys"
  3754 by (induct xs ys rule: splice.induct) auto
  3755 
  3756 
  3757 subsubsection {* Transpose *}
  3758 
  3759 function transpose where
  3760 "transpose []             = []" |
  3761 "transpose ([]     # xss) = transpose xss" |
  3762 "transpose ((x#xs) # xss) =
  3763   (x # [h. (h#t) \<leftarrow> xss]) # transpose (xs # [t. (h#t) \<leftarrow> xss])"
  3764 by pat_completeness auto
  3765 
  3766 lemma transpose_aux_filter_head:
  3767   "concat (map (list_case [] (\<lambda>h t. [h])) xss) =
  3768   map (\<lambda>xs. hd xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  3769   by (induct xss) (auto split: list.split)
  3770 
  3771 lemma transpose_aux_filter_tail:
  3772   "concat (map (list_case [] (\<lambda>h t. [t])) xss) =
  3773   map (\<lambda>xs. tl xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  3774   by (induct xss) (auto split: list.split)
  3775 
  3776 lemma transpose_aux_max:
  3777   "max (Suc (length xs)) (foldr (\<lambda>xs. max (length xs)) xss 0) =
  3778   Suc (max (length xs) (foldr (\<lambda>x. max (length x - Suc 0)) [ys\<leftarrow>xss . ys\<noteq>[]] 0))"
  3779   (is "max _ ?foldB = Suc (max _ ?foldA)")
  3780 proof (cases "[ys\<leftarrow>xss . ys\<noteq>[]] = []")
  3781   case True
  3782   hence "foldr (\<lambda>xs. max (length xs)) xss 0 = 0"
  3783   proof (induct xss)
  3784     case (Cons x xs)
  3785     moreover hence "x = []" by (cases x) auto
  3786     ultimately show ?case by auto
  3787   qed simp
  3788   thus ?thesis using True by simp
  3789 next
  3790   case False
  3791 
  3792   have foldA: "?foldA = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0 - 1"
  3793     by (induct xss) auto
  3794   have foldB: "?foldB = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0"
  3795     by (induct xss) auto
  3796 
  3797   have "0 < ?foldB"
  3798   proof -
  3799     from False
  3800     obtain z zs where zs: "[ys\<leftarrow>xss . ys \<noteq> []] = z#zs" by (auto simp: neq_Nil_conv)
  3801     hence "z \<in> set ([ys\<leftarrow>xss . ys \<noteq> []])" by auto
  3802     hence "z \<noteq> []" by auto
  3803     thus ?thesis
  3804       unfolding foldB zs
  3805       by (auto simp: max_def intro: less_le_trans)
  3806   qed
  3807   thus ?thesis
  3808     unfolding foldA foldB max_Suc_Suc[symmetric]
  3809     by simp
  3810 qed
  3811 
  3812 termination transpose
  3813   by (relation "measure (\<lambda>xs. foldr (\<lambda>xs. max (length xs)) xs 0 + length xs)")
  3814      (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max less_Suc_eq_le)
  3815 
  3816 lemma transpose_empty: "(transpose xs = []) \<longleftrightarrow> (\<forall>x \<in> set xs. x = [])"
  3817   by (induct rule: transpose.induct) simp_all
  3818 
  3819 lemma length_transpose:
  3820   fixes xs :: "'a list list"
  3821   shows "length (transpose xs) = foldr (\<lambda>xs. max (length xs)) xs 0"
  3822   by (induct rule: transpose.induct)
  3823     (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max
  3824                 max_Suc_Suc[symmetric] simp del: max_Suc_Suc)
  3825 
  3826 lemma nth_transpose:
  3827   fixes xs :: "'a list list"
  3828   assumes "i < length (transpose xs)"
  3829   shows "transpose xs ! i = map (\<lambda>xs. xs ! i) [ys \<leftarrow> xs. i < length ys]"
  3830 using assms proof (induct arbitrary: i rule: transpose.induct)
  3831   case (3 x xs xss)
  3832   def XS == "(x # xs) # xss"
  3833   hence [simp]: "XS \<noteq> []" by auto
  3834   thus ?case
  3835   proof (cases i)
  3836     case 0
  3837     thus ?thesis by (simp add: transpose_aux_filter_head hd_conv_nth)
  3838   next
  3839     case (Suc j)
  3840     have *: "\<And>xss. xs # map tl xss = map tl ((x#xs)#xss)" by simp
  3841     have **: "\<And>xss. (x#xs) # filter (\<lambda>ys. ys \<noteq> []) xss = filter (\<lambda>ys. ys \<noteq> []) ((x#xs)#xss)" by simp
  3842     { fix x have "Suc j < length x \<longleftrightarrow> x \<noteq> [] \<and> j < length x - Suc 0"
  3843       by (cases x) simp_all
  3844     } note *** = this
  3845 
  3846     have j_less: "j < length (transpose (xs # concat (map (list_case [] (\<lambda>h t. [t])) xss)))"
  3847       using "3.prems" by (simp add: transpose_aux_filter_tail length_transpose Suc)
  3848 
  3849     show ?thesis
  3850       unfolding transpose.simps `i = Suc j` nth_Cons_Suc "3.hyps"[OF j_less]
  3851       apply (auto simp: transpose_aux_filter_tail filter_map comp_def length_transpose * ** *** XS_def[symmetric])
  3852       apply (rule_tac y=x in list.exhaust)
  3853       by auto
  3854   qed
  3855 qed simp_all
  3856 
  3857 lemma transpose_map_map:
  3858   "transpose (map (map f) xs) = map (map f) (transpose xs)"
  3859 proof (rule nth_equalityI, safe)
  3860   have [simp]: "length (transpose (map (map f) xs)) = length (transpose xs)"
  3861     by (simp add: length_transpose foldr_map comp_def)
  3862   show "length (transpose (map (map f) xs)) = length (map (map f) (transpose xs))" by simp
  3863 
  3864   fix i assume "i < length (transpose (map (map f) xs))"
  3865   thus "transpose (map (map f) xs) ! i = map (map f) (transpose xs) ! i"
  3866     by (simp add: nth_transpose filter_map comp_def)
  3867 qed
  3868 
  3869 
  3870 subsubsection {* (In)finiteness *}
  3871 
  3872 lemma finite_maxlen:
  3873   "finite (M::'a list set) ==> EX n. ALL s:M. size s < n"
  3874 proof (induct rule: finite.induct)
  3875   case emptyI show ?case by simp
  3876 next
  3877   case (insertI M xs)
  3878   then obtain n where "\<forall>s\<in>M. length s < n" by blast
  3879   hence "ALL s:insert xs M. size s < max n (size xs) + 1" by auto
  3880   thus ?case ..
  3881 qed
  3882 
  3883 lemma lists_length_Suc_eq:
  3884   "{xs. set xs \<subseteq> A \<and> length xs = Suc n} =
  3885     (\<lambda>(xs, n). n#xs) ` ({xs. set xs \<subseteq> A \<and> length xs = n} \<times> A)"
  3886   by (auto simp: length_Suc_conv)
  3887 
  3888 lemma
  3889   assumes "finite A"
  3890   shows finite_lists_length_eq: "finite {xs. set xs \<subseteq> A \<and> length xs = n}"
  3891   and card_lists_length_eq: "card {xs. set xs \<subseteq> A \<and> length xs = n} = (card A)^n"
  3892   using `finite A`
  3893   by (induct n)
  3894      (auto simp: card_image inj_split_Cons lists_length_Suc_eq cong: conj_cong)
  3895 
  3896 lemma finite_lists_length_le:
  3897   assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs \<le> n}"
  3898  (is "finite ?S")
  3899 proof-
  3900   have "?S = (\<Union>n\<in>{0..n}. {xs. set xs \<subseteq> A \<and> length xs = n})" by auto
  3901   thus ?thesis by (auto intro: finite_lists_length_eq[OF `finite A`])
  3902 qed
  3903 
  3904 lemma card_lists_length_le:
  3905   assumes "finite A" shows "card {xs. set xs \<subseteq> A \<and> length xs \<le> n} = (\<Sum>i\<le>n. card A^i)"
  3906 proof -
  3907   have "(\<Sum>i\<le>n. card A^i) = card (\<Union>i\<le>n. {xs. set xs \<subseteq> A \<and> length xs = i})"
  3908     using `finite A`
  3909     by (subst card_UN_disjoint)
  3910        (auto simp add: card_lists_length_eq finite_lists_length_eq)
  3911   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}"
  3912     by auto
  3913   finally show ?thesis by simp
  3914 qed
  3915 
  3916 lemma card_lists_distinct_length_eq:
  3917   assumes "k < card A"
  3918   shows "card {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A} = \<Prod>{card A - k + 1 .. card A}"
  3919 using assms
  3920 proof (induct k)
  3921   case 0
  3922   then have "{xs. length xs = 0 \<and> distinct xs \<and> set xs \<subseteq> A} = {[]}" by auto
  3923   then show ?case by simp
  3924 next
  3925   case (Suc k)
  3926   let "?k_list" = "\<lambda>k xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A"
  3927   have inj_Cons: "\<And>A. inj_on (\<lambda>(xs, n). n # xs) A"  by (rule inj_onI) auto
  3928 
  3929   from Suc have "k < card A" by simp
  3930   moreover have "finite A" using assms by (simp add: card_ge_0_finite)
  3931   moreover have "finite {xs. ?k_list k xs}"
  3932     using finite_lists_length_eq[OF `finite A`, of k]
  3933     by - (rule finite_subset, auto)
  3934   moreover have "\<And>i j. i \<noteq> j \<longrightarrow> {i} \<times> (A - set i) \<inter> {j} \<times> (A - set j) = {}"
  3935     by auto
