src/HOL/List.thy
author wenzelm
Thu Jan 05 18:18:39 2012 +0100 (2012-01-05)
changeset 46125 00cd193a48dc
parent 46034 773c0c4994df
child 46133 d9fe85d3d2cd
permissions -rw-r--r--
improved case syntax: more careful treatment of position constraints, which enables PIDE markup;
tuned;
     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"} $ 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 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 nth_tl:
  1358   assumes "n < length (tl x)" shows "tl x ! n = x ! Suc n"
  1359 using assms by (induct x) auto
  1360 
  1361 lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
  1362 by(cases xs) simp_all
  1363 
  1364 
  1365 lemma list_eq_iff_nth_eq:
  1366  "(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
  1367 apply(induct xs arbitrary: ys)
  1368  apply force
  1369 apply(case_tac ys)
  1370  apply simp
  1371 apply(simp add:nth_Cons split:nat.split)apply blast
  1372 done
  1373 
  1374 lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
  1375 apply (induct xs, simp, simp)
  1376 apply safe
  1377 apply (metis nat_case_0 nth.simps zero_less_Suc)
  1378 apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
  1379 apply (case_tac i, simp)
  1380 apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff)
  1381 done
  1382 
  1383 lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
  1384 by(auto simp:set_conv_nth)
  1385 
  1386 lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
  1387 by (auto simp add: set_conv_nth)
  1388 
  1389 lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
  1390 by (auto simp add: set_conv_nth)
  1391 
  1392 lemma all_nth_imp_all_set:
  1393 "[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
  1394 by (auto simp add: set_conv_nth)
  1395 
  1396 lemma all_set_conv_all_nth:
  1397 "(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
  1398 by (auto simp add: set_conv_nth)
  1399 
  1400 lemma rev_nth:
  1401   "n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)"
  1402 proof (induct xs arbitrary: n)
  1403   case Nil thus ?case by simp
  1404 next
  1405   case (Cons x xs)
  1406   hence n: "n < Suc (length xs)" by simp
  1407   moreover
  1408   { assume "n < length xs"
  1409     with n obtain n' where "length xs - n = Suc n'"
  1410       by (cases "length xs - n", auto)
  1411     moreover
  1412     then have "length xs - Suc n = n'" by simp
  1413     ultimately
  1414     have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp
  1415   }
  1416   ultimately
  1417   show ?case by (clarsimp simp add: Cons nth_append)
  1418 qed
  1419 
  1420 lemma Skolem_list_nth:
  1421   "(ALL i<k. EX x. P i x) = (EX xs. size xs = k & (ALL i<k. P i (xs!i)))"
  1422   (is "_ = (EX xs. ?P k xs)")
  1423 proof(induct k)
  1424   case 0 show ?case by simp
  1425 next
  1426   case (Suc k)
  1427   show ?case (is "?L = ?R" is "_ = (EX xs. ?P' xs)")
  1428   proof
  1429     assume "?R" thus "?L" using Suc by auto
  1430   next
  1431     assume "?L"
  1432     with Suc obtain x xs where "?P k xs & P k x" by (metis less_Suc_eq)
  1433     hence "?P'(xs@[x])" by(simp add:nth_append less_Suc_eq)
  1434     thus "?R" ..
  1435   qed
  1436 qed
  1437 
  1438 
  1439 subsubsection {* @{text list_update} *}
  1440 
  1441 lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
  1442 by (induct xs arbitrary: i) (auto split: nat.split)
  1443 
  1444 lemma nth_list_update:
  1445 "i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
  1446 by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
  1447 
  1448 lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
  1449 by (simp add: nth_list_update)
  1450 
  1451 lemma nth_list_update_neq [simp]: "i \<noteq> j ==> xs[i:=x]!j = xs!j"
  1452 by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
  1453 
  1454 lemma list_update_id[simp]: "xs[i := xs!i] = xs"
  1455 by (induct xs arbitrary: i) (simp_all split:nat.splits)
  1456 
  1457 lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
  1458 apply (induct xs arbitrary: i)
  1459  apply simp
  1460 apply (case_tac i)
  1461 apply simp_all
  1462 done
  1463 
  1464 lemma list_update_nonempty[simp]: "xs[k:=x] = [] \<longleftrightarrow> xs=[]"
  1465 by(metis length_0_conv length_list_update)
  1466 
  1467 lemma list_update_same_conv:
  1468 "i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
  1469 by (induct xs arbitrary: i) (auto split: nat.split)
  1470 
  1471 lemma list_update_append1:
  1472  "i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
  1473 apply (induct xs arbitrary: i, simp)
  1474 apply(simp split:nat.split)
  1475 done
  1476 
  1477 lemma list_update_append:
  1478   "(xs @ ys) [n:= x] = 
  1479   (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
  1480 by (induct xs arbitrary: n) (auto split:nat.splits)
  1481 
  1482 lemma list_update_length [simp]:
  1483  "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
  1484 by (induct xs, auto)
  1485 
  1486 lemma map_update: "map f (xs[k:= y]) = (map f xs)[k := f y]"
  1487 by(induct xs arbitrary: k)(auto split:nat.splits)
  1488 
  1489 lemma rev_update:
  1490   "k < length xs \<Longrightarrow> rev (xs[k:= y]) = (rev xs)[length xs - k - 1 := y]"
  1491 by (induct xs arbitrary: k) (auto simp: list_update_append split:nat.splits)
  1492 
  1493 lemma update_zip:
  1494   "(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
  1495 by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split)
  1496 
  1497 lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)"
  1498 by (induct xs arbitrary: i) (auto split: nat.split)
  1499 
  1500 lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
  1501 by (blast dest!: set_update_subset_insert [THEN subsetD])
  1502 
  1503 lemma set_update_memI: "n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
  1504 by (induct xs arbitrary: n) (auto split:nat.splits)
  1505 
  1506 lemma list_update_overwrite[simp]:
  1507   "xs [i := x, i := y] = xs [i := y]"
  1508 apply (induct xs arbitrary: i) apply simp
  1509 apply (case_tac i, simp_all)
  1510 done
  1511 
  1512 lemma list_update_swap:
  1513   "i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]"
  1514 apply (induct xs arbitrary: i i')
  1515 apply simp
  1516 apply (case_tac i, case_tac i')
  1517 apply auto
  1518 apply (case_tac i')
  1519 apply auto
  1520 done
  1521 
  1522 lemma list_update_code [code]:
  1523   "[][i := y] = []"
  1524   "(x # xs)[0 := y] = y # xs"
  1525   "(x # xs)[Suc i := y] = x # xs[i := y]"
  1526   by simp_all
  1527 
  1528 
  1529 subsubsection {* @{text last} and @{text butlast} *}
  1530 
  1531 lemma last_snoc [simp]: "last (xs @ [x]) = x"
  1532 by (induct xs) auto
  1533 
  1534 lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
  1535 by (induct xs) auto
  1536 
  1537 lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
  1538   by simp
  1539 
  1540 lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
  1541   by simp
  1542 
  1543 lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
  1544 by (induct xs) (auto)
  1545 
  1546 lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
  1547 by(simp add:last_append)
  1548 
  1549 lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
  1550 by(simp add:last_append)
  1551 
  1552 lemma last_tl: "xs = [] \<or> tl xs \<noteq> [] \<Longrightarrow>last (tl xs) = last xs"
  1553 by (induct xs) simp_all
  1554 
  1555 lemma butlast_tl: "butlast (tl xs) = tl (butlast xs)"
  1556 by (induct xs) simp_all
  1557 
  1558 lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
  1559 by(rule rev_exhaust[of xs]) simp_all
  1560 
  1561 lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
  1562 by(cases xs) simp_all
  1563 
  1564 lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
  1565 by (induct as) auto
  1566 
  1567 lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
  1568 by (induct xs rule: rev_induct) auto
  1569 
  1570 lemma butlast_append:
  1571   "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
  1572 by (induct xs arbitrary: ys) auto
  1573 
  1574 lemma append_butlast_last_id [simp]:
  1575 "xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
  1576 by (induct xs) auto
  1577 
  1578 lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
  1579 by (induct xs) (auto split: split_if_asm)
  1580 
  1581 lemma in_set_butlast_appendI:
  1582 "x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
  1583 by (auto dest: in_set_butlastD simp add: butlast_append)
  1584 
  1585 lemma last_drop[simp]: "n < length xs \<Longrightarrow> last (drop n xs) = last xs"
  1586 apply (induct xs arbitrary: n)
  1587  apply simp
  1588 apply (auto split:nat.split)
  1589 done
  1590 
  1591 lemma nth_butlast:
  1592   assumes "n < length (butlast xs)" shows "butlast xs ! n = xs ! n"
  1593 proof (cases xs)
  1594   case (Cons y ys)
  1595   moreover from assms have "butlast xs ! n = (butlast xs @ [last xs]) ! n"
  1596     by (simp add: nth_append)
  1597   ultimately show ?thesis using append_butlast_last_id by simp
  1598 qed simp
  1599 
  1600 lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
  1601 by(induct xs)(auto simp:neq_Nil_conv)
  1602 
  1603 lemma butlast_conv_take: "butlast xs = take (length xs - 1) xs"
  1604 by (induct xs, simp, case_tac xs, simp_all)
  1605 
  1606 lemma last_list_update:
  1607   "xs \<noteq> [] \<Longrightarrow> last(xs[k:=x]) = (if k = size xs - 1 then x else last xs)"
  1608 by (auto simp: last_conv_nth)
  1609 
  1610 lemma butlast_list_update:
  1611   "butlast(xs[k:=x]) =
  1612  (if k = size xs - 1 then butlast xs else (butlast xs)[k:=x])"
  1613 apply(cases xs rule:rev_cases)
  1614 apply simp
  1615 apply(simp add:list_update_append split:nat.splits)
  1616 done
  1617 
  1618 lemma last_map:
  1619   "xs \<noteq> [] \<Longrightarrow> last (map f xs) = f (last xs)"
  1620   by (cases xs rule: rev_cases) simp_all
  1621 
  1622 lemma map_butlast:
  1623   "map f (butlast xs) = butlast (map f xs)"
  1624   by (induct xs) simp_all
  1625 
  1626 lemma snoc_eq_iff_butlast:
  1627   "xs @ [x] = ys \<longleftrightarrow> (ys \<noteq> [] & butlast ys = xs & last ys = x)"
  1628 by (metis append_butlast_last_id append_is_Nil_conv butlast_snoc last_snoc not_Cons_self)
  1629 
  1630 
  1631 subsubsection {* @{text take} and @{text drop} *}
  1632 
  1633 lemma take_0 [simp]: "take 0 xs = []"
  1634 by (induct xs) auto
  1635 
  1636 lemma drop_0 [simp]: "drop 0 xs = xs"
  1637 by (induct xs) auto
  1638 
  1639 lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
  1640 by simp
  1641 
  1642 lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
  1643 by simp
  1644 
  1645 declare take_Cons [simp del] and drop_Cons [simp del]
  1646 
  1647 lemma take_1_Cons [simp]: "take 1 (x # xs) = [x]"
  1648   unfolding One_nat_def by simp
  1649 
  1650 lemma drop_1_Cons [simp]: "drop 1 (x # xs) = xs"
  1651   unfolding One_nat_def by simp
  1652 
  1653 lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
  1654 by(clarsimp simp add:neq_Nil_conv)
  1655 
  1656 lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
  1657 by(cases xs, simp_all)
  1658 
  1659 lemma take_tl: "take n (tl xs) = tl (take (Suc n) xs)"
  1660 by (induct xs arbitrary: n) simp_all
  1661 
  1662 lemma drop_tl: "drop n (tl xs) = tl(drop n xs)"
  1663 by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split)
  1664 
  1665 lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)"
  1666 by (cases n, simp, cases xs, auto)
  1667 
  1668 lemma tl_drop: "tl (drop n xs) = drop n (tl xs)"
