src/HOL/List.thy
author hoelzl
Thu Dec 01 15:41:58 2011 +0100 (2011-12-01)
changeset 45714 ad4242285560
parent 45607 16b4f5774621
child 45789 36ea69266e61
permissions -rw-r--r--
cardinality of sets of lists
     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 primrec
    53   map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
    54     "map f [] = []"
    55   | "map f (x # xs) = f x # map f xs"
    56 
    57 primrec
    58   append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65) where
    59     append_Nil:"[] @ ys = ys"
    60   | append_Cons: "(x#xs) @ ys = x # xs @ ys"
    61 
    62 primrec
    63   rev :: "'a list \<Rightarrow> 'a list" where
    64     "rev [] = []"
    65   | "rev (x # xs) = rev xs @ [x]"
    66 
    67 primrec
    68   filter:: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
    69     "filter P [] = []"
    70   | "filter P (x # xs) = (if P x then x # filter P xs else filter P xs)"
    71 
    72 syntax
    73   -- {* Special syntax for filter *}
    74   "_filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_<-_./ _])")
    75 
    76 translations
    77   "[x<-xs . P]"== "CONST filter (%x. P) xs"
    78 
    79 syntax (xsymbols)
    80   "_filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
    81 syntax (HTML output)
    82   "_filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
    83 
    84 primrec
    85   foldl :: "('b \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b" where
    86     foldl_Nil: "foldl f a [] = a"
    87   | foldl_Cons: "foldl f a (x # xs) = foldl f (f a x) xs"
    88 
    89 primrec
    90   foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where
    91     "foldr f [] a = a"
    92   | "foldr f (x # xs) a = f x (foldr f xs a)"
    93 
    94 primrec
    95   concat:: "'a list list \<Rightarrow> 'a list" where
    96     "concat [] = []"
    97   | "concat (x # xs) = x @ concat xs"
    98 
    99 definition (in monoid_add)
   100   listsum :: "'a list \<Rightarrow> 'a" where
   101   "listsum xs = foldr plus xs 0"
   102 
   103 primrec
   104   drop:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   105     drop_Nil: "drop n [] = []"
   106   | drop_Cons: "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
   107   -- {*Warning: simpset does not contain this definition, but separate
   108        theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
   109 
   110 primrec
   111   take:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   112     take_Nil:"take n [] = []"
   113   | take_Cons: "take n (x # xs) = (case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # take 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   nth :: "'a list => nat => 'a" (infixl "!" 100) where
   119   nth_Cons: "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
   120   -- {*Warning: simpset does not contain this definition, but separate
   121        theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
   122 
   123 primrec
   124   list_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
   125     "list_update [] i v = []"
   126   | "list_update (x # xs) i v = (case i of 0 \<Rightarrow> v # xs | Suc j \<Rightarrow> x # list_update xs j v)"
   127 
   128 nonterminal lupdbinds and lupdbind
   129 
   130 syntax
   131   "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
   132   "" :: "lupdbind => lupdbinds"    ("_")
   133   "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
   134   "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
   135 
   136 translations
   137   "_LUpdate xs (_lupdbinds b bs)" == "_LUpdate (_LUpdate xs b) bs"
   138   "xs[i:=x]" == "CONST list_update xs i x"
   139 
   140 primrec
   141   takeWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   142     "takeWhile P [] = []"
   143   | "takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])"
   144 
   145 primrec
   146   dropWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   147     "dropWhile P [] = []"
   148   | "dropWhile P (x # xs) = (if P x then dropWhile P xs else x # xs)"
   149 
   150 primrec
   151   zip :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
   152     "zip xs [] = []"
   153   | zip_Cons: "zip xs (y # ys) = (case xs of [] => [] | z # zs => (z, y) # zip zs ys)"
   154   -- {*Warning: simpset does not contain this definition, but separate
   155        theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
   156 
   157 primrec 
   158   upt :: "nat \<Rightarrow> nat \<Rightarrow> nat list" ("(1[_..</_'])") where
   159     upt_0: "[i..<0] = []"
   160   | upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
   161 
   162 definition
   163   insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   164   "insert x xs = (if x \<in> set xs then xs else x # xs)"
   165 
   166 hide_const (open) insert
   167 hide_fact (open) insert_def
   168 
   169 primrec
   170   remove1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   171     "remove1 x [] = []"
   172   | "remove1 x (y # xs) = (if x = y then xs else y # remove1 x xs)"
   173 
   174 primrec
   175   removeAll :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   176     "removeAll x [] = []"
   177   | "removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)"
   178 
   179 primrec
   180   distinct :: "'a list \<Rightarrow> bool" where
   181     "distinct [] \<longleftrightarrow> True"
   182   | "distinct (x # xs) \<longleftrightarrow> x \<notin> set xs \<and> distinct xs"
   183 
   184 primrec
   185   remdups :: "'a list \<Rightarrow> 'a list" where
   186     "remdups [] = []"
   187   | "remdups (x # xs) = (if x \<in> set xs then remdups xs else x # remdups xs)"
   188 
   189 primrec
   190   replicate :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
   191     replicate_0: "replicate 0 x = []"
   192   | replicate_Suc: "replicate (Suc n) x = x # replicate n x"
   193 
   194 text {*
   195   Function @{text size} is overloaded for all datatypes. Users may
   196   refer to the list version as @{text length}. *}
   197 
   198 abbreviation
   199   length :: "'a list \<Rightarrow> nat" where
   200   "length \<equiv> size"
   201 
   202 definition
   203   rotate1 :: "'a list \<Rightarrow> 'a list" where
   204   "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"
   205 
   206 definition
   207   rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   208   "rotate n = rotate1 ^^ n"
   209 
   210 definition
   211   list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
   212   "list_all2 P xs ys =
   213     (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"
   214 
   215 definition
   216   sublist :: "'a list => nat set => 'a list" where
   217   "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"
   218 
   219 fun splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   220 "splice [] ys = ys" |
   221 "splice xs [] = xs" |
   222 "splice (x#xs) (y#ys) = x # y # splice xs ys"
   223 
   224 text{*
   225 \begin{figure}[htbp]
   226 \fbox{
   227 \begin{tabular}{l}
   228 @{lemma "[a,b]@[c,d] = [a,b,c,d]" by simp}\\
   229 @{lemma "length [a,b,c] = 3" by simp}\\
   230 @{lemma "set [a,b,c] = {a,b,c}" by simp}\\
   231 @{lemma "map f [a,b,c] = [f a, f b, f c]" by simp}\\
   232 @{lemma "rev [a,b,c] = [c,b,a]" by simp}\\
   233 @{lemma "hd [a,b,c,d] = a" by simp}\\
   234 @{lemma "tl [a,b,c,d] = [b,c,d]" by simp}\\
   235 @{lemma "last [a,b,c,d] = d" by simp}\\
   236 @{lemma "butlast [a,b,c,d] = [a,b,c]" by simp}\\
   237 @{lemma[source] "filter (\<lambda>n::nat. n<2) [0,2,1] = [0,1]" by simp}\\
   238 @{lemma "concat [[a,b],[c,d,e],[],[f]] = [a,b,c,d,e,f]" by simp}\\
   239 @{lemma "foldl f x [a,b,c] = f (f (f x a) b) c" by simp}\\
   240 @{lemma "foldr f [a,b,c] x = f a (f b (f c x))" by simp}\\
   241 @{lemma "zip [a,b,c] [x,y,z] = [(a,x),(b,y),(c,z)]" by simp}\\
   242 @{lemma "zip [a,b] [x,y,z] = [(a,x),(b,y)]" by simp}\\
   243 @{lemma "splice [a,b,c] [x,y,z] = [a,x,b,y,c,z]" by simp}\\
   244 @{lemma "splice [a,b,c,d] [x,y] = [a,x,b,y,c,d]" by simp}\\
   245 @{lemma "take 2 [a,b,c,d] = [a,b]" by simp}\\
   246 @{lemma "take 6 [a,b,c,d] = [a,b,c,d]" by simp}\\
   247 @{lemma "drop 2 [a,b,c,d] = [c,d]" by simp}\\
   248 @{lemma "drop 6 [a,b,c,d] = []" by simp}\\
   249 @{lemma "takeWhile (%n::nat. n<3) [1,2,3,0] = [1,2]" by simp}\\
   250 @{lemma "dropWhile (%n::nat. n<3) [1,2,3,0] = [3,0]" by simp}\\
   251 @{lemma "distinct [2,0,1::nat]" by simp}\\
   252 @{lemma "remdups [2,0,2,1::nat,2] = [0,1,2]" by simp}\\
   253 @{lemma "List.insert 2 [0::nat,1,2] = [0,1,2]" by (simp add: List.insert_def)}\\
   254 @{lemma "List.insert 3 [0::nat,1,2] = [3,0,1,2]" by (simp add: List.insert_def)}\\
   255 @{lemma "remove1 2 [2,0,2,1::nat,2] = [0,2,1,2]" by simp}\\
   256 @{lemma "removeAll 2 [2,0,2,1::nat,2] = [0,1]" by simp}\\
   257 @{lemma "nth [a,b,c,d] 2 = c" by simp}\\
   258 @{lemma "[a,b,c,d][2 := x] = [a,b,x,d]" by simp}\\
   259 @{lemma "sublist [a,b,c,d,e] {0,2,3} = [a,c,d]" by (simp add:sublist_def)}\\
   260 @{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by (simp add:rotate1_def)}\\
   261 @{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate1_def rotate_def eval_nat_numeral)}\\
   262 @{lemma "replicate 4 a = [a,a,a,a]" by (simp add:eval_nat_numeral)}\\
   263 @{lemma "[2..<5] = [2,3,4]" by (simp add:eval_nat_numeral)}\\
   264 @{lemma "listsum [1,2,3::nat] = 6" by (simp add: listsum_def)}
   265 \end{tabular}}
   266 \caption{Characteristic examples}
   267 \label{fig:Characteristic}
   268 \end{figure}
   269 Figure~\ref{fig:Characteristic} shows characteristic examples
   270 that should give an intuitive understanding of the above functions.
   271 *}
   272 
   273 text{* The following simple sort functions are intended for proofs,
   274 not for efficient implementations. *}
   275 
   276 context linorder
   277 begin
   278 
   279 inductive sorted :: "'a list \<Rightarrow> bool" where
   280   Nil [iff]: "sorted []"
   281 | Cons: "\<forall>y\<in>set xs. x \<le> y \<Longrightarrow> sorted xs \<Longrightarrow> sorted (x # xs)"
   282 
   283 lemma sorted_single [iff]:
   284   "sorted [x]"
   285   by (rule sorted.Cons) auto
   286 
   287 lemma sorted_many:
   288   "x \<le> y \<Longrightarrow> sorted (y # zs) \<Longrightarrow> sorted (x # y # zs)"
   289   by (rule sorted.Cons) (cases "y # zs" rule: sorted.cases, auto)
   290 
   291 lemma sorted_many_eq [simp, code]:
   292   "sorted (x # y # zs) \<longleftrightarrow> x \<le> y \<and> sorted (y # zs)"
   293   by (auto intro: sorted_many elim: sorted.cases)
   294 
   295 lemma [code]:
   296   "sorted [] \<longleftrightarrow> True"
   297   "sorted [x] \<longleftrightarrow> True"
   298   by simp_all
   299 
   300 primrec insort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
   301 "insort_key f x [] = [x]" |
   302 "insort_key f x (y#ys) = (if f x \<le> f y then (x#y#ys) else y#(insort_key f x ys))"
   303 
   304 definition sort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b list \<Rightarrow> 'b list" where
   305 "sort_key f xs = foldr (insort_key f) xs []"
   306 
   307 definition insort_insert_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
   308   "insort_insert_key f x xs = (if f x \<in> f ` set xs then xs else insort_key f x xs)"
   309 
   310 abbreviation "sort \<equiv> sort_key (\<lambda>x. x)"
   311 abbreviation "insort \<equiv> insort_key (\<lambda>x. x)"
   312 abbreviation "insort_insert \<equiv> insort_insert_key (\<lambda>x. x)"
   313 
   314 end
   315 
   316 
   317 subsubsection {* List comprehension *}
   318 
   319 text{* Input syntax for Haskell-like list comprehension notation.
   320 Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
   321 the list of all pairs of distinct elements from @{text xs} and @{text ys}.
   322 The syntax is as in Haskell, except that @{text"|"} becomes a dot
   323 (like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
   324 \verb![e| x <- xs, ...]!.
   325 
   326 The qualifiers after the dot are
   327 \begin{description}
   328 \item[generators] @{text"p \<leftarrow> xs"},
   329  where @{text p} is a pattern and @{text xs} an expression of list type, or
   330 \item[guards] @{text"b"}, where @{text b} is a boolean expression.
   331 %\item[local bindings] @ {text"let x = e"}.
   332 \end{description}
   333 
   334 Just like in Haskell, list comprehension is just a shorthand. To avoid
   335 misunderstandings, the translation into desugared form is not reversed
   336 upon output. Note that the translation of @{text"[e. x \<leftarrow> xs]"} is
   337 optmized to @{term"map (%x. e) xs"}.
   338 
   339 It is easy to write short list comprehensions which stand for complex
   340 expressions. During proofs, they may become unreadable (and
   341 mangled). In such cases it can be advisable to introduce separate
   342 definitions for the list comprehensions in question.  *}
   343 
   344 nonterminal lc_gen and lc_qual and lc_quals
   345 
   346 syntax
   347 "_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
   348 "_lc_gen" :: "lc_gen \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ <- _")
   349 "_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
   350 (*"_lc_let" :: "letbinds => lc_qual"  ("let _")*)
   351 "_lc_end" :: "lc_quals" ("]")
   352 "_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals" (", __")
   353 "_lc_abs" :: "'a => 'b list => 'b list"
   354 "_strip_positions" :: "'a \<Rightarrow> lc_gen"  ("_")
   355 
   356 (* These are easier than ML code but cannot express the optimized
   357    translation of [e. p<-xs]
   358 translations
   359 "[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
   360 "_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
   361  => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
   362 "[e. P]" => "if P then [e] else []"
   363 "_listcompr e (_lc_test P) (_lc_quals Q Qs)"
   364  => "if P then (_listcompr e Q Qs) else []"
   365 "_listcompr e (_lc_let b) (_lc_quals Q Qs)"
   366  => "_Let b (_listcompr e Q Qs)"
   367 *)
   368 
   369 syntax (xsymbols)
   370 "_lc_gen" :: "lc_gen \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
   371 syntax (HTML output)
   372 "_lc_gen" :: "lc_gen \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
   373 
   374 parse_translation (advanced) {*
   375 let
   376   val NilC = Syntax.const @{const_syntax Nil};
   377   val ConsC = Syntax.const @{const_syntax Cons};
   378   val mapC = Syntax.const @{const_syntax map};
   379   val concatC = Syntax.const @{const_syntax concat};
   380   val IfC = Syntax.const @{const_syntax If};
   381 
   382   fun singl x = ConsC $ x $ NilC;
   383 
   384   fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
   385     let
   386       (* FIXME proper name context!? *)
   387       val x = Free (singleton (Name.variant_list (fold Term.add_free_names [p, e] [])) "x", dummyT);
   388       val e = if opti then singl e else e;
   389       val case1 = Syntax.const @{syntax_const "_case1"} $ Term_Position.strip_positions p $ e;
   390       val case2 =
   391         Syntax.const @{syntax_const "_case1"} $
   392           Syntax.const @{const_syntax dummy_pattern} $ NilC;
   393       val cs = Syntax.const @{syntax_const "_case2"} $ case1 $ case2;
   394       val ft = Datatype_Case.case_tr false Datatype.info_of_constr_permissive ctxt [x, cs];
   395     in lambda x ft end;
   396 
   397   fun abs_tr ctxt (p as Free (s, T)) e opti =
   398         let
   399           val thy = Proof_Context.theory_of ctxt;
   400           val s' = Proof_Context.intern_const ctxt s;
   401         in
   402           if Sign.declared_const thy s'
   403           then (pat_tr ctxt p e opti, false)
   404           else (lambda p e, true)
   405         end
   406     | abs_tr ctxt p e opti = (pat_tr ctxt p e opti, false);
   407 
   408   fun lc_tr ctxt [e, Const (@{syntax_const "_lc_test"}, _) $ b, qs] =
   409         let
   410           val res =
   411             (case qs of
   412               Const (@{syntax_const "_lc_end"}, _) => singl e
   413             | Const (@{syntax_const "_lc_quals"}, _) $ q $ qs => lc_tr ctxt [e, q, qs]);
   414         in IfC $ b $ res $ NilC end
   415     | lc_tr ctxt
   416           [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
   417             Const(@{syntax_const "_lc_end"}, _)] =
   418         (case abs_tr ctxt p e true of
   419           (f, true) => mapC $ f $ es
   420         | (f, false) => concatC $ (mapC $ f $ es))
   421     | lc_tr ctxt
   422           [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
   423             Const (@{syntax_const "_lc_quals"}, _) $ q $ qs] =
   424         let val e' = lc_tr ctxt [e, q, qs];
   425         in concatC $ (mapC $ (fst (abs_tr ctxt p e' false)) $ es) end;
   426 
   427 in [(@{syntax_const "_listcompr"}, lc_tr)] end
   428 *}
   429 
   430 ML {*
   431   let
   432     val read = Syntax.read_term @{context};
   433     fun check s1 s2 = read s1 aconv read s2 orelse error ("Check failed: " ^ quote s1);
   434   in
   435     check "[(x,y,z). b]" "if b then [(x, y, z)] else []";
   436     check "[(x,y,z). x\<leftarrow>xs]" "map (\<lambda>x. (x, y, z)) xs";
   437     check "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]" "concat (map (\<lambda>x. map (\<lambda>y. e x y) ys) xs)";
   438     check "[(x,y,z). x<a, x>b]" "if x < a then if b < x then [(x, y, z)] else [] else []";
   439     check "[(x,y,z). x\<leftarrow>xs, x>b]" "concat (map (\<lambda>x. if b < x then [(x, y, z)] else []) xs)";
   440     check "[(x,y,z). x<a, x\<leftarrow>xs]" "if x < a then map (\<lambda>x. (x, y, z)) xs else []";
   441     check "[(x,y). Cons True x \<leftarrow> xs]"
   442       "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | True # x \<Rightarrow> [(x, y)] | False # x \<Rightarrow> []) xs)";
   443     check "[(x,y,z). Cons x [] \<leftarrow> xs]"
   444       "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | [x] \<Rightarrow> [(x, y, z)] | x # aa # lista \<Rightarrow> []) xs)";
   445     check "[(x,y,z). x<a, x>b, x=d]"
   446       "if x < a then if b < x then if x = d then [(x, y, z)] else [] else [] else []";
   447     check "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
   448       "if x < a then if b < x then map (\<lambda>y. (x, y, z)) ys else [] else []";
   449     check "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
   450       "if x < a then concat (map (\<lambda>x. if b < y then [(x, y, z)] else []) xs) else []";
   451     check "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
   452       "if x < a then concat (map (\<lambda>x. map (\<lambda>y. (x, y, z)) ys) xs) else []";
   453     check "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
   454       "concat (map (\<lambda>x. if b < x then if y < a then [(x, y, z)] else [] else []) xs)";
   455     check "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
   456       "concat (map (\<lambda>x. if b < x then map (\<lambda>y. (x, y, z)) ys else []) xs)";
   457     check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
   458       "concat (map (\<lambda>x. concat (map (\<lambda>y. if x < y then [(x, y, z)] else []) ys)) xs)";
   459     check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
   460       "concat (map (\<lambda>x. concat (map (\<lambda>y. map (\<lambda>z. (x, y, z)) zs) ys)) xs)"
   461   end;
   462 *}
   463 
   464 (*
   465 term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
   466 *)
   467 
   468 
   469 use "Tools/list_to_set_comprehension.ML"
   470 
   471 simproc_setup list_to_set_comprehension ("set xs") = {* K List_to_Set_Comprehension.simproc *}
   472 
   473 
   474 subsubsection {* @{const Nil} and @{const Cons} *}
   475 
   476 lemma not_Cons_self [simp]:
   477   "xs \<noteq> x # xs"
   478 by (induct xs) auto
   479 
   480 lemma not_Cons_self2 [simp]:
   481   "x # xs \<noteq> xs"
   482 by (rule not_Cons_self [symmetric])
   483 
   484 lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
   485 by (induct xs) auto
   486 
   487 lemma length_induct:
   488   "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
   489 by (rule measure_induct [of length]) iprover
   490 
   491 lemma list_nonempty_induct [consumes 1, case_names single cons]:
   492   assumes "xs \<noteq> []"
   493   assumes single: "\<And>x. P [x]"
   494   assumes cons: "\<And>x xs. xs \<noteq> [] \<Longrightarrow> P xs \<Longrightarrow> P (x # xs)"
   495   shows "P xs"
   496 using `xs \<noteq> []` proof (induct xs)
   497   case Nil then show ?case by simp
   498 next
   499   case (Cons x xs) show ?case proof (cases xs)
   500     case Nil with single show ?thesis by simp
   501   next
   502     case Cons then have "xs \<noteq> []" by simp
   503     moreover with Cons.hyps have "P xs" .
   504     ultimately show ?thesis by (rule cons)
   505   qed
   506 qed
   507 
   508 lemma inj_split_Cons: "inj_on (\<lambda>(xs, n). n#xs) X"
   509   by (auto intro!: inj_onI)
   510 
   511 subsubsection {* @{const length} *}
   512 
   513 text {*
   514   Needs to come before @{text "@"} because of theorem @{text
   515   append_eq_append_conv}.
   516 *}
   517 
   518 lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
   519 by (induct xs) auto
   520 
   521 lemma length_map [simp]: "length (map f xs) = length xs"
   522 by (induct xs) auto
   523 
   524 lemma length_rev [simp]: "length (rev xs) = length xs"
   525 by (induct xs) auto
   526 
   527 lemma length_tl [simp]: "length (tl xs) = length xs - 1"
   528 by (cases xs) auto
   529 
   530 lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
   531 by (induct xs) auto
   532 
   533 lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
   534 by (induct xs) auto
   535 
   536 lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
   537 by auto
   538 
   539 lemma length_Suc_conv:
   540 "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
   541 by (induct xs) auto
   542 
   543 lemma Suc_length_conv:
   544 "(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
   545 apply (induct xs, simp, simp)
   546 apply blast
   547 done
   548 
   549 lemma impossible_Cons: "length xs <= length ys ==> xs = x # ys = False"
   550   by (induct xs) auto
   551 
   552 lemma list_induct2 [consumes 1, case_names Nil Cons]:
   553   "length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
   554    (\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
   555    \<Longrightarrow> P xs ys"
   556 proof (induct xs arbitrary: ys)
   557   case Nil then show ?case by simp
   558 next
   559   case (Cons x xs ys) then show ?case by (cases ys) simp_all
   560 qed
   561 
   562 lemma list_induct3 [consumes 2, case_names Nil Cons]:
   563   "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
   564    (\<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))
   565    \<Longrightarrow> P xs ys zs"
   566 proof (induct xs arbitrary: ys zs)
   567   case Nil then show ?case by simp
   568 next
   569   case (Cons x xs ys zs) then show ?case by (cases ys, simp_all)
   570     (cases zs, simp_all)
   571 qed
   572 
   573 lemma list_induct4 [consumes 3, case_names Nil Cons]:
   574   "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow>
   575    P [] [] [] [] \<Longrightarrow> (\<And>x xs y ys z zs w ws. length xs = length ys \<Longrightarrow>
   576    length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow> P xs ys zs ws \<Longrightarrow>
   577    P (x#xs) (y#ys) (z#zs) (w#ws)) \<Longrightarrow> P xs ys zs ws"
   578 proof (induct xs arbitrary: ys zs ws)
   579   case Nil then show ?case by simp
   580 next
   581   case (Cons x xs ys zs ws) then show ?case by ((cases ys, simp_all), (cases zs,simp_all)) (cases ws, simp_all)
   582 qed
   583 
   584 lemma list_induct2': 
   585   "\<lbrakk> P [] [];
   586   \<And>x xs. P (x#xs) [];
   587   \<And>y ys. P [] (y#ys);
   588    \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
   589  \<Longrightarrow> P xs ys"
   590 by (induct xs arbitrary: ys) (case_tac x, auto)+
   591 
   592 lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
   593 by (rule Eq_FalseI) auto
   594 
   595 simproc_setup list_neq ("(xs::'a list) = ys") = {*
   596 (*
   597 Reduces xs=ys to False if xs and ys cannot be of the same length.
   598 This is the case if the atomic sublists of one are a submultiset
   599 of those of the other list and there are fewer Cons's in one than the other.
   600 *)
   601 
   602 let
   603 
   604 fun len (Const(@{const_name Nil},_)) acc = acc
   605   | len (Const(@{const_name Cons},_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
   606   | len (Const(@{const_name append},_) $ xs $ ys) acc = len xs (len ys acc)
   607   | len (Const(@{const_name rev},_) $ xs) acc = len xs acc
   608   | len (Const(@{const_name map},_) $ _ $ xs) acc = len xs acc
   609   | len t (ts,n) = (t::ts,n);
   610 
   611 fun list_neq _ ss ct =
   612   let
   613     val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
   614     val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
   615     fun prove_neq() =
   616       let
   617         val Type(_,listT::_) = eqT;
   618         val size = HOLogic.size_const listT;
   619         val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
   620         val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
   621         val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
   622           (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
   623       in SOME (thm RS @{thm neq_if_length_neq}) end
   624   in
   625     if m < n andalso submultiset (op aconv) (ls,rs) orelse
   626        n < m andalso submultiset (op aconv) (rs,ls)
   627     then prove_neq() else NONE
   628   end;
   629 in list_neq end;
   630 *}
   631 
   632 
   633 subsubsection {* @{text "@"} -- append *}
   634 
   635 lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
   636 by (induct xs) auto
   637 
   638 lemma append_Nil2 [simp]: "xs @ [] = xs"
   639 by (induct xs) auto
   640 
   641 lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
   642 by (induct xs) auto
   643 
   644 lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
   645 by (induct xs) auto
   646 
   647 lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
   648 by (induct xs) auto
   649 
   650 lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
   651 by (induct xs) auto
   652 
   653 lemma append_eq_append_conv [simp, no_atp]:
   654  "length xs = length ys \<or> length us = length vs
   655  ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
   656 apply (induct xs arbitrary: ys)
   657  apply (case_tac ys, simp, force)
   658 apply (case_tac ys, force, simp)
   659 done
   660 
   661 lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
   662   (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
   663 apply (induct xs arbitrary: ys zs ts)
   664  apply fastforce
   665 apply(case_tac zs)
   666  apply simp
   667 apply fastforce
   668 done
   669 
   670 lemma same_append_eq [iff, induct_simp]: "(xs @ ys = xs @ zs) = (ys = zs)"
   671 by simp
   672 
   673 lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
   674 by simp
   675 
   676 lemma append_same_eq [iff, induct_simp]: "(ys @ xs = zs @ xs) = (ys = zs)"
   677 by simp
   678 
   679 lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
   680 using append_same_eq [of _ _ "[]"] by auto
   681 
   682 lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
   683 using append_same_eq [of "[]"] by auto
   684 
   685 lemma hd_Cons_tl [simp,no_atp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
   686 by (induct xs) auto
   687 
   688 lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
   689 by (induct xs) auto
   690 
   691 lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
   692 by (simp add: hd_append split: list.split)
   693 
   694 lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
   695 by (simp split: list.split)
   696 
   697 lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
   698 by (simp add: tl_append split: list.split)
   699 
   700 
   701 lemma Cons_eq_append_conv: "x#xs = ys@zs =
   702  (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
   703 by(cases ys) auto
   704 
   705 lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
   706  (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
   707 by(cases ys) auto
   708 
   709 
   710 text {* Trivial rules for solving @{text "@"}-equations automatically. *}
   711 
   712 lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
   713 by simp
   714 
   715 lemma Cons_eq_appendI:
   716 "[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
   717 by (drule sym) simp
   718 
   719 lemma append_eq_appendI:
   720 "[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
   721 by (drule sym) simp
   722 
   723 
   724 text {*
   725 Simplification procedure for all list equalities.
   726 Currently only tries to rearrange @{text "@"} to see if
   727 - both lists end in a singleton list,
   728 - or both lists end in the same list.