  3936   moreover have "\<And>i. i \<in>Collect (?k_list k) \<Longrightarrow> card (A - set i) = card A - k"
  3937     by (simp add: card_Diff_subset distinct_card)
  3938   moreover have "{xs. ?k_list (Suc k) xs} =
  3939       (\<lambda>(xs, n). n#xs) ` \<Union>(\<lambda>xs. {xs} \<times> (A - set xs)) ` {xs. ?k_list k xs}"
  3940     by (auto simp: length_Suc_conv)
  3941   moreover
  3942   have "Suc (card A - Suc k) = card A - k" using Suc.prems by simp
  3943   then have "(card A - k) * \<Prod>{Suc (card A - k)..card A} = \<Prod>{Suc (card A - Suc k)..card A}"
  3944     by (subst setprod_insert[symmetric]) (simp add: atLeastAtMost_insertL)+
  3945   ultimately show ?case
  3946     by (simp add: card_image inj_Cons card_UN_disjoint Suc.hyps algebra_simps)
  3947 qed
  3948 
  3949 lemma infinite_UNIV_listI: "~ finite(UNIV::'a list set)"
  3950 apply(rule notI)
  3951 apply(drule finite_maxlen)
  3952 apply (metis UNIV_I length_replicate less_not_refl)
  3953 done
  3954 
  3955 
  3956 subsection {* Sorting *}
  3957 
  3958 text{* Currently it is not shown that @{const sort} returns a
  3959 permutation of its input because the nicest proof is via multisets,
  3960 which are not yet available. Alternatively one could define a function
  3961 that counts the number of occurrences of an element in a list and use
  3962 that instead of multisets to state the correctness property. *}
  3963 
  3964 context linorder
  3965 begin
  3966 
  3967 lemma length_insort [simp]:
  3968   "length (insort_key f x xs) = Suc (length xs)"
  3969   by (induct xs) simp_all
  3970 
  3971 lemma insort_key_left_comm:
  3972   assumes "f x \<noteq> f y"
  3973   shows "insort_key f y (insort_key f x xs) = insort_key f x (insort_key f y xs)"
  3974   by (induct xs) (auto simp add: assms dest: antisym)
  3975 
  3976 lemma insort_left_comm:
  3977   "insort x (insort y xs) = insort y (insort x xs)"
  3978   by (cases "x = y") (auto intro: insort_key_left_comm)
  3979 
  3980 lemma comp_fun_commute_insort:
  3981   "comp_fun_commute insort"
  3982 proof
  3983 qed (simp add: insort_left_comm fun_eq_iff)
  3984 
  3985 lemma sort_key_simps [simp]:
  3986   "sort_key f [] = []"
  3987   "sort_key f (x#xs) = insort_key f x (sort_key f xs)"
  3988   by (simp_all add: sort_key_def)
  3989 
  3990 lemma (in linorder) sort_key_conv_fold:
  3991   assumes "inj_on f (set xs)"
  3992   shows "sort_key f xs = fold (insort_key f) xs []"
  3993 proof -
  3994   have "fold (insort_key f) (rev xs) = fold (insort_key f) xs"
  3995   proof (rule fold_rev, rule ext)
  3996     fix zs
  3997     fix x y
  3998     assume "x \<in> set xs" "y \<in> set xs"
  3999     with assms have *: "f y = f x \<Longrightarrow> y = x" by (auto dest: inj_onD)
  4000     have **: "x = y \<longleftrightarrow> y = x" by auto
  4001     show "(insort_key f y \<circ> insort_key f x) zs = (insort_key f x \<circ> insort_key f y) zs"
  4002       by (induct zs) (auto intro: * simp add: **)
  4003   qed
  4004   then show ?thesis by (simp add: sort_key_def foldr_def)
  4005 qed
  4006 
  4007 lemma (in linorder) sort_conv_fold:
  4008   "sort xs = fold insort xs []"
  4009   by (rule sort_key_conv_fold) simp
  4010 
  4011 lemma length_sort[simp]: "length (sort_key f xs) = length xs"
  4012 by (induct xs, auto)
  4013 
  4014 lemma sorted_Cons: "sorted (x#xs) = (sorted xs & (ALL y:set xs. x <= y))"
  4015 apply(induct xs arbitrary: x) apply simp
  4016 by simp (blast intro: order_trans)
  4017 
  4018 lemma sorted_tl:
  4019   "sorted xs \<Longrightarrow> sorted (tl xs)"
  4020   by (cases xs) (simp_all add: sorted_Cons)
  4021 
  4022 lemma sorted_append:
  4023   "sorted (xs@ys) = (sorted xs & sorted ys & (\<forall>x \<in> set xs. \<forall>y \<in> set ys. x\<le>y))"
  4024 by (induct xs) (auto simp add:sorted_Cons)
  4025 
  4026 lemma sorted_nth_mono:
  4027   "sorted xs \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!i \<le> xs!j"
  4028 by (induct xs arbitrary: i j) (auto simp:nth_Cons' sorted_Cons)
  4029 
  4030 lemma sorted_rev_nth_mono:
  4031   "sorted (rev xs) \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!j \<le> xs!i"
  4032 using sorted_nth_mono[ of "rev xs" "length xs - j - 1" "length xs - i - 1"]
  4033       rev_nth[of "length xs - i - 1" "xs"] rev_nth[of "length xs - j - 1" "xs"]
  4034 by auto
  4035 
  4036 lemma sorted_nth_monoI:
  4037   "(\<And> i j. \<lbrakk> i \<le> j ; j < length xs \<rbrakk> \<Longrightarrow> xs ! i \<le> xs ! j) \<Longrightarrow> sorted xs"
  4038 proof (induct xs)
  4039   case (Cons x xs)
  4040   have "sorted xs"
  4041   proof (rule Cons.hyps)
  4042     fix i j assume "i \<le> j" and "j < length xs"
  4043     with Cons.prems[of "Suc i" "Suc j"]
  4044     show "xs ! i \<le> xs ! j" by auto
  4045   qed
  4046   moreover
  4047   {
  4048     fix y assume "y \<in> set xs"
  4049     then obtain j where "j < length xs" and "xs ! j = y"
  4050       unfolding in_set_conv_nth by blast
  4051     with Cons.prems[of 0 "Suc j"]
  4052     have "x \<le> y"
  4053       by auto
  4054   }
  4055   ultimately
  4056   show ?case
  4057     unfolding sorted_Cons by auto
  4058 qed simp
  4059 
  4060 lemma sorted_equals_nth_mono:
  4061   "sorted xs = (\<forall>j < length xs. \<forall>i \<le> j. xs ! i \<le> xs ! j)"
  4062 by (auto intro: sorted_nth_monoI sorted_nth_mono)
  4063 
  4064 lemma set_insort: "set(insort_key f x xs) = insert x (set xs)"
  4065 by (induct xs) auto
  4066 
  4067 lemma set_sort[simp]: "set(sort_key f xs) = set xs"
  4068 by (induct xs) (simp_all add:set_insort)
  4069 
  4070 lemma distinct_insort: "distinct (insort_key f x xs) = (x \<notin> set xs \<and> distinct xs)"
  4071 by(induct xs)(auto simp:set_insort)
  4072 
  4073 lemma distinct_sort[simp]: "distinct (sort_key f xs) = distinct xs"
  4074   by (induct xs) (simp_all add: distinct_insort)
  4075 
  4076 lemma sorted_insort_key: "sorted (map f (insort_key f x xs)) = sorted (map f xs)"
  4077   by (induct xs) (auto simp:sorted_Cons set_insort)
  4078 
  4079 lemma sorted_insort: "sorted (insort x xs) = sorted xs"
  4080   using sorted_insort_key [where f="\<lambda>x. x"] by simp
  4081 
  4082 theorem sorted_sort_key [simp]: "sorted (map f (sort_key f xs))"
  4083   by (induct xs) (auto simp:sorted_insort_key)
  4084 
  4085 theorem sorted_sort [simp]: "sorted (sort xs)"
  4086   using sorted_sort_key [where f="\<lambda>x. x"] by simp
  4087 
  4088 lemma sorted_butlast:
  4089   assumes "xs \<noteq> []" and "sorted xs"
  4090   shows "sorted (butlast xs)"
  4091 proof -
  4092   from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  4093   with `sorted xs` show ?thesis by (simp add: sorted_append)
  4094 qed
  4095   
  4096 lemma insort_not_Nil [simp]:
  4097   "insort_key f a xs \<noteq> []"
  4098   by (induct xs) simp_all
  4099 
  4100 lemma insort_is_Cons: "\<forall>x\<in>set xs. f a \<le> f x \<Longrightarrow> insort_key f a xs = a # xs"
  4101 by (cases xs) auto
  4102 
  4103 lemma sorted_sort_id: "sorted xs \<Longrightarrow> sort xs = xs"
  4104   by (induct xs) (auto simp add: sorted_Cons insort_is_Cons)
  4105 
  4106 lemma sorted_map_remove1:
  4107   "sorted (map f xs) \<Longrightarrow> sorted (map f (remove1 x xs))"
  4108   by (induct xs) (auto simp add: sorted_Cons)
  4109 
  4110 lemma sorted_remove1: "sorted xs \<Longrightarrow> sorted (remove1 a xs)"
  4111   using sorted_map_remove1 [of "\<lambda>x. x"] by simp
  4112 
  4113 lemma insort_key_remove1:
  4114   assumes "a \<in> set xs" and "sorted (map f xs)" and "hd (filter (\<lambda>x. f a = f x) xs) = a"
  4115   shows "insort_key f a (remove1 a xs) = xs"
  4116 using assms proof (induct xs)
  4117   case (Cons x xs)
  4118   then show ?case
  4119   proof (cases "x = a")
  4120     case False
  4121     then have "f x \<noteq> f a" using Cons.prems by auto
  4122     then have "f x < f a" using Cons.prems by (auto simp: sorted_Cons)
  4123     with `f x \<noteq> f a` show ?thesis using Cons by (auto simp: sorted_Cons insort_is_Cons)
  4124   qed (auto simp: sorted_Cons insort_is_Cons)
  4125 qed simp
  4126 
  4127 lemma insort_remove1:
  4128   assumes "a \<in> set xs" and "sorted xs"
  4129   shows "insort a (remove1 a xs) = xs"
  4130 proof (rule insort_key_remove1)
  4131   from `a \<in> set xs` show "a \<in> set xs" .
