src/HOL/List.thy
author bulwahn
Thu Feb 16 09:18:23 2012 +0100 (2012-02-16)
changeset 46500 0196966d6d2d
parent 46448 f1201fac7398
child 46635 cde737f9c911
permissions -rw-r--r--
removing unnecessary premises in theorems of List theory
     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 declare Inf_set_foldr [where 'a = "'a Predicate.pred", code] Sup_set_foldr [where 'a = "'a Predicate.pred", code]
  2666 
  2667 lemma (in complete_lattice) INF_set_foldr [code]:
  2668   "INFI (set xs) f = foldr (inf \<circ> f) xs top"
  2669   by (simp only: INF_def Inf_set_foldr foldr_map set_map [symmetric])
  2670 
  2671 lemma (in complete_lattice) SUP_set_foldr [code]:
  2672   "SUPR (set xs) f = foldr (sup \<circ> f) xs bot"
  2673   by (simp only: SUP_def Sup_set_foldr foldr_map set_map [symmetric])
  2674 
  2675 (* FIXME: better implement conversion by bisection *)
  2676 
  2677 lemma pred_of_set_fold_sup:
  2678   assumes "finite A"
  2679   shows "pred_of_set A = Finite_Set.fold sup bot (Predicate.single ` A)" (is "?lhs = ?rhs")
  2680 proof (rule sym)
  2681   interpret comp_fun_idem "sup :: 'a Predicate.pred \<Rightarrow> 'a Predicate.pred \<Rightarrow> 'a Predicate.pred"
  2682     by (fact comp_fun_idem_sup)
  2683   from `finite A` show "?rhs = ?lhs" by (induct A) (auto intro!: pred_eqI)
  2684 qed
  2685 
  2686 lemma pred_of_set_set_fold_sup:
  2687   "pred_of_set (set xs) = fold sup (map Predicate.single xs) bot"
  2688 proof -
  2689   interpret comp_fun_idem "sup :: 'a Predicate.pred \<Rightarrow> 'a Predicate.pred \<Rightarrow> 'a Predicate.pred"
  2690     by (fact comp_fun_idem_sup)
  2691   show ?thesis by (simp add: pred_of_set_fold_sup fold_set_fold [symmetric])
  2692 qed
  2693 
  2694 lemma pred_of_set_set_foldr_sup [code]:
  2695   "pred_of_set (set xs) = foldr sup (map Predicate.single xs) bot"
  2696   by (simp add: pred_of_set_set_fold_sup ac_simps foldr_fold fun_eq_iff)
  2697 
  2698 
  2699 subsubsection {* @{text upt} *}
  2700 
  2701 lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
  2702 -- {* simp does not terminate! *}
  2703 by (induct j) auto
  2704 
  2705 lemmas upt_rec_number_of[simp] = upt_rec[of "number_of m" "number_of n"] for m n
  2706 
  2707 lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
  2708 by (subst upt_rec) simp
  2709 
  2710 lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
  2711 by(induct j)simp_all
  2712 
  2713 lemma upt_eq_Cons_conv:
  2714  "([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
  2715 apply(induct j arbitrary: x xs)
  2716  apply simp
  2717 apply(clarsimp simp add: append_eq_Cons_conv)
  2718 apply arith
  2719 done
  2720 
  2721 lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
  2722 -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
  2723 by simp
  2724 
  2725 lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
  2726   by (simp add: upt_rec)
  2727 
  2728 lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
  2729 -- {* LOOPS as a simprule, since @{text "j <= j"}. *}
  2730 by (induct k) auto
  2731 
  2732 lemma length_upt [simp]: "length [i..<j] = j - i"
  2733 by (induct j) (auto simp add: Suc_diff_le)
  2734 
  2735 lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
  2736 apply (induct j)
  2737 apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
  2738 done
  2739 
  2740 
  2741 lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
  2742 by(simp add:upt_conv_Cons)
  2743 
  2744 lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
  2745 apply(cases j)
  2746  apply simp
  2747 by(simp add:upt_Suc_append)
  2748 
  2749 lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]"
  2750 apply (induct m arbitrary: i, simp)
  2751 apply (subst upt_rec)
  2752 apply (rule sym)
  2753 apply (subst upt_rec)
  2754 apply (simp del: upt.simps)
  2755 done
  2756 
  2757 lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
  2758 apply(induct j)
  2759 apply auto
  2760 done
  2761 
  2762 lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]"
  2763 by (induct n) auto
  2764 
  2765 lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
  2766 apply (induct n m  arbitrary: i rule: diff_induct)
  2767 prefer 3 apply (subst map_Suc_upt[symmetric])
  2768 apply (auto simp add: less_diff_conv)
  2769 done
  2770 
  2771 lemma nth_take_lemma:
  2772   "k <= length xs ==> k <= length ys ==>
  2773      (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
  2774 apply (atomize, induct k arbitrary: xs ys)
  2775 apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
  2776 txt {* Both lists must be non-empty *}
  2777 apply (case_tac xs, simp)
  2778 apply (case_tac ys, clarify)
  2779  apply (simp (no_asm_use))
  2780 apply clarify
  2781 txt {* prenexing's needed, not miniscoping *}
  2782 apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
  2783 apply blast
  2784 done
  2785 
  2786 lemma nth_equalityI:
  2787  "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
  2788   by (frule nth_take_lemma [OF le_refl eq_imp_le]) simp_all
  2789 
  2790 lemma map_nth:
  2791   "map (\<lambda>i. xs ! i) [0..<length xs] = xs"
  2792   by (rule nth_equalityI, auto)
  2793 
  2794 (* needs nth_equalityI *)
  2795 lemma list_all2_antisym:
  2796   "\<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> 
  2797   \<Longrightarrow> xs = ys"
  2798   apply (simp add: list_all2_conv_all_nth) 
  2799   apply (rule nth_equalityI, blast, simp)
  2800   done
  2801 
  2802 lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
  2803 -- {* The famous take-lemma. *}
  2804 apply (drule_tac x = "max (length xs) (length ys)" in spec)
  2805 apply (simp add: le_max_iff_disj)
  2806 done
  2807 
  2808 
  2809 lemma take_Cons':
  2810      "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
  2811 by (cases n) simp_all
  2812 
  2813 lemma drop_Cons':
  2814      "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
  2815 by (cases n) simp_all
  2816 
  2817 lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
  2818 by (cases n) simp_all
  2819 
  2820 lemmas take_Cons_number_of = take_Cons'[of "number_of v"] for v
  2821 lemmas drop_Cons_number_of = drop_Cons'[of "number_of v"] for v
  2822 lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v"] for v
  2823 
  2824 declare take_Cons_number_of [simp] 
  2825         drop_Cons_number_of [simp] 
  2826         nth_Cons_number_of [simp] 
  2827 
  2828 
  2829 subsubsection {* @{text upto}: interval-list on @{typ int} *}
  2830 
  2831 (* FIXME make upto tail recursive? *)
  2832 
  2833 function upto :: "int \<Rightarrow> int \<Rightarrow> int list" ("(1[_../_])") where
  2834 "upto i j = (if i \<le> j then i # [i+1..j] else [])"
  2835 by auto
  2836 termination
  2837 by(relation "measure(%(i::int,j). nat(j - i + 1))") auto
  2838 
  2839 declare upto.simps[code, simp del]
  2840 
  2841 lemmas upto_rec_number_of[simp] = upto.simps[of "number_of m" "number_of n"] for m n
  2842 
  2843 lemma upto_empty[simp]: "j < i \<Longrightarrow> [i..j] = []"
  2844 by(simp add: upto.simps)
  2845 
  2846 lemma set_upto[simp]: "set[i..j] = {i..j}"
  2847 proof(induct i j rule:upto.induct)
  2848   case (1 i j)
  2849   from this show ?case
  2850     unfolding upto.simps[of i j] simp_from_to[of i j] by auto
  2851 qed
  2852 
  2853 
  2854 subsubsection {* @{text "distinct"} and @{text remdups} *}
  2855 
  2856 lemma distinct_tl:
  2857   "distinct xs \<Longrightarrow> distinct (tl xs)"
  2858   by (cases xs) simp_all
  2859 
  2860 lemma distinct_append [simp]:
  2861 "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
  2862 by (induct xs) auto
  2863 
  2864 lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
  2865 by(induct xs) auto
  2866 
  2867 lemma set_remdups [simp]: "set (remdups xs) = set xs"
  2868 by (induct xs) (auto simp add: insert_absorb)
  2869 
  2870 lemma distinct_remdups [iff]: "distinct (remdups xs)"
  2871 by (induct xs) auto
  2872 
  2873 lemma distinct_remdups_id: "distinct xs ==> remdups xs = xs"
  2874 by (induct xs, auto)
  2875 
  2876 lemma remdups_id_iff_distinct [simp]: "remdups xs = xs \<longleftrightarrow> distinct xs"
  2877 by (metis distinct_remdups distinct_remdups_id)
  2878 
  2879 lemma finite_distinct_list: "finite A \<Longrightarrow> EX xs. set xs = A & distinct xs"
  2880 by (metis distinct_remdups finite_list set_remdups)
  2881 
  2882 lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
  2883 by (induct x, auto)
  2884 
  2885 lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
  2886 by (induct x, auto)
  2887 
  2888 lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
  2889 by (induct xs) auto
  2890 
  2891 lemma length_remdups_eq[iff]:
  2892   "(length (remdups xs) = length xs) = (remdups xs = xs)"
  2893 apply(induct xs)
  2894  apply auto
  2895 apply(subgoal_tac "length (remdups xs) <= length xs")
  2896  apply arith
  2897 apply(rule length_remdups_leq)
  2898 done
  2899 
  2900 lemma remdups_filter: "remdups(filter P xs) = filter P (remdups xs)"
  2901 apply(induct xs)
  2902 apply auto
  2903 done
  2904 
  2905 lemma distinct_map:
  2906   "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
  2907 by (induct xs) auto
  2908 
  2909 
  2910 lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
  2911 by (induct xs) auto
  2912 
  2913 lemma distinct_upt[simp]: "distinct[i..<j]"
  2914 by (induct j) auto
  2915 
  2916 lemma distinct_upto[simp]: "distinct[i..j]"
  2917 apply(induct i j rule:upto.induct)
  2918 apply(subst upto.simps)
  2919 apply(simp)
  2920 done
  2921 
  2922 lemma distinct_take[simp]: "distinct xs \<Longrightarrow> distinct (take i xs)"
  2923 apply(induct xs arbitrary: i)
  2924  apply simp
  2925 apply (case_tac i)
  2926  apply simp_all
  2927 apply(blast dest:in_set_takeD)
  2928 done
  2929 
  2930 lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)"
  2931 apply(induct xs arbitrary: i)
  2932  apply simp
  2933 apply (case_tac i)
  2934  apply simp_all
  2935 done
  2936 
  2937 lemma distinct_list_update:
  2938 assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
  2939 shows "distinct (xs[i:=a])"
  2940 proof (cases "i < length xs")
  2941   case True
  2942   with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
  2943     apply (drule_tac id_take_nth_drop) by simp
  2944   with d True show ?thesis
  2945     apply (simp add: upd_conv_take_nth_drop)
  2946     apply (drule subst [OF id_take_nth_drop]) apply assumption
  2947     apply simp apply (cases "a = xs!i") apply simp by blast
  2948 next
  2949   case False with d show ?thesis by auto
  2950 qed
  2951 
  2952 lemma distinct_concat:
  2953   assumes "distinct xs"
  2954   and "\<And> ys. ys \<in> set xs \<Longrightarrow> distinct ys"
  2955   and "\<And> ys zs. \<lbrakk> ys \<in> set xs ; zs \<in> set xs ; ys \<noteq> zs \<rbrakk> \<Longrightarrow> set ys \<inter> set zs = {}"
  2956   shows "distinct (concat xs)"
  2957   using assms by (induct xs) auto
  2958 
  2959 text {* It is best to avoid this indexed version of distinct, but
  2960 sometimes it is useful. *}
  2961 
  2962 lemma distinct_conv_nth:
  2963 "distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
  2964 apply (induct xs, simp, simp)
  2965 apply (rule iffI, clarsimp)
  2966  apply (case_tac i)
  2967 apply (case_tac j, simp)
  2968 apply (simp add: set_conv_nth)
  2969  apply (case_tac j)
  2970 apply (clarsimp simp add: set_conv_nth, simp)
  2971 apply (rule conjI)
  2972 (*TOO SLOW
  2973 apply (metis Zero_neq_Suc gr0_conv_Suc in_set_conv_nth lessI less_trans_Suc nth_Cons' nth_Cons_Suc)
  2974 *)
  2975  apply (clarsimp simp add: set_conv_nth)
  2976  apply (erule_tac x = 0 in allE, simp)
  2977  apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
  2978 (*TOO SLOW
  2979 apply (metis Suc_Suc_eq lessI less_trans_Suc nth_Cons_Suc)
  2980 *)
  2981 apply (erule_tac x = "Suc i" in allE, simp)
  2982 apply (erule_tac x = "Suc j" in allE, simp)
  2983 done
  2984 
  2985 lemma nth_eq_iff_index_eq:
  2986  "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
  2987 by(auto simp: distinct_conv_nth)
  2988 
  2989 lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
  2990 by (induct xs) auto
  2991 
  2992 lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
  2993 proof (induct xs)
  2994   case Nil thus ?case by simp
  2995 next
  2996   case (Cons x xs)
  2997   show ?case
  2998   proof (cases "x \<in> set xs")
  2999     case False with Cons show ?thesis by simp
  3000   next
  3001     case True with Cons.prems
  3002     have "card (set xs) = Suc (length xs)"
  3003       by (simp add: card_insert_if split: split_if_asm)
  3004     moreover have "card (set xs) \<le> length xs" by (rule card_length)
  3005     ultimately have False by simp
  3006     thus ?thesis ..
  3007   qed
  3008 qed
  3009 
  3010 lemma distinct_length_filter: "distinct xs \<Longrightarrow> length (filter P xs) = card ({x. P x} Int set xs)"
  3011 by (induct xs) (auto)
  3012 
  3013 lemma not_distinct_decomp: "~ distinct ws ==> EX xs ys zs y. ws = xs@[y]@ys@[y]@zs"
  3014 apply (induct n == "length ws" arbitrary:ws) apply simp
  3015 apply(case_tac ws) apply simp
  3016 apply (simp split:split_if_asm)
  3017 apply (metis Cons_eq_appendI eq_Nil_appendI split_list)
  3018 done
  3019 
  3020 lemma not_distinct_conv_prefix:
  3021   defines "dec as xs y ys \<equiv> y \<in> set xs \<and> distinct xs \<and> as = xs @ y # ys"
  3022   shows "\<not>distinct as \<longleftrightarrow> (\<exists>xs y ys. dec as xs y ys)" (is "?L = ?R")
  3023 proof
  3024   assume "?L" then show "?R"
  3025   proof (induct "length as" arbitrary: as rule: less_induct)
  3026     case less
  3027     obtain xs ys zs y where decomp: "as = (xs @ y # ys) @ y # zs"
  3028       using not_distinct_decomp[OF less.prems] by auto
  3029     show ?case
  3030     proof (cases "distinct (xs @ y # ys)")
  3031       case True
  3032       with decomp have "dec as (xs @ y # ys) y zs" by (simp add: dec_def)
  3033       then show ?thesis by blast
  3034     next
  3035       case False
  3036       with less decomp obtain xs' y' ys' where "dec (xs @ y # ys) xs' y' ys'"
  3037         by atomize_elim auto
  3038       with decomp have "dec as xs' y' (ys' @ y # zs)" by (simp add: dec_def)
  3039       then show ?thesis by blast
  3040     qed
  3041   qed
  3042 qed (auto simp: dec_def)
  3043 
  3044 lemma length_remdups_concat:
  3045   "length (remdups (concat xss)) = card (\<Union>xs\<in>set xss. set xs)"
  3046   by (simp add: distinct_card [symmetric])
  3047 
  3048 lemma length_remdups_card_conv: "length(remdups xs) = card(set xs)"
  3049 proof -
  3050   have xs: "concat[xs] = xs" by simp
  3051   from length_remdups_concat[of "[xs]"] show ?thesis unfolding xs by simp
  3052 qed
  3053 
  3054 lemma remdups_remdups:
  3055   "remdups (remdups xs) = remdups xs"
  3056   by (induct xs) simp_all
  3057 
  3058 lemma distinct_butlast:
  3059   assumes "distinct xs"
  3060   shows "distinct (butlast xs)"
  3061 proof (cases "xs = []")
  3062   case False
  3063     from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  3064     with `distinct xs` show ?thesis by simp
  3065 qed (auto)
  3066 
  3067 lemma remdups_map_remdups:
  3068   "remdups (map f (remdups xs)) = remdups (map f xs)"
  3069   by (induct xs) simp_all
  3070 
  3071 lemma distinct_zipI1:
  3072   assumes "distinct xs"
  3073   shows "distinct (zip xs ys)"
  3074 proof (rule zip_obtain_same_length)
  3075   fix xs' :: "'a list" and ys' :: "'b list" and n
  3076   assume "length xs' = length ys'"
  3077   assume "xs' = take n xs"
  3078   with assms have "distinct xs'" by simp
  3079   with `length xs' = length ys'` show "distinct (zip xs' ys')"
  3080     by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
  3081 qed
  3082 
  3083 lemma distinct_zipI2:
  3084   assumes "distinct ys"
  3085   shows "distinct (zip xs ys)"
  3086 proof (rule zip_obtain_same_length)
  3087   fix xs' :: "'b list" and ys' :: "'a list" and n
  3088   assume "length xs' = length ys'"
  3089   assume "ys' = take n ys"
  3090   with assms have "distinct ys'" by simp
  3091   with `length xs' = length ys'` show "distinct (zip xs' ys')"
  3092     by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
  3093 qed
  3094 
  3095 (* The next two lemmas help Sledgehammer. *)
  3096 
  3097 lemma distinct_singleton: "distinct [x]" by simp
  3098 
  3099 lemma distinct_length_2_or_more:
  3100 "distinct (a # b # xs) \<longleftrightarrow> (a \<noteq> b \<and> distinct (a # xs) \<and> distinct (b # xs))"
  3101 by (metis distinct.simps(2) hd.simps hd_in_set list.simps(2) set_ConsD set_rev_mp set_subset_Cons)
  3102 
  3103 subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
  3104 
  3105 lemma (in monoid_add) listsum_simps [simp]:
  3106   "listsum [] = 0"
  3107   "listsum (x # xs) = x + listsum xs"
  3108   by (simp_all add: listsum_def)
  3109 
  3110 lemma (in monoid_add) listsum_append [simp]:
  3111   "listsum (xs @ ys) = listsum xs + listsum ys"
  3112   by (induct xs) (simp_all add: add.assoc)
  3113 
  3114 lemma (in comm_monoid_add) listsum_rev [simp]:
  3115   "listsum (rev xs) = listsum xs"
  3116   by (simp add: listsum_def foldr_def fold_rev fun_eq_iff add_ac)
  3117 
  3118 lemma (in monoid_add) fold_plus_listsum_rev:
  3119   "fold plus xs = plus (listsum (rev xs))"
  3120 proof
  3121   fix x
  3122   have "fold plus xs x = fold plus xs (x + 0)" by simp
  3123   also have "\<dots> = fold plus (x # xs) 0" by simp
  3124   also have "\<dots> = foldr plus (rev xs @ [x]) 0" by (simp add: foldr_def)
  3125   also have "\<dots> = listsum (rev xs @ [x])" by (simp add: listsum_def)
  3126   also have "\<dots> = listsum (rev xs) + listsum [x]" by simp
  3127   finally show "fold plus xs x = listsum (rev xs) + x" by simp
  3128 qed
  3129 
  3130 lemma (in semigroup_add) foldl_assoc:
  3131   "foldl plus (x + y) zs = x + foldl plus y zs"
  3132   by (simp add: foldl_def fold_commute_apply [symmetric] fun_eq_iff add_assoc)
  3133 
  3134 lemma (in ab_semigroup_add) foldr_conv_foldl:
  3135   "foldr plus xs a = foldl plus a xs"
  3136   by (simp add: foldl_def foldr_fold fun_eq_iff add_ac)
  3137 
  3138 text {*
  3139   Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
  3140   difficult to use because it requires an additional transitivity step.
