src/HOL/List.thy
author haftmann
Fri Jan 06 22:16:01 2012 +0100 (2012-01-06)
changeset 46147 2c4d8de91c4c
parent 46143 c932c80d3eae
child 46149 54ca5b2775a8
permissions -rw-r--r--
moved lemmas about List.set and set operations to 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 definition
   210   rotate1 :: "'a list \<Rightarrow> 'a list" where
   211   "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> 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 add:rotate1_def)}\\
   269 @{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate1_def 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]: "(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: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<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  "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<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; length ys = length vs |] ==>
  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 union_set_foldr:
  2609   "set xs \<union> A = foldr Set.insert xs A"
  2610 proof -
  2611   have "\<And>x y :: 'a. insert y \<circ> insert x = insert x \<circ> insert y"
  2612     by auto
  2613   then show ?thesis by (simp add: union_set_fold foldr_fold)
  2614 qed
  2615 
  2616 lemma minus_set_foldr:
  2617   "A - set xs = foldr Set.remove xs A"
  2618 proof -
  2619   have "\<And>x y :: 'a. Set.remove y \<circ> Set.remove x = Set.remove x \<circ> Set.remove y"
  2620     by (auto simp add: remove_def)
  2621   then show ?thesis by (simp add: minus_set_fold foldr_fold)
  2622 qed
  2623 
  2624 lemma (in lattice) Inf_fin_set_foldr [code]:
  2625   "Inf_fin (set (x # xs)) = foldr inf xs x"
  2626   by (simp add: Inf_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
  2627 
  2628 lemma (in lattice) Sup_fin_set_foldr [code]:
  2629   "Sup_fin (set (x # xs)) = foldr sup xs x"
  2630   by (simp add: Sup_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
  2631 
  2632 lemma (in linorder) Min_fin_set_foldr [code]:
  2633   "Min (set (x # xs)) = foldr min xs x"
  2634   by (simp add: Min_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
  2635 
  2636 lemma (in linorder) Max_fin_set_foldr [code]:
  2637   "Max (set (x # xs)) = foldr max xs x"
  2638   by (simp add: Max_fin_set_fold ac_simps foldr_fold fun_eq_iff del: set.simps)
  2639 
  2640 lemma (in complete_lattice) Inf_set_foldr:
  2641   "Inf (set xs) = foldr inf xs top"
  2642   by (simp add: Inf_set_fold ac_simps foldr_fold fun_eq_iff)
  2643 
  2644 lemma (in complete_lattice) Sup_set_foldr:
  2645   "Sup (set xs) = foldr sup xs bot"
  2646   by (simp add: Sup_set_fold ac_simps foldr_fold fun_eq_iff)
  2647 
  2648 lemma (in complete_lattice) INF_set_foldr [code]:
  2649   "INFI (set xs) f = foldr (inf \<circ> f) xs top"
  2650   by (simp only: INF_def Inf_set_foldr foldr_map set_map [symmetric])
  2651 
  2652 lemma (in complete_lattice) SUP_set_foldr [code]:
  2653   "SUPR (set xs) f = foldr (sup \<circ> f) xs bot"
  2654   by (simp only: SUP_def Sup_set_foldr foldr_map set_map [symmetric])
  2655 
  2656 
  2657 subsubsection {* @{text upt} *}
  2658 
  2659 lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
  2660 -- {* simp does not terminate! *}
  2661 by (induct j) auto
  2662 
  2663 lemmas upt_rec_number_of[simp] = upt_rec[of "number_of m" "number_of n"] for m n
  2664 
  2665 lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
  2666 by (subst upt_rec) simp
  2667 
  2668 lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
  2669 by(induct j)simp_all
  2670 
  2671 lemma upt_eq_Cons_conv:
  2672  "([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
  2673 apply(induct j arbitrary: x xs)
  2674  apply simp
  2675 apply(clarsimp simp add: append_eq_Cons_conv)
  2676 apply arith
  2677 done
  2678 
  2679 lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
  2680 -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
  2681 by simp
  2682 
  2683 lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
  2684   by (simp add: upt_rec)
  2685 
  2686 lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
  2687 -- {* LOOPS as a simprule, since @{text "j <= j"}. *}
  2688 by (induct k) auto
  2689 
  2690 lemma length_upt [simp]: "length [i..<j] = j - i"
  2691 by (induct j) (auto simp add: Suc_diff_le)
  2692 
  2693 lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
  2694 apply (induct j)
  2695 apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
  2696 done
  2697 
  2698 
  2699 lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
  2700 by(simp add:upt_conv_Cons)
  2701 
  2702 lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
  2703 apply(cases j)
  2704  apply simp
  2705 by(simp add:upt_Suc_append)
  2706 
  2707 lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]"
  2708 apply (induct m arbitrary: i, simp)
  2709 apply (subst upt_rec)
  2710 apply (rule sym)
  2711 apply (subst upt_rec)
  2712 apply (simp del: upt.simps)
  2713 done
  2714 
  2715 lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
  2716 apply(induct j)
  2717 apply auto
  2718 done
  2719 
  2720 lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]"
  2721 by (induct n) auto
  2722 
  2723 lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
  2724 apply (induct n m  arbitrary: i rule: diff_induct)
  2725 prefer 3 apply (subst map_Suc_upt[symmetric])
  2726 apply (auto simp add: less_diff_conv)
  2727 done
  2728 
  2729 lemma nth_take_lemma:
  2730   "k <= length xs ==> k <= length ys ==>
  2731      (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
  2732 apply (atomize, induct k arbitrary: xs ys)
  2733 apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
  2734 txt {* Both lists must be non-empty *}
  2735 apply (case_tac xs, simp)
  2736 apply (case_tac ys, clarify)
  2737  apply (simp (no_asm_use))
  2738 apply clarify
  2739 txt {* prenexing's needed, not miniscoping *}
  2740 apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
  2741 apply blast
  2742 done
  2743 
  2744 lemma nth_equalityI:
  2745  "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
  2746   by (frule nth_take_lemma [OF le_refl eq_imp_le]) simp_all
  2747 
  2748 lemma map_nth:
  2749   "map (\<lambda>i. xs ! i) [0..<length xs] = xs"
  2750   by (rule nth_equalityI, auto)
  2751 
  2752 (* needs nth_equalityI *)
  2753 lemma list_all2_antisym:
  2754   "\<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> 
  2755   \<Longrightarrow> xs = ys"
  2756   apply (simp add: list_all2_conv_all_nth) 
  2757   apply (rule nth_equalityI, blast, simp)
  2758   done
  2759 
  2760 lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
  2761 -- {* The famous take-lemma. *}
  2762 apply (drule_tac x = "max (length xs) (length ys)" in spec)
  2763 apply (simp add: le_max_iff_disj)
  2764 done
  2765 
  2766 
  2767 lemma take_Cons':
  2768      "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
  2769 by (cases n) simp_all
  2770 
  2771 lemma drop_Cons':
  2772      "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
  2773 by (cases n) simp_all
  2774 
  2775 lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
  2776 by (cases n) simp_all
  2777 
  2778 lemmas take_Cons_number_of = take_Cons'[of "number_of v"] for v
  2779 lemmas drop_Cons_number_of = drop_Cons'[of "number_of v"] for v
  2780 lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v"] for v
  2781 
  2782 declare take_Cons_number_of [simp] 
  2783         drop_Cons_number_of [simp] 
  2784         nth_Cons_number_of [simp] 
  2785 
  2786 
  2787 subsubsection {* @{text upto}: interval-list on @{typ int} *}
  2788 
  2789 (* FIXME make upto tail recursive? *)
  2790 
  2791 function upto :: "int \<Rightarrow> int \<Rightarrow> int list" ("(1[_../_])") where
  2792 "upto i j = (if i \<le> j then i # [i+1..j] else [])"
  2793 by auto
  2794 termination
  2795 by(relation "measure(%(i::int,j). nat(j - i + 1))") auto
  2796 
  2797 declare upto.simps[code, simp del]
  2798 
  2799 lemmas upto_rec_number_of[simp] = upto.simps[of "number_of m" "number_of n"] for m n
  2800 
  2801 lemma upto_empty[simp]: "j < i \<Longrightarrow> [i..j] = []"
  2802 by(simp add: upto.simps)
  2803 
  2804 lemma set_upto[simp]: "set[i..j] = {i..j}"
  2805 proof(induct i j rule:upto.induct)
  2806   case (1 i j)
  2807   from this show ?case
  2808     unfolding upto.simps[of i j] simp_from_to[of i j] by auto
  2809 qed
  2810 
  2811 
  2812 subsubsection {* @{text "distinct"} and @{text remdups} *}
  2813 
  2814 lemma distinct_tl:
  2815   "distinct xs \<Longrightarrow> distinct (tl xs)"
  2816   by (cases xs) simp_all
  2817 
  2818 lemma distinct_append [simp]:
  2819 "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
  2820 by (induct xs) auto
  2821 
  2822 lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
  2823 by(induct xs) auto
  2824 
  2825 lemma set_remdups [simp]: "set (remdups xs) = set xs"
  2826 by (induct xs) (auto simp add: insert_absorb)
  2827 
  2828 lemma distinct_remdups [iff]: "distinct (remdups xs)"
  2829 by (induct xs) auto
  2830 
  2831 lemma distinct_remdups_id: "distinct xs ==> remdups xs = xs"
  2832 by (induct xs, auto)
  2833 
  2834 lemma remdups_id_iff_distinct [simp]: "remdups xs = xs \<longleftrightarrow> distinct xs"
  2835 by (metis distinct_remdups distinct_remdups_id)
  2836 
  2837 lemma finite_distinct_list: "finite A \<Longrightarrow> EX xs. set xs = A & distinct xs"
  2838 by (metis distinct_remdups finite_list set_remdups)
  2839 
  2840 lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
  2841 by (induct x, auto) 
  2842 
  2843 lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
  2844 by (induct x, auto)
  2845 
  2846 lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
  2847 by (induct xs) auto
  2848 
  2849 lemma length_remdups_eq[iff]:
  2850   "(length (remdups xs) = length xs) = (remdups xs = xs)"
  2851 apply(induct xs)
  2852  apply auto
  2853 apply(subgoal_tac "length (remdups xs) <= length xs")
  2854  apply arith
  2855 apply(rule length_remdups_leq)
  2856 done
  2857 
  2858 lemma remdups_filter: "remdups(filter P xs) = filter P (remdups xs)"
  2859 apply(induct xs)
  2860 apply auto
  2861 done
  2862 
  2863 lemma distinct_map:
  2864   "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
  2865 by (induct xs) auto
  2866 
  2867 
  2868 lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
  2869 by (induct xs) auto
  2870 
  2871 lemma distinct_upt[simp]: "distinct[i..<j]"
  2872 by (induct j) auto
  2873 
  2874 lemma distinct_upto[simp]: "distinct[i..j]"
  2875 apply(induct i j rule:upto.induct)
  2876 apply(subst upto.simps)
  2877 apply(simp)
  2878 done
  2879 
  2880 lemma distinct_take[simp]: "distinct xs \<Longrightarrow> distinct (take i xs)"
  2881 apply(induct xs arbitrary: i)
  2882  apply simp
  2883 apply (case_tac i)
  2884  apply simp_all
  2885 apply(blast dest:in_set_takeD)
  2886 done
  2887 
  2888 lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)"
  2889 apply(induct xs arbitrary: i)
  2890  apply simp
  2891 apply (case_tac i)
  2892  apply simp_all
  2893 done
  2894 
  2895 lemma distinct_list_update:
  2896 assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
  2897 shows "distinct (xs[i:=a])"
  2898 proof (cases "i < length xs")
  2899   case True
  2900   with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
  2901     apply (drule_tac id_take_nth_drop) by simp
  2902   with d True show ?thesis
  2903     apply (simp add: upd_conv_take_nth_drop)
  2904     apply (drule subst [OF id_take_nth_drop]) apply assumption
  2905     apply simp apply (cases "a = xs!i") apply simp by blast
  2906 next
  2907   case False with d show ?thesis by auto
  2908 qed
  2909 
  2910 lemma distinct_concat:
  2911   assumes "distinct xs"
  2912   and "\<And> ys. ys \<in> set xs \<Longrightarrow> distinct ys"
  2913   and "\<And> ys zs. \<lbrakk> ys \<in> set xs ; zs \<in> set xs ; ys \<noteq> zs \<rbrakk> \<Longrightarrow> set ys \<inter> set zs = {}"
  2914   shows "distinct (concat xs)"
  2915   using assms by (induct xs) auto
  2916 
  2917 text {* It is best to avoid this indexed version of distinct, but
  2918 sometimes it is useful. *}
  2919 
  2920 lemma distinct_conv_nth:
  2921 "distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
  2922 apply (induct xs, simp, simp)
  2923 apply (rule iffI, clarsimp)
  2924  apply (case_tac i)
  2925 apply (case_tac j, simp)
  2926 apply (simp add: set_conv_nth)
  2927  apply (case_tac j)
  2928 apply (clarsimp simp add: set_conv_nth, simp) 
  2929 apply (rule conjI)
  2930 (*TOO SLOW
  2931 apply (metis Zero_neq_Suc gr0_conv_Suc in_set_conv_nth lessI less_trans_Suc nth_Cons' nth_Cons_Suc)
  2932 *)
  2933  apply (clarsimp simp add: set_conv_nth)
  2934  apply (erule_tac x = 0 in allE, simp)
  2935  apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
  2936 (*TOO SLOW
  2937 apply (metis Suc_Suc_eq lessI less_trans_Suc nth_Cons_Suc)
  2938 *)
  2939 apply (erule_tac x = "Suc i" in allE, simp)
  2940 apply (erule_tac x = "Suc j" in allE, simp)
  2941 done
  2942 
  2943 lemma nth_eq_iff_index_eq:
  2944  "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
  2945 by(auto simp: distinct_conv_nth)
  2946 
  2947 lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
  2948 by (induct xs) auto
  2949 
  2950 lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
  2951 proof (induct xs)
  2952   case Nil thus ?case by simp
  2953 next
  2954   case (Cons x xs)
  2955   show ?case
  2956   proof (cases "x \<in> set xs")
  2957     case False with Cons show ?thesis by simp
  2958   next
  2959     case True with Cons.prems
  2960     have "card (set xs) = Suc (length xs)" 
  2961       by (simp add: card_insert_if split: split_if_asm)
  2962     moreover have "card (set xs) \<le> length xs" by (rule card_length)
  2963     ultimately have False by simp
  2964     thus ?thesis ..