   729 *}
   730 
   731 simproc_setup list_eq ("(xs::'a list) = ys")  = {*
   732   let
   733     fun last (cons as Const (@{const_name Cons}, _) $ _ $ xs) =
   734           (case xs of Const (@{const_name Nil}, _) => cons | _ => last xs)
   735       | last (Const(@{const_name append},_) $ _ $ ys) = last ys
   736       | last t = t;
   737     
   738     fun list1 (Const(@{const_name Cons},_) $ _ $ Const(@{const_name Nil},_)) = true
   739       | list1 _ = false;
   740     
   741     fun butlast ((cons as Const(@{const_name Cons},_) $ x) $ xs) =
   742           (case xs of Const (@{const_name Nil}, _) => xs | _ => cons $ butlast xs)
   743       | butlast ((app as Const (@{const_name append}, _) $ xs) $ ys) = app $ butlast ys
   744       | butlast xs = Const(@{const_name Nil}, fastype_of xs);
   745     
   746     val rearr_ss =
   747       HOL_basic_ss addsimps [@{thm append_assoc}, @{thm append_Nil}, @{thm append_Cons}];
   748     
   749     fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
   750       let
   751         val lastl = last lhs and lastr = last rhs;
   752         fun rearr conv =
   753           let
   754             val lhs1 = butlast lhs and rhs1 = butlast rhs;
   755             val Type(_,listT::_) = eqT
   756             val appT = [listT,listT] ---> listT
   757             val app = Const(@{const_name append},appT)
   758             val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
   759             val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
   760             val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
   761               (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
   762           in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
   763       in
   764         if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
   765         else if lastl aconv lastr then rearr @{thm append_same_eq}
   766         else NONE
   767       end;
   768   in fn _ => fn ss => fn ct => list_eq ss (term_of ct) end;
   769 *}
   770 
   771 
   772 subsubsection {* @{text map} *}
   773 
   774 lemma hd_map:
   775   "xs \<noteq> [] \<Longrightarrow> hd (map f xs) = f (hd xs)"
   776   by (cases xs) simp_all
   777 
   778 lemma map_tl:
   779   "map f (tl xs) = tl (map f xs)"
   780   by (cases xs) simp_all
   781 
   782 lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
   783 by (induct xs) simp_all
   784 
   785 lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
   786 by (rule ext, induct_tac xs) auto
   787 
   788 lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
   789 by (induct xs) auto
   790 
   791 lemma map_map [simp]: "map f (map g xs) = map (f \<circ> g) xs"
   792 by (induct xs) auto
   793 
   794 lemma map_comp_map[simp]: "((map f) o (map g)) = map(f o g)"
   795 apply(rule ext)
   796 apply(simp)
   797 done
   798 
   799 lemma rev_map: "rev (map f xs) = map f (rev xs)"
   800 by (induct xs) auto
   801 
   802 lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
   803 by (induct xs) auto
   804 
   805 lemma map_cong [fundef_cong]:
   806   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> map f xs = map g ys"
   807   by simp
   808 
   809 lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
   810 by (cases xs) auto
   811 
   812 lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
   813 by (cases xs) auto
   814 
   815 lemma map_eq_Cons_conv:
   816  "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
   817 by (cases xs) auto
   818 
   819 lemma Cons_eq_map_conv:
   820  "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
   821 by (cases ys) auto
   822 
   823 lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
   824 lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
   825 declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
   826 
   827 lemma ex_map_conv:
   828   "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
   829 by(induct ys, auto simp add: Cons_eq_map_conv)
   830 
   831 lemma map_eq_imp_length_eq:
   832   assumes "map f xs = map g ys"
   833   shows "length xs = length ys"
   834 using assms proof (induct ys arbitrary: xs)
   835   case Nil then show ?case by simp
   836 next
   837   case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto
   838   from Cons xs have "map f zs = map g ys" by simp
   839   moreover with Cons have "length zs = length ys" by blast
   840   with xs show ?case by simp
   841 qed
   842   
   843 lemma map_inj_on:
   844  "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
   845   ==> xs = ys"
   846 apply(frule map_eq_imp_length_eq)
   847 apply(rotate_tac -1)
   848 apply(induct rule:list_induct2)
   849  apply simp
   850 apply(simp)
   851 apply (blast intro:sym)
   852 done
   853 
   854 lemma inj_on_map_eq_map:
   855  "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
   856 by(blast dest:map_inj_on)
   857 
   858 lemma map_injective:
   859  "map f xs = map f ys ==> inj f ==> xs = ys"
   860 by (induct ys arbitrary: xs) (auto dest!:injD)
   861 
   862 lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
   863 by(blast dest:map_injective)
   864 
   865 lemma inj_mapI: "inj f ==> inj (map f)"
   866 by (iprover dest: map_injective injD intro: inj_onI)
   867 
   868 lemma inj_mapD: "inj (map f) ==> inj f"
   869 apply (unfold inj_on_def, clarify)
   870 apply (erule_tac x = "[x]" in ballE)
   871  apply (erule_tac x = "[y]" in ballE, simp, blast)
   872 apply blast
   873 done
   874 
   875 lemma inj_map[iff]: "inj (map f) = inj f"
   876 by (blast dest: inj_mapD intro: inj_mapI)
   877 
   878 lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
   879 apply(rule inj_onI)
   880 apply(erule map_inj_on)
   881 apply(blast intro:inj_onI dest:inj_onD)
   882 done
   883 
   884 lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
   885 by (induct xs, auto)
   886 
   887 lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
   888 by (induct xs) auto
   889 
   890 lemma map_fst_zip[simp]:
   891   "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
   892 by (induct rule:list_induct2, simp_all)
   893 
   894 lemma map_snd_zip[simp]:
   895   "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
   896 by (induct rule:list_induct2, simp_all)
   897 
   898 enriched_type map: map
   899   by (simp_all add: fun_eq_iff id_def)
   900 
   901 
   902 subsubsection {* @{text rev} *}
   903 
   904 lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
   905 by (induct xs) auto
   906 
   907 lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
   908 by (induct xs) auto
   909 
   910 lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
   911 by auto
   912 
   913 lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
   914 by (induct xs) auto
   915 
   916 lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
   917 by (induct xs) auto
   918 
   919 lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
   920 by (cases xs) auto
   921 
   922 lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
   923 by (cases xs) auto
   924 
   925 lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
   926 apply (induct xs arbitrary: ys, force)
   927 apply (case_tac ys, simp, force)
   928 done
   929 
   930 lemma inj_on_rev[iff]: "inj_on rev A"
   931 by(simp add:inj_on_def)
   932 
   933 lemma rev_induct [case_names Nil snoc]:
   934   "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
   935 apply(simplesubst rev_rev_ident[symmetric])
   936 apply(rule_tac list = "rev xs" in list.induct, simp_all)
   937 done
   938 
   939 lemma rev_exhaust [case_names Nil snoc]:
   940   "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
   941 by (induct xs rule: rev_induct) auto
   942 
   943 lemmas rev_cases = rev_exhaust
   944 
   945 lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
   946 by(rule rev_cases[of xs]) auto
   947 
   948 
   949 subsubsection {* @{text set} *}
   950 
   951 lemma finite_set [iff]: "finite (set xs)"
   952 by (induct xs) auto
   953 
   954 lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
   955 by (induct xs) auto
   956 
   957 lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
   958 by(cases xs) auto
   959 
   960 lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
   961 by auto
   962 
   963 lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
   964 by auto
   965 
   966 lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
   967 by (induct xs) auto
   968 
   969 lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
   970 by(induct xs) auto
   971 
   972 lemma set_rev [simp]: "set (rev xs) = set xs"
   973 by (induct xs) auto
   974 
   975 lemma set_map [simp]: "set (map f xs) = f`(set xs)"
   976 by (induct xs) auto
   977 
   978 lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
   979 by (induct xs) auto
   980 
   981 lemma set_upt [simp]: "set[i..<j] = {i..<j}"
   982 by (induct j) auto
   983 
   984 
   985 lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs"
   986 proof (induct xs)
   987   case Nil thus ?case by simp
   988 next
   989   case Cons thus ?case by (auto intro: Cons_eq_appendI)
   990 qed
   991 
   992 lemma in_set_conv_decomp: "x \<in> set xs \<longleftrightarrow> (\<exists>ys zs. xs = ys @ x # zs)"
   993   by (auto elim: split_list)
   994 
   995 lemma split_list_first: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
   996 proof (induct xs)
   997   case Nil thus ?case by simp
   998 next
   999   case (Cons a xs)
  1000   show ?case
  1001   proof cases
  1002     assume "x = a" thus ?case using Cons by fastforce
  1003   next
  1004     assume "x \<noteq> a" thus ?case using Cons by(fastforce intro!: Cons_eq_appendI)
  1005   qed
  1006 qed
  1007 
  1008 lemma in_set_conv_decomp_first:
  1009   "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
  1010   by (auto dest!: split_list_first)
  1011 
  1012 lemma split_list_last: "x \<in> set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
  1013 proof (induct xs rule: rev_induct)
  1014   case Nil thus ?case by simp
  1015 next
  1016   case (snoc a xs)
  1017   show ?case
  1018   proof cases
  1019     assume "x = a" thus ?case using snoc by (metis List.set.simps(1) emptyE)
  1020   next
  1021     assume "x \<noteq> a" thus ?case using snoc by fastforce
  1022   qed
  1023 qed
  1024 
  1025 lemma in_set_conv_decomp_last:
  1026   "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
  1027   by (auto dest!: split_list_last)
  1028 
  1029 lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs & P x"
  1030 proof (induct xs)
  1031   case Nil thus ?case by simp
  1032 next
  1033   case Cons thus ?case
  1034     by(simp add:Bex_def)(metis append_Cons append.simps(1))
  1035 qed
  1036 
  1037 lemma split_list_propE:
  1038   assumes "\<exists>x \<in> set xs. P x"
  1039   obtains ys x zs where "xs = ys @ x # zs" and "P x"
  1040 using split_list_prop [OF assms] by blast
  1041 
  1042 lemma split_list_first_prop:
  1043   "\<exists>x \<in> set xs. P x \<Longrightarrow>
  1044    \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)"
  1045 proof (induct xs)
  1046   case Nil thus ?case by simp
  1047 next
  1048   case (Cons x xs)
  1049   show ?case
  1050   proof cases
  1051     assume "P x"
  1052     thus ?thesis by simp (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
  1053   next
  1054     assume "\<not> P x"
  1055     hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
  1056     thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
  1057   qed
  1058 qed
  1059 
  1060 lemma split_list_first_propE:
  1061   assumes "\<exists>x \<in> set xs. P x"
  1062   obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y"
  1063 using split_list_first_prop [OF assms] by blast
  1064 
  1065 lemma split_list_first_prop_iff:
  1066   "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
  1067    (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))"
  1068 by (rule, erule split_list_first_prop) auto
  1069 
  1070 lemma split_list_last_prop:
  1071   "\<exists>x \<in> set xs. P x \<Longrightarrow>
  1072    \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)"
  1073 proof(induct xs rule:rev_induct)
  1074   case Nil thus ?case by simp
  1075 next
  1076   case (snoc x xs)
  1077   show ?case
  1078   proof cases
  1079     assume "P x" thus ?thesis by (metis emptyE set_empty)
  1080   next
  1081     assume "\<not> P x"
  1082     hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
  1083     thus ?thesis using `\<not> P x` snoc(1) by fastforce
  1084   qed
  1085 qed
  1086 
  1087 lemma split_list_last_propE:
  1088   assumes "\<exists>x \<in> set xs. P x"
  1089   obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z"
  1090 using split_list_last_prop [OF assms] by blast
  1091 
  1092 lemma split_list_last_prop_iff:
  1093   "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
  1094    (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
  1095 by (metis split_list_last_prop [where P=P] in_set_conv_decomp)
  1096 
  1097 lemma finite_list: "finite A ==> EX xs. set xs = A"
  1098   by (erule finite_induct)
  1099     (auto simp add: set.simps(2) [symmetric] simp del: set.simps(2))
  1100 
  1101 lemma card_length: "card (set xs) \<le> length xs"
  1102 by (induct xs) (auto simp add: card_insert_if)
  1103 
  1104 lemma set_minus_filter_out:
  1105   "set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)"
  1106   by (induct xs) auto
  1107 
  1108 
  1109 subsubsection {* @{text filter} *}
  1110 
  1111 lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
  1112 by (induct xs) auto
  1113 
  1114 lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
  1115 by (induct xs) simp_all
  1116 
  1117 lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
  1118 by (induct xs) auto
  1119 
  1120 lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
  1121 by (induct xs) (auto simp add: le_SucI)
  1122 
  1123 lemma sum_length_filter_compl:
  1124   "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
  1125 by(induct xs) simp_all
  1126 
  1127 lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
  1128 by (induct xs) auto
  1129 
  1130 lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
  1131 by (induct xs) auto
  1132 
  1133 lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
  1134 by (induct xs) simp_all
  1135 
  1136 lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
  1137 apply (induct xs)
  1138  apply auto
  1139 apply(cut_tac P=P and xs=xs in length_filter_le)
  1140 apply simp
  1141 done
  1142 
  1143 lemma filter_map:
  1144   "filter P (map f xs) = map f (filter (P o f) xs)"
  1145 by (induct xs) simp_all
  1146 
  1147 lemma length_filter_map[simp]:
  1148   "length (filter P (map f xs)) = length(filter (P o f) xs)"
  1149 by (simp add:filter_map)
  1150 
  1151 lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
  1152 by auto
  1153 
  1154 lemma length_filter_less:
  1155   "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
  1156 proof (induct xs)
  1157   case Nil thus ?case by simp
  1158 next
  1159   case (Cons x xs) thus ?case
  1160     apply (auto split:split_if_asm)
  1161     using length_filter_le[of P xs] apply arith
  1162   done
  1163 qed
  1164 
  1165 lemma length_filter_conv_card:
  1166  "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
  1167 proof (induct xs)
  1168   case Nil thus ?case by simp
  1169 next
  1170   case (Cons x xs)
  1171   let ?S = "{i. i < length xs & p(xs!i)}"
  1172   have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
  1173   show ?case (is "?l = card ?S'")
  1174   proof (cases)
  1175     assume "p x"
  1176     hence eq: "?S' = insert 0 (Suc ` ?S)"
  1177       by(auto simp: image_def split:nat.split dest:gr0_implies_Suc)
  1178     have "length (filter p (x # xs)) = Suc(card ?S)"
  1179       using Cons `p x` by simp
  1180     also have "\<dots> = Suc(card(Suc ` ?S))" using fin
  1181       by (simp add: card_image)
  1182     also have "\<dots> = card ?S'" using eq fin
  1183       by (simp add:card_insert_if) (simp add:image_def)
  1184     finally show ?thesis .
  1185   next
  1186     assume "\<not> p x"
  1187     hence eq: "?S' = Suc ` ?S"
  1188       by(auto simp add: image_def split:nat.split elim:lessE)
  1189     have "length (filter p (x # xs)) = card ?S"
  1190       using Cons `\<not> p x` by simp
  1191     also have "\<dots> = card(Suc ` ?S)" using fin
  1192       by (simp add: card_image)
  1193     also have "\<dots> = card ?S'" using eq fin
  1194       by (simp add:card_insert_if)
  1195     finally show ?thesis .
  1196   qed
  1197 qed
  1198 
  1199 lemma Cons_eq_filterD:
  1200  "x#xs = filter P ys \<Longrightarrow>
  1201   \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
  1202   (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
  1203 proof(induct ys)
  1204   case Nil thus ?case by simp
  1205 next
  1206   case (Cons y ys)
  1207   show ?case (is "\<exists>x. ?Q x")
  1208   proof cases
  1209     assume Py: "P y"
  1210     show ?thesis
  1211     proof cases
  1212       assume "x = y"
  1213       with Py Cons.prems have "?Q []" by simp
  1214       then show ?thesis ..
  1215     next
  1216       assume "x \<noteq> y"
  1217       with Py Cons.prems show ?thesis by simp
  1218     qed
  1219   next
  1220     assume "\<not> P y"
  1221     with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastforce
  1222     then have "?Q (y#us)" by simp
  1223     then show ?thesis ..
  1224   qed
  1225 qed
  1226 
  1227 lemma filter_eq_ConsD:
  1228  "filter P ys = x#xs \<Longrightarrow>
  1229   \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
  1230 by(rule Cons_eq_filterD) simp
  1231 
  1232 lemma filter_eq_Cons_iff:
  1233  "(filter P ys = x#xs) =
  1234   (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
  1235 by(auto dest:filter_eq_ConsD)
  1236 
  1237 lemma Cons_eq_filter_iff:
  1238  "(x#xs = filter P ys) =
  1239   (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
  1240 by(auto dest:Cons_eq_filterD)
  1241 
  1242 lemma filter_cong[fundef_cong]:
  1243  "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
  1244 apply simp
  1245 apply(erule thin_rl)
  1246 by (induct ys) simp_all
  1247 
  1248 
  1249 subsubsection {* List partitioning *}
  1250 
  1251 primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where
  1252   "partition P [] = ([], [])"
  1253   | "partition P (x # xs) = 
  1254       (let (yes, no) = partition P xs
  1255       in if P x then (x # yes, no) else (yes, x # no))"
  1256 
  1257 lemma partition_filter1:
  1258     "fst (partition P xs) = filter P xs"
  1259 by (induct xs) (auto simp add: Let_def split_def)
  1260 
  1261 lemma partition_filter2:
  1262     "snd (partition P xs) = filter (Not o P) xs"
  1263 by (induct xs) (auto simp add: Let_def split_def)
  1264 
  1265 lemma partition_P:
  1266   assumes "partition P xs = (yes, no)"
  1267   shows "(\<forall>p \<in> set yes.  P p) \<and> (\<forall>p  \<in> set no. \<not> P p)"
  1268 proof -
  1269   from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
  1270     by simp_all
  1271   then show ?thesis by (simp_all add: partition_filter1 partition_filter2)
  1272 qed
  1273 
  1274 lemma partition_set:
  1275   assumes "partition P xs = (yes, no)"
  1276   shows "set yes \<union> set no = set xs"
  1277 proof -
  1278   from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
  1279     by simp_all
  1280   then show ?thesis by (auto simp add: partition_filter1 partition_filter2) 
  1281 qed
  1282 
  1283 lemma partition_filter_conv[simp]:
  1284   "partition f xs = (filter f xs,filter (Not o f) xs)"
  1285 unfolding partition_filter2[symmetric]
  1286 unfolding partition_filter1[symmetric] by simp
  1287 
  1288 declare partition.simps[simp del]
  1289 
  1290 
  1291 subsubsection {* @{text concat} *}
  1292 
  1293 lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
  1294 by (induct xs) auto
  1295 
  1296 lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
  1297 by (induct xss) auto
  1298 
  1299 lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
  1300 by (induct xss) auto
  1301 
  1302 lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
  1303 by (induct xs) auto
  1304 
  1305 lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs"
  1306 by (induct xs) auto
  1307 
  1308 lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
  1309 by (induct xs) auto
  1310 
  1311 lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
  1312 by (induct xs) auto
  1313 
  1314 lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
  1315 by (induct xs) auto
  1316 
  1317 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)"
  1318 proof (induct xs arbitrary: ys)
  1319   case (Cons x xs ys)
  1320   thus ?case by (cases ys) auto
  1321 qed (auto)
  1322 
  1323 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"
  1324 by (simp add: concat_eq_concat_iff)
  1325 
  1326 
  1327 subsubsection {* @{text nth} *}
  1328 
  1329 lemma nth_Cons_0 [simp, code]: "(x # xs)!0 = x"
  1330 by auto
  1331 
  1332 lemma nth_Cons_Suc [simp, code]: "(x # xs)!(Suc n) = xs!n"
  1333 by auto
  1334 
  1335 declare nth.simps [simp del]
  1336 
  1337 lemma nth_Cons_pos[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
  1338 by(auto simp: Nat.gr0_conv_Suc)
  1339 
  1340 lemma nth_append:
  1341   "(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
  1342 apply (induct xs arbitrary: n, simp)
  1343 apply (case_tac n, auto)
  1344 done
  1345 
  1346 lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
  1347 by (induct xs) auto
  1348 
  1349 lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
  1350 by (induct xs) auto
  1351 
  1352 lemma nth_map [simp]: "n < length xs ==> (map f xs)!n = f(xs!n)"
  1353 apply (induct xs arbitrary: n, simp)
  1354 apply (case_tac n, auto)
  1355 done
  1356 
  1357 lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
  1358 by(cases xs) simp_all
  1359 
  1360 
  1361 lemma list_eq_iff_nth_eq:
  1362  "(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
  1363 apply(induct xs arbitrary: ys)
  1364  apply force
  1365 apply(case_tac ys)
  1366  apply simp
  1367 apply(simp add:nth_Cons split:nat.split)apply blast
  1368 done
  1369 
  1370 lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
  1371 apply (induct xs, simp, simp)
  1372 apply safe
  1373 apply (metis nat_case_0 nth.simps zero_less_Suc)
  1374 apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
  1375 apply (case_tac i, simp)
  1376 apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff)
  1377 done
  1378 
  1379 lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
  1380 by(auto simp:set_conv_nth)
  1381 
  1382 lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
  1383 by (auto simp add: set_conv_nth)
  1384 
  1385 lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
  1386 by (auto simp add: set_conv_nth)
  1387 
  1388 lemma all_nth_imp_all_set:
  1389 "[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
  1390 by (auto simp add: set_conv_nth)
  1391 
  1392 lemma all_set_conv_all_nth:
  1393 "(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
  1394 by (auto simp add: set_conv_nth)
  1395 
  1396 lemma rev_nth:
  1397   "n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)"
  1398 proof (induct xs arbitrary: n)
  1399   case Nil thus ?case by simp
  1400 next
  1401   case (Cons x xs)
  1402   hence n: "n < Suc (length xs)" by simp
  1403   moreover
  1404   { assume "n < length xs"
  1405     with n obtain n' where "length xs - n = Suc n'"
  1406       by (cases "length xs - n", auto)
  1407     moreover
  1408     then have "length xs - Suc n = n'" by simp
  1409     ultimately
  1410     have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp
  1411   }
  1412   ultimately
  1413   show ?case by (clarsimp simp add: Cons nth_append)
  1414 qed
  1415 
  1416 lemma Skolem_list_nth:
  1417   "(ALL i<k. EX x. P i x) = (EX xs. size xs = k & (ALL i<k. P i (xs!i)))"
  1418   (is "_ = (EX xs. ?P k xs)")
  1419 proof(induct k)
  1420   case 0 show ?case by simp
  1421 next
  1422   case (Suc k)
  1423   show ?case (is "?L = ?R" is "_ = (EX xs. ?P' xs)")
  1424   proof
  1425     assume "?R" thus "?L" using Suc by auto
  1426   next
  1427     assume "?L"
  1428     with Suc obtain x xs where "?P k xs & P k x" by (metis less_Suc_eq)
  1429     hence "?P'(xs@[x])" by(simp add:nth_append less_Suc_eq)
  1430     thus "?R" ..