  3141 *}
  3142 
  3143 lemma start_le_sum:
  3144   fixes m n :: nat
  3145   shows "m \<le> n \<Longrightarrow> m \<le> foldl plus n ns"
  3146   by (simp add: foldl_def add_commute fold_plus_listsum_rev)
  3147 
  3148 lemma elem_le_sum:
  3149   fixes m n :: nat 
  3150   shows "n \<in> set ns \<Longrightarrow> n \<le> foldl plus 0 ns"
  3151   by (force intro: start_le_sum simp add: in_set_conv_decomp)
  3152 
  3153 lemma sum_eq_0_conv [iff]:
  3154   fixes m :: nat
  3155   shows "foldl plus m ns = 0 \<longleftrightarrow> m = 0 \<and> (\<forall>n \<in> set ns. n = 0)"
  3156   by (induct ns arbitrary: m) auto
  3157 
  3158 text{* Some syntactic sugar for summing a function over a list: *}
  3159 
  3160 syntax
  3161   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
  3162 syntax (xsymbols)
  3163   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
  3164 syntax (HTML output)
  3165   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
  3166 
  3167 translations -- {* Beware of argument permutation! *}
  3168   "SUM x<-xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  3169   "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  3170 
  3171 lemma (in comm_monoid_add) listsum_map_remove1:
  3172   "x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))"
  3173   by (induct xs) (auto simp add: ac_simps)
  3174 
  3175 lemma (in monoid_add) list_size_conv_listsum:
  3176   "list_size f xs = listsum (map f xs) + size xs"
  3177   by (induct xs) auto
  3178 
  3179 lemma (in monoid_add) length_concat:
  3180   "length (concat xss) = listsum (map length xss)"
  3181   by (induct xss) simp_all
  3182 
  3183 lemma (in monoid_add) listsum_map_filter:
  3184   assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
  3185   shows "listsum (map f (filter P xs)) = listsum (map f xs)"
  3186   using assms by (induct xs) auto
  3187 
  3188 lemma (in monoid_add) distinct_listsum_conv_Setsum:
  3189   "distinct xs \<Longrightarrow> listsum xs = Setsum (set xs)"
  3190   by (induct xs) simp_all
  3191 
  3192 lemma listsum_eq_0_nat_iff_nat [simp]:
  3193   "listsum ns = (0::nat) \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
  3194   by (simp add: listsum_def foldr_conv_foldl)
  3195 
  3196 lemma elem_le_listsum_nat:
  3197   "k < size ns \<Longrightarrow> ns ! k \<le> listsum (ns::nat list)"
  3198 apply(induct ns arbitrary: k)
  3199  apply simp
  3200 apply(fastforce simp add:nth_Cons split: nat.split)
  3201 done
  3202 
  3203 lemma listsum_update_nat:
  3204   "k<size ns \<Longrightarrow> listsum (ns[k := (n::nat)]) = listsum ns + n - ns ! k"
  3205 apply(induct ns arbitrary:k)
  3206  apply (auto split:nat.split)
  3207 apply(drule elem_le_listsum_nat)
  3208 apply arith
  3209 done
  3210 
  3211 lemma (in monoid_add) listsum_triv:
  3212   "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
  3213   by (induct xs) (simp_all add: left_distrib)
  3214 
  3215 lemma (in monoid_add) listsum_0 [simp]:
  3216   "(\<Sum>x\<leftarrow>xs. 0) = 0"
  3217   by (induct xs) (simp_all add: left_distrib)
  3218 
  3219 text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
  3220 lemma (in ab_group_add) uminus_listsum_map:
  3221   "- listsum (map f xs) = listsum (map (uminus \<circ> f) xs)"
  3222   by (induct xs) simp_all
  3223 
  3224 lemma (in comm_monoid_add) listsum_addf:
  3225   "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
  3226   by (induct xs) (simp_all add: algebra_simps)
  3227 
  3228 lemma (in ab_group_add) listsum_subtractf:
  3229   "(\<Sum>x\<leftarrow>xs. f x - g x) = listsum (map f xs) - listsum (map g xs)"
  3230   by (induct xs) (simp_all add: algebra_simps)
  3231 
  3232 lemma (in semiring_0) listsum_const_mult:
  3233   "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
  3234   by (induct xs) (simp_all add: algebra_simps)
  3235 
  3236 lemma (in semiring_0) listsum_mult_const:
  3237   "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
  3238   by (induct xs) (simp_all add: algebra_simps)
  3239 
  3240 lemma (in ordered_ab_group_add_abs) listsum_abs:
  3241   "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
  3242   by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])
  3243 
  3244 lemma listsum_mono:
  3245   fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}"
  3246   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)"
  3247   by (induct xs) (simp, simp add: add_mono)
  3248 
  3249 lemma (in monoid_add) listsum_distinct_conv_setsum_set:
  3250   "distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)"
  3251   by (induct xs) simp_all
  3252 
  3253 lemma (in monoid_add) interv_listsum_conv_setsum_set_nat:
  3254   "listsum (map f [m..<n]) = setsum f (set [m..<n])"
  3255   by (simp add: listsum_distinct_conv_setsum_set)
  3256 
  3257 lemma (in monoid_add) interv_listsum_conv_setsum_set_int:
  3258   "listsum (map f [k..l]) = setsum f (set [k..l])"
  3259   by (simp add: listsum_distinct_conv_setsum_set)
  3260 
  3261 text {* General equivalence between @{const listsum} and @{const setsum} *}
  3262 lemma (in monoid_add) listsum_setsum_nth:
  3263   "listsum xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
  3264   using interv_listsum_conv_setsum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth)
  3265 
  3266 
  3267 subsubsection {* @{const insert} *}
  3268 
  3269 lemma in_set_insert [simp]:
  3270   "x \<in> set xs \<Longrightarrow> List.insert x xs = xs"
  3271   by (simp add: List.insert_def)
  3272 
  3273 lemma not_in_set_insert [simp]:
  3274   "x \<notin> set xs \<Longrightarrow> List.insert x xs = x # xs"
  3275   by (simp add: List.insert_def)
  3276 
  3277 lemma insert_Nil [simp]:
  3278   "List.insert x [] = [x]"
  3279   by simp
  3280 
  3281 lemma set_insert [simp]:
  3282   "set (List.insert x xs) = insert x (set xs)"
  3283   by (auto simp add: List.insert_def)
  3284 
  3285 lemma distinct_insert [simp]:
  3286   "distinct xs \<Longrightarrow> distinct (List.insert x xs)"
  3287   by (simp add: List.insert_def)
  3288 
  3289 lemma insert_remdups:
  3290   "List.insert x (remdups xs) = remdups (List.insert x xs)"
  3291   by (simp add: List.insert_def)
  3292 
  3293 
  3294 subsubsection {* @{text remove1} *}
  3295 
  3296 lemma remove1_append:
  3297   "remove1 x (xs @ ys) =
  3298   (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
  3299 by (induct xs) auto
  3300 
  3301 lemma remove1_commute: "remove1 x (remove1 y zs) = remove1 y (remove1 x zs)"
  3302 by (induct zs) auto
  3303 
  3304 lemma in_set_remove1[simp]:
  3305   "a \<noteq> b \<Longrightarrow> a : set(remove1 b xs) = (a : set xs)"
  3306 apply (induct xs)
  3307 apply auto
  3308 done
  3309 
  3310 lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
  3311 apply(induct xs)
  3312  apply simp
  3313 apply simp
  3314 apply blast
  3315 done
  3316 
  3317 lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
  3318 apply(induct xs)
  3319  apply simp
  3320 apply simp
  3321 apply blast
  3322 done
  3323 
  3324 lemma length_remove1:
  3325   "length(remove1 x xs) = (if x : set xs then length xs - 1 else length xs)"
  3326 apply (induct xs)
  3327  apply (auto dest!:length_pos_if_in_set)
  3328 done
  3329 
  3330 lemma remove1_filter_not[simp]:
  3331   "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
  3332 by(induct xs) auto
  3333 
  3334 lemma filter_remove1:
  3335   "filter Q (remove1 x xs) = remove1 x (filter Q xs)"
  3336 by (induct xs) auto
  3337 
  3338 lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
  3339 apply(insert set_remove1_subset)
  3340 apply fast
  3341 done
  3342 
  3343 lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
  3344 by (induct xs) simp_all
  3345 
  3346 lemma remove1_remdups:
  3347   "distinct xs \<Longrightarrow> remove1 x (remdups xs) = remdups (remove1 x xs)"
  3348   by (induct xs) simp_all
  3349 
  3350 lemma remove1_idem:
  3351   assumes "x \<notin> set xs"
  3352   shows "remove1 x xs = xs"
  3353   using assms by (induct xs) simp_all
  3354 
  3355 
  3356 subsubsection {* @{text removeAll} *}
  3357 
  3358 lemma removeAll_filter_not_eq:
  3359   "removeAll x = filter (\<lambda>y. x \<noteq> y)"
  3360 proof
  3361   fix xs
  3362   show "removeAll x xs = filter (\<lambda>y. x \<noteq> y) xs"
  3363     by (induct xs) auto
  3364 qed
  3365 
  3366 lemma removeAll_append[simp]:
  3367   "removeAll x (xs @ ys) = removeAll x xs @ removeAll x ys"
  3368 by (induct xs) auto
  3369 
  3370 lemma set_removeAll[simp]: "set(removeAll x xs) = set xs - {x}"
  3371 by (induct xs) auto
  3372 
  3373 lemma removeAll_id[simp]: "x \<notin> set xs \<Longrightarrow> removeAll x xs = xs"
  3374 by (induct xs) auto
  3375 
  3376 (* Needs count:: 'a \<Rightarrow> 'a list \<Rightarrow> nat
  3377 lemma length_removeAll:
  3378   "length(removeAll x xs) = length xs - count x xs"
  3379 *)
  3380 
  3381 lemma removeAll_filter_not[simp]:
  3382   "\<not> P x \<Longrightarrow> removeAll x (filter P xs) = filter P xs"
  3383 by(induct xs) auto
  3384 
  3385 lemma distinct_removeAll:
  3386   "distinct xs \<Longrightarrow> distinct (removeAll x xs)"
  3387   by (simp add: removeAll_filter_not_eq)
  3388 
  3389 lemma distinct_remove1_removeAll:
  3390   "distinct xs ==> remove1 x xs = removeAll x xs"
  3391 by (induct xs) simp_all
  3392 
  3393 lemma map_removeAll_inj_on: "inj_on f (insert x (set xs)) \<Longrightarrow>
  3394   map f (removeAll x xs) = removeAll (f x) (map f xs)"
  3395 by (induct xs) (simp_all add:inj_on_def)
  3396 
  3397 lemma map_removeAll_inj: "inj f \<Longrightarrow>
  3398   map f (removeAll x xs) = removeAll (f x) (map f xs)"
  3399 by(metis map_removeAll_inj_on subset_inj_on subset_UNIV)
  3400 
  3401 
  3402 subsubsection {* @{text replicate} *}
  3403 
  3404 lemma length_replicate [simp]: "length (replicate n x) = n"
  3405 by (induct n) auto
  3406 
  3407 lemma Ex_list_of_length: "\<exists>xs. length xs = n"
  3408 by (rule exI[of _ "replicate n undefined"]) simp
  3409 
  3410 lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
  3411 by (induct n) auto
  3412 
  3413 lemma map_replicate_const:
  3414   "map (\<lambda> x. k) lst = replicate (length lst) k"
  3415   by (induct lst) auto
  3416 
  3417 lemma replicate_app_Cons_same:
  3418 "(replicate n x) @ (x # xs) = x # replicate n x @ xs"
  3419 by (induct n) auto
  3420 
  3421 lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
  3422 apply (induct n, simp)
  3423 apply (simp add: replicate_app_Cons_same)
  3424 done
  3425 
  3426 lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
  3427 by (induct n) auto
  3428 
  3429 text{* Courtesy of Matthias Daum: *}
  3430 lemma append_replicate_commute:
  3431   "replicate n x @ replicate k x = replicate k x @ replicate n x"
  3432 apply (simp add: replicate_add [THEN sym])
  3433 apply (simp add: add_commute)
  3434 done
  3435 
  3436 text{* Courtesy of Andreas Lochbihler: *}
  3437 lemma filter_replicate:
  3438   "filter P (replicate n x) = (if P x then replicate n x else [])"
  3439 by(induct n) auto
  3440 
  3441 lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
  3442 by (induct n) auto
  3443 
  3444 lemma tl_replicate [simp]: "tl (replicate n x) = replicate (n - 1) x"
  3445 by (induct n) auto
  3446 
  3447 lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
  3448 by (atomize (full), induct n) auto
  3449 
  3450 lemma nth_replicate[simp]: "i < n ==> (replicate n x)!i = x"
  3451 apply (induct n arbitrary: i, simp)
  3452 apply (simp add: nth_Cons split: nat.split)
  3453 done
  3454 
  3455 text{* Courtesy of Matthias Daum (2 lemmas): *}
  3456 lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
  3457 apply (case_tac "k \<le> i")
  3458  apply  (simp add: min_def)
  3459 apply (drule not_leE)
  3460 apply (simp add: min_def)
  3461 apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
  3462  apply  simp
  3463 apply (simp add: replicate_add [symmetric])
  3464 done
  3465 
  3466 lemma drop_replicate[simp]: "drop i (replicate k x) = replicate (k-i) x"
  3467 apply (induct k arbitrary: i)
  3468  apply simp
  3469 apply clarsimp
  3470 apply (case_tac i)
  3471  apply simp
  3472 apply clarsimp
  3473 done
  3474 
  3475 
  3476 lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
  3477 by (induct n) auto
  3478 
  3479 lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
  3480 by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
  3481 
  3482 lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
  3483 by auto
  3484 
  3485 lemma in_set_replicate[simp]: "(x : set (replicate n y)) = (x = y & n \<noteq> 0)"
  3486 by (simp add: set_replicate_conv_if)
  3487 
  3488 lemma Ball_set_replicate[simp]:
  3489   "(ALL x : set(replicate n a). P x) = (P a | n=0)"
  3490 by(simp add: set_replicate_conv_if)
  3491 
  3492 lemma Bex_set_replicate[simp]:
  3493   "(EX x : set(replicate n a). P x) = (P a & n\<noteq>0)"
  3494 by(simp add: set_replicate_conv_if)
  3495 
  3496 lemma replicate_append_same:
  3497   "replicate i x @ [x] = x # replicate i x"
  3498   by (induct i) simp_all
  3499 
  3500 lemma map_replicate_trivial:
  3501   "map (\<lambda>i. x) [0..<i] = replicate i x"
  3502   by (induct i) (simp_all add: replicate_append_same)
  3503 
  3504 lemma concat_replicate_trivial[simp]:
  3505   "concat (replicate i []) = []"
  3506   by (induct i) (auto simp add: map_replicate_const)
  3507 
  3508 lemma replicate_empty[simp]: "(replicate n x = []) \<longleftrightarrow> n=0"
  3509 by (induct n) auto
  3510 
  3511 lemma empty_replicate[simp]: "([] = replicate n x) \<longleftrightarrow> n=0"
  3512 by (induct n) auto
  3513 
  3514 lemma replicate_eq_replicate[simp]:
  3515   "(replicate m x = replicate n y) \<longleftrightarrow> (m=n & (m\<noteq>0 \<longrightarrow> x=y))"
  3516 apply(induct m arbitrary: n)
  3517  apply simp
  3518 apply(induct_tac n)
  3519 apply auto
  3520 done
  3521 
  3522 lemma replicate_length_filter:
  3523   "replicate (length (filter (\<lambda>y. x = y) xs)) x = filter (\<lambda>y. x = y) xs"
  3524   by (induct xs) auto
  3525 
  3526 lemma comm_append_are_replicate:
  3527   fixes xs ys :: "'a list"
  3528   assumes "xs \<noteq> []" "ys \<noteq> []"
  3529   assumes "xs @ ys = ys @ xs"
  3530   shows "\<exists>m n zs. concat (replicate m zs) = xs \<and> concat (replicate n zs) = ys"
  3531   using assms
  3532 proof (induct "length (xs @ ys)" arbitrary: xs ys rule: less_induct)
  3533   case less
  3534 