  2965   qed
  2966 qed
  2967 
  2968 lemma distinct_length_filter: "distinct xs \<Longrightarrow> length (filter P xs) = card ({x. P x} Int set xs)"
  2969 by (induct xs) (auto)
  2970 
  2971 lemma not_distinct_decomp: "~ distinct ws ==> EX xs ys zs y. ws = xs@[y]@ys@[y]@zs"
  2972 apply (induct n == "length ws" arbitrary:ws) apply simp
  2973 apply(case_tac ws) apply simp
  2974 apply (simp split:split_if_asm)
  2975 apply (metis Cons_eq_appendI eq_Nil_appendI split_list)
  2976 done
  2977 
  2978 lemma not_distinct_conv_prefix:
  2979   defines "dec as xs y ys \<equiv> y \<in> set xs \<and> distinct xs \<and> as = xs @ y # ys"
  2980   shows "\<not>distinct as \<longleftrightarrow> (\<exists>xs y ys. dec as xs y ys)" (is "?L = ?R")
  2981 proof
  2982   assume "?L" then show "?R"
  2983   proof (induct "length as" arbitrary: as rule: less_induct)
  2984     case less
  2985     obtain xs ys zs y where decomp: "as = (xs @ y # ys) @ y # zs"
  2986       using not_distinct_decomp[OF less.prems] by auto
  2987     show ?case
  2988     proof (cases "distinct (xs @ y # ys)")
  2989       case True
  2990       with decomp have "dec as (xs @ y # ys) y zs" by (simp add: dec_def)
  2991       then show ?thesis by blast
  2992     next
  2993       case False
  2994       with less decomp obtain xs' y' ys' where "dec (xs @ y # ys) xs' y' ys'"
  2995         by atomize_elim auto
  2996       with decomp have "dec as xs' y' (ys' @ y # zs)" by (simp add: dec_def)
  2997       then show ?thesis by blast
  2998     qed
  2999   qed
  3000 qed (auto simp: dec_def)
  3001 
  3002 lemma length_remdups_concat:
  3003   "length (remdups (concat xss)) = card (\<Union>xs\<in>set xss. set xs)"
  3004   by (simp add: distinct_card [symmetric])
  3005 
  3006 lemma length_remdups_card_conv: "length(remdups xs) = card(set xs)"
  3007 proof -
  3008   have xs: "concat[xs] = xs" by simp
  3009   from length_remdups_concat[of "[xs]"] show ?thesis unfolding xs by simp
  3010 qed
  3011 
  3012 lemma remdups_remdups:
  3013   "remdups (remdups xs) = remdups xs"
  3014   by (induct xs) simp_all
  3015 
  3016 lemma distinct_butlast:
  3017   assumes "xs \<noteq> []" and "distinct xs"
  3018   shows "distinct (butlast xs)"
  3019 proof -
  3020   from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  3021   with `distinct xs` show ?thesis by simp
  3022 qed
  3023 
  3024 lemma remdups_map_remdups:
  3025   "remdups (map f (remdups xs)) = remdups (map f xs)"
  3026   by (induct xs) simp_all
  3027 
  3028 lemma distinct_zipI1:
  3029   assumes "distinct xs"
  3030   shows "distinct (zip xs ys)"
  3031 proof (rule zip_obtain_same_length)
  3032   fix xs' :: "'a list" and ys' :: "'b list" and n
  3033   assume "length xs' = length ys'"
  3034   assume "xs' = take n xs"
  3035   with assms have "distinct xs'" by simp
  3036   with `length xs' = length ys'` show "distinct (zip xs' ys')"
  3037     by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
  3038 qed
  3039 
  3040 lemma distinct_zipI2:
  3041   assumes "distinct ys"
  3042   shows "distinct (zip xs ys)"
  3043 proof (rule zip_obtain_same_length)
  3044   fix xs' :: "'b list" and ys' :: "'a list" and n
  3045   assume "length xs' = length ys'"
  3046   assume "ys' = take n ys"
  3047   with assms have "distinct ys'" by simp
  3048   with `length xs' = length ys'` show "distinct (zip xs' ys')"
  3049     by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
  3050 qed
  3051 
  3052 (* The next two lemmas help Sledgehammer. *)
  3053 
  3054 lemma distinct_singleton: "distinct [x]" by simp
  3055 
  3056 lemma distinct_length_2_or_more:
  3057 "distinct (a # b # xs) \<longleftrightarrow> (a \<noteq> b \<and> distinct (a # xs) \<and> distinct (b # xs))"
  3058 by (metis distinct.simps(2) hd.simps hd_in_set list.simps(2) set_ConsD set_rev_mp set_subset_Cons)
  3059 
  3060 subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
  3061 
  3062 lemma (in monoid_add) listsum_simps [simp]:
  3063   "listsum [] = 0"
  3064   "listsum (x # xs) = x + listsum xs"
  3065   by (simp_all add: listsum_def)
  3066 
  3067 lemma (in monoid_add) listsum_append [simp]:
  3068   "listsum (xs @ ys) = listsum xs + listsum ys"
  3069   by (induct xs) (simp_all add: add.assoc)
  3070 
  3071 lemma (in comm_monoid_add) listsum_rev [simp]:
  3072   "listsum (rev xs) = listsum xs"
  3073   by (simp add: listsum_def foldr_def fold_rev fun_eq_iff add_ac)
  3074 
  3075 lemma (in monoid_add) fold_plus_listsum_rev:
  3076   "fold plus xs = plus (listsum (rev xs))"
  3077 proof
  3078   fix x
  3079   have "fold plus xs x = fold plus xs (x + 0)" by simp
  3080   also have "\<dots> = fold plus (x # xs) 0" by simp
  3081   also have "\<dots> = foldr plus (rev xs @ [x]) 0" by (simp add: foldr_def)
  3082   also have "\<dots> = listsum (rev xs @ [x])" by (simp add: listsum_def)
  3083   also have "\<dots> = listsum (rev xs) + listsum [x]" by simp
  3084   finally show "fold plus xs x = listsum (rev xs) + x" by simp
  3085 qed
  3086 
  3087 lemma (in semigroup_add) foldl_assoc:
  3088   "foldl plus (x + y) zs = x + foldl plus y zs"
  3089   by (simp add: foldl_def fold_commute_apply [symmetric] fun_eq_iff add_assoc)
  3090 
  3091 lemma (in ab_semigroup_add) foldr_conv_foldl:
  3092   "foldr plus xs a = foldl plus a xs"
  3093   by (simp add: foldl_def foldr_fold fun_eq_iff add_ac)
  3094 
  3095 text {*
  3096   Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
  3097   difficult to use because it requires an additional transitivity step.
  3098 *}
  3099 
  3100 lemma start_le_sum:
  3101   fixes m n :: nat
  3102   shows "m \<le> n \<Longrightarrow> m \<le> foldl plus n ns"
  3103   by (simp add: foldl_def add_commute fold_plus_listsum_rev)
  3104 
  3105 lemma elem_le_sum:
  3106   fixes m n :: nat 
  3107   shows "n \<in> set ns \<Longrightarrow> n \<le> foldl plus 0 ns"
  3108   by (force intro: start_le_sum simp add: in_set_conv_decomp)
  3109 
  3110 lemma sum_eq_0_conv [iff]:
  3111   fixes m :: nat
  3112   shows "foldl plus m ns = 0 \<longleftrightarrow> m = 0 \<and> (\<forall>n \<in> set ns. n = 0)"
  3113   by (induct ns arbitrary: m) auto
  3114 
  3115 text{* Some syntactic sugar for summing a function over a list: *}
  3116 
  3117 syntax
  3118   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
  3119 syntax (xsymbols)
  3120   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
  3121 syntax (HTML output)
  3122   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
  3123 
  3124 translations -- {* Beware of argument permutation! *}
  3125   "SUM x<-xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  3126   "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  3127 
  3128 lemma (in comm_monoid_add) listsum_map_remove1:
  3129   "x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))"
  3130   by (induct xs) (auto simp add: ac_simps)
  3131 
  3132 lemma (in monoid_add) list_size_conv_listsum:
  3133   "list_size f xs = listsum (map f xs) + size xs"
  3134   by (induct xs) auto
  3135 
  3136 lemma (in monoid_add) length_concat:
  3137   "length (concat xss) = listsum (map length xss)"
  3138   by (induct xss) simp_all
  3139 
  3140 lemma (in monoid_add) listsum_map_filter:
  3141   assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
  3142   shows "listsum (map f (filter P xs)) = listsum (map f xs)"
  3143   using assms by (induct xs) auto
  3144 
  3145 lemma (in monoid_add) distinct_listsum_conv_Setsum:
  3146   "distinct xs \<Longrightarrow> listsum xs = Setsum (set xs)"
  3147   by (induct xs) simp_all
  3148 
  3149 lemma listsum_eq_0_nat_iff_nat [simp]:
  3150   "listsum ns = (0::nat) \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
  3151   by (simp add: listsum_def foldr_conv_foldl)
  3152 
  3153 lemma elem_le_listsum_nat:
  3154   "k < size ns \<Longrightarrow> ns ! k \<le> listsum (ns::nat list)"
  3155 apply(induct ns arbitrary: k)
  3156  apply simp
  3157 apply(fastforce simp add:nth_Cons split: nat.split)
  3158 done
  3159 
  3160 lemma listsum_update_nat:
  3161   "k<size ns \<Longrightarrow> listsum (ns[k := (n::nat)]) = listsum ns + n - ns ! k"
  3162 apply(induct ns arbitrary:k)
  3163  apply (auto split:nat.split)
  3164 apply(drule elem_le_listsum_nat)
  3165 apply arith
  3166 done
  3167 
  3168 lemma (in monoid_add) listsum_triv:
  3169   "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
  3170   by (induct xs) (simp_all add: left_distrib)
  3171 
  3172 lemma (in monoid_add) listsum_0 [simp]:
  3173   "(\<Sum>x\<leftarrow>xs. 0) = 0"
  3174   by (induct xs) (simp_all add: left_distrib)
  3175 
  3176 text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
  3177 lemma (in ab_group_add) uminus_listsum_map:
  3178   "- listsum (map f xs) = listsum (map (uminus \<circ> f) xs)"
  3179   by (induct xs) simp_all
  3180 
  3181 lemma (in comm_monoid_add) listsum_addf:
  3182   "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
  3183   by (induct xs) (simp_all add: algebra_simps)
  3184 
  3185 lemma (in ab_group_add) listsum_subtractf:
  3186   "(\<Sum>x\<leftarrow>xs. f x - g x) = listsum (map f xs) - listsum (map g xs)"
  3187   by (induct xs) (simp_all add: algebra_simps)
  3188 
  3189 lemma (in semiring_0) listsum_const_mult:
  3190   "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
  3191   by (induct xs) (simp_all add: algebra_simps)
  3192 
  3193 lemma (in semiring_0) listsum_mult_const:
  3194   "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
  3195   by (induct xs) (simp_all add: algebra_simps)
  3196 
  3197 lemma (in ordered_ab_group_add_abs) listsum_abs:
  3198   "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
  3199   by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])
  3200 
  3201 lemma listsum_mono:
  3202   fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}"
  3203   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)"
  3204   by (induct xs) (simp, simp add: add_mono)
  3205 
  3206 lemma (in monoid_add) listsum_distinct_conv_setsum_set:
  3207   "distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)"
  3208   by (induct xs) simp_all
  3209 
  3210 lemma (in monoid_add) interv_listsum_conv_setsum_set_nat:
  3211   "listsum (map f [m..<n]) = setsum f (set [m..<n])"
  3212   by (simp add: listsum_distinct_conv_setsum_set)
  3213 
  3214 lemma (in monoid_add) interv_listsum_conv_setsum_set_int:
  3215   "listsum (map f [k..l]) = setsum f (set [k..l])"
  3216   by (simp add: listsum_distinct_conv_setsum_set)
  3217 
  3218 text {* General equivalence between @{const listsum} and @{const setsum} *}
  3219 lemma (in monoid_add) listsum_setsum_nth:
  3220   "listsum xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
  3221   using interv_listsum_conv_setsum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth)
  3222 
  3223 
  3224 subsubsection {* @{const insert} *}
  3225 
  3226 lemma in_set_insert [simp]:
  3227   "x \<in> set xs \<Longrightarrow> List.insert x xs = xs"
  3228   by (simp add: List.insert_def)
  3229 
  3230 lemma not_in_set_insert [simp]:
  3231   "x \<notin> set xs \<Longrightarrow> List.insert x xs = x # xs"
  3232   by (simp add: List.insert_def)
  3233 
  3234 lemma insert_Nil [simp]:
  3235   "List.insert x [] = [x]"
  3236   by simp
  3237 
  3238 lemma set_insert [simp]:
  3239   "set (List.insert x xs) = insert x (set xs)"
  3240   by (auto simp add: List.insert_def)
  3241 
  3242 lemma distinct_insert [simp]:
  3243   "distinct xs \<Longrightarrow> distinct (List.insert x xs)"
  3244   by (simp add: List.insert_def)
  3245 
  3246 lemma insert_remdups:
  3247   "List.insert x (remdups xs) = remdups (List.insert x xs)"
  3248   by (simp add: List.insert_def)
  3249 
  3250 
  3251 subsubsection {* @{text remove1} *}
  3252 
  3253 lemma remove1_append:
  3254   "remove1 x (xs @ ys) =
  3255   (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
  3256 by (induct xs) auto
  3257 
  3258 lemma remove1_commute: "remove1 x (remove1 y zs) = remove1 y (remove1 x zs)"
  3259 by (induct zs) auto
  3260 
  3261 lemma in_set_remove1[simp]:
  3262   "a \<noteq> b \<Longrightarrow> a : set(remove1 b xs) = (a : set xs)"
  3263 apply (induct xs)
  3264 apply auto
  3265 done
  3266 
  3267 lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
  3268 apply(induct xs)
  3269  apply simp
  3270 apply simp
  3271 apply blast
  3272 done
  3273 
  3274 lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
  3275 apply(induct xs)
  3276  apply simp
  3277 apply simp
  3278 apply blast
  3279 done
  3280 
  3281 lemma length_remove1:
  3282   "length(remove1 x xs) = (if x : set xs then length xs - 1 else length xs)"
  3283 apply (induct xs)
  3284  apply (auto dest!:length_pos_if_in_set)
  3285 done
  3286 
  3287 lemma remove1_filter_not[simp]:
  3288   "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
  3289 by(induct xs) auto
  3290 
  3291 lemma filter_remove1:
  3292   "filter Q (remove1 x xs) = remove1 x (filter Q xs)"
  3293 by (induct xs) auto
  3294 
  3295 lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
  3296 apply(insert set_remove1_subset)
  3297 apply fast
  3298 done
  3299 
  3300 lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
  3301 by (induct xs) simp_all
  3302 
  3303 lemma remove1_remdups:
  3304   "distinct xs \<Longrightarrow> remove1 x (remdups xs) = remdups (remove1 x xs)"
  3305   by (induct xs) simp_all
  3306 
  3307 lemma remove1_idem:
  3308   assumes "x \<notin> set xs"
  3309   shows "remove1 x xs = xs"
  3310   using assms by (induct xs) simp_all
  3311 
  3312 
  3313 subsubsection {* @{text removeAll} *}
  3314 
  3315 lemma removeAll_filter_not_eq:
  3316   "removeAll x = filter (\<lambda>y. x \<noteq> y)"
  3317 proof
  3318   fix xs
  3319   show "removeAll x xs = filter (\<lambda>y. x \<noteq> y) xs"
  3320     by (induct xs) auto
  3321 qed
  3322 
  3323 lemma removeAll_append[simp]:
  3324   "removeAll x (xs @ ys) = removeAll x xs @ removeAll x ys"
  3325 by (induct xs) auto
  3326 
  3327 lemma set_removeAll[simp]: "set(removeAll x xs) = set xs - {x}"
  3328 by (induct xs) auto
  3329 
  3330 lemma removeAll_id[simp]: "x \<notin> set xs \<Longrightarrow> removeAll x xs = xs"
  3331 by (induct xs) auto
  3332 
  3333 (* Needs count:: 'a \<Rightarrow> a' list \<Rightarrow> nat
  3334 lemma length_removeAll:
  3335   "length(removeAll x xs) = length xs - count x xs"
  3336 *)
  3337 
  3338 lemma removeAll_filter_not[simp]:
  3339   "\<not> P x \<Longrightarrow> removeAll x (filter P xs) = filter P xs"
  3340 by(induct xs) auto
  3341 
  3342 lemma distinct_removeAll:
  3343   "distinct xs \<Longrightarrow> distinct (removeAll x xs)"
  3344   by (simp add: removeAll_filter_not_eq)
  3345 
  3346 lemma distinct_remove1_removeAll:
  3347   "distinct xs ==> remove1 x xs = removeAll x xs"
  3348 by (induct xs) simp_all
  3349 
  3350 lemma map_removeAll_inj_on: "inj_on f (insert x (set xs)) \<Longrightarrow>
  3351   map f (removeAll x xs) = removeAll (f x) (map f xs)"
  3352 by (induct xs) (simp_all add:inj_on_def)
  3353 
  3354 lemma map_removeAll_inj: "inj f \<Longrightarrow>
  3355   map f (removeAll x xs) = removeAll (f x) (map f xs)"
  3356 by(metis map_removeAll_inj_on subset_inj_on subset_UNIV)
  3357 
  3358 
  3359 subsubsection {* @{text replicate} *}
  3360 
  3361 lemma length_replicate [simp]: "length (replicate n x) = n"
  3362 by (induct n) auto
  3363 
  3364 lemma Ex_list_of_length: "\<exists>xs. length xs = n"
  3365 by (rule exI[of _ "replicate n undefined"]) simp
  3366 
  3367 lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
  3368 by (induct n) auto
  3369 
  3370 lemma map_replicate_const:
  3371   "map (\<lambda> x. k) lst = replicate (length lst) k"
  3372   by (induct lst) auto
  3373 
  3374 lemma replicate_app_Cons_same:
  3375 "(replicate n x) @ (x # xs) = x # replicate n x @ xs"
  3376 by (induct n) auto
  3377 
  3378 lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
  3379 apply (induct n, simp)
  3380 apply (simp add: replicate_app_Cons_same)
  3381 done
  3382 
  3383 lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
  3384 by (induct n) auto
  3385 
  3386 text{* Courtesy of Matthias Daum: *}
  3387 lemma append_replicate_commute:
  3388   "replicate n x @ replicate k x = replicate k x @ replicate n x"
  3389 apply (simp add: replicate_add [THEN sym])
  3390 apply (simp add: add_commute)
  3391 done
  3392 
  3393 text{* Courtesy of Andreas Lochbihler: *}
  3394 lemma filter_replicate:
  3395   "filter P (replicate n x) = (if P x then replicate n x else [])"
  3396 by(induct n) auto
  3397 
  3398 lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
  3399 by (induct n) auto
  3400 
  3401 lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
  3402 by (induct n) auto
  3403 
  3404 lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
  3405 by (atomize (full), induct n) auto
  3406 
  3407 lemma nth_replicate[simp]: "i < n ==> (replicate n x)!i = x"
  3408 apply (induct n arbitrary: i, simp)
  3409 apply (simp add: nth_Cons split: nat.split)
  3410 done
  3411 
  3412 text{* Courtesy of Matthias Daum (2 lemmas): *}
  3413 lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
  3414 apply (case_tac "k \<le> i")
  3415  apply  (simp add: min_def)
  3416 apply (drule not_leE)
  3417 apply (simp add: min_def)