  4132   from `sorted xs` show "sorted (map (\<lambda>x. x) xs)" by simp
  4133   from `a \<in> set xs` have "a \<in> set (filter (op = a) xs)" by auto
  4134   then have "set (filter (op = a) xs) \<noteq> {}" by auto
  4135   then have "filter (op = a) xs \<noteq> []" by (auto simp only: set_empty)
  4136   then have "length (filter (op = a) xs) > 0" by simp
  4137   then obtain n where n: "Suc n = length (filter (op = a) xs)"
  4138     by (cases "length (filter (op = a) xs)") simp_all
  4139   moreover have "replicate (Suc n) a = a # replicate n a"
  4140     by simp
  4141   ultimately show "hd (filter (op = a) xs) = a" by (simp add: replicate_length_filter)
  4142 qed
  4143 
  4144 lemma sorted_remdups[simp]:
  4145   "sorted l \<Longrightarrow> sorted (remdups l)"
  4146 by (induct l) (auto simp: sorted_Cons)
  4147 
  4148 lemma sorted_distinct_set_unique:
  4149 assumes "sorted xs" "distinct xs" "sorted ys" "distinct ys" "set xs = set ys"
  4150 shows "xs = ys"
  4151 proof -
  4152   from assms have 1: "length xs = length ys" by (auto dest!: distinct_card)
  4153   from assms show ?thesis
  4154   proof(induct rule:list_induct2[OF 1])
  4155     case 1 show ?case by simp
  4156   next
  4157     case 2 thus ?case by (simp add:sorted_Cons)
  4158        (metis Diff_insert_absorb antisym insertE insert_iff)
  4159   qed
  4160 qed
  4161 
  4162 lemma map_sorted_distinct_set_unique:
  4163   assumes "inj_on f (set xs \<union> set ys)"
  4164   assumes "sorted (map f xs)" "distinct (map f xs)"
  4165     "sorted (map f ys)" "distinct (map f ys)"
  4166   assumes "set xs = set ys"
  4167   shows "xs = ys"
  4168 proof -
  4169   from assms have "map f xs = map f ys"
  4170     by (simp add: sorted_distinct_set_unique)
  4171   moreover with `inj_on f (set xs \<union> set ys)` show "xs = ys"
  4172     by (blast intro: map_inj_on)
  4173 qed
  4174 
  4175 lemma finite_sorted_distinct_unique:
  4176 shows "finite A \<Longrightarrow> EX! xs. set xs = A & sorted xs & distinct xs"
  4177 apply(drule finite_distinct_list)
  4178 apply clarify
  4179 apply(rule_tac a="sort xs" in ex1I)
  4180 apply (auto simp: sorted_distinct_set_unique)
  4181 done
  4182 
  4183 lemma
  4184   assumes "sorted xs"
  4185   shows sorted_take: "sorted (take n xs)"
  4186   and sorted_drop: "sorted (drop n xs)"
  4187 proof -
  4188   from assms have "sorted (take n xs @ drop n xs)" by simp
  4189   then show "sorted (take n xs)" and "sorted (drop n xs)"
  4190     unfolding sorted_append by simp_all
  4191 qed
  4192 
  4193 lemma sorted_dropWhile: "sorted xs \<Longrightarrow> sorted (dropWhile P xs)"
  4194   by (auto dest: sorted_drop simp add: dropWhile_eq_drop)
  4195 
  4196 lemma sorted_takeWhile: "sorted xs \<Longrightarrow> sorted (takeWhile P xs)"
  4197   by (subst takeWhile_eq_take) (auto dest: sorted_take)
  4198 
  4199 lemma sorted_filter:
  4200   "sorted (map f xs) \<Longrightarrow> sorted (map f (filter P xs))"
  4201   by (induct xs) (simp_all add: sorted_Cons)
  4202 
  4203 lemma foldr_max_sorted:
  4204   assumes "sorted (rev xs)"
  4205   shows "foldr max xs y = (if xs = [] then y else max (xs ! 0) y)"
  4206 using assms proof (induct xs)
  4207   case (Cons x xs)
  4208   moreover hence "sorted (rev xs)" using sorted_append by auto
  4209   ultimately show ?case
  4210     by (cases xs, auto simp add: sorted_append max_def)
  4211 qed simp
  4212 
  4213 lemma filter_equals_takeWhile_sorted_rev:
  4214   assumes sorted: "sorted (rev (map f xs))"
  4215   shows "[x \<leftarrow> xs. t < f x] = takeWhile (\<lambda> x. t < f x) xs"
  4216     (is "filter ?P xs = ?tW")
  4217 proof (rule takeWhile_eq_filter[symmetric])
  4218   let "?dW" = "dropWhile ?P xs"
  4219   fix x assume "x \<in> set ?dW"
  4220   then obtain i where i: "i < length ?dW" and nth_i: "x = ?dW ! i"
  4221     unfolding in_set_conv_nth by auto
  4222   hence "length ?tW + i < length (?tW @ ?dW)"
  4223     unfolding length_append by simp
  4224   hence i': "length (map f ?tW) + i < length (map f xs)" by simp
  4225   have "(map f ?tW @ map f ?dW) ! (length (map f ?tW) + i) \<le>
  4226         (map f ?tW @ map f ?dW) ! (length (map f ?tW) + 0)"
  4227     using sorted_rev_nth_mono[OF sorted _ i', of "length ?tW"]
  4228     unfolding map_append[symmetric] by simp
  4229   hence "f x \<le> f (?dW ! 0)"
  4230     unfolding nth_append_length_plus nth_i
  4231     using i preorder_class.le_less_trans[OF le0 i] by simp
  4232   also have "... \<le> t"
  4233     using hd_dropWhile[of "?P" xs] le0[THEN preorder_class.le_less_trans, OF i]
  4234     using hd_conv_nth[of "?dW"] by simp
  4235   finally show "\<not> t < f x" by simp
  4236 qed
  4237 
  4238 lemma insort_insert_key_triv:
  4239   "f x \<in> f ` set xs \<Longrightarrow> insort_insert_key f x xs = xs"
  4240   by (simp add: insort_insert_key_def)
  4241 
  4242 lemma insort_insert_triv:
  4243   "x \<in> set xs \<Longrightarrow> insort_insert x xs = xs"
  4244   using insort_insert_key_triv [of "\<lambda>x. x"] by simp
  4245 
  4246 lemma insort_insert_insort_key:
  4247   "f x \<notin> f ` set xs \<Longrightarrow> insort_insert_key f x xs = insort_key f x xs"
  4248   by (simp add: insort_insert_key_def)
  4249 
  4250 lemma insort_insert_insort:
  4251   "x \<notin> set xs \<Longrightarrow> insort_insert x xs = insort x xs"
  4252   using insort_insert_insort_key [of "\<lambda>x. x"] by simp
  4253 
  4254 lemma set_insort_insert:
  4255   "set (insort_insert x xs) = insert x (set xs)"
  4256   by (auto simp add: insort_insert_key_def set_insort)
  4257 
  4258 lemma distinct_insort_insert:
  4259   assumes "distinct xs"
  4260   shows "distinct (insort_insert_key f x xs)"
  4261   using assms by (induct xs) (auto simp add: insort_insert_key_def set_insort)
  4262 
  4263 lemma sorted_insort_insert_key:
  4264   assumes "sorted (map f xs)"
  4265   shows "sorted (map f (insort_insert_key f x xs))"
  4266   using assms by (simp add: insort_insert_key_def sorted_insort_key)
  4267 
  4268 lemma sorted_insort_insert:
  4269   assumes "sorted xs"
  4270   shows "sorted (insort_insert x xs)"
  4271   using assms sorted_insort_insert_key [of "\<lambda>x. x"] by simp
  4272 
  4273 lemma filter_insort_triv:
  4274   "\<not> P x \<Longrightarrow> filter P (insort_key f x xs) = filter P xs"
  4275   by (induct xs) simp_all
  4276 
  4277 lemma filter_insort:
  4278   "sorted (map f xs) \<Longrightarrow> P x \<Longrightarrow> filter P (insort_key f x xs) = insort_key f x (filter P xs)"
  4279   using assms by (induct xs)
  4280     (auto simp add: sorted_Cons, subst insort_is_Cons, auto)
  4281 
  4282 lemma filter_sort:
  4283   "filter P (sort_key f xs) = sort_key f (filter P xs)"
  4284   by (induct xs) (simp_all add: filter_insort_triv filter_insort)
  4285 
  4286 lemma sorted_map_same:
  4287   "sorted (map f [x\<leftarrow>xs. f x = g xs])"
  4288 proof (induct xs arbitrary: g)
  4289   case Nil then show ?case by simp
  4290 next
  4291   case (Cons x xs)
  4292   then have "sorted (map f [y\<leftarrow>xs . f y = (\<lambda>xs. f x) xs])" .
  4293   moreover from Cons have "sorted (map f [y\<leftarrow>xs . f y = (g \<circ> Cons x) xs])" .
  4294   ultimately show ?case by (simp_all add: sorted_Cons)
  4295 qed
  4296 
  4297 lemma sorted_same:
  4298   "sorted [x\<leftarrow>xs. x = g xs]"
  4299   using sorted_map_same [of "\<lambda>x. x"] by simp
  4300 
  4301 lemma remove1_insort [simp]:
  4302   "remove1 x (insort x xs) = xs"
  4303   by (induct xs) simp_all
  4304 
  4305 end
  4306 
  4307 lemma sorted_upt[simp]: "sorted[i..<j]"
  4308 by (induct j) (simp_all add:sorted_append)
  4309 
  4310 lemma sorted_upto[simp]: "sorted[i..j]"
  4311 apply(induct i j rule:upto.induct)
  4312 apply(subst upto.simps)
  4313 apply(simp add:sorted_Cons)
  4314 done
  4315 
  4316 
  4317 subsubsection {* @{const transpose} on sorted lists *}
  4318 
  4319 lemma sorted_transpose[simp]:
  4320   shows "sorted (rev (map length (transpose xs)))"
  4321   by (auto simp: sorted_equals_nth_mono rev_nth nth_transpose
  4322     length_filter_conv_card intro: card_mono)
  4323 
  4324 lemma transpose_max_length:
  4325   "foldr (\<lambda>xs. max (length xs)) (transpose xs) 0 = length [x \<leftarrow> xs. x \<noteq> []]"
  4326   (is "?L = ?R")
  4327 proof (cases "transpose xs = []")
  4328   case False
  4329   have "?L = foldr max (map length (transpose xs)) 0"
  4330     by (simp add: foldr_map comp_def)
  4331   also have "... = length (transpose xs ! 0)"
  4332     using False sorted_transpose by (simp add: foldr_max_sorted)
  4333   finally show ?thesis
  4334     using False by (simp add: nth_transpose)
  4335 next
  4336   case True
  4337   hence "[x \<leftarrow> xs. x \<noteq> []] = []"
  4338     by (auto intro!: filter_False simp: transpose_empty)
  4339   thus ?thesis by (simp add: transpose_empty True)
  4340 qed
  4341 
  4342 lemma length_transpose_sorted:
  4343   fixes xs :: "'a list list"
  4344   assumes sorted: "sorted (rev (map length xs))"
  4345   shows "length (transpose xs) = (if xs = [] then 0 else length (xs ! 0))"
  4346 proof (cases "xs = []")
  4347   case False
  4348   thus ?thesis
  4349     using foldr_max_sorted[OF sorted] False
  4350     unfolding length_transpose foldr_map comp_def
  4351     by simp
  4352 qed simp
  4353 
  4354 lemma nth_nth_transpose_sorted[simp]:
  4355   fixes xs :: "'a list list"
  4356   assumes sorted: "sorted (rev (map length xs))"
  4357   and i: "i < length (transpose xs)"
  4358   and j: "j < length [ys \<leftarrow> xs. i < length ys]"
  4359   shows "transpose xs ! i ! j = xs ! j  ! i"
  4360   using j filter_equals_takeWhile_sorted_rev[OF sorted, of i]
  4361     nth_transpose[OF i] nth_map[OF j]
  4362   by (simp add: takeWhile_nth)
  4363 
  4364 lemma transpose_column_length:
  4365   fixes xs :: "'a list list"
  4366   assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  4367   shows "length (filter (\<lambda>ys. i < length ys) (transpose xs)) = length (xs ! i)"
  4368 proof -
  4369   have "xs \<noteq> []" using `i < length xs` by auto
  4370   note filter_equals_takeWhile_sorted_rev[OF sorted, simp]
  4371   { fix j assume "j \<le> i"
  4372     note sorted_rev_nth_mono[OF sorted, of j i, simplified, OF this `i < length xs`]
  4373   } note sortedE = this[consumes 1]
  4374 
  4375   have "{j. j < length (transpose xs) \<and> i < length (transpose xs ! j)}
  4376     = {..< length (xs ! i)}"
  4377   proof safe
  4378     fix j
  4379     assume "j < length (transpose xs)" and "i < length (transpose xs ! j)"
  4380     with this(2) nth_transpose[OF this(1)]
  4381     have "i < length (takeWhile (\<lambda>ys. j < length ys) xs)" by simp
  4382     from nth_mem[OF this] takeWhile_nth[OF this]
  4383     show "j < length (xs ! i)" by (auto dest: set_takeWhileD)
  4384   next
  4385     fix j assume "j < length (xs ! i)"
  4386     thus "j < length (transpose xs)"
  4387       using foldr_max_sorted[OF sorted] `xs \<noteq> []` sortedE[OF le0]
  4388       by (auto simp: length_transpose comp_def foldr_map)
  4389 
  4390     have "Suc i \<le> length (takeWhile (\<lambda>ys. j < length ys) xs)"
  4391       using `i < length xs` `j < length (xs ! i)` less_Suc_eq_le
  4392       by (auto intro!: length_takeWhile_less_P_nth dest!: sortedE)
  4393     with nth_transpose[OF `j < length (transpose xs)`]
  4394     show "i < length (transpose xs ! j)" by simp
  4395   qed
  4396   thus ?thesis by (simp add: length_filter_conv_card)
  4397 qed
  4398 
  4399 lemma transpose_column:
  4400   fixes xs :: "'a list list"
  4401   assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  4402   shows "map (\<lambda>ys. ys ! i) (filter (\<lambda>ys. i < length ys) (transpose xs))
  4403     = xs ! i" (is "?R = _")
  4404 proof (rule nth_equalityI, safe)
  4405   show length: "length ?R = length (xs ! i)"
  4406     using transpose_column_length[OF assms] by simp
  4407 
  4408   fix j assume j: "j < length ?R"
  4409   note * = less_le_trans[OF this, unfolded length_map, OF length_filter_le]
  4410   from j have j_less: "j < length (xs ! i)" using length by simp
  4411   have i_less_tW: "Suc i \<le> length (takeWhile (\<lambda>ys. Suc j \<le> length ys) xs)"