  3535   def xs' \<equiv> "if (length xs \<le> length ys) then xs else ys"
  3536     and ys' \<equiv> "if (length xs \<le> length ys) then ys else xs"
  3537   then have
  3538     prems': "length xs' \<le> length ys'"
  3539             "xs' @ ys' = ys' @ xs'"
  3540       and "xs' \<noteq> []"
  3541       and len: "length (xs @ ys) = length (xs' @ ys')"
  3542     using less by (auto intro: less.hyps)
  3543 
  3544   from prems'
  3545   obtain ws where "ys' = xs' @ ws"
  3546     by (auto simp: append_eq_append_conv2)
  3547 
  3548   have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ys'"
  3549   proof (cases "ws = []")
  3550     case True
  3551     then have "concat (replicate 1 xs') = xs'"
  3552       and "concat (replicate 1 xs') = ys'"
  3553       using `ys' = xs' @ ws` by auto
  3554     then show ?thesis by blast
  3555   next
  3556     case False
  3557     from `ys' = xs' @ ws` and `xs' @ ys' = ys' @ xs'`
  3558     have "xs' @ ws = ws @ xs'" by simp
  3559     then have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ws"
  3560       using False and `xs' \<noteq> []` and `ys' = xs' @ ws` and len
  3561       by (intro less.hyps) auto
  3562     then obtain m n zs where "concat (replicate m zs) = xs'"
  3563       and "concat (replicate n zs) = ws" by blast
  3564     moreover
  3565     then have "concat (replicate (m + n) zs) = ys'"
  3566       using `ys' = xs' @ ws`
  3567       by (simp add: replicate_add)
  3568     ultimately
  3569     show ?thesis by blast
  3570   qed
  3571   then show ?case
  3572     using xs'_def ys'_def by metis
  3573 qed
  3574 
  3575 lemma comm_append_is_replicate:
  3576   fixes xs ys :: "'a list"
  3577   assumes "xs \<noteq> []" "ys \<noteq> []"
  3578   assumes "xs @ ys = ys @ xs"
  3579   shows "\<exists>n zs. n > 1 \<and> concat (replicate n zs) = xs @ ys"
  3580 
  3581 proof -
  3582   obtain m n zs where "concat (replicate m zs) = xs"
  3583     and "concat (replicate n zs) = ys"
  3584     using assms by (metis comm_append_are_replicate)
  3585   then have "m + n > 1" and "concat (replicate (m+n) zs) = xs @ ys"
  3586     using `xs \<noteq> []` and `ys \<noteq> []`
  3587     by (auto simp: replicate_add)
  3588   then show ?thesis by blast
  3589 qed
  3590 
  3591 
  3592 subsubsection{*@{text rotate1} and @{text rotate}*}
  3593 
  3594 lemma rotate0[simp]: "rotate 0 = id"
  3595 by(simp add:rotate_def)
  3596 
  3597 lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
  3598 by(simp add:rotate_def)
  3599 
  3600 lemma rotate_add:
  3601   "rotate (m+n) = rotate m o rotate n"
  3602 by(simp add:rotate_def funpow_add)
  3603 
  3604 lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
  3605 by(simp add:rotate_add)
  3606 
  3607 lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
  3608 by(simp add:rotate_def funpow_swap1)
  3609 
  3610 lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
  3611 by(cases xs) simp_all
  3612 
  3613 lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
  3614 apply(induct n)
  3615  apply simp
  3616 apply (simp add:rotate_def)
  3617 done
  3618 
  3619 lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
  3620 by (cases xs) simp_all
  3621 
  3622 lemma rotate_drop_take:
  3623   "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
  3624 apply(induct n)
  3625  apply simp
  3626 apply(simp add:rotate_def)
  3627 apply(cases "xs = []")
  3628  apply (simp)
  3629 apply(case_tac "n mod length xs = 0")
  3630  apply(simp add:mod_Suc)
  3631  apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
  3632 apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
  3633                 take_hd_drop linorder_not_le)
  3634 done
  3635 
  3636 lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
  3637 by(simp add:rotate_drop_take)
  3638 
  3639 lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
  3640 by(simp add:rotate_drop_take)
  3641 
  3642 lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
  3643 by (cases xs) simp_all
  3644 
  3645 lemma length_rotate[simp]: "length(rotate n xs) = length xs"
  3646 by (induct n arbitrary: xs) (simp_all add:rotate_def)
  3647 
  3648 lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
  3649 by (cases xs) auto
  3650 
  3651 lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
  3652 by (induct n) (simp_all add:rotate_def)
  3653 
  3654 lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
  3655 by(simp add:rotate_drop_take take_map drop_map)
  3656 
  3657 lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
  3658 by (cases xs) auto
  3659 
  3660 lemma set_rotate[simp]: "set(rotate n xs) = set xs"
  3661 by (induct n) (simp_all add:rotate_def)
  3662 
  3663 lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
  3664 by (cases xs) auto
  3665 
  3666 lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
  3667 by (induct n) (simp_all add:rotate_def)
  3668 
  3669 lemma rotate_rev:
  3670   "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
  3671 apply(simp add:rotate_drop_take rev_drop rev_take)
  3672 apply(cases "length xs = 0")
  3673  apply simp
  3674 apply(cases "n mod length xs = 0")
  3675  apply simp
  3676 apply(simp add:rotate_drop_take rev_drop rev_take)
  3677 done
  3678 
  3679 lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
  3680 apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
  3681 apply(subgoal_tac "length xs \<noteq> 0")
  3682  prefer 2 apply simp
  3683 using mod_less_divisor[of "length xs" n] by arith
  3684 
  3685 
  3686 subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
  3687 
  3688 lemma sublist_empty [simp]: "sublist xs {} = []"
  3689 by (auto simp add: sublist_def)
  3690 
  3691 lemma sublist_nil [simp]: "sublist [] A = []"
  3692 by (auto simp add: sublist_def)
  3693 
  3694 lemma length_sublist:
  3695   "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
  3696 by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
  3697 
  3698 lemma sublist_shift_lemma_Suc:
  3699   "map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
  3700    map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
  3701 apply(induct xs arbitrary: "is")
  3702  apply simp
  3703 apply (case_tac "is")
  3704  apply simp
  3705 apply simp
  3706 done
  3707 
  3708 lemma sublist_shift_lemma:
  3709      "map fst [p<-zip xs [i..<i + length xs] . snd p : A] =
  3710       map fst [p<-zip xs [0..<length xs] . snd p + i : A]"
  3711 by (induct xs rule: rev_induct) (simp_all add: add_commute)
  3712 
  3713 lemma sublist_append:
  3714      "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
  3715 apply (unfold sublist_def)
  3716 apply (induct l' rule: rev_induct, simp)
  3717 apply (simp add: upt_add_eq_append[of 0] sublist_shift_lemma)
  3718 apply (simp add: add_commute)
  3719 done
  3720 
  3721 lemma sublist_Cons:
  3722 "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
  3723 apply (induct l rule: rev_induct)
  3724  apply (simp add: sublist_def)
  3725 apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
  3726 done
  3727 
  3728 lemma set_sublist: "set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
  3729 apply(induct xs arbitrary: I)
  3730 apply(auto simp: sublist_Cons nth_Cons split:nat.split dest!: gr0_implies_Suc)
  3731 done
  3732 
  3733 lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
  3734 by(auto simp add:set_sublist)
  3735 
  3736 lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
  3737 by(auto simp add:set_sublist)
  3738 
  3739 lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
  3740 by(auto simp add:set_sublist)
  3741 
  3742 lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
  3743 by (simp add: sublist_Cons)
  3744 
  3745 
  3746 lemma distinct_sublistI[simp]: "distinct xs \<Longrightarrow> distinct(sublist xs I)"
  3747 apply(induct xs arbitrary: I)
  3748  apply simp
  3749 apply(auto simp add:sublist_Cons)
  3750 done
  3751 
  3752 
  3753 lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
  3754 apply (induct l rule: rev_induct, simp)
  3755 apply (simp split: nat_diff_split add: sublist_append)
  3756 done
  3757 
  3758 lemma filter_in_sublist:
  3759  "distinct xs \<Longrightarrow> filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
  3760 proof (induct xs arbitrary: s)
  3761   case Nil thus ?case by simp
  3762 next
  3763   case (Cons a xs)
  3764   moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
  3765   ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
  3766 qed
  3767 
  3768 
  3769 subsubsection {* @{const splice} *}
  3770 
  3771 lemma splice_Nil2 [simp, code]: "splice xs [] = xs"
  3772 by (cases xs) simp_all
  3773 
  3774 declare splice.simps(1,3)[code]
  3775 declare splice.simps(2)[simp del]
  3776 
  3777 lemma length_splice[simp]: "length(splice xs ys) = length xs + length ys"
  3778 by (induct xs ys rule: splice.induct) auto
  3779 
  3780 
  3781 subsubsection {* Transpose *}
  3782 
  3783 function transpose where
  3784 "transpose []             = []" |
  3785 "transpose ([]     # xss) = transpose xss" |
  3786 "transpose ((x#xs) # xss) =
  3787   (x # [h. (h#t) \<leftarrow> xss]) # transpose (xs # [t. (h#t) \<leftarrow> xss])"
  3788 by pat_completeness auto
  3789 
  3790 lemma transpose_aux_filter_head:
  3791   "concat (map (list_case [] (\<lambda>h t. [h])) xss) =
  3792   map (\<lambda>xs. hd xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  3793   by (induct xss) (auto split: list.split)
  3794 
  3795 lemma transpose_aux_filter_tail:
  3796   "concat (map (list_case [] (\<lambda>h t. [t])) xss) =
  3797   map (\<lambda>xs. tl xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  3798   by (induct xss) (auto split: list.split)
  3799 
  3800 lemma transpose_aux_max:
  3801   "max (Suc (length xs)) (foldr (\<lambda>xs. max (length xs)) xss 0) =
  3802   Suc (max (length xs) (foldr (\<lambda>x. max (length x - Suc 0)) [ys\<leftarrow>xss . ys\<noteq>[]] 0))"
  3803   (is "max _ ?foldB = Suc (max _ ?foldA)")
  3804 proof (cases "[ys\<leftarrow>xss . ys\<noteq>[]] = []")
  3805   case True
  3806   hence "foldr (\<lambda>xs. max (length xs)) xss 0 = 0"
  3807   proof (induct xss)
  3808     case (Cons x xs)
  3809     moreover hence "x = []" by (cases x) auto
  3810     ultimately show ?case by auto
  3811   qed simp
  3812   thus ?thesis using True by simp
  3813 next
  3814   case False
  3815 
  3816   have foldA: "?foldA = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0 - 1"
  3817     by (induct xss) auto
  3818   have foldB: "?foldB = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0"
  3819     by (induct xss) auto
  3820 
  3821   have "0 < ?foldB"
  3822   proof -
  3823     from False
  3824     obtain z zs where zs: "[ys\<leftarrow>xss . ys \<noteq> []] = z#zs" by (auto simp: neq_Nil_conv)
  3825     hence "z \<in> set ([ys\<leftarrow>xss . ys \<noteq> []])" by auto
  3826     hence "z \<noteq> []" by auto
  3827     thus ?thesis
  3828       unfolding foldB zs
  3829       by (auto simp: max_def intro: less_le_trans)
  3830   qed
  3831   thus ?thesis
  3832     unfolding foldA foldB max_Suc_Suc[symmetric]
  3833     by simp
  3834 qed
  3835 
  3836 termination transpose
  3837   by (relation "measure (\<lambda>xs. foldr (\<lambda>xs. max (length xs)) xs 0 + length xs)")
  3838      (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max less_Suc_eq_le)
  3839 
  3840 lemma transpose_empty: "(transpose xs = []) \<longleftrightarrow> (\<forall>x \<in> set xs. x = [])"
  3841   by (induct rule: transpose.induct) simp_all
  3842 
  3843 lemma length_transpose:
  3844   fixes xs :: "'a list list"
  3845   shows "length (transpose xs) = foldr (\<lambda>xs. max (length xs)) xs 0"
  3846   by (induct rule: transpose.induct)
  3847     (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max
  3848                 max_Suc_Suc[symmetric] simp del: max_Suc_Suc)
  3849 
  3850 lemma nth_transpose:
  3851   fixes xs :: "'a list list"
  3852   assumes "i < length (transpose xs)"
  3853   shows "transpose xs ! i = map (\<lambda>xs. xs ! i) [ys \<leftarrow> xs. i < length ys]"
  3854 using assms proof (induct arbitrary: i rule: transpose.induct)
  3855   case (3 x xs xss)
  3856   def XS == "(x # xs) # xss"
  3857   hence [simp]: "XS \<noteq> []" by auto
  3858   thus ?case
  3859   proof (cases i)
  3860     case 0
  3861     thus ?thesis by (simp add: transpose_aux_filter_head hd_conv_nth)
  3862   next
  3863     case (Suc j)
  3864     have *: "\<And>xss. xs # map tl xss = map tl ((x#xs)#xss)" by simp
  3865     have **: "\<And>xss. (x#xs) # filter (\<lambda>ys. ys \<noteq> []) xss = filter (\<lambda>ys. ys \<noteq> []) ((x#xs)#xss)" by simp
  3866     { fix x have "Suc j < length x \<longleftrightarrow> x \<noteq> [] \<and> j < length x - Suc 0"
  3867       by (cases x) simp_all
  3868     } note *** = this
  3869 
  3870     have j_less: "j < length (transpose (xs # concat (map (list_case [] (\<lambda>h t. [t])) xss)))"
  3871       using "3.prems" by (simp add: transpose_aux_filter_tail length_transpose Suc)
  3872 
  3873     show ?thesis
  3874       unfolding transpose.simps `i = Suc j` nth_Cons_Suc "3.hyps"[OF j_less]
  3875       apply (auto simp: transpose_aux_filter_tail filter_map comp_def length_transpose * ** *** XS_def[symmetric])
  3876       apply (rule_tac y=x in list.exhaust)
  3877       by auto
  3878   qed
  3879 qed simp_all
  3880 
  3881 lemma transpose_map_map:
  3882   "transpose (map (map f) xs) = map (map f) (transpose xs)"
  3883 proof (rule nth_equalityI, safe)
  3884   have [simp]: "length (transpose (map (map f) xs)) = length (transpose xs)"
  3885     by (simp add: length_transpose foldr_map comp_def)
  3886   show "length (transpose (map (map f) xs)) = length (map (map f) (transpose xs))" by simp
  3887 
  3888   fix i assume "i < length (transpose (map (map f) xs))"
  3889   thus "transpose (map (map f) xs) ! i = map (map f) (transpose xs) ! i"
  3890     by (simp add: nth_transpose filter_map comp_def)
  3891 qed
  3892 
  3893 
  3894 subsubsection {* (In)finiteness *}
  3895 
  3896 lemma finite_maxlen:
  3897   "finite (M::'a list set) ==> EX n. ALL s:M. size s < n"
  3898 proof (induct rule: finite.induct)
  3899   case emptyI show ?case by simp
  3900 next
  3901   case (insertI M xs)
  3902   then obtain n where "\<forall>s\<in>M. length s < n" by blast
  3903   hence "ALL s:insert xs M. size s < max n (size xs) + 1" by auto
  3904   thus ?case ..