  3418 apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
  3419  apply  simp
  3420 apply (simp add: replicate_add [symmetric])
  3421 done
  3422 
  3423 lemma drop_replicate[simp]: "drop i (replicate k x) = replicate (k-i) x"
  3424 apply (induct k arbitrary: i)
  3425  apply simp
  3426 apply clarsimp
  3427 apply (case_tac i)
  3428  apply simp
  3429 apply clarsimp
  3430 done
  3431 
  3432 
  3433 lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
  3434 by (induct n) auto
  3435 
  3436 lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
  3437 by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
  3438 
  3439 lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
  3440 by auto
  3441 
  3442 lemma in_set_replicate[simp]: "(x : set (replicate n y)) = (x = y & n \<noteq> 0)"
  3443 by (simp add: set_replicate_conv_if)
  3444 
  3445 lemma Ball_set_replicate[simp]:
  3446   "(ALL x : set(replicate n a). P x) = (P a | n=0)"
  3447 by(simp add: set_replicate_conv_if)
  3448 
  3449 lemma Bex_set_replicate[simp]:
  3450   "(EX x : set(replicate n a). P x) = (P a & n\<noteq>0)"
  3451 by(simp add: set_replicate_conv_if)
  3452 
  3453 lemma replicate_append_same:
  3454   "replicate i x @ [x] = x # replicate i x"
  3455   by (induct i) simp_all
  3456 
  3457 lemma map_replicate_trivial:
  3458   "map (\<lambda>i. x) [0..<i] = replicate i x"
  3459   by (induct i) (simp_all add: replicate_append_same)
  3460 
  3461 lemma concat_replicate_trivial[simp]:
  3462   "concat (replicate i []) = []"
  3463   by (induct i) (auto simp add: map_replicate_const)
  3464 
  3465 lemma replicate_empty[simp]: "(replicate n x = []) \<longleftrightarrow> n=0"
  3466 by (induct n) auto
  3467 
  3468 lemma empty_replicate[simp]: "([] = replicate n x) \<longleftrightarrow> n=0"
  3469 by (induct n) auto
  3470 
  3471 lemma replicate_eq_replicate[simp]:
  3472   "(replicate m x = replicate n y) \<longleftrightarrow> (m=n & (m\<noteq>0 \<longrightarrow> x=y))"
  3473 apply(induct m arbitrary: n)
  3474  apply simp
  3475 apply(induct_tac n)
  3476 apply auto
  3477 done
  3478 
  3479 lemma replicate_length_filter:
  3480   "replicate (length (filter (\<lambda>y. x = y) xs)) x = filter (\<lambda>y. x = y) xs"
  3481   by (induct xs) auto
  3482 
  3483 lemma comm_append_are_replicate:
  3484   fixes xs ys :: "'a list"
  3485   assumes "xs \<noteq> []" "ys \<noteq> []"
  3486   assumes "xs @ ys = ys @ xs"
  3487   shows "\<exists>m n zs. concat (replicate m zs) = xs \<and> concat (replicate n zs) = ys"
  3488   using assms
  3489 proof (induct "length (xs @ ys)" arbitrary: xs ys rule: less_induct)
  3490   case less
  3491 
  3492   def xs' \<equiv> "if (length xs \<le> length ys) then xs else ys"
  3493     and ys' \<equiv> "if (length xs \<le> length ys) then ys else xs"
  3494   then have
  3495     prems': "length xs' \<le> length ys'"
  3496             "xs' @ ys' = ys' @ xs'"
  3497       and "xs' \<noteq> []"
  3498       and len: "length (xs @ ys) = length (xs' @ ys')"
  3499     using less by (auto intro: less.hyps)
  3500 
  3501   from prems'
  3502   obtain ws where "ys' = xs' @ ws"
  3503     by (auto simp: append_eq_append_conv2)
  3504 
  3505   have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ys'"
  3506   proof (cases "ws = []")
  3507     case True
  3508     then have "concat (replicate 1 xs') = xs'"
  3509       and "concat (replicate 1 xs') = ys'"
  3510       using `ys' = xs' @ ws` by auto
  3511     then show ?thesis by blast
  3512   next
  3513     case False
  3514     from `ys' = xs' @ ws` and `xs' @ ys' = ys' @ xs'`
  3515     have "xs' @ ws = ws @ xs'" by simp
  3516     then have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ws"
  3517       using False and `xs' \<noteq> []` and `ys' = xs' @ ws` and len
  3518       by (intro less.hyps) auto
  3519     then obtain m n zs where "concat (replicate m zs) = xs'"
  3520       and "concat (replicate n zs) = ws" by blast
  3521     moreover
  3522     then have "concat (replicate (m + n) zs) = ys'"
  3523       using `ys' = xs' @ ws`
  3524       by (simp add: replicate_add)
  3525     ultimately
  3526     show ?thesis by blast
  3527   qed
  3528   then show ?case
  3529     using xs'_def ys'_def by metis
  3530 qed
  3531 
  3532 lemma comm_append_is_replicate:
  3533   fixes xs ys :: "'a list"
  3534   assumes "xs \<noteq> []" "ys \<noteq> []"
  3535   assumes "xs @ ys = ys @ xs"
  3536   shows "\<exists>n zs. n > 1 \<and> concat (replicate n zs) = xs @ ys"
  3537 
  3538 proof -
  3539   obtain m n zs where "concat (replicate m zs) = xs"
  3540     and "concat (replicate n zs) = ys"
  3541     using assms by (metis comm_append_are_replicate)
  3542   then have "m + n > 1" and "concat (replicate (m+n) zs) = xs @ ys"
  3543     using `xs \<noteq> []` and `ys \<noteq> []`
  3544     by (auto simp: replicate_add)
  3545   then show ?thesis by blast
  3546 qed
  3547 
  3548 
  3549 subsubsection{*@{text rotate1} and @{text rotate}*}
  3550 
  3551 lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
  3552 by(simp add:rotate1_def)
  3553 
  3554 lemma rotate0[simp]: "rotate 0 = id"
  3555 by(simp add:rotate_def)
  3556 
  3557 lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
  3558 by(simp add:rotate_def)
  3559 
  3560 lemma rotate_add:
  3561   "rotate (m+n) = rotate m o rotate n"
  3562 by(simp add:rotate_def funpow_add)
  3563 
  3564 lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
  3565 by(simp add:rotate_add)
  3566 
  3567 lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
  3568 by(simp add:rotate_def funpow_swap1)
  3569 
  3570 lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
  3571 by(cases xs) simp_all
  3572 
  3573 lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
  3574 apply(induct n)
  3575  apply simp
  3576 apply (simp add:rotate_def)
  3577 done
  3578 
  3579 lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
  3580 by(simp add:rotate1_def split:list.split)
  3581 
  3582 lemma rotate_drop_take:
  3583   "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
  3584 apply(induct n)
  3585  apply simp
  3586 apply(simp add:rotate_def)
  3587 apply(cases "xs = []")
  3588  apply (simp)
  3589 apply(case_tac "n mod length xs = 0")
  3590  apply(simp add:mod_Suc)
  3591  apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
  3592 apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
  3593                 take_hd_drop linorder_not_le)
  3594 done
  3595 
  3596 lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
  3597 by(simp add:rotate_drop_take)
  3598 
  3599 lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
  3600 by(simp add:rotate_drop_take)
  3601 
  3602 lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
  3603 by(simp add:rotate1_def split:list.split)
  3604 
  3605 lemma length_rotate[simp]: "length(rotate n xs) = length xs"
  3606 by (induct n arbitrary: xs) (simp_all add:rotate_def)
  3607 
  3608 lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
  3609 by(simp add:rotate1_def split:list.split) blast
  3610 
  3611 lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
  3612 by (induct n) (simp_all add:rotate_def)
  3613 
  3614 lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
  3615 by(simp add:rotate_drop_take take_map drop_map)
  3616 
  3617 lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
  3618 by (cases xs) (auto simp add:rotate1_def)
  3619 
  3620 lemma set_rotate[simp]: "set(rotate n xs) = set xs"
  3621 by (induct n) (simp_all add:rotate_def)
  3622 
  3623 lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
  3624 by(simp add:rotate1_def split:list.split)
  3625 
  3626 lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
  3627 by (induct n) (simp_all add:rotate_def)
  3628 
  3629 lemma rotate_rev:
  3630   "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
  3631 apply(simp add:rotate_drop_take rev_drop rev_take)
  3632 apply(cases "length xs = 0")
  3633  apply simp
  3634 apply(cases "n mod length xs = 0")
  3635  apply simp
  3636 apply(simp add:rotate_drop_take rev_drop rev_take)
  3637 done
  3638 
  3639 lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
  3640 apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
  3641 apply(subgoal_tac "length xs \<noteq> 0")
  3642  prefer 2 apply simp
  3643 using mod_less_divisor[of "length xs" n] by arith
  3644 
  3645 
  3646 subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
  3647 
  3648 lemma sublist_empty [simp]: "sublist xs {} = []"
  3649 by (auto simp add: sublist_def)
  3650 
  3651 lemma sublist_nil [simp]: "sublist [] A = []"
  3652 by (auto simp add: sublist_def)
  3653 
  3654 lemma length_sublist:
  3655   "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
  3656 by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
  3657 
  3658 lemma sublist_shift_lemma_Suc:
  3659   "map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
  3660    map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
  3661 apply(induct xs arbitrary: "is")
  3662  apply simp
  3663 apply (case_tac "is")
  3664  apply simp
  3665 apply simp
  3666 done
  3667 
  3668 lemma sublist_shift_lemma:
  3669      "map fst [p<-zip xs [i..<i + length xs] . snd p : A] =
  3670       map fst [p<-zip xs [0..<length xs] . snd p + i : A]"
  3671 by (induct xs rule: rev_induct) (simp_all add: add_commute)
  3672 
  3673 lemma sublist_append:
  3674      "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
  3675 apply (unfold sublist_def)
  3676 apply (induct l' rule: rev_induct, simp)
  3677 apply (simp add: upt_add_eq_append[of 0] sublist_shift_lemma)
  3678 apply (simp add: add_commute)
  3679 done
  3680 
  3681 lemma sublist_Cons:
  3682 "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
  3683 apply (induct l rule: rev_induct)
  3684  apply (simp add: sublist_def)
  3685 apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
  3686 done
  3687 
  3688 lemma set_sublist: "set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
  3689 apply(induct xs arbitrary: I)
  3690 apply(auto simp: sublist_Cons nth_Cons split:nat.split dest!: gr0_implies_Suc)
  3691 done
  3692 
  3693 lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
  3694 by(auto simp add:set_sublist)
  3695 
  3696 lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
  3697 by(auto simp add:set_sublist)
  3698 
  3699 lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
  3700 by(auto simp add:set_sublist)
  3701 
  3702 lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
  3703 by (simp add: sublist_Cons)
  3704 
  3705 
  3706 lemma distinct_sublistI[simp]: "distinct xs \<Longrightarrow> distinct(sublist xs I)"
  3707 apply(induct xs arbitrary: I)
  3708  apply simp
  3709 apply(auto simp add:sublist_Cons)
  3710 done
  3711 
  3712 
  3713 lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
  3714 apply (induct l rule: rev_induct, simp)
  3715 apply (simp split: nat_diff_split add: sublist_append)
  3716 done
  3717 
  3718 lemma filter_in_sublist:
  3719  "distinct xs \<Longrightarrow> filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
  3720 proof (induct xs arbitrary: s)
  3721   case Nil thus ?case by simp
  3722 next
  3723   case (Cons a xs)
  3724   moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
  3725   ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
  3726 qed
  3727 
  3728 
  3729 subsubsection {* @{const splice} *}
  3730 
  3731 lemma splice_Nil2 [simp, code]: "splice xs [] = xs"
  3732 by (cases xs) simp_all
  3733 
  3734 declare splice.simps(1,3)[code]
  3735 declare splice.simps(2)[simp del]
  3736 
  3737 lemma length_splice[simp]: "length(splice xs ys) = length xs + length ys"
  3738 by (induct xs ys rule: splice.induct) auto
  3739 
  3740 
  3741 subsubsection {* Transpose *}
  3742 
  3743 function transpose where
  3744 "transpose []             = []" |
  3745 "transpose ([]     # xss) = transpose xss" |
  3746 "transpose ((x#xs) # xss) =
  3747   (x # [h. (h#t) \<leftarrow> xss]) # transpose (xs # [t. (h#t) \<leftarrow> xss])"
  3748 by pat_completeness auto
  3749 
  3750 lemma transpose_aux_filter_head:
  3751   "concat (map (list_case [] (\<lambda>h t. [h])) xss) =
  3752   map (\<lambda>xs. hd xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  3753   by (induct xss) (auto split: list.split)
  3754 
  3755 lemma transpose_aux_filter_tail:
  3756   "concat (map (list_case [] (\<lambda>h t. [t])) xss) =
  3757   map (\<lambda>xs. tl xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  3758   by (induct xss) (auto split: list.split)
  3759 
  3760 lemma transpose_aux_max:
  3761   "max (Suc (length xs)) (foldr (\<lambda>xs. max (length xs)) xss 0) =
  3762   Suc (max (length xs) (foldr (\<lambda>x. max (length x - Suc 0)) [ys\<leftarrow>xss . ys\<noteq>[]] 0))"
  3763   (is "max _ ?foldB = Suc (max _ ?foldA)")
  3764 proof (cases "[ys\<leftarrow>xss . ys\<noteq>[]] = []")
  3765   case True
  3766   hence "foldr (\<lambda>xs. max (length xs)) xss 0 = 0"
  3767   proof (induct xss)
  3768     case (Cons x xs)
  3769     moreover hence "x = []" by (cases x) auto
  3770     ultimately show ?case by auto
  3771   qed simp
  3772   thus ?thesis using True by simp
  3773 next
  3774   case False
  3775 
  3776   have foldA: "?foldA = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0 - 1"
  3777     by (induct xss) auto
  3778   have foldB: "?foldB = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0"
  3779     by (induct xss) auto
  3780 
  3781   have "0 < ?foldB"
  3782   proof -
  3783     from False
  3784     obtain z zs where zs: "[ys\<leftarrow>xss . ys \<noteq> []] = z#zs" by (auto simp: neq_Nil_conv)
  3785     hence "z \<in> set ([ys\<leftarrow>xss . ys \<noteq> []])" by auto
  3786     hence "z \<noteq> []" by auto
  3787     thus ?thesis
  3788       unfolding foldB zs
  3789       by (auto simp: max_def intro: less_le_trans)
  3790   qed
  3791   thus ?thesis
  3792     unfolding foldA foldB max_Suc_Suc[symmetric]
  3793     by simp
  3794 qed
  3795 
  3796 termination transpose
  3797   by (relation "measure (\<lambda>xs. foldr (\<lambda>xs. max (length xs)) xs 0 + length xs)")
  3798      (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max less_Suc_eq_le)
  3799 
  3800 lemma transpose_empty: "(transpose xs = []) \<longleftrightarrow> (\<forall>x \<in> set xs. x = [])"
  3801   by (induct rule: transpose.induct) simp_all
  3802 
  3803 lemma length_transpose:
  3804   fixes xs :: "'a list list"
  3805   shows "length (transpose xs) = foldr (\<lambda>xs. max (length xs)) xs 0"
  3806   by (induct rule: transpose.induct)
  3807     (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max
  3808                 max_Suc_Suc[symmetric] simp del: max_Suc_Suc)
  3809 
  3810 lemma nth_transpose:
  3811   fixes xs :: "'a list list"
  3812   assumes "i < length (transpose xs)"
  3813   shows "transpose xs ! i = map (\<lambda>xs. xs ! i) [ys \<leftarrow> xs. i < length ys]"
  3814 using assms proof (induct arbitrary: i rule: transpose.induct)
  3815   case (3 x xs xss)
  3816   def XS == "(x # xs) # xss"
  3817   hence [simp]: "XS \<noteq> []" by auto
  3818   thus ?case
  3819   proof (cases i)
  3820     case 0
  3821     thus ?thesis by (simp add: transpose_aux_filter_head hd_conv_nth)
  3822   next
  3823     case (Suc j)
  3824     have *: "\<And>xss. xs # map tl xss = map tl ((x#xs)#xss)" by simp
  3825     have **: "\<And>xss. (x#xs) # filter (\<lambda>ys. ys \<noteq> []) xss = filter (\<lambda>ys. ys \<noteq> []) ((x#xs)#xss)" by simp
  3826     { fix x have "Suc j < length x \<longleftrightarrow> x \<noteq> [] \<and> j < length x - Suc 0"
  3827       by (cases x) simp_all
  3828     } note *** = this
  3829 
  3830     have j_less: "j < length (transpose (xs # concat (map (list_case [] (\<lambda>h t. [t])) xss)))"
  3831       using "3.prems" by (simp add: transpose_aux_filter_tail length_transpose Suc)
  3832 
  3833     show ?thesis
  3834       unfolding transpose.simps `i = Suc j` nth_Cons_Suc "3.hyps"[OF j_less]
  3835       apply (auto simp: transpose_aux_filter_tail filter_map comp_def length_transpose * ** *** XS_def[symmetric])
  3836       apply (rule_tac y=x in list.exhaust)
  3837       by auto
  3838   qed
  3839 qed simp_all
  3840 
  3841 lemma transpose_map_map:
  3842   "transpose (map (map f) xs) = map (map f) (transpose xs)"
  3843 proof (rule nth_equalityI, safe)
  3844   have [simp]: "length (transpose (map (map f) xs)) = length (transpose xs)"
  3845     by (simp add: length_transpose foldr_map comp_def)
  3846   show "length (transpose (map (map f) xs)) = length (map (map f) (transpose xs))" by simp
  3847 
  3848   fix i assume "i < length (transpose (map (map f) xs))"
  3849   thus "transpose (map (map f) xs) ! i = map (map f) (transpose xs) ! i"
  3850     by (simp add: nth_transpose filter_map comp_def)
  3851 qed
  3852 
  3853 
  3854 subsubsection {* (In)finiteness *}
  3855 
  3856 lemma finite_maxlen:
  3857   "finite (M::'a list set) ==> EX n. ALL s:M. size s < n"
  3858 proof (induct rule: finite.induct)
  3859   case emptyI show ?case by simp
  3860 next
  3861   case (insertI M xs)
  3862   then obtain n where "\<forall>s\<in>M. length s < n" by blast
  3863   hence "ALL s:insert xs M. size s < max n (size xs) + 1" by auto
  3864   thus ?case ..