  1669 by (simp only: drop_tl)
  1670 
  1671 lemma nth_via_drop: "drop n xs = y#ys \<Longrightarrow> xs!n = y"
  1672 apply (induct xs arbitrary: n, simp)
  1673 apply(simp add:drop_Cons nth_Cons split:nat.splits)
  1674 done
  1675 
  1676 lemma take_Suc_conv_app_nth:
  1677   "i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
  1678 apply (induct xs arbitrary: i, simp)
  1679 apply (case_tac i, auto)
  1680 done
  1681 
  1682 lemma drop_Suc_conv_tl:
  1683   "i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
  1684 apply (induct xs arbitrary: i, simp)
  1685 apply (case_tac i, auto)
  1686 done
  1687 
  1688 lemma length_take [simp]: "length (take n xs) = min (length xs) n"
  1689 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1690 
  1691 lemma length_drop [simp]: "length (drop n xs) = (length xs - n)"
  1692 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1693 
  1694 lemma take_all [simp]: "length xs <= n ==> take n xs = xs"
  1695 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1696 
  1697 lemma drop_all [simp]: "length xs <= n ==> drop n xs = []"
  1698 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1699 
  1700 lemma take_append [simp]:
  1701   "take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
  1702 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1703 
  1704 lemma drop_append [simp]:
  1705   "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
  1706 by (induct n arbitrary: xs) (auto, case_tac xs, auto)
  1707 
  1708 lemma take_take [simp]: "take n (take m xs) = take (min n m) xs"
  1709 apply (induct m arbitrary: xs n, auto)
  1710 apply (case_tac xs, auto)
  1711 apply (case_tac n, auto)
  1712 done
  1713 
  1714 lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs"
  1715 apply (induct m arbitrary: xs, auto)
  1716 apply (case_tac xs, auto)
  1717 done
  1718 
  1719 lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)"
  1720 apply (induct m arbitrary: xs n, auto)
  1721 apply (case_tac xs, auto)
  1722 done
  1723 
  1724 lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)"
  1725 apply(induct xs arbitrary: m n)
  1726  apply simp
  1727 apply(simp add: take_Cons drop_Cons split:nat.split)
  1728 done
  1729 
  1730 lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs"
  1731 apply (induct n arbitrary: xs, auto)
  1732 apply (case_tac xs, auto)
  1733 done
  1734 
  1735 lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])"
  1736 apply(induct xs arbitrary: n)
  1737  apply simp
  1738 apply(simp add:take_Cons split:nat.split)
  1739 done
  1740 
  1741 lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)"
  1742 apply(induct xs arbitrary: n)
  1743 apply simp
  1744 apply(simp add:drop_Cons split:nat.split)
  1745 done
  1746 
  1747 lemma take_map: "take n (map f xs) = map f (take n xs)"
  1748 apply (induct n arbitrary: xs, auto)
  1749 apply (case_tac xs, auto)
  1750 done
  1751 
  1752 lemma drop_map: "drop n (map f xs) = map f (drop n xs)"
  1753 apply (induct n arbitrary: xs, auto)
  1754 apply (case_tac xs, auto)
  1755 done
  1756 
  1757 lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)"
  1758 apply (induct xs arbitrary: i, auto)
  1759 apply (case_tac i, auto)
  1760 done
  1761 
  1762 lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)"
  1763 apply (induct xs arbitrary: i, auto)
  1764 apply (case_tac i, auto)
  1765 done
  1766 
  1767 lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i"
  1768 apply (induct xs arbitrary: i n, auto)
  1769 apply (case_tac n, blast)
  1770 apply (case_tac i, auto)
  1771 done
  1772 
  1773 lemma nth_drop [simp]:
  1774   "n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
  1775 apply (induct n arbitrary: xs i, auto)
  1776 apply (case_tac xs, auto)
  1777 done
  1778 
  1779 lemma butlast_take:
  1780   "n <= length xs ==> butlast (take n xs) = take (n - 1) xs"
  1781 by (simp add: butlast_conv_take min_max.inf_absorb1 min_max.inf_absorb2)
  1782 
  1783 lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)"
  1784 by (simp add: butlast_conv_take drop_take add_ac)
  1785 
  1786 lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs"
  1787 by (simp add: butlast_conv_take min_max.inf_absorb1)
  1788 
  1789 lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)"
  1790 by (simp add: butlast_conv_take drop_take add_ac)
  1791 
  1792 lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
  1793 by(simp add: hd_conv_nth)
  1794 
  1795 lemma set_take_subset_set_take:
  1796   "m <= n \<Longrightarrow> set(take m xs) <= set(take n xs)"
  1797 apply (induct xs arbitrary: m n)
  1798 apply simp
  1799 apply (case_tac n)
  1800 apply (auto simp: take_Cons)
  1801 done
  1802 
  1803 lemma set_take_subset: "set(take n xs) \<subseteq> set xs"
  1804 by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split)
  1805 
  1806 lemma set_drop_subset: "set(drop n xs) \<subseteq> set xs"
  1807 by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split)
  1808 
  1809 lemma set_drop_subset_set_drop:
  1810   "m >= n \<Longrightarrow> set(drop m xs) <= set(drop n xs)"
  1811 apply(induct xs arbitrary: m n)
  1812 apply(auto simp:drop_Cons split:nat.split)
  1813 apply (metis set_drop_subset subset_iff)
  1814 done
  1815 
  1816 lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
  1817 using set_take_subset by fast
  1818 
  1819 lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
  1820 using set_drop_subset by fast
  1821 
  1822 lemma append_eq_conv_conj:
  1823   "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
  1824 apply (induct xs arbitrary: zs, simp, clarsimp)
  1825 apply (case_tac zs, auto)
  1826 done
  1827 
  1828 lemma take_add: 
  1829   "take (i+j) xs = take i xs @ take j (drop i xs)"
  1830 apply (induct xs arbitrary: i, auto) 
  1831 apply (case_tac i, simp_all)
  1832 done
  1833 
  1834 lemma append_eq_append_conv_if:
  1835  "(xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
  1836   (if size xs\<^isub>1 \<le> size ys\<^isub>1
  1837    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
  1838    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)"
  1839 apply(induct xs\<^isub>1 arbitrary: ys\<^isub>1)
  1840  apply simp
  1841 apply(case_tac ys\<^isub>1)
  1842 apply simp_all
  1843 done
  1844 
  1845 lemma take_hd_drop:
  1846   "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (Suc n) xs"
  1847 apply(induct xs arbitrary: n)
  1848 apply simp
  1849 apply(simp add:drop_Cons split:nat.split)
  1850 done
  1851 
  1852 lemma id_take_nth_drop:
  1853  "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
  1854 proof -
  1855   assume si: "i < length xs"
  1856   hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
  1857   moreover
  1858   from si have "take (Suc i) xs = take i xs @ [xs!i]"
  1859     apply (rule_tac take_Suc_conv_app_nth) by arith
  1860   ultimately show ?thesis by auto
  1861 qed
  1862   
  1863 lemma upd_conv_take_nth_drop:
  1864  "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
  1865 proof -
  1866   assume i: "i < length xs"
  1867   have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
  1868     by(rule arg_cong[OF id_take_nth_drop[OF i]])
  1869   also have "\<dots> = take i xs @ a # drop (Suc i) xs"
  1870     using i by (simp add: list_update_append)
  1871   finally show ?thesis .
  1872 qed
  1873 
  1874 lemma nth_drop':
  1875   "i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs"
  1876 apply (induct i arbitrary: xs)
  1877 apply (simp add: neq_Nil_conv)
  1878 apply (erule exE)+
  1879 apply simp
  1880 apply (case_tac xs)
  1881 apply simp_all
  1882 done
  1883 
  1884 
  1885 subsubsection {* @{text takeWhile} and @{text dropWhile} *}
  1886 
  1887 lemma length_takeWhile_le: "length (takeWhile P xs) \<le> length xs"
  1888   by (induct xs) auto
  1889 
  1890 lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
  1891 by (induct xs) auto
  1892 
  1893 lemma takeWhile_append1 [simp]:
  1894 "[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
  1895 by (induct xs) auto
  1896 
  1897 lemma takeWhile_append2 [simp]:
  1898 "(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
  1899 by (induct xs) auto
  1900 
  1901 lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
  1902 by (induct xs) auto
  1903 
  1904 lemma takeWhile_nth: "j < length (takeWhile P xs) \<Longrightarrow> takeWhile P xs ! j = xs ! j"
  1905 apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
  1906 
  1907 lemma dropWhile_nth: "j < length (dropWhile P xs) \<Longrightarrow> dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))"
  1908 apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
  1909 
  1910 lemma length_dropWhile_le: "length (dropWhile P xs) \<le> length xs"
  1911 by (induct xs) auto
  1912 
  1913 lemma dropWhile_append1 [simp]:
  1914 "[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
  1915 by (induct xs) auto
  1916 
  1917 lemma dropWhile_append2 [simp]:
  1918 "(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
  1919 by (induct xs) auto
  1920 
  1921 lemma dropWhile_append3:
  1922   "\<not> P y \<Longrightarrow>dropWhile P (xs @ y # ys) = dropWhile P xs @ y # ys"
  1923 by (induct xs) auto
  1924 
  1925 lemma dropWhile_last:
  1926   "x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> last (dropWhile P xs) = last xs"
  1927 by (auto simp add: dropWhile_append3 in_set_conv_decomp)
  1928 
  1929 lemma set_dropWhileD: "x \<in> set (dropWhile P xs) \<Longrightarrow> x \<in> set xs"
  1930 by (induct xs) (auto split: split_if_asm)
  1931 
  1932 lemma set_takeWhileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
  1933 by (induct xs) (auto split: split_if_asm)
  1934 
  1935 lemma takeWhile_eq_all_conv[simp]:
  1936  "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
  1937 by(induct xs, auto)
  1938 
  1939 lemma dropWhile_eq_Nil_conv[simp]:
  1940  "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
  1941 by(induct xs, auto)
  1942 
  1943 lemma dropWhile_eq_Cons_conv:
  1944  "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
  1945 by(induct xs, auto)
  1946 
  1947 lemma distinct_takeWhile[simp]: "distinct xs ==> distinct (takeWhile P xs)"
  1948 by (induct xs) (auto dest: set_takeWhileD)
  1949 
  1950 lemma distinct_dropWhile[simp]: "distinct xs ==> distinct (dropWhile P xs)"
  1951 by (induct xs) auto
  1952 
  1953 lemma takeWhile_map: "takeWhile P (map f xs) = map f (takeWhile (P \<circ> f) xs)"
  1954 by (induct xs) auto
  1955 
  1956 lemma dropWhile_map: "dropWhile P (map f xs) = map f (dropWhile (P \<circ> f) xs)"
  1957 by (induct xs) auto
  1958 
  1959 lemma takeWhile_eq_take: "takeWhile P xs = take (length (takeWhile P xs)) xs"
  1960 by (induct xs) auto
  1961 
  1962 lemma dropWhile_eq_drop: "dropWhile P xs = drop (length (takeWhile P xs)) xs"
  1963 by (induct xs) auto
  1964 
  1965 lemma hd_dropWhile:
  1966   "dropWhile P xs \<noteq> [] \<Longrightarrow> \<not> P (hd (dropWhile P xs))"
  1967 using assms by (induct xs) auto
  1968 
  1969 lemma takeWhile_eq_filter:
  1970   assumes "\<And> x. x \<in> set (dropWhile P xs) \<Longrightarrow> \<not> P x"
  1971   shows "takeWhile P xs = filter P xs"
  1972 proof -
  1973   have A: "filter P xs = filter P (takeWhile P xs @ dropWhile P xs)"
  1974     by simp
  1975   have B: "filter P (dropWhile P xs) = []"
  1976     unfolding filter_empty_conv using assms by blast
  1977   have "filter P xs = takeWhile P xs"
  1978     unfolding A filter_append B
  1979     by (auto simp add: filter_id_conv dest: set_takeWhileD)
  1980   thus ?thesis ..