  3905 qed
  3906 
  3907 lemma lists_length_Suc_eq:
  3908   "{xs. set xs \<subseteq> A \<and> length xs = Suc n} =
  3909     (\<lambda>(xs, n). n#xs) ` ({xs. set xs \<subseteq> A \<and> length xs = n} \<times> A)"
  3910   by (auto simp: length_Suc_conv)
  3911 
  3912 lemma
  3913   assumes "finite A"
  3914   shows finite_lists_length_eq: "finite {xs. set xs \<subseteq> A \<and> length xs = n}"
  3915   and card_lists_length_eq: "card {xs. set xs \<subseteq> A \<and> length xs = n} = (card A)^n"
  3916   using `finite A`
  3917   by (induct n)
  3918      (auto simp: card_image inj_split_Cons lists_length_Suc_eq cong: conj_cong)
  3919 
  3920 lemma finite_lists_length_le:
  3921   assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs \<le> n}"
  3922  (is "finite ?S")
  3923 proof-
  3924   have "?S = (\<Union>n\<in>{0..n}. {xs. set xs \<subseteq> A \<and> length xs = n})" by auto
  3925   thus ?thesis by (auto intro: finite_lists_length_eq[OF `finite A`])
  3926 qed
  3927 
  3928 lemma card_lists_length_le:
  3929   assumes "finite A" shows "card {xs. set xs \<subseteq> A \<and> length xs \<le> n} = (\<Sum>i\<le>n. card A^i)"
  3930 proof -
  3931   have "(\<Sum>i\<le>n. card A^i) = card (\<Union>i\<le>n. {xs. set xs \<subseteq> A \<and> length xs = i})"
  3932     using `finite A`
  3933     by (subst card_UN_disjoint)
  3934        (auto simp add: card_lists_length_eq finite_lists_length_eq)
  3935   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}"
  3936     by auto
  3937   finally show ?thesis by simp
  3938 qed
  3939 
  3940 lemma card_lists_distinct_length_eq:
  3941   assumes "k < card A"
  3942   shows "card {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A} = \<Prod>{card A - k + 1 .. card A}"
  3943 using assms
  3944 proof (induct k)
  3945   case 0
  3946   then have "{xs. length xs = 0 \<and> distinct xs \<and> set xs \<subseteq> A} = {[]}" by auto
  3947   then show ?case by simp
  3948 next
  3949   case (Suc k)
  3950   let "?k_list" = "\<lambda>k xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A"
  3951   have inj_Cons: "\<And>A. inj_on (\<lambda>(xs, n). n # xs) A"  by (rule inj_onI) auto
  3952 
  3953   from Suc have "k < card A" by simp
  3954   moreover have "finite A" using assms by (simp add: card_ge_0_finite)
  3955   moreover have "finite {xs. ?k_list k xs}"
  3956     using finite_lists_length_eq[OF `finite A`, of k]
  3957     by - (rule finite_subset, auto)
  3958   moreover have "\<And>i j. i \<noteq> j \<longrightarrow> {i} \<times> (A - set i) \<inter> {j} \<times> (A - set j) = {}"
  3959     by auto
  3960   moreover have "\<And>i. i \<in>Collect (?k_list k) \<Longrightarrow> card (A - set i) = card A - k"
  3961     by (simp add: card_Diff_subset distinct_card)
  3962   moreover have "{xs. ?k_list (Suc k) xs} =
  3963       (\<lambda>(xs, n). n#xs) ` \<Union>(\<lambda>xs. {xs} \<times> (A - set xs)) ` {xs. ?k_list k xs}"
  3964     by (auto simp: length_Suc_conv)
  3965   moreover
  3966   have "Suc (card A - Suc k) = card A - k" using Suc.prems by simp
  3967   then have "(card A - k) * \<Prod>{Suc (card A - k)..card A} = \<Prod>{Suc (card A - Suc k)..card A}"
  3968     by (subst setprod_insert[symmetric]) (simp add: atLeastAtMost_insertL)+
  3969   ultimately show ?case
  3970     by (simp add: card_image inj_Cons card_UN_disjoint Suc.hyps algebra_simps)
  3971 qed
  3972 
  3973 lemma infinite_UNIV_listI: "~ finite(UNIV::'a list set)"
  3974 apply(rule notI)
  3975 apply(drule finite_maxlen)
  3976 apply (metis UNIV_I length_replicate less_not_refl)
  3977 done
  3978 
  3979 
  3980 subsection {* Sorting *}
  3981 
  3982 text{* Currently it is not shown that @{const sort} returns a
  3983 permutation of its input because the nicest proof is via multisets,
  3984 which are not yet available. Alternatively one could define a function
  3985 that counts the number of occurrences of an element in a list and use
  3986 that instead of multisets to state the correctness property. *}
  3987 
  3988 context linorder
  3989 begin
  3990 
  3991 lemma length_insort [simp]:
  3992   "length (insort_key f x xs) = Suc (length xs)"
  3993   by (induct xs) simp_all
  3994 
  3995 lemma insort_key_left_comm:
  3996   assumes "f x \<noteq> f y"
  3997   shows "insort_key f y (insort_key f x xs) = insort_key f x (insort_key f y xs)"
  3998   by (induct xs) (auto simp add: assms dest: antisym)
  3999 
  4000 lemma insort_left_comm:
  4001   "insort x (insort y xs) = insort y (insort x xs)"
  4002   by (cases "x = y") (auto intro: insort_key_left_comm)
  4003 
  4004 lemma comp_fun_commute_insort:
  4005   "comp_fun_commute insort"
  4006 proof
  4007 qed (simp add: insort_left_comm fun_eq_iff)
  4008 
  4009 lemma sort_key_simps [simp]:
  4010   "sort_key f [] = []"
  4011   "sort_key f (x#xs) = insort_key f x (sort_key f xs)"
  4012   by (simp_all add: sort_key_def)
  4013 
  4014 lemma (in linorder) sort_key_conv_fold:
  4015   assumes "inj_on f (set xs)"
  4016   shows "sort_key f xs = fold (insort_key f) xs []"
  4017 proof -
  4018   have "fold (insort_key f) (rev xs) = fold (insort_key f) xs"
  4019   proof (rule fold_rev, rule ext)
  4020     fix zs
  4021     fix x y
  4022     assume "x \<in> set xs" "y \<in> set xs"
  4023     with assms have *: "f y = f x \<Longrightarrow> y = x" by (auto dest: inj_onD)
  4024     have **: "x = y \<longleftrightarrow> y = x" by auto
  4025     show "(insort_key f y \<circ> insort_key f x) zs = (insort_key f x \<circ> insort_key f y) zs"
  4026       by (induct zs) (auto intro: * simp add: **)
  4027   qed
  4028   then show ?thesis by (simp add: sort_key_def foldr_def)
  4029 qed
  4030 
  4031 lemma (in linorder) sort_conv_fold:
  4032   "sort xs = fold insort xs []"
  4033   by (rule sort_key_conv_fold) simp
  4034 
  4035 lemma length_sort[simp]: "length (sort_key f xs) = length xs"
  4036 by (induct xs, auto)
  4037 
  4038 lemma sorted_Cons: "sorted (x#xs) = (sorted xs & (ALL y:set xs. x <= y))"
  4039 apply(induct xs arbitrary: x) apply simp
  4040 by simp (blast intro: order_trans)
  4041 
  4042 lemma sorted_tl:
  4043   "sorted xs \<Longrightarrow> sorted (tl xs)"
  4044   by (cases xs) (simp_all add: sorted_Cons)
  4045 
  4046 lemma sorted_append:
  4047   "sorted (xs@ys) = (sorted xs & sorted ys & (\<forall>x \<in> set xs. \<forall>y \<in> set ys. x\<le>y))"
  4048 by (induct xs) (auto simp add:sorted_Cons)
  4049 
  4050 lemma sorted_nth_mono:
  4051   "sorted xs \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!i \<le> xs!j"
  4052 by (induct xs arbitrary: i j) (auto simp:nth_Cons' sorted_Cons)
  4053 
  4054 lemma sorted_rev_nth_mono:
  4055   "sorted (rev xs) \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!j \<le> xs!i"
  4056 using sorted_nth_mono[ of "rev xs" "length xs - j - 1" "length xs - i - 1"]
  4057       rev_nth[of "length xs - i - 1" "xs"] rev_nth[of "length xs - j - 1" "xs"]
  4058 by auto
  4059 
  4060 lemma sorted_nth_monoI:
  4061   "(\<And> i j. \<lbrakk> i \<le> j ; j < length xs \<rbrakk> \<Longrightarrow> xs ! i \<le> xs ! j) \<Longrightarrow> sorted xs"
  4062 proof (induct xs)
  4063   case (Cons x xs)
  4064   have "sorted xs"
  4065   proof (rule Cons.hyps)
  4066     fix i j assume "i \<le> j" and "j < length xs"
  4067     with Cons.prems[of "Suc i" "Suc j"]
  4068     show "xs ! i \<le> xs ! j" by auto
  4069   qed
  4070   moreover
  4071   {
  4072     fix y assume "y \<in> set xs"
  4073     then obtain j where "j < length xs" and "xs ! j = y"
  4074       unfolding in_set_conv_nth by blast
  4075     with Cons.prems[of 0 "Suc j"]
  4076     have "x \<le> y"
  4077       by auto
  4078   }
  4079   ultimately
  4080   show ?case
  4081     unfolding sorted_Cons by auto
  4082 qed simp
  4083 
  4084 lemma sorted_equals_nth_mono:
  4085   "sorted xs = (\<forall>j < length xs. \<forall>i \<le> j. xs ! i \<le> xs ! j)"
  4086 by (auto intro: sorted_nth_monoI sorted_nth_mono)
  4087 
  4088 lemma set_insort: "set(insort_key f x xs) = insert x (set xs)"
  4089 by (induct xs) auto
  4090 
  4091 lemma set_sort[simp]: "set(sort_key f xs) = set xs"
  4092 by (induct xs) (simp_all add:set_insort)
  4093 
  4094 lemma distinct_insort: "distinct (insort_key f x xs) = (x \<notin> set xs \<and> distinct xs)"
  4095 by(induct xs)(auto simp:set_insort)
  4096 
  4097 lemma distinct_sort[simp]: "distinct (sort_key f xs) = distinct xs"
  4098   by (induct xs) (simp_all add: distinct_insort)
  4099 
  4100 lemma sorted_insort_key: "sorted (map f (insort_key f x xs)) = sorted (map f xs)"
  4101   by (induct xs) (auto simp:sorted_Cons set_insort)
  4102 
  4103 lemma sorted_insort: "sorted (insort x xs) = sorted xs"
  4104   using sorted_insort_key [where f="\<lambda>x. x"] by simp
  4105 
  4106 theorem sorted_sort_key [simp]: "sorted (map f (sort_key f xs))"
  4107   by (induct xs) (auto simp:sorted_insort_key)
  4108 
  4109 theorem sorted_sort [simp]: "sorted (sort xs)"
  4110   using sorted_sort_key [where f="\<lambda>x. x"] by simp
  4111 
  4112 lemma sorted_butlast:
  4113   assumes "xs \<noteq> []" and "sorted xs"
  4114   shows "sorted (butlast xs)"
  4115 proof -
  4116   from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  4117   with `sorted xs` show ?thesis by (simp add: sorted_append)
  4118 qed
  4119   
  4120 lemma insort_not_Nil [simp]:
  4121   "insort_key f a xs \<noteq> []"
  4122   by (induct xs) simp_all
  4123 
  4124 lemma insort_is_Cons: "\<forall>x\<in>set xs. f a \<le> f x \<Longrightarrow> insort_key f a xs = a # xs"
  4125 by (cases xs) auto
  4126 
  4127 lemma sorted_sort_id: "sorted xs \<Longrightarrow> sort xs = xs"
  4128   by (induct xs) (auto simp add: sorted_Cons insort_is_Cons)
  4129 
  4130 lemma sorted_map_remove1:
  4131   "sorted (map f xs) \<Longrightarrow> sorted (map f (remove1 x xs))"
  4132   by (induct xs) (auto simp add: sorted_Cons)
  4133 
  4134 lemma sorted_remove1: "sorted xs \<Longrightarrow> sorted (remove1 a xs)"
  4135   using sorted_map_remove1 [of "\<lambda>x. x"] by simp
  4136 
  4137 lemma insort_key_remove1:
  4138   assumes "a \<in> set xs" and "sorted (map f xs)" and "hd (filter (\<lambda>x. f a = f x) xs) = a"
  4139   shows "insort_key f a (remove1 a xs) = xs"
  4140 using assms proof (induct xs)
  4141   case (Cons x xs)
  4142   then show ?case
  4143   proof (cases "x = a")
  4144     case False
  4145     then have "f x \<noteq> f a" using Cons.prems by auto
  4146     then have "f x < f a" using Cons.prems by (auto simp: sorted_Cons)
  4147     with `f x \<noteq> f a` show ?thesis using Cons by (auto simp: sorted_Cons insort_is_Cons)
  4148   qed (auto simp: sorted_Cons insort_is_Cons)
  4149 qed simp
  4150 
  4151 lemma insort_remove1:
  4152   assumes "a \<in> set xs" and "sorted xs"
  4153   shows "insort a (remove1 a xs) = xs"
  4154 proof (rule insort_key_remove1)
  4155   from `a \<in> set xs` show "a \<in> set xs" .