  3865 qed
  3866 
  3867 lemma lists_length_Suc_eq:
  3868   "{xs. set xs \<subseteq> A \<and> length xs = Suc n} =
  3869     (\<lambda>(xs, n). n#xs) ` ({xs. set xs \<subseteq> A \<and> length xs = n} \<times> A)"
  3870   by (auto simp: length_Suc_conv)
  3871 
  3872 lemma
  3873   assumes "finite A"
  3874   shows finite_lists_length_eq: "finite {xs. set xs \<subseteq> A \<and> length xs = n}"
  3875   and card_lists_length_eq: "card {xs. set xs \<subseteq> A \<and> length xs = n} = (card A)^n"
  3876   using `finite A`
  3877   by (induct n)
  3878      (auto simp: card_image inj_split_Cons lists_length_Suc_eq cong: conj_cong)
  3879 
  3880 lemma finite_lists_length_le:
  3881   assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs \<le> n}"
  3882  (is "finite ?S")
  3883 proof-
  3884   have "?S = (\<Union>n\<in>{0..n}. {xs. set xs \<subseteq> A \<and> length xs = n})" by auto
  3885   thus ?thesis by (auto intro: finite_lists_length_eq[OF `finite A`])
  3886 qed
  3887 
  3888 lemma card_lists_length_le:
  3889   assumes "finite A" shows "card {xs. set xs \<subseteq> A \<and> length xs \<le> n} = (\<Sum>i\<le>n. card A^i)"
  3890 proof -
  3891   have "(\<Sum>i\<le>n. card A^i) = card (\<Union>i\<le>n. {xs. set xs \<subseteq> A \<and> length xs = i})"
  3892     using `finite A`
  3893     by (subst card_UN_disjoint)
  3894        (auto simp add: card_lists_length_eq finite_lists_length_eq)
  3895   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}"
  3896     by auto
  3897   finally show ?thesis by simp
  3898 qed
  3899 
  3900 lemma card_lists_distinct_length_eq:
  3901   assumes "k < card A"
  3902   shows "card {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A} = \<Prod>{card A - k + 1 .. card A}"
  3903 using assms
  3904 proof (induct k)
  3905   case 0
  3906   then have "{xs. length xs = 0 \<and> distinct xs \<and> set xs \<subseteq> A} = {[]}" by auto
  3907   then show ?case by simp
  3908 next
  3909   case (Suc k)
  3910   let "?k_list" = "\<lambda>k xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A"
  3911   have inj_Cons: "\<And>A. inj_on (\<lambda>(xs, n). n # xs) A"  by (rule inj_onI) auto
  3912 
  3913   from Suc have "k < card A" by simp
  3914   moreover have "finite A" using assms by (simp add: card_ge_0_finite)
  3915   moreover have "finite {xs. ?k_list k xs}"
  3916     using finite_lists_length_eq[OF `finite A`, of k]
  3917     by - (rule finite_subset, auto)
  3918   moreover have "\<And>i j. i \<noteq> j \<longrightarrow> {i} \<times> (A - set i) \<inter> {j} \<times> (A - set j) = {}"
  3919     by auto
  3920   moreover have "\<And>i. i \<in>Collect (?k_list k) \<Longrightarrow> card (A - set i) = card A - k"
  3921     by (simp add: card_Diff_subset distinct_card)
  3922   moreover have "{xs. ?k_list (Suc k) xs} =
  3923       (\<lambda>(xs, n). n#xs) ` \<Union>(\<lambda>xs. {xs} \<times> (A - set xs)) ` {xs. ?k_list k xs}"
  3924     by (auto simp: length_Suc_conv)
  3925   moreover
  3926   have "Suc (card A - Suc k) = card A - k" using Suc.prems by simp
  3927   then have "(card A - k) * \<Prod>{Suc (card A - k)..card A} = \<Prod>{Suc (card A - Suc k)..card A}"
  3928     by (subst setprod_insert[symmetric]) (simp add: atLeastAtMost_insertL)+
  3929   ultimately show ?case
  3930     by (simp add: card_image inj_Cons card_UN_disjoint Suc.hyps algebra_simps)
  3931 qed
  3932 
  3933 lemma infinite_UNIV_listI: "~ finite(UNIV::'a list set)"
  3934 apply(rule notI)
  3935 apply(drule finite_maxlen)
  3936 apply (metis UNIV_I length_replicate less_not_refl)
  3937 done
  3938 
  3939 
  3940 subsection {* Sorting *}
  3941 
  3942 text{* Currently it is not shown that @{const sort} returns a
  3943 permutation of its input because the nicest proof is via multisets,
  3944 which are not yet available. Alternatively one could define a function
  3945 that counts the number of occurrences of an element in a list and use
  3946 that instead of multisets to state the correctness property. *}
  3947 
  3948 context linorder
  3949 begin
  3950 
  3951 lemma length_insort [simp]:
  3952   "length (insort_key f x xs) = Suc (length xs)"
  3953   by (induct xs) simp_all
  3954 
  3955 lemma insort_key_left_comm:
  3956   assumes "f x \<noteq> f y"
  3957   shows "insort_key f y (insort_key f x xs) = insort_key f x (insort_key f y xs)"
  3958   by (induct xs) (auto simp add: assms dest: antisym)
  3959 
  3960 lemma insort_left_comm:
  3961   "insort x (insort y xs) = insort y (insort x xs)"
  3962   by (cases "x = y") (auto intro: insort_key_left_comm)
  3963 
  3964 lemma comp_fun_commute_insort:
  3965   "comp_fun_commute insort"
  3966 proof
  3967 qed (simp add: insort_left_comm fun_eq_iff)
  3968 
  3969 lemma sort_key_simps [simp]:
  3970   "sort_key f [] = []"
  3971   "sort_key f (x#xs) = insort_key f x (sort_key f xs)"
  3972   by (simp_all add: sort_key_def)
  3973 
  3974 lemma (in linorder) sort_key_conv_fold:
  3975   assumes "inj_on f (set xs)"
  3976   shows "sort_key f xs = fold (insort_key f) xs []"
  3977 proof -
  3978   have "fold (insort_key f) (rev xs) = fold (insort_key f) xs"
  3979   proof (rule fold_rev, rule ext)
  3980     fix zs
  3981     fix x y
  3982     assume "x \<in> set xs" "y \<in> set xs"
  3983     with assms have *: "f y = f x \<Longrightarrow> y = x" by (auto dest: inj_onD)
  3984     have **: "x = y \<longleftrightarrow> y = x" by auto
  3985     show "(insort_key f y \<circ> insort_key f x) zs = (insort_key f x \<circ> insort_key f y) zs"
  3986       by (induct zs) (auto intro: * simp add: **)
  3987   qed
  3988   then show ?thesis by (simp add: sort_key_def foldr_def)
  3989 qed
  3990 
  3991 lemma (in linorder) sort_conv_fold:
  3992   "sort xs = fold insort xs []"
  3993   by (rule sort_key_conv_fold) simp
  3994 
  3995 lemma length_sort[simp]: "length (sort_key f xs) = length xs"
  3996 by (induct xs, auto)
  3997 
  3998 lemma sorted_Cons: "sorted (x#xs) = (sorted xs & (ALL y:set xs. x <= y))"
  3999 apply(induct xs arbitrary: x) apply simp
  4000 by simp (blast intro: order_trans)
  4001 
  4002 lemma sorted_tl:
  4003   "sorted xs \<Longrightarrow> sorted (tl xs)"
  4004   by (cases xs) (simp_all add: sorted_Cons)
  4005 
  4006 lemma sorted_append:
  4007   "sorted (xs@ys) = (sorted xs & sorted ys & (\<forall>x \<in> set xs. \<forall>y \<in> set ys. x\<le>y))"
  4008 by (induct xs) (auto simp add:sorted_Cons)
  4009 
  4010 lemma sorted_nth_mono:
  4011   "sorted xs \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!i \<le> xs!j"
  4012 by (induct xs arbitrary: i j) (auto simp:nth_Cons' sorted_Cons)
  4013 
  4014 lemma sorted_rev_nth_mono:
  4015   "sorted (rev xs) \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!j \<le> xs!i"
  4016 using sorted_nth_mono[ of "rev xs" "length xs - j - 1" "length xs - i - 1"]
  4017       rev_nth[of "length xs - i - 1" "xs"] rev_nth[of "length xs - j - 1" "xs"]
  4018 by auto
  4019 
  4020 lemma sorted_nth_monoI:
  4021   "(\<And> i j. \<lbrakk> i \<le> j ; j < length xs \<rbrakk> \<Longrightarrow> xs ! i \<le> xs ! j) \<Longrightarrow> sorted xs"
  4022 proof (induct xs)
  4023   case (Cons x xs)
  4024   have "sorted xs"
  4025   proof (rule Cons.hyps)
  4026     fix i j assume "i \<le> j" and "j < length xs"
  4027     with Cons.prems[of "Suc i" "Suc j"]
  4028     show "xs ! i \<le> xs ! j" by auto
  4029   qed
  4030   moreover
  4031   {
  4032     fix y assume "y \<in> set xs"
  4033     then obtain j where "j < length xs" and "xs ! j = y"
  4034       unfolding in_set_conv_nth by blast
  4035     with Cons.prems[of 0 "Suc j"]
  4036     have "x \<le> y"
  4037       by auto
  4038   }
  4039   ultimately
  4040   show ?case
  4041     unfolding sorted_Cons by auto
  4042 qed simp
  4043 
  4044 lemma sorted_equals_nth_mono:
  4045   "sorted xs = (\<forall>j < length xs. \<forall>i \<le> j. xs ! i \<le> xs ! j)"
  4046 by (auto intro: sorted_nth_monoI sorted_nth_mono)
  4047 
  4048 lemma set_insort: "set(insort_key f x xs) = insert x (set xs)"
  4049 by (induct xs) auto
  4050 
  4051 lemma set_sort[simp]: "set(sort_key f xs) = set xs"
  4052 by (induct xs) (simp_all add:set_insort)
  4053 
  4054 lemma distinct_insort: "distinct (insort_key f x xs) = (x \<notin> set xs \<and> distinct xs)"
  4055 by(induct xs)(auto simp:set_insort)
  4056 
  4057 lemma distinct_sort[simp]: "distinct (sort_key f xs) = distinct xs"
  4058   by (induct xs) (simp_all add: distinct_insort)
  4059 
  4060 lemma sorted_insort_key: "sorted (map f (insort_key f x xs)) = sorted (map f xs)"
  4061   by (induct xs) (auto simp:sorted_Cons set_insort)
  4062 
  4063 lemma sorted_insort: "sorted (insort x xs) = sorted xs"
  4064   using sorted_insort_key [where f="\<lambda>x. x"] by simp
  4065 
  4066 theorem sorted_sort_key [simp]: "sorted (map f (sort_key f xs))"
  4067   by (induct xs) (auto simp:sorted_insort_key)
  4068 
  4069 theorem sorted_sort [simp]: "sorted (sort xs)"
  4070   using sorted_sort_key [where f="\<lambda>x. x"] by simp
  4071 
  4072 lemma sorted_butlast:
  4073   assumes "xs \<noteq> []" and "sorted xs"
  4074   shows "sorted (butlast xs)"
  4075 proof -
  4076   from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  4077   with `sorted xs` show ?thesis by (simp add: sorted_append)
  4078 qed
  4079   
  4080 lemma insort_not_Nil [simp]:
  4081   "insort_key f a xs \<noteq> []"
  4082   by (induct xs) simp_all
  4083 
  4084 lemma insort_is_Cons: "\<forall>x\<in>set xs. f a \<le> f x \<Longrightarrow> insort_key f a xs = a # xs"
  4085 by (cases xs) auto
  4086 
  4087 lemma sorted_sort_id: "sorted xs \<Longrightarrow> sort xs = xs"
  4088   by (induct xs) (auto simp add: sorted_Cons insort_is_Cons)
  4089 
  4090 lemma sorted_map_remove1:
  4091   "sorted (map f xs) \<Longrightarrow> sorted (map f (remove1 x xs))"
  4092   by (induct xs) (auto simp add: sorted_Cons)
  4093 
  4094 lemma sorted_remove1: "sorted xs \<Longrightarrow> sorted (remove1 a xs)"
  4095   using sorted_map_remove1 [of "\<lambda>x. x"] by simp
  4096 
  4097 lemma insort_key_remove1:
  4098   assumes "a \<in> set xs" and "sorted (map f xs)" and "hd (filter (\<lambda>x. f a = f x) xs) = a"
  4099   shows "insort_key f a (remove1 a xs) = xs"
  4100 using assms proof (induct xs)
  4101   case (Cons x xs)
  4102   then show ?case
  4103   proof (cases "x = a")
  4104     case False
  4105     then have "f x \<noteq> f a" using Cons.prems by auto
  4106     then have "f x < f a" using Cons.prems by (auto simp: sorted_Cons)
  4107     with `f x \<noteq> f a` show ?thesis using Cons by (auto simp: sorted_Cons insort_is_Cons)
  4108   qed (auto simp: sorted_Cons insort_is_Cons)
  4109 qed simp
  4110 
  4111 lemma insort_remove1:
  4112   assumes "a \<in> set xs" and "sorted xs"
  4113   shows "insort a (remove1 a xs) = xs"
  4114 proof (rule insort_key_remove1)
  4115   from `a \<in> set xs` show "a \<in> set xs" .