  4412   proof (rule length_takeWhile_less_P_nth)
  4413     show "Suc i \<le> length xs" using `i < length xs` by simp
  4414     fix k assume "k < Suc i"
  4415     hence "k \<le> i" by auto
  4416     with sorted_rev_nth_mono[OF sorted this] `i < length xs`
  4417     have "length (xs ! i) \<le> length (xs ! k)" by simp
  4418     thus "Suc j \<le> length (xs ! k)" using j_less by simp
  4419   qed
  4420   have i_less_filter: "i < length [ys\<leftarrow>xs . j < length ys]"
  4421     unfolding filter_equals_takeWhile_sorted_rev[OF sorted, of j]
  4422     using i_less_tW by (simp_all add: Suc_le_eq)
  4423   from j show "?R ! j = xs ! i ! j"
  4424     unfolding filter_equals_takeWhile_sorted_rev[OF sorted_transpose, of i]
  4425     by (simp add: takeWhile_nth nth_nth_transpose_sorted[OF sorted * i_less_filter])
  4426 qed
  4427 
  4428 lemma transpose_transpose:
  4429   fixes xs :: "'a list list"
  4430   assumes sorted: "sorted (rev (map length xs))"
  4431   shows "transpose (transpose xs) = takeWhile (\<lambda>x. x \<noteq> []) xs" (is "?L = ?R")
  4432 proof -
  4433   have len: "length ?L = length ?R"
  4434     unfolding length_transpose transpose_max_length
  4435     using filter_equals_takeWhile_sorted_rev[OF sorted, of 0]
  4436     by simp
  4437 
  4438   { fix i assume "i < length ?R"
  4439     with less_le_trans[OF _ length_takeWhile_le[of _ xs]]
  4440     have "i < length xs" by simp
  4441   } note * = this
  4442   show ?thesis
  4443     by (rule nth_equalityI)
  4444        (simp_all add: len nth_transpose transpose_column[OF sorted] * takeWhile_nth)
  4445 qed
  4446 
  4447 theorem transpose_rectangle:
  4448   assumes "xs = [] \<Longrightarrow> n = 0"
  4449   assumes rect: "\<And> i. i < length xs \<Longrightarrow> length (xs ! i) = n"
  4450   shows "transpose xs = map (\<lambda> i. map (\<lambda> j. xs ! j ! i) [0..<length xs]) [0..<n]"
  4451     (is "?trans = ?map")
  4452 proof (rule nth_equalityI)
  4453   have "sorted (rev (map length xs))"
  4454     by (auto simp: rev_nth rect intro!: sorted_nth_monoI)
  4455   from foldr_max_sorted[OF this] assms
  4456   show len: "length ?trans = length ?map"
  4457     by (simp_all add: length_transpose foldr_map comp_def)
  4458   moreover
  4459   { fix i assume "i < n" hence "[ys\<leftarrow>xs . i < length ys] = xs"
  4460       using rect by (auto simp: in_set_conv_nth intro!: filter_True) }
  4461   ultimately show "\<forall>i < length ?trans. ?trans ! i = ?map ! i"
  4462     by (auto simp: nth_transpose intro: nth_equalityI)
  4463 qed
  4464 
  4465 
  4466 subsubsection {* @{text sorted_list_of_set} *}
  4467 
  4468 text{* This function maps (finite) linearly ordered sets to sorted
  4469 lists. Warning: in most cases it is not a good idea to convert from
  4470 sets to lists but one should convert in the other direction (via
  4471 @{const set}). *}
  4472 
  4473 context linorder
  4474 begin
  4475 
  4476 definition sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where
  4477   "sorted_list_of_set = Finite_Set.fold insort []"
  4478 
  4479 lemma sorted_list_of_set_empty [simp]:
  4480   "sorted_list_of_set {} = []"
  4481   by (simp add: sorted_list_of_set_def)
  4482 
  4483 lemma sorted_list_of_set_insert [simp]:
  4484   assumes "finite A"
  4485   shows "sorted_list_of_set (insert x A) = insort x (sorted_list_of_set (A - {x}))"
  4486 proof -
  4487   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  4488   with assms show ?thesis by (simp add: sorted_list_of_set_def fold_insert_remove)
  4489 qed
  4490 
  4491 lemma sorted_list_of_set [simp]:
  4492   "finite A \<Longrightarrow> set (sorted_list_of_set A) = A \<and> sorted (sorted_list_of_set A) 
  4493     \<and> distinct (sorted_list_of_set A)"
  4494   by (induct A rule: finite_induct) (simp_all add: set_insort sorted_insort distinct_insort)
  4495 
  4496 lemma sorted_list_of_set_sort_remdups:
  4497   "sorted_list_of_set (set xs) = sort (remdups xs)"
  4498 proof -
  4499   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  4500   show ?thesis by (simp add: sorted_list_of_set_def sort_conv_fold fold_set_fold_remdups)
  4501 qed
  4502 
  4503 lemma sorted_list_of_set_remove:
  4504   assumes "finite A"
  4505   shows "sorted_list_of_set (A - {x}) = remove1 x (sorted_list_of_set A)"
  4506 proof (cases "x \<in> A")
  4507   case False with assms have "x \<notin> set (sorted_list_of_set A)" by simp
  4508   with False show ?thesis by (simp add: remove1_idem)
  4509 next
  4510   case True then obtain B where A: "A = insert x B" by (rule Set.set_insert)
  4511   with assms show ?thesis by simp
  4512 qed
  4513 
  4514 end
  4515 
  4516 lemma sorted_list_of_set_range [simp]:
  4517   "sorted_list_of_set {m..<n} = [m..<n]"
  4518   by (rule sorted_distinct_set_unique) simp_all
  4519 
  4520 
  4521 subsubsection {* @{text lists}: the list-forming operator over sets *}
  4522 
  4523 inductive_set
  4524   lists :: "'a set => 'a list set"
  4525   for A :: "'a set"
  4526 where
  4527     Nil [intro!, simp]: "[]: lists A"
  4528   | Cons [intro!, simp, no_atp]: "[| a: A; l: lists A|] ==> a#l : lists A"
  4529 
  4530 inductive_cases listsE [elim!,no_atp]: "x#l : lists A"
  4531 inductive_cases listspE [elim!,no_atp]: "listsp A (x # l)"
  4532 
  4533 inductive_simps listsp_simps[code]:
  4534   "listsp A []"
  4535   "listsp A (x # xs)"
  4536 
  4537 lemma listsp_mono [mono]: "A \<le> B ==> listsp A \<le> listsp B"
  4538 by (rule predicate1I, erule listsp.induct, (blast dest: predicate1D)+)
  4539 
  4540 lemmas lists_mono = listsp_mono [to_set]
  4541 
  4542 lemma listsp_infI:
  4543   assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
  4544 by induct blast+
  4545 
  4546 lemmas lists_IntI = listsp_infI [to_set]
  4547 
  4548 lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
  4549 proof (rule mono_inf [where f=listsp, THEN order_antisym])
  4550   show "mono listsp" by (simp add: mono_def listsp_mono)
  4551   show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro!: listsp_infI predicate1I)
  4552 qed
  4553 
  4554 lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_def inf_bool_def]
  4555 
  4556 lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set]
  4557 
  4558 lemma Cons_in_lists_iff[simp]: "x#xs : lists A \<longleftrightarrow> x:A \<and> xs : lists A"
  4559 by auto
  4560 
  4561 lemma append_in_listsp_conv [iff]:
  4562      "(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
  4563 by (induct xs) auto
  4564 
  4565 lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]
  4566 
  4567 lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
  4568 -- {* eliminate @{text listsp} in favour of @{text set} *}
  4569 by (induct xs) auto
  4570 
  4571 lemmas in_lists_conv_set [code_unfold] = in_listsp_conv_set [to_set]
  4572 
  4573 lemma in_listspD [dest!,no_atp]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
  4574 by (rule in_listsp_conv_set [THEN iffD1])
  4575 
  4576 lemmas in_listsD [dest!,no_atp] = in_listspD [to_set]
  4577 
  4578 lemma in_listspI [intro!,no_atp]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
  4579 by (rule in_listsp_conv_set [THEN iffD2])
  4580 
  4581 lemmas in_listsI [intro!,no_atp] = in_listspI [to_set]
  4582 
  4583 lemma lists_eq_set: "lists A = {xs. set xs <= A}"
  4584 by auto
  4585 
  4586 lemma lists_empty [simp]: "lists {} = {[]}"
  4587 by auto
  4588 
  4589 lemma lists_UNIV [simp]: "lists UNIV = UNIV"
  4590 by auto
  4591 
  4592 
  4593 subsubsection {* Inductive definition for membership *}
  4594 
  4595 inductive ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
  4596 where
  4597     elem:  "ListMem x (x # xs)"
  4598   | insert:  "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"
  4599 
  4600 lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
  4601 apply (rule iffI)
  4602  apply (induct set: ListMem)
  4603   apply auto
  4604 apply (induct xs)
  4605  apply (auto intro: ListMem.intros)
  4606 done
  4607 
  4608 
  4609 subsubsection {* Lists as Cartesian products *}
  4610 
  4611 text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
  4612 @{term A} and tail drawn from @{term Xs}.*}
  4613 
  4614 definition
  4615   set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set" where
  4616   "set_Cons A XS = {z. \<exists>x xs. z = x # xs \<and> x \<in> A \<and> xs \<in> XS}"
  4617 
  4618 lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
  4619 by (auto simp add: set_Cons_def)
  4620 
  4621 text{*Yields the set of lists, all of the same length as the argument and
  4622 with elements drawn from the corresponding element of the argument.*}
  4623 
  4624 primrec
  4625   listset :: "'a set list \<Rightarrow> 'a list set" where
  4626      "listset [] = {[]}"
  4627   |  "listset (A # As) = set_Cons A (listset As)"
  4628 
  4629 
  4630 subsection {* Relations on Lists *}
  4631 
  4632 subsubsection {* Length Lexicographic Ordering *}
  4633 
  4634 text{*These orderings preserve well-foundedness: shorter lists 
  4635   precede longer lists. These ordering are not used in dictionaries.*}
  4636         
  4637 primrec -- {*The lexicographic ordering for lists of the specified length*}
  4638   lexn :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> ('a list \<times> 'a list) set" where
  4639     "lexn r 0 = {}"
  4640   | "lexn r (Suc n) = (map_pair (%(x, xs). x#xs) (%(x, xs). x#xs) ` (r <*lex*> lexn r n)) Int
  4641       {(xs, ys). length xs = Suc n \<and> length ys = Suc n}"
  4642 
  4643 definition
  4644   lex :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4645   "lex r = (\<Union>n. lexn r n)" -- {*Holds only between lists of the same length*}
  4646 
  4647 definition
  4648   lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" where
  4649   "lenlex r = inv_image (less_than <*lex*> lex r) (\<lambda>xs. (length xs, xs))"
  4650         -- {*Compares lists by their length and then lexicographically*}
  4651 
  4652 lemma wf_lexn: "wf r ==> wf (lexn r n)"
  4653 apply (induct n, simp, simp)
  4654 apply(rule wf_subset)
  4655  prefer 2 apply (rule Int_lower1)
  4656 apply(rule wf_map_pair_image)
  4657  prefer 2 apply (rule inj_onI, auto)
  4658 done
  4659 
  4660 lemma lexn_length:
  4661   "(xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
  4662 by (induct n arbitrary: xs ys) auto
  4663 
  4664 lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
  4665 apply (unfold lex_def)
  4666 apply (rule wf_UN)
  4667 apply (blast intro: wf_lexn, clarify)
  4668 apply (rename_tac m n)
  4669 apply (subgoal_tac "m \<noteq> n")
  4670  prefer 2 apply blast
  4671 apply (blast dest: lexn_length not_sym)
  4672 done
  4673 
  4674 lemma lexn_conv:
  4675   "lexn r n =
  4676     {(xs,ys). length xs = n \<and> length ys = n \<and>
  4677     (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
  4678 apply (induct n, simp)
  4679 apply (simp add: image_Collect lex_prod_def, safe, blast)
  4680  apply (rule_tac x = "ab # xys" in exI, simp)
  4681 apply (case_tac xys, simp_all, blast)
  4682 done
  4683 
  4684 lemma lex_conv:
  4685   "lex r =
  4686     {(xs,ys). length xs = length ys \<and>
  4687     (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
  4688 by (force simp add: lex_def lexn_conv)
  4689 
  4690 lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
  4691 by (unfold lenlex_def) blast
  4692 
  4693 lemma lenlex_conv:
  4694     "lenlex r = {(xs,ys). length xs < length ys |
  4695                  length xs = length ys \<and> (xs, ys) : lex r}"
  4696 by (simp add: lenlex_def Id_on_def lex_prod_def inv_image_def)
  4697 
  4698 lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
  4699 by (simp add: lex_conv)
  4700 
  4701 lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
  4702 by (simp add:lex_conv)
  4703 
  4704 lemma Cons_in_lex [simp]:
  4705     "((x # xs, y # ys) : lex r) =
  4706       ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
  4707 apply (simp add: lex_conv)
  4708 apply (rule iffI)
  4709  prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
  4710 apply (case_tac xys, simp, simp)