  4156   from `sorted xs` show "sorted (map (\<lambda>x. x) xs)" by simp
  4157   from `a \<in> set xs` have "a \<in> set (filter (op = a) xs)" by auto
  4158   then have "set (filter (op = a) xs) \<noteq> {}" by auto
  4159   then have "filter (op = a) xs \<noteq> []" by (auto simp only: set_empty)
  4160   then have "length (filter (op = a) xs) > 0" by simp
  4161   then obtain n where n: "Suc n = length (filter (op = a) xs)"
  4162     by (cases "length (filter (op = a) xs)") simp_all
  4163   moreover have "replicate (Suc n) a = a # replicate n a"
  4164     by simp
  4165   ultimately show "hd (filter (op = a) xs) = a" by (simp add: replicate_length_filter)
  4166 qed
  4167 
  4168 lemma sorted_remdups[simp]:
  4169   "sorted l \<Longrightarrow> sorted (remdups l)"
  4170 by (induct l) (auto simp: sorted_Cons)
  4171 
  4172 lemma sorted_distinct_set_unique:
  4173 assumes "sorted xs" "distinct xs" "sorted ys" "distinct ys" "set xs = set ys"
  4174 shows "xs = ys"
  4175 proof -
  4176   from assms have 1: "length xs = length ys" by (auto dest!: distinct_card)
  4177   from assms show ?thesis
  4178   proof(induct rule:list_induct2[OF 1])
  4179     case 1 show ?case by simp
  4180   next
  4181     case 2 thus ?case by (simp add:sorted_Cons)
  4182        (metis Diff_insert_absorb antisym insertE insert_iff)
  4183   qed
  4184 qed
  4185 
  4186 lemma map_sorted_distinct_set_unique:
  4187   assumes "inj_on f (set xs \<union> set ys)"
  4188   assumes "sorted (map f xs)" "distinct (map f xs)"
  4189     "sorted (map f ys)" "distinct (map f ys)"
  4190   assumes "set xs = set ys"
  4191   shows "xs = ys"
  4192 proof -
  4193   from assms have "map f xs = map f ys"
  4194     by (simp add: sorted_distinct_set_unique)
  4195   moreover with `inj_on f (set xs \<union> set ys)` show "xs = ys"
  4196     by (blast intro: map_inj_on)
  4197 qed
  4198 
  4199 lemma finite_sorted_distinct_unique:
  4200 shows "finite A \<Longrightarrow> EX! xs. set xs = A & sorted xs & distinct xs"
  4201 apply(drule finite_distinct_list)
  4202 apply clarify
  4203 apply(rule_tac a="sort xs" in ex1I)
  4204 apply (auto simp: sorted_distinct_set_unique)
  4205 done
  4206 
  4207 lemma
  4208   assumes "sorted xs"
  4209   shows sorted_take: "sorted (take n xs)"
  4210   and sorted_drop: "sorted (drop n xs)"
  4211 proof -
  4212   from assms have "sorted (take n xs @ drop n xs)" by simp
  4213   then show "sorted (take n xs)" and "sorted (drop n xs)"
  4214     unfolding sorted_append by simp_all
  4215 qed
  4216 
  4217 lemma sorted_dropWhile: "sorted xs \<Longrightarrow> sorted (dropWhile P xs)"
  4218   by (auto dest: sorted_drop simp add: dropWhile_eq_drop)
  4219 
  4220 lemma sorted_takeWhile: "sorted xs \<Longrightarrow> sorted (takeWhile P xs)"
  4221   by (subst takeWhile_eq_take) (auto dest: sorted_take)
  4222 
  4223 lemma sorted_filter:
  4224   "sorted (map f xs) \<Longrightarrow> sorted (map f (filter P xs))"
  4225   by (induct xs) (simp_all add: sorted_Cons)
  4226 
  4227 lemma foldr_max_sorted:
  4228   assumes "sorted (rev xs)"
  4229   shows "foldr max xs y = (if xs = [] then y else max (xs ! 0) y)"
  4230 using assms proof (induct xs)
  4231   case (Cons x xs)
  4232   moreover hence "sorted (rev xs)" using sorted_append by auto
  4233   ultimately show ?case
  4234     by (cases xs, auto simp add: sorted_append max_def)
  4235 qed simp
  4236 
  4237 lemma filter_equals_takeWhile_sorted_rev:
  4238   assumes sorted: "sorted (rev (map f xs))"
  4239   shows "[x \<leftarrow> xs. t < f x] = takeWhile (\<lambda> x. t < f x) xs"
  4240     (is "filter ?P xs = ?tW")
  4241 proof (rule takeWhile_eq_filter[symmetric])
  4242   let "?dW" = "dropWhile ?P xs"
  4243   fix x assume "x \<in> set ?dW"
  4244   then obtain i where i: "i < length ?dW" and nth_i: "x = ?dW ! i"
  4245     unfolding in_set_conv_nth by auto
  4246   hence "length ?tW + i < length (?tW @ ?dW)"
  4247     unfolding length_append by simp
  4248   hence i': "length (map f ?tW) + i < length (map f xs)" by simp
  4249   have "(map f ?tW @ map f ?dW) ! (length (map f ?tW) + i) \<le>
  4250         (map f ?tW @ map f ?dW) ! (length (map f ?tW) + 0)"
  4251     using sorted_rev_nth_mono[OF sorted _ i', of "length ?tW"]
  4252     unfolding map_append[symmetric] by simp
  4253   hence "f x \<le> f (?dW ! 0)"
  4254     unfolding nth_append_length_plus nth_i
  4255     using i preorder_class.le_less_trans[OF le0 i] by simp
  4256   also have "... \<le> t"
  4257     using hd_dropWhile[of "?P" xs] le0[THEN preorder_class.le_less_trans, OF i]
  4258     using hd_conv_nth[of "?dW"] by simp
  4259   finally show "\<not> t < f x" by simp
  4260 qed
  4261 
  4262 lemma insort_insert_key_triv:
  4263   "f x \<in> f ` set xs \<Longrightarrow> insort_insert_key f x xs = xs"
  4264   by (simp add: insort_insert_key_def)
  4265 
  4266 lemma insort_insert_triv:
  4267   "x \<in> set xs \<Longrightarrow> insort_insert x xs = xs"
  4268   using insort_insert_key_triv [of "\<lambda>x. x"] by simp
  4269 
  4270 lemma insort_insert_insort_key:
  4271   "f x \<notin> f ` set xs \<Longrightarrow> insort_insert_key f x xs = insort_key f x xs"
  4272   by (simp add: insort_insert_key_def)
  4273 
  4274 lemma insort_insert_insort:
  4275   "x \<notin> set xs \<Longrightarrow> insort_insert x xs = insort x xs"
  4276   using insort_insert_insort_key [of "\<lambda>x. x"] by simp
  4277 
  4278 lemma set_insort_insert:
  4279   "set (insort_insert x xs) = insert x (set xs)"
  4280   by (auto simp add: insort_insert_key_def set_insort)
  4281 
  4282 lemma distinct_insort_insert:
  4283   assumes "distinct xs"
  4284   shows "distinct (insort_insert_key f x xs)"
  4285   using assms by (induct xs) (auto simp add: insort_insert_key_def set_insort)
  4286 
  4287 lemma sorted_insort_insert_key:
  4288   assumes "sorted (map f xs)"
  4289   shows "sorted (map f (insort_insert_key f x xs))"
  4290   using assms by (simp add: insort_insert_key_def sorted_insort_key)
  4291 
  4292 lemma sorted_insort_insert:
  4293   assumes "sorted xs"
  4294   shows "sorted (insort_insert x xs)"
  4295   using assms sorted_insort_insert_key [of "\<lambda>x. x"] by simp
  4296 
  4297 lemma filter_insort_triv:
  4298   "\<not> P x \<Longrightarrow> filter P (insort_key f x xs) = filter P xs"
  4299   by (induct xs) simp_all
  4300 
  4301 lemma filter_insort:
  4302   "sorted (map f xs) \<Longrightarrow> P x \<Longrightarrow> filter P (insort_key f x xs) = insort_key f x (filter P xs)"
  4303   using assms by (induct xs)
  4304     (auto simp add: sorted_Cons, subst insort_is_Cons, auto)
  4305 
  4306 lemma filter_sort:
  4307   "filter P (sort_key f xs) = sort_key f (filter P xs)"
  4308   by (induct xs) (simp_all add: filter_insort_triv filter_insort)
  4309 
  4310 lemma sorted_map_same:
  4311   "sorted (map f [x\<leftarrow>xs. f x = g xs])"
  4312 proof (induct xs arbitrary: g)
  4313   case Nil then show ?case by simp
  4314 next
  4315   case (Cons x xs)
  4316   then have "sorted (map f [y\<leftarrow>xs . f y = (\<lambda>xs. f x) xs])" .
  4317   moreover from Cons have "sorted (map f [y\<leftarrow>xs . f y = (g \<circ> Cons x) xs])" .
  4318   ultimately show ?case by (simp_all add: sorted_Cons)
  4319 qed
  4320 
  4321 lemma sorted_same:
  4322   "sorted [x\<leftarrow>xs. x = g xs]"
  4323   using sorted_map_same [of "\<lambda>x. x"] by simp
  4324 
  4325 lemma remove1_insort [simp]:
  4326   "remove1 x (insort x xs) = xs"
  4327   by (induct xs) simp_all
  4328 
  4329 end
  4330 
  4331 lemma sorted_upt[simp]: "sorted[i..<j]"
  4332 by (induct j) (simp_all add:sorted_append)
  4333 
  4334 lemma sorted_upto[simp]: "sorted[i..j]"
  4335 apply(induct i j rule:upto.induct)
  4336 apply(subst upto.simps)
  4337 apply(simp add:sorted_Cons)
  4338 done
  4339 
  4340 
  4341 subsubsection {* @{const transpose} on sorted lists *}
  4342 
  4343 lemma sorted_transpose[simp]:
  4344   shows "sorted (rev (map length (transpose xs)))"
  4345   by (auto simp: sorted_equals_nth_mono rev_nth nth_transpose
  4346     length_filter_conv_card intro: card_mono)
  4347 
  4348 lemma transpose_max_length:
  4349   "foldr (\<lambda>xs. max (length xs)) (transpose xs) 0 = length [x \<leftarrow> xs. x \<noteq> []]"
  4350   (is "?L = ?R")
  4351 proof (cases "transpose xs = []")
  4352   case False
  4353   have "?L = foldr max (map length (transpose xs)) 0"
  4354     by (simp add: foldr_map comp_def)
  4355   also have "... = length (transpose xs ! 0)"
  4356     using False sorted_transpose by (simp add: foldr_max_sorted)
  4357   finally show ?thesis
  4358     using False by (simp add: nth_transpose)
  4359 next
  4360   case True
  4361   hence "[x \<leftarrow> xs. x \<noteq> []] = []"
  4362     by (auto intro!: filter_False simp: transpose_empty)
  4363   thus ?thesis by (simp add: transpose_empty True)
  4364 qed
  4365 
  4366 lemma length_transpose_sorted:
  4367   fixes xs :: "'a list list"
  4368   assumes sorted: "sorted (rev (map length xs))"
  4369   shows "length (transpose xs) = (if xs = [] then 0 else length (xs ! 0))"
  4370 proof (cases "xs = []")
  4371   case False
  4372   thus ?thesis
  4373     using foldr_max_sorted[OF sorted] False
  4374     unfolding length_transpose foldr_map comp_def
  4375     by simp
  4376 qed simp
  4377 
  4378 lemma nth_nth_transpose_sorted[simp]:
  4379   fixes xs :: "'a list list"
  4380   assumes sorted: "sorted (rev (map length xs))"
  4381   and i: "i < length (transpose xs)"
  4382   and j: "j < length [ys \<leftarrow> xs. i < length ys]"
  4383   shows "transpose xs ! i ! j = xs ! j  ! i"
  4384   using j filter_equals_takeWhile_sorted_rev[OF sorted, of i]
  4385     nth_transpose[OF i] nth_map[OF j]
  4386   by (simp add: takeWhile_nth)
  4387 
  4388 lemma transpose_column_length:
  4389   fixes xs :: "'a list list"
  4390   assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  4391   shows "length (filter (\<lambda>ys. i < length ys) (transpose xs)) = length (xs ! i)"
  4392 proof -
  4393   have "xs \<noteq> []" using `i < length xs` by auto
  4394   note filter_equals_takeWhile_sorted_rev[OF sorted, simp]
  4395   { fix j assume "j \<le> i"
  4396     note sorted_rev_nth_mono[OF sorted, of j i, simplified, OF this `i < length xs`]
  4397   } note sortedE = this[consumes 1]
  4398 
  4399   have "{j. j < length (transpose xs) \<and> i < length (transpose xs ! j)}
  4400     = {..< length (xs ! i)}"
  4401   proof safe
  4402     fix j
  4403     assume "j < length (transpose xs)" and "i < length (transpose xs ! j)"
  4404     with this(2) nth_transpose[OF this(1)]
  4405     have "i < length (takeWhile (\<lambda>ys. j < length ys) xs)" by simp
  4406     from nth_mem[OF this] takeWhile_nth[OF this]
  4407     show "j < length (xs ! i)" by (auto dest: set_takeWhileD)
  4408   next
  4409     fix j assume "j < length (xs ! i)"
  4410     thus "j < length (transpose xs)"
  4411       using foldr_max_sorted[OF sorted] `xs \<noteq> []` sortedE[OF le0]
  4412       by (auto simp: length_transpose comp_def foldr_map)
  4413 
  4414     have "Suc i \<le> length (takeWhile (\<lambda>ys. j < length ys) xs)"
  4415       using `i < length xs` `j < length (xs ! i)` less_Suc_eq_le
  4416       by (auto intro!: length_takeWhile_less_P_nth dest!: sortedE)
  4417     with nth_transpose[OF `j < length (transpose xs)`]
  4418     show "i < length (transpose xs ! j)" by simp
  4419   qed
  4420   thus ?thesis by (simp add: length_filter_conv_card)
  4421 qed
  4422 
  4423 lemma transpose_column:
  4424   fixes xs :: "'a list list"
  4425   assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  4426   shows "map (\<lambda>ys. ys ! i) (filter (\<lambda>ys. i < length ys) (transpose xs))
  4427     = xs ! i" (is "?R = _")
  4428 proof (rule nth_equalityI, safe)
  4429   show length: "length ?R = length (xs ! i)"
  4430     using transpose_column_length[OF assms] by simp
  4431 
  4432   fix j assume j: "j < length ?R"
  4433   note * = less_le_trans[OF this, unfolded length_map, OF length_filter_le]
  4434   from j have j_less: "j < length (xs ! i)" using length by simp
  4435   have i_less_tW: "Suc i \<le> length (takeWhile (\<lambda>ys. Suc j \<le> length ys) xs)"
  4436   proof (rule length_takeWhile_less_P_nth)
  4437     show "Suc i \<le> length xs" using `i < length xs` by simp
  4438     fix k assume "k < Suc i"
  4439     hence "k \<le> i" by auto
  4440     with sorted_rev_nth_mono[OF sorted this] `i < length xs`
  4441     have "length (xs ! i) \<le> length (xs ! k)" by simp
  4442     thus "Suc j \<le> length (xs ! k)" using j_less by simp
  4443   qed
  4444   have i_less_filter: "i < length [ys\<leftarrow>xs . j < length ys]"
  4445     unfolding filter_equals_takeWhile_sorted_rev[OF sorted, of j]
  4446     using i_less_tW by (simp_all add: Suc_le_eq)
  4447   from j show "?R ! j = xs ! i ! j"
  4448     unfolding filter_equals_takeWhile_sorted_rev[OF sorted_transpose, of i]
  4449     by (simp add: takeWhile_nth nth_nth_transpose_sorted[OF sorted * i_less_filter])
  4450 qed
  4451 
  4452 lemma transpose_transpose:
  4453   fixes xs :: "'a list list"
  4454   assumes sorted: "sorted (rev (map length xs))"
  4455   shows "transpose (transpose xs) = takeWhile (\<lambda>x. x \<noteq> []) xs" (is "?L = ?R")
  4456 proof -
  4457   have len: "length ?L = length ?R"
  4458     unfolding length_transpose transpose_max_length
  4459     using filter_equals_takeWhile_sorted_rev[OF sorted, of 0]
  4460     by simp
  4461 
  4462   { fix i assume "i < length ?R"
  4463     with less_le_trans[OF _ length_takeWhile_le[of _ xs]]
  4464     have "i < length xs" by simp
  4465   } note * = this
  4466   show ?thesis
  4467     by (rule nth_equalityI)
  4468        (simp_all add: len nth_transpose transpose_column[OF sorted] * takeWhile_nth)
  4469 qed
  4470 
  4471 theorem transpose_rectangle:
  4472   assumes "xs = [] \<Longrightarrow> n = 0"
  4473   assumes rect: "\<And> i. i < length xs \<Longrightarrow> length (xs ! i) = n"
  4474   shows "transpose xs = map (\<lambda> i. map (\<lambda> j. xs ! j ! i) [0..<length xs]) [0..<n]"
  4475     (is "?trans = ?map")
  4476 proof (rule nth_equalityI)
  4477   have "sorted (rev (map length xs))"
  4478     by (auto simp: rev_nth rect intro!: sorted_nth_monoI)
  4479   from foldr_max_sorted[OF this] assms
  4480   show len: "length ?trans = length ?map"
  4481     by (simp_all add: length_transpose foldr_map comp_def)
  4482   moreover
  4483   { fix i assume "i < n" hence "[ys\<leftarrow>xs . i < length ys] = xs"
  4484       using rect by (auto simp: in_set_conv_nth intro!: filter_True) }
  4485   ultimately show "\<forall>i < length ?trans. ?trans ! i = ?map ! i"
  4486     by (auto simp: nth_transpose intro: nth_equalityI)
  4487 qed
  4488 
  4489 
  4490 subsubsection {* @{text sorted_list_of_set} *}
  4491 
  4492 text{* This function maps (finite) linearly ordered sets to sorted
  4493 lists. Warning: in most cases it is not a good idea to convert from
  4494 sets to lists but one should convert in the other direction (via
  4495 @{const set}). *}
  4496 
  4497 context linorder
  4498 begin
  4499 
  4500 definition sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where