  4116   from `sorted xs` show "sorted (map (\<lambda>x. x) xs)" by simp
  4117   from `a \<in> set xs` have "a \<in> set (filter (op = a) xs)" by auto
  4118   then have "set (filter (op = a) xs) \<noteq> {}" by auto
  4119   then have "filter (op = a) xs \<noteq> []" by (auto simp only: set_empty)
  4120   then have "length (filter (op = a) xs) > 0" by simp
  4121   then obtain n where n: "Suc n = length (filter (op = a) xs)"
  4122     by (cases "length (filter (op = a) xs)") simp_all
  4123   moreover have "replicate (Suc n) a = a # replicate n a"
  4124     by simp
  4125   ultimately show "hd (filter (op = a) xs) = a" by (simp add: replicate_length_filter)
  4126 qed
  4127 
  4128 lemma sorted_remdups[simp]:
  4129   "sorted l \<Longrightarrow> sorted (remdups l)"
  4130 by (induct l) (auto simp: sorted_Cons)
  4131 
  4132 lemma sorted_distinct_set_unique:
  4133 assumes "sorted xs" "distinct xs" "sorted ys" "distinct ys" "set xs = set ys"
  4134 shows "xs = ys"
  4135 proof -
  4136   from assms have 1: "length xs = length ys" by (auto dest!: distinct_card)
  4137   from assms show ?thesis
  4138   proof(induct rule:list_induct2[OF 1])
  4139     case 1 show ?case by simp
  4140   next
  4141     case 2 thus ?case by (simp add:sorted_Cons)
  4142        (metis Diff_insert_absorb antisym insertE insert_iff)
  4143   qed
  4144 qed
  4145 
  4146 lemma map_sorted_distinct_set_unique:
  4147   assumes "inj_on f (set xs \<union> set ys)"
  4148   assumes "sorted (map f xs)" "distinct (map f xs)"
  4149     "sorted (map f ys)" "distinct (map f ys)"
  4150   assumes "set xs = set ys"
  4151   shows "xs = ys"
  4152 proof -
  4153   from assms have "map f xs = map f ys"
  4154     by (simp add: sorted_distinct_set_unique)
  4155   moreover with `inj_on f (set xs \<union> set ys)` show "xs = ys"
  4156     by (blast intro: map_inj_on)
  4157 qed
  4158 
  4159 lemma finite_sorted_distinct_unique:
  4160 shows "finite A \<Longrightarrow> EX! xs. set xs = A & sorted xs & distinct xs"
  4161 apply(drule finite_distinct_list)
  4162 apply clarify
  4163 apply(rule_tac a="sort xs" in ex1I)
  4164 apply (auto simp: sorted_distinct_set_unique)
  4165 done
  4166 
  4167 lemma
  4168   assumes "sorted xs"
  4169   shows sorted_take: "sorted (take n xs)"
  4170   and sorted_drop: "sorted (drop n xs)"
  4171 proof -
  4172   from assms have "sorted (take n xs @ drop n xs)" by simp
  4173   then show "sorted (take n xs)" and "sorted (drop n xs)"
  4174     unfolding sorted_append by simp_all
  4175 qed
  4176 
  4177 lemma sorted_dropWhile: "sorted xs \<Longrightarrow> sorted (dropWhile P xs)"
  4178   by (auto dest: sorted_drop simp add: dropWhile_eq_drop)
  4179 
  4180 lemma sorted_takeWhile: "sorted xs \<Longrightarrow> sorted (takeWhile P xs)"
  4181   by (subst takeWhile_eq_take) (auto dest: sorted_take)
  4182 
  4183 lemma sorted_filter:
  4184   "sorted (map f xs) \<Longrightarrow> sorted (map f (filter P xs))"
  4185   by (induct xs) (simp_all add: sorted_Cons)
  4186 
  4187 lemma foldr_max_sorted:
  4188   assumes "sorted (rev xs)"
  4189   shows "foldr max xs y = (if xs = [] then y else max (xs ! 0) y)"
  4190 using assms proof (induct xs)
  4191   case (Cons x xs)
  4192   moreover hence "sorted (rev xs)" using sorted_append by auto
  4193   ultimately show ?case
  4194     by (cases xs, auto simp add: sorted_append max_def)
  4195 qed simp
  4196 
  4197 lemma filter_equals_takeWhile_sorted_rev:
  4198   assumes sorted: "sorted (rev (map f xs))"
  4199   shows "[x \<leftarrow> xs. t < f x] = takeWhile (\<lambda> x. t < f x) xs"
  4200     (is "filter ?P xs = ?tW")
  4201 proof (rule takeWhile_eq_filter[symmetric])
  4202   let "?dW" = "dropWhile ?P xs"
  4203   fix x assume "x \<in> set ?dW"
  4204   then obtain i where i: "i < length ?dW" and nth_i: "x = ?dW ! i"
  4205     unfolding in_set_conv_nth by auto
  4206   hence "length ?tW + i < length (?tW @ ?dW)"
  4207     unfolding length_append by simp
  4208   hence i': "length (map f ?tW) + i < length (map f xs)" by simp
  4209   have "(map f ?tW @ map f ?dW) ! (length (map f ?tW) + i) \<le>
  4210         (map f ?tW @ map f ?dW) ! (length (map f ?tW) + 0)"
  4211     using sorted_rev_nth_mono[OF sorted _ i', of "length ?tW"]
  4212     unfolding map_append[symmetric] by simp
  4213   hence "f x \<le> f (?dW ! 0)"
  4214     unfolding nth_append_length_plus nth_i
  4215     using i preorder_class.le_less_trans[OF le0 i] by simp
  4216   also have "... \<le> t"
  4217     using hd_dropWhile[of "?P" xs] le0[THEN preorder_class.le_less_trans, OF i]
  4218     using hd_conv_nth[of "?dW"] by simp
  4219   finally show "\<not> t < f x" by simp
  4220 qed
  4221 
  4222 lemma insort_insert_key_triv:
  4223   "f x \<in> f ` set xs \<Longrightarrow> insort_insert_key f x xs = xs"
  4224   by (simp add: insort_insert_key_def)
  4225 
  4226 lemma insort_insert_triv:
  4227   "x \<in> set xs \<Longrightarrow> insort_insert x xs = xs"
  4228   using insort_insert_key_triv [of "\<lambda>x. x"] by simp
  4229 
  4230 lemma insort_insert_insort_key:
  4231   "f x \<notin> f ` set xs \<Longrightarrow> insort_insert_key f x xs = insort_key f x xs"
  4232   by (simp add: insort_insert_key_def)
  4233 
  4234 lemma insort_insert_insort:
  4235   "x \<notin> set xs \<Longrightarrow> insort_insert x xs = insort x xs"
  4236   using insort_insert_insort_key [of "\<lambda>x. x"] by simp
  4237 
  4238 lemma set_insort_insert:
  4239   "set (insort_insert x xs) = insert x (set xs)"
  4240   by (auto simp add: insort_insert_key_def set_insort)
  4241 
  4242 lemma distinct_insort_insert:
  4243   assumes "distinct xs"
  4244   shows "distinct (insort_insert_key f x xs)"
  4245   using assms by (induct xs) (auto simp add: insort_insert_key_def set_insort)
  4246 
  4247 lemma sorted_insort_insert_key:
  4248   assumes "sorted (map f xs)"
  4249   shows "sorted (map f (insort_insert_key f x xs))"
  4250   using assms by (simp add: insort_insert_key_def sorted_insort_key)
  4251 
  4252 lemma sorted_insort_insert:
  4253   assumes "sorted xs"
  4254   shows "sorted (insort_insert x xs)"
  4255   using assms sorted_insort_insert_key [of "\<lambda>x. x"] by simp
  4256 
  4257 lemma filter_insort_triv:
  4258   "\<not> P x \<Longrightarrow> filter P (insort_key f x xs) = filter P xs"
  4259   by (induct xs) simp_all
  4260 
  4261 lemma filter_insort:
  4262   "sorted (map f xs) \<Longrightarrow> P x \<Longrightarrow> filter P (insort_key f x xs) = insort_key f x (filter P xs)"
  4263   using assms by (induct xs)
  4264     (auto simp add: sorted_Cons, subst insort_is_Cons, auto)
  4265 
  4266 lemma filter_sort:
  4267   "filter P (sort_key f xs) = sort_key f (filter P xs)"
  4268   by (induct xs) (simp_all add: filter_insort_triv filter_insort)
  4269 
  4270 lemma sorted_map_same:
  4271   "sorted (map f [x\<leftarrow>xs. f x = g xs])"
  4272 proof (induct xs arbitrary: g)
  4273   case Nil then show ?case by simp
  4274 next
  4275   case (Cons x xs)
  4276   then have "sorted (map f [y\<leftarrow>xs . f y = (\<lambda>xs. f x) xs])" .
  4277   moreover from Cons have "sorted (map f [y\<leftarrow>xs . f y = (g \<circ> Cons x) xs])" .
  4278   ultimately show ?case by (simp_all add: sorted_Cons)
  4279 qed
  4280 
  4281 lemma sorted_same:
  4282   "sorted [x\<leftarrow>xs. x = g xs]"
  4283   using sorted_map_same [of "\<lambda>x. x"] by simp
  4284 
  4285 lemma remove1_insort [simp]:
  4286   "remove1 x (insort x xs) = xs"
  4287   by (induct xs) simp_all
  4288 
  4289 end
  4290 
  4291 lemma sorted_upt[simp]: "sorted[i..<j]"
  4292 by (induct j) (simp_all add:sorted_append)
  4293 
  4294 lemma sorted_upto[simp]: "sorted[i..j]"
  4295 apply(induct i j rule:upto.induct)
  4296 apply(subst upto.simps)
  4297 apply(simp add:sorted_Cons)
  4298 done
  4299 
  4300 
  4301 subsubsection {* @{const transpose} on sorted lists *}
  4302 
  4303 lemma sorted_transpose[simp]:
  4304   shows "sorted (rev (map length (transpose xs)))"
  4305   by (auto simp: sorted_equals_nth_mono rev_nth nth_transpose
  4306     length_filter_conv_card intro: card_mono)
  4307 
  4308 lemma transpose_max_length:
  4309   "foldr (\<lambda>xs. max (length xs)) (transpose xs) 0 = length [x \<leftarrow> xs. x \<noteq> []]"
  4310   (is "?L = ?R")
  4311 proof (cases "transpose xs = []")
  4312   case False
  4313   have "?L = foldr max (map length (transpose xs)) 0"
  4314     by (simp add: foldr_map comp_def)
  4315   also have "... = length (transpose xs ! 0)"
  4316     using False sorted_transpose by (simp add: foldr_max_sorted)
  4317   finally show ?thesis
  4318     using False by (simp add: nth_transpose)
  4319 next
  4320   case True
  4321   hence "[x \<leftarrow> xs. x \<noteq> []] = []"
  4322     by (auto intro!: filter_False simp: transpose_empty)
  4323   thus ?thesis by (simp add: transpose_empty True)
  4324 qed
  4325 
  4326 lemma length_transpose_sorted:
  4327   fixes xs :: "'a list list"
  4328   assumes sorted: "sorted (rev (map length xs))"
  4329   shows "length (transpose xs) = (if xs = [] then 0 else length (xs ! 0))"
  4330 proof (cases "xs = []")
  4331   case False
  4332   thus ?thesis
  4333     using foldr_max_sorted[OF sorted] False
  4334     unfolding length_transpose foldr_map comp_def
  4335     by simp
  4336 qed simp
  4337 
  4338 lemma nth_nth_transpose_sorted[simp]:
  4339   fixes xs :: "'a list list"
  4340   assumes sorted: "sorted (rev (map length xs))"
  4341   and i: "i < length (transpose xs)"
  4342   and j: "j < length [ys \<leftarrow> xs. i < length ys]"
  4343   shows "transpose xs ! i ! j = xs ! j  ! i"
  4344   using j filter_equals_takeWhile_sorted_rev[OF sorted, of i]
  4345     nth_transpose[OF i] nth_map[OF j]
  4346   by (simp add: takeWhile_nth)
  4347 
  4348 lemma transpose_column_length:
  4349   fixes xs :: "'a list list"
  4350   assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  4351   shows "length (filter (\<lambda>ys. i < length ys) (transpose xs)) = length (xs ! i)"
  4352 proof -
  4353   have "xs \<noteq> []" using `i < length xs` by auto
  4354   note filter_equals_takeWhile_sorted_rev[OF sorted, simp]
  4355   { fix j assume "j \<le> i"
  4356     note sorted_rev_nth_mono[OF sorted, of j i, simplified, OF this `i < length xs`]
  4357   } note sortedE = this[consumes 1]
  4358 
  4359   have "{j. j < length (transpose xs) \<and> i < length (transpose xs ! j)}
  4360     = {..< length (xs ! i)}"
  4361   proof safe
  4362     fix j
  4363     assume "j < length (transpose xs)" and "i < length (transpose xs ! j)"
  4364     with this(2) nth_transpose[OF this(1)]
  4365     have "i < length (takeWhile (\<lambda>ys. j < length ys) xs)" by simp
  4366     from nth_mem[OF this] takeWhile_nth[OF this]
  4367     show "j < length (xs ! i)" by (auto dest: set_takeWhileD)
  4368   next
  4369     fix j assume "j < length (xs ! i)"
  4370     thus "j < length (transpose xs)"
  4371       using foldr_max_sorted[OF sorted] `xs \<noteq> []` sortedE[OF le0]
  4372       by (auto simp: length_transpose comp_def foldr_map)
  4373 
  4374     have "Suc i \<le> length (takeWhile (\<lambda>ys. j < length ys) xs)"
  4375       using `i < length xs` `j < length (xs ! i)` less_Suc_eq_le
  4376       by (auto intro!: length_takeWhile_less_P_nth dest!: sortedE)
  4377     with nth_transpose[OF `j < length (transpose xs)`]
  4378     show "i < length (transpose xs ! j)" by simp
  4379   qed
  4380   thus ?thesis by (simp add: length_filter_conv_card)
  4381 qed
  4382 
  4383 lemma transpose_column:
  4384   fixes xs :: "'a list list"
  4385   assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  4386   shows "map (\<lambda>ys. ys ! i) (filter (\<lambda>ys. i < length ys) (transpose xs))
  4387     = xs ! i" (is "?R = _")
  4388 proof (rule nth_equalityI, safe)
  4389   show length: "length ?R = length (xs ! i)"
  4390     using transpose_column_length[OF assms] by simp
  4391 
  4392   fix j assume j: "j < length ?R"
  4393   note * = less_le_trans[OF this, unfolded length_map, OF length_filter_le]
  4394   from j have j_less: "j < length (xs ! i)" using length by simp
  4395   have i_less_tW: "Suc i \<le> length (takeWhile (\<lambda>ys. Suc j \<le> length ys) xs)"
  4396   proof (rule length_takeWhile_less_P_nth)
  4397     show "Suc i \<le> length xs" using `i < length xs` by simp
  4398     fix k assume "k < Suc i"
  4399     hence "k \<le> i" by auto
  4400     with sorted_rev_nth_mono[OF sorted this] `i < length xs`
  4401     have "length (xs ! i) \<le> length (xs ! k)" by simp
  4402     thus "Suc j \<le> length (xs ! k)" using j_less by simp
  4403   qed
  4404   have i_less_filter: "i < length [ys\<leftarrow>xs . j < length ys]"
  4405     unfolding filter_equals_takeWhile_sorted_rev[OF sorted, of j]
  4406     using i_less_tW by (simp_all add: Suc_le_eq)
  4407   from j show "?R ! j = xs ! i ! j"
  4408     unfolding filter_equals_takeWhile_sorted_rev[OF sorted_transpose, of i]
  4409     by (simp add: takeWhile_nth nth_nth_transpose_sorted[OF sorted * i_less_filter])
  4410 qed
  4411 
  4412 lemma transpose_transpose:
  4413   fixes xs :: "'a list list"
  4414   assumes sorted: "sorted (rev (map length xs))"
  4415   shows "transpose (transpose xs) = takeWhile (\<lambda>x. x \<noteq> []) xs" (is "?L = ?R")
  4416 proof -
  4417   have len: "length ?L = length ?R"
  4418     unfolding length_transpose transpose_max_length
  4419     using filter_equals_takeWhile_sorted_rev[OF sorted, of 0]
  4420     by simp
  4421 
  4422   { fix i assume "i < length ?R"
  4423     with less_le_trans[OF _ length_takeWhile_le[of _ xs]]
  4424     have "i < length xs" by simp
  4425   } note * = this
  4426   show ?thesis
  4427     by (rule nth_equalityI)
  4428        (simp_all add: len nth_transpose transpose_column[OF sorted] * takeWhile_nth)
  4429 qed
  4430 
  4431 theorem transpose_rectangle:
  4432   assumes "xs = [] \<Longrightarrow> n = 0"
  4433   assumes rect: "\<And> i. i < length xs \<Longrightarrow> length (xs ! i) = n"
  4434   shows "transpose xs = map (\<lambda> i. map (\<lambda> j. xs ! j ! i) [0..<length xs]) [0..<n]"
  4435     (is "?trans = ?map")
  4436 proof (rule nth_equalityI)
  4437   have "sorted (rev (map length xs))"
  4438     by (auto simp: rev_nth rect intro!: sorted_nth_monoI)
  4439   from foldr_max_sorted[OF this] assms
  4440   show len: "length ?trans = length ?map"
  4441     by (simp_all add: length_transpose foldr_map comp_def)
  4442   moreover
  4443   { fix i assume "i < n" hence "[ys\<leftarrow>xs . i < length ys] = xs"
  4444       using rect by (auto simp: in_set_conv_nth intro!: filter_True) }
  4445   ultimately show "\<forall>i < length ?trans. ?trans ! i = ?map ! i"
  4446     by (auto simp: nth_transpose intro: nth_equalityI)
  4447 qed
  4448 
  4449 
  4450 subsubsection {* @{text sorted_list_of_set} *}
  4451 
  4452 text{* This function maps (finite) linearly ordered sets to sorted
  4453 lists. Warning: in most cases it is not a good idea to convert from
  4454 sets to lists but one should convert in the other direction (via
  4455 @{const set}). *}
  4456 
  4457 context linorder
  4458 begin
  4459 