  4711 apply blast
  4712 done
  4713 
  4714 
  4715 subsubsection {* Lexicographic Ordering *}
  4716 
  4717 text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
  4718     This ordering does \emph{not} preserve well-foundedness.
  4719      Author: N. Voelker, March 2005. *} 
  4720 
  4721 definition
  4722   lexord :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4723   "lexord r = {(x,y ). \<exists> a v. y = x @ a # v \<or>
  4724             (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
  4725 
  4726 lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
  4727 by (unfold lexord_def, induct_tac y, auto) 
  4728 
  4729 lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
  4730 by (unfold lexord_def, induct_tac x, auto)
  4731 
  4732 lemma lexord_cons_cons[simp]:
  4733      "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
  4734   apply (unfold lexord_def, safe, simp_all)
  4735   apply (case_tac u, simp, simp)
  4736   apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
  4737   apply (erule_tac x="b # u" in allE)
  4738   by force
  4739 
  4740 lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
  4741 
  4742 lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
  4743 by (induct_tac x, auto)  
  4744 
  4745 lemma lexord_append_left_rightI:
  4746      "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
  4747 by (induct_tac u, auto)
  4748 
  4749 lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
  4750 by (induct x, auto)
  4751 
  4752 lemma lexord_append_leftD:
  4753      "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
  4754 by (erule rev_mp, induct_tac x, auto)
  4755 
  4756 lemma lexord_take_index_conv: 
  4757    "((x,y) : lexord r) = 
  4758     ((length x < length y \<and> take (length x) y = x) \<or> 
  4759      (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
  4760   apply (unfold lexord_def Let_def, clarsimp) 
  4761   apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
  4762   apply auto 
  4763   apply (rule_tac x="hd (drop (length x) y)" in exI)
  4764   apply (rule_tac x="tl (drop (length x) y)" in exI)
  4765   apply (erule subst, simp add: min_def) 
  4766   apply (rule_tac x ="length u" in exI, simp) 
  4767   apply (rule_tac x ="take i x" in exI) 
  4768   apply (rule_tac x ="x ! i" in exI) 
  4769   apply (rule_tac x ="y ! i" in exI, safe) 
  4770   apply (rule_tac x="drop (Suc i) x" in exI)
  4771   apply (drule sym, simp add: drop_Suc_conv_tl) 
  4772   apply (rule_tac x="drop (Suc i) y" in exI)
  4773   by (simp add: drop_Suc_conv_tl) 
  4774 
  4775 -- {* lexord is extension of partial ordering List.lex *} 
  4776 lemma lexord_lex: "(x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
  4777   apply (rule_tac x = y in spec) 
  4778   apply (induct_tac x, clarsimp) 
  4779   by (clarify, case_tac x, simp, force)
  4780 
  4781 lemma lexord_irreflexive: "ALL x. (x,x) \<notin> r \<Longrightarrow> (xs,xs) \<notin> lexord r"
  4782 by (induct xs) auto
  4783 
  4784 text{* By Ren\'e Thiemann: *}
  4785 lemma lexord_partial_trans: 
  4786   "(\<And>x y z. x \<in> set xs \<Longrightarrow> (x,y) \<in> r \<Longrightarrow> (y,z) \<in> r \<Longrightarrow> (x,z) \<in> r)
  4787    \<Longrightarrow>  (xs,ys) \<in> lexord r  \<Longrightarrow>  (ys,zs) \<in> lexord r \<Longrightarrow>  (xs,zs) \<in> lexord r"
  4788 proof (induct xs arbitrary: ys zs)
  4789   case Nil
  4790   from Nil(3) show ?case unfolding lexord_def by (cases zs, auto)
  4791 next
  4792   case (Cons x xs yys zzs)
  4793   from Cons(3) obtain y ys where yys: "yys = y # ys" unfolding lexord_def
  4794     by (cases yys, auto)
  4795   note Cons = Cons[unfolded yys]
  4796   from Cons(3) have one: "(x,y) \<in> r \<or> x = y \<and> (xs,ys) \<in> lexord r" by auto
  4797   from Cons(4) obtain z zs where zzs: "zzs = z # zs" unfolding lexord_def
  4798     by (cases zzs, auto)
  4799   note Cons = Cons[unfolded zzs]
  4800   from Cons(4) have two: "(y,z) \<in> r \<or> y = z \<and> (ys,zs) \<in> lexord r" by auto
  4801   {
  4802     assume "(xs,ys) \<in> lexord r" and "(ys,zs) \<in> lexord r"
  4803     from Cons(1)[OF _ this] Cons(2)
  4804     have "(xs,zs) \<in> lexord r" by auto
  4805   } note ind1 = this
  4806   {
  4807     assume "(x,y) \<in> r" and "(y,z) \<in> r"
  4808     from Cons(2)[OF _ this] have "(x,z) \<in> r" by auto
  4809   } note ind2 = this
  4810   from one two ind1 ind2
  4811   have "(x,z) \<in> r \<or> x = z \<and> (xs,zs) \<in> lexord r" by blast
  4812   thus ?case unfolding zzs by auto
  4813 qed
  4814 
  4815 lemma lexord_trans: 
  4816     "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
  4817 by(auto simp: trans_def intro:lexord_partial_trans)
  4818 
  4819 lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
  4820 by (rule transI, drule lexord_trans, blast) 
  4821 
  4822 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"
  4823   apply (rule_tac x = y in spec) 
  4824   apply (induct_tac x, rule allI) 
  4825   apply (case_tac x, simp, simp) 
  4826   apply (rule allI, case_tac x, simp, simp) 
  4827   by blast
  4828 
  4829 
  4830 subsubsection {* Lexicographic combination of measure functions *}
  4831 
  4832 text {* These are useful for termination proofs *}
  4833 
  4834 definition
  4835   "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"
  4836 
  4837 lemma wf_measures[simp]: "wf (measures fs)"
  4838 unfolding measures_def
  4839 by blast
  4840 
  4841 lemma in_measures[simp]: 
  4842   "(x, y) \<in> measures [] = False"
  4843   "(x, y) \<in> measures (f # fs)
  4844          = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"  
  4845 unfolding measures_def
  4846 by auto
  4847 
  4848 lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
  4849 by simp
  4850 
  4851 lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
  4852 by auto
  4853 
  4854 
  4855 subsubsection {* Lifting Relations to Lists: one element *}
  4856 
  4857 definition listrel1 :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4858 "listrel1 r = {(xs,ys).
  4859    \<exists>us z z' vs. xs = us @ z # vs \<and> (z,z') \<in> r \<and> ys = us @ z' # vs}"
  4860 
  4861 lemma listrel1I:
  4862   "\<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow>
  4863   (xs, ys) \<in> listrel1 r"
  4864 unfolding listrel1_def by auto
  4865 
  4866 lemma listrel1E:
  4867   "\<lbrakk> (xs, ys) \<in> listrel1 r;
  4868      !!x y us vs. \<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow> P
  4869    \<rbrakk> \<Longrightarrow> P"
  4870 unfolding listrel1_def by auto
  4871 
  4872 lemma not_Nil_listrel1 [iff]: "([], xs) \<notin> listrel1 r"
  4873 unfolding listrel1_def by blast
  4874 
  4875 lemma not_listrel1_Nil [iff]: "(xs, []) \<notin> listrel1 r"
  4876 unfolding listrel1_def by blast
  4877 
  4878 lemma Cons_listrel1_Cons [iff]:
  4879   "(x # xs, y # ys) \<in> listrel1 r \<longleftrightarrow>
  4880    (x,y) \<in> r \<and> xs = ys \<or> x = y \<and> (xs, ys) \<in> listrel1 r"
  4881 by (simp add: listrel1_def Cons_eq_append_conv) (blast)
  4882 
  4883 lemma listrel1I1: "(x,y) \<in> r \<Longrightarrow> (x # xs, y # xs) \<in> listrel1 r"
  4884 by (metis Cons_listrel1_Cons)
  4885 
  4886 lemma listrel1I2: "(xs, ys) \<in> listrel1 r \<Longrightarrow> (x # xs, x # ys) \<in> listrel1 r"
  4887 by (metis Cons_listrel1_Cons)
  4888 
  4889 lemma append_listrel1I:
  4890   "(xs, ys) \<in> listrel1 r \<and> us = vs \<or> xs = ys \<and> (us, vs) \<in> listrel1 r
  4891     \<Longrightarrow> (xs @ us, ys @ vs) \<in> listrel1 r"
  4892 unfolding listrel1_def
  4893 by auto (blast intro: append_eq_appendI)+
  4894 
  4895 lemma Cons_listrel1E1[elim!]:
  4896   assumes "(x # xs, ys) \<in> listrel1 r"
  4897     and "\<And>y. ys = y # xs \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
  4898     and "\<And>zs. ys = x # zs \<Longrightarrow> (xs, zs) \<in> listrel1 r \<Longrightarrow> R"
  4899   shows R
  4900 using assms by (cases ys) blast+
  4901 
  4902 lemma Cons_listrel1E2[elim!]:
  4903   assumes "(xs, y # ys) \<in> listrel1 r"
  4904     and "\<And>x. xs = x # ys \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
  4905     and "\<And>zs. xs = y # zs \<Longrightarrow> (zs, ys) \<in> listrel1 r \<Longrightarrow> R"
  4906   shows R
  4907 using assms by (cases xs) blast+
  4908 
  4909 lemma snoc_listrel1_snoc_iff:
  4910   "(xs @ [x], ys @ [y]) \<in> listrel1 r
  4911     \<longleftrightarrow> (xs, ys) \<in> listrel1 r \<and> x = y \<or> xs = ys \<and> (x,y) \<in> r" (is "?L \<longleftrightarrow> ?R")
  4912 proof
  4913   assume ?L thus ?R
  4914     by (fastforce simp: listrel1_def snoc_eq_iff_butlast butlast_append)
  4915 next
  4916   assume ?R then show ?L unfolding listrel1_def by force
  4917 qed
  4918 
  4919 lemma listrel1_eq_len: "(xs,ys) \<in> listrel1 r \<Longrightarrow> length xs = length ys"
  4920 unfolding listrel1_def by auto
  4921 
  4922 lemma listrel1_mono:
  4923   "r \<subseteq> s \<Longrightarrow> listrel1 r \<subseteq> listrel1 s"
  4924 unfolding listrel1_def by blast
  4925 
  4926 
  4927 lemma listrel1_converse: "listrel1 (r^-1) = (listrel1 r)^-1"
  4928 unfolding listrel1_def by blast
  4929 
  4930 lemma in_listrel1_converse:
  4931   "(x,y) : listrel1 (r^-1) \<longleftrightarrow> (x,y) : (listrel1 r)^-1"
  4932 unfolding listrel1_def by blast
  4933 
  4934 lemma listrel1_iff_update:
  4935   "(xs,ys) \<in> (listrel1 r)
  4936    \<longleftrightarrow> (\<exists>y n. (xs ! n, y) \<in> r \<and> n < length xs \<and> ys = xs[n:=y])" (is "?L \<longleftrightarrow> ?R")
  4937 proof
  4938   assume "?L"
  4939   then obtain x y u v where "xs = u @ x # v"  "ys = u @ y # v"  "(x,y) \<in> r"
  4940     unfolding listrel1_def by auto
  4941   then have "ys = xs[length u := y]" and "length u < length xs"
  4942     and "(xs ! length u, y) \<in> r" by auto
  4943   then show "?R" by auto
  4944 next
  4945   assume "?R"
  4946   then obtain x y n where "(xs!n, y) \<in> r" "n < size xs" "ys = xs[n:=y]" "x = xs!n"
  4947     by auto
  4948   then obtain u v where "xs = u @ x # v" and "ys = u @ y # v" and "(x, y) \<in> r"
  4949     by (auto intro: upd_conv_take_nth_drop id_take_nth_drop)
  4950   then show "?L" by (auto simp: listrel1_def)
  4951 qed
  4952 
  4953 
  4954 text{* Accessible part and wellfoundedness: *}
  4955 
  4956 lemma Cons_acc_listrel1I [intro!]:
  4957   "x \<in> acc r \<Longrightarrow> xs \<in> acc (listrel1 r) \<Longrightarrow> (x # xs) \<in> acc (listrel1 r)"