  1431   qed
  1432 qed
  1433 
  1434 
  1435 subsubsection {* @{text list_update} *}
  1436 
  1437 lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
  1438 by (induct xs arbitrary: i) (auto split: nat.split)
  1439 
  1440 lemma nth_list_update:
  1441 "i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
  1442 by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
  1443 
  1444 lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
  1445 by (simp add: nth_list_update)
  1446 
  1447 lemma nth_list_update_neq [simp]: "i \<noteq> j ==> xs[i:=x]!j = xs!j"
  1448 by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
  1449 
  1450 lemma list_update_id[simp]: "xs[i := xs!i] = xs"
  1451 by (induct xs arbitrary: i) (simp_all split:nat.splits)
  1452 
  1453 lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
  1454 apply (induct xs arbitrary: i)
  1455  apply simp
  1456 apply (case_tac i)
  1457 apply simp_all
  1458 done
  1459 
  1460 lemma list_update_nonempty[simp]: "xs[k:=x] = [] \<longleftrightarrow> xs=[]"
  1461 by(metis length_0_conv length_list_update)
  1462 
  1463 lemma list_update_same_conv:
  1464 "i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
  1465 by (induct xs arbitrary: i) (auto split: nat.split)
  1466 
  1467 lemma list_update_append1:
  1468  "i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
  1469 apply (induct xs arbitrary: i, simp)
  1470 apply(simp split:nat.split)
  1471 done
  1472 
  1473 lemma list_update_append:
  1474   "(xs @ ys) [n:= x] = 
  1475   (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
  1476 by (induct xs arbitrary: n) (auto split:nat.splits)
  1477 
  1478 lemma list_update_length [simp]:
  1479  "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
  1480 by (induct xs, auto)
  1481 
  1482 lemma map_update: "map f (xs[k:= y]) = (map f xs)[k := f y]"
  1483 by(induct xs arbitrary: k)(auto split:nat.splits)
  1484 
  1485 lemma rev_update:
  1486   "k < length xs \<Longrightarrow> rev (xs[k:= y]) = (rev xs)[length xs - k - 1 := y]"
  1487 by (induct xs arbitrary: k) (auto simp: list_update_append split:nat.splits)
  1488 
  1489 lemma update_zip:
  1490   "(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
  1491 by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split)
  1492 
  1493 lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)"
  1494 by (induct xs arbitrary: i) (auto split: nat.split)
  1495 
  1496 lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
  1497 by (blast dest!: set_update_subset_insert [THEN subsetD])
  1498 
  1499 lemma set_update_memI: "n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
  1500 by (induct xs arbitrary: n) (auto split:nat.splits)
  1501 
  1502 lemma list_update_overwrite[simp]:
  1503   "xs [i := x, i := y] = xs [i := y]"
  1504 apply (induct xs arbitrary: i) apply simp
  1505 apply (case_tac i, simp_all)
  1506 done
  1507 
  1508 lemma list_update_swap:
  1509   "i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]"
  1510 apply (induct xs arbitrary: i i')
  1511 apply simp
  1512 apply (case_tac i, case_tac i')
  1513 apply auto
  1514 apply (case_tac i')
  1515 apply auto
  1516 done
  1517 
  1518 lemma list_update_code [code]:
  1519   "[][i := y] = []"
  1520   "(x # xs)[0 := y] = y # xs"
  1521   "(x # xs)[Suc i := y] = x # xs[i := y]"
  1522   by simp_all
  1523 
  1524 
  1525 subsubsection {* @{text last} and @{text butlast} *}
  1526 
  1527 lemma last_snoc [simp]: "last (xs @ [x]) = x"
  1528 by (induct xs) auto
  1529 
  1530 lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
  1531 by (induct xs) auto
  1532 
  1533 lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
  1534   by simp
  1535 
  1536 lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
  1537   by simp
  1538 
  1539 lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
  1540 by (induct xs) (auto)
  1541 
  1542 lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
  1543 by(simp add:last_append)
  1544 
  1545 lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
  1546 by(simp add:last_append)
  1547 
  1548 lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
  1549 by(rule rev_exhaust[of xs]) simp_all
  1550 
  1551 lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
  1552 by(cases xs) simp_all
  1553 
  1554 lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
  1555 by (induct as) auto
  1556 
  1557 lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
  1558 by (induct xs rule: rev_induct) auto
  1559 
  1560 lemma butlast_append:
  1561   "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
  1562 by (induct xs arbitrary: ys) auto
  1563 
  1564 lemma append_butlast_last_id [simp]:
  1565 "xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
  1566 by (induct xs) auto
  1567 
  1568 lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
  1569 by (induct xs) (auto split: split_if_asm)
  1570 
  1571 lemma in_set_butlast_appendI:
  1572 "x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
  1573 by (auto dest: in_set_butlastD simp add: butlast_append)
  1574 
  1575 lemma last_drop[simp]: "n < length xs \<Longrightarrow> last (drop n xs) = last xs"
  1576 apply (induct xs arbitrary: n)
  1577  apply simp
  1578 apply (auto split:nat.split)
  1579 done
  1580 
  1581 lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
  1582 by(induct xs)(auto simp:neq_Nil_conv)
  1583 
  1584 lemma butlast_conv_take: "butlast xs = take (length xs - 1) xs"
  1585 by (induct xs, simp, case_tac xs, simp_all)
  1586 
  1587 lemma last_list_update:
  1588   "xs \<noteq> [] \<Longrightarrow> last(xs[k:=x]) = (if k = size xs - 1 then x else last xs)"
  1589 by (auto simp: last_conv_nth)
  1590 
  1591 lemma butlast_list_update:
  1592   "butlast(xs[k:=x]) =
  1593  (if k = size xs - 1 then butlast xs else (butlast xs)[k:=x])"
  1594 apply(cases xs rule:rev_cases)
  1595 apply simp
  1596 apply(simp add:list_update_append split:nat.splits)
  1597 done
  1598 
  1599 lemma last_map:
  1600   "xs \<noteq> [] \<Longrightarrow> last (map f xs) = f (last xs)"
  1601   by (cases xs rule: rev_cases) simp_all
  1602 
  1603 lemma map_butlast:
  1604   "map f (butlast xs) = butlast (map f xs)"
  1605   by (induct xs) simp_all
  1606 
  1607 lemma snoc_eq_iff_butlast:
  1608   "xs @ [x] = ys \<longleftrightarrow> (ys \<noteq> [] & butlast ys = xs & last ys = x)"
  1609 by (metis append_butlast_last_id append_is_Nil_conv butlast_snoc last_snoc not_Cons_self)
  1610 
  1611 
  1612 subsubsection {* @{text take} and @{text drop} *}
  1613 
  1614 lemma take_0 [simp]: "take 0 xs = []"
  1615 by (induct xs) auto
  1616 
  1617 lemma drop_0 [simp]: "drop 0 xs = xs"
  1618 by (induct xs) auto
  1619 
  1620 lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
  1621 by simp
  1622 
  1623 lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
  1624 by simp
  1625 
  1626 declare take_Cons [simp del] and drop_Cons [simp del]
  1627 
  1628 lemma take_1_Cons [simp]: "take 1 (x # xs) = [x]"
  1629   unfolding One_nat_def by simp
  1630 
  1631 lemma drop_1_Cons [simp]: "drop 1 (x # xs) = xs"
  1632   unfolding One_nat_def by simp
  1633 
  1634 lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
  1635 by(clarsimp simp add:neq_Nil_conv)
  1636 
  1637 lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
  1638 by(cases xs, simp_all)
  1639 
  1640 lemma take_tl: "take n (tl xs) = tl (take (Suc n) xs)"
  1641 by (induct xs arbitrary: n) simp_all
  1642 
  1643 lemma drop_tl: "drop n (tl xs) = tl(drop n xs)"
  1644 by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split)
  1645 
  1646 lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)"
  1647 by (cases n, simp, cases xs, auto)
  1648 
  1649 lemma tl_drop: "tl (drop n xs) = drop n (tl xs)"
  1650 by (simp only: drop_tl)
  1651 
  1652 lemma nth_via_drop: "drop n xs = y#ys \<Longrightarrow> xs!n = y"
  1653 apply (induct xs arbitrary: n, simp)
  1654 apply(simp add:drop_Cons nth_Cons split:nat.splits)
  1655 done
  1656 
  1657 lemma take_Suc_conv_app_nth:
  1658   "i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
  1659 apply (induct xs arbitrary: i, simp)
  1660 apply (case_tac i, auto)
  1661 done
  1662 
  1663 lemma drop_Suc_conv_tl:
  1664   "i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
  1665 apply (induct xs arbitrary: i, simp)
  1666 apply (case_tac i, auto)
  1667 done
  1668 
  1669 lemma length_take [simp]: "length (take n xs) = min (length xs) n"
  1670 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1671 
  1672 lemma length_drop [simp]: "length (drop n xs) = (length xs - n)"
  1673 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1674 
  1675 lemma take_all [simp]: "length xs <= n ==> take n xs = xs"
  1676 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1677 
  1678 lemma drop_all [simp]: "length xs <= n ==> drop n xs = []"
  1679 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1680 
  1681 lemma take_append [simp]:
  1682   "take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
  1683 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1684 
  1685 lemma drop_append [simp]:
  1686   "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
  1687 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1688 
  1689 lemma take_take [simp]: "take n (take m xs) = take (min n m) xs"
  1690 apply (induct m arbitrary: xs n, auto)
  1691 apply (case_tac xs, auto)
  1692 apply (case_tac n, auto)
  1693 done
  1694 
  1695 lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs"
  1696 apply (induct m arbitrary: xs, auto)
  1697 apply (case_tac xs, auto)
  1698 done
  1699 
  1700 lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)"
  1701 apply (induct m arbitrary: xs n, auto)
  1702 apply (case_tac xs, auto)
  1703 done
  1704 
  1705 lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)"
  1706 apply(induct xs arbitrary: m n)
  1707  apply simp
  1708 apply(simp add: take_Cons drop_Cons split:nat.split)
  1709 done
  1710 
  1711 lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs"
  1712 apply (induct n arbitrary: xs, auto)
  1713 apply (case_tac xs, auto)
  1714 done
  1715 
  1716 lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])"
  1717 apply(induct xs arbitrary: n)
  1718  apply simp
  1719 apply(simp add:take_Cons split:nat.split)
  1720 done
  1721 
  1722 lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)"
  1723 apply(induct xs arbitrary: n)
  1724 apply simp
  1725 apply(simp add:drop_Cons split:nat.split)
  1726 done
  1727 
  1728 lemma take_map: "take n (map f xs) = map f (take n xs)"
  1729 apply (induct n arbitrary: xs, auto)
  1730 apply (case_tac xs, auto)
  1731 done
  1732 
  1733 lemma drop_map: "drop n (map f xs) = map f (drop n xs)"
  1734 apply (induct n arbitrary: xs, auto)
  1735 apply (case_tac xs, auto)
  1736 done
  1737 
  1738 lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)"
  1739 apply (induct xs arbitrary: i, auto)
  1740 apply (case_tac i, auto)
  1741 done
  1742 
  1743 lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)"
  1744 apply (induct xs arbitrary: i, auto)
  1745 apply (case_tac i, auto)
  1746 done
  1747 
  1748 lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i"
  1749 apply (induct xs arbitrary: i n, auto)
  1750 apply (case_tac n, blast)
  1751 apply (case_tac i, auto)
  1752 done
  1753 
  1754 lemma nth_drop [simp]:
  1755   "n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
  1756 apply (induct n arbitrary: xs i, auto)
  1757 apply (case_tac xs, auto)
  1758 done
  1759 
  1760 lemma butlast_take:
  1761   "n <= length xs ==> butlast (take n xs) = take (n - 1) xs"
  1762 by (simp add: butlast_conv_take min_max.inf_absorb1 min_max.inf_absorb2)
  1763 
  1764 lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)"
  1765 by (simp add: butlast_conv_take drop_take add_ac)
  1766 
  1767 lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs"
  1768 by (simp add: butlast_conv_take min_max.inf_absorb1)
  1769 
  1770 lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)"
  1771 by (simp add: butlast_conv_take drop_take add_ac)
  1772 
  1773 lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
  1774 by(simp add: hd_conv_nth)
  1775 
  1776 lemma set_take_subset_set_take:
  1777   "m <= n \<Longrightarrow> set(take m xs) <= set(take n xs)"
  1778 apply (induct xs arbitrary: m n)
  1779 apply simp
  1780 apply (case_tac n)
  1781 apply (auto simp: take_Cons)
  1782 done
  1783 
  1784 lemma set_take_subset: "set(take n xs) \<subseteq> set xs"
  1785 by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split)
  1786 
  1787 lemma set_drop_subset: "set(drop n xs) \<subseteq> set xs"
  1788 by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split)
  1789 
  1790 lemma set_drop_subset_set_drop:
  1791   "m >= n \<Longrightarrow> set(drop m xs) <= set(drop n xs)"
  1792 apply(induct xs arbitrary: m n)
  1793 apply(auto simp:drop_Cons split:nat.split)
  1794 apply (metis set_drop_subset subset_iff)
  1795 done
  1796 
  1797 lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
  1798 using set_take_subset by fast
  1799 
  1800 lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
  1801 using set_drop_subset by fast
  1802 
  1803 lemma append_eq_conv_conj:
  1804   "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
  1805 apply (induct xs arbitrary: zs, simp, clarsimp)
  1806 apply (case_tac zs, auto)
  1807 done
  1808 
  1809 lemma take_add: 
  1810   "take (i+j) xs = take i xs @ take j (drop i xs)"
  1811 apply (induct xs arbitrary: i, auto) 
  1812 apply (case_tac i, simp_all)
  1813 done
  1814 
  1815 lemma append_eq_append_conv_if:
  1816  "(xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
  1817   (if size xs\<^isub>1 \<le> size ys\<^isub>1
  1818    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
  1819    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)"
  1820 apply(induct xs\<^isub>1 arbitrary: ys\<^isub>1)
  1821  apply simp
  1822 apply(case_tac ys\<^isub>1)
  1823 apply simp_all
  1824 done
  1825 
  1826 lemma take_hd_drop:
  1827   "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (Suc n) xs"
  1828 apply(induct xs arbitrary: n)
  1829 apply simp
  1830 apply(simp add:drop_Cons split:nat.split)
  1831 done
  1832 
  1833 lemma id_take_nth_drop:
  1834  "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
  1835 proof -
  1836   assume si: "i < length xs"
  1837   hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
  1838   moreover
  1839   from si have "take (Suc i) xs = take i xs @ [xs!i]"
  1840     apply (rule_tac take_Suc_conv_app_nth) by arith
  1841   ultimately show ?thesis by auto
  1842 qed
  1843   
  1844 lemma upd_conv_take_nth_drop:
  1845  "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
  1846 proof -
  1847   assume i: "i < length xs"
  1848   have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
  1849     by(rule arg_cong[OF id_take_nth_drop[OF i]])
  1850   also have "\<dots> = take i xs @ a # drop (Suc i) xs"
  1851     using i by (simp add: list_update_append)
  1852   finally show ?thesis .
  1853 qed
  1854 
  1855 lemma nth_drop':
  1856   "i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs"
  1857 apply (induct i arbitrary: xs)
  1858 apply (simp add: neq_Nil_conv)
  1859 apply (erule exE)+
  1860 apply simp
  1861 apply (case_tac xs)
  1862 apply simp_all
  1863 done
  1864 
  1865 
  1866 subsubsection {* @{text takeWhile} and @{text dropWhile} *}
  1867 
  1868 lemma length_takeWhile_le: "length (takeWhile P xs) \<le> length xs"
  1869   by (induct xs) auto
  1870 
  1871 lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
  1872 by (induct xs) auto
  1873 
  1874 lemma takeWhile_append1 [simp]:
  1875 "[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
  1876 by (induct xs) auto
  1877 
  1878 lemma takeWhile_append2 [simp]:
  1879 "(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
  1880 by (induct xs) auto
  1881 
  1882 lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
  1883 by (induct xs) auto
  1884 
  1885 lemma takeWhile_nth: "j < length (takeWhile P xs) \<Longrightarrow> takeWhile P xs ! j = xs ! j"
  1886 apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
  1887 
  1888 lemma dropWhile_nth: "j < length (dropWhile P xs) \<Longrightarrow> dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))"
  1889 apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
  1890 
  1891 lemma length_dropWhile_le: "length (dropWhile P xs) \<le> length xs"
  1892 by (induct xs) auto
  1893 
  1894 lemma dropWhile_append1 [simp]:
  1895 "[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
  1896 by (induct xs) auto
  1897 
  1898 lemma dropWhile_append2 [simp]:
  1899 "(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
  1900 by (induct xs) auto
  1901 
  1902 lemma set_takeWhileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
  1903 by (induct xs) (auto split: split_if_asm)
  1904 
  1905 lemma takeWhile_eq_all_conv[simp]:
  1906  "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
  1907 by(induct xs, auto)
  1908 
  1909 lemma dropWhile_eq_Nil_conv[simp]:
  1910  "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
  1911 by(induct xs, auto)
  1912 
  1913 lemma dropWhile_eq_Cons_conv:
  1914  "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
  1915 by(induct xs, auto)
  1916 
  1917 lemma distinct_takeWhile[simp]: "distinct xs ==> distinct (takeWhile P xs)"
  1918 by (induct xs) (auto dest: set_takeWhileD)
  1919 
  1920 lemma distinct_dropWhile[simp]: "distinct xs ==> distinct (dropWhile P xs)"
  1921 by (induct xs) auto
  1922 
  1923 lemma takeWhile_map: "takeWhile P (map f xs) = map f (takeWhile (P \<circ> f) xs)"
  1924 by (induct xs) auto
  1925 
  1926 lemma dropWhile_map: "dropWhile P (map f xs) = map f (dropWhile (P \<circ> f) xs)"
  1927 by (induct xs) auto
  1928 
  1929 lemma takeWhile_eq_take: "takeWhile P xs = take (length (takeWhile P xs)) xs"
  1930 by (induct xs) auto
  1931 
  1932 lemma dropWhile_eq_drop: "dropWhile P xs = drop (length (takeWhile P xs)) xs"
  1933 by (induct xs) auto
  1934 
  1935 lemma hd_dropWhile:
  1936   "dropWhile P xs \<noteq> [] \<Longrightarrow> \<not> P (hd (dropWhile P xs))"
  1937 using assms by (induct xs) auto
  1938 
  1939 lemma takeWhile_eq_filter:
  1940   assumes "\<And> x. x \<in> set (dropWhile P xs) \<Longrightarrow> \<not> P x"
  1941   shows "takeWhile P xs = filter P xs"
  1942 proof -
  1943   have A: "filter P xs = filter P (takeWhile P xs @ dropWhile P xs)"
  1944     by simp
  1945   have B: "filter P (dropWhile P xs) = []"
  1946     unfolding filter_empty_conv using assms by blast
  1947   have "filter P xs = takeWhile P xs"
  1948     unfolding A filter_append B
  1949     by (auto simp add: filter_id_conv dest: set_takeWhileD)
  1950   thus ?thesis ..
  1951 qed
  1952 
  1953 lemma takeWhile_eq_take_P_nth:
  1954   "\<lbrakk> \<And> i. \<lbrakk> i < n ; i < length xs \<rbrakk> \<Longrightarrow> P (xs ! i) ; n < length xs \<Longrightarrow> \<not> P (xs ! n) \<rbrakk> \<Longrightarrow>
  1955   takeWhile P xs = take n xs"
  1956 proof (induct xs arbitrary: n)
  1957   case (Cons x xs)
  1958   thus ?case
  1959   proof (cases n)
  1960     case (Suc n') note this[simp]
  1961     have "P x" using Cons.prems(1)[of 0] by simp
  1962     moreover have "takeWhile P xs = take n' xs"
  1963     proof (rule Cons.hyps)
  1964       case goal1 thus "P (xs ! i)" using Cons.prems(1)[of "Suc i"] by simp
  1965     next case goal2 thus ?case using Cons by auto
  1966     qed
  1967     ultimately show ?thesis by simp
  1968    qed simp
  1969 qed simp
  1970 
  1971 lemma nth_length_takeWhile:
  1972   "length (takeWhile P xs) < length xs \<Longrightarrow> \<not> P (xs ! length (takeWhile P xs))"
  1973 by (induct xs) auto
  1974 
  1975 lemma length_takeWhile_less_P_nth:
  1976   assumes all: "\<And> i. i < j \<Longrightarrow> P (xs ! i)" and "j \<le> length xs"
  1977   shows "j \<le> length (takeWhile P xs)"
  1978 proof (rule classical)
  1979   assume "\<not> ?thesis"
  1980   hence "length (takeWhile P xs) < length xs" using assms by simp
  1981   thus ?thesis using all `\<not> ?thesis` nth_length_takeWhile[of P xs] by auto
  1982 qed
  1983 
  1984 text{* The following two lemmmas could be generalized to an arbitrary
  1985 property. *}
  1986 
  1987 lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
  1988  takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
  1989 by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
  1990 
  1991 lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
  1992   dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
  1993 apply(induct xs)
  1994  apply simp
  1995 apply auto
  1996 apply(subst dropWhile_append2)
  1997 apply auto
  1998 done
  1999 
  2000 lemma takeWhile_not_last:
  2001  "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
  2002 apply(induct xs)
  2003  apply simp
  2004 apply(case_tac xs)
  2005 apply(auto)
  2006 done
  2007 
  2008 lemma takeWhile_cong [fundef_cong]:
  2009   "[| l = k; !!x. x : set l ==> P x = Q x |] 
  2010   ==> takeWhile P l = takeWhile Q k"
  2011 by (induct k arbitrary: l) (simp_all)
  2012 
  2013 lemma dropWhile_cong [fundef_cong]:
  2014   "[| l = k; !!x. x : set l ==> P x = Q x |] 
  2015   ==> dropWhile P l = dropWhile Q k"
  2016 by (induct k arbitrary: l, simp_all)
  2017 
  2018 
  2019 subsubsection {* @{text zip} *}
  2020 
  2021 lemma zip_Nil [simp]: "zip [] ys = []"
  2022 by (induct ys) auto
  2023 
  2024 lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
  2025 by simp
  2026 
  2027 declare zip_Cons [simp del]
  2028 
  2029 lemma [code]:
  2030   "zip [] ys = []"
  2031   "zip xs [] = []"
  2032   "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
  2033   by (fact zip_Nil zip.simps(1) zip_Cons_Cons)+
  2034 
  2035 lemma zip_Cons1:
  2036  "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
  2037 by(auto split:list.split)
  2038 
  2039 lemma length_zip [simp]:
  2040 "length (zip xs ys) = min (length xs) (length ys)"
  2041 by (induct xs ys rule:list_induct2') auto
  2042 
  2043 lemma zip_obtain_same_length:
  2044   assumes "\<And>zs ws n. length zs = length ws \<Longrightarrow> n = min (length xs) (length ys)
  2045     \<Longrightarrow> zs = take n xs \<Longrightarrow> ws = take n ys \<Longrightarrow> P (zip zs ws)"
  2046   shows "P (zip xs ys)"
  2047 proof -
  2048   let ?n = "min (length xs) (length ys)"
  2049   have "P (zip (take ?n xs) (take ?n ys))"
  2050     by (rule assms) simp_all
  2051   moreover have "zip xs ys = zip (take ?n xs) (take ?n ys)"
  2052   proof (induct xs arbitrary: ys)
  2053     case Nil then show ?case by simp
  2054   next
  2055     case (Cons x xs) then show ?case by (cases ys) simp_all
  2056   qed
  2057   ultimately show ?thesis by simp
  2058 qed
  2059 
  2060 lemma zip_append1:
  2061 "zip (xs @ ys) zs =
  2062 zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
  2063 by (induct xs zs rule:list_induct2') auto
  2064 
  2065 lemma zip_append2:
  2066 "zip xs (ys @ zs) =
  2067 zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
  2068 by (induct xs ys rule:list_induct2') auto
  2069 
  2070 lemma zip_append [simp]:
  2071  "[| length xs = length us; length ys = length vs |] ==>
  2072 zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
  2073 by (simp add: zip_append1)
  2074 
  2075 lemma zip_rev:
  2076 "length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
  2077 by (induct rule:list_induct2, simp_all)
  2078 
  2079 lemma zip_map_map:
  2080   "zip (map f xs) (map g ys) = map (\<lambda> (x, y). (f x, g y)) (zip xs ys)"
  2081 proof (induct xs arbitrary: ys)
  2082   case (Cons x xs) note Cons_x_xs = Cons.hyps
  2083   show ?case
  2084   proof (cases ys)
  2085     case (Cons y ys')
  2086     show ?thesis unfolding Cons using Cons_x_xs by simp
  2087   qed simp
  2088 qed simp
  2089 
  2090 lemma zip_map1:
  2091   "zip (map f xs) ys = map (\<lambda>(x, y). (f x, y)) (zip xs ys)"
  2092 using zip_map_map[of f xs "\<lambda>x. x" ys] by simp
  2093 
  2094 lemma zip_map2:
  2095   "zip xs (map f ys) = map (\<lambda>(x, y). (x, f y)) (zip xs ys)"
  2096 using zip_map_map[of "\<lambda>x. x" xs f ys] by simp
  2097 
  2098 lemma map_zip_map:
  2099   "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
  2100 unfolding zip_map1 by auto
  2101 
  2102 lemma map_zip_map2:
  2103   "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
  2104 unfolding zip_map2 by auto
  2105 
  2106 text{* Courtesy of Andreas Lochbihler: *}
  2107 lemma zip_same_conv_map: "zip xs xs = map (\<lambda>x. (x, x)) xs"
  2108 by(induct xs) auto
  2109 
  2110 lemma nth_zip [simp]:
  2111 "[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
  2112 apply (induct ys arbitrary: i xs, simp)
  2113 apply (case_tac xs)
  2114  apply (simp_all add: nth.simps split: nat.split)
  2115 done
  2116 
  2117 lemma set_zip:
  2118 "set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
  2119 by(simp add: set_conv_nth cong: rev_conj_cong)
  2120 
  2121 lemma zip_same: "((a,b) \<in> set (zip xs xs)) = (a \<in> set xs \<and> a = b)"
  2122 by(induct xs) auto
  2123 
  2124 lemma zip_update:
  2125   "zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
  2126 by(rule sym, simp add: update_zip)
  2127 
  2128 lemma zip_replicate [simp]:
  2129   "zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
  2130 apply (induct i arbitrary: j, auto)
  2131 apply (case_tac j, auto)
  2132 done
  2133 
  2134 lemma take_zip:
  2135   "take n (zip xs ys) = zip (take n xs) (take n ys)"
  2136 apply (induct n arbitrary: xs ys)
  2137  apply simp
  2138 apply (case_tac xs, simp)
  2139 apply (case_tac ys, simp_all)
  2140 done
  2141 
  2142 lemma drop_zip:
  2143   "drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
  2144 apply (induct n arbitrary: xs ys)
  2145  apply simp
  2146 apply (case_tac xs, simp)
  2147 apply (case_tac ys, simp_all)
  2148 done
  2149 
  2150 lemma zip_takeWhile_fst: "zip (takeWhile P xs) ys = takeWhile (P \<circ> fst) (zip xs ys)"
  2151 proof (induct xs arbitrary: ys)
  2152   case (Cons x xs) thus ?case by (cases ys) auto
  2153 qed simp
  2154 
  2155 lemma zip_takeWhile_snd: "zip xs (takeWhile P ys) = takeWhile (P \<circ> snd) (zip xs ys)"
  2156 proof (induct xs arbitrary: ys)
  2157   case (Cons x xs) thus ?case by (cases ys) auto
  2158 qed simp
  2159 
  2160 lemma set_zip_leftD:
  2161   "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
  2162 by (induct xs ys rule:list_induct2') auto
  2163 
  2164 lemma set_zip_rightD:
  2165   "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
  2166 by (induct xs ys rule:list_induct2') auto
  2167 
  2168 lemma in_set_zipE:
  2169   "(x,y) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
  2170 by(blast dest: set_zip_leftD set_zip_rightD)
  2171 
  2172 lemma zip_map_fst_snd:
  2173   "zip (map fst zs) (map snd zs) = zs"
  2174   by (induct zs) simp_all
  2175 
  2176 lemma zip_eq_conv:
  2177   "length xs = length ys \<Longrightarrow> zip xs ys = zs \<longleftrightarrow> map fst zs = xs \<and> map snd zs = ys"
  2178   by (auto simp add: zip_map_fst_snd)
  2179 
  2180 
  2181 subsubsection {* @{text list_all2} *}
  2182 
  2183 lemma list_all2_lengthD [intro?]: 
  2184   "list_all2 P xs ys ==> length xs = length ys"
  2185 by (simp add: list_all2_def)
  2186 
  2187 lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
  2188 by (simp add: list_all2_def)
  2189 
  2190 lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
  2191 by (simp add: list_all2_def)
  2192 
  2193 lemma list_all2_Cons [iff, code]:
  2194   "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
  2195 by (auto simp add: list_all2_def)
  2196 
  2197 lemma list_all2_Cons1:
  2198 "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
  2199 by (cases ys) auto
  2200 
  2201 lemma list_all2_Cons2:
  2202 "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
  2203 by (cases xs) auto
  2204 
  2205 lemma list_all2_rev [iff]:
  2206 "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
  2207 by (simp add: list_all2_def zip_rev cong: conj_cong)
  2208 
  2209 lemma list_all2_rev1:
  2210 "list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
  2211 by (subst list_all2_rev [symmetric]) simp
  2212 
  2213 lemma list_all2_append1:
  2214 "list_all2 P (xs @ ys) zs =
  2215 (EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
  2216 list_all2 P xs us \<and> list_all2 P ys vs)"
  2217 apply (simp add: list_all2_def zip_append1)
  2218 apply (rule iffI)
  2219  apply (rule_tac x = "take (length xs) zs" in exI)
  2220  apply (rule_tac x = "drop (length xs) zs" in exI)
  2221  apply (force split: nat_diff_split simp add: min_def, clarify)
  2222 apply (simp add: ball_Un)
  2223 done
  2224 
  2225 lemma list_all2_append2:
  2226 "list_all2 P xs (ys @ zs) =
  2227 (EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
  2228 list_all2 P us ys \<and> list_all2 P vs zs)"
  2229 apply (simp add: list_all2_def zip_append2)
  2230 apply (rule iffI)
  2231  apply (rule_tac x = "take (length ys) xs" in exI)
  2232  apply (rule_tac x = "drop (length ys) xs" in exI)
  2233  apply (force split: nat_diff_split simp add: min_def, clarify)
  2234 apply (simp add: ball_Un)
  2235 done
  2236 
  2237 lemma list_all2_append:
  2238   "length xs = length ys \<Longrightarrow>
  2239   list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
  2240 by (induct rule:list_induct2, simp_all)
  2241 
  2242 lemma list_all2_appendI [intro?, trans]:
  2243   "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
  2244 by (simp add: list_all2_append list_all2_lengthD)
  2245 
  2246 lemma list_all2_conv_all_nth:
  2247 "list_all2 P xs ys =
  2248 (length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
  2249 by (force simp add: list_all2_def set_zip)
  2250 
  2251 lemma list_all2_trans:
  2252   assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
  2253   shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
  2254         (is "!!bs cs. PROP ?Q as bs cs")
  2255 proof (induct as)
  2256   fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
  2257   show "!!cs. PROP ?Q (x # xs) bs cs"
  2258   proof (induct bs)
  2259     fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
  2260     show "PROP ?Q (x # xs) (y # ys) cs"
  2261       by (induct cs) (auto intro: tr I1 I2)
  2262   qed simp
  2263 qed simp
  2264 
  2265 lemma list_all2_all_nthI [intro?]:
  2266   "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
  2267 by (simp add: list_all2_conv_all_nth)
  2268 
  2269 lemma list_all2I:
  2270   "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
  2271 by (simp add: list_all2_def)
  2272 
  2273 lemma list_all2_nthD:
  2274   "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
  2275 by (simp add: list_all2_conv_all_nth)
  2276 
  2277 lemma list_all2_nthD2:
  2278   "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
  2279 by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
  2280 
  2281 lemma list_all2_map1: 
  2282   "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
  2283 by (simp add: list_all2_conv_all_nth)
  2284 
  2285 lemma list_all2_map2: 
  2286   "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
  2287 by (auto simp add: list_all2_conv_all_nth)
  2288 
  2289 lemma list_all2_refl [intro?]:
  2290   "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
  2291 by (simp add: list_all2_conv_all_nth)
  2292 
  2293 lemma list_all2_update_cong:
  2294   "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
  2295 by (simp add: list_all2_conv_all_nth nth_list_update)
  2296 
  2297 lemma list_all2_update_cong2:
  2298   "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
  2299 by (simp add: list_all2_lengthD list_all2_update_cong)
  2300 
  2301 lemma list_all2_takeI [simp,intro?]:
  2302   "list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
  2303 apply (induct xs arbitrary: n ys)
  2304  apply simp
  2305 apply (clarsimp simp add: list_all2_Cons1)
  2306 apply (case_tac n)
  2307 apply auto
  2308 done
  2309 
  2310 lemma list_all2_dropI [simp,intro?]:
  2311   "list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
  2312 apply (induct as arbitrary: n bs, simp)
  2313 apply (clarsimp simp add: list_all2_Cons1)
  2314 apply (case_tac n, simp, simp)
  2315 done
  2316 
  2317 lemma list_all2_mono [intro?]:
  2318   "list_all2 P xs ys \<Longrightarrow> (\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys) \<Longrightarrow> list_all2 Q xs ys"
  2319 apply (induct xs arbitrary: ys, simp)
  2320 apply (case_tac ys, auto)
  2321 done
  2322 
  2323 lemma list_all2_eq:
  2324   "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
  2325 by (induct xs ys rule: list_induct2') auto
  2326 
  2327 lemma list_eq_iff_zip_eq:
  2328   "xs = ys \<longleftrightarrow> length xs = length ys \<and> (\<forall>(x,y) \<in> set (zip xs ys). x = y)"
  2329 by(auto simp add: set_zip list_all2_eq list_all2_conv_all_nth cong: conj_cong)
  2330 
  2331 
  2332 subsubsection {* @{text foldl} and @{text foldr} *}
  2333 
  2334 lemma foldl_append [simp]:
  2335   "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
  2336 by (induct xs arbitrary: a) auto
  2337 
  2338 lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
  2339 by (induct xs) auto
  2340 
  2341 lemma foldr_map: "foldr g (map f xs) a = foldr (g o f) xs a"
  2342 by(induct xs) simp_all
  2343 
  2344 text{* For efficient code generation: avoid intermediate list. *}
  2345 lemma foldl_map[code_unfold]:
  2346   "foldl g a (map f xs) = foldl (%a x. g a (f x)) a xs"
  2347 by(induct xs arbitrary:a) simp_all
  2348 
  2349 lemma foldl_apply:
  2350   assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x \<circ> h = h \<circ> g x"
  2351   shows "foldl (\<lambda>s x. f x s) (h s) xs = h (foldl (\<lambda>s x. g x s) s xs)"
  2352   by (rule sym, insert assms, induct xs arbitrary: s) (simp_all add: fun_eq_iff)
  2353 
  2354 lemma foldl_cong [fundef_cong]:
  2355   "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] 
  2356   ==> foldl f a l = foldl g b k"
  2357 by (induct k arbitrary: a b l) simp_all
  2358 
  2359 lemma foldr_cong [fundef_cong]:
  2360   "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] 
  2361   ==> foldr f l a = foldr g k b"
  2362 by (induct k arbitrary: a b l) simp_all
  2363 
  2364 lemma foldl_fun_comm:
  2365   assumes "\<And>x y s. f (f s x) y = f (f s y) x"
  2366   shows "f (foldl f s xs) x = foldl f (f s x) xs"
  2367   by (induct xs arbitrary: s)
  2368     (simp_all add: assms)
  2369 
  2370 lemma (in semigroup_add) foldl_assoc:
  2371 shows "foldl op+ (x+y) zs = x + (foldl op+ y zs)"
  2372 by (induct zs arbitrary: y) (simp_all add:add_assoc)
  2373 
  2374 lemma (in monoid_add) foldl_absorb0:
  2375 shows "x + (foldl op+ 0 zs) = foldl op+ x zs"
  2376 by (induct zs) (simp_all add:foldl_assoc)
  2377 
  2378 lemma foldl_rev:
  2379   assumes "\<And>x y s. f (f s x) y = f (f s y) x"
  2380   shows "foldl f s (rev xs) = foldl f s xs"
  2381 proof (induct xs arbitrary: s)
  2382   case Nil then show ?case by simp
  2383 next
  2384   case (Cons x xs) with assms show ?case by (simp add: foldl_fun_comm)
  2385 qed
  2386 
  2387 lemma rev_foldl_cons [code]:
  2388   "rev xs = foldl (\<lambda>xs x. x # xs) [] xs"
  2389 proof (induct xs)
  2390   case Nil then show ?case by simp
  2391 next
  2392   case Cons
  2393   {
  2394     fix x xs ys
  2395     have "foldl (\<lambda>xs x. x # xs) ys xs @ [x]
  2396       = foldl (\<lambda>xs x. x # xs) (ys @ [x]) xs"
  2397     by (induct xs arbitrary: ys) auto
  2398   }
  2399   note aux = this
  2400   show ?case by (induct xs) (auto simp add: Cons aux)
  2401 qed
  2402 
  2403 
  2404 text{* The ``Third Duality Theorem'' in Bird \& Wadler: *}
  2405 
  2406 lemma foldr_foldl:
  2407   "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
  2408   by (induct xs) auto
  2409 
  2410 lemma foldl_foldr:
  2411   "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
  2412   by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
  2413 
  2414 
  2415 text{* The ``First Duality Theorem'' in Bird \& Wadler: *}
  2416 
  2417 lemma (in monoid_add) foldl_foldr1_lemma:
  2418   "foldl op + a xs = a + foldr op + xs 0"
  2419   by (induct xs arbitrary: a) (auto simp: add_assoc)
  2420 
  2421 corollary (in monoid_add) foldl_foldr1:
  2422   "foldl op + 0 xs = foldr op + xs 0"
  2423   by (simp add: foldl_foldr1_lemma)
  2424 
  2425 lemma (in ab_semigroup_add) foldr_conv_foldl:
  2426   "foldr op + xs a = foldl op + a xs"
  2427   by (induct xs) (simp_all add: foldl_assoc add.commute)
  2428 
  2429 text {*
  2430 Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
  2431 difficult to use because it requires an additional transitivity step.