  1981 qed
  1982 
  1983 lemma takeWhile_eq_take_P_nth:
  1984   "\<lbrakk> \<And> i. \<lbrakk> i < n ; i < length xs \<rbrakk> \<Longrightarrow> P (xs ! i) ; n < length xs \<Longrightarrow> \<not> P (xs ! n) \<rbrakk> \<Longrightarrow>
  1985   takeWhile P xs = take n xs"
  1986 proof (induct xs arbitrary: n)
  1987   case (Cons x xs)
  1988   thus ?case
  1989   proof (cases n)
  1990     case (Suc n') note this[simp]
  1991     have "P x" using Cons.prems(1)[of 0] by simp
  1992     moreover have "takeWhile P xs = take n' xs"
  1993     proof (rule Cons.hyps)
  1994       case goal1 thus "P (xs ! i)" using Cons.prems(1)[of "Suc i"] by simp
  1995     next case goal2 thus ?case using Cons by auto
  1996     qed
  1997     ultimately show ?thesis by simp
  1998    qed simp
  1999 qed simp
  2000 
  2001 lemma nth_length_takeWhile:
  2002   "length (takeWhile P xs) < length xs \<Longrightarrow> \<not> P (xs ! length (takeWhile P xs))"
  2003 by (induct xs) auto
  2004 
  2005 lemma length_takeWhile_less_P_nth:
  2006   assumes all: "\<And> i. i < j \<Longrightarrow> P (xs ! i)" and "j \<le> length xs"
  2007   shows "j \<le> length (takeWhile P xs)"
  2008 proof (rule classical)
  2009   assume "\<not> ?thesis"
  2010   hence "length (takeWhile P xs) < length xs" using assms by simp
  2011   thus ?thesis using all `\<not> ?thesis` nth_length_takeWhile[of P xs] by auto
  2012 qed
  2013 
  2014 text{* The following two lemmmas could be generalized to an arbitrary
  2015 property. *}
  2016 
  2017 lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
  2018  takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
  2019 by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
  2020 
  2021 lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
  2022   dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
  2023 apply(induct xs)
  2024  apply simp
  2025 apply auto
  2026 apply(subst dropWhile_append2)
  2027 apply auto
  2028 done
  2029 
  2030 lemma takeWhile_not_last:
  2031  "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
  2032 apply(induct xs)
  2033  apply simp
  2034 apply(case_tac xs)
  2035 apply(auto)
  2036 done
  2037 
  2038 lemma takeWhile_cong [fundef_cong]:
  2039   "[| l = k; !!x. x : set l ==> P x = Q x |] 
  2040   ==> takeWhile P l = takeWhile Q k"
  2041 by (induct k arbitrary: l) (simp_all)
  2042 
  2043 lemma dropWhile_cong [fundef_cong]:
  2044   "[| l = k; !!x. x : set l ==> P x = Q x |] 
  2045   ==> dropWhile P l = dropWhile Q k"
  2046 by (induct k arbitrary: l, simp_all)
  2047 
  2048 
  2049 subsubsection {* @{text zip} *}
  2050 
  2051 lemma zip_Nil [simp]: "zip [] ys = []"
  2052 by (induct ys) auto
  2053 
  2054 lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
  2055 by simp
  2056 
  2057 declare zip_Cons [simp del]
  2058 
  2059 lemma [code]:
  2060   "zip [] ys = []"
  2061   "zip xs [] = []"
  2062   "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
  2063   by (fact zip_Nil zip.simps(1) zip_Cons_Cons)+
  2064 
  2065 lemma zip_Cons1:
  2066  "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
  2067 by(auto split:list.split)
  2068 
  2069 lemma length_zip [simp]:
  2070 "length (zip xs ys) = min (length xs) (length ys)"
  2071 by (induct xs ys rule:list_induct2') auto
  2072 
  2073 lemma zip_obtain_same_length:
  2074   assumes "\<And>zs ws n. length zs = length ws \<Longrightarrow> n = min (length xs) (length ys)
  2075     \<Longrightarrow> zs = take n xs \<Longrightarrow> ws = take n ys \<Longrightarrow> P (zip zs ws)"
  2076   shows "P (zip xs ys)"
  2077 proof -
  2078   let ?n = "min (length xs) (length ys)"
  2079   have "P (zip (take ?n xs) (take ?n ys))"
  2080     by (rule assms) simp_all
  2081   moreover have "zip xs ys = zip (take ?n xs) (take ?n ys)"
  2082   proof (induct xs arbitrary: ys)
  2083     case Nil then show ?case by simp
  2084   next
  2085     case (Cons x xs) then show ?case by (cases ys) simp_all
  2086   qed
  2087   ultimately show ?thesis by simp
  2088 qed
  2089 
  2090 lemma zip_append1:
  2091 "zip (xs @ ys) zs =
  2092 zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
  2093 by (induct xs zs rule:list_induct2') auto
  2094 
  2095 lemma zip_append2:
  2096 "zip xs (ys @ zs) =
  2097 zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
  2098 by (induct xs ys rule:list_induct2') auto
  2099 
  2100 lemma zip_append [simp]:
  2101  "[| length xs = length us; length ys = length vs |] ==>
  2102 zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
  2103 by (simp add: zip_append1)
  2104 
  2105 lemma zip_rev:
  2106 "length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
  2107 by (induct rule:list_induct2, simp_all)
  2108 
  2109 lemma zip_map_map:
  2110   "zip (map f xs) (map g ys) = map (\<lambda> (x, y). (f x, g y)) (zip xs ys)"
  2111 proof (induct xs arbitrary: ys)
  2112   case (Cons x xs) note Cons_x_xs = Cons.hyps
  2113   show ?case
  2114   proof (cases ys)
  2115     case (Cons y ys')
  2116     show ?thesis unfolding Cons using Cons_x_xs by simp
  2117   qed simp
  2118 qed simp
  2119 
  2120 lemma zip_map1:
  2121   "zip (map f xs) ys = map (\<lambda>(x, y). (f x, y)) (zip xs ys)"
  2122 using zip_map_map[of f xs "\<lambda>x. x" ys] by simp
  2123 
  2124 lemma zip_map2:
  2125   "zip xs (map f ys) = map (\<lambda>(x, y). (x, f y)) (zip xs ys)"
  2126 using zip_map_map[of "\<lambda>x. x" xs f ys] by simp
  2127 
  2128 lemma map_zip_map:
  2129   "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
  2130 unfolding zip_map1 by auto
  2131 
  2132 lemma map_zip_map2:
  2133   "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
  2134 unfolding zip_map2 by auto
  2135 
  2136 text{* Courtesy of Andreas Lochbihler: *}
  2137 lemma zip_same_conv_map: "zip xs xs = map (\<lambda>x. (x, x)) xs"
  2138 by(induct xs) auto
  2139 
  2140 lemma nth_zip [simp]:
  2141 "[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
  2142 apply (induct ys arbitrary: i xs, simp)
  2143 apply (case_tac xs)
  2144  apply (simp_all add: nth.simps split: nat.split)
  2145 done
  2146 
  2147 lemma set_zip:
  2148 "set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
  2149 by(simp add: set_conv_nth cong: rev_conj_cong)
  2150 
  2151 lemma zip_same: "((a,b) \<in> set (zip xs xs)) = (a \<in> set xs \<and> a = b)"
  2152 by(induct xs) auto
  2153 
  2154 lemma zip_update:
  2155   "zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
  2156 by(rule sym, simp add: update_zip)
  2157 
  2158 lemma zip_replicate [simp]:
  2159   "zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
  2160 apply (induct i arbitrary: j, auto)
  2161 apply (case_tac j, auto)
  2162 done
  2163 
  2164 lemma take_zip:
  2165   "take n (zip xs ys) = zip (take n xs) (take n ys)"
  2166 apply (induct n arbitrary: xs ys)
  2167  apply simp
  2168 apply (case_tac xs, simp)
  2169 apply (case_tac ys, simp_all)
  2170 done
  2171 
  2172 lemma drop_zip:
  2173   "drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
  2174 apply (induct n arbitrary: xs ys)
  2175  apply simp
  2176 apply (case_tac xs, simp)
  2177 apply (case_tac ys, simp_all)
  2178 done
  2179 
  2180 lemma zip_takeWhile_fst: "zip (takeWhile P xs) ys = takeWhile (P \<circ> fst) (zip xs ys)"
  2181 proof (induct xs arbitrary: ys)
  2182   case (Cons x xs) thus ?case by (cases ys) auto
  2183 qed simp
  2184 
  2185 lemma zip_takeWhile_snd: "zip xs (takeWhile P ys) = takeWhile (P \<circ> snd) (zip xs ys)"
  2186 proof (induct xs arbitrary: ys)
  2187   case (Cons x xs) thus ?case by (cases ys) auto
  2188 qed simp
  2189 
  2190 lemma set_zip_leftD:
  2191   "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
  2192 by (induct xs ys rule:list_induct2') auto
  2193 
  2194 lemma set_zip_rightD:
  2195   "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
  2196 by (induct xs ys rule:list_induct2') auto
  2197 
  2198 lemma in_set_zipE:
  2199   "(x,y) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
  2200 by(blast dest: set_zip_leftD set_zip_rightD)
  2201 
  2202 lemma zip_map_fst_snd:
  2203   "zip (map fst zs) (map snd zs) = zs"
  2204   by (induct zs) simp_all
  2205 
  2206 lemma zip_eq_conv:
  2207   "length xs = length ys \<Longrightarrow> zip xs ys = zs \<longleftrightarrow> map fst zs = xs \<and> map snd zs = ys"
  2208   by (auto simp add: zip_map_fst_snd)
  2209 
  2210 
  2211 subsubsection {* @{text list_all2} *}
  2212 
  2213 lemma list_all2_lengthD [intro?]: 
  2214   "list_all2 P xs ys ==> length xs = length ys"
  2215 by (simp add: list_all2_def)
  2216 
  2217 lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
  2218 by (simp add: list_all2_def)
  2219 
  2220 lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
  2221 by (simp add: list_all2_def)
  2222 
  2223 lemma list_all2_Cons [iff, code]:
  2224   "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
  2225 by (auto simp add: list_all2_def)
  2226 
  2227 lemma list_all2_Cons1:
  2228 "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
  2229 by (cases ys) auto
  2230 
  2231 lemma list_all2_Cons2:
  2232 "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
  2233 by (cases xs) auto
  2234 
  2235 lemma list_all2_induct
  2236   [consumes 1, case_names Nil Cons, induct set: list_all2]:
  2237   assumes P: "list_all2 P xs ys"
  2238   assumes Nil: "R [] []"
  2239   assumes Cons: "\<And>x xs y ys. \<lbrakk>P x y; R xs ys\<rbrakk> \<Longrightarrow> R (x # xs) (y # ys)"
  2240   shows "R xs ys"
  2241 using P
  2242 by (induct xs arbitrary: ys) (auto simp add: list_all2_Cons1 Nil Cons)
  2243 
  2244 lemma list_all2_rev [iff]:
  2245 "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
  2246 by (simp add: list_all2_def zip_rev cong: conj_cong)
  2247 
  2248 lemma list_all2_rev1:
  2249 "list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
  2250 by (subst list_all2_rev [symmetric]) simp
  2251 
  2252 lemma list_all2_append1:
  2253 "list_all2 P (xs @ ys) zs =
  2254 (EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
  2255 list_all2 P xs us \<and> list_all2 P ys vs)"
  2256 apply (simp add: list_all2_def zip_append1)
  2257 apply (rule iffI)
  2258  apply (rule_tac x = "take (length xs) zs" in exI)
  2259  apply (rule_tac x = "drop (length xs) zs" in exI)
  2260  apply (force split: nat_diff_split simp add: min_def, clarify)
  2261 apply (simp add: ball_Un)
  2262 done
  2263 
  2264 lemma list_all2_append2:
  2265 "list_all2 P xs (ys @ zs) =
  2266 (EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
  2267 list_all2 P us ys \<and> list_all2 P vs zs)"
  2268 apply (simp add: list_all2_def zip_append2)
  2269 apply (rule iffI)
  2270  apply (rule_tac x = "take (length ys) xs" in exI)
  2271  apply (rule_tac x = "drop (length ys) xs" in exI)
  2272  apply (force split: nat_diff_split simp add: min_def, clarify)
  2273 apply (simp add: ball_Un)
  2274 done
  2275 
  2276 lemma list_all2_append:
  2277   "length xs = length ys \<Longrightarrow>
  2278   list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
  2279 by (induct rule:list_induct2, simp_all)
  2280 
  2281 lemma list_all2_appendI [intro?, trans]:
  2282   "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
  2283 by (simp add: list_all2_append list_all2_lengthD)
  2284 
  2285 lemma list_all2_conv_all_nth:
  2286 "list_all2 P xs ys =
  2287 (length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
  2288 by (force simp add: list_all2_def set_zip)
  2289 
  2290 lemma list_all2_trans:
  2291   assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
  2292   shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
  2293         (is "!!bs cs. PROP ?Q as bs cs")
  2294 proof (induct as)
  2295   fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
  2296   show "!!cs. PROP ?Q (x # xs) bs cs"
  2297   proof (induct bs)
  2298     fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
  2299     show "PROP ?Q (x # xs) (y # ys) cs"
  2300       by (induct cs) (auto intro: tr I1 I2)
  2301   qed simp
  2302 qed simp
  2303 
  2304 lemma list_all2_all_nthI [intro?]:
  2305   "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
  2306 by (simp add: list_all2_conv_all_nth)
  2307 
  2308 lemma list_all2I:
  2309   "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
  2310 by (simp add: list_all2_def)
  2311 
  2312 lemma list_all2_nthD:
  2313   "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
  2314 by (simp add: list_all2_conv_all_nth)
  2315 
  2316 lemma list_all2_nthD2:
  2317   "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
  2318 by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
  2319 
  2320 lemma list_all2_map1: 
  2321   "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
  2322 by (simp add: list_all2_conv_all_nth)
  2323 
  2324 lemma list_all2_map2: 
  2325   "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
  2326 by (auto simp add: list_all2_conv_all_nth)
  2327 
  2328 lemma list_all2_refl [intro?]:
  2329   "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