  4958 apply (induct arbitrary: xs set: acc)
  4959 apply (erule thin_rl)
  4960 apply (erule acc_induct)
  4961 apply (rule accI)
  4962 apply (blast)
  4963 done
  4964 
  4965 lemma lists_accD: "xs \<in> lists (acc r) \<Longrightarrow> xs \<in> acc (listrel1 r)"
  4966 apply (induct set: lists)
  4967  apply (rule accI)
  4968  apply simp
  4969 apply (rule accI)
  4970 apply (fast dest: acc_downward)
  4971 done
  4972 
  4973 lemma lists_accI: "xs \<in> acc (listrel1 r) \<Longrightarrow> xs \<in> lists (acc r)"
  4974 apply (induct set: acc)
  4975 apply clarify
  4976 apply (rule accI)
  4977 apply (fastforce dest!: in_set_conv_decomp[THEN iffD1] simp: listrel1_def)
  4978 done
  4979 
  4980 lemma wf_listrel1_iff[simp]: "wf(listrel1 r) = wf r"
  4981 by(metis wf_acc_iff in_lists_conv_set lists_accI lists_accD Cons_in_lists_iff)
  4982 
  4983 
  4984 subsubsection {* Lifting Relations to Lists: all elements *}
  4985 
  4986 inductive_set
  4987   listrel :: "('a \<times> 'b) set \<Rightarrow> ('a list \<times> 'b list) set"
  4988   for r :: "('a \<times> 'b) set"
  4989 where
  4990     Nil:  "([],[]) \<in> listrel r"
  4991   | Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
  4992 
  4993 inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
  4994 inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
  4995 inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
  4996 inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
  4997 
  4998 
  4999 lemma listrel_eq_len:  "(xs, ys) \<in> listrel r \<Longrightarrow> length xs = length ys"
  5000 by(induct rule: listrel.induct) auto
  5001 
  5002 lemma listrel_iff_zip [code_unfold]: "(xs,ys) : listrel r \<longleftrightarrow>
  5003   length xs = length ys & (\<forall>(x,y) \<in> set(zip xs ys). (x,y) \<in> r)" (is "?L \<longleftrightarrow> ?R")
  5004 proof
  5005   assume ?L thus ?R by induct (auto intro: listrel_eq_len)
  5006 next
  5007   assume ?R thus ?L
  5008     apply (clarify)
  5009     by (induct rule: list_induct2) (auto intro: listrel.intros)
  5010 qed
  5011 
  5012 lemma listrel_iff_nth: "(xs,ys) : listrel r \<longleftrightarrow>
  5013   length xs = length ys & (\<forall>n < length xs. (xs!n, ys!n) \<in> r)" (is "?L \<longleftrightarrow> ?R")
  5014 by (auto simp add: all_set_conv_all_nth listrel_iff_zip)
  5015 
  5016 
  5017 lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
  5018 apply clarify  
  5019 apply (erule listrel.induct)
  5020 apply (blast intro: listrel.intros)+
  5021 done
  5022 
  5023 lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
  5024 apply clarify 
  5025 apply (erule listrel.induct, auto) 
  5026 done
  5027 
  5028 lemma listrel_refl_on: "refl_on A r \<Longrightarrow> refl_on (lists A) (listrel r)" 
  5029 apply (simp add: refl_on_def listrel_subset Ball_def)
  5030 apply (rule allI) 
  5031 apply (induct_tac x) 
  5032 apply (auto intro: listrel.intros)
  5033 done
  5034 
  5035 lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
  5036 apply (auto simp add: sym_def)
  5037 apply (erule listrel.induct) 
  5038 apply (blast intro: listrel.intros)+
  5039 done
  5040 
  5041 lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
  5042 apply (simp add: trans_def)
  5043 apply (intro allI) 
  5044 apply (rule impI) 
  5045 apply (erule listrel.induct) 
  5046 apply (blast intro: listrel.intros)+
  5047 done
  5048 
  5049 theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
  5050 by (simp add: equiv_def listrel_refl_on listrel_sym listrel_trans) 
  5051 
  5052 lemma listrel_rtrancl_refl[iff]: "(xs,xs) : listrel(r^*)"
  5053 using listrel_refl_on[of UNIV, OF refl_rtrancl]
  5054 by(auto simp: refl_on_def)
  5055 
  5056 lemma listrel_rtrancl_trans:
  5057   "\<lbrakk> (xs,ys) : listrel(r^*);  (ys,zs) : listrel(r^*) \<rbrakk>
  5058   \<Longrightarrow> (xs,zs) : listrel(r^*)"
  5059 by (metis listrel_trans trans_def trans_rtrancl)
  5060 
  5061 
  5062 lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
  5063 by (blast intro: listrel.intros)
  5064 
  5065 lemma listrel_Cons:
  5066      "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})"
  5067 by (auto simp add: set_Cons_def intro: listrel.intros)
  5068 
  5069 text {* Relating @{term listrel1}, @{term listrel} and closures: *}
  5070 
  5071 lemma listrel1_rtrancl_subset_rtrancl_listrel1:
  5072   "listrel1 (r^*) \<subseteq> (listrel1 r)^*"
  5073 proof (rule subrelI)
  5074   fix xs ys assume 1: "(xs,ys) \<in> listrel1 (r^*)"
  5075   { fix x y us vs
  5076     have "(x,y) : r^* \<Longrightarrow> (us @ x # vs, us @ y # vs) : (listrel1 r)^*"
  5077     proof(induct rule: rtrancl.induct)
  5078       case rtrancl_refl show ?case by simp
  5079     next
  5080       case rtrancl_into_rtrancl thus ?case
  5081         by (metis listrel1I rtrancl.rtrancl_into_rtrancl)
  5082     qed }
  5083   thus "(xs,ys) \<in> (listrel1 r)^*" using 1 by(blast elim: listrel1E)
  5084 qed
  5085 
  5086 lemma rtrancl_listrel1_eq_len: "(x,y) \<in> (listrel1 r)^* \<Longrightarrow> length x = length y"
  5087 by (induct rule: rtrancl.induct) (auto intro: listrel1_eq_len)
  5088 
  5089 lemma rtrancl_listrel1_ConsI1:
  5090   "(xs,ys) : (listrel1 r)^* \<Longrightarrow> (x#xs,x#ys) : (listrel1 r)^*"
  5091 apply(induct rule: rtrancl.induct)
  5092  apply simp
  5093 by (metis listrel1I2 rtrancl.rtrancl_into_rtrancl)
  5094 
  5095 lemma rtrancl_listrel1_ConsI2:
  5096   "(x,y) \<in> r^* \<Longrightarrow> (xs, ys) \<in> (listrel1 r)^*
  5097   \<Longrightarrow> (x # xs, y # ys) \<in> (listrel1 r)^*"
  5098   by (blast intro: rtrancl_trans rtrancl_listrel1_ConsI1 
  5099     subsetD[OF listrel1_rtrancl_subset_rtrancl_listrel1 listrel1I1])
  5100 
  5101 lemma listrel1_subset_listrel:
  5102   "r \<subseteq> r' \<Longrightarrow> refl r' \<Longrightarrow> listrel1 r \<subseteq> listrel(r')"
  5103 by(auto elim!: listrel1E simp add: listrel_iff_zip set_zip refl_on_def)
  5104 
  5105 lemma listrel_reflcl_if_listrel1:
  5106   "(xs,ys) : listrel1 r \<Longrightarrow> (xs,ys) : listrel(r^*)"
  5107 by(erule listrel1E)(auto simp add: listrel_iff_zip set_zip)
  5108 
  5109 lemma listrel_rtrancl_eq_rtrancl_listrel1: "listrel (r^*) = (listrel1 r)^*"
  5110 proof
  5111   { fix x y assume "(x,y) \<in> listrel (r^*)"
  5112     then have "(x,y) \<in> (listrel1 r)^*"
  5113     by induct (auto intro: rtrancl_listrel1_ConsI2) }
  5114   then show "listrel (r^*) \<subseteq> (listrel1 r)^*"
  5115     by (rule subrelI)
  5116 next
  5117   show "listrel (r^*) \<supseteq> (listrel1 r)^*"
  5118   proof(rule subrelI)
  5119     fix xs ys assume "(xs,ys) \<in> (listrel1 r)^*"
  5120     then show "(xs,ys) \<in> listrel (r^*)"
  5121     proof induct
  5122       case base show ?case by(auto simp add: listrel_iff_zip set_zip)
  5123     next
  5124       case (step ys zs)
  5125       thus ?case  by (metis listrel_reflcl_if_listrel1 listrel_rtrancl_trans)
  5126     qed
  5127   qed
  5128 qed
  5129 
  5130 lemma rtrancl_listrel1_if_listrel:
  5131   "(xs,ys) : listrel r \<Longrightarrow> (xs,ys) : (listrel1 r)^*"
  5132 by(metis listrel_rtrancl_eq_rtrancl_listrel1 subsetD[OF listrel_mono] r_into_rtrancl subsetI)
  5133 
  5134 lemma listrel_subset_rtrancl_listrel1: "listrel r \<subseteq> (listrel1 r)^*"
  5135 by(fast intro:rtrancl_listrel1_if_listrel)
  5136 
  5137 
  5138 subsection {* Size function *}
  5139 
  5140 lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (list_size f)"
  5141 by (rule is_measure_trivial)
  5142 
  5143 lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (option_size f)"
  5144 by (rule is_measure_trivial)
  5145 
  5146 lemma list_size_estimation[termination_simp]: 
  5147   "x \<in> set xs \<Longrightarrow> y < f x \<Longrightarrow> y < list_size f xs"
  5148 by (induct xs) auto
  5149 
  5150 lemma list_size_estimation'[termination_simp]: 
  5151   "x \<in> set xs \<Longrightarrow> y \<le> f x \<Longrightarrow> y \<le> list_size f xs"
  5152 by (induct xs) auto
  5153 
  5154 lemma list_size_map[simp]: "list_size f (map g xs) = list_size (f o g) xs"
  5155 by (induct xs) auto
  5156 
  5157 lemma list_size_append[simp]: "list_size f (xs @ ys) = list_size f xs + list_size f ys"
  5158 by (induct xs, auto)
  5159 
  5160 lemma list_size_pointwise[termination_simp]: 
  5161   "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> list_size f xs \<le> list_size g xs"
  5162 by (induct xs) force+
  5163 
  5164 
  5165 subsection {* Monad operation *}
  5166 
  5167 definition bind :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b list) \<Rightarrow> 'b list" where
  5168   "bind xs f = concat (map f xs)"
  5169 
  5170 hide_const (open) bind
  5171 
  5172 lemma bind_simps [simp]:
  5173   "List.bind [] f = []"
  5174   "List.bind (x # xs) f = f x @ List.bind xs f"
  5175   by (simp_all add: bind_def)
  5176 
  5177 
  5178 subsection {* Transfer *}
  5179 
  5180 definition
  5181   embed_list :: "nat list \<Rightarrow> int list"
  5182 where
  5183   "embed_list l = map int l"
  5184 
  5185 definition
  5186   nat_list :: "int list \<Rightarrow> bool"
  5187 where
  5188   "nat_list l = nat_set (set l)"
  5189 
  5190 definition
  5191   return_list :: "int list \<Rightarrow> nat list"
  5192 where
  5193   "return_list l = map nat l"
  5194 
  5195 lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow>
  5196     embed_list (return_list l) = l"
  5197   unfolding embed_list_def return_list_def nat_list_def nat_set_def
  5198   apply (induct l)
  5199   apply auto
  5200 done
  5201 
  5202 lemma transfer_nat_int_list_functions:
  5203   "l @ m = return_list (embed_list l @ embed_list m)"
  5204   "[] = return_list []"
  5205   unfolding return_list_def embed_list_def
  5206   apply auto
  5207   apply (induct l, auto)
  5208   apply (induct m, auto)
  5209 done
  5210 
  5211 (*
  5212 lemma transfer_nat_int_fold1: "fold f l x =
  5213     fold (%x. f (nat x)) (embed_list l) x";
  5214 *)
  5215 
  5216 
  5217 subsection {* Code generation *}
  5218 
  5219 subsubsection {* Counterparts for set-related operations *}
  5220 
  5221 definition member :: "'a list \<Rightarrow> 'a \<Rightarrow> bool" where
  5222   [code_abbrev]: "member xs x \<longleftrightarrow> x \<in> set xs"
  5223 
  5224 text {*
  5225   Use @{text member} only for generating executable code.  Otherwise use
  5226   @{prop "x \<in> set xs"} instead --- it is much easier to reason about.