  2432 *}
  2433 
  2434 lemma start_le_sum: "(m::nat) <= n ==> m <= foldl (op +) n ns"
  2435 by (induct ns arbitrary: n) auto
  2436 
  2437 lemma elem_le_sum: "(n::nat) : set ns ==> n <= foldl (op +) 0 ns"
  2438 by (force intro: start_le_sum simp add: in_set_conv_decomp)
  2439 
  2440 lemma sum_eq_0_conv [iff]:
  2441   "(foldl (op +) (m::nat) ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
  2442 by (induct ns arbitrary: m) auto
  2443 
  2444 lemma foldr_invariant: 
  2445   "\<lbrakk>Q x ; \<forall> x\<in> set xs. P x; \<forall> x y. P x \<and> Q y \<longrightarrow> Q (f x y) \<rbrakk> \<Longrightarrow> Q (foldr f xs x)"
  2446   by (induct xs, simp_all)
  2447 
  2448 lemma foldl_invariant: 
  2449   "\<lbrakk>Q x ; \<forall> x\<in> set xs. P x; \<forall> x y. P x \<and> Q y \<longrightarrow> Q (f y x) \<rbrakk> \<Longrightarrow> Q (foldl f x xs)"
  2450   by (induct xs arbitrary: x, simp_all)
  2451 
  2452 lemma foldl_weak_invariant:
  2453   assumes "P s"
  2454     and "\<And>s x. x \<in> set xs \<Longrightarrow> P s \<Longrightarrow> P (f s x)"
  2455   shows "P (foldl f s xs)"
  2456   using assms by (induct xs arbitrary: s) simp_all
  2457 
  2458 text {* @{const foldl} and @{const concat} *}
  2459 
  2460 lemma foldl_conv_concat:
  2461   "foldl (op @) xs xss = xs @ concat xss"
  2462 proof (induct xss arbitrary: xs)
  2463   case Nil show ?case by simp
  2464 next
  2465   interpret monoid_add "op @" "[]" proof qed simp_all
  2466   case Cons then show ?case by (simp add: foldl_absorb0)
  2467 qed
  2468 
  2469 lemma concat_conv_foldl: "concat xss = foldl (op @) [] xss"
  2470   by (simp add: foldl_conv_concat)
  2471 
  2472 text {* @{const Finite_Set.fold} and @{const foldl} *}
  2473 
  2474 lemma (in comp_fun_commute) fold_set_remdups:
  2475   "fold f y (set xs) = foldl (\<lambda>y x. f x y) y (remdups xs)"
  2476   by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm insert_absorb)
  2477 
  2478 lemma (in comp_fun_idem) fold_set:
  2479   "fold f y (set xs) = foldl (\<lambda>y x. f x y) y xs"
  2480   by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm)
  2481 
  2482 lemma (in ab_semigroup_idem_mult) fold1_set:
  2483   assumes "xs \<noteq> []"
  2484   shows "fold1 times (set xs) = foldl times (hd xs) (tl xs)"
  2485 proof -
  2486   interpret comp_fun_idem times by (fact comp_fun_idem)
  2487   from assms obtain y ys where xs: "xs = y # ys"
  2488     by (cases xs) auto
  2489   show ?thesis
  2490   proof (cases "set ys = {}")
  2491     case True with xs show ?thesis by simp
  2492   next
  2493     case False
  2494     then have "fold1 times (insert y (set ys)) = fold times y (set ys)"
  2495       by (simp only: finite_set fold1_eq_fold_idem)
  2496     with xs show ?thesis by (simp add: fold_set mult_commute)
  2497   qed
  2498 qed
  2499 
  2500 lemma (in lattice) Inf_fin_set_fold [code_unfold]:
  2501   "Inf_fin (set (x # xs)) = foldl inf x xs"
  2502 proof -
  2503   interpret ab_semigroup_idem_mult "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2504     by (fact ab_semigroup_idem_mult_inf)
  2505   show ?thesis
  2506     by (simp add: Inf_fin_def fold1_set del: set.simps)
  2507 qed
  2508 
  2509 lemma (in lattice) Sup_fin_set_fold [code_unfold]:
  2510   "Sup_fin (set (x # xs)) = foldl sup x xs"
  2511 proof -
  2512   interpret ab_semigroup_idem_mult "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2513     by (fact ab_semigroup_idem_mult_sup)
  2514   show ?thesis
  2515     by (simp add: Sup_fin_def fold1_set del: set.simps)
  2516 qed
  2517 
  2518 lemma (in linorder) Min_fin_set_fold [code_unfold]:
  2519   "Min (set (x # xs)) = foldl min x xs"
  2520 proof -
  2521   interpret ab_semigroup_idem_mult "min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2522     by (fact ab_semigroup_idem_mult_min)
  2523   show ?thesis
  2524     by (simp add: Min_def fold1_set del: set.simps)
  2525 qed
  2526 
  2527 lemma (in linorder) Max_fin_set_fold [code_unfold]:
  2528   "Max (set (x # xs)) = foldl max x xs"
  2529 proof -
  2530   interpret ab_semigroup_idem_mult "max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2531     by (fact ab_semigroup_idem_mult_max)
  2532   show ?thesis
  2533     by (simp add: Max_def fold1_set del: set.simps)
  2534 qed
  2535 
  2536 lemma (in complete_lattice) Inf_set_fold [code_unfold]:
  2537   "Inf (set xs) = foldl inf top xs"
  2538 proof -
  2539   interpret comp_fun_idem "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2540     by (fact comp_fun_idem_inf)
  2541   show ?thesis by (simp add: Inf_fold_inf fold_set inf_commute)
  2542 qed
  2543 
  2544 lemma (in complete_lattice) Sup_set_fold [code_unfold]:
  2545   "Sup (set xs) = foldl sup bot xs"
  2546 proof -
  2547   interpret comp_fun_idem "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
  2548     by (fact comp_fun_idem_sup)
  2549   show ?thesis by (simp add: Sup_fold_sup fold_set sup_commute)
  2550 qed
  2551 
  2552 lemma (in complete_lattice) INFI_set_fold:
  2553   "INFI (set xs) f = foldl (\<lambda>y x. inf (f x) y) top xs"
  2554   unfolding INF_def set_map [symmetric] Inf_set_fold foldl_map
  2555     by (simp add: inf_commute)
  2556 
  2557 lemma (in complete_lattice) SUPR_set_fold:
  2558   "SUPR (set xs) f = foldl (\<lambda>y x. sup (f x) y) bot xs"
  2559   unfolding SUP_def set_map [symmetric] Sup_set_fold foldl_map
  2560     by (simp add: sup_commute)
  2561 
  2562 
  2563 subsubsection {* @{text upt} *}
  2564 
  2565 lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
  2566 -- {* simp does not terminate! *}
  2567 by (induct j) auto
  2568 
  2569 lemmas upt_rec_number_of[simp] = upt_rec[of "number_of m" "number_of n"] for m n
  2570 
  2571 lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
  2572 by (subst upt_rec) simp
  2573 
  2574 lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
  2575 by(induct j)simp_all
  2576 
  2577 lemma upt_eq_Cons_conv:
  2578  "([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
  2579 apply(induct j arbitrary: x xs)
  2580  apply simp
  2581 apply(clarsimp simp add: append_eq_Cons_conv)
  2582 apply arith
  2583 done
  2584 
  2585 lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
  2586 -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
  2587 by simp
  2588 
  2589 lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
  2590   by (simp add: upt_rec)
  2591 
  2592 lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
  2593 -- {* LOOPS as a simprule, since @{text "j <= j"}. *}
  2594 by (induct k) auto
  2595 
  2596 lemma length_upt [simp]: "length [i..<j] = j - i"
  2597 by (induct j) (auto simp add: Suc_diff_le)
  2598 
  2599 lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
  2600 apply (induct j)
  2601 apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
  2602 done
  2603 
  2604 
  2605 lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
  2606 by(simp add:upt_conv_Cons)
  2607 
  2608 lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
  2609 apply(cases j)
  2610  apply simp
  2611 by(simp add:upt_Suc_append)
  2612 
  2613 lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]"
  2614 apply (induct m arbitrary: i, simp)
  2615 apply (subst upt_rec)
  2616 apply (rule sym)
  2617 apply (subst upt_rec)
  2618 apply (simp del: upt.simps)
  2619 done
  2620 
  2621 lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
  2622 apply(induct j)
  2623 apply auto
  2624 done
  2625 
  2626 lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]"
  2627 by (induct n) auto
  2628 
  2629 lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
  2630 apply (induct n m  arbitrary: i rule: diff_induct)
  2631 prefer 3 apply (subst map_Suc_upt[symmetric])
  2632 apply (auto simp add: less_diff_conv)
  2633 done
  2634 
  2635 lemma nth_take_lemma:
  2636   "k <= length xs ==> k <= length ys ==>
  2637      (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
  2638 apply (atomize, induct k arbitrary: xs ys)
  2639 apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
  2640 txt {* Both lists must be non-empty *}
  2641 apply (case_tac xs, simp)
  2642 apply (case_tac ys, clarify)
  2643  apply (simp (no_asm_use))
  2644 apply clarify
  2645 txt {* prenexing's needed, not miniscoping *}
  2646 apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
  2647 apply blast
  2648 done
  2649 
  2650 lemma nth_equalityI:
  2651  "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
  2652   by (frule nth_take_lemma [OF le_refl eq_imp_le]) simp_all
  2653 
  2654 lemma map_nth:
  2655   "map (\<lambda>i. xs ! i) [0..<length xs] = xs"
  2656   by (rule nth_equalityI, auto)
  2657 
  2658 (* needs nth_equalityI *)
  2659 lemma list_all2_antisym:
  2660   "\<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> 
  2661   \<Longrightarrow> xs = ys"
  2662   apply (simp add: list_all2_conv_all_nth) 
  2663   apply (rule nth_equalityI, blast, simp)
  2664   done
  2665 
  2666 lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
  2667 -- {* The famous take-lemma. *}
  2668 apply (drule_tac x = "max (length xs) (length ys)" in spec)
  2669 apply (simp add: le_max_iff_disj)
  2670 done
  2671 
  2672 
  2673 lemma take_Cons':
  2674      "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
  2675 by (cases n) simp_all
  2676 
  2677 lemma drop_Cons':
  2678      "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
  2679 by (cases n) simp_all
  2680 
  2681 lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
  2682 by (cases n) simp_all
  2683 
  2684 lemmas take_Cons_number_of = take_Cons'[of "number_of v"] for v
  2685 lemmas drop_Cons_number_of = drop_Cons'[of "number_of v"] for v
  2686 lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v"] for v
  2687 
  2688 declare take_Cons_number_of [simp] 
  2689         drop_Cons_number_of [simp] 
  2690         nth_Cons_number_of [simp] 
  2691 
  2692 
  2693 subsubsection {* @{text upto}: interval-list on @{typ int} *}
  2694 
  2695 (* FIXME make upto tail recursive? *)
  2696 
  2697 function upto :: "int \<Rightarrow> int \<Rightarrow> int list" ("(1[_../_])") where
  2698 "upto i j = (if i \<le> j then i # [i+1..j] else [])"
  2699 by auto
  2700 termination
  2701 by(relation "measure(%(i::int,j). nat(j - i + 1))") auto
  2702 
  2703 declare upto.simps[code, simp del]
  2704 
  2705 lemmas upto_rec_number_of[simp] = upto.simps[of "number_of m" "number_of n"] for m n
  2706 
  2707 lemma upto_empty[simp]: "j < i \<Longrightarrow> [i..j] = []"
  2708 by(simp add: upto.simps)
  2709 
  2710 lemma set_upto[simp]: "set[i..j] = {i..j}"
  2711 proof(induct i j rule:upto.induct)
  2712   case (1 i j)
  2713   from this show ?case
  2714     unfolding upto.simps[of i j] simp_from_to[of i j] by auto
  2715 qed
  2716 
  2717 
  2718 subsubsection {* @{text "distinct"} and @{text remdups} *}
  2719 
  2720 lemma distinct_tl:
  2721   "distinct xs \<Longrightarrow> distinct (tl xs)"
  2722   by (cases xs) simp_all
  2723 
  2724 lemma distinct_append [simp]:
  2725 "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
  2726 by (induct xs) auto
  2727 
  2728 lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
  2729 by(induct xs) auto
  2730 
  2731 lemma set_remdups [simp]: "set (remdups xs) = set xs"
  2732 by (induct xs) (auto simp add: insert_absorb)
  2733 
  2734 lemma distinct_remdups [iff]: "distinct (remdups xs)"
  2735 by (induct xs) auto
  2736 
  2737 lemma distinct_remdups_id: "distinct xs ==> remdups xs = xs"
  2738 by (induct xs, auto)
  2739 
  2740 lemma remdups_id_iff_distinct [simp]: "remdups xs = xs \<longleftrightarrow> distinct xs"
  2741 by (metis distinct_remdups distinct_remdups_id)
  2742 
  2743 lemma finite_distinct_list: "finite A \<Longrightarrow> EX xs. set xs = A & distinct xs"
  2744 by (metis distinct_remdups finite_list set_remdups)
  2745 
  2746 lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
  2747 by (induct x, auto) 
  2748 
  2749 lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
  2750 by (induct x, auto)
  2751 
  2752 lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
  2753 by (induct xs) auto
  2754 
  2755 lemma length_remdups_eq[iff]:
  2756   "(length (remdups xs) = length xs) = (remdups xs = xs)"
  2757 apply(induct xs)
  2758  apply auto
  2759 apply(subgoal_tac "length (remdups xs) <= length xs")
  2760  apply arith
  2761 apply(rule length_remdups_leq)
  2762 done
  2763 
  2764 lemma remdups_filter: "remdups(filter P xs) = filter P (remdups xs)"
  2765 apply(induct xs)
  2766 apply auto
  2767 done
  2768 
  2769 lemma distinct_map:
  2770   "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
  2771 by (induct xs) auto
  2772 
  2773 
  2774 lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
  2775 by (induct xs) auto
  2776 
  2777 lemma distinct_upt[simp]: "distinct[i..<j]"
  2778 by (induct j) auto
  2779 
  2780 lemma distinct_upto[simp]: "distinct[i..j]"
  2781 apply(induct i j rule:upto.induct)
  2782 apply(subst upto.simps)
  2783 apply(simp)
  2784 done
  2785 
  2786 lemma distinct_take[simp]: "distinct xs \<Longrightarrow> distinct (take i xs)"
  2787 apply(induct xs arbitrary: i)
  2788  apply simp
  2789 apply (case_tac i)
  2790  apply simp_all
  2791 apply(blast dest:in_set_takeD)
  2792 done
  2793 
  2794 lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)"
  2795 apply(induct xs arbitrary: i)
  2796  apply simp
  2797 apply (case_tac i)
  2798  apply simp_all
  2799 done
  2800 
  2801 lemma distinct_list_update:
  2802 assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
  2803 shows "distinct (xs[i:=a])"
  2804 proof (cases "i < length xs")
  2805   case True
  2806   with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
  2807     apply (drule_tac id_take_nth_drop) by simp
  2808   with d True show ?thesis
  2809     apply (simp add: upd_conv_take_nth_drop)
  2810     apply (drule subst [OF id_take_nth_drop]) apply assumption
  2811     apply simp apply (cases "a = xs!i") apply simp by blast
  2812 next
  2813   case False with d show ?thesis by auto
  2814 qed
  2815 
  2816 lemma distinct_concat:
  2817   assumes "distinct xs"
  2818   and "\<And> ys. ys \<in> set xs \<Longrightarrow> distinct ys"
  2819   and "\<And> ys zs. \<lbrakk> ys \<in> set xs ; zs \<in> set xs ; ys \<noteq> zs \<rbrakk> \<Longrightarrow> set ys \<inter> set zs = {}"
  2820   shows "distinct (concat xs)"
  2821   using assms by (induct xs) auto
  2822 
  2823 text {* It is best to avoid this indexed version of distinct, but
  2824 sometimes it is useful. *}
  2825 
  2826 lemma distinct_conv_nth:
  2827 "distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
  2828 apply (induct xs, simp, simp)
  2829 apply (rule iffI, clarsimp)
  2830  apply (case_tac i)
  2831 apply (case_tac j, simp)
  2832 apply (simp add: set_conv_nth)
  2833  apply (case_tac j)
  2834 apply (clarsimp simp add: set_conv_nth, simp) 
  2835 apply (rule conjI)
  2836 (*TOO SLOW
  2837 apply (metis Zero_neq_Suc gr0_conv_Suc in_set_conv_nth lessI less_trans_Suc nth_Cons' nth_Cons_Suc)
  2838 *)
  2839  apply (clarsimp simp add: set_conv_nth)
  2840  apply (erule_tac x = 0 in allE, simp)
  2841  apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
  2842 (*TOO SLOW
  2843 apply (metis Suc_Suc_eq lessI less_trans_Suc nth_Cons_Suc)
  2844 *)
  2845 apply (erule_tac x = "Suc i" in allE, simp)
  2846 apply (erule_tac x = "Suc j" in allE, simp)
  2847 done
  2848 
  2849 lemma nth_eq_iff_index_eq:
  2850  "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
  2851 by(auto simp: distinct_conv_nth)
  2852 
  2853 lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
  2854 by (induct xs) auto
  2855 
  2856 lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
  2857 proof (induct xs)
  2858   case Nil thus ?case by simp
  2859 next
  2860   case (Cons x xs)
  2861   show ?case
  2862   proof (cases "x \<in> set xs")
  2863     case False with Cons show ?thesis by simp
  2864   next
  2865     case True with Cons.prems
  2866     have "card (set xs) = Suc (length xs)" 
  2867       by (simp add: card_insert_if split: split_if_asm)
  2868     moreover have "card (set xs) \<le> length xs" by (rule card_length)
  2869     ultimately have False by simp
  2870     thus ?thesis ..