  2330 by (simp add: list_all2_conv_all_nth)
  2331 
  2332 lemma list_all2_update_cong:
  2333   "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
  2334 by (simp add: list_all2_conv_all_nth nth_list_update)
  2335 
  2336 lemma list_all2_update_cong2:
  2337   "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
  2338 by (simp add: list_all2_lengthD list_all2_update_cong)
  2339 
  2340 lemma list_all2_takeI [simp,intro?]:
  2341   "list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
  2342 apply (induct xs arbitrary: n ys)
  2343  apply simp
  2344 apply (clarsimp simp add: list_all2_Cons1)
  2345 apply (case_tac n)
  2346 apply auto
  2347 done
  2348 
  2349 lemma list_all2_dropI [simp,intro?]:
  2350   "list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
  2351 apply (induct as arbitrary: n bs, simp)
  2352 apply (clarsimp simp add: list_all2_Cons1)
  2353 apply (case_tac n, simp, simp)
  2354 done
  2355 
  2356 lemma list_all2_mono [intro?]:
  2357   "list_all2 P xs ys \<Longrightarrow> (\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys) \<Longrightarrow> list_all2 Q xs ys"
  2358 apply (induct xs arbitrary: ys, simp)
  2359 apply (case_tac ys, auto)
  2360 done
  2361 
  2362 lemma list_all2_eq:
  2363   "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
  2364 by (induct xs ys rule: list_induct2') auto
  2365 
  2366 lemma list_eq_iff_zip_eq:
  2367   "xs = ys \<longleftrightarrow> length xs = length ys \<and> (\<forall>(x,y) \<in> set (zip xs ys). x = y)"
  2368 by(auto simp add: set_zip list_all2_eq list_all2_conv_all_nth cong: conj_cong)
  2369 
  2370 
  2371 subsubsection {* @{text foldl} and @{text foldr} *}
  2372 
  2373 lemma foldl_append [simp]:
  2374   "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
  2375 by (induct xs arbitrary: a) auto
  2376 
  2377 lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
  2378 by (induct xs) auto
  2379 
  2380 lemma foldr_map: "foldr g (map f xs) a = foldr (g o f) xs a"
  2381 by(induct xs) simp_all
  2382 
  2383 text{* For efficient code generation: avoid intermediate list. *}
  2384 lemma foldl_map[code_unfold]:
  2385   "foldl g a (map f xs) = foldl (%a x. g a (f x)) a xs"
  2386 by(induct xs arbitrary:a) simp_all
  2387 
  2388 lemma foldl_apply:
  2389   assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x \<circ> h = h \<circ> g x"
  2390   shows "foldl (\<lambda>s x. f x s) (h s) xs = h (foldl (\<lambda>s x. g x s) s xs)"
  2391   by (rule sym, insert assms, induct xs arbitrary: s) (simp_all add: fun_eq_iff)
  2392 
  2393 lemma foldl_cong [fundef_cong]:
  2394   "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] 
  2395   ==> foldl f a l = foldl g b k"
  2396 by (induct k arbitrary: a b l) simp_all
  2397 
  2398 lemma foldr_cong [fundef_cong]:
  2399   "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] 
  2400   ==> foldr f l a = foldr g k b"
  2401 by (induct k arbitrary: a b l) simp_all
  2402 
  2403 lemma foldl_fun_comm:
  2404   assumes "\<And>x y s. f (f s x) y = f (f s y) x"
  2405   shows "f (foldl f s xs) x = foldl f (f s x) xs"
  2406   by (induct xs arbitrary: s)
  2407     (simp_all add: assms)
  2408 
  2409 lemma (in semigroup_add) foldl_assoc:
  2410 shows "foldl op+ (x+y) zs = x + (foldl op+ y zs)"
  2411 by (induct zs arbitrary: y) (simp_all add:add_assoc)
  2412 
  2413 lemma (in monoid_add) foldl_absorb0:
  2414 shows "x + (foldl op+ 0 zs) = foldl op+ x zs"
  2415 by (induct zs) (simp_all add:foldl_assoc)
  2416 
  2417 lemma foldl_rev:
  2418   assumes "\<And>x y s. f (f s x) y = f (f s y) x"
  2419   shows "foldl f s (rev xs) = foldl f s xs"
  2420 proof (induct xs arbitrary: s)
  2421   case Nil then show ?case by simp
  2422 next
  2423   case (Cons x xs) with assms show ?case by (simp add: foldl_fun_comm)
  2424 qed
  2425 
  2426 lemma rev_foldl_cons [code]:
  2427   "rev xs = foldl (\<lambda>xs x. x # xs) [] xs"
  2428 proof (induct xs)
  2429   case Nil then show ?case by simp
  2430 next
  2431   case Cons
  2432   {
  2433     fix x xs ys
  2434     have "foldl (\<lambda>xs x. x # xs) ys xs @ [x]
  2435       = foldl (\<lambda>xs x. x # xs) (ys @ [x]) xs"
  2436     by (induct xs arbitrary: ys) auto
  2437   }
  2438   note aux = this
  2439   show ?case by (induct xs) (auto simp add: Cons aux)
  2440 qed
  2441 
  2442 
  2443 text{* The ``Third Duality Theorem'' in Bird \& Wadler: *}
  2444 
  2445 lemma foldr_foldl:
  2446   "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
  2447   by (induct xs) auto
  2448 
  2449 lemma foldl_foldr:
  2450   "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
  2451   by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
  2452 
  2453 
  2454 text{* The ``First Duality Theorem'' in Bird \& Wadler: *}
  2455 
  2456 lemma (in monoid_add) foldl_foldr1_lemma:
  2457   "foldl op + a xs = a + foldr op + xs 0"
  2458   by (induct xs arbitrary: a) (auto simp: add_assoc)
  2459 
  2460 corollary (in monoid_add) foldl_foldr1:
  2461   "foldl op + 0 xs = foldr op + xs 0"
  2462   by (simp add: foldl_foldr1_lemma)
  2463 
  2464 lemma (in ab_semigroup_add) foldr_conv_foldl:
  2465   "foldr op + xs a = foldl op + a xs"
  2466   by (induct xs) (simp_all add: foldl_assoc add.commute)
  2467 
  2468 text {*
  2469 Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
  2470 difficult to use because it requires an additional transitivity step.
  2471 *}
  2472 
  2473 lemma start_le_sum: "(m::nat) <= n ==> m <= foldl (op +) n ns"
  2474 by (induct ns arbitrary: n) auto
  2475 
  2476 lemma elem_le_sum: "(n::nat) : set ns ==> n <= foldl (op +) 0 ns"
  2477 by (force intro: start_le_sum simp add: in_set_conv_decomp)
  2478 
  2479 lemma sum_eq_0_conv [iff]:
  2480   "(foldl (op +) (m::nat) ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
  2481 by (induct ns arbitrary: m) auto
  2482 
  2483 lemma foldr_invariant: 
  2484   "\<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)"
  2485   by (induct xs, simp_all)
  2486 
  2487 lemma foldl_invariant: 
  2488   "\<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)"
  2489   by (induct xs arbitrary: x, simp_all)
  2490 
  2491 lemma foldl_weak_invariant:
  2492   assumes "P s"
  2493     and "\<And>s x. x \<in> set xs \<Longrightarrow> P s \<Longrightarrow> P (f s x)"
  2494   shows "P (foldl f s xs)"
  2495   using assms by (induct xs arbitrary: s) simp_all
  2496 
  2497 text {* @{const foldl} and @{const concat} *}
  2498 
  2499 lemma foldl_conv_concat:
  2500   "foldl (op @) xs xss = xs @ concat xss"
  2501 proof (induct xss arbitrary: xs)
  2502   case Nil show ?case by simp
  2503 next
  2504   interpret monoid_add "op @" "[]" proof qed simp_all
  2505   case Cons then show ?case by (simp add: foldl_absorb0)
  2506 qed
  2507 
  2508 lemma concat_conv_foldl: "concat xss = foldl (op @) [] xss"
  2509   by (simp add: foldl_conv_concat)
  2510 
  2511 text {* @{const Finite_Set.fold} and @{const foldl} *}
  2512 
  2513 lemma (in comp_fun_commute) fold_set_remdups:
  2514   "Finite_Set.fold f y (set xs) = foldl (\<lambda>y x. f x y) y (remdups xs)"
  2515   by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm insert_absorb)
  2516 
  2517 lemma (in comp_fun_idem) fold_set:
  2518   "Finite_Set.fold f y (set xs) = foldl (\<lambda>y x. f x y) y xs"
  2519   by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm)
  2520 
  2521 lemma (in ab_semigroup_idem_mult) fold1_set:
  2522   assumes "xs \<noteq> []"
  2523   shows "fold1 times (set xs) = foldl times (hd xs) (tl xs)"
  2524 proof -
  2525   interpret comp_fun_idem times by (fact comp_fun_idem)
  2526   from assms obtain y ys where xs: "xs = y # ys"
  2527     by (cases xs) auto
  2528   show ?thesis
  2529   proof (cases "set ys = {}")
  2530     case True with xs show ?thesis by simp
  2531   next
  2532     case False
  2533     then have "fold1 times (insert y (set ys)) = Finite_Set.fold times y (set ys)"
  2534       by (simp only: finite_set fold1_eq_fold_idem)
  2535     with xs show ?thesis by (simp add: fold_set mult_commute)
  2536   qed
  2537 qed
  2538 
  2539 
  2540 subsubsection {* @{text upt} *}
  2541 
  2542 lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
  2543 -- {* simp does not terminate! *}
  2544 by (induct j) auto
  2545 
  2546 lemmas upt_rec_number_of[simp] = upt_rec[of "number_of m" "number_of n"] for m n
  2547 
  2548 lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
  2549 by (subst upt_rec) simp
  2550 
  2551 lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
  2552 by(induct j)simp_all
  2553 
  2554 lemma upt_eq_Cons_conv:
  2555  "([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
  2556 apply(induct j arbitrary: x xs)
  2557  apply simp
  2558 apply(clarsimp simp add: append_eq_Cons_conv)
  2559 apply arith
  2560 done
  2561 
  2562 lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
  2563 -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
  2564 by simp
  2565 
  2566 lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
  2567   by (simp add: upt_rec)
  2568 
  2569 lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
  2570 -- {* LOOPS as a simprule, since @{text "j <= j"}. *}
  2571 by (induct k) auto
  2572 
  2573 lemma length_upt [simp]: "length [i..<j] = j - i"
  2574 by (induct j) (auto simp add: Suc_diff_le)
  2575 
  2576 lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
  2577 apply (induct j)
  2578 apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
  2579 done
  2580 
  2581 
  2582 lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
  2583 by(simp add:upt_conv_Cons)
  2584 
  2585 lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
  2586 apply(cases j)
  2587  apply simp
  2588 by(simp add:upt_Suc_append)
  2589 
  2590 lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]"
  2591 apply (induct m arbitrary: i, simp)
  2592 apply (subst upt_rec)
  2593 apply (rule sym)
  2594 apply (subst upt_rec)
  2595 apply (simp del: upt.simps)
  2596 done
  2597 
  2598 lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
  2599 apply(induct j)
  2600 apply auto
  2601 done
  2602 
  2603 lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]"
  2604 by (induct n) auto
  2605 
  2606 lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
  2607 apply (induct n m  arbitrary: i rule: diff_induct)
  2608 prefer 3 apply (subst map_Suc_upt[symmetric])
  2609 apply (auto simp add: less_diff_conv)
  2610 done
  2611 
  2612 lemma nth_take_lemma:
  2613   "k <= length xs ==> k <= length ys ==>
  2614      (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
  2615 apply (atomize, induct k arbitrary: xs ys)
  2616 apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
  2617 txt {* Both lists must be non-empty *}
  2618 apply (case_tac xs, simp)
  2619 apply (case_tac ys, clarify)
  2620  apply (simp (no_asm_use))
  2621 apply clarify
  2622 txt {* prenexing's needed, not miniscoping *}
  2623 apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
  2624 apply blast
  2625 done
  2626 
  2627 lemma nth_equalityI:
  2628  "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
  2629   by (frule nth_take_lemma [OF le_refl eq_imp_le]) simp_all
  2630 
  2631 lemma map_nth:
  2632   "map (\<lambda>i. xs ! i) [0..<length xs] = xs"
  2633   by (rule nth_equalityI, auto)
  2634 
  2635 (* needs nth_equalityI *)
  2636 lemma list_all2_antisym:
  2637   "\<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> 
  2638   \<Longrightarrow> xs = ys"
  2639   apply (simp add: list_all2_conv_all_nth) 
  2640   apply (rule nth_equalityI, blast, simp)
  2641   done
  2642 
  2643 lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
  2644 -- {* The famous take-lemma. *}
  2645 apply (drule_tac x = "max (length xs) (length ys)" in spec)
  2646 apply (simp add: le_max_iff_disj)
  2647 done
  2648 
  2649 
  2650 lemma take_Cons':
  2651      "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
  2652 by (cases n) simp_all
  2653 
  2654 lemma drop_Cons':
  2655      "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
  2656 by (cases n) simp_all
  2657 
  2658 lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
  2659 by (cases n) simp_all
  2660 
  2661 lemmas take_Cons_number_of = take_Cons'[of "number_of v"] for v
  2662 lemmas drop_Cons_number_of = drop_Cons'[of "number_of v"] for v
  2663 lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v"] for v
  2664 
  2665 declare take_Cons_number_of [simp] 
  2666         drop_Cons_number_of [simp] 
  2667         nth_Cons_number_of [simp] 
  2668 
  2669 
  2670 subsubsection {* @{text upto}: interval-list on @{typ int} *}
  2671 
  2672 (* FIXME make upto tail recursive? *)
  2673 
  2674 function upto :: "int \<Rightarrow> int \<Rightarrow> int list" ("(1[_../_])") where
  2675 "upto i j = (if i \<le> j then i # [i+1..j] else [])"
  2676 by auto
  2677 termination
  2678 by(relation "measure(%(i::int,j). nat(j - i + 1))") auto
  2679 
  2680 declare upto.simps[code, simp del]
  2681 