  4460 definition sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where
  4461   "sorted_list_of_set = Finite_Set.fold insort []"
  4462 
  4463 lemma sorted_list_of_set_empty [simp]:
  4464   "sorted_list_of_set {} = []"
  4465   by (simp add: sorted_list_of_set_def)
  4466 
  4467 lemma sorted_list_of_set_insert [simp]:
  4468   assumes "finite A"
  4469   shows "sorted_list_of_set (insert x A) = insort x (sorted_list_of_set (A - {x}))"
  4470 proof -
  4471   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  4472   with assms show ?thesis by (simp add: sorted_list_of_set_def fold_insert_remove)
  4473 qed
  4474 
  4475 lemma sorted_list_of_set [simp]:
  4476   "finite A \<Longrightarrow> set (sorted_list_of_set A) = A \<and> sorted (sorted_list_of_set A) 
  4477     \<and> distinct (sorted_list_of_set A)"
  4478   by (induct A rule: finite_induct) (simp_all add: set_insort sorted_insort distinct_insort)
  4479 
  4480 lemma sorted_list_of_set_sort_remdups:
  4481   "sorted_list_of_set (set xs) = sort (remdups xs)"
  4482 proof -
  4483   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  4484   show ?thesis by (simp add: sorted_list_of_set_def sort_conv_fold fold_set_fold_remdups)
  4485 qed
  4486 
  4487 lemma sorted_list_of_set_remove:
  4488   assumes "finite A"
  4489   shows "sorted_list_of_set (A - {x}) = remove1 x (sorted_list_of_set A)"
  4490 proof (cases "x \<in> A")
  4491   case False with assms have "x \<notin> set (sorted_list_of_set A)" by simp
  4492   with False show ?thesis by (simp add: remove1_idem)
  4493 next
  4494   case True then obtain B where A: "A = insert x B" by (rule Set.set_insert)
  4495   with assms show ?thesis by simp
  4496 qed
  4497 
  4498 end
  4499 
  4500 lemma sorted_list_of_set_range [simp]:
  4501   "sorted_list_of_set {m..<n} = [m..<n]"
  4502   by (rule sorted_distinct_set_unique) simp_all
  4503 
  4504 
  4505 subsubsection {* @{text lists}: the list-forming operator over sets *}
  4506 
  4507 inductive_set
  4508   lists :: "'a set => 'a list set"
  4509   for A :: "'a set"
  4510 where
  4511     Nil [intro!, simp]: "[]: lists A"
  4512   | Cons [intro!, simp, no_atp]: "[| a: A; l: lists A|] ==> a#l : lists A"
  4513 
  4514 inductive_cases listsE [elim!,no_atp]: "x#l : lists A"
  4515 inductive_cases listspE [elim!,no_atp]: "listsp A (x # l)"
  4516 
  4517 lemma listsp_mono [mono]: "A \<le> B ==> listsp A \<le> listsp B"
  4518 by (rule predicate1I, erule listsp.induct, (blast dest: predicate1D)+)
  4519 
  4520 lemmas lists_mono = listsp_mono [to_set pred_subset_eq]
  4521 
  4522 lemma listsp_infI:
  4523   assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
  4524 by induct blast+
  4525 
  4526 lemmas lists_IntI = listsp_infI [to_set]
  4527 
  4528 lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
  4529 proof (rule mono_inf [where f=listsp, THEN order_antisym])
  4530   show "mono listsp" by (simp add: mono_def listsp_mono)
  4531   show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro!: listsp_infI predicate1I)
  4532 qed
  4533 
  4534 lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_def inf_bool_def]
  4535 
  4536 lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set pred_equals_eq]
  4537 
  4538 lemma Cons_in_lists_iff[simp]: "x#xs : lists A \<longleftrightarrow> x:A \<and> xs : lists A"
  4539 by auto
  4540 
  4541 lemma append_in_listsp_conv [iff]:
  4542      "(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
  4543 by (induct xs) auto
  4544 
  4545 lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]
  4546 
  4547 lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
  4548 -- {* eliminate @{text listsp} in favour of @{text set} *}
  4549 by (induct xs) auto
  4550 
  4551 lemmas in_lists_conv_set = in_listsp_conv_set [to_set]
  4552 
  4553 lemma in_listspD [dest!,no_atp]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
  4554 by (rule in_listsp_conv_set [THEN iffD1])
  4555 
  4556 lemmas in_listsD [dest!,no_atp] = in_listspD [to_set]
  4557 
  4558 lemma in_listspI [intro!,no_atp]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
  4559 by (rule in_listsp_conv_set [THEN iffD2])
  4560 
  4561 lemmas in_listsI [intro!,no_atp] = in_listspI [to_set]
  4562 
  4563 lemma lists_eq_set: "lists A = {xs. set xs <= A}"
  4564 by auto
  4565 
  4566 lemma lists_empty [simp]: "lists {} = {[]}"
  4567 by auto
  4568 
  4569 lemma lists_UNIV [simp]: "lists UNIV = UNIV"
  4570 by auto
  4571 
  4572 
  4573 subsubsection {* Inductive definition for membership *}
  4574 
  4575 inductive ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
  4576 where
  4577     elem:  "ListMem x (x # xs)"
  4578   | insert:  "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"
  4579 
  4580 lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
  4581 apply (rule iffI)
  4582  apply (induct set: ListMem)
  4583   apply auto
  4584 apply (induct xs)
  4585  apply (auto intro: ListMem.intros)
  4586 done
  4587 
  4588 
  4589 subsubsection {* Lists as Cartesian products *}
  4590 
  4591 text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
  4592 @{term A} and tail drawn from @{term Xs}.*}
  4593 
  4594 definition
  4595   set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set" where
  4596   "set_Cons A XS = {z. \<exists>x xs. z = x # xs \<and> x \<in> A \<and> xs \<in> XS}"
  4597 
  4598 lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
  4599 by (auto simp add: set_Cons_def)
  4600 
  4601 text{*Yields the set of lists, all of the same length as the argument and
  4602 with elements drawn from the corresponding element of the argument.*}
  4603 
  4604 primrec
  4605   listset :: "'a set list \<Rightarrow> 'a list set" where
  4606      "listset [] = {[]}"
  4607   |  "listset (A # As) = set_Cons A (listset As)"
  4608 
  4609 
  4610 subsection {* Relations on Lists *}
  4611 
  4612 subsubsection {* Length Lexicographic Ordering *}
  4613 
  4614 text{*These orderings preserve well-foundedness: shorter lists 
  4615   precede longer lists. These ordering are not used in dictionaries.*}
  4616         
  4617 primrec -- {*The lexicographic ordering for lists of the specified length*}
  4618   lexn :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> ('a list \<times> 'a list) set" where
  4619     "lexn r 0 = {}"
  4620   | "lexn r (Suc n) = (map_pair (%(x, xs). x#xs) (%(x, xs). x#xs) ` (r <*lex*> lexn r n)) Int
  4621       {(xs, ys). length xs = Suc n \<and> length ys = Suc n}"
  4622 
  4623 definition
  4624   lex :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4625   "lex r = (\<Union>n. lexn r n)" -- {*Holds only between lists of the same length*}
  4626 
  4627 definition
  4628   lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" where
  4629   "lenlex r = inv_image (less_than <*lex*> lex r) (\<lambda>xs. (length xs, xs))"
  4630         -- {*Compares lists by their length and then lexicographically*}
  4631 
  4632 lemma wf_lexn: "wf r ==> wf (lexn r n)"
  4633 apply (induct n, simp, simp)
  4634 apply(rule wf_subset)
  4635  prefer 2 apply (rule Int_lower1)
  4636 apply(rule wf_map_pair_image)
  4637  prefer 2 apply (rule inj_onI, auto)
  4638 done
  4639 
  4640 lemma lexn_length:
  4641   "(xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
  4642 by (induct n arbitrary: xs ys) auto
  4643 
  4644 lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
  4645 apply (unfold lex_def)
  4646 apply (rule wf_UN)
  4647 apply (blast intro: wf_lexn, clarify)
  4648 apply (rename_tac m n)
  4649 apply (subgoal_tac "m \<noteq> n")
  4650  prefer 2 apply blast
  4651 apply (blast dest: lexn_length not_sym)
  4652 done
  4653 
  4654 lemma lexn_conv:
  4655   "lexn r n =
  4656     {(xs,ys). length xs = n \<and> length ys = n \<and>
  4657     (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
  4658 apply (induct n, simp)
  4659 apply (simp add: image_Collect lex_prod_def, safe, blast)
  4660  apply (rule_tac x = "ab # xys" in exI, simp)
  4661 apply (case_tac xys, simp_all, blast)
  4662 done
  4663 
  4664 lemma lex_conv:
  4665   "lex r =
  4666     {(xs,ys). length xs = length ys \<and>
  4667     (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
  4668 by (force simp add: lex_def lexn_conv)
  4669 
  4670 lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
  4671 by (unfold lenlex_def) blast
  4672 
  4673 lemma lenlex_conv:
  4674     "lenlex r = {(xs,ys). length xs < length ys |
  4675                  length xs = length ys \<and> (xs, ys) : lex r}"
  4676 by (simp add: lenlex_def Id_on_def lex_prod_def inv_image_def)
  4677 
  4678 lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
  4679 by (simp add: lex_conv)
  4680 
  4681 lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
  4682 by (simp add:lex_conv)
  4683 
  4684 lemma Cons_in_lex [simp]:
  4685     "((x # xs, y # ys) : lex r) =
  4686       ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
  4687 apply (simp add: lex_conv)
  4688 apply (rule iffI)
  4689  prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
  4690 apply (case_tac xys, simp, simp)
  4691 apply blast
  4692 done
  4693 
  4694 
  4695 subsubsection {* Lexicographic Ordering *}
  4696 
  4697 text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
  4698     This ordering does \emph{not} preserve well-foundedness.
  4699      Author: N. Voelker, March 2005. *} 
  4700 
  4701 definition
  4702   lexord :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4703   "lexord r = {(x,y ). \<exists> a v. y = x @ a # v \<or>
  4704             (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
  4705 
  4706 lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
  4707 by (unfold lexord_def, induct_tac y, auto) 
  4708 
  4709 lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
  4710 by (unfold lexord_def, induct_tac x, auto)
  4711 
  4712 lemma lexord_cons_cons[simp]:
  4713      "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
  4714   apply (unfold lexord_def, safe, simp_all)
  4715   apply (case_tac u, simp, simp)
  4716   apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
  4717   apply (erule_tac x="b # u" in allE)
  4718   by force
  4719 
  4720 lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
  4721 
  4722 lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
  4723 by (induct_tac x, auto)  
  4724 
  4725 lemma lexord_append_left_rightI:
  4726      "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
  4727 by (induct_tac u, auto)
  4728 
  4729 lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
  4730 by (induct x, auto)
  4731 
  4732 lemma lexord_append_leftD:
  4733      "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
  4734 by (erule rev_mp, induct_tac x, auto)
  4735 
  4736 lemma lexord_take_index_conv: 
  4737    "((x,y) : lexord r) = 
  4738     ((length x < length y \<and> take (length x) y = x) \<or> 
  4739      (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
  4740   apply (unfold lexord_def Let_def, clarsimp) 
  4741   apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
  4742   apply auto 
  4743   apply (rule_tac x="hd (drop (length x) y)" in exI)
  4744   apply (rule_tac x="tl (drop (length x) y)" in exI)
  4745   apply (erule subst, simp add: min_def) 
  4746   apply (rule_tac x ="length u" in exI, simp) 
  4747   apply (rule_tac x ="take i x" in exI) 
  4748   apply (rule_tac x ="x ! i" in exI) 
  4749   apply (rule_tac x ="y ! i" in exI, safe) 
  4750   apply (rule_tac x="drop (Suc i) x" in exI)
  4751   apply (drule sym, simp add: drop_Suc_conv_tl) 
  4752   apply (rule_tac x="drop (Suc i) y" in exI)
  4753   by (simp add: drop_Suc_conv_tl) 
  4754 
  4755 -- {* lexord is extension of partial ordering List.lex *} 
  4756 lemma lexord_lex: "(x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
  4757   apply (rule_tac x = y in spec) 
  4758   apply (induct_tac x, clarsimp) 
  4759   by (clarify, case_tac x, simp, force)
  4760 
  4761 lemma lexord_irreflexive: "ALL x. (x,x) \<notin> r \<Longrightarrow> (xs,xs) \<notin> lexord r"
  4762 by (induct xs) auto
  4763 
  4764 text{* By Ren\'e Thiemann: *}
  4765 lemma lexord_partial_trans: 
  4766   "(\<And>x y z. x \<in> set xs \<Longrightarrow> (x,y) \<in> r \<Longrightarrow> (y,z) \<in> r \<Longrightarrow> (x,z) \<in> r)
  4767    \<Longrightarrow>  (xs,ys) \<in> lexord r  \<Longrightarrow>  (ys,zs) \<in> lexord r \<Longrightarrow>  (xs,zs) \<in> lexord r"
  4768 proof (induct xs arbitrary: ys zs)
  4769   case Nil
  4770   from Nil(3) show ?case unfolding lexord_def by (cases zs, auto)
  4771 next
  4772   case (Cons x xs yys zzs)
  4773   from Cons(3) obtain y ys where yys: "yys = y # ys" unfolding lexord_def
  4774     by (cases yys, auto)
  4775   note Cons = Cons[unfolded yys]
  4776   from Cons(3) have one: "(x,y) \<in> r \<or> x = y \<and> (xs,ys) \<in> lexord r" by auto
  4777   from Cons(4) obtain z zs where zzs: "zzs = z # zs" unfolding lexord_def
  4778     by (cases zzs, auto)
  4779   note Cons = Cons[unfolded zzs]
  4780   from Cons(4) have two: "(y,z) \<in> r \<or> y = z \<and> (ys,zs) \<in> lexord r" by auto
  4781   {
  4782     assume "(xs,ys) \<in> lexord r" and "(ys,zs) \<in> lexord r"
  4783     from Cons(1)[OF _ this] Cons(2)
  4784     have "(xs,zs) \<in> lexord r" by auto
  4785   } note ind1 = this
  4786   {
  4787     assume "(x,y) \<in> r" and "(y,z) \<in> r"
  4788     from Cons(2)[OF _ this] have "(x,z) \<in> r" by auto
  4789   } note ind2 = this
  4790   from one two ind1 ind2
  4791   have "(x,z) \<in> r \<or> x = z \<and> (xs,zs) \<in> lexord r" by blast
  4792   thus ?case unfolding zzs by auto
  4793 qed
  4794 
  4795 lemma lexord_trans: 
  4796     "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
  4797 by(auto simp: trans_def intro:lexord_partial_trans)
  4798 
  4799 lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
  4800 by (rule transI, drule lexord_trans, blast) 
  4801 
  4802 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"
  4803   apply (rule_tac x = y in spec) 
  4804   apply (induct_tac x, rule allI) 
  4805   apply (case_tac x, simp, simp) 
  4806   apply (rule allI, case_tac x, simp, simp) 
  4807   by blast
  4808 
  4809 
  4810 subsubsection {* Lexicographic combination of measure functions *}
  4811 
  4812 text {* These are useful for termination proofs *}
  4813 
  4814 definition
  4815   "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"
  4816 
  4817 lemma wf_measures[simp]: "wf (measures fs)"
  4818 unfolding measures_def
  4819 by blast
  4820 
  4821 lemma in_measures[simp]: 
  4822   "(x, y) \<in> measures [] = False"
  4823   "(x, y) \<in> measures (f # fs)
  4824          = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"  
  4825 unfolding measures_def
  4826 by auto
  4827 
  4828 lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
  4829 by simp
  4830 
  4831 lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
  4832 by auto
  4833 
  4834 
  4835 subsubsection {* Lifting Relations to Lists: one element *}
  4836 
  4837 definition listrel1 :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4838 "listrel1 r = {(xs,ys).