  2871   qed
  2872 qed
  2873 
  2874 lemma distinct_length_filter: "distinct xs \<Longrightarrow> length (filter P xs) = card ({x. P x} Int set xs)"
  2875 by (induct xs) (auto)
  2876 
  2877 lemma not_distinct_decomp: "~ distinct ws ==> EX xs ys zs y. ws = xs@[y]@ys@[y]@zs"
  2878 apply (induct n == "length ws" arbitrary:ws) apply simp
  2879 apply(case_tac ws) apply simp
  2880 apply (simp split:split_if_asm)
  2881 apply (metis Cons_eq_appendI eq_Nil_appendI split_list)
  2882 done
  2883 
  2884 lemma length_remdups_concat:
  2885   "length (remdups (concat xss)) = card (\<Union>xs\<in>set xss. set xs)"
  2886   by (simp add: distinct_card [symmetric])
  2887 
  2888 lemma length_remdups_card_conv: "length(remdups xs) = card(set xs)"
  2889 proof -
  2890   have xs: "concat[xs] = xs" by simp
  2891   from length_remdups_concat[of "[xs]"] show ?thesis unfolding xs by simp
  2892 qed
  2893 
  2894 lemma remdups_remdups:
  2895   "remdups (remdups xs) = remdups xs"
  2896   by (induct xs) simp_all
  2897 
  2898 lemma distinct_butlast:
  2899   assumes "xs \<noteq> []" and "distinct xs"
  2900   shows "distinct (butlast xs)"
  2901 proof -
  2902   from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  2903   with `distinct xs` show ?thesis by simp
  2904 qed
  2905 
  2906 lemma remdups_map_remdups:
  2907   "remdups (map f (remdups xs)) = remdups (map f xs)"
  2908   by (induct xs) simp_all
  2909 
  2910 lemma distinct_zipI1:
  2911   assumes "distinct xs"
  2912   shows "distinct (zip xs ys)"
  2913 proof (rule zip_obtain_same_length)
  2914   fix xs' :: "'a list" and ys' :: "'b list" and n
  2915   assume "length xs' = length ys'"
  2916   assume "xs' = take n xs"
  2917   with assms have "distinct xs'" by simp
  2918   with `length xs' = length ys'` show "distinct (zip xs' ys')"
  2919     by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
  2920 qed
  2921 
  2922 lemma distinct_zipI2:
  2923   assumes "distinct ys"
  2924   shows "distinct (zip xs ys)"
  2925 proof (rule zip_obtain_same_length)
  2926   fix xs' :: "'b list" and ys' :: "'a list" and n
  2927   assume "length xs' = length ys'"
  2928   assume "ys' = take n ys"
  2929   with assms have "distinct ys'" by simp
  2930   with `length xs' = length ys'` show "distinct (zip xs' ys')"
  2931     by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
  2932 qed
  2933 
  2934 (* The next two lemmas help Sledgehammer. *)
  2935 
  2936 lemma distinct_singleton: "distinct [x]" by simp
  2937 
  2938 lemma distinct_length_2_or_more:
  2939 "distinct (a # b # xs) \<longleftrightarrow> (a \<noteq> b \<and> distinct (a # xs) \<and> distinct (b # xs))"
  2940 by (metis distinct.simps(2) hd.simps hd_in_set list.simps(2) set_ConsD set_rev_mp set_subset_Cons)
  2941 
  2942 
  2943 subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
  2944 
  2945 lemma (in monoid_add) listsum_foldl [code]:
  2946   "listsum = foldl (op +) 0"
  2947   by (simp add: listsum_def foldl_foldr1 fun_eq_iff)
  2948 
  2949 lemma (in monoid_add) listsum_simps [simp]:
  2950   "listsum [] = 0"
  2951   "listsum (x#xs) = x + listsum xs"
  2952   by (simp_all add: listsum_def)
  2953 
  2954 lemma (in monoid_add) listsum_append [simp]:
  2955   "listsum (xs @ ys) = listsum xs + listsum ys"
  2956   by (induct xs) (simp_all add: add.assoc)
  2957 
  2958 lemma (in comm_monoid_add) listsum_rev [simp]:
  2959   "listsum (rev xs) = listsum xs"
  2960   by (simp add: listsum_def [of "rev xs"]) (simp add: listsum_foldl foldr_foldl add.commute)
  2961 
  2962 lemma (in comm_monoid_add) listsum_map_remove1:
  2963   "x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))"
  2964   by (induct xs) (auto simp add: ac_simps)
  2965 
  2966 lemma (in monoid_add) list_size_conv_listsum:
  2967   "list_size f xs = listsum (map f xs) + size xs"
  2968   by (induct xs) auto
  2969 
  2970 lemma (in monoid_add) length_concat:
  2971   "length (concat xss) = listsum (map length xss)"
  2972   by (induct xss) simp_all
  2973 
  2974 lemma (in monoid_add) listsum_map_filter:
  2975   assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
  2976   shows "listsum (map f (filter P xs)) = listsum (map f xs)"
  2977   using assms by (induct xs) auto
  2978 
  2979 lemma (in monoid_add) distinct_listsum_conv_Setsum:
  2980   "distinct xs \<Longrightarrow> listsum xs = Setsum (set xs)"
  2981   by (induct xs) simp_all
  2982 
  2983 lemma listsum_eq_0_nat_iff_nat [simp]:
  2984   "listsum ns = (0::nat) \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
  2985   by (simp add: listsum_foldl)
  2986 
  2987 lemma elem_le_listsum_nat:
  2988   "k < size ns \<Longrightarrow> ns ! k \<le> listsum (ns::nat list)"
  2989 apply(induct ns arbitrary: k)
  2990  apply simp
  2991 apply(fastforce simp add:nth_Cons split: nat.split)
  2992 done
  2993 
  2994 lemma listsum_update_nat:
  2995   "k<size ns \<Longrightarrow> listsum (ns[k := (n::nat)]) = listsum ns + n - ns ! k"
  2996 apply(induct ns arbitrary:k)
  2997  apply (auto split:nat.split)
  2998 apply(drule elem_le_listsum_nat)
  2999 apply arith
  3000 done
  3001 
  3002 text{* Some syntactic sugar for summing a function over a list: *}
  3003 
  3004 syntax
  3005   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
  3006 syntax (xsymbols)
  3007   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
  3008 syntax (HTML output)
  3009   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
  3010 
  3011 translations -- {* Beware of argument permutation! *}
  3012   "SUM x<-xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  3013   "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  3014 
  3015 lemma (in monoid_add) listsum_triv:
  3016   "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
  3017   by (induct xs) (simp_all add: left_distrib)
  3018 
  3019 lemma (in monoid_add) listsum_0 [simp]:
  3020   "(\<Sum>x\<leftarrow>xs. 0) = 0"
  3021   by (induct xs) (simp_all add: left_distrib)
  3022 
  3023 text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
  3024 lemma (in ab_group_add) uminus_listsum_map:
  3025   "- listsum (map f xs) = listsum (map (uminus \<circ> f) xs)"
  3026   by (induct xs) simp_all
  3027 
  3028 lemma (in comm_monoid_add) listsum_addf:
  3029   "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
  3030   by (induct xs) (simp_all add: algebra_simps)
  3031 
  3032 lemma (in ab_group_add) listsum_subtractf:
  3033   "(\<Sum>x\<leftarrow>xs. f x - g x) = listsum (map f xs) - listsum (map g xs)"
  3034   by (induct xs) (simp_all add: algebra_simps)
  3035 
  3036 lemma (in semiring_0) listsum_const_mult:
  3037   "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
  3038   by (induct xs) (simp_all add: algebra_simps)
  3039 
  3040 lemma (in semiring_0) listsum_mult_const:
  3041   "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
  3042   by (induct xs) (simp_all add: algebra_simps)
  3043 
  3044 lemma (in ordered_ab_group_add_abs) listsum_abs:
  3045   "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
  3046   by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])
  3047 
  3048 lemma listsum_mono:
  3049   fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}"
  3050   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)"
  3051   by (induct xs) (simp, simp add: add_mono)
  3052 
  3053 lemma (in monoid_add) listsum_distinct_conv_setsum_set:
  3054   "distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)"
  3055   by (induct xs) simp_all
  3056 
  3057 lemma (in monoid_add) interv_listsum_conv_setsum_set_nat:
  3058   "listsum (map f [m..<n]) = setsum f (set [m..<n])"
  3059   by (simp add: listsum_distinct_conv_setsum_set)
  3060 
  3061 lemma (in monoid_add) interv_listsum_conv_setsum_set_int:
  3062   "listsum (map f [k..l]) = setsum f (set [k..l])"
  3063   by (simp add: listsum_distinct_conv_setsum_set)
  3064 
  3065 text {* General equivalence between @{const listsum} and @{const setsum} *}
  3066 lemma (in monoid_add) listsum_setsum_nth:
  3067   "listsum xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
  3068   using interv_listsum_conv_setsum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth)
  3069 
  3070 
  3071 subsubsection {* @{const insert} *}
  3072 
  3073 lemma in_set_insert [simp]:
  3074   "x \<in> set xs \<Longrightarrow> List.insert x xs = xs"
  3075   by (simp add: List.insert_def)
  3076 
  3077 lemma not_in_set_insert [simp]:
  3078   "x \<notin> set xs \<Longrightarrow> List.insert x xs = x # xs"
  3079   by (simp add: List.insert_def)
  3080 
  3081 lemma insert_Nil [simp]:
  3082   "List.insert x [] = [x]"
  3083   by simp
  3084 
  3085 lemma set_insert [simp]:
  3086   "set (List.insert x xs) = insert x (set xs)"
  3087   by (auto simp add: List.insert_def)
  3088 
  3089 lemma distinct_insert [simp]:
  3090   "distinct xs \<Longrightarrow> distinct (List.insert x xs)"
  3091   by (simp add: List.insert_def)
  3092 
  3093 lemma insert_remdups:
  3094   "List.insert x (remdups xs) = remdups (List.insert x xs)"
  3095   by (simp add: List.insert_def)
  3096 
  3097 
  3098 subsubsection {* @{text remove1} *}
  3099 
  3100 lemma remove1_append:
  3101   "remove1 x (xs @ ys) =
  3102   (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
  3103 by (induct xs) auto
  3104 
  3105 lemma remove1_commute: "remove1 x (remove1 y zs) = remove1 y (remove1 x zs)"
  3106 by (induct zs) auto
  3107 
  3108 lemma in_set_remove1[simp]:
  3109   "a \<noteq> b \<Longrightarrow> a : set(remove1 b xs) = (a : set xs)"
  3110 apply (induct xs)
  3111 apply auto
  3112 done
  3113 
  3114 lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
  3115 apply(induct xs)
  3116  apply simp
  3117 apply simp
  3118 apply blast
  3119 done
  3120 
  3121 lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
  3122 apply(induct xs)
  3123  apply simp
  3124 apply simp
  3125 apply blast
  3126 done
  3127 
  3128 lemma length_remove1:
  3129   "length(remove1 x xs) = (if x : set xs then length xs - 1 else length xs)"
  3130 apply (induct xs)
  3131  apply (auto dest!:length_pos_if_in_set)
  3132 done
  3133 
  3134 lemma remove1_filter_not[simp]:
  3135   "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
  3136 by(induct xs) auto
  3137 
  3138 lemma filter_remove1:
  3139   "filter Q (remove1 x xs) = remove1 x (filter Q xs)"
  3140 by (induct xs) auto
  3141 
  3142 lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
  3143 apply(insert set_remove1_subset)
  3144 apply fast
  3145 done
  3146 
  3147 lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
  3148 by (induct xs) simp_all
  3149 
  3150 lemma remove1_remdups:
  3151   "distinct xs \<Longrightarrow> remove1 x (remdups xs) = remdups (remove1 x xs)"
  3152   by (induct xs) simp_all
  3153 
  3154 lemma remove1_idem:
  3155   assumes "x \<notin> set xs"
  3156   shows "remove1 x xs = xs"
  3157   using assms by (induct xs) simp_all
  3158 
  3159 
  3160 subsubsection {* @{text removeAll} *}
  3161 
  3162 lemma removeAll_filter_not_eq:
  3163   "removeAll x = filter (\<lambda>y. x \<noteq> y)"
  3164 proof
  3165   fix xs
  3166   show "removeAll x xs = filter (\<lambda>y. x \<noteq> y) xs"
  3167     by (induct xs) auto
  3168 qed
  3169 
  3170 lemma removeAll_append[simp]:
  3171   "removeAll x (xs @ ys) = removeAll x xs @ removeAll x ys"
  3172 by (induct xs) auto
  3173 
  3174 lemma set_removeAll[simp]: "set(removeAll x xs) = set xs - {x}"
  3175 by (induct xs) auto
  3176 
  3177 lemma removeAll_id[simp]: "x \<notin> set xs \<Longrightarrow> removeAll x xs = xs"
  3178 by (induct xs) auto
  3179 
  3180 (* Needs count:: 'a \<Rightarrow> a' list \<Rightarrow> nat
  3181 lemma length_removeAll:
  3182   "length(removeAll x xs) = length xs - count x xs"
  3183 *)
  3184 
  3185 lemma removeAll_filter_not[simp]:
  3186   "\<not> P x \<Longrightarrow> removeAll x (filter P xs) = filter P xs"
  3187 by(induct xs) auto
  3188 
  3189 lemma distinct_removeAll:
  3190   "distinct xs \<Longrightarrow> distinct (removeAll x xs)"
  3191   by (simp add: removeAll_filter_not_eq)
  3192 
  3193 lemma distinct_remove1_removeAll:
  3194   "distinct xs ==> remove1 x xs = removeAll x xs"
  3195 by (induct xs) simp_all
  3196 
  3197 lemma map_removeAll_inj_on: "inj_on f (insert x (set xs)) \<Longrightarrow>
  3198   map f (removeAll x xs) = removeAll (f x) (map f xs)"
  3199 by (induct xs) (simp_all add:inj_on_def)
  3200 
  3201 lemma map_removeAll_inj: "inj f \<Longrightarrow>
  3202   map f (removeAll x xs) = removeAll (f x) (map f xs)"
  3203 by(metis map_removeAll_inj_on subset_inj_on subset_UNIV)
  3204 
  3205 
  3206 subsubsection {* @{text replicate} *}
  3207 
  3208 lemma length_replicate [simp]: "length (replicate n x) = n"
  3209 by (induct n) auto
  3210 
  3211 lemma Ex_list_of_length: "\<exists>xs. length xs = n"
  3212 by (rule exI[of _ "replicate n undefined"]) simp
  3213 
  3214 lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
  3215 by (induct n) auto
  3216 
  3217 lemma map_replicate_const:
  3218   "map (\<lambda> x. k) lst = replicate (length lst) k"
  3219   by (induct lst) auto
  3220 
  3221 lemma replicate_app_Cons_same:
  3222 "(replicate n x) @ (x # xs) = x # replicate n x @ xs"
  3223 by (induct n) auto
  3224 
  3225 lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
  3226 apply (induct n, simp)
  3227 apply (simp add: replicate_app_Cons_same)
  3228 done
  3229 
  3230 lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
  3231 by (induct n) auto
  3232 
  3233 text{* Courtesy of Matthias Daum: *}
  3234 lemma append_replicate_commute:
  3235   "replicate n x @ replicate k x = replicate k x @ replicate n x"
  3236 apply (simp add: replicate_add [THEN sym])
  3237 apply (simp add: add_commute)
  3238 done
  3239 
  3240 text{* Courtesy of Andreas Lochbihler: *}
  3241 lemma filter_replicate:
  3242   "filter P (replicate n x) = (if P x then replicate n x else [])"
  3243 by(induct n) auto
  3244 
  3245 lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
  3246 by (induct n) auto
  3247 
  3248 lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
  3249 by (induct n) auto
  3250 
  3251 lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
  3252 by (atomize (full), induct n) auto
  3253 
  3254 lemma nth_replicate[simp]: "i < n ==> (replicate n x)!i = x"
  3255 apply (induct n arbitrary: i, simp)
  3256 apply (simp add: nth_Cons split: nat.split)
  3257 done
  3258 
  3259 text{* Courtesy of Matthias Daum (2 lemmas): *}
  3260 lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
  3261 apply (case_tac "k \<le> i")
  3262  apply  (simp add: min_def)
  3263 apply (drule not_leE)
  3264 apply (simp add: min_def)
  3265 apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
  3266  apply  simp
  3267 apply (simp add: replicate_add [symmetric])
  3268 done
  3269 
  3270 lemma drop_replicate[simp]: "drop i (replicate k x) = replicate (k-i) x"
  3271 apply (induct k arbitrary: i)
  3272  apply simp
  3273 apply clarsimp
  3274 apply (case_tac i)
  3275  apply simp
  3276 apply clarsimp
  3277 done
  3278 
  3279 
  3280 lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
  3281 by (induct n) auto
  3282 
  3283 lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
  3284 by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
  3285 
  3286 lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
  3287 by auto
  3288 
  3289 lemma in_set_replicate[simp]: "(x : set (replicate n y)) = (x = y & n \<noteq> 0)"
  3290 by (simp add: set_replicate_conv_if)
  3291 
  3292 lemma Ball_set_replicate[simp]:
  3293   "(ALL x : set(replicate n a). P x) = (P a | n=0)"
  3294 by(simp add: set_replicate_conv_if)
  3295 
  3296 lemma Bex_set_replicate[simp]:
  3297   "(EX x : set(replicate n a). P x) = (P a & n\<noteq>0)"
  3298 by(simp add: set_replicate_conv_if)
  3299 
  3300 lemma replicate_append_same:
  3301   "replicate i x @ [x] = x # replicate i x"
  3302   by (induct i) simp_all
  3303 
  3304 lemma map_replicate_trivial:
  3305   "map (\<lambda>i. x) [0..<i] = replicate i x"
  3306   by (induct i) (simp_all add: replicate_append_same)
  3307 
  3308 lemma concat_replicate_trivial[simp]:
  3309   "concat (replicate i []) = []"
  3310   by (induct i) (auto simp add: map_replicate_const)
  3311 
  3312 lemma replicate_empty[simp]: "(replicate n x = []) \<longleftrightarrow> n=0"
  3313 by (induct n) auto
  3314 
  3315 lemma empty_replicate[simp]: "([] = replicate n x) \<longleftrightarrow> n=0"
  3316 by (induct n) auto
  3317 
  3318 lemma replicate_eq_replicate[simp]:
  3319   "(replicate m x = replicate n y) \<longleftrightarrow> (m=n & (m\<noteq>0 \<longrightarrow> x=y))"
  3320 apply(induct m arbitrary: n)
  3321  apply simp
  3322 apply(induct_tac n)
  3323 apply auto
  3324 done
  3325 
  3326 lemma replicate_length_filter:
  3327   "replicate (length (filter (\<lambda>y. x = y) xs)) x = filter (\<lambda>y. x = y) xs"
  3328   by (induct xs) auto
  3329 
  3330 lemma comm_append_are_replicate:
  3331   fixes xs ys :: "'a list"
  3332   assumes "xs \<noteq> []" "ys \<noteq> []"
  3333   assumes "xs @ ys = ys @ xs"
  3334   shows "\<exists>m n zs. concat (replicate m zs) = xs \<and> concat (replicate n zs) = ys"
  3335   using assms
  3336 proof (induct "length (xs @ ys)" arbitrary: xs ys rule: less_induct)
  3337   case less
  3338 
  3339   def xs' \<equiv> "if (length xs \<le> length ys) then xs else ys"
  3340     and ys' \<equiv> "if (length xs \<le> length ys) then ys else xs"
  3341   then have
  3342     prems': "length xs' \<le> length ys'"
  3343             "xs' @ ys' = ys' @ xs'"
  3344       and "xs' \<noteq> []"
  3345       and len: "length (xs @ ys) = length (xs' @ ys')"
  3346     using less by (auto intro: less.hyps)
  3347 
  3348   from prems'
  3349   obtain ws where "ys' = xs' @ ws"
  3350     by (auto simp: append_eq_append_conv2)
  3351 
  3352   have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ys'"
  3353   proof (cases "ws = []")
  3354     case True
  3355     then have "concat (replicate 1 xs') = xs'"
  3356       and "concat (replicate 1 xs') = ys'"
  3357       using `ys' = xs' @ ws` by auto
  3358     then show ?thesis by blast
  3359   next
  3360     case False
  3361     from `ys' = xs' @ ws` and `xs' @ ys' = ys' @ xs'`
  3362     have "xs' @ ws = ws @ xs'" by simp
  3363     then have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ws"
  3364       using False and `xs' \<noteq> []` and `ys' = xs' @ ws` and len
  3365       by (intro less.hyps) auto
  3366     then obtain m n zs where "concat (replicate m zs) = xs'"
  3367       and "concat (replicate n zs) = ws" by blast
  3368     moreover
  3369     then have "concat (replicate (m + n) zs) = ys'"
  3370       using `ys' = xs' @ ws`
  3371       by (simp add: replicate_add)
  3372     ultimately
  3373     show ?thesis by blast
  3374   qed
  3375   then show ?case
  3376     using xs'_def ys'_def by metis
  3377 qed
  3378 
  3379 lemma comm_append_is_replicate:
  3380   fixes xs ys :: "'a list"
  3381   assumes "xs \<noteq> []" "ys \<noteq> []"
  3382   assumes "xs @ ys = ys @ xs"
  3383   shows "\<exists>n zs. n > 1 \<and> concat (replicate n zs) = xs @ ys"
  3384 
  3385 proof -
  3386   obtain m n zs where "concat (replicate m zs) = xs"
  3387     and "concat (replicate n zs) = ys"
  3388     using assms by (metis comm_append_are_replicate)
  3389   then have "m + n > 1" and "concat (replicate (m+n) zs) = xs @ ys"
  3390     using `xs \<noteq> []` and `ys \<noteq> []`
  3391     by (auto simp: replicate_add)
  3392   then show ?thesis by blast
  3393 qed
  3394 
  3395 
  3396 subsubsection{*@{text rotate1} and @{text rotate}*}
  3397 
  3398 lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
  3399 by(simp add:rotate1_def)
  3400 
  3401 lemma rotate0[simp]: "rotate 0 = id"
  3402 by(simp add:rotate_def)
  3403 
  3404 lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
  3405 by(simp add:rotate_def)
  3406 
  3407 lemma rotate_add:
  3408   "rotate (m+n) = rotate m o rotate n"
  3409 by(simp add:rotate_def funpow_add)
  3410 
  3411 lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
  3412 by(simp add:rotate_add)
  3413 
  3414 lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
  3415 by(simp add:rotate_def funpow_swap1)
  3416 
  3417 lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
  3418 by(cases xs) simp_all
  3419 
  3420 lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
  3421 apply(induct n)
  3422  apply simp
  3423 apply (simp add:rotate_def)
  3424 done
  3425 
  3426 lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
  3427 by(simp add:rotate1_def split:list.split)
  3428 
  3429 lemma rotate_drop_take:
  3430   "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
  3431 apply(induct n)
  3432  apply simp
  3433 apply(simp add:rotate_def)
  3434 apply(cases "xs = []")
  3435  apply (simp)
  3436 apply(case_tac "n mod length xs = 0")
  3437  apply(simp add:mod_Suc)
  3438  apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
  3439 apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
  3440                 take_hd_drop linorder_not_le)
  3441 done
  3442 
  3443 lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
  3444 by(simp add:rotate_drop_take)
  3445 
  3446 lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
  3447 by(simp add:rotate_drop_take)
  3448 
  3449 lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
  3450 by(simp add:rotate1_def split:list.split)
  3451 
  3452 lemma length_rotate[simp]: "length(rotate n xs) = length xs"
  3453 by (induct n arbitrary: xs) (simp_all add:rotate_def)
  3454 
  3455 lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
  3456 by(simp add:rotate1_def split:list.split) blast
  3457 
  3458 lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
  3459 by (induct n) (simp_all add:rotate_def)
  3460 
  3461 lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
  3462 by(simp add:rotate_drop_take take_map drop_map)
  3463 
  3464 lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
  3465 by (cases xs) (auto simp add:rotate1_def)
  3466 
  3467 lemma set_rotate[simp]: "set(rotate n xs) = set xs"
  3468 by (induct n) (simp_all add:rotate_def)
  3469 
  3470 lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
  3471 by(simp add:rotate1_def split:list.split)
  3472 
  3473 lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
  3474 by (induct n) (simp_all add:rotate_def)
  3475 
  3476 lemma rotate_rev:
  3477   "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
  3478 apply(simp add:rotate_drop_take rev_drop rev_take)
  3479 apply(cases "length xs = 0")
  3480  apply simp
  3481 apply(cases "n mod length xs = 0")
  3482  apply simp
  3483 apply(simp add:rotate_drop_take rev_drop rev_take)
  3484 done
  3485 
  3486 lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
  3487 apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
  3488 apply(subgoal_tac "length xs \<noteq> 0")
  3489  prefer 2 apply simp
  3490 using mod_less_divisor[of "length xs" n] by arith
  3491 
  3492 
  3493 subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
  3494 
  3495 lemma sublist_empty [simp]: "sublist xs {} = []"
  3496 by (auto simp add: sublist_def)
  3497 
  3498 lemma sublist_nil [simp]: "sublist [] A = []"
  3499 by (auto simp add: sublist_def)
  3500 
  3501 lemma length_sublist:
  3502   "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
  3503 by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
  3504 
  3505 lemma sublist_shift_lemma_Suc:
  3506   "map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
  3507    map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
  3508 apply(induct xs arbitrary: "is")
  3509  apply simp
  3510 apply (case_tac "is")
  3511  apply simp
  3512 apply simp
  3513 done
  3514 
  3515 lemma sublist_shift_lemma:
  3516      "map fst [p<-zip xs [i..<i + length xs] . snd p : A] =
  3517       map fst [p<-zip xs [0..<length xs] . snd p + i : A]"
  3518 by (induct xs rule: rev_induct) (simp_all add: add_commute)
  3519 
  3520 lemma sublist_append:
  3521      "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
  3522 apply (unfold sublist_def)
  3523 apply (induct l' rule: rev_induct, simp)
  3524 apply (simp add: upt_add_eq_append[of 0] sublist_shift_lemma)
  3525 apply (simp add: add_commute)
  3526 done
  3527 
  3528 lemma sublist_Cons:
  3529 "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
  3530 apply (induct l rule: rev_induct)
  3531  apply (simp add: sublist_def)
  3532 apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
  3533 done
  3534 
  3535 lemma set_sublist: "set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
  3536 apply(induct xs arbitrary: I)
  3537 apply(auto simp: sublist_Cons nth_Cons split:nat.split dest!: gr0_implies_Suc)
  3538 done
  3539 
  3540 lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
  3541 by(auto simp add:set_sublist)
  3542 
  3543 lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
  3544 by(auto simp add:set_sublist)
  3545 
  3546 lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
  3547 by(auto simp add:set_sublist)
  3548 
  3549 lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
  3550 by (simp add: sublist_Cons)
  3551 
  3552 
  3553 lemma distinct_sublistI[simp]: "distinct xs \<Longrightarrow> distinct(sublist xs I)"
  3554 apply(induct xs arbitrary: I)
  3555  apply simp
  3556 apply(auto simp add:sublist_Cons)
  3557 done
  3558 
  3559 
  3560 lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
  3561 apply (induct l rule: rev_induct, simp)
  3562 apply (simp split: nat_diff_split add: sublist_append)
  3563 done
  3564 
  3565 lemma filter_in_sublist:
  3566  "distinct xs \<Longrightarrow> filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
  3567 proof (induct xs arbitrary: s)
  3568   case Nil thus ?case by simp
  3569 next
  3570   case (Cons a xs)
  3571   moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
  3572   ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
  3573 qed
  3574 
  3575 
  3576 subsubsection {* @{const splice} *}
  3577 
  3578 lemma splice_Nil2 [simp, code]: "splice xs [] = xs"
  3579 by (cases xs) simp_all
  3580 
  3581 declare splice.simps(1,3)[code]
  3582 declare splice.simps(2)[simp del]
  3583 
  3584 lemma length_splice[simp]: "length(splice xs ys) = length xs + length ys"
  3585 by (induct xs ys rule: splice.induct) auto
  3586 
  3587 
  3588 subsubsection {* Transpose *}
  3589 
  3590 function transpose where
  3591 "transpose []             = []" |
  3592 "transpose ([]     # xss) = transpose xss" |
  3593 "transpose ((x#xs) # xss) =
  3594   (x # [h. (h#t) \<leftarrow> xss]) # transpose (xs # [t. (h#t) \<leftarrow> xss])"
  3595 by pat_completeness auto
  3596 
  3597 lemma transpose_aux_filter_head:
  3598   "concat (map (list_case [] (\<lambda>h t. [h])) xss) =
  3599   map (\<lambda>xs. hd xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  3600   by (induct xss) (auto split: list.split)
  3601 
  3602 lemma transpose_aux_filter_tail:
  3603   "concat (map (list_case [] (\<lambda>h t. [t])) xss) =
  3604   map (\<lambda>xs. tl xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  3605   by (induct xss) (auto split: list.split)
  3606 
  3607 lemma transpose_aux_max:
  3608   "max (Suc (length xs)) (foldr (\<lambda>xs. max (length xs)) xss 0) =
  3609   Suc (max (length xs) (foldr (\<lambda>x. max (length x - Suc 0)) [ys\<leftarrow>xss . ys\<noteq>[]] 0))"
  3610   (is "max _ ?foldB = Suc (max _ ?foldA)")
  3611 proof (cases "[ys\<leftarrow>xss . ys\<noteq>[]] = []")
  3612   case True
  3613   hence "foldr (\<lambda>xs. max (length xs)) xss 0 = 0"
  3614   proof (induct xss)
  3615     case (Cons x xs)
  3616     moreover hence "x = []" by (cases x) auto
  3617     ultimately show ?case by auto
  3618   qed simp
  3619   thus ?thesis using True by simp
  3620 next
  3621   case False
  3622 
  3623   have foldA: "?foldA = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0 - 1"
  3624     by (induct xss) auto
  3625   have foldB: "?foldB = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0"
  3626     by (induct xss) auto
  3627 
  3628   have "0 < ?foldB"
  3629   proof -
  3630     from False
  3631     obtain z zs where zs: "[ys\<leftarrow>xss . ys \<noteq> []] = z#zs" by (auto simp: neq_Nil_conv)
  3632     hence "z \<in> set ([ys\<leftarrow>xss . ys \<noteq> []])" by auto
  3633     hence "z \<noteq> []" by auto
  3634     thus ?thesis
  3635       unfolding foldB zs
  3636       by (auto simp: max_def intro: less_le_trans)
  3637   qed
  3638   thus ?thesis
  3639     unfolding foldA foldB max_Suc_Suc[symmetric]
  3640     by simp
  3641 qed
  3642 
  3643 termination transpose
  3644   by (relation "measure (\<lambda>xs. foldr (\<lambda>xs. max (length xs)) xs 0 + length xs)")
  3645      (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max less_Suc_eq_le)
  3646 
  3647 lemma transpose_empty: "(transpose xs = []) \<longleftrightarrow> (\<forall>x \<in> set xs. x = [])"
  3648   by (induct rule: transpose.induct) simp_all
  3649 
  3650 lemma length_transpose:
  3651   fixes xs :: "'a list list"
  3652   shows "length (transpose xs) = foldr (\<lambda>xs. max (length xs)) xs 0"
  3653   by (induct rule: transpose.induct)
  3654     (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max
  3655                 max_Suc_Suc[symmetric] simp del: max_Suc_Suc)
  3656 
  3657 lemma nth_transpose:
  3658   fixes xs :: "'a list list"
  3659   assumes "i < length (transpose xs)"
  3660   shows "transpose xs ! i = map (\<lambda>xs. xs ! i) [ys \<leftarrow> xs. i < length ys]"
  3661 using assms proof (induct arbitrary: i rule: transpose.induct)
  3662   case (3 x xs xss)
  3663   def XS == "(x # xs) # xss"
  3664   hence [simp]: "XS \<noteq> []" by auto
  3665   thus ?case
  3666   proof (cases i)
  3667     case 0
  3668     thus ?thesis by (simp add: transpose_aux_filter_head hd_conv_nth)
  3669   next
  3670     case (Suc j)
  3671     have *: "\<And>xss. xs # map tl xss = map tl ((x#xs)#xss)" by simp
  3672     have **: "\<And>xss. (x#xs) # filter (\<lambda>ys. ys \<noteq> []) xss = filter (\<lambda>ys. ys \<noteq> []) ((x#xs)#xss)" by simp
  3673     { fix x have "Suc j < length x \<longleftrightarrow> x \<noteq> [] \<and> j < length x - Suc 0"
  3674       by (cases x) simp_all
  3675     } note *** = this
  3676 
  3677     have j_less: "j < length (transpose (xs # concat (map (list_case [] (\<lambda>h t. [t])) xss)))"
  3678       using "3.prems" by (simp add: transpose_aux_filter_tail length_transpose Suc)
  3679 
  3680     show ?thesis
  3681       unfolding transpose.simps `i = Suc j` nth_Cons_Suc "3.hyps"[OF j_less]
  3682       apply (auto simp: transpose_aux_filter_tail filter_map comp_def length_transpose * ** *** XS_def[symmetric])
  3683       apply (rule_tac y=x in list.exhaust)
  3684       by auto
  3685   qed
  3686 qed simp_all
  3687 
  3688 lemma transpose_map_map:
  3689   "transpose (map (map f) xs) = map (map f) (transpose xs)"
  3690 proof (rule nth_equalityI, safe)
  3691   have [simp]: "length (transpose (map (map f) xs)) = length (transpose xs)"
  3692     by (simp add: length_transpose foldr_map comp_def)
  3693   show "length (transpose (map (map f) xs)) = length (map (map f) (transpose xs))" by simp
  3694 
  3695   fix i assume "i < length (transpose (map (map f) xs))"
  3696   thus "transpose (map (map f) xs) ! i = map (map f) (transpose xs) ! i"
  3697     by (simp add: nth_transpose filter_map comp_def)
  3698 qed
  3699 
  3700 
  3701 subsubsection {* (In)finiteness *}
  3702 
  3703 lemma finite_maxlen:
  3704   "finite (M::'a list set) ==> EX n. ALL s:M. size s < n"
  3705 proof (induct rule: finite.induct)
  3706   case emptyI show ?case by simp
  3707 next
  3708   case (insertI M xs)
  3709   then obtain n where "\<forall>s\<in>M. length s < n" by blast
  3710   hence "ALL s:insert xs M. size s < max n (size xs) + 1" by auto
  3711   thus ?case ..