  2682 lemmas upto_rec_number_of[simp] = upto.simps[of "number_of m" "number_of n"] for m n
  2683 
  2684 lemma upto_empty[simp]: "j < i \<Longrightarrow> [i..j] = []"
  2685 by(simp add: upto.simps)
  2686 
  2687 lemma set_upto[simp]: "set[i..j] = {i..j}"
  2688 proof(induct i j rule:upto.induct)
  2689   case (1 i j)
  2690   from this show ?case
  2691     unfolding upto.simps[of i j] simp_from_to[of i j] by auto
  2692 qed
  2693 
  2694 
  2695 subsubsection {* @{text "distinct"} and @{text remdups} *}
  2696 
  2697 lemma distinct_tl:
  2698   "distinct xs \<Longrightarrow> distinct (tl xs)"
  2699   by (cases xs) simp_all
  2700 
  2701 lemma distinct_append [simp]:
  2702 "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
  2703 by (induct xs) auto
  2704 
  2705 lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
  2706 by(induct xs) auto
  2707 
  2708 lemma set_remdups [simp]: "set (remdups xs) = set xs"
  2709 by (induct xs) (auto simp add: insert_absorb)
  2710 
  2711 lemma distinct_remdups [iff]: "distinct (remdups xs)"
  2712 by (induct xs) auto
  2713 
  2714 lemma distinct_remdups_id: "distinct xs ==> remdups xs = xs"
  2715 by (induct xs, auto)
  2716 
  2717 lemma remdups_id_iff_distinct [simp]: "remdups xs = xs \<longleftrightarrow> distinct xs"
  2718 by (metis distinct_remdups distinct_remdups_id)
  2719 
  2720 lemma finite_distinct_list: "finite A \<Longrightarrow> EX xs. set xs = A & distinct xs"
  2721 by (metis distinct_remdups finite_list set_remdups)
  2722 
  2723 lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
  2724 by (induct x, auto) 
  2725 
  2726 lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
  2727 by (induct x, auto)
  2728 
  2729 lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
  2730 by (induct xs) auto
  2731 
  2732 lemma length_remdups_eq[iff]:
  2733   "(length (remdups xs) = length xs) = (remdups xs = xs)"
  2734 apply(induct xs)
  2735  apply auto
  2736 apply(subgoal_tac "length (remdups xs) <= length xs")
  2737  apply arith
  2738 apply(rule length_remdups_leq)
  2739 done
  2740 
  2741 lemma remdups_filter: "remdups(filter P xs) = filter P (remdups xs)"
  2742 apply(induct xs)
  2743 apply auto
  2744 done
  2745 
  2746 lemma distinct_map:
  2747   "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
  2748 by (induct xs) auto
  2749 
  2750 
  2751 lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
  2752 by (induct xs) auto
  2753 
  2754 lemma distinct_upt[simp]: "distinct[i..<j]"
  2755 by (induct j) auto
  2756 
  2757 lemma distinct_upto[simp]: "distinct[i..j]"
  2758 apply(induct i j rule:upto.induct)
  2759 apply(subst upto.simps)
  2760 apply(simp)
  2761 done
  2762 
  2763 lemma distinct_take[simp]: "distinct xs \<Longrightarrow> distinct (take i xs)"
  2764 apply(induct xs arbitrary: i)
  2765  apply simp
  2766 apply (case_tac i)
  2767  apply simp_all
  2768 apply(blast dest:in_set_takeD)
  2769 done
  2770 
  2771 lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)"
  2772 apply(induct xs arbitrary: i)
  2773  apply simp
  2774 apply (case_tac i)
  2775  apply simp_all
  2776 done
  2777 
  2778 lemma distinct_list_update:
  2779 assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
  2780 shows "distinct (xs[i:=a])"
  2781 proof (cases "i < length xs")
  2782   case True
  2783   with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
  2784     apply (drule_tac id_take_nth_drop) by simp
  2785   with d True show ?thesis
  2786     apply (simp add: upd_conv_take_nth_drop)
  2787     apply (drule subst [OF id_take_nth_drop]) apply assumption
  2788     apply simp apply (cases "a = xs!i") apply simp by blast
  2789 next
  2790   case False with d show ?thesis by auto
  2791 qed
  2792 
  2793 lemma distinct_concat:
  2794   assumes "distinct xs"
  2795   and "\<And> ys. ys \<in> set xs \<Longrightarrow> distinct ys"
  2796   and "\<And> ys zs. \<lbrakk> ys \<in> set xs ; zs \<in> set xs ; ys \<noteq> zs \<rbrakk> \<Longrightarrow> set ys \<inter> set zs = {}"
  2797   shows "distinct (concat xs)"
  2798   using assms by (induct xs) auto
  2799 
  2800 text {* It is best to avoid this indexed version of distinct, but
  2801 sometimes it is useful. *}
  2802 
  2803 lemma distinct_conv_nth:
  2804 "distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
  2805 apply (induct xs, simp, simp)
  2806 apply (rule iffI, clarsimp)
  2807  apply (case_tac i)
  2808 apply (case_tac j, simp)
  2809 apply (simp add: set_conv_nth)
  2810  apply (case_tac j)
  2811 apply (clarsimp simp add: set_conv_nth, simp) 
  2812 apply (rule conjI)
  2813 (*TOO SLOW
  2814 apply (metis Zero_neq_Suc gr0_conv_Suc in_set_conv_nth lessI less_trans_Suc nth_Cons' nth_Cons_Suc)
  2815 *)
  2816  apply (clarsimp simp add: set_conv_nth)
  2817  apply (erule_tac x = 0 in allE, simp)
  2818  apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
  2819 (*TOO SLOW
  2820 apply (metis Suc_Suc_eq lessI less_trans_Suc nth_Cons_Suc)
  2821 *)
  2822 apply (erule_tac x = "Suc i" in allE, simp)
  2823 apply (erule_tac x = "Suc j" in allE, simp)
  2824 done
  2825 
  2826 lemma nth_eq_iff_index_eq:
  2827  "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
  2828 by(auto simp: distinct_conv_nth)
  2829 
  2830 lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
  2831 by (induct xs) auto
  2832 
  2833 lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
  2834 proof (induct xs)
  2835   case Nil thus ?case by simp
  2836 next
  2837   case (Cons x xs)
  2838   show ?case
  2839   proof (cases "x \<in> set xs")
  2840     case False with Cons show ?thesis by simp
  2841   next
  2842     case True with Cons.prems
  2843     have "card (set xs) = Suc (length xs)" 
  2844       by (simp add: card_insert_if split: split_if_asm)
  2845     moreover have "card (set xs) \<le> length xs" by (rule card_length)
  2846     ultimately have False by simp
  2847     thus ?thesis ..
  2848   qed
  2849 qed
  2850 
  2851 lemma distinct_length_filter: "distinct xs \<Longrightarrow> length (filter P xs) = card ({x. P x} Int set xs)"
  2852 by (induct xs) (auto)
  2853 
  2854 lemma not_distinct_decomp: "~ distinct ws ==> EX xs ys zs y. ws = xs@[y]@ys@[y]@zs"
  2855 apply (induct n == "length ws" arbitrary:ws) apply simp
  2856 apply(case_tac ws) apply simp
  2857 apply (simp split:split_if_asm)
  2858 apply (metis Cons_eq_appendI eq_Nil_appendI split_list)
  2859 done
  2860 
  2861 lemma not_distinct_conv_prefix:
  2862   defines "dec as xs y ys \<equiv> y \<in> set xs \<and> distinct xs \<and> as = xs @ y # ys"
  2863   shows "\<not>distinct as \<longleftrightarrow> (\<exists>xs y ys. dec as xs y ys)" (is "?L = ?R")
  2864 proof
  2865   assume "?L" then show "?R"
  2866   proof (induct "length as" arbitrary: as rule: less_induct)
  2867     case less
  2868     obtain xs ys zs y where decomp: "as = (xs @ y # ys) @ y # zs"
  2869       using not_distinct_decomp[OF less.prems] by auto
  2870     show ?case
  2871     proof (cases "distinct (xs @ y # ys)")
  2872       case True
  2873       with decomp have "dec as (xs @ y # ys) y zs" by (simp add: dec_def)
  2874       then show ?thesis by blast
  2875     next
  2876       case False
  2877       with less decomp obtain xs' y' ys' where "dec (xs @ y # ys) xs' y' ys'"
  2878         by atomize_elim auto
  2879       with decomp have "dec as xs' y' (ys' @ y # zs)" by (simp add: dec_def)
  2880       then show ?thesis by blast
  2881     qed
  2882   qed
  2883 qed (auto simp: dec_def)
  2884 
  2885 lemma length_remdups_concat:
  2886   "length (remdups (concat xss)) = card (\<Union>xs\<in>set xss. set xs)"
  2887   by (simp add: distinct_card [symmetric])
  2888 
  2889 lemma length_remdups_card_conv: "length(remdups xs) = card(set xs)"
  2890 proof -
  2891   have xs: "concat[xs] = xs" by simp
  2892   from length_remdups_concat[of "[xs]"] show ?thesis unfolding xs by simp
  2893 qed
  2894 
  2895 lemma remdups_remdups:
  2896   "remdups (remdups xs) = remdups xs"
  2897   by (induct xs) simp_all
  2898 
  2899 lemma distinct_butlast:
  2900   assumes "xs \<noteq> []" and "distinct xs"
  2901   shows "distinct (butlast xs)"
  2902 proof -
  2903   from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  2904   with `distinct xs` show ?thesis by simp
  2905 qed
  2906 
  2907 lemma remdups_map_remdups:
  2908   "remdups (map f (remdups xs)) = remdups (map f xs)"
  2909   by (induct xs) simp_all
  2910 
  2911 lemma distinct_zipI1:
  2912   assumes "distinct xs"
  2913   shows "distinct (zip xs ys)"
  2914 proof (rule zip_obtain_same_length)
  2915   fix xs' :: "'a list" and ys' :: "'b list" and n
  2916   assume "length xs' = length ys'"
  2917   assume "xs' = take n xs"
  2918   with assms have "distinct xs'" by simp
  2919   with `length xs' = length ys'` show "distinct (zip xs' ys')"
  2920     by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
  2921 qed
  2922 
  2923 lemma distinct_zipI2:
  2924   assumes "distinct ys"
  2925   shows "distinct (zip xs ys)"
  2926 proof (rule zip_obtain_same_length)
  2927   fix xs' :: "'b list" and ys' :: "'a list" and n
  2928   assume "length xs' = length ys'"
  2929   assume "ys' = take n ys"
  2930   with assms have "distinct ys'" by simp
  2931   with `length xs' = length ys'` show "distinct (zip xs' ys')"
  2932     by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE)
  2933 qed
  2934 
  2935 (* The next two lemmas help Sledgehammer. *)
  2936 
  2937 lemma distinct_singleton: "distinct [x]" by simp
  2938 
  2939 lemma distinct_length_2_or_more:
  2940 "distinct (a # b # xs) \<longleftrightarrow> (a \<noteq> b \<and> distinct (a # xs) \<and> distinct (b # xs))"
  2941 by (metis distinct.simps(2) hd.simps hd_in_set list.simps(2) set_ConsD set_rev_mp set_subset_Cons)
  2942 
  2943 
  2944 subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
  2945 
  2946 lemma (in monoid_add) listsum_foldl [code]:
  2947   "listsum = foldl (op +) 0"
  2948   by (simp add: listsum_def foldl_foldr1 fun_eq_iff)
  2949 
  2950 lemma (in monoid_add) listsum_simps [simp]:
  2951   "listsum [] = 0"
  2952   "listsum (x#xs) = x + listsum xs"
  2953   by (simp_all add: listsum_def)
  2954 
  2955 lemma (in monoid_add) listsum_append [simp]:
  2956   "listsum (xs @ ys) = listsum xs + listsum ys"
  2957   by (induct xs) (simp_all add: add.assoc)
  2958 
  2959 lemma (in comm_monoid_add) listsum_rev [simp]:
  2960   "listsum (rev xs) = listsum xs"
  2961   by (simp add: listsum_def [of "rev xs"]) (simp add: listsum_foldl foldr_foldl add.commute)
  2962 
  2963 lemma (in comm_monoid_add) listsum_map_remove1:
  2964   "x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))"
  2965   by (induct xs) (auto simp add: ac_simps)
  2966 
  2967 lemma (in monoid_add) list_size_conv_listsum:
  2968   "list_size f xs = listsum (map f xs) + size xs"
  2969   by (induct xs) auto
  2970 
  2971 lemma (in monoid_add) length_concat:
  2972   "length (concat xss) = listsum (map length xss)"
  2973   by (induct xss) simp_all
  2974 
  2975 lemma (in monoid_add) listsum_map_filter:
  2976   assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
  2977   shows "listsum (map f (filter P xs)) = listsum (map f xs)"
  2978   using assms by (induct xs) auto
  2979 
  2980 lemma (in monoid_add) distinct_listsum_conv_Setsum:
  2981   "distinct xs \<Longrightarrow> listsum xs = Setsum (set xs)"
  2982   by (induct xs) simp_all
  2983 
  2984 lemma listsum_eq_0_nat_iff_nat [simp]:
  2985   "listsum ns = (0::nat) \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
  2986   by (simp add: listsum_foldl)
  2987 
  2988 lemma elem_le_listsum_nat:
  2989   "k < size ns \<Longrightarrow> ns ! k \<le> listsum (ns::nat list)"
  2990 apply(induct ns arbitrary: k)
  2991  apply simp
  2992 apply(fastforce simp add:nth_Cons split: nat.split)
  2993 done
  2994 
  2995 lemma listsum_update_nat:
  2996   "k<size ns \<Longrightarrow> listsum (ns[k := (n::nat)]) = listsum ns + n - ns ! k"
  2997 apply(induct ns arbitrary:k)
  2998  apply (auto split:nat.split)
  2999 apply(drule elem_le_listsum_nat)
  3000 apply arith
  3001 done
  3002 
  3003 text{* Some syntactic sugar for summing a function over a list: *}
  3004 
  3005 syntax
  3006   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
  3007 syntax (xsymbols)
  3008   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
  3009 syntax (HTML output)
  3010   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
  3011 
  3012 translations -- {* Beware of argument permutation! *}
  3013   "SUM x<-xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  3014   "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)"
  3015 
  3016 lemma (in monoid_add) listsum_triv:
  3017   "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
  3018   by (induct xs) (simp_all add: left_distrib)
  3019 
  3020 lemma (in monoid_add) listsum_0 [simp]:
  3021   "(\<Sum>x\<leftarrow>xs. 0) = 0"
  3022   by (induct xs) (simp_all add: left_distrib)
  3023 