  4839    \<exists>us z z' vs. xs = us @ z # vs \<and> (z,z') \<in> r \<and> ys = us @ z' # vs}"
  4840 
  4841 lemma listrel1I:
  4842   "\<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow>
  4843   (xs, ys) \<in> listrel1 r"
  4844 unfolding listrel1_def by auto
  4845 
  4846 lemma listrel1E:
  4847   "\<lbrakk> (xs, ys) \<in> listrel1 r;
  4848      !!x y us vs. \<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow> P
  4849    \<rbrakk> \<Longrightarrow> P"
  4850 unfolding listrel1_def by auto
  4851 
  4852 lemma not_Nil_listrel1 [iff]: "([], xs) \<notin> listrel1 r"
  4853 unfolding listrel1_def by blast
  4854 
  4855 lemma not_listrel1_Nil [iff]: "(xs, []) \<notin> listrel1 r"
  4856 unfolding listrel1_def by blast
  4857 
  4858 lemma Cons_listrel1_Cons [iff]:
  4859   "(x # xs, y # ys) \<in> listrel1 r \<longleftrightarrow>
  4860    (x,y) \<in> r \<and> xs = ys \<or> x = y \<and> (xs, ys) \<in> listrel1 r"
  4861 by (simp add: listrel1_def Cons_eq_append_conv) (blast)
  4862 
  4863 lemma listrel1I1: "(x,y) \<in> r \<Longrightarrow> (x # xs, y # xs) \<in> listrel1 r"
  4864 by (metis Cons_listrel1_Cons)
  4865 
  4866 lemma listrel1I2: "(xs, ys) \<in> listrel1 r \<Longrightarrow> (x # xs, x # ys) \<in> listrel1 r"
  4867 by (metis Cons_listrel1_Cons)
  4868 
  4869 lemma append_listrel1I:
  4870   "(xs, ys) \<in> listrel1 r \<and> us = vs \<or> xs = ys \<and> (us, vs) \<in> listrel1 r
  4871     \<Longrightarrow> (xs @ us, ys @ vs) \<in> listrel1 r"
  4872 unfolding listrel1_def
  4873 by auto (blast intro: append_eq_appendI)+
  4874 
  4875 lemma Cons_listrel1E1[elim!]:
  4876   assumes "(x # xs, ys) \<in> listrel1 r"
  4877     and "\<And>y. ys = y # xs \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
  4878     and "\<And>zs. ys = x # zs \<Longrightarrow> (xs, zs) \<in> listrel1 r \<Longrightarrow> R"
  4879   shows R
  4880 using assms by (cases ys) blast+
  4881 
  4882 lemma Cons_listrel1E2[elim!]:
  4883   assumes "(xs, y # ys) \<in> listrel1 r"
  4884     and "\<And>x. xs = x # ys \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
  4885     and "\<And>zs. xs = y # zs \<Longrightarrow> (zs, ys) \<in> listrel1 r \<Longrightarrow> R"
  4886   shows R
  4887 using assms by (cases xs) blast+
  4888 
  4889 lemma snoc_listrel1_snoc_iff:
  4890   "(xs @ [x], ys @ [y]) \<in> listrel1 r
  4891     \<longleftrightarrow> (xs, ys) \<in> listrel1 r \<and> x = y \<or> xs = ys \<and> (x,y) \<in> r" (is "?L \<longleftrightarrow> ?R")
  4892 proof
  4893   assume ?L thus ?R
  4894     by (fastforce simp: listrel1_def snoc_eq_iff_butlast butlast_append)
  4895 next
  4896   assume ?R then show ?L unfolding listrel1_def by force
  4897 qed
  4898 
  4899 lemma listrel1_eq_len: "(xs,ys) \<in> listrel1 r \<Longrightarrow> length xs = length ys"
  4900 unfolding listrel1_def by auto
  4901 
  4902 lemma listrel1_mono:
  4903   "r \<subseteq> s \<Longrightarrow> listrel1 r \<subseteq> listrel1 s"
  4904 unfolding listrel1_def by blast
  4905 
  4906 
  4907 lemma listrel1_converse: "listrel1 (r^-1) = (listrel1 r)^-1"
  4908 unfolding listrel1_def by blast
  4909 
  4910 lemma in_listrel1_converse:
  4911   "(x,y) : listrel1 (r^-1) \<longleftrightarrow> (x,y) : (listrel1 r)^-1"
  4912 unfolding listrel1_def by blast
  4913 
  4914 lemma listrel1_iff_update:
  4915   "(xs,ys) \<in> (listrel1 r)
  4916    \<longleftrightarrow> (\<exists>y n. (xs ! n, y) \<in> r \<and> n < length xs \<and> ys = xs[n:=y])" (is "?L \<longleftrightarrow> ?R")
  4917 proof
  4918   assume "?L"
  4919   then obtain x y u v where "xs = u @ x # v"  "ys = u @ y # v"  "(x,y) \<in> r"
  4920     unfolding listrel1_def by auto
  4921   then have "ys = xs[length u := y]" and "length u < length xs"
  4922     and "(xs ! length u, y) \<in> r" by auto
  4923   then show "?R" by auto
  4924 next
  4925   assume "?R"
  4926   then obtain x y n where "(xs!n, y) \<in> r" "n < size xs" "ys = xs[n:=y]" "x = xs!n"
  4927     by auto
  4928   then obtain u v where "xs = u @ x # v" and "ys = u @ y # v" and "(x, y) \<in> r"
  4929     by (auto intro: upd_conv_take_nth_drop id_take_nth_drop)
  4930   then show "?L" by (auto simp: listrel1_def)
  4931 qed
  4932 
  4933 
  4934 text{* Accessible part and wellfoundedness: *}
  4935 
  4936 lemma Cons_acc_listrel1I [intro!]:
  4937   "x \<in> acc r \<Longrightarrow> xs \<in> acc (listrel1 r) \<Longrightarrow> (x # xs) \<in> acc (listrel1 r)"
  4938 apply (induct arbitrary: xs set: acc)
  4939 apply (erule thin_rl)
  4940 apply (erule acc_induct)
  4941 apply (rule accI)
  4942 apply (blast)
  4943 done
  4944 
  4945 lemma lists_accD: "xs \<in> lists (acc r) \<Longrightarrow> xs \<in> acc (listrel1 r)"
  4946 apply (induct set: lists)
  4947  apply (rule accI)
  4948  apply simp
  4949 apply (rule accI)
  4950 apply (fast dest: acc_downward)
  4951 done
  4952 
  4953 lemma lists_accI: "xs \<in> acc (listrel1 r) \<Longrightarrow> xs \<in> lists (acc r)"
  4954 apply (induct set: acc)
  4955 apply clarify
  4956 apply (rule accI)
  4957 apply (fastforce dest!: in_set_conv_decomp[THEN iffD1] simp: listrel1_def)
  4958 done
  4959 
  4960 lemma wf_listrel1_iff[simp]: "wf(listrel1 r) = wf r"
  4961 by(metis wf_acc_iff in_lists_conv_set lists_accI lists_accD Cons_in_lists_iff)
  4962 
  4963 
  4964 subsubsection {* Lifting Relations to Lists: all elements *}
  4965 
  4966 inductive_set
  4967   listrel :: "('a * 'a)set => ('a list * 'a list)set"
  4968   for r :: "('a * 'a)set"
  4969 where
  4970     Nil:  "([],[]) \<in> listrel r"
  4971   | Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
  4972 
  4973 inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
  4974 inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
  4975 inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
  4976 inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
  4977 
  4978 
  4979 lemma listrel_eq_len:  "(xs, ys) \<in> listrel r \<Longrightarrow> length xs = length ys"
  4980 by(induct rule: listrel.induct) auto
  4981 
  4982 lemma listrel_iff_zip: "(xs,ys) : listrel r \<longleftrightarrow>
  4983   length xs = length ys & (\<forall>(x,y) \<in> set(zip xs ys). (x,y) \<in> r)" (is "?L \<longleftrightarrow> ?R")
  4984 proof
  4985   assume ?L thus ?R by induct (auto intro: listrel_eq_len)
  4986 next
  4987   assume ?R thus ?L
  4988     apply (clarify)
  4989     by (induct rule: list_induct2) (auto intro: listrel.intros)
  4990 qed
  4991 
  4992 lemma listrel_iff_nth: "(xs,ys) : listrel r \<longleftrightarrow>
  4993   length xs = length ys & (\<forall>n < length xs. (xs!n, ys!n) \<in> r)" (is "?L \<longleftrightarrow> ?R")
  4994 by (auto simp add: all_set_conv_all_nth listrel_iff_zip)
  4995 
  4996 
  4997 lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
  4998 apply clarify  
  4999 apply (erule listrel.induct)
  5000 apply (blast intro: listrel.intros)+
  5001 done
  5002 
  5003 lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
  5004 apply clarify 
  5005 apply (erule listrel.induct, auto) 
  5006 done
  5007 
  5008 lemma listrel_refl_on: "refl_on A r \<Longrightarrow> refl_on (lists A) (listrel r)" 
  5009 apply (simp add: refl_on_def listrel_subset Ball_def)
  5010 apply (rule allI) 
  5011 apply (induct_tac x) 
  5012 apply (auto intro: listrel.intros)
  5013 done
  5014 
  5015 lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
  5016 apply (auto simp add: sym_def)
  5017 apply (erule listrel.induct) 
  5018 apply (blast intro: listrel.intros)+
  5019 done
  5020 
  5021 lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
  5022 apply (simp add: trans_def)
  5023 apply (intro allI) 
  5024 apply (rule impI) 
  5025 apply (erule listrel.induct) 
  5026 apply (blast intro: listrel.intros)+
  5027 done
  5028 
  5029 theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
  5030 by (simp add: equiv_def listrel_refl_on listrel_sym listrel_trans) 
  5031 
  5032 lemma listrel_rtrancl_refl[iff]: "(xs,xs) : listrel(r^*)"
  5033 using listrel_refl_on[of UNIV, OF refl_rtrancl]
  5034 by(auto simp: refl_on_def)
  5035 
  5036 lemma listrel_rtrancl_trans:
  5037   "\<lbrakk> (xs,ys) : listrel(r^*);  (ys,zs) : listrel(r^*) \<rbrakk>
  5038   \<Longrightarrow> (xs,zs) : listrel(r^*)"
  5039 by (metis listrel_trans trans_def trans_rtrancl)
  5040 
  5041 
  5042 lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
  5043 by (blast intro: listrel.intros)
  5044 
  5045 lemma listrel_Cons:
  5046      "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})"
  5047 by (auto simp add: set_Cons_def intro: listrel.intros)
  5048 
  5049 text {* Relating @{term listrel1}, @{term listrel} and closures: *}
  5050 
  5051 lemma listrel1_rtrancl_subset_rtrancl_listrel1:
  5052   "listrel1 (r^*) \<subseteq> (listrel1 r)^*"
  5053 proof (rule subrelI)
  5054   fix xs ys assume 1: "(xs,ys) \<in> listrel1 (r^*)"
  5055   { fix x y us vs
  5056     have "(x,y) : r^* \<Longrightarrow> (us @ x # vs, us @ y # vs) : (listrel1 r)^*"
  5057     proof(induct rule: rtrancl.induct)
  5058       case rtrancl_refl show ?case by simp
  5059     next
  5060       case rtrancl_into_rtrancl thus ?case
  5061         by (metis listrel1I rtrancl.rtrancl_into_rtrancl)
  5062     qed }
  5063   thus "(xs,ys) \<in> (listrel1 r)^*" using 1 by(blast elim: listrel1E)
  5064 qed
  5065 
  5066 lemma rtrancl_listrel1_eq_len: "(x,y) \<in> (listrel1 r)^* \<Longrightarrow> length x = length y"
  5067 by (induct rule: rtrancl.induct) (auto intro: listrel1_eq_len)
  5068 
  5069 lemma rtrancl_listrel1_ConsI1:
  5070   "(xs,ys) : (listrel1 r)^* \<Longrightarrow> (x#xs,x#ys) : (listrel1 r)^*"
  5071 apply(induct rule: rtrancl.induct)
  5072  apply simp
  5073 by (metis listrel1I2 rtrancl.rtrancl_into_rtrancl)
  5074 
  5075 lemma rtrancl_listrel1_ConsI2:
  5076   "(x,y) \<in> r^* \<Longrightarrow> (xs, ys) \<in> (listrel1 r)^*
  5077   \<Longrightarrow> (x # xs, y # ys) \<in> (listrel1 r)^*"
  5078   by (blast intro: rtrancl_trans rtrancl_listrel1_ConsI1 
  5079     subsetD[OF listrel1_rtrancl_subset_rtrancl_listrel1 listrel1I1])
  5080 
  5081 lemma listrel1_subset_listrel:
  5082   "r \<subseteq> r' \<Longrightarrow> refl r' \<Longrightarrow> listrel1 r \<subseteq> listrel(r')"
  5083 by(auto elim!: listrel1E simp add: listrel_iff_zip set_zip refl_on_def)
  5084 
  5085 lemma listrel_reflcl_if_listrel1:
  5086   "(xs,ys) : listrel1 r \<Longrightarrow> (xs,ys) : listrel(r^*)"
  5087 by(erule listrel1E)(auto simp add: listrel_iff_zip set_zip)
  5088 
  5089 lemma listrel_rtrancl_eq_rtrancl_listrel1: "listrel (r^*) = (listrel1 r)^*"
  5090 proof
  5091   { fix x y assume "(x,y) \<in> listrel (r^*)"
  5092     then have "(x,y) \<in> (listrel1 r)^*"
  5093     by induct (auto intro: rtrancl_listrel1_ConsI2) }
  5094   then show "listrel (r^*) \<subseteq> (listrel1 r)^*"
  5095     by (rule subrelI)
  5096 next
  5097   show "listrel (r^*) \<supseteq> (listrel1 r)^*"
  5098   proof(rule subrelI)
  5099     fix xs ys assume "(xs,ys) \<in> (listrel1 r)^*"
  5100     then show "(xs,ys) \<in> listrel (r^*)"
  5101     proof induct
  5102       case base show ?case by(auto simp add: listrel_iff_zip set_zip)
  5103     next
  5104       case (step ys zs)
  5105       thus ?case  by (metis listrel_reflcl_if_listrel1 listrel_rtrancl_trans)
  5106     qed
  5107   qed
  5108 qed
  5109 
  5110 lemma rtrancl_listrel1_if_listrel:
  5111   "(xs,ys) : listrel r \<Longrightarrow> (xs,ys) : (listrel1 r)^*"
  5112 by(metis listrel_rtrancl_eq_rtrancl_listrel1 subsetD[OF listrel_mono] r_into_rtrancl subsetI)
  5113 
  5114 lemma listrel_subset_rtrancl_listrel1: "listrel r \<subseteq> (listrel1 r)^*"
  5115 by(fast intro:rtrancl_listrel1_if_listrel)
  5116 
  5117 
  5118 subsection {* Size function *}
  5119 
  5120 lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (list_size f)"
  5121 by (rule is_measure_trivial)
  5122 
  5123 lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (option_size f)"
  5124 by (rule is_measure_trivial)
  5125 
  5126 lemma list_size_estimation[termination_simp]: 
  5127   "x \<in> set xs \<Longrightarrow> y < f x \<Longrightarrow> y < list_size f xs"
  5128 by (induct xs) auto
  5129 
  5130 lemma list_size_estimation'[termination_simp]: 
  5131   "x \<in> set xs \<Longrightarrow> y \<le> f x \<Longrightarrow> y \<le> list_size f xs"
  5132 by (induct xs) auto
  5133 
  5134 lemma list_size_map[simp]: "list_size f (map g xs) = list_size (f o g) xs"
  5135 by (induct xs) auto
  5136 
  5137 lemma list_size_append[simp]: "list_size f (xs @ ys) = list_size f xs + list_size f ys"
  5138 by (induct xs, auto)
  5139 
  5140 lemma list_size_pointwise[termination_simp]: 
  5141   "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> list_size f xs \<le> list_size g xs"
  5142 by (induct xs) force+
  5143 
  5144 
  5145 subsection {* Monad operation *}
  5146 
  5147 definition bind :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b list) \<Rightarrow> 'b list" where
  5148   "bind xs f = concat (map f xs)"
  5149 
  5150 hide_const (open) bind
  5151 
  5152 lemma bind_simps [simp]:
  5153   "List.bind [] f = []"
  5154   "List.bind (x # xs) f = f x @ List.bind xs f"
  5155   by (simp_all add: bind_def)
  5156 
  5157 
  5158 subsection {* Transfer *}
  5159 
  5160 definition
  5161   embed_list :: "nat list \<Rightarrow> int list"
  5162 where
  5163   "embed_list l = map int l"
  5164 
  5165 definition
  5166   nat_list :: "int list \<Rightarrow> bool"
  5167 where
  5168   "nat_list l = nat_set (set l)"
  5169 
  5170 definition
  5171   return_list :: "int list \<Rightarrow> nat list"
  5172 where
  5173   "return_list l = map nat l"
  5174 
  5175 lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow>
  5176     embed_list (return_list l) = l"
  5177   unfolding embed_list_def return_list_def nat_list_def nat_set_def
  5178   apply (induct l)
  5179   apply auto
  5180 done
  5181 
  5182 lemma transfer_nat_int_list_functions:
  5183   "l @ m = return_list (embed_list l @ embed_list m)"
  5184   "[] = return_list []"
  5185   unfolding return_list_def embed_list_def
  5186   apply auto
  5187   apply (induct l, auto)
  5188   apply (induct m, auto)
  5189 done
  5190 
  5191 (*
  5192 lemma transfer_nat_int_fold1: "fold f l x =
  5193     fold (%x. f (nat x)) (embed_list l) x";
  5194 *)
  5195 
  5196 
  5197 subsection {* Code generation *}
  5198 
  5199 subsubsection {* Counterparts for set-related operations *}
  5200 
  5201 definition member :: "'a list \<Rightarrow> 'a \<Rightarrow> bool" where
  5202   "member xs x \<longleftrightarrow> x \<in> set xs"
  5203 
  5204 text {*
  5205   Use @{text member} only for generating executable code.  Otherwise use
  5206   @{prop "x \<in> set xs"} instead --- it is much easier to reason about.