  3712 qed
  3713 
  3714 lemma lists_length_Suc_eq:
  3715   "{xs. set xs \<subseteq> A \<and> length xs = Suc n} =
  3716     (\<lambda>(xs, n). n#xs) ` ({xs. set xs \<subseteq> A \<and> length xs = n} \<times> A)"
  3717   by (auto simp: length_Suc_conv)
  3718 
  3719 lemma
  3720   assumes "finite A"
  3721   shows finite_lists_length_eq: "finite {xs. set xs \<subseteq> A \<and> length xs = n}"
  3722   and card_lists_length_eq: "card {xs. set xs \<subseteq> A \<and> length xs = n} = (card A)^n"
  3723   using `finite A`
  3724   by (induct n)
  3725      (auto simp: card_image inj_split_Cons lists_length_Suc_eq cong: conj_cong)
  3726 
  3727 lemma finite_lists_length_le:
  3728   assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs \<le> n}"
  3729  (is "finite ?S")
  3730 proof-
  3731   have "?S = (\<Union>n\<in>{0..n}. {xs. set xs \<subseteq> A \<and> length xs = n})" by auto
  3732   thus ?thesis by (auto intro: finite_lists_length_eq[OF `finite A`])
  3733 qed
  3734 
  3735 lemma card_lists_length_le:
  3736   assumes "finite A" shows "card {xs. set xs \<subseteq> A \<and> length xs \<le> n} = (\<Sum>i\<le>n. card A^i)"
  3737 proof -
  3738   have "(\<Sum>i\<le>n. card A^i) = card (\<Union>i\<le>n. {xs. set xs \<subseteq> A \<and> length xs = i})"
  3739     using `finite A`
  3740     by (subst card_UN_disjoint)
  3741        (auto simp add: card_lists_length_eq finite_lists_length_eq)
  3742   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}"
  3743     by auto
  3744   finally show ?thesis by simp
  3745 qed
  3746 
  3747 lemma infinite_UNIV_listI: "~ finite(UNIV::'a list set)"
  3748 apply(rule notI)
  3749 apply(drule finite_maxlen)
  3750 apply (metis UNIV_I length_replicate less_not_refl)
  3751 done
  3752 
  3753 
  3754 subsection {* Sorting *}
  3755 
  3756 text{* Currently it is not shown that @{const sort} returns a
  3757 permutation of its input because the nicest proof is via multisets,
  3758 which are not yet available. Alternatively one could define a function
  3759 that counts the number of occurrences of an element in a list and use
  3760 that instead of multisets to state the correctness property. *}
  3761 
  3762 context linorder
  3763 begin
  3764 
  3765 lemma length_insort [simp]:
  3766   "length (insort_key f x xs) = Suc (length xs)"
  3767   by (induct xs) simp_all
  3768 
  3769 lemma insort_key_left_comm:
  3770   assumes "f x \<noteq> f y"
  3771   shows "insort_key f y (insort_key f x xs) = insort_key f x (insort_key f y xs)"
  3772   by (induct xs) (auto simp add: assms dest: antisym)
  3773 
  3774 lemma insort_left_comm:
  3775   "insort x (insort y xs) = insort y (insort x xs)"
  3776   by (cases "x = y") (auto intro: insort_key_left_comm)
  3777 
  3778 lemma comp_fun_commute_insort:
  3779   "comp_fun_commute insort"
  3780 proof
  3781 qed (simp add: insort_left_comm fun_eq_iff)
  3782 
  3783 lemma sort_key_simps [simp]:
  3784   "sort_key f [] = []"
  3785   "sort_key f (x#xs) = insort_key f x (sort_key f xs)"
  3786   by (simp_all add: sort_key_def)
  3787 
  3788 lemma sort_foldl_insort:
  3789   "sort xs = foldl (\<lambda>ys x. insort x ys) [] xs"
  3790   by (simp add: sort_key_def foldr_foldl foldl_rev insort_left_comm)
  3791 
  3792 lemma length_sort[simp]: "length (sort_key f xs) = length xs"
  3793 by (induct xs, auto)
  3794 
  3795 lemma sorted_Cons: "sorted (x#xs) = (sorted xs & (ALL y:set xs. x <= y))"
  3796 apply(induct xs arbitrary: x) apply simp
  3797 by simp (blast intro: order_trans)
  3798 
  3799 lemma sorted_tl:
  3800   "sorted xs \<Longrightarrow> sorted (tl xs)"
  3801   by (cases xs) (simp_all add: sorted_Cons)
  3802 
  3803 lemma sorted_append:
  3804   "sorted (xs@ys) = (sorted xs & sorted ys & (\<forall>x \<in> set xs. \<forall>y \<in> set ys. x\<le>y))"
  3805 by (induct xs) (auto simp add:sorted_Cons)
  3806 
  3807 lemma sorted_nth_mono:
  3808   "sorted xs \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!i \<le> xs!j"
  3809 by (induct xs arbitrary: i j) (auto simp:nth_Cons' sorted_Cons)
  3810 
  3811 lemma sorted_rev_nth_mono:
  3812   "sorted (rev xs) \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!j \<le> xs!i"
  3813 using sorted_nth_mono[ of "rev xs" "length xs - j - 1" "length xs - i - 1"]
  3814       rev_nth[of "length xs - i - 1" "xs"] rev_nth[of "length xs - j - 1" "xs"]
  3815 by auto
  3816 
  3817 lemma sorted_nth_monoI:
  3818   "(\<And> i j. \<lbrakk> i \<le> j ; j < length xs \<rbrakk> \<Longrightarrow> xs ! i \<le> xs ! j) \<Longrightarrow> sorted xs"
  3819 proof (induct xs)
  3820   case (Cons x xs)
  3821   have "sorted xs"
  3822   proof (rule Cons.hyps)
  3823     fix i j assume "i \<le> j" and "j < length xs"
  3824     with Cons.prems[of "Suc i" "Suc j"]
  3825     show "xs ! i \<le> xs ! j" by auto
  3826   qed
  3827   moreover
  3828   {
  3829     fix y assume "y \<in> set xs"
  3830     then obtain j where "j < length xs" and "xs ! j = y"
  3831       unfolding in_set_conv_nth by blast
  3832     with Cons.prems[of 0 "Suc j"]
  3833     have "x \<le> y"
  3834       by auto
  3835   }
  3836   ultimately
  3837   show ?case
  3838     unfolding sorted_Cons by auto
  3839 qed simp
  3840 
  3841 lemma sorted_equals_nth_mono:
  3842   "sorted xs = (\<forall>j < length xs. \<forall>i \<le> j. xs ! i \<le> xs ! j)"
  3843 by (auto intro: sorted_nth_monoI sorted_nth_mono)
  3844 
  3845 lemma set_insort: "set(insort_key f x xs) = insert x (set xs)"
  3846 by (induct xs) auto
  3847 
  3848 lemma set_sort[simp]: "set(sort_key f xs) = set xs"
  3849 by (induct xs) (simp_all add:set_insort)
  3850 
  3851 lemma distinct_insort: "distinct (insort_key f x xs) = (x \<notin> set xs \<and> distinct xs)"
  3852 by(induct xs)(auto simp:set_insort)
  3853 
  3854 lemma distinct_sort[simp]: "distinct (sort_key f xs) = distinct xs"
  3855   by (induct xs) (simp_all add: distinct_insort)
  3856 
  3857 lemma sorted_insort_key: "sorted (map f (insort_key f x xs)) = sorted (map f xs)"
  3858   by (induct xs) (auto simp:sorted_Cons set_insort)
  3859 
  3860 lemma sorted_insort: "sorted (insort x xs) = sorted xs"
  3861   using sorted_insort_key [where f="\<lambda>x. x"] by simp
  3862 
  3863 theorem sorted_sort_key [simp]: "sorted (map f (sort_key f xs))"
  3864   by (induct xs) (auto simp:sorted_insort_key)
  3865 
  3866 theorem sorted_sort [simp]: "sorted (sort xs)"
  3867   using sorted_sort_key [where f="\<lambda>x. x"] by simp
  3868 
  3869 lemma sorted_butlast:
  3870   assumes "xs \<noteq> []" and "sorted xs"
  3871   shows "sorted (butlast xs)"
  3872 proof -
  3873   from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  3874   with `sorted xs` show ?thesis by (simp add: sorted_append)
  3875 qed
  3876   
  3877 lemma insort_not_Nil [simp]:
  3878   "insort_key f a xs \<noteq> []"
  3879   by (induct xs) simp_all
  3880 
  3881 lemma insort_is_Cons: "\<forall>x\<in>set xs. f a \<le> f x \<Longrightarrow> insort_key f a xs = a # xs"
  3882 by (cases xs) auto
  3883 
  3884 lemma sorted_sort_id: "sorted xs \<Longrightarrow> sort xs = xs"
  3885   by (induct xs) (auto simp add: sorted_Cons insort_is_Cons)
  3886 
  3887 lemma sorted_map_remove1:
  3888   "sorted (map f xs) \<Longrightarrow> sorted (map f (remove1 x xs))"
  3889   by (induct xs) (auto simp add: sorted_Cons)
  3890 
  3891 lemma sorted_remove1: "sorted xs \<Longrightarrow> sorted (remove1 a xs)"
  3892   using sorted_map_remove1 [of "\<lambda>x. x"] by simp
  3893 
  3894 lemma insort_key_remove1:
  3895   assumes "a \<in> set xs" and "sorted (map f xs)" and "hd (filter (\<lambda>x. f a = f x) xs) = a"
  3896   shows "insort_key f a (remove1 a xs) = xs"
  3897 using assms proof (induct xs)
  3898   case (Cons x xs)
  3899   then show ?case
  3900   proof (cases "x = a")
  3901     case False
  3902     then have "f x \<noteq> f a" using Cons.prems by auto
  3903     then have "f x < f a" using Cons.prems by (auto simp: sorted_Cons)
  3904     with `f x \<noteq> f a` show ?thesis using Cons by (auto simp: sorted_Cons insort_is_Cons)
  3905   qed (auto simp: sorted_Cons insort_is_Cons)
  3906 qed simp
  3907 
  3908 lemma insort_remove1:
  3909   assumes "a \<in> set xs" and "sorted xs"
  3910   shows "insort a (remove1 a xs) = xs"
  3911 proof (rule insort_key_remove1)
  3912   from `a \<in> set xs` show "a \<in> set xs" .
  3913   from `sorted xs` show "sorted (map (\<lambda>x. x) xs)" by simp
  3914   from `a \<in> set xs` have "a \<in> set (filter (op = a) xs)" by auto
  3915   then have "set (filter (op = a) xs) \<noteq> {}" by auto
  3916   then have "filter (op = a) xs \<noteq> []" by (auto simp only: set_empty)
  3917   then have "length (filter (op = a) xs) > 0" by simp
  3918   then obtain n where n: "Suc n = length (filter (op = a) xs)"
  3919     by (cases "length (filter (op = a) xs)") simp_all
  3920   moreover have "replicate (Suc n) a = a # replicate n a"
  3921     by simp
  3922   ultimately show "hd (filter (op = a) xs) = a" by (simp add: replicate_length_filter)
  3923 qed
  3924 
  3925 lemma sorted_remdups[simp]:
  3926   "sorted l \<Longrightarrow> sorted (remdups l)"
  3927 by (induct l) (auto simp: sorted_Cons)
  3928 
  3929 lemma sorted_distinct_set_unique:
  3930 assumes "sorted xs" "distinct xs" "sorted ys" "distinct ys" "set xs = set ys"
  3931 shows "xs = ys"
  3932 proof -
  3933   from assms have 1: "length xs = length ys" by (auto dest!: distinct_card)
  3934   from assms show ?thesis
  3935   proof(induct rule:list_induct2[OF 1])
  3936     case 1 show ?case by simp
  3937   next
  3938     case 2 thus ?case by (simp add:sorted_Cons)
  3939        (metis Diff_insert_absorb antisym insertE insert_iff)
  3940   qed
  3941 qed
  3942 
  3943 lemma map_sorted_distinct_set_unique:
  3944   assumes "inj_on f (set xs \<union> set ys)"
  3945   assumes "sorted (map f xs)" "distinct (map f xs)"
  3946     "sorted (map f ys)" "distinct (map f ys)"
  3947   assumes "set xs = set ys"
  3948   shows "xs = ys"
  3949 proof -
  3950   from assms have "map f xs = map f ys"
  3951     by (simp add: sorted_distinct_set_unique)
  3952   moreover with `inj_on f (set xs \<union> set ys)` show "xs = ys"
  3953     by (blast intro: map_inj_on)
  3954 qed
  3955 
  3956 lemma finite_sorted_distinct_unique:
  3957 shows "finite A \<Longrightarrow> EX! xs. set xs = A & sorted xs & distinct xs"
  3958 apply(drule finite_distinct_list)
  3959 apply clarify
  3960 apply(rule_tac a="sort xs" in ex1I)
  3961 apply (auto simp: sorted_distinct_set_unique)
  3962 done
  3963 
  3964 lemma
  3965   assumes "sorted xs"
  3966   shows sorted_take: "sorted (take n xs)"
  3967   and sorted_drop: "sorted (drop n xs)"
  3968 proof -
  3969   from assms have "sorted (take n xs @ drop n xs)" by simp
  3970   then show "sorted (take n xs)" and "sorted (drop n xs)"
  3971     unfolding sorted_append by simp_all
  3972 qed
  3973 
  3974 lemma sorted_dropWhile: "sorted xs \<Longrightarrow> sorted (dropWhile P xs)"
  3975   by (auto dest: sorted_drop simp add: dropWhile_eq_drop)
  3976 
  3977 lemma sorted_takeWhile: "sorted xs \<Longrightarrow> sorted (takeWhile P xs)"
  3978   by (subst takeWhile_eq_take) (auto dest: sorted_take)
  3979 
  3980 lemma sorted_filter:
  3981   "sorted (map f xs) \<Longrightarrow> sorted (map f (filter P xs))"
  3982   by (induct xs) (simp_all add: sorted_Cons)
  3983 
  3984 lemma foldr_max_sorted:
  3985   assumes "sorted (rev xs)"
  3986   shows "foldr max xs y = (if xs = [] then y else max (xs ! 0) y)"
  3987 using assms proof (induct xs)
  3988   case (Cons x xs)
  3989   moreover hence "sorted (rev xs)" using sorted_append by auto
  3990   ultimately show ?case
  3991     by (cases xs, auto simp add: sorted_append max_def)
  3992 qed simp
  3993 
  3994 lemma filter_equals_takeWhile_sorted_rev:
  3995   assumes sorted: "sorted (rev (map f xs))"
  3996   shows "[x \<leftarrow> xs. t < f x] = takeWhile (\<lambda> x. t < f x) xs"
  3997     (is "filter ?P xs = ?tW")
  3998 proof (rule takeWhile_eq_filter[symmetric])
  3999   let "?dW" = "dropWhile ?P xs"
  4000   fix x assume "x \<in> set ?dW"
  4001   then obtain i where i: "i < length ?dW" and nth_i: "x = ?dW ! i"
  4002     unfolding in_set_conv_nth by auto
  4003   hence "length ?tW + i < length (?tW @ ?dW)"
  4004     unfolding length_append by simp
  4005   hence i': "length (map f ?tW) + i < length (map f xs)" by simp
  4006   have "(map f ?tW @ map f ?dW) ! (length (map f ?tW) + i) \<le>
  4007         (map f ?tW @ map f ?dW) ! (length (map f ?tW) + 0)"
  4008     using sorted_rev_nth_mono[OF sorted _ i', of "length ?tW"]
  4009     unfolding map_append[symmetric] by simp
  4010   hence "f x \<le> f (?dW ! 0)"
  4011     unfolding nth_append_length_plus nth_i
  4012     using i preorder_class.le_less_trans[OF le0 i] by simp
  4013   also have "... \<le> t"
  4014     using hd_dropWhile[of "?P" xs] le0[THEN preorder_class.le_less_trans, OF i]
  4015     using hd_conv_nth[of "?dW"] by simp
  4016   finally show "\<not> t < f x" by simp
  4017 qed
  4018 
  4019 lemma insort_insert_key_triv:
  4020   "f x \<in> f ` set xs \<Longrightarrow> insort_insert_key f x xs = xs"
  4021   by (simp add: insort_insert_key_def)
  4022 
  4023 lemma insort_insert_triv:
  4024   "x \<in> set xs \<Longrightarrow> insort_insert x xs = xs"
  4025   using insort_insert_key_triv [of "\<lambda>x. x"] by simp
  4026 
  4027 lemma insort_insert_insort_key:
  4028   "f x \<notin> f ` set xs \<Longrightarrow> insort_insert_key f x xs = insort_key f x xs"
  4029   by (simp add: insort_insert_key_def)
  4030 
  4031 lemma insort_insert_insort:
  4032   "x \<notin> set xs \<Longrightarrow> insort_insert x xs = insort x xs"
  4033   using insort_insert_insort_key [of "\<lambda>x. x"] by simp
  4034 
  4035 lemma set_insort_insert:
  4036   "set (insort_insert x xs) = insert x (set xs)"
  4037   by (auto simp add: insort_insert_key_def set_insort)
  4038 
  4039 lemma distinct_insort_insert:
  4040   assumes "distinct xs"
  4041   shows "distinct (insort_insert_key f x xs)"
  4042   using assms by (induct xs) (auto simp add: insort_insert_key_def set_insort)
  4043 
  4044 lemma sorted_insort_insert_key:
  4045   assumes "sorted (map f xs)"
  4046   shows "sorted (map f (insort_insert_key f x xs))"
  4047   using assms by (simp add: insort_insert_key_def sorted_insort_key)
  4048 
  4049 lemma sorted_insort_insert:
  4050   assumes "sorted xs"
  4051   shows "sorted (insort_insert x xs)"
  4052   using assms sorted_insort_insert_key [of "\<lambda>x. x"] by simp
  4053 
  4054 lemma filter_insort_triv:
  4055   "\<not> P x \<Longrightarrow> filter P (insort_key f x xs) = filter P xs"
  4056   by (induct xs) simp_all
  4057 
  4058 lemma filter_insort:
  4059   "sorted (map f xs) \<Longrightarrow> P x \<Longrightarrow> filter P (insort_key f x xs) = insort_key f x (filter P xs)"
  4060   using assms by (induct xs)
  4061     (auto simp add: sorted_Cons, subst insort_is_Cons, auto)
  4062 
  4063 lemma filter_sort:
  4064   "filter P (sort_key f xs) = sort_key f (filter P xs)"
  4065   by (induct xs) (simp_all add: filter_insort_triv filter_insort)
  4066 
  4067 lemma sorted_map_same:
  4068   "sorted (map f [x\<leftarrow>xs. f x = g xs])"
  4069 proof (induct xs arbitrary: g)
  4070   case Nil then show ?case by simp
  4071 next
  4072   case (Cons x xs)
  4073   then have "sorted (map f [y\<leftarrow>xs . f y = (\<lambda>xs. f x) xs])" .
  4074   moreover from Cons have "sorted (map f [y\<leftarrow>xs . f y = (g \<circ> Cons x) xs])" .