  5227 *}
  5228 
  5229 lemma member_rec [code]:
  5230   "member (x # xs) y \<longleftrightarrow> x = y \<or> member xs y"
  5231   "member [] y \<longleftrightarrow> False"
  5232   by (auto simp add: member_def)
  5233 
  5234 lemma in_set_member (* FIXME delete candidate *):
  5235   "x \<in> set xs \<longleftrightarrow> member xs x"
  5236   by (simp add: member_def)
  5237 
  5238 definition list_all :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5239   list_all_iff [code_abbrev]: "list_all P xs \<longleftrightarrow> Ball (set xs) P"
  5240 
  5241 definition list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5242   list_ex_iff [code_abbrev]: "list_ex P xs \<longleftrightarrow> Bex (set xs) P"
  5243 
  5244 definition list_ex1 :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5245   list_ex1_iff [code_abbrev]: "list_ex1 P xs \<longleftrightarrow> (\<exists>! x. x \<in> set xs \<and> P x)"
  5246 
  5247 text {*
  5248   Usually you should prefer @{text "\<forall>x\<in>set xs"}, @{text "\<exists>x\<in>set xs"}
  5249   and @{text "\<exists>!x. x\<in>set xs \<and> _"} over @{const list_all}, @{const list_ex}
  5250   and @{const list_ex1} in specifications.
  5251 *}
  5252 
  5253 lemma list_all_simps [simp, code]:
  5254   "list_all P (x # xs) \<longleftrightarrow> P x \<and> list_all P xs"
  5255   "list_all P [] \<longleftrightarrow> True"
  5256   by (simp_all add: list_all_iff)
  5257 
  5258 lemma list_ex_simps [simp, code]:
  5259   "list_ex P (x # xs) \<longleftrightarrow> P x \<or> list_ex P xs"
  5260   "list_ex P [] \<longleftrightarrow> False"
  5261   by (simp_all add: list_ex_iff)
  5262 
  5263 lemma list_ex1_simps [simp, code]:
  5264   "list_ex1 P [] = False"
  5265   "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)"
  5266   by (auto simp add: list_ex1_iff list_all_iff)
  5267 
  5268 lemma Ball_set_list_all: (* FIXME delete candidate *)
  5269   "Ball (set xs) P \<longleftrightarrow> list_all P xs"
  5270   by (simp add: list_all_iff)
  5271 
  5272 lemma Bex_set_list_ex: (* FIXME delete candidate *)
  5273   "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
  5274   by (simp add: list_ex_iff)
  5275 
  5276 lemma list_all_append [simp]:
  5277   "list_all P (xs @ ys) \<longleftrightarrow> list_all P xs \<and> list_all P ys"
  5278   by (auto simp add: list_all_iff)
  5279 
  5280 lemma list_ex_append [simp]:
  5281   "list_ex P (xs @ ys) \<longleftrightarrow> list_ex P xs \<or> list_ex P ys"
  5282   by (auto simp add: list_ex_iff)
  5283 
  5284 lemma list_all_rev [simp]:
  5285   "list_all P (rev xs) \<longleftrightarrow> list_all P xs"
  5286   by (simp add: list_all_iff)
  5287 
  5288 lemma list_ex_rev [simp]:
  5289   "list_ex P (rev xs) \<longleftrightarrow> list_ex P xs"
  5290   by (simp add: list_ex_iff)
  5291 
  5292 lemma list_all_length:
  5293   "list_all P xs \<longleftrightarrow> (\<forall>n < length xs. P (xs ! n))"
  5294   by (auto simp add: list_all_iff set_conv_nth)
  5295 
  5296 lemma list_ex_length:
  5297   "list_ex P xs \<longleftrightarrow> (\<exists>n < length xs. P (xs ! n))"
  5298   by (auto simp add: list_ex_iff set_conv_nth)
  5299 
  5300 lemma list_all_cong [fundef_cong]:
  5301   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_all f xs = list_all g ys"
  5302   by (simp add: list_all_iff)
  5303 
  5304 lemma list_any_cong [fundef_cong]:
  5305   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_ex f xs = list_ex g ys"
  5306   by (simp add: list_ex_iff)
  5307 
  5308 text {* Executable checks for relations on sets *}
  5309 
  5310 definition listrel1p :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
  5311 "listrel1p r xs ys = ((xs, ys) \<in> listrel1 {(x, y). r x y})"
  5312 
  5313 lemma [code_unfold]:
  5314   "(xs, ys) \<in> listrel1 r = listrel1p (\<lambda>x y. (x, y) \<in> r) xs ys"
  5315 unfolding listrel1p_def by auto
  5316 
  5317 lemma [code]:
  5318   "listrel1p r [] xs = False"
  5319   "listrel1p r xs [] =  False"
  5320   "listrel1p r (x # xs) (y # ys) \<longleftrightarrow>
  5321      r x y \<and> xs = ys \<or> x = y \<and> listrel1p r xs ys"
  5322 by (simp add: listrel1p_def)+
  5323 
  5324 definition
  5325   lexordp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
  5326   "lexordp r xs ys = ((xs, ys) \<in> lexord {(x, y). r x y})"
  5327 
  5328 lemma [code_unfold]:
  5329   "(xs, ys) \<in> lexord r = lexordp (\<lambda>x y. (x, y) \<in> r) xs ys"
  5330 unfolding lexordp_def by auto
  5331 
  5332 lemma [code]:
  5333   "lexordp r xs [] = False"
  5334   "lexordp r [] (y#ys) = True"
  5335   "lexordp r (x # xs) (y # ys) = (r x y | (x = y & lexordp r xs ys))"
  5336 unfolding lexordp_def by auto
  5337 
  5338 text {* Bounded quantification and summation over nats. *}
  5339 
  5340 lemma atMost_upto [code_unfold]:
  5341   "{..n} = set [0..<Suc n]"
  5342   by auto
  5343 
  5344 lemma atLeast_upt [code_unfold]:
  5345   "{..<n} = set [0..<n]"
  5346   by auto
  5347 
  5348 lemma greaterThanLessThan_upt [code_unfold]:
  5349   "{n<..<m} = set [Suc n..<m]"
  5350   by auto
  5351 
  5352 lemmas atLeastLessThan_upt [code_unfold] = set_upt [symmetric]
  5353 
  5354 lemma greaterThanAtMost_upt [code_unfold]:
  5355   "{n<..m} = set [Suc n..<Suc m]"
  5356   by auto
  5357 
  5358 lemma atLeastAtMost_upt [code_unfold]:
  5359   "{n..m} = set [n..<Suc m]"
  5360   by auto
  5361 
  5362 lemma all_nat_less_eq [code_unfold]:
  5363   "(\<forall>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..<n}. P m)"
  5364   by auto
  5365 
  5366 lemma ex_nat_less_eq [code_unfold]:
  5367   "(\<exists>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..<n}. P m)"
  5368   by auto
  5369 
  5370 lemma all_nat_less [code_unfold]:
  5371   "(\<forall>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..n}. P m)"
  5372   by auto
  5373 
  5374 lemma ex_nat_less [code_unfold]:
  5375   "(\<exists>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..n}. P m)"
  5376   by auto
  5377 
  5378 lemma setsum_set_upt_conv_listsum_nat [code_unfold]:
  5379   "setsum f (set [m..<n]) = listsum (map f [m..<n])"
  5380   by (simp add: interv_listsum_conv_setsum_set_nat)
  5381 
  5382 text {* Summation over ints. *}
  5383 
  5384 lemma greaterThanLessThan_upto [code_unfold]:
  5385   "{i<..<j::int} = set [i+1..j - 1]"
  5386 by auto
  5387 
  5388 lemma atLeastLessThan_upto [code_unfold]:
  5389   "{i..<j::int} = set [i..j - 1]"
  5390 by auto
  5391 
  5392 lemma greaterThanAtMost_upto [code_unfold]:
  5393   "{i<..j::int} = set [i+1..j]"
  5394 by auto
  5395 
  5396 lemmas atLeastAtMost_upto [code_unfold] = set_upto [symmetric]
  5397 
  5398 lemma setsum_set_upto_conv_listsum_int [code_unfold]:
  5399   "setsum f (set [i..j::int]) = listsum (map f [i..j])"
  5400   by (simp add: interv_listsum_conv_setsum_set_int)
  5401 
  5402 
  5403 subsubsection {* Optimizing by rewriting *}
  5404 
  5405 definition null :: "'a list \<Rightarrow> bool" where
  5406   [code_abbrev]: "null xs \<longleftrightarrow> xs = []"
  5407 
  5408 text {*
  5409   Efficient emptyness check is implemented by @{const null}.
  5410 *}
  5411 
  5412 lemma null_rec [code]:
  5413   "null (x # xs) \<longleftrightarrow> False"
  5414   "null [] \<longleftrightarrow> True"
  5415   by (simp_all add: null_def)
  5416 
  5417 lemma eq_Nil_null: (* FIXME delete candidate *)
  5418   "xs = [] \<longleftrightarrow> null xs"
  5419   by (simp add: null_def)
  5420 
  5421 lemma equal_Nil_null [code_unfold]:
  5422   "HOL.equal xs [] \<longleftrightarrow> null xs"
  5423   by (simp add: equal eq_Nil_null)
  5424 
  5425 definition maps :: "('a \<Rightarrow> 'b list) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  5426   [code_abbrev]: "maps f xs = concat (map f xs)"
  5427 
  5428 definition map_filter :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  5429   [code_post]: "map_filter f xs = map (the \<circ> f) (filter (\<lambda>x. f x \<noteq> None) xs)"
  5430 
  5431 text {*
  5432   Operations @{const maps} and @{const map_filter} avoid
  5433   intermediate lists on execution -- do not use for proving.