  3024 text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
  3025 lemma (in ab_group_add) uminus_listsum_map:
  3026   "- listsum (map f xs) = listsum (map (uminus \<circ> f) xs)"
  3027   by (induct xs) simp_all
  3028 
  3029 lemma (in comm_monoid_add) listsum_addf:
  3030   "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)"
  3031   by (induct xs) (simp_all add: algebra_simps)
  3032 
  3033 lemma (in ab_group_add) listsum_subtractf:
  3034   "(\<Sum>x\<leftarrow>xs. f x - g x) = listsum (map f xs) - listsum (map g xs)"
  3035   by (induct xs) (simp_all add: algebra_simps)
  3036 
  3037 lemma (in semiring_0) listsum_const_mult:
  3038   "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
  3039   by (induct xs) (simp_all add: algebra_simps)
  3040 
  3041 lemma (in semiring_0) listsum_mult_const:
  3042   "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
  3043   by (induct xs) (simp_all add: algebra_simps)
  3044 
  3045 lemma (in ordered_ab_group_add_abs) listsum_abs:
  3046   "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)"
  3047   by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])
  3048 
  3049 lemma listsum_mono:
  3050   fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}"
  3051   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)"
  3052   by (induct xs) (simp, simp add: add_mono)
  3053 
  3054 lemma (in monoid_add) listsum_distinct_conv_setsum_set:
  3055   "distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)"
  3056   by (induct xs) simp_all
  3057 
  3058 lemma (in monoid_add) interv_listsum_conv_setsum_set_nat:
  3059   "listsum (map f [m..<n]) = setsum f (set [m..<n])"
  3060   by (simp add: listsum_distinct_conv_setsum_set)
  3061 
  3062 lemma (in monoid_add) interv_listsum_conv_setsum_set_int:
  3063   "listsum (map f [k..l]) = setsum f (set [k..l])"
  3064   by (simp add: listsum_distinct_conv_setsum_set)
  3065 
  3066 text {* General equivalence between @{const listsum} and @{const setsum} *}
  3067 lemma (in monoid_add) listsum_setsum_nth:
  3068   "listsum xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
  3069   using interv_listsum_conv_setsum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth)
  3070 
  3071 
  3072 subsubsection {* @{const insert} *}
  3073 
  3074 lemma in_set_insert [simp]:
  3075   "x \<in> set xs \<Longrightarrow> List.insert x xs = xs"
  3076   by (simp add: List.insert_def)
  3077 
  3078 lemma not_in_set_insert [simp]:
  3079   "x \<notin> set xs \<Longrightarrow> List.insert x xs = x # xs"
  3080   by (simp add: List.insert_def)
  3081 
  3082 lemma insert_Nil [simp]:
  3083   "List.insert x [] = [x]"
  3084   by simp
  3085 
  3086 lemma set_insert [simp]:
  3087   "set (List.insert x xs) = insert x (set xs)"
  3088   by (auto simp add: List.insert_def)
  3089 
  3090 lemma distinct_insert [simp]:
  3091   "distinct xs \<Longrightarrow> distinct (List.insert x xs)"
  3092   by (simp add: List.insert_def)
  3093 
  3094 lemma insert_remdups:
  3095   "List.insert x (remdups xs) = remdups (List.insert x xs)"
  3096   by (simp add: List.insert_def)
  3097 
  3098 
  3099 subsubsection {* @{text remove1} *}
  3100 
  3101 lemma remove1_append:
  3102   "remove1 x (xs @ ys) =
  3103   (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
  3104 by (induct xs) auto
  3105 
  3106 lemma remove1_commute: "remove1 x (remove1 y zs) = remove1 y (remove1 x zs)"
  3107 by (induct zs) auto
  3108 
  3109 lemma in_set_remove1[simp]:
  3110   "a \<noteq> b \<Longrightarrow> a : set(remove1 b xs) = (a : set xs)"
  3111 apply (induct xs)
  3112 apply auto
  3113 done
  3114 
  3115 lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
  3116 apply(induct xs)
  3117  apply simp
  3118 apply simp
  3119 apply blast
  3120 done
  3121 
  3122 lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
  3123 apply(induct xs)
  3124  apply simp
  3125 apply simp
  3126 apply blast
  3127 done
  3128 
  3129 lemma length_remove1:
  3130   "length(remove1 x xs) = (if x : set xs then length xs - 1 else length xs)"
  3131 apply (induct xs)
  3132  apply (auto dest!:length_pos_if_in_set)
  3133 done
  3134 
  3135 lemma remove1_filter_not[simp]:
  3136   "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
  3137 by(induct xs) auto
  3138 
  3139 lemma filter_remove1:
  3140   "filter Q (remove1 x xs) = remove1 x (filter Q xs)"
  3141 by (induct xs) auto
  3142 
  3143 lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
  3144 apply(insert set_remove1_subset)
  3145 apply fast
  3146 done
  3147 
  3148 lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
  3149 by (induct xs) simp_all
  3150 
  3151 lemma remove1_remdups:
  3152   "distinct xs \<Longrightarrow> remove1 x (remdups xs) = remdups (remove1 x xs)"
  3153   by (induct xs) simp_all
  3154 
  3155 lemma remove1_idem:
  3156   assumes "x \<notin> set xs"
  3157   shows "remove1 x xs = xs"
  3158   using assms by (induct xs) simp_all
  3159 
  3160 
  3161 subsubsection {* @{text removeAll} *}
  3162 
  3163 lemma removeAll_filter_not_eq:
  3164   "removeAll x = filter (\<lambda>y. x \<noteq> y)"
  3165 proof
  3166   fix xs
  3167   show "removeAll x xs = filter (\<lambda>y. x \<noteq> y) xs"
  3168     by (induct xs) auto
  3169 qed
  3170 
  3171 lemma removeAll_append[simp]:
  3172   "removeAll x (xs @ ys) = removeAll x xs @ removeAll x ys"
  3173 by (induct xs) auto
  3174 
  3175 lemma set_removeAll[simp]: "set(removeAll x xs) = set xs - {x}"
  3176 by (induct xs) auto
  3177 
  3178 lemma removeAll_id[simp]: "x \<notin> set xs \<Longrightarrow> removeAll x xs = xs"
  3179 by (induct xs) auto
  3180 
  3181 (* Needs count:: 'a \<Rightarrow> a' list \<Rightarrow> nat
  3182 lemma length_removeAll:
  3183   "length(removeAll x xs) = length xs - count x xs"
  3184 *)
  3185 
  3186 lemma removeAll_filter_not[simp]:
  3187   "\<not> P x \<Longrightarrow> removeAll x (filter P xs) = filter P xs"
  3188 by(induct xs) auto
  3189 
  3190 lemma distinct_removeAll:
  3191   "distinct xs \<Longrightarrow> distinct (removeAll x xs)"
  3192   by (simp add: removeAll_filter_not_eq)
  3193 
  3194 lemma distinct_remove1_removeAll:
  3195   "distinct xs ==> remove1 x xs = removeAll x xs"
  3196 by (induct xs) simp_all
  3197 
  3198 lemma map_removeAll_inj_on: "inj_on f (insert x (set xs)) \<Longrightarrow>
  3199   map f (removeAll x xs) = removeAll (f x) (map f xs)"
  3200 by (induct xs) (simp_all add:inj_on_def)
  3201 
  3202 lemma map_removeAll_inj: "inj f \<Longrightarrow>
  3203   map f (removeAll x xs) = removeAll (f x) (map f xs)"
  3204 by(metis map_removeAll_inj_on subset_inj_on subset_UNIV)
  3205 
  3206 
  3207 subsubsection {* @{text replicate} *}
  3208 
  3209 lemma length_replicate [simp]: "length (replicate n x) = n"
  3210 by (induct n) auto
  3211 
  3212 lemma Ex_list_of_length: "\<exists>xs. length xs = n"
  3213 by (rule exI[of _ "replicate n undefined"]) simp
  3214 
  3215 lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
  3216 by (induct n) auto
  3217 
  3218 lemma map_replicate_const:
  3219   "map (\<lambda> x. k) lst = replicate (length lst) k"
  3220   by (induct lst) auto
  3221 
  3222 lemma replicate_app_Cons_same:
  3223 "(replicate n x) @ (x # xs) = x # replicate n x @ xs"
  3224 by (induct n) auto
  3225 
  3226 lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
  3227 apply (induct n, simp)
  3228 apply (simp add: replicate_app_Cons_same)
  3229 done
  3230 
  3231 lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
  3232 by (induct n) auto
  3233 
  3234 text{* Courtesy of Matthias Daum: *}
  3235 lemma append_replicate_commute:
  3236   "replicate n x @ replicate k x = replicate k x @ replicate n x"
  3237 apply (simp add: replicate_add [THEN sym])
  3238 apply (simp add: add_commute)
  3239 done
  3240 
  3241 text{* Courtesy of Andreas Lochbihler: *}
  3242 lemma filter_replicate:
  3243   "filter P (replicate n x) = (if P x then replicate n x else [])"
  3244 by(induct n) auto
  3245 
  3246 lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
  3247 by (induct n) auto
  3248 
  3249 lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
  3250 by (induct n) auto
  3251 
  3252 lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
  3253 by (atomize (full), induct n) auto
  3254 
  3255 lemma nth_replicate[simp]: "i < n ==> (replicate n x)!i = x"
  3256 apply (induct n arbitrary: i, simp)
  3257 apply (simp add: nth_Cons split: nat.split)
  3258 done
  3259 
  3260 text{* Courtesy of Matthias Daum (2 lemmas): *}
  3261 lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
  3262 apply (case_tac "k \<le> i")
  3263  apply  (simp add: min_def)
  3264 apply (drule not_leE)
  3265 apply (simp add: min_def)
  3266 apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
  3267  apply  simp
  3268 apply (simp add: replicate_add [symmetric])
  3269 done
  3270 
  3271 lemma drop_replicate[simp]: "drop i (replicate k x) = replicate (k-i) x"
  3272 apply (induct k arbitrary: i)
  3273  apply simp
  3274 apply clarsimp
  3275 apply (case_tac i)
  3276  apply simp
  3277 apply clarsimp
  3278 done
  3279 
  3280 
  3281 lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
  3282 by (induct n) auto
  3283 
  3284 lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
  3285 by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
  3286 
  3287 lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
  3288 by auto
  3289 
  3290 lemma in_set_replicate[simp]: "(x : set (replicate n y)) = (x = y & n \<noteq> 0)"
  3291 by (simp add: set_replicate_conv_if)
  3292 
  3293 lemma Ball_set_replicate[simp]:
  3294   "(ALL x : set(replicate n a). P x) = (P a | n=0)"
  3295 by(simp add: set_replicate_conv_if)
  3296 
  3297 lemma Bex_set_replicate[simp]:
  3298   "(EX x : set(replicate n a). P x) = (P a & n\<noteq>0)"
  3299 by(simp add: set_replicate_conv_if)
  3300 
  3301 lemma replicate_append_same:
  3302   "replicate i x @ [x] = x # replicate i x"
  3303   by (induct i) simp_all
  3304 
  3305 lemma map_replicate_trivial:
  3306   "map (\<lambda>i. x) [0..<i] = replicate i x"
  3307   by (induct i) (simp_all add: replicate_append_same)
  3308 
  3309 lemma concat_replicate_trivial[simp]:
  3310   "concat (replicate i []) = []"
  3311   by (induct i) (auto simp add: map_replicate_const)
  3312 
  3313 lemma replicate_empty[simp]: "(replicate n x = []) \<longleftrightarrow> n=0"
  3314 by (induct n) auto
  3315 
  3316 lemma empty_replicate[simp]: "([] = replicate n x) \<longleftrightarrow> n=0"
  3317 by (induct n) auto
  3318 
  3319 lemma replicate_eq_replicate[simp]:
  3320   "(replicate m x = replicate n y) \<longleftrightarrow> (m=n & (m\<noteq>0 \<longrightarrow> x=y))"
  3321 apply(induct m arbitrary: n)
  3322  apply simp
  3323 apply(induct_tac n)
  3324 apply auto
  3325 done
  3326 
  3327 lemma replicate_length_filter:
  3328   "replicate (length (filter (\<lambda>y. x = y) xs)) x = filter (\<lambda>y. x = y) xs"
  3329   by (induct xs) auto
  3330 
  3331 lemma comm_append_are_replicate:
  3332   fixes xs ys :: "'a list"
  3333   assumes "xs \<noteq> []" "ys \<noteq> []"
  3334   assumes "xs @ ys = ys @ xs"
  3335   shows "\<exists>m n zs. concat (replicate m zs) = xs \<and> concat (replicate n zs) = ys"
  3336   using assms
  3337 proof (induct "length (xs @ ys)" arbitrary: xs ys rule: less_induct)
  3338   case less
  3339 
  3340   def xs' \<equiv> "if (length xs \<le> length ys) then xs else ys"
  3341     and ys' \<equiv> "if (length xs \<le> length ys) then ys else xs"
  3342   then have
  3343     prems': "length xs' \<le> length ys'"
  3344             "xs' @ ys' = ys' @ xs'"
  3345       and "xs' \<noteq> []"
  3346       and len: "length (xs @ ys) = length (xs' @ ys')"
  3347     using less by (auto intro: less.hyps)
  3348 
  3349   from prems'
  3350   obtain ws where "ys' = xs' @ ws"
  3351     by (auto simp: append_eq_append_conv2)
  3352 
  3353   have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ys'"
  3354   proof (cases "ws = []")
  3355     case True
  3356     then have "concat (replicate 1 xs') = xs'"
  3357       and "concat (replicate 1 xs') = ys'"
  3358       using `ys' = xs' @ ws` by auto
  3359     then show ?thesis by blast
  3360   next
  3361     case False
  3362     from `ys' = xs' @ ws` and `xs' @ ys' = ys' @ xs'`
  3363     have "xs' @ ws = ws @ xs'" by simp
  3364     then have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ws"
  3365       using False and `xs' \<noteq> []` and `ys' = xs' @ ws` and len
  3366       by (intro less.hyps) auto
  3367     then obtain m n zs where "concat (replicate m zs) = xs'"
  3368       and "concat (replicate n zs) = ws" by blast
  3369     moreover
  3370     then have "concat (replicate (m + n) zs) = ys'"
  3371       using `ys' = xs' @ ws`
  3372       by (simp add: replicate_add)
  3373     ultimately
  3374     show ?thesis by blast
  3375   qed
  3376   then show ?case
  3377     using xs'_def ys'_def by metis
  3378 qed
  3379 
  3380 lemma comm_append_is_replicate:
  3381   fixes xs ys :: "'a list"
  3382   assumes "xs \<noteq> []" "ys \<noteq> []"
  3383   assumes "xs @ ys = ys @ xs"
  3384   shows "\<exists>n zs. n > 1 \<and> concat (replicate n zs) = xs @ ys"
  3385 
  3386 proof -
  3387   obtain m n zs where "concat (replicate m zs) = xs"
  3388     and "concat (replicate n zs) = ys"
  3389     using assms by (metis comm_append_are_replicate)
  3390   then have "m + n > 1" and "concat (replicate (m+n) zs) = xs @ ys"
  3391     using `xs \<noteq> []` and `ys \<noteq> []`
  3392     by (auto simp: replicate_add)
  3393   then show ?thesis by blast
  3394 qed
  3395 
  3396 
  3397 subsubsection{*@{text rotate1} and @{text rotate}*}
  3398 
  3399 lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
  3400 by(simp add:rotate1_def)
  3401 
  3402 lemma rotate0[simp]: "rotate 0 = id"
  3403 by(simp add:rotate_def)
  3404 
  3405 lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
  3406 by(simp add:rotate_def)
  3407 
  3408 lemma rotate_add:
  3409   "rotate (m+n) = rotate m o rotate n"
  3410 by(simp add:rotate_def funpow_add)
  3411 
  3412 lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
  3413 by(simp add:rotate_add)
  3414 
  3415 lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
  3416 by(simp add:rotate_def funpow_swap1)
  3417 
  3418 lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
  3419 by(cases xs) simp_all
  3420 
  3421 lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
  3422 apply(induct n)
  3423  apply simp
  3424 apply (simp add:rotate_def)
  3425 done
  3426 
  3427 lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
  3428 by(simp add:rotate1_def split:list.split)
  3429 
  3430 lemma rotate_drop_take:
  3431   "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
  3432 apply(induct n)
  3433  apply simp
  3434 apply(simp add:rotate_def)
  3435 apply(cases "xs = []")
  3436  apply (simp)
  3437 apply(case_tac "n mod length xs = 0")
  3438  apply(simp add:mod_Suc)
  3439  apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
  3440 apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
  3441                 take_hd_drop linorder_not_le)
  3442 done
  3443 
  3444 lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
  3445 by(simp add:rotate_drop_take)
  3446 
  3447 lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
  3448 by(simp add:rotate_drop_take)
  3449 
  3450 lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
  3451 by(simp add:rotate1_def split:list.split)
  3452 
  3453 lemma length_rotate[simp]: "length(rotate n xs) = length xs"
  3454 by (induct n arbitrary: xs) (simp_all add:rotate_def)
  3455 
  3456 lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
  3457 by(simp add:rotate1_def split:list.split) blast
  3458 
  3459 lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
  3460 by (induct n) (simp_all add:rotate_def)
  3461 
  3462 lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
  3463 by(simp add:rotate_drop_take take_map drop_map)
  3464 
  3465 lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
  3466 by (cases xs) (auto simp add:rotate1_def)
  3467 
  3468 lemma set_rotate[simp]: "set(rotate n xs) = set xs"
  3469 by (induct n) (simp_all add:rotate_def)
  3470 
  3471 lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
  3472 by(simp add:rotate1_def split:list.split)
  3473 
  3474 lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
  3475 by (induct n) (simp_all add:rotate_def)
  3476 
  3477 lemma rotate_rev:
  3478   "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
  3479 apply(simp add:rotate_drop_take rev_drop rev_take)
  3480 apply(cases "length xs = 0")
  3481  apply simp
  3482 apply(cases "n mod length xs = 0")
  3483  apply simp
  3484 apply(simp add:rotate_drop_take rev_drop rev_take)
  3485 done
  3486 
  3487 lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
  3488 apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
  3489 apply(subgoal_tac "length xs \<noteq> 0")
  3490  prefer 2 apply simp
  3491 using mod_less_divisor[of "length xs" n] by arith
  3492 
  3493 
  3494 subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
  3495 
  3496 lemma sublist_empty [simp]: "sublist xs {} = []"
  3497 by (auto simp add: sublist_def)
  3498 
  3499 lemma sublist_nil [simp]: "sublist [] A = []"
  3500 by (auto simp add: sublist_def)
  3501 
  3502 lemma length_sublist:
  3503   "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
  3504 by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
  3505 
  3506 lemma sublist_shift_lemma_Suc:
  3507   "map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
  3508    map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
  3509 apply(induct xs arbitrary: "is")
  3510  apply simp
  3511 apply (case_tac "is")
  3512  apply simp
  3513 apply simp
  3514 done
  3515 
  3516 lemma sublist_shift_lemma:
  3517      "map fst [p<-zip xs [i..<i + length xs] . snd p : A] =
  3518       map fst [p<-zip xs [0..<length xs] . snd p + i : A]"
  3519 by (induct xs rule: rev_induct) (simp_all add: add_commute)
  3520 
  3521 lemma sublist_append:
  3522      "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
  3523 apply (unfold sublist_def)
  3524 apply (induct l' rule: rev_induct, simp)
  3525 apply (simp add: upt_add_eq_append[of 0] sublist_shift_lemma)
  3526 apply (simp add: add_commute)
  3527 done
  3528 
  3529 lemma sublist_Cons:
  3530 "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
  3531 apply (induct l rule: rev_induct)
  3532  apply (simp add: sublist_def)
  3533 apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
  3534 done
  3535 
  3536 lemma set_sublist: "set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
  3537 apply(induct xs arbitrary: I)
  3538 apply(auto simp: sublist_Cons nth_Cons split:nat.split dest!: gr0_implies_Suc)
  3539 done
  3540 
  3541 lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
  3542 by(auto simp add:set_sublist)
  3543 
  3544 lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
  3545 by(auto simp add:set_sublist)
  3546 
  3547 lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
  3548 by(auto simp add:set_sublist)
  3549 
  3550 lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
  3551 by (simp add: sublist_Cons)
  3552 
  3553 
  3554 lemma distinct_sublistI[simp]: "distinct xs \<Longrightarrow> distinct(sublist xs I)"
  3555 apply(induct xs arbitrary: I)
  3556  apply simp
  3557 apply(auto simp add:sublist_Cons)
  3558 done
  3559 
  3560 
  3561 lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
  3562 apply (induct l rule: rev_induct, simp)
  3563 apply (simp split: nat_diff_split add: sublist_append)
  3564 done
  3565 
  3566 lemma filter_in_sublist:
  3567  "distinct xs \<Longrightarrow> filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
  3568 proof (induct xs arbitrary: s)
  3569   case Nil thus ?case by simp
  3570 next
  3571   case (Cons a xs)
  3572   moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
  3573   ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
  3574 qed
  3575 
  3576 
  3577 subsubsection {* @{const splice} *}
  3578 
  3579 lemma splice_Nil2 [simp, code]: "splice xs [] = xs"
  3580 by (cases xs) simp_all
  3581 
  3582 declare splice.simps(1,3)[code]
  3583 declare splice.simps(2)[simp del]
  3584 
  3585 lemma length_splice[simp]: "length(splice xs ys) = length xs + length ys"
  3586 by (induct xs ys rule: splice.induct) auto
  3587 
  3588 
  3589 subsubsection {* Transpose *}
  3590 
  3591 function transpose where
  3592 "transpose []             = []" |
  3593 "transpose ([]     # xss) = transpose xss" |
  3594 "transpose ((x#xs) # xss) =
  3595   (x # [h. (h#t) \<leftarrow> xss]) # transpose (xs # [t. (h#t) \<leftarrow> xss])"
  3596 by pat_completeness auto
  3597 
  3598 lemma transpose_aux_filter_head:
  3599   "concat (map (list_case [] (\<lambda>h t. [h])) xss) =
  3600   map (\<lambda>xs. hd xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  3601   by (induct xss) (auto split: list.split)
  3602 
  3603 lemma transpose_aux_filter_tail:
  3604   "concat (map (list_case [] (\<lambda>h t. [t])) xss) =
  3605   map (\<lambda>xs. tl xs) [ys\<leftarrow>xss . ys \<noteq> []]"
  3606   by (induct xss) (auto split: list.split)
  3607 
  3608 lemma transpose_aux_max:
  3609   "max (Suc (length xs)) (foldr (\<lambda>xs. max (length xs)) xss 0) =
  3610   Suc (max (length xs) (foldr (\<lambda>x. max (length x - Suc 0)) [ys\<leftarrow>xss . ys\<noteq>[]] 0))"
  3611   (is "max _ ?foldB = Suc (max _ ?foldA)")
  3612 proof (cases "[ys\<leftarrow>xss . ys\<noteq>[]] = []")
  3613   case True
  3614   hence "foldr (\<lambda>xs. max (length xs)) xss 0 = 0"
  3615   proof (induct xss)
  3616     case (Cons x xs)
  3617     moreover hence "x = []" by (cases x) auto
  3618     ultimately show ?case by auto
  3619   qed simp
  3620   thus ?thesis using True by simp
  3621 next
  3622   case False
  3623 
  3624   have foldA: "?foldA = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0 - 1"
  3625     by (induct xss) auto
  3626   have foldB: "?foldB = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0"
  3627     by (induct xss) auto
  3628 
  3629   have "0 < ?foldB"
  3630   proof -
  3631     from False
  3632     obtain z zs where zs: "[ys\<leftarrow>xss . ys \<noteq> []] = z#zs" by (auto simp: neq_Nil_conv)
  3633     hence "z \<in> set ([ys\<leftarrow>xss . ys \<noteq> []])" by auto
  3634     hence "z \<noteq> []" by auto
  3635     thus ?thesis
  3636       unfolding foldB zs
  3637       by (auto simp: max_def intro: less_le_trans)
  3638   qed
  3639   thus ?thesis
  3640     unfolding foldA foldB max_Suc_Suc[symmetric]
  3641     by simp
  3642 qed
  3643 
  3644 termination transpose
  3645   by (relation "measure (\<lambda>xs. foldr (\<lambda>xs. max (length xs)) xs 0 + length xs)")
  3646      (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max less_Suc_eq_le)
  3647 
  3648 lemma transpose_empty: "(transpose xs = []) \<longleftrightarrow> (\<forall>x \<in> set xs. x = [])"
  3649   by (induct rule: transpose.induct) simp_all
  3650 
  3651 lemma length_transpose:
  3652   fixes xs :: "'a list list"
  3653   shows "length (transpose xs) = foldr (\<lambda>xs. max (length xs)) xs 0"
  3654   by (induct rule: transpose.induct)
  3655     (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max
  3656                 max_Suc_Suc[symmetric] simp del: max_Suc_Suc)
  3657 
  3658 lemma nth_transpose:
  3659   fixes xs :: "'a list list"
  3660   assumes "i < length (transpose xs)"
  3661   shows "transpose xs ! i = map (\<lambda>xs. xs ! i) [ys \<leftarrow> xs. i < length ys]"
  3662 using assms proof (induct arbitrary: i rule: transpose.induct)
  3663   case (3 x xs xss)
  3664   def XS == "(x # xs) # xss"
  3665   hence [simp]: "XS \<noteq> []" by auto
  3666   thus ?case
  3667   proof (cases i)
  3668     case 0
  3669     thus ?thesis by (simp add: transpose_aux_filter_head hd_conv_nth)
  3670   next
  3671     case (Suc j)
  3672     have *: "\<And>xss. xs # map tl xss = map tl ((x#xs)#xss)" by simp
  3673     have **: "\<And>xss. (x#xs) # filter (\<lambda>ys. ys \<noteq> []) xss = filter (\<lambda>ys. ys \<noteq> []) ((x#xs)#xss)" by simp
  3674     { fix x have "Suc j < length x \<longleftrightarrow> x \<noteq> [] \<and> j < length x - Suc 0"
  3675       by (cases x) simp_all
  3676     } note *** = this
  3677 
  3678     have j_less: "j < length (transpose (xs # concat (map (list_case [] (\<lambda>h t. [t])) xss)))"
  3679       using "3.prems" by (simp add: transpose_aux_filter_tail length_transpose Suc)
  3680 
  3681     show ?thesis
  3682       unfolding transpose.simps `i = Suc j` nth_Cons_Suc "3.hyps"[OF j_less]
  3683       apply (auto simp: transpose_aux_filter_tail filter_map comp_def length_transpose * ** *** XS_def[symmetric])
  3684       apply (rule_tac y=x in list.exhaust)
  3685       by auto
  3686   qed
  3687 qed simp_all
  3688 
  3689 lemma transpose_map_map:
  3690   "transpose (map (map f) xs) = map (map f) (transpose xs)"
  3691 proof (rule nth_equalityI, safe)
  3692   have [simp]: "length (transpose (map (map f) xs)) = length (transpose xs)"
  3693     by (simp add: length_transpose foldr_map comp_def)
  3694   show "length (transpose (map (map f) xs)) = length (map (map f) (transpose xs))" by simp
  3695 
  3696   fix i assume "i < length (transpose (map (map f) xs))"
  3697   thus "transpose (map (map f) xs) ! i = map (map f) (transpose xs) ! i"
  3698     by (simp add: nth_transpose filter_map comp_def)
  3699 qed
  3700 
  3701 
  3702 subsubsection {* (In)finiteness *}
  3703 
  3704 lemma finite_maxlen:
  3705   "finite (M::'a list set) ==> EX n. ALL s:M. size s < n"
  3706 proof (induct rule: finite.induct)
  3707   case emptyI show ?case by simp
  3708 next
  3709   case (insertI M xs)
  3710   then obtain n where "\<forall>s\<in>M. length s < n" by blast
  3711   hence "ALL s:insert xs M. size s < max n (size xs) + 1" by auto
  3712   thus ?case ..