  4075   ultimately show ?case by (simp_all add: sorted_Cons)
  4076 qed
  4077 
  4078 lemma sorted_same:
  4079   "sorted [x\<leftarrow>xs. x = g xs]"
  4080   using sorted_map_same [of "\<lambda>x. x"] by simp
  4081 
  4082 lemma remove1_insort [simp]:
  4083   "remove1 x (insort x xs) = xs"
  4084   by (induct xs) simp_all
  4085 
  4086 end
  4087 
  4088 lemma sorted_upt[simp]: "sorted[i..<j]"
  4089 by (induct j) (simp_all add:sorted_append)
  4090 
  4091 lemma sorted_upto[simp]: "sorted[i..j]"
  4092 apply(induct i j rule:upto.induct)
  4093 apply(subst upto.simps)
  4094 apply(simp add:sorted_Cons)
  4095 done
  4096 
  4097 
  4098 subsubsection {* @{const transpose} on sorted lists *}
  4099 
  4100 lemma sorted_transpose[simp]:
  4101   shows "sorted (rev (map length (transpose xs)))"
  4102   by (auto simp: sorted_equals_nth_mono rev_nth nth_transpose
  4103     length_filter_conv_card intro: card_mono)
  4104 
  4105 lemma transpose_max_length:
  4106   "foldr (\<lambda>xs. max (length xs)) (transpose xs) 0 = length [x \<leftarrow> xs. x \<noteq> []]"
  4107   (is "?L = ?R")
  4108 proof (cases "transpose xs = []")
  4109   case False
  4110   have "?L = foldr max (map length (transpose xs)) 0"
  4111     by (simp add: foldr_map comp_def)
  4112   also have "... = length (transpose xs ! 0)"
  4113     using False sorted_transpose by (simp add: foldr_max_sorted)
  4114   finally show ?thesis
  4115     using False by (simp add: nth_transpose)
  4116 next
  4117   case True
  4118   hence "[x \<leftarrow> xs. x \<noteq> []] = []"
  4119     by (auto intro!: filter_False simp: transpose_empty)
  4120   thus ?thesis by (simp add: transpose_empty True)
  4121 qed
  4122 
  4123 lemma length_transpose_sorted:
  4124   fixes xs :: "'a list list"
  4125   assumes sorted: "sorted (rev (map length xs))"
  4126   shows "length (transpose xs) = (if xs = [] then 0 else length (xs ! 0))"
  4127 proof (cases "xs = []")
  4128   case False
  4129   thus ?thesis
  4130     using foldr_max_sorted[OF sorted] False
  4131     unfolding length_transpose foldr_map comp_def
  4132     by simp
  4133 qed simp
  4134 
  4135 lemma nth_nth_transpose_sorted[simp]:
  4136   fixes xs :: "'a list list"
  4137   assumes sorted: "sorted (rev (map length xs))"
  4138   and i: "i < length (transpose xs)"
  4139   and j: "j < length [ys \<leftarrow> xs. i < length ys]"
  4140   shows "transpose xs ! i ! j = xs ! j  ! i"
  4141   using j filter_equals_takeWhile_sorted_rev[OF sorted, of i]
  4142     nth_transpose[OF i] nth_map[OF j]
  4143   by (simp add: takeWhile_nth)
  4144 
  4145 lemma transpose_column_length:
  4146   fixes xs :: "'a list list"
  4147   assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  4148   shows "length (filter (\<lambda>ys. i < length ys) (transpose xs)) = length (xs ! i)"
  4149 proof -
  4150   have "xs \<noteq> []" using `i < length xs` by auto
  4151   note filter_equals_takeWhile_sorted_rev[OF sorted, simp]
  4152   { fix j assume "j \<le> i"
  4153     note sorted_rev_nth_mono[OF sorted, of j i, simplified, OF this `i < length xs`]
  4154   } note sortedE = this[consumes 1]
  4155 
  4156   have "{j. j < length (transpose xs) \<and> i < length (transpose xs ! j)}
  4157     = {..< length (xs ! i)}"
  4158   proof safe
  4159     fix j
  4160     assume "j < length (transpose xs)" and "i < length (transpose xs ! j)"
  4161     with this(2) nth_transpose[OF this(1)]
  4162     have "i < length (takeWhile (\<lambda>ys. j < length ys) xs)" by simp
  4163     from nth_mem[OF this] takeWhile_nth[OF this]
  4164     show "j < length (xs ! i)" by (auto dest: set_takeWhileD)
  4165   next
  4166     fix j assume "j < length (xs ! i)"
  4167     thus "j < length (transpose xs)"
  4168       using foldr_max_sorted[OF sorted] `xs \<noteq> []` sortedE[OF le0]
  4169       by (auto simp: length_transpose comp_def foldr_map)
  4170 
  4171     have "Suc i \<le> length (takeWhile (\<lambda>ys. j < length ys) xs)"
  4172       using `i < length xs` `j < length (xs ! i)` less_Suc_eq_le
  4173       by (auto intro!: length_takeWhile_less_P_nth dest!: sortedE)
  4174     with nth_transpose[OF `j < length (transpose xs)`]
  4175     show "i < length (transpose xs ! j)" by simp
  4176   qed
  4177   thus ?thesis by (simp add: length_filter_conv_card)
  4178 qed
  4179 
  4180 lemma transpose_column:
  4181   fixes xs :: "'a list list"
  4182   assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  4183   shows "map (\<lambda>ys. ys ! i) (filter (\<lambda>ys. i < length ys) (transpose xs))
  4184     = xs ! i" (is "?R = _")
  4185 proof (rule nth_equalityI, safe)
  4186   show length: "length ?R = length (xs ! i)"
  4187     using transpose_column_length[OF assms] by simp
  4188 
  4189   fix j assume j: "j < length ?R"
  4190   note * = less_le_trans[OF this, unfolded length_map, OF length_filter_le]
  4191   from j have j_less: "j < length (xs ! i)" using length by simp
  4192   have i_less_tW: "Suc i \<le> length (takeWhile (\<lambda>ys. Suc j \<le> length ys) xs)"
  4193   proof (rule length_takeWhile_less_P_nth)
  4194     show "Suc i \<le> length xs" using `i < length xs` by simp
  4195     fix k assume "k < Suc i"
  4196     hence "k \<le> i" by auto
  4197     with sorted_rev_nth_mono[OF sorted this] `i < length xs`
  4198     have "length (xs ! i) \<le> length (xs ! k)" by simp
  4199     thus "Suc j \<le> length (xs ! k)" using j_less by simp
  4200   qed
  4201   have i_less_filter: "i < length [ys\<leftarrow>xs . j < length ys]"
  4202     unfolding filter_equals_takeWhile_sorted_rev[OF sorted, of j]
  4203     using i_less_tW by (simp_all add: Suc_le_eq)
  4204   from j show "?R ! j = xs ! i ! j"
  4205     unfolding filter_equals_takeWhile_sorted_rev[OF sorted_transpose, of i]
  4206     by (simp add: takeWhile_nth nth_nth_transpose_sorted[OF sorted * i_less_filter])
  4207 qed
  4208 
  4209 lemma transpose_transpose:
  4210   fixes xs :: "'a list list"
  4211   assumes sorted: "sorted (rev (map length xs))"
  4212   shows "transpose (transpose xs) = takeWhile (\<lambda>x. x \<noteq> []) xs" (is "?L = ?R")
  4213 proof -
  4214   have len: "length ?L = length ?R"
  4215     unfolding length_transpose transpose_max_length
  4216     using filter_equals_takeWhile_sorted_rev[OF sorted, of 0]
  4217     by simp
  4218 
  4219   { fix i assume "i < length ?R"
  4220     with less_le_trans[OF _ length_takeWhile_le[of _ xs]]
  4221     have "i < length xs" by simp
  4222   } note * = this
  4223   show ?thesis
  4224     by (rule nth_equalityI)
  4225        (simp_all add: len nth_transpose transpose_column[OF sorted] * takeWhile_nth)
  4226 qed
  4227 
  4228 theorem transpose_rectangle:
  4229   assumes "xs = [] \<Longrightarrow> n = 0"
  4230   assumes rect: "\<And> i. i < length xs \<Longrightarrow> length (xs ! i) = n"
  4231   shows "transpose xs = map (\<lambda> i. map (\<lambda> j. xs ! j ! i) [0..<length xs]) [0..<n]"
  4232     (is "?trans = ?map")
  4233 proof (rule nth_equalityI)
  4234   have "sorted (rev (map length xs))"
  4235     by (auto simp: rev_nth rect intro!: sorted_nth_monoI)
  4236   from foldr_max_sorted[OF this] assms
  4237   show len: "length ?trans = length ?map"
  4238     by (simp_all add: length_transpose foldr_map comp_def)
  4239   moreover
  4240   { fix i assume "i < n" hence "[ys\<leftarrow>xs . i < length ys] = xs"
  4241       using rect by (auto simp: in_set_conv_nth intro!: filter_True) }
  4242   ultimately show "\<forall>i < length ?trans. ?trans ! i = ?map ! i"
  4243     by (auto simp: nth_transpose intro: nth_equalityI)
  4244 qed
  4245 
  4246 
  4247 subsubsection {* @{text sorted_list_of_set} *}
  4248 
  4249 text{* This function maps (finite) linearly ordered sets to sorted
  4250 lists. Warning: in most cases it is not a good idea to convert from
  4251 sets to lists but one should convert in the other direction (via
  4252 @{const set}). *}
  4253 
  4254 context linorder
  4255 begin
  4256 
  4257 definition sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where
  4258   "sorted_list_of_set = Finite_Set.fold insort []"
  4259 
  4260 lemma sorted_list_of_set_empty [simp]:
  4261   "sorted_list_of_set {} = []"
  4262   by (simp add: sorted_list_of_set_def)
  4263 
  4264 lemma sorted_list_of_set_insert [simp]:
  4265   assumes "finite A"
  4266   shows "sorted_list_of_set (insert x A) = insort x (sorted_list_of_set (A - {x}))"
  4267 proof -
  4268   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  4269   with assms show ?thesis by (simp add: sorted_list_of_set_def fold_insert_remove)
  4270 qed
  4271 
  4272 lemma sorted_list_of_set [simp]:
  4273   "finite A \<Longrightarrow> set (sorted_list_of_set A) = A \<and> sorted (sorted_list_of_set A) 
  4274     \<and> distinct (sorted_list_of_set A)"
  4275   by (induct A rule: finite_induct) (simp_all add: set_insort sorted_insort distinct_insort)
  4276 
  4277 lemma sorted_list_of_set_sort_remdups:
  4278   "sorted_list_of_set (set xs) = sort (remdups xs)"
  4279 proof -
  4280   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  4281   show ?thesis by (simp add: sort_foldl_insort sorted_list_of_set_def fold_set_remdups)
  4282 qed
  4283 
  4284 lemma sorted_list_of_set_remove:
  4285   assumes "finite A"
  4286   shows "sorted_list_of_set (A - {x}) = remove1 x (sorted_list_of_set A)"
  4287 proof (cases "x \<in> A")
  4288   case False with assms have "x \<notin> set (sorted_list_of_set A)" by simp
  4289   with False show ?thesis by (simp add: remove1_idem)
  4290 next
  4291   case True then obtain B where A: "A = insert x B" by (rule Set.set_insert)
  4292   with assms show ?thesis by simp
  4293 qed
  4294 
  4295 end
  4296 
  4297 lemma sorted_list_of_set_range [simp]:
  4298   "sorted_list_of_set {m..<n} = [m..<n]"
  4299   by (rule sorted_distinct_set_unique) simp_all
  4300 
  4301 
  4302 subsubsection {* @{text lists}: the list-forming operator over sets *}
  4303 
  4304 inductive_set
  4305   lists :: "'a set => 'a list set"
  4306   for A :: "'a set"
  4307 where
  4308     Nil [intro!, simp]: "[]: lists A"
  4309   | Cons [intro!, simp, no_atp]: "[| a: A; l: lists A|] ==> a#l : lists A"
  4310 
  4311 inductive_cases listsE [elim!,no_atp]: "x#l : lists A"
  4312 inductive_cases listspE [elim!,no_atp]: "listsp A (x # l)"
  4313 
  4314 lemma listsp_mono [mono]: "A \<le> B ==> listsp A \<le> listsp B"
  4315 by (rule predicate1I, erule listsp.induct, (blast dest: predicate1D)+)
  4316 
  4317 lemmas lists_mono = listsp_mono [to_set pred_subset_eq]
  4318 
  4319 lemma listsp_infI:
  4320   assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
  4321 by induct blast+
  4322 
  4323 lemmas lists_IntI = listsp_infI [to_set]
  4324 
  4325 lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
  4326 proof (rule mono_inf [where f=listsp, THEN order_antisym])
  4327   show "mono listsp" by (simp add: mono_def listsp_mono)
  4328   show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro!: listsp_infI predicate1I)
  4329 qed
  4330 
  4331 lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_def inf_bool_def]
  4332 
  4333 lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set pred_equals_eq]
  4334 
  4335 lemma Cons_in_lists_iff[simp]: "x#xs : lists A \<longleftrightarrow> x:A \<and> xs : lists A"
  4336 by auto
  4337 
  4338 lemma append_in_listsp_conv [iff]:
  4339      "(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
  4340 by (induct xs) auto
  4341 
  4342 lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]
  4343 
  4344 lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
  4345 -- {* eliminate @{text listsp} in favour of @{text set} *}
  4346 by (induct xs) auto
  4347 
  4348 lemmas in_lists_conv_set = in_listsp_conv_set [to_set]
  4349 
  4350 lemma in_listspD [dest!,no_atp]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
  4351 by (rule in_listsp_conv_set [THEN iffD1])
  4352 
  4353 lemmas in_listsD [dest!,no_atp] = in_listspD [to_set]
  4354 
  4355 lemma in_listspI [intro!,no_atp]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
  4356 by (rule in_listsp_conv_set [THEN iffD2])
  4357 
  4358 lemmas in_listsI [intro!,no_atp] = in_listspI [to_set]
  4359 
  4360 lemma lists_eq_set: "lists A = {xs. set xs <= A}"
  4361 by auto
  4362 
  4363 lemma lists_empty [simp]: "lists {} = {[]}"
  4364 by auto
  4365 
  4366 lemma lists_UNIV [simp]: "lists UNIV = UNIV"
  4367 by auto
  4368 
  4369 
  4370 subsubsection {* Inductive definition for membership *}
  4371 
  4372 inductive ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
  4373 where
  4374     elem:  "ListMem x (x # xs)"
  4375   | insert:  "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"
  4376 
  4377 lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
  4378 apply (rule iffI)
  4379  apply (induct set: ListMem)
  4380   apply auto
  4381 apply (induct xs)
  4382  apply (auto intro: ListMem.intros)
  4383 done
  4384 
  4385 
  4386 subsubsection {* Lists as Cartesian products *}
  4387 
  4388 text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
  4389 @{term A} and tail drawn from @{term Xs}.*}
  4390 
  4391 definition
  4392   set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set" where
  4393   "set_Cons A XS = {z. \<exists>x xs. z = x # xs \<and> x \<in> A \<and> xs \<in> XS}"
  4394 
  4395 lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
  4396 by (auto simp add: set_Cons_def)
  4397 
  4398 text{*Yields the set of lists, all of the same length as the argument and
  4399 with elements drawn from the corresponding element of the argument.*}
  4400 
  4401 primrec
  4402   listset :: "'a set list \<Rightarrow> 'a list set" where
  4403      "listset [] = {[]}"
  4404   |  "listset (A # As) = set_Cons A (listset As)"
  4405 
  4406 
  4407 subsection {* Relations on Lists *}
  4408 
  4409 subsubsection {* Length Lexicographic Ordering *}
  4410 
  4411 text{*These orderings preserve well-foundedness: shorter lists 
  4412   precede longer lists. These ordering are not used in dictionaries.*}
  4413         
  4414 primrec -- {*The lexicographic ordering for lists of the specified length*}
  4415   lexn :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> ('a list \<times> 'a list) set" where
  4416     "lexn r 0 = {}"
  4417   | "lexn r (Suc n) = (map_pair (%(x, xs). x#xs) (%(x, xs). x#xs) ` (r <*lex*> lexn r n)) Int
  4418       {(xs, ys). length xs = Suc n \<and> length ys = Suc n}"
  4419 
  4420 definition
  4421   lex :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4422   "lex r = (\<Union>n. lexn r n)" -- {*Holds only between lists of the same length*}
  4423 
  4424 definition
  4425   lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" where
  4426   "lenlex r = inv_image (less_than <*lex*> lex r) (\<lambda>xs. (length xs, xs))"
  4427         -- {*Compares lists by their length and then lexicographically*}
  4428 
  4429 lemma wf_lexn: "wf r ==> wf (lexn r n)"
  4430 apply (induct n, simp, simp)
  4431 apply(rule wf_subset)
  4432  prefer 2 apply (rule Int_lower1)
  4433 apply(rule wf_map_pair_image)
  4434  prefer 2 apply (rule inj_onI, auto)
  4435 done
  4436 
  4437 lemma lexn_length:
  4438   "(xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
  4439 by (induct n arbitrary: xs ys) auto
  4440 
  4441 lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
  4442 apply (unfold lex_def)
  4443 apply (rule wf_UN)
  4444 apply (blast intro: wf_lexn, clarify)
  4445 apply (rename_tac m n)
  4446 apply (subgoal_tac "m \<noteq> n")
  4447  prefer 2 apply blast
  4448 apply (blast dest: lexn_length not_sym)
  4449 done
  4450 
  4451 lemma lexn_conv:
  4452   "lexn r n =
  4453     {(xs,ys). length xs = n \<and> length ys = n \<and>
  4454     (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
  4455 apply (induct n, simp)
  4456 apply (simp add: image_Collect lex_prod_def, safe, blast)
  4457  apply (rule_tac x = "ab # xys" in exI, simp)
  4458 apply (case_tac xys, simp_all, blast)
  4459 done
  4460 
  4461 lemma lex_conv:
  4462   "lex r =
  4463     {(xs,ys). length xs = length ys \<and>
  4464     (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
  4465 by (force simp add: lex_def lexn_conv)
  4466 
  4467 lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
  4468 by (unfold lenlex_def) blast
  4469 
  4470 lemma lenlex_conv:
  4471     "lenlex r = {(xs,ys). length xs < length ys |
  4472                  length xs = length ys \<and> (xs, ys) : lex r}"
  4473 by (simp add: lenlex_def Id_on_def lex_prod_def inv_image_def)
  4474 
  4475 lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
  4476 by (simp add: lex_conv)
  4477 
  4478 lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
  4479 by (simp add:lex_conv)
  4480 
  4481 lemma Cons_in_lex [simp]:
  4482     "((x # xs, y # ys) : lex r) =
  4483       ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
  4484 apply (simp add: lex_conv)
  4485 apply (rule iffI)
  4486  prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
  4487 apply (case_tac xys, simp, simp)
  4488 apply blast
  4489 done
  4490 
  4491 
  4492 subsubsection {* Lexicographic Ordering *}
  4493 
  4494 text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
  4495     This ordering does \emph{not} preserve well-foundedness.
  4496      Author: N. Voelker, March 2005. *} 
  4497 
  4498 definition
  4499   lexord :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4500   "lexord r = {(x,y ). \<exists> a v. y = x @ a # v \<or>
  4501             (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
  4502 
  4503 lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
  4504 by (unfold lexord_def, induct_tac y, auto) 
  4505 
  4506 lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
  4507 by (unfold lexord_def, induct_tac x, auto)
  4508 
  4509 lemma lexord_cons_cons[simp]:
  4510      "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
  4511   apply (unfold lexord_def, safe, simp_all)
  4512   apply (case_tac u, simp, simp)
  4513   apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
  4514   apply (erule_tac x="b # u" in allE)
  4515   by force
  4516 
  4517 lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
  4518 
  4519 lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
  4520 by (induct_tac x, auto)  
  4521 
  4522 lemma lexord_append_left_rightI:
  4523      "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
  4524 by (induct_tac u, auto)
  4525 
  4526 lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
  4527 by (induct x, auto)
  4528 
  4529 lemma lexord_append_leftD:
  4530      "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
  4531 by (erule rev_mp, induct_tac x, auto)
  4532 
  4533 lemma lexord_take_index_conv: 
  4534    "((x,y) : lexord r) = 
  4535     ((length x < length y \<and> take (length x) y = x) \<or> 
  4536      (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
  4537   apply (unfold lexord_def Let_def, clarsimp) 
  4538   apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
  4539   apply auto 
  4540   apply (rule_tac x="hd (drop (length x) y)" in exI)
  4541   apply (rule_tac x="tl (drop (length x) y)" in exI)
  4542   apply (erule subst, simp add: min_def) 
  4543   apply (rule_tac x ="length u" in exI, simp) 
  4544   apply (rule_tac x ="take i x" in exI) 
  4545   apply (rule_tac x ="x ! i" in exI) 
  4546   apply (rule_tac x ="y ! i" in exI, safe) 
  4547   apply (rule_tac x="drop (Suc i) x" in exI)
  4548   apply (drule sym, simp add: drop_Suc_conv_tl) 
  4549   apply (rule_tac x="drop (Suc i) y" in exI)
  4550   by (simp add: drop_Suc_conv_tl) 
  4551 
  4552 -- {* lexord is extension of partial ordering List.lex *} 
  4553 lemma lexord_lex: "(x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
  4554   apply (rule_tac x = y in spec) 
  4555   apply (induct_tac x, clarsimp) 
  4556   by (clarify, case_tac x, simp, force)
  4557 
  4558 lemma lexord_irreflexive: "ALL x. (x,x) \<notin> r \<Longrightarrow> (xs,xs) \<notin> lexord r"
  4559 by (induct xs) auto
  4560 
  4561 text{* By Ren\'e Thiemann: *}
  4562 lemma lexord_partial_trans: 
  4563   "(\<And>x y z. x \<in> set xs \<Longrightarrow> (x,y) \<in> r \<Longrightarrow> (y,z) \<in> r \<Longrightarrow> (x,z) \<in> r)
  4564    \<Longrightarrow>  (xs,ys) \<in> lexord r  \<Longrightarrow>  (ys,zs) \<in> lexord r \<Longrightarrow>  (xs,zs) \<in> lexord r"
  4565 proof (induct xs arbitrary: ys zs)
  4566   case Nil
  4567   from Nil(3) show ?case unfolding lexord_def by (cases zs, auto)
  4568 next
  4569   case (Cons x xs yys zzs)
  4570   from Cons(3) obtain y ys where yys: "yys = y # ys" unfolding lexord_def
  4571     by (cases yys, auto)
  4572   note Cons = Cons[unfolded yys]
  4573   from Cons(3) have one: "(x,y) \<in> r \<or> x = y \<and> (xs,ys) \<in> lexord r" by auto
  4574   from Cons(4) obtain z zs where zzs: "zzs = z # zs" unfolding lexord_def
  4575     by (cases zzs, auto)
  4576   note Cons = Cons[unfolded zzs]
  4577   from Cons(4) have two: "(y,z) \<in> r \<or> y = z \<and> (ys,zs) \<in> lexord r" by auto
  4578   {
  4579     assume "(xs,ys) \<in> lexord r" and "(ys,zs) \<in> lexord r"
  4580     from Cons(1)[OF _ this] Cons(2)
  4581     have "(xs,zs) \<in> lexord r" by auto
  4582   } note ind1 = this
  4583   {
  4584     assume "(x,y) \<in> r" and "(y,z) \<in> r"
  4585     from Cons(2)[OF _ this] have "(x,z) \<in> r" by auto
  4586   } note ind2 = this
  4587   from one two ind1 ind2
  4588   have "(x,z) \<in> r \<or> x = z \<and> (xs,zs) \<in> lexord r" by blast
  4589   thus ?case unfolding zzs by auto
  4590 qed
  4591 
  4592 lemma lexord_trans: 
  4593     "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
  4594 by(auto simp: trans_def intro:lexord_partial_trans)
  4595 
  4596 lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
  4597 by (rule transI, drule lexord_trans, blast) 
  4598 
  4599 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"
  4600   apply (rule_tac x = y in spec) 
  4601   apply (induct_tac x, rule allI) 
  4602   apply (case_tac x, simp, simp) 
  4603   apply (rule allI, case_tac x, simp, simp) 
  4604   by blast
  4605 
  4606 
  4607 subsubsection {* Lexicographic combination of measure functions *}
  4608 
  4609 text {* These are useful for termination proofs *}
  4610 
  4611 definition
  4612   "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"
  4613 
  4614 lemma wf_measures[simp]: "wf (measures fs)"
  4615 unfolding measures_def
  4616 by blast
  4617 
  4618 lemma in_measures[simp]: 
  4619   "(x, y) \<in> measures [] = False"
  4620   "(x, y) \<in> measures (f # fs)
  4621          = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"  
  4622 unfolding measures_def
  4623 by auto
  4624 
  4625 lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
  4626 by simp
  4627 
  4628 lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
  4629 by auto
  4630 
  4631 
  4632 subsubsection {* Lifting Relations to Lists: one element *}
  4633 
  4634 definition listrel1 :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4635 "listrel1 r = {(xs,ys).