  4501   "sorted_list_of_set = Finite_Set.fold insort []"
  4502 
  4503 lemma sorted_list_of_set_empty [simp]:
  4504   "sorted_list_of_set {} = []"
  4505   by (simp add: sorted_list_of_set_def)
  4506 
  4507 lemma sorted_list_of_set_insert [simp]:
  4508   assumes "finite A"
  4509   shows "sorted_list_of_set (insert x A) = insort x (sorted_list_of_set (A - {x}))"
  4510 proof -
  4511   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  4512   with assms show ?thesis by (simp add: sorted_list_of_set_def fold_insert_remove)
  4513 qed
  4514 
  4515 lemma sorted_list_of_set [simp]:
  4516   "finite A \<Longrightarrow> set (sorted_list_of_set A) = A \<and> sorted (sorted_list_of_set A) 
  4517     \<and> distinct (sorted_list_of_set A)"
  4518   by (induct A rule: finite_induct) (simp_all add: set_insort sorted_insort distinct_insort)
  4519 
  4520 lemma sorted_list_of_set_sort_remdups:
  4521   "sorted_list_of_set (set xs) = sort (remdups xs)"
  4522 proof -
  4523   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  4524   show ?thesis by (simp add: sorted_list_of_set_def sort_conv_fold fold_set_fold_remdups)
  4525 qed
  4526 
  4527 lemma sorted_list_of_set_remove:
  4528   assumes "finite A"
  4529   shows "sorted_list_of_set (A - {x}) = remove1 x (sorted_list_of_set A)"
  4530 proof (cases "x \<in> A")
  4531   case False with assms have "x \<notin> set (sorted_list_of_set A)" by simp
  4532   with False show ?thesis by (simp add: remove1_idem)
  4533 next
  4534   case True then obtain B where A: "A = insert x B" by (rule Set.set_insert)
  4535   with assms show ?thesis by simp
  4536 qed
  4537 
  4538 end
  4539 
  4540 lemma sorted_list_of_set_range [simp]:
  4541   "sorted_list_of_set {m..<n} = [m..<n]"
  4542   by (rule sorted_distinct_set_unique) simp_all
  4543 
  4544 
  4545 subsubsection {* @{text lists}: the list-forming operator over sets *}
  4546 
  4547 inductive_set
  4548   lists :: "'a set => 'a list set"
  4549   for A :: "'a set"
  4550 where
  4551     Nil [intro!, simp]: "[]: lists A"
  4552   | Cons [intro!, simp, no_atp]: "[| a: A; l: lists A|] ==> a#l : lists A"
  4553 
  4554 inductive_cases listsE [elim!,no_atp]: "x#l : lists A"
  4555 inductive_cases listspE [elim!,no_atp]: "listsp A (x # l)"
  4556 
  4557 inductive_simps listsp_simps[code]:
  4558   "listsp A []"
  4559   "listsp A (x # xs)"
  4560 
  4561 lemma listsp_mono [mono]: "A \<le> B ==> listsp A \<le> listsp B"
  4562 by (rule predicate1I, erule listsp.induct, (blast dest: predicate1D)+)
  4563 
  4564 lemmas lists_mono = listsp_mono [to_set]
  4565 
  4566 lemma listsp_infI:
  4567   assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
  4568 by induct blast+
  4569 
  4570 lemmas lists_IntI = listsp_infI [to_set]
  4571 
  4572 lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
  4573 proof (rule mono_inf [where f=listsp, THEN order_antisym])
  4574   show "mono listsp" by (simp add: mono_def listsp_mono)
  4575   show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro!: listsp_infI predicate1I)
  4576 qed
  4577 
  4578 lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_def inf_bool_def]
  4579 
  4580 lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set]
  4581 
  4582 lemma Cons_in_lists_iff[simp]: "x#xs : lists A \<longleftrightarrow> x:A \<and> xs : lists A"
  4583 by auto
  4584 
  4585 lemma append_in_listsp_conv [iff]:
  4586      "(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
  4587 by (induct xs) auto
  4588 
  4589 lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]
  4590 
  4591 lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
  4592 -- {* eliminate @{text listsp} in favour of @{text set} *}
  4593 by (induct xs) auto
  4594 
  4595 lemmas in_lists_conv_set [code_unfold] = in_listsp_conv_set [to_set]
  4596 
  4597 lemma in_listspD [dest!,no_atp]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
  4598 by (rule in_listsp_conv_set [THEN iffD1])
  4599 
  4600 lemmas in_listsD [dest!,no_atp] = in_listspD [to_set]
  4601 
  4602 lemma in_listspI [intro!,no_atp]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
  4603 by (rule in_listsp_conv_set [THEN iffD2])
  4604 
  4605 lemmas in_listsI [intro!,no_atp] = in_listspI [to_set]
  4606 
  4607 lemma lists_eq_set: "lists A = {xs. set xs <= A}"
  4608 by auto
  4609 
  4610 lemma lists_empty [simp]: "lists {} = {[]}"
  4611 by auto
  4612 
  4613 lemma lists_UNIV [simp]: "lists UNIV = UNIV"
  4614 by auto
  4615 
  4616 
  4617 subsubsection {* Inductive definition for membership *}
  4618 
  4619 inductive ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
  4620 where
  4621     elem:  "ListMem x (x # xs)"
  4622   | insert:  "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"
  4623 
  4624 lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
  4625 apply (rule iffI)
  4626  apply (induct set: ListMem)
  4627   apply auto
  4628 apply (induct xs)
  4629  apply (auto intro: ListMem.intros)
  4630 done
  4631 
  4632 
  4633 subsubsection {* Lists as Cartesian products *}
  4634 
  4635 text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
  4636 @{term A} and tail drawn from @{term Xs}.*}
  4637 
  4638 definition
  4639   set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set" where
  4640   "set_Cons A XS = {z. \<exists>x xs. z = x # xs \<and> x \<in> A \<and> xs \<in> XS}"
  4641 
  4642 lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
  4643 by (auto simp add: set_Cons_def)
  4644 
  4645 text{*Yields the set of lists, all of the same length as the argument and
  4646 with elements drawn from the corresponding element of the argument.*}
  4647 
  4648 primrec
  4649   listset :: "'a set list \<Rightarrow> 'a list set" where
  4650      "listset [] = {[]}"
  4651   |  "listset (A # As) = set_Cons A (listset As)"
  4652 
  4653 
  4654 subsection {* Relations on Lists *}
  4655 
  4656 subsubsection {* Length Lexicographic Ordering *}
  4657 
  4658 text{*These orderings preserve well-foundedness: shorter lists 
  4659   precede longer lists. These ordering are not used in dictionaries.*}
  4660         
  4661 primrec -- {*The lexicographic ordering for lists of the specified length*}
  4662   lexn :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> ('a list \<times> 'a list) set" where
  4663     "lexn r 0 = {}"
  4664   | "lexn r (Suc n) = (map_pair (%(x, xs). x#xs) (%(x, xs). x#xs) ` (r <*lex*> lexn r n)) Int
  4665       {(xs, ys). length xs = Suc n \<and> length ys = Suc n}"
  4666 
  4667 definition
  4668   lex :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4669   "lex r = (\<Union>n. lexn r n)" -- {*Holds only between lists of the same length*}
  4670 
  4671 definition
  4672   lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" where
  4673   "lenlex r = inv_image (less_than <*lex*> lex r) (\<lambda>xs. (length xs, xs))"
  4674         -- {*Compares lists by their length and then lexicographically*}
  4675 
  4676 lemma wf_lexn: "wf r ==> wf (lexn r n)"
  4677 apply (induct n, simp, simp)
  4678 apply(rule wf_subset)
  4679  prefer 2 apply (rule Int_lower1)
  4680 apply(rule wf_map_pair_image)
  4681  prefer 2 apply (rule inj_onI, auto)
  4682 done
  4683 
  4684 lemma lexn_length:
  4685   "(xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
  4686 by (induct n arbitrary: xs ys) auto
  4687 
  4688 lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
  4689 apply (unfold lex_def)
  4690 apply (rule wf_UN)
  4691 apply (blast intro: wf_lexn, clarify)
  4692 apply (rename_tac m n)
  4693 apply (subgoal_tac "m \<noteq> n")
  4694  prefer 2 apply blast
  4695 apply (blast dest: lexn_length not_sym)
  4696 done
  4697 
  4698 lemma lexn_conv:
  4699   "lexn r n =
  4700     {(xs,ys). length xs = n \<and> length ys = n \<and>
  4701     (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
  4702 apply (induct n, simp)
  4703 apply (simp add: image_Collect lex_prod_def, safe, blast)
  4704  apply (rule_tac x = "ab # xys" in exI, simp)
  4705 apply (case_tac xys, simp_all, blast)
  4706 done
  4707 
  4708 lemma lex_conv:
  4709   "lex r =
  4710     {(xs,ys). length xs = length ys \<and>
  4711     (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
  4712 by (force simp add: lex_def lexn_conv)
  4713 
  4714 lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
  4715 by (unfold lenlex_def) blast
  4716 
  4717 lemma lenlex_conv:
  4718     "lenlex r = {(xs,ys). length xs < length ys |
  4719                  length xs = length ys \<and> (xs, ys) : lex r}"
  4720 by (simp add: lenlex_def Id_on_def lex_prod_def inv_image_def)
  4721 
  4722 lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
  4723 by (simp add: lex_conv)
  4724 
  4725 lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
  4726 by (simp add:lex_conv)
  4727 
  4728 lemma Cons_in_lex [simp]:
  4729     "((x # xs, y # ys) : lex r) =
  4730       ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
  4731 apply (simp add: lex_conv)
  4732 apply (rule iffI)
  4733  prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
  4734 apply (case_tac xys, simp, simp)
  4735 apply blast
  4736 done
  4737 
  4738 
  4739 subsubsection {* Lexicographic Ordering *}
  4740 
  4741 text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
  4742     This ordering does \emph{not} preserve well-foundedness.
  4743      Author: N. Voelker, March 2005. *} 
  4744 
  4745 definition
  4746   lexord :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4747   "lexord r = {(x,y ). \<exists> a v. y = x @ a # v \<or>
  4748             (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
  4749 
  4750 lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
  4751 by (unfold lexord_def, induct_tac y, auto) 
  4752 
  4753 lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
  4754 by (unfold lexord_def, induct_tac x, auto)
  4755 
  4756 lemma lexord_cons_cons[simp]:
  4757      "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
  4758   apply (unfold lexord_def, safe, simp_all)
  4759   apply (case_tac u, simp, simp)
  4760   apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
  4761   apply (erule_tac x="b # u" in allE)
  4762   by force
  4763 
  4764 lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
  4765 
  4766 lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
  4767 by (induct_tac x, auto)  
  4768 
  4769 lemma lexord_append_left_rightI:
  4770      "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
  4771 by (induct_tac u, auto)
  4772 
  4773 lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
  4774 by (induct x, auto)
  4775 
  4776 lemma lexord_append_leftD:
  4777      "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
  4778 by (erule rev_mp, induct_tac x, auto)
  4779 
  4780 lemma lexord_take_index_conv: 
  4781    "((x,y) : lexord r) = 
  4782     ((length x < length y \<and> take (length x) y = x) \<or> 
  4783      (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
  4784   apply (unfold lexord_def Let_def, clarsimp) 
  4785   apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
  4786   apply auto 
  4787   apply (rule_tac x="hd (drop (length x) y)" in exI)
  4788   apply (rule_tac x="tl (drop (length x) y)" in exI)
  4789   apply (erule subst, simp add: min_def) 
  4790   apply (rule_tac x ="length u" in exI, simp) 
  4791   apply (rule_tac x ="take i x" in exI) 
  4792   apply (rule_tac x ="x ! i" in exI) 
  4793   apply (rule_tac x ="y ! i" in exI, safe) 
  4794   apply (rule_tac x="drop (Suc i) x" in exI)
  4795   apply (drule sym, simp add: drop_Suc_conv_tl) 
  4796   apply (rule_tac x="drop (Suc i) y" in exI)
  4797   by (simp add: drop_Suc_conv_tl) 
  4798 
  4799 -- {* lexord is extension of partial ordering List.lex *} 
  4800 lemma lexord_lex: "(x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
  4801   apply (rule_tac x = y in spec) 
  4802   apply (induct_tac x, clarsimp) 
  4803   by (clarify, case_tac x, simp, force)
  4804 
  4805 lemma lexord_irreflexive: "ALL x. (x,x) \<notin> r \<Longrightarrow> (xs,xs) \<notin> lexord r"
  4806 by (induct xs) auto
  4807 
  4808 text{* By Ren\'e Thiemann: *}
  4809 lemma lexord_partial_trans: 
  4810   "(\<And>x y z. x \<in> set xs \<Longrightarrow> (x,y) \<in> r \<Longrightarrow> (y,z) \<in> r \<Longrightarrow> (x,z) \<in> r)
  4811    \<Longrightarrow>  (xs,ys) \<in> lexord r  \<Longrightarrow>  (ys,zs) \<in> lexord r \<Longrightarrow>  (xs,zs) \<in> lexord r"
  4812 proof (induct xs arbitrary: ys zs)
  4813   case Nil
  4814   from Nil(3) show ?case unfolding lexord_def by (cases zs, auto)
  4815 next
  4816   case (Cons x xs yys zzs)
  4817   from Cons(3) obtain y ys where yys: "yys = y # ys" unfolding lexord_def
  4818     by (cases yys, auto)
  4819   note Cons = Cons[unfolded yys]
  4820   from Cons(3) have one: "(x,y) \<in> r \<or> x = y \<and> (xs,ys) \<in> lexord r" by auto
  4821   from Cons(4) obtain z zs where zzs: "zzs = z # zs" unfolding lexord_def
  4822     by (cases zzs, auto)
  4823   note Cons = Cons[unfolded zzs]
  4824   from Cons(4) have two: "(y,z) \<in> r \<or> y = z \<and> (ys,zs) \<in> lexord r" by auto
  4825   {
  4826     assume "(xs,ys) \<in> lexord r" and "(ys,zs) \<in> lexord r"
  4827     from Cons(1)[OF _ this] Cons(2)
  4828     have "(xs,zs) \<in> lexord r" by auto
  4829   } note ind1 = this
  4830   {
  4831     assume "(x,y) \<in> r" and "(y,z) \<in> r"
  4832     from Cons(2)[OF _ this] have "(x,z) \<in> r" by auto
  4833   } note ind2 = this
  4834   from one two ind1 ind2
  4835   have "(x,z) \<in> r \<or> x = z \<and> (xs,zs) \<in> lexord r" by blast
  4836   thus ?case unfolding zzs by auto
  4837 qed
  4838 
  4839 lemma lexord_trans: 
  4840     "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
  4841 by(auto simp: trans_def intro:lexord_partial_trans)
  4842 
  4843 lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
  4844 by (rule transI, drule lexord_trans, blast) 
  4845 
  4846 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"
  4847   apply (rule_tac x = y in spec) 
  4848   apply (induct_tac x, rule allI) 
  4849   apply (case_tac x, simp, simp) 
  4850   apply (rule allI, case_tac x, simp, simp) 
  4851   by blast
  4852 
  4853 
  4854 subsubsection {* Lexicographic combination of measure functions *}
  4855 
  4856 text {* These are useful for termination proofs *}
  4857 
  4858 definition
  4859   "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"
  4860 
  4861 lemma wf_measures[simp]: "wf (measures fs)"
  4862 unfolding measures_def
  4863 by blast
  4864 
  4865 lemma in_measures[simp]: 
  4866   "(x, y) \<in> measures [] = False"
  4867   "(x, y) \<in> measures (f # fs)
  4868          = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"  
  4869 unfolding measures_def
  4870 by auto
  4871 
  4872 lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
  4873 by simp
  4874 
  4875 lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
  4876 by auto
  4877 
  4878 
  4879 subsubsection {* Lifting Relations to Lists: one element *}
  4880 
  4881 definition listrel1 :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4882 "listrel1 r = {(xs,ys).