  5434 *}
  5435 
  5436 lemma maps_simps [code]:
  5437   "maps f (x # xs) = f x @ maps f xs"
  5438   "maps f [] = []"
  5439   by (simp_all add: maps_def)
  5440 
  5441 lemma map_filter_simps [code]:
  5442   "map_filter f (x # xs) = (case f x of None \<Rightarrow> map_filter f xs | Some y \<Rightarrow> y # map_filter f xs)"
  5443   "map_filter f [] = []"
  5444   by (simp_all add: map_filter_def split: option.split)
  5445 
  5446 lemma concat_map_maps: (* FIXME delete candidate *)
  5447   "concat (map f xs) = maps f xs"
  5448   by (simp add: maps_def)
  5449 
  5450 lemma map_filter_map_filter [code_unfold]:
  5451   "map f (filter P xs) = map_filter (\<lambda>x. if P x then Some (f x) else None) xs"
  5452   by (simp add: map_filter_def)
  5453 
  5454 text {* Optimized code for @{text"\<forall>i\<in>{a..b::int}"} and @{text"\<forall>n:{a..<b::nat}"}
  5455 and similiarly for @{text"\<exists>"}. *}
  5456 
  5457 definition all_interval_nat :: "(nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
  5458   "all_interval_nat P i j \<longleftrightarrow> (\<forall>n \<in> {i..<j}. P n)"
  5459 
  5460 lemma [code]:
  5461   "all_interval_nat P i j \<longleftrightarrow> i \<ge> j \<or> P i \<and> all_interval_nat P (Suc i) j"
  5462 proof -
  5463   have *: "\<And>n. P i \<Longrightarrow> \<forall>n\<in>{Suc i..<j}. P n \<Longrightarrow> i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n"
  5464   proof -
  5465     fix n
  5466     assume "P i" "\<forall>n\<in>{Suc i..<j}. P n" "i \<le> n" "n < j"
  5467     then show "P n" by (cases "n = i") simp_all
  5468   qed
  5469   show ?thesis by (auto simp add: all_interval_nat_def intro: *)
  5470 qed
  5471 
  5472 lemma list_all_iff_all_interval_nat [code_unfold]:
  5473   "list_all P [i..<j] \<longleftrightarrow> all_interval_nat P i j"
  5474   by (simp add: list_all_iff all_interval_nat_def)
  5475 
  5476 lemma list_ex_iff_not_all_inverval_nat [code_unfold]:
  5477   "list_ex P [i..<j] \<longleftrightarrow> \<not> (all_interval_nat (Not \<circ> P) i j)"
  5478   by (simp add: list_ex_iff all_interval_nat_def)
  5479 
  5480 definition all_interval_int :: "(int \<Rightarrow> bool) \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool" where
  5481   "all_interval_int P i j \<longleftrightarrow> (\<forall>k \<in> {i..j}. P k)"
  5482 
  5483 lemma [code]:
  5484   "all_interval_int P i j \<longleftrightarrow> i > j \<or> P i \<and> all_interval_int P (i + 1) j"
  5485 proof -
  5486   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"
  5487   proof -
  5488     fix k
  5489     assume "P i" "\<forall>k\<in>{i+1..j}. P k" "i \<le> k" "k \<le> j"
  5490     then show "P k" by (cases "k = i") simp_all
  5491   qed
  5492   show ?thesis by (auto simp add: all_interval_int_def intro: *)
  5493 qed
  5494 
  5495 lemma list_all_iff_all_interval_int [code_unfold]:
  5496   "list_all P [i..j] \<longleftrightarrow> all_interval_int P i j"
  5497   by (simp add: list_all_iff all_interval_int_def)
  5498 
  5499 lemma list_ex_iff_not_all_inverval_int [code_unfold]:
  5500   "list_ex P [i..j] \<longleftrightarrow> \<not> (all_interval_int (Not \<circ> P) i j)"
  5501   by (simp add: list_ex_iff all_interval_int_def)
  5502 
  5503 hide_const (open) member null maps map_filter all_interval_nat all_interval_int
  5504 
  5505 
  5506 subsubsection {* Pretty lists *}
  5507 
  5508 use "Tools/list_code.ML"
  5509 
  5510 code_type list
  5511   (SML "_ list")
  5512   (OCaml "_ list")
  5513   (Haskell "![(_)]")
  5514   (Scala "List[(_)]")
  5515 
  5516 code_const Nil
  5517   (SML "[]")
  5518   (OCaml "[]")
  5519   (Haskell "[]")
  5520   (Scala "!Nil")
  5521 
  5522 code_instance list :: equal
  5523   (Haskell -)
  5524 
  5525 code_const "HOL.equal \<Colon> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
  5526   (Haskell infix 4 "==")
  5527 
  5528 code_reserved SML
  5529   list
  5530 
  5531 code_reserved OCaml
  5532   list
  5533 
  5534 setup {* fold (List_Code.add_literal_list) ["SML", "OCaml", "Haskell", "Scala"] *}
  5535 
  5536 
  5537 subsubsection {* Use convenient predefined operations *}
  5538 
  5539 code_const "op @"
  5540   (SML infixr 7 "@")
  5541   (OCaml infixr 6 "@")
  5542   (Haskell infixr 5 "++")
  5543   (Scala infixl 7 "++")
  5544 
  5545 code_const map
  5546   (Haskell "map")
  5547 
  5548 code_const filter
  5549   (Haskell "filter")
  5550 
  5551 code_const concat
  5552   (Haskell "concat")
  5553 
  5554 code_const List.maps
  5555   (Haskell "concatMap")
  5556 
  5557 code_const rev
  5558   (Haskell "reverse")
  5559 
  5560 code_const zip
  5561   (Haskell "zip")
  5562 
  5563 code_const List.null
  5564   (Haskell "null")
  5565 
  5566 code_const takeWhile
  5567   (Haskell "takeWhile")
  5568 
  5569 code_const dropWhile
  5570   (Haskell "dropWhile")
  5571 
  5572 code_const list_all
  5573   (Haskell "all")
  5574 
  5575 code_const list_ex
  5576   (Haskell "any")
  5577 
  5578 
  5579 subsubsection {* Implementation of sets by lists *}
  5580 
  5581 text {* Basic operations *}
  5582 
  5583 lemma is_empty_set [code]:
  5584   "Set.is_empty (set xs) \<longleftrightarrow> List.null xs"
  5585   by (simp add: Set.is_empty_def null_def)
  5586 
  5587 lemma empty_set [code]:
  5588   "{} = set []"
  5589   by simp
  5590 
  5591 lemma UNIV_coset [code]:
  5592   "UNIV = List.coset []"
  5593   by simp
  5594 
  5595 lemma compl_set [code]:
  5596   "- set xs = List.coset xs"
  5597   by simp
  5598 
  5599 lemma compl_coset [code]:
  5600   "- List.coset xs = set xs"
  5601   by simp
  5602 
  5603 lemma [code]:
  5604   "x \<in> set xs \<longleftrightarrow> List.member xs x"
  5605   "x \<in> List.coset xs \<longleftrightarrow> \<not> List.member xs x"
  5606   by (simp_all add: member_def)
  5607 
  5608 lemma insert_code [code]:
  5609   "insert x (set xs) = set (List.insert x xs)"
  5610   "insert x (List.coset xs) = List.coset (removeAll x xs)"
  5611   by simp_all
  5612 
  5613 lemma remove_code [code]:
  5614   "Set.remove x (set xs) = set (removeAll x xs)"
  5615   "Set.remove x (List.coset xs) = List.coset (List.insert x xs)"
  5616   by (simp_all add: remove_def Compl_insert)
  5617 
  5618 lemma project_set [code]:
  5619   "Set.project P (set xs) = set (filter P xs)"
  5620   by auto
  5621 
  5622 lemma image_set [code]:
  5623   "image f (set xs) = set (map f xs)"
  5624   by simp
  5625 
  5626 lemma Ball_set [code]:
  5627   "Ball (set xs) P \<longleftrightarrow> list_all P xs"
  5628   by (simp add: list_all_iff)
  5629 
  5630 lemma Bex_set [code]:
  5631   "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
  5632   by (simp add: list_ex_iff)
  5633 
  5634 lemma card_set [code]:
  5635   "card (set xs) = length (remdups xs)"
  5636 proof -
  5637   have "card (set (remdups xs)) = length (remdups xs)"
  5638     by (rule distinct_card) simp
  5639   then show ?thesis by simp
  5640 qed
  5641 
  5642 lemma the_elem_set [code]:
  5643   "the_elem (set [x]) = x"
  5644   by simp
  5645 
  5646 lemma Pow_set [code]:
  5647   "Pow (set []) = {{}}"
  5648   "Pow (set (x # xs)) = (let A = Pow (set xs) in A \<union> insert x ` A)"
  5649   by (simp_all add: Pow_insert Let_def)
  5650 
  5651 text {* Further operations on sets *}
  5652 
  5653 (* Minimal refinement of equality on sets *)
  5654 declare subset_eq[code del]
  5655 lemma subset_code [code]:
  5656   "set xs <= B \<longleftrightarrow> (ALL x : set xs. x : B)"
  5657   "List.coset xs <= List.coset ys \<longleftrightarrow> set ys <= set xs"
  5658   "List.coset [] <= set [] \<longleftrightarrow> False"
  5659 by auto
  5660 
  5661 lemma setsum_code [code]:
  5662   "setsum f (set xs) = listsum (map f (remdups xs))"
  5663 by (simp add: listsum_distinct_conv_setsum_set)
  5664 
  5665 definition map_project :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a set \<Rightarrow> 'b set" where
  5666   "map_project f A = {b. \<exists> a \<in> A. f a = Some b}"
  5667 
  5668 lemma [code]:
  5669   "map_project f (set xs) = set (List.map_filter f xs)"
  5670 unfolding map_project_def map_filter_def
  5671 by auto (metis (lifting, mono_tags) CollectI image_eqI o_apply the.simps)
  5672 
  5673 hide_const (open) map_project
  5674 
  5675 text {* Operations on relations *}
  5676 
  5677 lemma product_code [code]:
  5678   "Product_Type.product (set xs) (set ys) = set [(x, y). x \<leftarrow> xs, y \<leftarrow> ys]"
  5679   by (auto simp add: Product_Type.product_def)
  5680 
  5681 lemma Id_on_set [code]:
  5682   "Id_on (set xs) = set [(x, x). x \<leftarrow> xs]"
  5683   by (auto simp add: Id_on_def)
  5684 
  5685 lemma [code]:
  5686   "R `` S = List.map_project (%(x, y). if x : S then Some y else None) R"
  5687 unfolding map_project_def by (auto split: prod.split split_if_asm)
  5688 
  5689 lemma trancl_set_ntrancl [code]:
  5690   "trancl (set xs) = ntrancl (card (set xs) - 1) (set xs)"
  5691   by (simp add: finite_trancl_ntranl)
  5692 
  5693 lemma set_rel_comp [code]:
  5694   "set xys O set yzs = set ([(fst xy, snd yz). xy \<leftarrow> xys, yz \<leftarrow> yzs, snd xy = fst yz])"
  5695   by (auto simp add: Bex_def)
  5696 
  5697 lemma wf_set [code]:
  5698   "wf (set xs) = acyclic (set xs)"
  5699   by (simp add: wf_iff_acyclic_if_finite)
  5700 
  5701 end