  5207 *}
  5208 
  5209 lemma member_rec [code]:
  5210   "member (x # xs) y \<longleftrightarrow> x = y \<or> member xs y"
  5211   "member [] y \<longleftrightarrow> False"
  5212   by (auto simp add: member_def)
  5213 
  5214 lemma in_set_member (* FIXME delete candidate *):
  5215   "x \<in> set xs \<longleftrightarrow> member xs x"
  5216   by (simp add: member_def)
  5217 
  5218 definition list_all :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5219   list_all_iff: "list_all P xs \<longleftrightarrow> Ball (set xs) P"
  5220 
  5221 definition list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5222   list_ex_iff: "list_ex P xs \<longleftrightarrow> Bex (set xs) P"
  5223 
  5224 definition list_ex1 :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5225   list_ex1_iff [code_abbrev]: "list_ex1 P xs \<longleftrightarrow> (\<exists>! x. x \<in> set xs \<and> P x)"
  5226 
  5227 text {*
  5228   Usually you should prefer @{text "\<forall>x\<in>set xs"}, @{text "\<exists>x\<in>set xs"}
  5229   and @{text "\<exists>!x. x\<in>set xs \<and> _"} over @{const list_all}, @{const list_ex}
  5230   and @{const list_ex1} in specifications.
  5231 *}
  5232 
  5233 lemma list_all_simps [simp, code]:
  5234   "list_all P (x # xs) \<longleftrightarrow> P x \<and> list_all P xs"
  5235   "list_all P [] \<longleftrightarrow> True"
  5236   by (simp_all add: list_all_iff)
  5237 
  5238 lemma list_ex_simps [simp, code]:
  5239   "list_ex P (x # xs) \<longleftrightarrow> P x \<or> list_ex P xs"
  5240   "list_ex P [] \<longleftrightarrow> False"
  5241   by (simp_all add: list_ex_iff)
  5242 
  5243 lemma list_ex1_simps [simp, code]:
  5244   "list_ex1 P [] = False"
  5245   "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)"
  5246   by (auto simp add: list_ex1_iff list_all_iff)
  5247 
  5248 lemma Ball_set_list_all: (* FIXME delete candidate *)
  5249   "Ball (set xs) P \<longleftrightarrow> list_all P xs"
  5250   by (simp add: list_all_iff)
  5251 
  5252 lemma Bex_set_list_ex: (* FIXME delete candidate *)
  5253   "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
  5254   by (simp add: list_ex_iff)
  5255 
  5256 lemma list_all_append [simp]:
  5257   "list_all P (xs @ ys) \<longleftrightarrow> list_all P xs \<and> list_all P ys"
  5258   by (auto simp add: list_all_iff)
  5259 
  5260 lemma list_ex_append [simp]:
  5261   "list_ex P (xs @ ys) \<longleftrightarrow> list_ex P xs \<or> list_ex P ys"
  5262   by (auto simp add: list_ex_iff)
  5263 
  5264 lemma list_all_rev [simp]:
  5265   "list_all P (rev xs) \<longleftrightarrow> list_all P xs"
  5266   by (simp add: list_all_iff)
  5267 
  5268 lemma list_ex_rev [simp]:
  5269   "list_ex P (rev xs) \<longleftrightarrow> list_ex P xs"
  5270   by (simp add: list_ex_iff)
  5271 
  5272 lemma list_all_length:
  5273   "list_all P xs \<longleftrightarrow> (\<forall>n < length xs. P (xs ! n))"
  5274   by (auto simp add: list_all_iff set_conv_nth)
  5275 
  5276 lemma list_ex_length:
  5277   "list_ex P xs \<longleftrightarrow> (\<exists>n < length xs. P (xs ! n))"
  5278   by (auto simp add: list_ex_iff set_conv_nth)
  5279 
  5280 lemma list_all_cong [fundef_cong]:
  5281   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_all f xs = list_all g ys"
  5282   by (simp add: list_all_iff)
  5283 
  5284 lemma list_any_cong [fundef_cong]:
  5285   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_ex f xs = list_ex g ys"
  5286   by (simp add: list_ex_iff)
  5287 
  5288 text {* Bounded quantification and summation over nats. *}
  5289 
  5290 lemma atMost_upto [code_unfold]:
  5291   "{..n} = set [0..<Suc n]"
  5292   by auto
  5293 
  5294 lemma atLeast_upt [code_unfold]:
  5295   "{..<n} = set [0..<n]"
  5296   by auto
  5297 
  5298 lemma greaterThanLessThan_upt [code_unfold]:
  5299   "{n<..<m} = set [Suc n..<m]"
  5300   by auto
  5301 
  5302 lemmas atLeastLessThan_upt [code_unfold] = set_upt [symmetric]
  5303 
  5304 lemma greaterThanAtMost_upt [code_unfold]:
  5305   "{n<..m} = set [Suc n..<Suc m]"
  5306   by auto
  5307 
  5308 lemma atLeastAtMost_upt [code_unfold]:
  5309   "{n..m} = set [n..<Suc m]"
  5310   by auto
  5311 
  5312 lemma all_nat_less_eq [code_unfold]:
  5313   "(\<forall>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..<n}. P m)"
  5314   by auto
  5315 
  5316 lemma ex_nat_less_eq [code_unfold]:
  5317   "(\<exists>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..<n}. P m)"
  5318   by auto
  5319 
  5320 lemma all_nat_less [code_unfold]:
  5321   "(\<forall>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..n}. P m)"
  5322   by auto
  5323 
  5324 lemma ex_nat_less [code_unfold]:
  5325   "(\<exists>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..n}. P m)"
  5326   by auto
  5327 
  5328 lemma setsum_set_upt_conv_listsum_nat [code_unfold]:
  5329   "setsum f (set [m..<n]) = listsum (map f [m..<n])"
  5330   by (simp add: interv_listsum_conv_setsum_set_nat)
  5331 
  5332 text {* Summation over ints. *}
  5333 
  5334 lemma greaterThanLessThan_upto [code_unfold]:
  5335   "{i<..<j::int} = set [i+1..j - 1]"
  5336 by auto
  5337 
  5338 lemma atLeastLessThan_upto [code_unfold]:
  5339   "{i..<j::int} = set [i..j - 1]"
  5340 by auto
  5341 
  5342 lemma greaterThanAtMost_upto [code_unfold]:
  5343   "{i<..j::int} = set [i+1..j]"
  5344 by auto
  5345 
  5346 lemmas atLeastAtMost_upto [code_unfold] = set_upto [symmetric]
  5347 
  5348 lemma setsum_set_upto_conv_listsum_int [code_unfold]:
  5349   "setsum f (set [i..j::int]) = listsum (map f [i..j])"
  5350   by (simp add: interv_listsum_conv_setsum_set_int)
  5351 
  5352 
  5353 subsubsection {* Optimizing by rewriting *}
  5354 
  5355 definition null :: "'a list \<Rightarrow> bool" where
  5356   [code_abbrev]: "null xs \<longleftrightarrow> xs = []"
  5357 
  5358 text {*
  5359   Efficient emptyness check is implemented by @{const null}.
  5360 *}
  5361 
  5362 lemma null_rec [code]:
  5363   "null (x # xs) \<longleftrightarrow> False"
  5364   "null [] \<longleftrightarrow> True"
  5365   by (simp_all add: null_def)
  5366 
  5367 lemma eq_Nil_null: (* FIXME delete candidate *)
  5368   "xs = [] \<longleftrightarrow> null xs"
  5369   by (simp add: null_def)
  5370 
  5371 lemma equal_Nil_null [code_unfold]:
  5372   "HOL.equal xs [] \<longleftrightarrow> null xs"
  5373   by (simp add: equal eq_Nil_null)
  5374 
  5375 definition maps :: "('a \<Rightarrow> 'b list) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  5376   [code_abbrev]: "maps f xs = concat (map f xs)"
  5377 
  5378 definition map_filter :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  5379   [code_post]: "map_filter f xs = map (the \<circ> f) (filter (\<lambda>x. f x \<noteq> None) xs)"
  5380 
  5381 text {*
  5382   Operations @{const maps} and @{const map_filter} avoid
  5383   intermediate lists on execution -- do not use for proving.
  5384 *}
  5385 
  5386 lemma maps_simps [code]:
  5387   "maps f (x # xs) = f x @ maps f xs"
  5388   "maps f [] = []"
  5389   by (simp_all add: maps_def)
  5390 
  5391 lemma map_filter_simps [code]:
  5392   "map_filter f (x # xs) = (case f x of None \<Rightarrow> map_filter f xs | Some y \<Rightarrow> y # map_filter f xs)"
  5393   "map_filter f [] = []"
  5394   by (simp_all add: map_filter_def split: option.split)
  5395 
  5396 lemma concat_map_maps: (* FIXME delete candidate *)
  5397   "concat (map f xs) = maps f xs"
  5398   by (simp add: maps_def)
  5399 
  5400 lemma map_filter_map_filter [code_unfold]:
  5401   "map f (filter P xs) = map_filter (\<lambda>x. if P x then Some (f x) else None) xs"
  5402   by (simp add: map_filter_def)
  5403 
  5404 text {* Optimized code for @{text"\<forall>i\<in>{a..b::int}"} and @{text"\<forall>n:{a..<b::nat}"}
  5405 and similiarly for @{text"\<exists>"}. *}
  5406 
  5407 definition all_interval_nat :: "(nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
  5408   "all_interval_nat P i j \<longleftrightarrow> (\<forall>n \<in> {i..<j}. P n)"
  5409 
  5410 lemma [code]:
  5411   "all_interval_nat P i j \<longleftrightarrow> i \<ge> j \<or> P i \<and> all_interval_nat P (Suc i) j"
  5412 proof -
  5413   have *: "\<And>n. P i \<Longrightarrow> \<forall>n\<in>{Suc i..<j}. P n \<Longrightarrow> i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n"
  5414   proof -
  5415     fix n
  5416     assume "P i" "\<forall>n\<in>{Suc i..<j}. P n" "i \<le> n" "n < j"
  5417     then show "P n" by (cases "n = i") simp_all
  5418   qed
  5419   show ?thesis by (auto simp add: all_interval_nat_def intro: *)
  5420 qed
  5421 
  5422 lemma list_all_iff_all_interval_nat [code_unfold]:
  5423   "list_all P [i..<j] \<longleftrightarrow> all_interval_nat P i j"
  5424   by (simp add: list_all_iff all_interval_nat_def)
  5425 
  5426 lemma list_ex_iff_not_all_inverval_nat [code_unfold]:
  5427   "list_ex P [i..<j] \<longleftrightarrow> \<not> (all_interval_nat (Not \<circ> P) i j)"
  5428   by (simp add: list_ex_iff all_interval_nat_def)
  5429 
  5430 definition all_interval_int :: "(int \<Rightarrow> bool) \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool" where
  5431   "all_interval_int P i j \<longleftrightarrow> (\<forall>k \<in> {i..j}. P k)"
  5432 
  5433 lemma [code]:
  5434   "all_interval_int P i j \<longleftrightarrow> i > j \<or> P i \<and> all_interval_int P (i + 1) j"
  5435 proof -
  5436   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"
  5437   proof -
  5438     fix k
  5439     assume "P i" "\<forall>k\<in>{i+1..j}. P k" "i \<le> k" "k \<le> j"
  5440     then show "P k" by (cases "k = i") simp_all
  5441   qed
  5442   show ?thesis by (auto simp add: all_interval_int_def intro: *)
  5443 qed
  5444 
  5445 lemma list_all_iff_all_interval_int [code_unfold]:
  5446   "list_all P [i..j] \<longleftrightarrow> all_interval_int P i j"
  5447   by (simp add: list_all_iff all_interval_int_def)
  5448 
  5449 lemma list_ex_iff_not_all_inverval_int [code_unfold]:
  5450   "list_ex P [i..j] \<longleftrightarrow> \<not> (all_interval_int (Not \<circ> P) i j)"
  5451   by (simp add: list_ex_iff all_interval_int_def)
  5452 
  5453 hide_const (open) member null maps map_filter all_interval_nat all_interval_int
  5454 
  5455 
  5456 subsubsection {* Pretty lists *}
  5457 
  5458 use "Tools/list_code.ML"
  5459 
  5460 code_type list
  5461   (SML "_ list")
  5462   (OCaml "_ list")
  5463   (Haskell "![(_)]")
  5464   (Scala "List[(_)]")
  5465 
  5466 code_const Nil
  5467   (SML "[]")
  5468   (OCaml "[]")
  5469   (Haskell "[]")
  5470   (Scala "!Nil")
  5471 
  5472 code_instance list :: equal
  5473   (Haskell -)
  5474 
  5475 code_const "HOL.equal \<Colon> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
  5476   (Haskell infix 4 "==")
  5477 
  5478 code_reserved SML
  5479   list
  5480 
  5481 code_reserved OCaml
  5482   list
  5483 
  5484 setup {* fold (List_Code.add_literal_list) ["SML", "OCaml", "Haskell", "Scala"] *}
  5485 
  5486 
  5487 subsubsection {* Use convenient predefined operations *}
  5488 
  5489 code_const "op @"
  5490   (SML infixr 7 "@")
  5491   (OCaml infixr 6 "@")
  5492   (Haskell infixr 5 "++")
  5493   (Scala infixl 7 "++")
  5494 
  5495 code_const map
  5496   (Haskell "map")
  5497 
  5498 code_const filter
  5499   (Haskell "filter")
  5500 
  5501 code_const concat
  5502   (Haskell "concat")
  5503 
  5504 code_const List.maps
  5505   (Haskell "concatMap")
  5506 
  5507 code_const rev
  5508   (Haskell "reverse")
  5509 
  5510 code_const zip
  5511   (Haskell "zip")
  5512 
  5513 code_const List.null
  5514   (Haskell "null")
  5515 
  5516 code_const takeWhile
  5517   (Haskell "takeWhile")
  5518 
  5519 code_const dropWhile
  5520   (Haskell "dropWhile")
  5521 
  5522 code_const list_all
  5523   (Haskell "all")
  5524 
  5525 code_const list_ex
  5526   (Haskell "any")
  5527 
  5528 
  5529 subsubsection {* Implementation of sets by lists *}
  5530 
  5531 text {* Basic operations *}
  5532 
  5533 lemma is_empty_set [code]:
  5534   "Set.is_empty (set xs) \<longleftrightarrow> List.null xs"
  5535   by (simp add: Set.is_empty_def null_def)
  5536 
  5537 lemma empty_set [code]:
  5538   "{} = set []"
  5539   by simp
  5540 
  5541 lemma [code]:
  5542   "x \<in> set xs \<longleftrightarrow> List.member xs x"
  5543   "x \<in> List.coset xs \<longleftrightarrow> \<not> List.member xs x"
  5544   by (simp_all add: member_def)
  5545 
  5546 lemma UNIV_coset [code]:
  5547   "UNIV = List.coset []"
  5548   by simp
  5549 
  5550 lemma insert_code [code]:
  5551   "insert x (set xs) = set (List.insert x xs)"
  5552   "insert x (List.coset xs) = List.coset (removeAll x xs)"
  5553   by simp_all
  5554 
  5555 lemma remove_code [code]:
  5556   "Set.remove x (set xs) = set (removeAll x xs)"
  5557   "Set.remove x (List.coset xs) = List.coset (List.insert x xs)"
  5558   by (simp_all add: remove_def Compl_insert)
  5559 
  5560 lemma Ball_set [code]:
  5561   "Ball (set xs) P \<longleftrightarrow> list_all P xs"
  5562   by (simp add: list_all_iff)
  5563 
  5564 lemma Bex_set [code]:
  5565   "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
  5566   by (simp add: list_ex_iff)
  5567 
  5568 lemma card_set [code]:
  5569   "card (set xs) = length (remdups xs)"
  5570 proof -
  5571   have "card (set (remdups xs)) = length (remdups xs)"
  5572     by (rule distinct_card) simp
  5573   then show ?thesis by simp
  5574 qed
  5575 
  5576 
  5577 text {* Operations on relations *}
  5578 
  5579 lemma product_code [code]:
  5580   "Product_Type.product (set xs) (set ys) = set [(x, y). x \<leftarrow> xs, y \<leftarrow> ys]"
  5581   by (auto simp add: Product_Type.product_def)
  5582 
  5583 lemma Id_on_set [code]:
  5584   "Id_on (set xs) = set [(x, x). x \<leftarrow> xs]"
  5585   by (auto simp add: Id_on_def)
  5586 
  5587 lemma trancl_set_ntrancl [code]:
  5588   "trancl (set xs) = ntrancl (card (set xs) - 1) (set xs)"
  5589   by (simp add: finite_trancl_ntranl)
  5590 
  5591 lemma set_rel_comp [code]:
  5592   "set xys O set yzs = set ([(fst xy, snd yz). xy \<leftarrow> xys, yz \<leftarrow> yzs, snd xy = fst yz])"
  5593   by (auto simp add: Bex_def)
  5594 
  5595 lemma wf_set [code]:
  5596   "wf (set xs) = acyclic (set xs)"
  5597   by (simp add: wf_iff_acyclic_if_finite)
  5598 
  5599 end