  4883    \<exists>us z z' vs. xs = us @ z # vs \<and> (z,z') \<in> r \<and> ys = us @ z' # vs}"
  4884 
  4885 lemma listrel1I:
  4886   "\<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow>
  4887   (xs, ys) \<in> listrel1 r"
  4888 unfolding listrel1_def by auto
  4889 
  4890 lemma listrel1E:
  4891   "\<lbrakk> (xs, ys) \<in> listrel1 r;
  4892      !!x y us vs. \<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow> P
  4893    \<rbrakk> \<Longrightarrow> P"
  4894 unfolding listrel1_def by auto
  4895 
  4896 lemma not_Nil_listrel1 [iff]: "([], xs) \<notin> listrel1 r"
  4897 unfolding listrel1_def by blast
  4898 
  4899 lemma not_listrel1_Nil [iff]: "(xs, []) \<notin> listrel1 r"
  4900 unfolding listrel1_def by blast
  4901 
  4902 lemma Cons_listrel1_Cons [iff]:
  4903   "(x # xs, y # ys) \<in> listrel1 r \<longleftrightarrow>
  4904    (x,y) \<in> r \<and> xs = ys \<or> x = y \<and> (xs, ys) \<in> listrel1 r"
  4905 by (simp add: listrel1_def Cons_eq_append_conv) (blast)
  4906 
  4907 lemma listrel1I1: "(x,y) \<in> r \<Longrightarrow> (x # xs, y # xs) \<in> listrel1 r"
  4908 by (metis Cons_listrel1_Cons)
  4909 
  4910 lemma listrel1I2: "(xs, ys) \<in> listrel1 r \<Longrightarrow> (x # xs, x # ys) \<in> listrel1 r"
  4911 by (metis Cons_listrel1_Cons)
  4912 
  4913 lemma append_listrel1I:
  4914   "(xs, ys) \<in> listrel1 r \<and> us = vs \<or> xs = ys \<and> (us, vs) \<in> listrel1 r
  4915     \<Longrightarrow> (xs @ us, ys @ vs) \<in> listrel1 r"
  4916 unfolding listrel1_def
  4917 by auto (blast intro: append_eq_appendI)+
  4918 
  4919 lemma Cons_listrel1E1[elim!]:
  4920   assumes "(x # xs, ys) \<in> listrel1 r"
  4921     and "\<And>y. ys = y # xs \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
  4922     and "\<And>zs. ys = x # zs \<Longrightarrow> (xs, zs) \<in> listrel1 r \<Longrightarrow> R"
  4923   shows R
  4924 using assms by (cases ys) blast+
  4925 
  4926 lemma Cons_listrel1E2[elim!]:
  4927   assumes "(xs, y # ys) \<in> listrel1 r"
  4928     and "\<And>x. xs = x # ys \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
  4929     and "\<And>zs. xs = y # zs \<Longrightarrow> (zs, ys) \<in> listrel1 r \<Longrightarrow> R"
  4930   shows R
  4931 using assms by (cases xs) blast+
  4932 
  4933 lemma snoc_listrel1_snoc_iff:
  4934   "(xs @ [x], ys @ [y]) \<in> listrel1 r
  4935     \<longleftrightarrow> (xs, ys) \<in> listrel1 r \<and> x = y \<or> xs = ys \<and> (x,y) \<in> r" (is "?L \<longleftrightarrow> ?R")
  4936 proof
  4937   assume ?L thus ?R
  4938     by (fastforce simp: listrel1_def snoc_eq_iff_butlast butlast_append)
  4939 next
  4940   assume ?R then show ?L unfolding listrel1_def by force
  4941 qed
  4942 
  4943 lemma listrel1_eq_len: "(xs,ys) \<in> listrel1 r \<Longrightarrow> length xs = length ys"
  4944 unfolding listrel1_def by auto
  4945 
  4946 lemma listrel1_mono:
  4947   "r \<subseteq> s \<Longrightarrow> listrel1 r \<subseteq> listrel1 s"
  4948 unfolding listrel1_def by blast
  4949 
  4950 
  4951 lemma listrel1_converse: "listrel1 (r^-1) = (listrel1 r)^-1"
  4952 unfolding listrel1_def by blast
  4953 
  4954 lemma in_listrel1_converse:
  4955   "(x,y) : listrel1 (r^-1) \<longleftrightarrow> (x,y) : (listrel1 r)^-1"
  4956 unfolding listrel1_def by blast
  4957 
  4958 lemma listrel1_iff_update:
  4959   "(xs,ys) \<in> (listrel1 r)
  4960    \<longleftrightarrow> (\<exists>y n. (xs ! n, y) \<in> r \<and> n < length xs \<and> ys = xs[n:=y])" (is "?L \<longleftrightarrow> ?R")
  4961 proof
  4962   assume "?L"
  4963   then obtain x y u v where "xs = u @ x # v"  "ys = u @ y # v"  "(x,y) \<in> r"
  4964     unfolding listrel1_def by auto
  4965   then have "ys = xs[length u := y]" and "length u < length xs"
  4966     and "(xs ! length u, y) \<in> r" by auto
  4967   then show "?R" by auto
  4968 next
  4969   assume "?R"
  4970   then obtain x y n where "(xs!n, y) \<in> r" "n < size xs" "ys = xs[n:=y]" "x = xs!n"
  4971     by auto
  4972   then obtain u v where "xs = u @ x # v" and "ys = u @ y # v" and "(x, y) \<in> r"
  4973     by (auto intro: upd_conv_take_nth_drop id_take_nth_drop)
  4974   then show "?L" by (auto simp: listrel1_def)
  4975 qed
  4976 
  4977 
  4978 text{* Accessible part and wellfoundedness: *}
  4979 
  4980 lemma Cons_acc_listrel1I [intro!]:
  4981   "x \<in> acc r \<Longrightarrow> xs \<in> acc (listrel1 r) \<Longrightarrow> (x # xs) \<in> acc (listrel1 r)"
  4982 apply (induct arbitrary: xs set: acc)
  4983 apply (erule thin_rl)
  4984 apply (erule acc_induct)
  4985 apply (rule accI)
  4986 apply (blast)
  4987 done
  4988 
  4989 lemma lists_accD: "xs \<in> lists (acc r) \<Longrightarrow> xs \<in> acc (listrel1 r)"
  4990 apply (induct set: lists)
  4991  apply (rule accI)
  4992  apply simp
  4993 apply (rule accI)
  4994 apply (fast dest: acc_downward)
  4995 done
  4996 
  4997 lemma lists_accI: "xs \<in> acc (listrel1 r) \<Longrightarrow> xs \<in> lists (acc r)"
  4998 apply (induct set: acc)
  4999 apply clarify
  5000 apply (rule accI)
  5001 apply (fastforce dest!: in_set_conv_decomp[THEN iffD1] simp: listrel1_def)
  5002 done
  5003 
  5004 lemma wf_listrel1_iff[simp]: "wf(listrel1 r) = wf r"
  5005 by(metis wf_acc_iff in_lists_conv_set lists_accI lists_accD Cons_in_lists_iff)
  5006 
  5007 
  5008 subsubsection {* Lifting Relations to Lists: all elements *}
  5009 
  5010 inductive_set
  5011   listrel :: "('a \<times> 'b) set \<Rightarrow> ('a list \<times> 'b list) set"
  5012   for r :: "('a \<times> 'b) set"
  5013 where
  5014     Nil:  "([],[]) \<in> listrel r"
  5015   | Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
  5016 
  5017 inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
  5018 inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
  5019 inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
  5020 inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
  5021 
  5022 
  5023 lemma listrel_eq_len:  "(xs, ys) \<in> listrel r \<Longrightarrow> length xs = length ys"
  5024 by(induct rule: listrel.induct) auto
  5025 
  5026 lemma listrel_iff_zip [code_unfold]: "(xs,ys) : listrel r \<longleftrightarrow>
  5027   length xs = length ys & (\<forall>(x,y) \<in> set(zip xs ys). (x,y) \<in> r)" (is "?L \<longleftrightarrow> ?R")
  5028 proof
  5029   assume ?L thus ?R by induct (auto intro: listrel_eq_len)
  5030 next
  5031   assume ?R thus ?L
  5032     apply (clarify)
  5033     by (induct rule: list_induct2) (auto intro: listrel.intros)
  5034 qed
  5035 
  5036 lemma listrel_iff_nth: "(xs,ys) : listrel r \<longleftrightarrow>
  5037   length xs = length ys & (\<forall>n < length xs. (xs!n, ys!n) \<in> r)" (is "?L \<longleftrightarrow> ?R")
  5038 by (auto simp add: all_set_conv_all_nth listrel_iff_zip)
  5039 
  5040 
  5041 lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
  5042 apply clarify  
  5043 apply (erule listrel.induct)
  5044 apply (blast intro: listrel.intros)+
  5045 done
  5046 
  5047 lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
  5048 apply clarify 
  5049 apply (erule listrel.induct, auto) 
  5050 done
  5051 
  5052 lemma listrel_refl_on: "refl_on A r \<Longrightarrow> refl_on (lists A) (listrel r)" 
  5053 apply (simp add: refl_on_def listrel_subset Ball_def)
  5054 apply (rule allI) 
  5055 apply (induct_tac x) 
  5056 apply (auto intro: listrel.intros)
  5057 done
  5058 
  5059 lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
  5060 apply (auto simp add: sym_def)
  5061 apply (erule listrel.induct) 
  5062 apply (blast intro: listrel.intros)+
  5063 done
  5064 
  5065 lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
  5066 apply (simp add: trans_def)
  5067 apply (intro allI) 
  5068 apply (rule impI) 
  5069 apply (erule listrel.induct) 
  5070 apply (blast intro: listrel.intros)+
  5071 done
  5072 
  5073 theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
  5074 by (simp add: equiv_def listrel_refl_on listrel_sym listrel_trans) 
  5075 
  5076 lemma listrel_rtrancl_refl[iff]: "(xs,xs) : listrel(r^*)"
  5077 using listrel_refl_on[of UNIV, OF refl_rtrancl]
  5078 by(auto simp: refl_on_def)
  5079 
  5080 lemma listrel_rtrancl_trans:
  5081   "\<lbrakk> (xs,ys) : listrel(r^*);  (ys,zs) : listrel(r^*) \<rbrakk>
  5082   \<Longrightarrow> (xs,zs) : listrel(r^*)"
  5083 by (metis listrel_trans trans_def trans_rtrancl)
  5084 
  5085 
  5086 lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
  5087 by (blast intro: listrel.intros)
  5088 
  5089 lemma listrel_Cons:
  5090      "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})"
  5091 by (auto simp add: set_Cons_def intro: listrel.intros)
  5092 
  5093 text {* Relating @{term listrel1}, @{term listrel} and closures: *}
  5094 
  5095 lemma listrel1_rtrancl_subset_rtrancl_listrel1:
  5096   "listrel1 (r^*) \<subseteq> (listrel1 r)^*"
  5097 proof (rule subrelI)
  5098   fix xs ys assume 1: "(xs,ys) \<in> listrel1 (r^*)"
  5099   { fix x y us vs
  5100     have "(x,y) : r^* \<Longrightarrow> (us @ x # vs, us @ y # vs) : (listrel1 r)^*"
  5101     proof(induct rule: rtrancl.induct)
  5102       case rtrancl_refl show ?case by simp
  5103     next
  5104       case rtrancl_into_rtrancl thus ?case
  5105         by (metis listrel1I rtrancl.rtrancl_into_rtrancl)
  5106     qed }
  5107   thus "(xs,ys) \<in> (listrel1 r)^*" using 1 by(blast elim: listrel1E)
  5108 qed
  5109 
  5110 lemma rtrancl_listrel1_eq_len: "(x,y) \<in> (listrel1 r)^* \<Longrightarrow> length x = length y"
  5111 by (induct rule: rtrancl.induct) (auto intro: listrel1_eq_len)
  5112 
  5113 lemma rtrancl_listrel1_ConsI1:
  5114   "(xs,ys) : (listrel1 r)^* \<Longrightarrow> (x#xs,x#ys) : (listrel1 r)^*"
  5115 apply(induct rule: rtrancl.induct)
  5116  apply simp
  5117 by (metis listrel1I2 rtrancl.rtrancl_into_rtrancl)
  5118 
  5119 lemma rtrancl_listrel1_ConsI2:
  5120   "(x,y) \<in> r^* \<Longrightarrow> (xs, ys) \<in> (listrel1 r)^*
  5121   \<Longrightarrow> (x # xs, y # ys) \<in> (listrel1 r)^*"
  5122   by (blast intro: rtrancl_trans rtrancl_listrel1_ConsI1 
  5123     subsetD[OF listrel1_rtrancl_subset_rtrancl_listrel1 listrel1I1])
  5124 
  5125 lemma listrel1_subset_listrel:
  5126   "r \<subseteq> r' \<Longrightarrow> refl r' \<Longrightarrow> listrel1 r \<subseteq> listrel(r')"
  5127 by(auto elim!: listrel1E simp add: listrel_iff_zip set_zip refl_on_def)
  5128 
  5129 lemma listrel_reflcl_if_listrel1:
  5130   "(xs,ys) : listrel1 r \<Longrightarrow> (xs,ys) : listrel(r^*)"
  5131 by(erule listrel1E)(auto simp add: listrel_iff_zip set_zip)
  5132 
  5133 lemma listrel_rtrancl_eq_rtrancl_listrel1: "listrel (r^*) = (listrel1 r)^*"
  5134 proof
  5135   { fix x y assume "(x,y) \<in> listrel (r^*)"
  5136     then have "(x,y) \<in> (listrel1 r)^*"
  5137     by induct (auto intro: rtrancl_listrel1_ConsI2) }
  5138   then show "listrel (r^*) \<subseteq> (listrel1 r)^*"
  5139     by (rule subrelI)
  5140 next
  5141   show "listrel (r^*) \<supseteq> (listrel1 r)^*"
  5142   proof(rule subrelI)
  5143     fix xs ys assume "(xs,ys) \<in> (listrel1 r)^*"
  5144     then show "(xs,ys) \<in> listrel (r^*)"
  5145     proof induct
  5146       case base show ?case by(auto simp add: listrel_iff_zip set_zip)
  5147     next
  5148       case (step ys zs)
  5149       thus ?case  by (metis listrel_reflcl_if_listrel1 listrel_rtrancl_trans)
  5150     qed
  5151   qed
  5152 qed
  5153 
  5154 lemma rtrancl_listrel1_if_listrel:
  5155   "(xs,ys) : listrel r \<Longrightarrow> (xs,ys) : (listrel1 r)^*"
  5156 by(metis listrel_rtrancl_eq_rtrancl_listrel1 subsetD[OF listrel_mono] r_into_rtrancl subsetI)
  5157 
  5158 lemma listrel_subset_rtrancl_listrel1: "listrel r \<subseteq> (listrel1 r)^*"
  5159 by(fast intro:rtrancl_listrel1_if_listrel)
  5160 
  5161 
  5162 subsection {* Size function *}
  5163 
  5164 lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (list_size f)"
  5165 by (rule is_measure_trivial)
  5166 
  5167 lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (option_size f)"
  5168 by (rule is_measure_trivial)
  5169 
  5170 lemma list_size_estimation[termination_simp]: 
  5171   "x \<in> set xs \<Longrightarrow> y < f x \<Longrightarrow> y < list_size f xs"
  5172 by (induct xs) auto
  5173 
  5174 lemma list_size_estimation'[termination_simp]: 
  5175   "x \<in> set xs \<Longrightarrow> y \<le> f x \<Longrightarrow> y \<le> list_size f xs"
  5176 by (induct xs) auto
  5177 
  5178 lemma list_size_map[simp]: "list_size f (map g xs) = list_size (f o g) xs"
  5179 by (induct xs) auto
  5180 
  5181 lemma list_size_append[simp]: "list_size f (xs @ ys) = list_size f xs + list_size f ys"
  5182 by (induct xs, auto)
  5183 
  5184 lemma list_size_pointwise[termination_simp]: 
  5185   "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> list_size f xs \<le> list_size g xs"
  5186 by (induct xs) force+
  5187 
  5188 
  5189 subsection {* Monad operation *}
  5190 
  5191 definition bind :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b list) \<Rightarrow> 'b list" where
  5192   "bind xs f = concat (map f xs)"
  5193 
  5194 hide_const (open) bind
  5195 
  5196 lemma bind_simps [simp]:
  5197   "List.bind [] f = []"
  5198   "List.bind (x # xs) f = f x @ List.bind xs f"
  5199   by (simp_all add: bind_def)
  5200 
  5201 
  5202 subsection {* Transfer *}
  5203 
  5204 definition
  5205   embed_list :: "nat list \<Rightarrow> int list"
  5206 where
  5207   "embed_list l = map int l"
  5208 
  5209 definition
  5210   nat_list :: "int list \<Rightarrow> bool"
  5211 where
  5212   "nat_list l = nat_set (set l)"
  5213 
  5214 definition
  5215   return_list :: "int list \<Rightarrow> nat list"
  5216 where
  5217   "return_list l = map nat l"
  5218 
  5219 lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow>
  5220     embed_list (return_list l) = l"
  5221   unfolding embed_list_def return_list_def nat_list_def nat_set_def
  5222   apply (induct l)
  5223   apply auto
  5224 done
  5225 
  5226 lemma transfer_nat_int_list_functions:
  5227   "l @ m = return_list (embed_list l @ embed_list m)"
  5228   "[] = return_list []"
  5229   unfolding return_list_def embed_list_def
  5230   apply auto
  5231   apply (induct l, auto)
  5232   apply (induct m, auto)
  5233 done
  5234 
  5235 (*
  5236 lemma transfer_nat_int_fold1: "fold f l x =
  5237     fold (%x. f (nat x)) (embed_list l) x";
  5238 *)
  5239 
  5240 
  5241 subsection {* Code generation *}
  5242 
  5243 subsubsection {* Counterparts for set-related operations *}
  5244 
  5245 definition member :: "'a list \<Rightarrow> 'a \<Rightarrow> bool" where
  5246   [code_abbrev]: "member xs x \<longleftrightarrow> x \<in> set xs"
  5247 
  5248 text {*
  5249   Use @{text member} only for generating executable code.  Otherwise use
  5250   @{prop "x \<in> set xs"} instead --- it is much easier to reason about.
  5251 *}
  5252 
  5253 lemma member_rec [code]:
  5254   "member (x # xs) y \<longleftrightarrow> x = y \<or> member xs y"
  5255   "member [] y \<longleftrightarrow> False"
  5256   by (auto simp add: member_def)
  5257 
  5258 lemma in_set_member (* FIXME delete candidate *):
  5259   "x \<in> set xs \<longleftrightarrow> member xs x"
  5260   by (simp add: member_def)
  5261 
  5262 definition list_all :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5263   list_all_iff [code_abbrev]: "list_all P xs \<longleftrightarrow> Ball (set xs) P"
  5264 
  5265 definition list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5266   list_ex_iff [code_abbrev]: "list_ex P xs \<longleftrightarrow> Bex (set xs) P"
  5267 
  5268 definition list_ex1 :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5269   list_ex1_iff [code_abbrev]: "list_ex1 P xs \<longleftrightarrow> (\<exists>! x. x \<in> set xs \<and> P x)"
  5270 
  5271 text {*
  5272   Usually you should prefer @{text "\<forall>x\<in>set xs"}, @{text "\<exists>x\<in>set xs"}
  5273   and @{text "\<exists>!x. x\<in>set xs \<and> _"} over @{const list_all}, @{const list_ex}
  5274   and @{const list_ex1} in specifications.