  4636    \<exists>us z z' vs. xs = us @ z # vs \<and> (z,z') \<in> r \<and> ys = us @ z' # vs}"
  4637 
  4638 lemma listrel1I:
  4639   "\<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow>
  4640   (xs, ys) \<in> listrel1 r"
  4641 unfolding listrel1_def by auto
  4642 
  4643 lemma listrel1E:
  4644   "\<lbrakk> (xs, ys) \<in> listrel1 r;
  4645      !!x y us vs. \<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow> P
  4646    \<rbrakk> \<Longrightarrow> P"
  4647 unfolding listrel1_def by auto
  4648 
  4649 lemma not_Nil_listrel1 [iff]: "([], xs) \<notin> listrel1 r"
  4650 unfolding listrel1_def by blast
  4651 
  4652 lemma not_listrel1_Nil [iff]: "(xs, []) \<notin> listrel1 r"
  4653 unfolding listrel1_def by blast
  4654 
  4655 lemma Cons_listrel1_Cons [iff]:
  4656   "(x # xs, y # ys) \<in> listrel1 r \<longleftrightarrow>
  4657    (x,y) \<in> r \<and> xs = ys \<or> x = y \<and> (xs, ys) \<in> listrel1 r"
  4658 by (simp add: listrel1_def Cons_eq_append_conv) (blast)
  4659 
  4660 lemma listrel1I1: "(x,y) \<in> r \<Longrightarrow> (x # xs, y # xs) \<in> listrel1 r"
  4661 by (metis Cons_listrel1_Cons)
  4662 
  4663 lemma listrel1I2: "(xs, ys) \<in> listrel1 r \<Longrightarrow> (x # xs, x # ys) \<in> listrel1 r"
  4664 by (metis Cons_listrel1_Cons)
  4665 
  4666 lemma append_listrel1I:
  4667   "(xs, ys) \<in> listrel1 r \<and> us = vs \<or> xs = ys \<and> (us, vs) \<in> listrel1 r
  4668     \<Longrightarrow> (xs @ us, ys @ vs) \<in> listrel1 r"
  4669 unfolding listrel1_def
  4670 by auto (blast intro: append_eq_appendI)+
  4671 
  4672 lemma Cons_listrel1E1[elim!]:
  4673   assumes "(x # xs, ys) \<in> listrel1 r"
  4674     and "\<And>y. ys = y # xs \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
  4675     and "\<And>zs. ys = x # zs \<Longrightarrow> (xs, zs) \<in> listrel1 r \<Longrightarrow> R"
  4676   shows R
  4677 using assms by (cases ys) blast+
  4678 
  4679 lemma Cons_listrel1E2[elim!]:
  4680   assumes "(xs, y # ys) \<in> listrel1 r"
  4681     and "\<And>x. xs = x # ys \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
  4682     and "\<And>zs. xs = y # zs \<Longrightarrow> (zs, ys) \<in> listrel1 r \<Longrightarrow> R"
  4683   shows R
  4684 using assms by (cases xs) blast+
  4685 
  4686 lemma snoc_listrel1_snoc_iff:
  4687   "(xs @ [x], ys @ [y]) \<in> listrel1 r
  4688     \<longleftrightarrow> (xs, ys) \<in> listrel1 r \<and> x = y \<or> xs = ys \<and> (x,y) \<in> r" (is "?L \<longleftrightarrow> ?R")
  4689 proof
  4690   assume ?L thus ?R
  4691     by (fastforce simp: listrel1_def snoc_eq_iff_butlast butlast_append)
  4692 next
  4693   assume ?R then show ?L unfolding listrel1_def by force
  4694 qed
  4695 
  4696 lemma listrel1_eq_len: "(xs,ys) \<in> listrel1 r \<Longrightarrow> length xs = length ys"
  4697 unfolding listrel1_def by auto
  4698 
  4699 lemma listrel1_mono:
  4700   "r \<subseteq> s \<Longrightarrow> listrel1 r \<subseteq> listrel1 s"
  4701 unfolding listrel1_def by blast
  4702 
  4703 
  4704 lemma listrel1_converse: "listrel1 (r^-1) = (listrel1 r)^-1"
  4705 unfolding listrel1_def by blast
  4706 
  4707 lemma in_listrel1_converse:
  4708   "(x,y) : listrel1 (r^-1) \<longleftrightarrow> (x,y) : (listrel1 r)^-1"
  4709 unfolding listrel1_def by blast
  4710 
  4711 lemma listrel1_iff_update:
  4712   "(xs,ys) \<in> (listrel1 r)
  4713    \<longleftrightarrow> (\<exists>y n. (xs ! n, y) \<in> r \<and> n < length xs \<and> ys = xs[n:=y])" (is "?L \<longleftrightarrow> ?R")
  4714 proof
  4715   assume "?L"
  4716   then obtain x y u v where "xs = u @ x # v"  "ys = u @ y # v"  "(x,y) \<in> r"
  4717     unfolding listrel1_def by auto
  4718   then have "ys = xs[length u := y]" and "length u < length xs"
  4719     and "(xs ! length u, y) \<in> r" by auto
  4720   then show "?R" by auto
  4721 next
  4722   assume "?R"
  4723   then obtain x y n where "(xs!n, y) \<in> r" "n < size xs" "ys = xs[n:=y]" "x = xs!n"
  4724     by auto
  4725   then obtain u v where "xs = u @ x # v" and "ys = u @ y # v" and "(x, y) \<in> r"
  4726     by (auto intro: upd_conv_take_nth_drop id_take_nth_drop)
  4727   then show "?L" by (auto simp: listrel1_def)
  4728 qed
  4729 
  4730 
  4731 text{* Accessible part and wellfoundedness: *}
  4732 
  4733 lemma Cons_acc_listrel1I [intro!]:
  4734   "x \<in> acc r \<Longrightarrow> xs \<in> acc (listrel1 r) \<Longrightarrow> (x # xs) \<in> acc (listrel1 r)"
  4735 apply (induct arbitrary: xs set: acc)
  4736 apply (erule thin_rl)
  4737 apply (erule acc_induct)
  4738 apply (rule accI)
  4739 apply (blast)
  4740 done
  4741 
  4742 lemma lists_accD: "xs \<in> lists (acc r) \<Longrightarrow> xs \<in> acc (listrel1 r)"
  4743 apply (induct set: lists)
  4744  apply (rule accI)
  4745  apply simp
  4746 apply (rule accI)
  4747 apply (fast dest: acc_downward)
  4748 done
  4749 
  4750 lemma lists_accI: "xs \<in> acc (listrel1 r) \<Longrightarrow> xs \<in> lists (acc r)"
  4751 apply (induct set: acc)
  4752 apply clarify
  4753 apply (rule accI)
  4754 apply (fastforce dest!: in_set_conv_decomp[THEN iffD1] simp: listrel1_def)
  4755 done
  4756 
  4757 lemma wf_listrel1_iff[simp]: "wf(listrel1 r) = wf r"
  4758 by(metis wf_acc_iff in_lists_conv_set lists_accI lists_accD Cons_in_lists_iff)
  4759 
  4760 
  4761 subsubsection {* Lifting Relations to Lists: all elements *}
  4762 
  4763 inductive_set
  4764   listrel :: "('a * 'a)set => ('a list * 'a list)set"
  4765   for r :: "('a * 'a)set"
  4766 where
  4767     Nil:  "([],[]) \<in> listrel r"
  4768   | Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
  4769 
  4770 inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
  4771 inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
  4772 inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
  4773 inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
  4774 
  4775 
  4776 lemma listrel_eq_len:  "(xs, ys) \<in> listrel r \<Longrightarrow> length xs = length ys"
  4777 by(induct rule: listrel.induct) auto
  4778 
  4779 lemma listrel_iff_zip: "(xs,ys) : listrel r \<longleftrightarrow>
  4780   length xs = length ys & (\<forall>(x,y) \<in> set(zip xs ys). (x,y) \<in> r)" (is "?L \<longleftrightarrow> ?R")
  4781 proof
  4782   assume ?L thus ?R by induct (auto intro: listrel_eq_len)
  4783 next
  4784   assume ?R thus ?L
  4785     apply (clarify)
  4786     by (induct rule: list_induct2) (auto intro: listrel.intros)
  4787 qed
  4788 
  4789 lemma listrel_iff_nth: "(xs,ys) : listrel r \<longleftrightarrow>
  4790   length xs = length ys & (\<forall>n < length xs. (xs!n, ys!n) \<in> r)" (is "?L \<longleftrightarrow> ?R")
  4791 by (auto simp add: all_set_conv_all_nth listrel_iff_zip)
  4792 
  4793 
  4794 lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
  4795 apply clarify  
  4796 apply (erule listrel.induct)
  4797 apply (blast intro: listrel.intros)+
  4798 done
  4799 
  4800 lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
  4801 apply clarify 
  4802 apply (erule listrel.induct, auto) 
  4803 done
  4804 
  4805 lemma listrel_refl_on: "refl_on A r \<Longrightarrow> refl_on (lists A) (listrel r)" 
  4806 apply (simp add: refl_on_def listrel_subset Ball_def)
  4807 apply (rule allI) 
  4808 apply (induct_tac x) 
  4809 apply (auto intro: listrel.intros)
  4810 done
  4811 
  4812 lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
  4813 apply (auto simp add: sym_def)
  4814 apply (erule listrel.induct) 
  4815 apply (blast intro: listrel.intros)+
  4816 done
  4817 
  4818 lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
  4819 apply (simp add: trans_def)
  4820 apply (intro allI) 
  4821 apply (rule impI) 
  4822 apply (erule listrel.induct) 
  4823 apply (blast intro: listrel.intros)+
  4824 done
  4825 
  4826 theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
  4827 by (simp add: equiv_def listrel_refl_on listrel_sym listrel_trans) 
  4828 
  4829 lemma listrel_rtrancl_refl[iff]: "(xs,xs) : listrel(r^*)"
  4830 using listrel_refl_on[of UNIV, OF refl_rtrancl]
  4831 by(auto simp: refl_on_def)
  4832 
  4833 lemma listrel_rtrancl_trans:
  4834   "\<lbrakk> (xs,ys) : listrel(r^*);  (ys,zs) : listrel(r^*) \<rbrakk>
  4835   \<Longrightarrow> (xs,zs) : listrel(r^*)"
  4836 by (metis listrel_trans trans_def trans_rtrancl)
  4837 
  4838 
  4839 lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
  4840 by (blast intro: listrel.intros)
  4841 
  4842 lemma listrel_Cons:
  4843      "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})"
  4844 by (auto simp add: set_Cons_def intro: listrel.intros)
  4845 
  4846 text {* Relating @{term listrel1}, @{term listrel} and closures: *}
  4847 
  4848 lemma listrel1_rtrancl_subset_rtrancl_listrel1:
  4849   "listrel1 (r^*) \<subseteq> (listrel1 r)^*"
  4850 proof (rule subrelI)
  4851   fix xs ys assume 1: "(xs,ys) \<in> listrel1 (r^*)"
  4852   { fix x y us vs
  4853     have "(x,y) : r^* \<Longrightarrow> (us @ x # vs, us @ y # vs) : (listrel1 r)^*"
  4854     proof(induct rule: rtrancl.induct)
  4855       case rtrancl_refl show ?case by simp
  4856     next
  4857       case rtrancl_into_rtrancl thus ?case
  4858         by (metis listrel1I rtrancl.rtrancl_into_rtrancl)
  4859     qed }
  4860   thus "(xs,ys) \<in> (listrel1 r)^*" using 1 by(blast elim: listrel1E)
  4861 qed
  4862 
  4863 lemma rtrancl_listrel1_eq_len: "(x,y) \<in> (listrel1 r)^* \<Longrightarrow> length x = length y"
  4864 by (induct rule: rtrancl.induct) (auto intro: listrel1_eq_len)
  4865 
  4866 lemma rtrancl_listrel1_ConsI1:
  4867   "(xs,ys) : (listrel1 r)^* \<Longrightarrow> (x#xs,x#ys) : (listrel1 r)^*"
  4868 apply(induct rule: rtrancl.induct)
  4869  apply simp
  4870 by (metis listrel1I2 rtrancl.rtrancl_into_rtrancl)
  4871 
  4872 lemma rtrancl_listrel1_ConsI2:
  4873   "(x,y) \<in> r^* \<Longrightarrow> (xs, ys) \<in> (listrel1 r)^*
  4874   \<Longrightarrow> (x # xs, y # ys) \<in> (listrel1 r)^*"
  4875   by (blast intro: rtrancl_trans rtrancl_listrel1_ConsI1 
  4876     subsetD[OF listrel1_rtrancl_subset_rtrancl_listrel1 listrel1I1])
  4877 
  4878 lemma listrel1_subset_listrel:
  4879   "r \<subseteq> r' \<Longrightarrow> refl r' \<Longrightarrow> listrel1 r \<subseteq> listrel(r')"
  4880 by(auto elim!: listrel1E simp add: listrel_iff_zip set_zip refl_on_def)
  4881 
  4882 lemma listrel_reflcl_if_listrel1:
  4883   "(xs,ys) : listrel1 r \<Longrightarrow> (xs,ys) : listrel(r^*)"
  4884 by(erule listrel1E)(auto simp add: listrel_iff_zip set_zip)
  4885 
  4886 lemma listrel_rtrancl_eq_rtrancl_listrel1: "listrel (r^*) = (listrel1 r)^*"
  4887 proof
  4888   { fix x y assume "(x,y) \<in> listrel (r^*)"
  4889     then have "(x,y) \<in> (listrel1 r)^*"
  4890     by induct (auto intro: rtrancl_listrel1_ConsI2) }
  4891   then show "listrel (r^*) \<subseteq> (listrel1 r)^*"
  4892     by (rule subrelI)
  4893 next
  4894   show "listrel (r^*) \<supseteq> (listrel1 r)^*"
  4895   proof(rule subrelI)
  4896     fix xs ys assume "(xs,ys) \<in> (listrel1 r)^*"
  4897     then show "(xs,ys) \<in> listrel (r^*)"
  4898     proof induct
  4899       case base show ?case by(auto simp add: listrel_iff_zip set_zip)
  4900     next
  4901       case (step ys zs)
  4902       thus ?case  by (metis listrel_reflcl_if_listrel1 listrel_rtrancl_trans)
  4903     qed
  4904   qed
  4905 qed
  4906 
  4907 lemma rtrancl_listrel1_if_listrel:
  4908   "(xs,ys) : listrel r \<Longrightarrow> (xs,ys) : (listrel1 r)^*"
  4909 by(metis listrel_rtrancl_eq_rtrancl_listrel1 subsetD[OF listrel_mono] r_into_rtrancl subsetI)
  4910 
  4911 lemma listrel_subset_rtrancl_listrel1: "listrel r \<subseteq> (listrel1 r)^*"
  4912 by(fast intro:rtrancl_listrel1_if_listrel)
  4913 
  4914 
  4915 subsection {* Size function *}
  4916 
  4917 lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (list_size f)"
  4918 by (rule is_measure_trivial)
  4919 
  4920 lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (option_size f)"
  4921 by (rule is_measure_trivial)
  4922 
  4923 lemma list_size_estimation[termination_simp]: 
  4924   "x \<in> set xs \<Longrightarrow> y < f x \<Longrightarrow> y < list_size f xs"
  4925 by (induct xs) auto
  4926 
  4927 lemma list_size_estimation'[termination_simp]: 
  4928   "x \<in> set xs \<Longrightarrow> y \<le> f x \<Longrightarrow> y \<le> list_size f xs"
  4929 by (induct xs) auto
  4930 
  4931 lemma list_size_map[simp]: "list_size f (map g xs) = list_size (f o g) xs"
  4932 by (induct xs) auto
  4933 
  4934 lemma list_size_append[simp]: "list_size f (xs @ ys) = list_size f xs + list_size f ys"
  4935 by (induct xs, auto)
  4936 
  4937 lemma list_size_pointwise[termination_simp]: 
  4938   "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> list_size f xs \<le> list_size g xs"
  4939 by (induct xs) force+
  4940 
  4941 
  4942 subsection {* Transfer *}
  4943 
  4944 definition
  4945   embed_list :: "nat list \<Rightarrow> int list"
  4946 where
  4947   "embed_list l = map int l"
  4948 
  4949 definition
  4950   nat_list :: "int list \<Rightarrow> bool"
  4951 where
  4952   "nat_list l = nat_set (set l)"
  4953 
  4954 definition
  4955   return_list :: "int list \<Rightarrow> nat list"
  4956 where
  4957   "return_list l = map nat l"
  4958 
  4959 lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow>
  4960     embed_list (return_list l) = l"
  4961   unfolding embed_list_def return_list_def nat_list_def nat_set_def
  4962   apply (induct l)
  4963   apply auto
  4964 done
  4965 
  4966 lemma transfer_nat_int_list_functions:
  4967   "l @ m = return_list (embed_list l @ embed_list m)"
  4968   "[] = return_list []"
  4969   unfolding return_list_def embed_list_def
  4970   apply auto
  4971   apply (induct l, auto)
  4972   apply (induct m, auto)
  4973 done
  4974 
  4975 (*
  4976 lemma transfer_nat_int_fold1: "fold f l x =
  4977     fold (%x. f (nat x)) (embed_list l) x";
  4978 *)
  4979 
  4980 
  4981 subsection {* Code generation *}
  4982 
  4983 subsubsection {* Counterparts for set-related operations *}
  4984 
  4985 definition member :: "'a list \<Rightarrow> 'a \<Rightarrow> bool" where
  4986   [code_post]: "member xs x \<longleftrightarrow> x \<in> set xs"
  4987 
  4988 text {*
  4989   Only use @{text member} for generating executable code.  Otherwise use
  4990   @{prop "x \<in> set xs"} instead --- it is much easier to reason about.
  4991 *}
  4992 
  4993 lemma member_set:
  4994   "member = set"
  4995   by (simp add: fun_eq_iff member_def mem_def)
  4996 
  4997 lemma member_rec [code]:
  4998   "member (x # xs) y \<longleftrightarrow> x = y \<or> member xs y"
  4999   "member [] y \<longleftrightarrow> False"
  5000   by (auto simp add: member_def)
  5001 
  5002 lemma in_set_member [code_unfold]:
  5003   "x \<in> set xs \<longleftrightarrow> member xs x"
  5004   by (simp add: member_def)
  5005 
  5006 declare INF_def [code_unfold]
  5007 declare SUP_def [code_unfold]
  5008 
  5009 declare set_map [symmetric, code_unfold]
  5010 
  5011 definition list_all :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5012   list_all_iff [code_post]: "list_all P xs \<longleftrightarrow> (\<forall>x \<in> set xs. P x)"
  5013 
  5014 definition list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5015   list_ex_iff [code_post]: "list_ex P xs \<longleftrightarrow> (\<exists>x \<in> set xs. P x)"
  5016 
  5017 definition list_ex1
  5018 where
  5019   list_ex1_iff: "list_ex1 P xs \<longleftrightarrow> (\<exists>! x. x \<in> set xs \<and> P x)"
  5020 
  5021 text {*
  5022   Usually you should prefer @{text "\<forall>x\<in>set xs"} and @{text "\<exists>x\<in>set xs"}
  5023   over @{const list_all} and @{const list_ex} in specifications.
  5024 *}
  5025 
  5026 lemma list_all_simps [simp, code]:
  5027   "list_all P (x # xs) \<longleftrightarrow> P x \<and> list_all P xs"
  5028   "list_all P [] \<longleftrightarrow> True"
  5029   by (simp_all add: list_all_iff)
  5030 
  5031 lemma list_ex_simps [simp, code]:
  5032   "list_ex P (x # xs) \<longleftrightarrow> P x \<or> list_ex P xs"
  5033   "list_ex P [] \<longleftrightarrow> False"
  5034   by (simp_all add: list_ex_iff)
  5035 
  5036 lemma list_ex1_simps [simp, code]:
  5037   "list_ex1 P [] = False"
  5038   "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)"
  5039 unfolding list_ex1_iff list_all_iff by auto
  5040 
  5041 lemma Ball_set_list_all [code_unfold]:
  5042   "Ball (set xs) P \<longleftrightarrow> list_all P xs"
  5043   by (simp add: list_all_iff)
  5044 
  5045 lemma Bex_set_list_ex [code_unfold]:
  5046   "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
  5047   by (simp add: list_ex_iff)
  5048 
  5049 lemma list_all_append [simp]:
  5050   "list_all P (xs @ ys) \<longleftrightarrow> list_all P xs \<and> list_all P ys"
  5051   by (auto simp add: list_all_iff)
  5052 
  5053 lemma list_ex_append [simp]:
  5054   "list_ex P (xs @ ys) \<longleftrightarrow> list_ex P xs \<or> list_ex P ys"
  5055   by (auto simp add: list_ex_iff)
  5056 
  5057 lemma list_all_rev [simp]:
  5058   "list_all P (rev xs) \<longleftrightarrow> list_all P xs"
  5059   by (simp add: list_all_iff)
  5060 
  5061 lemma list_ex_rev [simp]:
  5062   "list_ex P (rev xs) \<longleftrightarrow> list_ex P xs"
  5063   by (simp add: list_ex_iff)
  5064 
  5065 lemma list_all_length:
  5066   "list_all P xs \<longleftrightarrow> (\<forall>n < length xs. P (xs ! n))"
  5067   by (auto simp add: list_all_iff set_conv_nth)
  5068 
  5069 lemma list_ex_length:
  5070   "list_ex P xs \<longleftrightarrow> (\<exists>n < length xs. P (xs ! n))"
  5071   by (auto simp add: list_ex_iff set_conv_nth)
  5072 
  5073 lemma list_all_cong [fundef_cong]:
  5074   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_all f xs = list_all g ys"
  5075   by (simp add: list_all_iff)
  5076 
  5077 lemma list_any_cong [fundef_cong]:
  5078   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_ex f xs = list_ex g ys"
  5079   by (simp add: list_ex_iff)
  5080 
  5081 text {* Bounded quantification and summation over nats. *}
  5082 
  5083 lemma atMost_upto [code_unfold]:
  5084   "{..n} = set [0..<Suc n]"
  5085   by auto
  5086 
  5087 lemma atLeast_upt [code_unfold]:
  5088   "{..<n} = set [0..<n]"
  5089   by auto
  5090 
  5091 lemma greaterThanLessThan_upt [code_unfold]:
  5092   "{n<..<m} = set [Suc n..<m]"
  5093   by auto
  5094 
  5095 lemmas atLeastLessThan_upt [code_unfold] = set_upt [symmetric]
  5096 
  5097 lemma greaterThanAtMost_upt [code_unfold]:
  5098   "{n<..m} = set [Suc n..<Suc m]"
  5099   by auto
  5100 
  5101 lemma atLeastAtMost_upt [code_unfold]:
  5102   "{n..m} = set [n..<Suc m]"
  5103   by auto
  5104 
  5105 lemma all_nat_less_eq [code_unfold]:
  5106   "(\<forall>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..<n}. P m)"
  5107   by auto
  5108 
  5109 lemma ex_nat_less_eq [code_unfold]:
  5110   "(\<exists>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..<n}. P m)"
  5111   by auto
  5112 
  5113 lemma all_nat_less [code_unfold]:
  5114   "(\<forall>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..n}. P m)"
  5115   by auto
  5116 
  5117 lemma ex_nat_less [code_unfold]:
  5118   "(\<exists>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..n}. P m)"
  5119   by auto
  5120 
  5121 lemma setsum_set_upt_conv_listsum_nat [code_unfold]:
  5122   "setsum f (set [m..<n]) = listsum (map f [m..<n])"
  5123   by (simp add: interv_listsum_conv_setsum_set_nat)
  5124 
  5125 text {* Summation over ints. *}
  5126 
  5127 lemma greaterThanLessThan_upto [code_unfold]:
  5128   "{i<..<j::int} = set [i+1..j - 1]"
  5129 by auto
  5130 
  5131 lemma atLeastLessThan_upto [code_unfold]:
  5132   "{i..<j::int} = set [i..j - 1]"
  5133 by auto
  5134 
  5135 lemma greaterThanAtMost_upto [code_unfold]:
  5136   "{i<..j::int} = set [i+1..j]"
  5137 by auto
  5138 
  5139 lemmas atLeastAtMost_upto [code_unfold] = set_upto [symmetric]
  5140 
  5141 lemma setsum_set_upto_conv_listsum_int [code_unfold]:
  5142   "setsum f (set [i..j::int]) = listsum (map f [i..j])"
  5143   by (simp add: interv_listsum_conv_setsum_set_int)
  5144 
  5145 
  5146 subsubsection {* Optimizing by rewriting *}
  5147 
  5148 definition null :: "'a list \<Rightarrow> bool" where
  5149   [code_post]: "null xs \<longleftrightarrow> xs = []"
  5150 
  5151 text {*
  5152   Efficient emptyness check is implemented by @{const null}.
  5153 *}
  5154 
  5155 lemma null_rec [code]:
  5156   "null (x # xs) \<longleftrightarrow> False"
  5157   "null [] \<longleftrightarrow> True"
  5158   by (simp_all add: null_def)
  5159 
  5160 lemma eq_Nil_null [code_unfold]:
  5161   "xs = [] \<longleftrightarrow> null xs"
  5162   by (simp add: null_def)
  5163 
  5164 lemma equal_Nil_null [code_unfold]:
  5165   "HOL.equal xs [] \<longleftrightarrow> null xs"
  5166   by (simp add: equal eq_Nil_null)
  5167 
  5168 definition maps :: "('a \<Rightarrow> 'b list) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  5169   [code_post]: "maps f xs = concat (map f xs)"
  5170 
  5171 definition map_filter :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  5172   [code_post]: "map_filter f xs = map (the \<circ> f) (filter (\<lambda>x. f x \<noteq> None) xs)"
  5173 
  5174 text {*
  5175   Operations @{const maps} and @{const map_filter} avoid
  5176   intermediate lists on execution -- do not use for proving.
  5177 *}
  5178 
  5179 lemma maps_simps [code]:
  5180   "maps f (x # xs) = f x @ maps f xs"
  5181   "maps f [] = []"
  5182   by (simp_all add: maps_def)
  5183 
  5184 lemma map_filter_simps [code]:
  5185   "map_filter f (x # xs) = (case f x of None \<Rightarrow> map_filter f xs | Some y \<Rightarrow> y # map_filter f xs)"
  5186   "map_filter f [] = []"
  5187   by (simp_all add: map_filter_def split: option.split)
  5188 
  5189 lemma concat_map_maps [code_unfold]:
  5190   "concat (map f xs) = maps f xs"
  5191   by (simp add: maps_def)
  5192 
  5193 lemma map_filter_map_filter [code_unfold]:
  5194   "map f (filter P xs) = map_filter (\<lambda>x. if P x then Some (f x) else None) xs"
  5195   by (simp add: map_filter_def)
  5196 
  5197 text {* Optimized code for @{text"\<forall>i\<in>{a..b::int}"} and @{text"\<forall>n:{a..<b::nat}"}
  5198 and similiarly for @{text"\<exists>"}. *}
  5199 
  5200 definition all_interval_nat :: "(nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
  5201   "all_interval_nat P i j \<longleftrightarrow> (\<forall>n \<in> {i..<j}. P n)"
  5202 
  5203 lemma [code]:
  5204   "all_interval_nat P i j \<longleftrightarrow> i \<ge> j \<or> P i \<and> all_interval_nat P (Suc i) j"
  5205 proof -
  5206   have *: "\<And>n. P i \<Longrightarrow> \<forall>n\<in>{Suc i..<j}. P n \<Longrightarrow> i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n"
  5207   proof -
  5208     fix n
  5209     assume "P i" "\<forall>n\<in>{Suc i..<j}. P n" "i \<le> n" "n < j"
  5210     then show "P n" by (cases "n = i") simp_all
  5211   qed
  5212   show ?thesis by (auto simp add: all_interval_nat_def intro: *)
  5213 qed
  5214 
  5215 lemma list_all_iff_all_interval_nat [code_unfold]:
  5216   "list_all P [i..<j] \<longleftrightarrow> all_interval_nat P i j"
  5217   by (simp add: list_all_iff all_interval_nat_def)
  5218 
  5219 lemma list_ex_iff_not_all_inverval_nat [code_unfold]:
  5220   "list_ex P [i..<j] \<longleftrightarrow> \<not> (all_interval_nat (Not \<circ> P) i j)"
  5221   by (simp add: list_ex_iff all_interval_nat_def)
  5222 
  5223 definition all_interval_int :: "(int \<Rightarrow> bool) \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool" where
  5224   "all_interval_int P i j \<longleftrightarrow> (\<forall>k \<in> {i..j}. P k)"
  5225 
  5226 lemma [code]:
  5227   "all_interval_int P i j \<longleftrightarrow> i > j \<or> P i \<and> all_interval_int P (i + 1) j"
  5228 proof -
  5229   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"
  5230   proof -
  5231     fix k
  5232     assume "P i" "\<forall>k\<in>{i+1..j}. P k" "i \<le> k" "k \<le> j"
  5233     then show "P k" by (cases "k = i") simp_all
  5234   qed
  5235   show ?thesis by (auto simp add: all_interval_int_def intro: *)
  5236 qed
  5237 
  5238 lemma list_all_iff_all_interval_int [code_unfold]:
  5239   "list_all P [i..j] \<longleftrightarrow> all_interval_int P i j"
  5240   by (simp add: list_all_iff all_interval_int_def)
  5241 
  5242 lemma list_ex_iff_not_all_inverval_int [code_unfold]:
  5243   "list_ex P [i..j] \<longleftrightarrow> \<not> (all_interval_int (Not \<circ> P) i j)"
  5244   by (simp add: list_ex_iff all_interval_int_def)
  5245 
  5246 hide_const (open) member null maps map_filter all_interval_nat all_interval_int
  5247 
  5248 
  5249 subsubsection {* Pretty lists *}
  5250 
  5251 use "Tools/list_code.ML"
  5252 
  5253 code_type list
  5254   (SML "_ list")
  5255   (OCaml "_ list")
  5256   (Haskell "![(_)]")
  5257   (Scala "List[(_)]")
  5258 
  5259 code_const Nil
  5260   (SML "[]")
  5261   (OCaml "[]")
  5262   (Haskell "[]")
  5263   (Scala "!Nil")
  5264 
  5265 code_instance list :: equal
  5266   (Haskell -)
  5267 
  5268 code_const "HOL.equal \<Colon> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
  5269   (Haskell infix 4 "==")
  5270 
  5271 code_reserved SML
  5272   list
  5273 
  5274 code_reserved OCaml
  5275   list
  5276 
  5277 setup {* fold (List_Code.add_literal_list) ["SML", "OCaml", "Haskell", "Scala"] *}
  5278 
  5279 
  5280 subsubsection {* Use convenient predefined operations *}
  5281 
  5282 code_const "op @"
  5283   (SML infixr 7 "@")
  5284   (OCaml infixr 6 "@")
  5285   (Haskell infixr 5 "++")
  5286   (Scala infixl 7 "++")
  5287 
  5288 code_const map
  5289   (Haskell "map")
  5290 
  5291 code_const filter
  5292   (Haskell "filter")
  5293 
  5294 code_const concat
  5295   (Haskell "concat")
  5296 
  5297 code_const List.maps
  5298   (Haskell "concatMap")
  5299 
  5300 code_const rev
  5301   (Haskell "reverse")
  5302 
  5303 code_const zip
  5304   (Haskell "zip")
  5305 
  5306 code_const List.null
  5307   (Haskell "null")
  5308 
  5309 code_const takeWhile
  5310   (Haskell "takeWhile")
  5311 
  5312 code_const dropWhile
  5313   (Haskell "dropWhile")
  5314 
  5315 code_const hd
  5316   (Haskell "head")
  5317 
  5318 code_const last
  5319   (Haskell "last")
  5320 
  5321 code_const list_all
  5322   (Haskell "all")
  5323 
  5324 code_const list_ex
  5325   (Haskell "any")
  5326 
  5327 end