  5275 *}
  5276 
  5277 lemma list_all_simps [simp, code]:
  5278   "list_all P (x # xs) \<longleftrightarrow> P x \<and> list_all P xs"
  5279   "list_all P [] \<longleftrightarrow> True"
  5280   by (simp_all add: list_all_iff)
  5281 
  5282 lemma list_ex_simps [simp, code]:
  5283   "list_ex P (x # xs) \<longleftrightarrow> P x \<or> list_ex P xs"
  5284   "list_ex P [] \<longleftrightarrow> False"
  5285   by (simp_all add: list_ex_iff)
  5286 
  5287 lemma list_ex1_simps [simp, code]:
  5288   "list_ex1 P [] = False"
  5289   "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)"
  5290   by (auto simp add: list_ex1_iff list_all_iff)
  5291 
  5292 lemma Ball_set_list_all: (* FIXME delete candidate *)
  5293   "Ball (set xs) P \<longleftrightarrow> list_all P xs"
  5294   by (simp add: list_all_iff)
  5295 
  5296 lemma Bex_set_list_ex: (* FIXME delete candidate *)
  5297   "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
  5298   by (simp add: list_ex_iff)
  5299 
  5300 lemma list_all_append [simp]:
  5301   "list_all P (xs @ ys) \<longleftrightarrow> list_all P xs \<and> list_all P ys"
  5302   by (auto simp add: list_all_iff)
  5303 
  5304 lemma list_ex_append [simp]:
  5305   "list_ex P (xs @ ys) \<longleftrightarrow> list_ex P xs \<or> list_ex P ys"
  5306   by (auto simp add: list_ex_iff)
  5307 
  5308 lemma list_all_rev [simp]:
  5309   "list_all P (rev xs) \<longleftrightarrow> list_all P xs"
  5310   by (simp add: list_all_iff)
  5311 
  5312 lemma list_ex_rev [simp]:
  5313   "list_ex P (rev xs) \<longleftrightarrow> list_ex P xs"
  5314   by (simp add: list_ex_iff)
  5315 
  5316 lemma list_all_length:
  5317   "list_all P xs \<longleftrightarrow> (\<forall>n < length xs. P (xs ! n))"
  5318   by (auto simp add: list_all_iff set_conv_nth)
  5319 
  5320 lemma list_ex_length:
  5321   "list_ex P xs \<longleftrightarrow> (\<exists>n < length xs. P (xs ! n))"
  5322   by (auto simp add: list_ex_iff set_conv_nth)
  5323 
  5324 lemma list_all_cong [fundef_cong]:
  5325   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_all f xs = list_all g ys"
  5326   by (simp add: list_all_iff)
  5327 
  5328 lemma list_any_cong [fundef_cong]:
  5329   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_ex f xs = list_ex g ys"
  5330   by (simp add: list_ex_iff)
  5331 
  5332 text {* Executable checks for relations on sets *}
  5333 
  5334 definition listrel1p :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
  5335 "listrel1p r xs ys = ((xs, ys) \<in> listrel1 {(x, y). r x y})"
  5336 
  5337 lemma [code_unfold]:
  5338   "(xs, ys) \<in> listrel1 r = listrel1p (\<lambda>x y. (x, y) \<in> r) xs ys"
  5339 unfolding listrel1p_def by auto
  5340 
  5341 lemma [code]:
  5342   "listrel1p r [] xs = False"
  5343   "listrel1p r xs [] =  False"
  5344   "listrel1p r (x # xs) (y # ys) \<longleftrightarrow>
  5345      r x y \<and> xs = ys \<or> x = y \<and> listrel1p r xs ys"
  5346 by (simp add: listrel1p_def)+
  5347 
  5348 definition
  5349   lexordp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
  5350   "lexordp r xs ys = ((xs, ys) \<in> lexord {(x, y). r x y})"
  5351 
  5352 lemma [code_unfold]:
  5353   "(xs, ys) \<in> lexord r = lexordp (\<lambda>x y. (x, y) \<in> r) xs ys"
  5354 unfolding lexordp_def by auto
  5355 
  5356 lemma [code]:
  5357   "lexordp r xs [] = False"
  5358   "lexordp r [] (y#ys) = True"
  5359   "lexordp r (x # xs) (y # ys) = (r x y | (x = y & lexordp r xs ys))"
  5360 unfolding lexordp_def by auto
  5361 
  5362 text {* Bounded quantification and summation over nats. *}
  5363 
  5364 lemma atMost_upto [code_unfold]:
  5365   "{..n} = set [0..<Suc n]"
  5366   by auto
  5367 
  5368 lemma atLeast_upt [code_unfold]:
  5369   "{..<n} = set [0..<n]"
  5370   by auto
  5371 
  5372 lemma greaterThanLessThan_upt [code_unfold]:
  5373   "{n<..<m} = set [Suc n..<m]"
  5374   by auto
  5375 
  5376 lemmas atLeastLessThan_upt [code_unfold] = set_upt [symmetric]
  5377 
  5378 lemma greaterThanAtMost_upt [code_unfold]:
  5379   "{n<..m} = set [Suc n..<Suc m]"
  5380   by auto
  5381 
  5382 lemma atLeastAtMost_upt [code_unfold]:
  5383   "{n..m} = set [n..<Suc m]"
  5384   by auto
  5385 
  5386 lemma all_nat_less_eq [code_unfold]:
  5387   "(\<forall>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..<n}. P m)"
  5388   by auto
  5389 
  5390 lemma ex_nat_less_eq [code_unfold]:
  5391   "(\<exists>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..<n}. P m)"
  5392   by auto
  5393 
  5394 lemma all_nat_less [code_unfold]:
  5395   "(\<forall>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..n}. P m)"
  5396   by auto
  5397 
  5398 lemma ex_nat_less [code_unfold]:
  5399   "(\<exists>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..n}. P m)"
  5400   by auto
  5401 
  5402 lemma setsum_set_upt_conv_listsum_nat [code_unfold]:
  5403   "setsum f (set [m..<n]) = listsum (map f [m..<n])"
  5404   by (simp add: interv_listsum_conv_setsum_set_nat)
  5405 
  5406 text {* Summation over ints. *}
  5407 
  5408 lemma greaterThanLessThan_upto [code_unfold]:
  5409   "{i<..<j::int} = set [i+1..j - 1]"
  5410 by auto
  5411 
  5412 lemma atLeastLessThan_upto [code_unfold]:
  5413   "{i..<j::int} = set [i..j - 1]"
  5414 by auto
  5415 
  5416 lemma greaterThanAtMost_upto [code_unfold]:
  5417   "{i<..j::int} = set [i+1..j]"
  5418 by auto
  5419 
  5420 lemmas atLeastAtMost_upto [code_unfold] = set_upto [symmetric]
  5421 
  5422 lemma setsum_set_upto_conv_listsum_int [code_unfold]:
  5423   "setsum f (set [i..j::int]) = listsum (map f [i..j])"
  5424   by (simp add: interv_listsum_conv_setsum_set_int)
  5425 
  5426 
  5427 subsubsection {* Optimizing by rewriting *}
  5428 
  5429 definition null :: "'a list \<Rightarrow> bool" where
  5430   [code_abbrev]: "null xs \<longleftrightarrow> xs = []"
  5431 
  5432 text {*
  5433   Efficient emptyness check is implemented by @{const null}.
  5434 *}
  5435 
  5436 lemma null_rec [code]:
  5437   "null (x # xs) \<longleftrightarrow> False"
  5438   "null [] \<longleftrightarrow> True"
  5439   by (simp_all add: null_def)
  5440 
  5441 lemma eq_Nil_null: (* FIXME delete candidate *)
  5442   "xs = [] \<longleftrightarrow> null xs"
  5443   by (simp add: null_def)
  5444 
  5445 lemma equal_Nil_null [code_unfold]:
  5446   "HOL.equal xs [] \<longleftrightarrow> null xs"
  5447   by (simp add: equal eq_Nil_null)
  5448 
  5449 definition maps :: "('a \<Rightarrow> 'b list) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  5450   [code_abbrev]: "maps f xs = concat (map f xs)"
  5451 
  5452 definition map_filter :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  5453   [code_post]: "map_filter f xs = map (the \<circ> f) (filter (\<lambda>x. f x \<noteq> None) xs)"
  5454 
  5455 text {*
  5456   Operations @{const maps} and @{const map_filter} avoid
  5457   intermediate lists on execution -- do not use for proving.
  5458 *}
  5459 
  5460 lemma maps_simps [code]:
  5461   "maps f (x # xs) = f x @ maps f xs"
  5462   "maps f [] = []"
  5463   by (simp_all add: maps_def)
  5464 
  5465 lemma map_filter_simps [code]:
  5466   "map_filter f (x # xs) = (case f x of None \<Rightarrow> map_filter f xs | Some y \<Rightarrow> y # map_filter f xs)"
  5467   "map_filter f [] = []"
  5468   by (simp_all add: map_filter_def split: option.split)
  5469 
  5470 lemma concat_map_maps: (* FIXME delete candidate *)
  5471   "concat (map f xs) = maps f xs"
  5472   by (simp add: maps_def)
  5473 
  5474 lemma map_filter_map_filter [code_unfold]:
  5475   "map f (filter P xs) = map_filter (\<lambda>x. if P x then Some (f x) else None) xs"
  5476   by (simp add: map_filter_def)
  5477 
  5478 text {* Optimized code for @{text"\<forall>i\<in>{a..b::int}"} and @{text"\<forall>n:{a..<b::nat}"}
  5479 and similiarly for @{text"\<exists>"}. *}
  5480 
  5481 definition all_interval_nat :: "(nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
  5482   "all_interval_nat P i j \<longleftrightarrow> (\<forall>n \<in> {i..<j}. P n)"
  5483 
  5484 lemma [code]:
  5485   "all_interval_nat P i j \<longleftrightarrow> i \<ge> j \<or> P i \<and> all_interval_nat P (Suc i) j"
  5486 proof -
  5487   have *: "\<And>n. P i \<Longrightarrow> \<forall>n\<in>{Suc i..<j}. P n \<Longrightarrow> i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n"
  5488   proof -
  5489     fix n
  5490     assume "P i" "\<forall>n\<in>{Suc i..<j}. P n" "i \<le> n" "n < j"
  5491     then show "P n" by (cases "n = i") simp_all
  5492   qed
  5493   show ?thesis by (auto simp add: all_interval_nat_def intro: *)
  5494 qed
  5495 
  5496 lemma list_all_iff_all_interval_nat [code_unfold]:
  5497   "list_all P [i..<j] \<longleftrightarrow> all_interval_nat P i j"
  5498   by (simp add: list_all_iff all_interval_nat_def)
  5499 
  5500 lemma list_ex_iff_not_all_inverval_nat [code_unfold]:
  5501   "list_ex P [i..<j] \<longleftrightarrow> \<not> (all_interval_nat (Not \<circ> P) i j)"
  5502   by (simp add: list_ex_iff all_interval_nat_def)
  5503 
  5504 definition all_interval_int :: "(int \<Rightarrow> bool) \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool" where
  5505   "all_interval_int P i j \<longleftrightarrow> (\<forall>k \<in> {i..j}. P k)"
  5506 
  5507 lemma [code]:
  5508   "all_interval_int P i j \<longleftrightarrow> i > j \<or> P i \<and> all_interval_int P (i + 1) j"
  5509 proof -
  5510   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"
  5511   proof -
  5512     fix k
  5513     assume "P i" "\<forall>k\<in>{i+1..j}. P k" "i \<le> k" "k \<le> j"
  5514     then show "P k" by (cases "k = i") simp_all
  5515   qed
  5516   show ?thesis by (auto simp add: all_interval_int_def intro: *)
  5517 qed
  5518 
  5519 lemma list_all_iff_all_interval_int [code_unfold]:
  5520   "list_all P [i..j] \<longleftrightarrow> all_interval_int P i j"
  5521   by (simp add: list_all_iff all_interval_int_def)
  5522 
  5523 lemma list_ex_iff_not_all_inverval_int [code_unfold]:
  5524   "list_ex P [i..j] \<longleftrightarrow> \<not> (all_interval_int (Not \<circ> P) i j)"
  5525   by (simp add: list_ex_iff all_interval_int_def)
  5526 
  5527 hide_const (open) member null maps map_filter all_interval_nat all_interval_int
  5528 
  5529 
  5530 subsubsection {* Pretty lists *}
  5531 
  5532 use "Tools/list_code.ML"
  5533 
  5534 code_type list
  5535   (SML "_ list")
  5536   (OCaml "_ list")
  5537   (Haskell "![(_)]")
  5538   (Scala "List[(_)]")
  5539 
  5540 code_const Nil
  5541   (SML "[]")
  5542   (OCaml "[]")
  5543   (Haskell "[]")
  5544   (Scala "!Nil")
  5545 
  5546 code_instance list :: equal
  5547   (Haskell -)
  5548 
  5549 code_const "HOL.equal \<Colon> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
  5550   (Haskell infix 4 "==")
  5551 
  5552 code_reserved SML
  5553   list
  5554 
  5555 code_reserved OCaml
  5556   list
  5557 
  5558 setup {* fold (List_Code.add_literal_list) ["SML", "OCaml", "Haskell", "Scala"] *}
  5559 
  5560 
  5561 subsubsection {* Use convenient predefined operations *}
  5562 
  5563 code_const "op @"
  5564   (SML infixr 7 "@")
  5565   (OCaml infixr 6 "@")
  5566   (Haskell infixr 5 "++")
  5567   (Scala infixl 7 "++")
  5568 
  5569 code_const map
  5570   (Haskell "map")
  5571 
  5572 code_const filter
  5573   (Haskell "filter")
  5574 
  5575 code_const concat
  5576   (Haskell "concat")
  5577 
  5578 code_const List.maps
  5579   (Haskell "concatMap")
  5580 
  5581 code_const rev
  5582   (Haskell "reverse")
  5583 
  5584 code_const zip
  5585   (Haskell "zip")
  5586 
  5587 code_const List.null
  5588   (Haskell "null")
  5589 
  5590 code_const takeWhile
  5591   (Haskell "takeWhile")
  5592 
  5593 code_const dropWhile
  5594   (Haskell "dropWhile")
  5595 
  5596 code_const list_all
  5597   (Haskell "all")
  5598 
  5599 code_const list_ex
  5600   (Haskell "any")
  5601 
  5602 
  5603 subsubsection {* Implementation of sets by lists *}
  5604 
  5605 text {* Basic operations *}
  5606 
  5607 lemma is_empty_set [code]:
  5608   "Set.is_empty (set xs) \<longleftrightarrow> List.null xs"
  5609   by (simp add: Set.is_empty_def null_def)
  5610 
  5611 lemma empty_set [code]:
  5612   "{} = set []"
  5613   by simp
  5614 
  5615 lemma UNIV_coset [code]:
  5616   "UNIV = List.coset []"
  5617   by simp
  5618 
  5619 lemma compl_set [code]:
  5620   "- set xs = List.coset xs"
  5621   by simp
  5622 
  5623 lemma compl_coset [code]:
  5624   "- List.coset xs = set xs"
  5625   by simp
  5626 
  5627 lemma [code]:
  5628   "x \<in> set xs \<longleftrightarrow> List.member xs x"
  5629   "x \<in> List.coset xs \<longleftrightarrow> \<not> List.member xs x"
  5630   by (simp_all add: member_def)
  5631 
  5632 lemma insert_code [code]:
  5633   "insert x (set xs) = set (List.insert x xs)"
  5634   "insert x (List.coset xs) = List.coset (removeAll x xs)"
  5635   by simp_all
  5636 
  5637 lemma remove_code [code]:
  5638   "Set.remove x (set xs) = set (removeAll x xs)"
  5639   "Set.remove x (List.coset xs) = List.coset (List.insert x xs)"
  5640   by (simp_all add: remove_def Compl_insert)
  5641 
  5642 lemma project_set [code]:
  5643   "Set.project P (set xs) = set (filter P xs)"
  5644   by auto
  5645 
  5646 lemma image_set [code]:
  5647   "image f (set xs) = set (map f xs)"
  5648   by simp
  5649 
  5650 lemma Ball_set [code]:
  5651   "Ball (set xs) P \<longleftrightarrow> list_all P xs"
  5652   by (simp add: list_all_iff)
  5653 
  5654 lemma Bex_set [code]:
  5655   "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
  5656   by (simp add: list_ex_iff)
  5657 
  5658 lemma card_set [code]:
  5659   "card (set xs) = length (remdups xs)"
  5660 proof -
  5661   have "card (set (remdups xs)) = length (remdups xs)"
  5662     by (rule distinct_card) simp
  5663   then show ?thesis by simp
  5664 qed
  5665 
  5666 lemma the_elem_set [code]:
  5667   "the_elem (set [x]) = x"
  5668   by simp
  5669 
  5670 lemma Pow_set [code]:
  5671   "Pow (set []) = {{}}"
  5672   "Pow (set (x # xs)) = (let A = Pow (set xs) in A \<union> insert x ` A)"
  5673   by (simp_all add: Pow_insert Let_def)
  5674 
  5675 text {* Further operations on sets *}
  5676 
  5677 (* Minimal refinement of equality on sets *)
  5678 declare subset_eq[code del]
  5679 lemma subset_code [code]:
  5680   "set xs <= B \<longleftrightarrow> (ALL x : set xs. x : B)"
  5681   "List.coset xs <= List.coset ys \<longleftrightarrow> set ys <= set xs"
  5682   "List.coset [] <= set [] \<longleftrightarrow> False"
  5683 by auto
  5684 
  5685 lemma setsum_code [code]:
  5686   "setsum f (set xs) = listsum (map f (remdups xs))"
  5687 by (simp add: listsum_distinct_conv_setsum_set)
  5688 
  5689 definition map_project :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a set \<Rightarrow> 'b set" where
  5690   "map_project f A = {b. \<exists> a \<in> A. f a = Some b}"
  5691 
  5692 lemma [code]:
  5693   "map_project f (set xs) = set (List.map_filter f xs)"
  5694 unfolding map_project_def map_filter_def
  5695 by auto (metis (lifting, mono_tags) CollectI image_eqI o_apply the.simps)
  5696 
  5697 hide_const (open) map_project
  5698 
  5699 text {* Operations on relations *}
  5700 
  5701 lemma product_code [code]:
  5702   "Product_Type.product (set xs) (set ys) = set [(x, y). x \<leftarrow> xs, y \<leftarrow> ys]"
  5703   by (auto simp add: Product_Type.product_def)
  5704 
  5705 lemma Id_on_set [code]:
  5706   "Id_on (set xs) = set [(x, x). x \<leftarrow> xs]"
  5707   by (auto simp add: Id_on_def)
  5708 
  5709 lemma [code]:
  5710   "R `` S = List.map_project (%(x, y). if x : S then Some y else None) R"
  5711 unfolding map_project_def by (auto split: prod.split split_if_asm)
  5712 
  5713 lemma trancl_set_ntrancl [code]:
  5714   "trancl (set xs) = ntrancl (card (set xs) - 1) (set xs)"
  5715   by (simp add: finite_trancl_ntranl)
  5716 
  5717 lemma set_rel_comp [code]:
  5718   "set xys O set yzs = set ([(fst xy, snd yz). xy \<leftarrow> xys, yz \<leftarrow> yzs, snd xy = fst yz])"
  5719   by (auto simp add: Bex_def)
  5720 
  5721 lemma wf_set [code]:
  5722   "wf (set xs) = acyclic (set xs)"
  5723   by (simp add: wf_iff_acyclic_if_finite)
  5724 
  5725 end