  3713 qed
  3714 
  3715 lemma lists_length_Suc_eq:
  3716   "{xs. set xs \<subseteq> A \<and> length xs = Suc n} =
  3717     (\<lambda>(xs, n). n#xs) ` ({xs. set xs \<subseteq> A \<and> length xs = n} \<times> A)"
  3718   by (auto simp: length_Suc_conv)
  3719 
  3720 lemma
  3721   assumes "finite A"
  3722   shows finite_lists_length_eq: "finite {xs. set xs \<subseteq> A \<and> length xs = n}"
  3723   and card_lists_length_eq: "card {xs. set xs \<subseteq> A \<and> length xs = n} = (card A)^n"
  3724   using `finite A`
  3725   by (induct n)
  3726      (auto simp: card_image inj_split_Cons lists_length_Suc_eq cong: conj_cong)
  3727 
  3728 lemma finite_lists_length_le:
  3729   assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs \<le> n}"
  3730  (is "finite ?S")
  3731 proof-
  3732   have "?S = (\<Union>n\<in>{0..n}. {xs. set xs \<subseteq> A \<and> length xs = n})" by auto
  3733   thus ?thesis by (auto intro: finite_lists_length_eq[OF `finite A`])
  3734 qed
  3735 
  3736 lemma card_lists_length_le:
  3737   assumes "finite A" shows "card {xs. set xs \<subseteq> A \<and> length xs \<le> n} = (\<Sum>i\<le>n. card A^i)"
  3738 proof -
  3739   have "(\<Sum>i\<le>n. card A^i) = card (\<Union>i\<le>n. {xs. set xs \<subseteq> A \<and> length xs = i})"
  3740     using `finite A`
  3741     by (subst card_UN_disjoint)
  3742        (auto simp add: card_lists_length_eq finite_lists_length_eq)
  3743   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}"
  3744     by auto
  3745   finally show ?thesis by simp
  3746 qed
  3747 
  3748 lemma card_lists_distinct_length_eq:
  3749   assumes "k < card A"
  3750   shows "card {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A} = \<Prod>{card A - k + 1 .. card A}"
  3751 using assms
  3752 proof (induct k)
  3753   case 0
  3754   then have "{xs. length xs = 0 \<and> distinct xs \<and> set xs \<subseteq> A} = {[]}" by auto
  3755   then show ?case by simp
  3756 next
  3757   case (Suc k)
  3758   let "?k_list" = "\<lambda>k xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A"
  3759   have inj_Cons: "\<And>A. inj_on (\<lambda>(xs, n). n # xs) A"  by (rule inj_onI) auto
  3760 
  3761   from Suc have "k < card A" by simp
  3762   moreover have "finite A" using assms by (simp add: card_ge_0_finite)
  3763   moreover have "finite {xs. ?k_list k xs}"
  3764     using finite_lists_length_eq[OF `finite A`, of k]
  3765     by - (rule finite_subset, auto)
  3766   moreover have "\<And>i j. i \<noteq> j \<longrightarrow> {i} \<times> (A - set i) \<inter> {j} \<times> (A - set j) = {}"
  3767     by auto
  3768   moreover have "\<And>i. i \<in>Collect (?k_list k) \<Longrightarrow> card (A - set i) = card A - k"
  3769     by (simp add: card_Diff_subset distinct_card)
  3770   moreover have "{xs. ?k_list (Suc k) xs} =
  3771       (\<lambda>(xs, n). n#xs) ` \<Union>(\<lambda>xs. {xs} \<times> (A - set xs)) ` {xs. ?k_list k xs}"
  3772     by (auto simp: length_Suc_conv)
  3773   moreover
  3774   have "Suc (card A - Suc k) = card A - k" using Suc.prems by simp
  3775   then have "(card A - k) * \<Prod>{Suc (card A - k)..card A} = \<Prod>{Suc (card A - Suc k)..card A}"
  3776     by (subst setprod_insert[symmetric]) (simp add: atLeastAtMost_insertL)+
  3777   ultimately show ?case
  3778     by (simp add: card_image inj_Cons card_UN_disjoint Suc.hyps algebra_simps)
  3779 qed
  3780 
  3781 lemma infinite_UNIV_listI: "~ finite(UNIV::'a list set)"
  3782 apply(rule notI)
  3783 apply(drule finite_maxlen)
  3784 apply (metis UNIV_I length_replicate less_not_refl)
  3785 done
  3786 
  3787 
  3788 subsection {* Sorting *}
  3789 
  3790 text{* Currently it is not shown that @{const sort} returns a
  3791 permutation of its input because the nicest proof is via multisets,
  3792 which are not yet available. Alternatively one could define a function
  3793 that counts the number of occurrences of an element in a list and use
  3794 that instead of multisets to state the correctness property. *}
  3795 
  3796 context linorder
  3797 begin
  3798 
  3799 lemma length_insort [simp]:
  3800   "length (insort_key f x xs) = Suc (length xs)"
  3801   by (induct xs) simp_all
  3802 
  3803 lemma insort_key_left_comm:
  3804   assumes "f x \<noteq> f y"
  3805   shows "insort_key f y (insort_key f x xs) = insort_key f x (insort_key f y xs)"
  3806   by (induct xs) (auto simp add: assms dest: antisym)
  3807 
  3808 lemma insort_left_comm:
  3809   "insort x (insort y xs) = insort y (insort x xs)"
  3810   by (cases "x = y") (auto intro: insort_key_left_comm)
  3811 
  3812 lemma comp_fun_commute_insort:
  3813   "comp_fun_commute insort"
  3814 proof
  3815 qed (simp add: insort_left_comm fun_eq_iff)
  3816 
  3817 lemma sort_key_simps [simp]:
  3818   "sort_key f [] = []"
  3819   "sort_key f (x#xs) = insort_key f x (sort_key f xs)"
  3820   by (simp_all add: sort_key_def)
  3821 
  3822 lemma sort_foldl_insort:
  3823   "sort xs = foldl (\<lambda>ys x. insort x ys) [] xs"
  3824   by (simp add: sort_key_def foldr_foldl foldl_rev insort_left_comm)
  3825 
  3826 lemma length_sort[simp]: "length (sort_key f xs) = length xs"
  3827 by (induct xs, auto)
  3828 
  3829 lemma sorted_Cons: "sorted (x#xs) = (sorted xs & (ALL y:set xs. x <= y))"
  3830 apply(induct xs arbitrary: x) apply simp
  3831 by simp (blast intro: order_trans)
  3832 
  3833 lemma sorted_tl:
  3834   "sorted xs \<Longrightarrow> sorted (tl xs)"
  3835   by (cases xs) (simp_all add: sorted_Cons)
  3836 
  3837 lemma sorted_append:
  3838   "sorted (xs@ys) = (sorted xs & sorted ys & (\<forall>x \<in> set xs. \<forall>y \<in> set ys. x\<le>y))"
  3839 by (induct xs) (auto simp add:sorted_Cons)
  3840 
  3841 lemma sorted_nth_mono:
  3842   "sorted xs \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!i \<le> xs!j"
  3843 by (induct xs arbitrary: i j) (auto simp:nth_Cons' sorted_Cons)
  3844 
  3845 lemma sorted_rev_nth_mono:
  3846   "sorted (rev xs) \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!j \<le> xs!i"
  3847 using sorted_nth_mono[ of "rev xs" "length xs - j - 1" "length xs - i - 1"]
  3848       rev_nth[of "length xs - i - 1" "xs"] rev_nth[of "length xs - j - 1" "xs"]
  3849 by auto
  3850 
  3851 lemma sorted_nth_monoI:
  3852   "(\<And> i j. \<lbrakk> i \<le> j ; j < length xs \<rbrakk> \<Longrightarrow> xs ! i \<le> xs ! j) \<Longrightarrow> sorted xs"
  3853 proof (induct xs)
  3854   case (Cons x xs)
  3855   have "sorted xs"
  3856   proof (rule Cons.hyps)
  3857     fix i j assume "i \<le> j" and "j < length xs"
  3858     with Cons.prems[of "Suc i" "Suc j"]
  3859     show "xs ! i \<le> xs ! j" by auto
  3860   qed
  3861   moreover
  3862   {
  3863     fix y assume "y \<in> set xs"
  3864     then obtain j where "j < length xs" and "xs ! j = y"
  3865       unfolding in_set_conv_nth by blast
  3866     with Cons.prems[of 0 "Suc j"]
  3867     have "x \<le> y"
  3868       by auto
  3869   }
  3870   ultimately
  3871   show ?case
  3872     unfolding sorted_Cons by auto
  3873 qed simp
  3874 
  3875 lemma sorted_equals_nth_mono:
  3876   "sorted xs = (\<forall>j < length xs. \<forall>i \<le> j. xs ! i \<le> xs ! j)"
  3877 by (auto intro: sorted_nth_monoI sorted_nth_mono)
  3878 
  3879 lemma set_insort: "set(insort_key f x xs) = insert x (set xs)"
  3880 by (induct xs) auto
  3881 
  3882 lemma set_sort[simp]: "set(sort_key f xs) = set xs"
  3883 by (induct xs) (simp_all add:set_insort)
  3884 
  3885 lemma distinct_insort: "distinct (insort_key f x xs) = (x \<notin> set xs \<and> distinct xs)"
  3886 by(induct xs)(auto simp:set_insort)
  3887 
  3888 lemma distinct_sort[simp]: "distinct (sort_key f xs) = distinct xs"
  3889   by (induct xs) (simp_all add: distinct_insort)
  3890 
  3891 lemma sorted_insort_key: "sorted (map f (insort_key f x xs)) = sorted (map f xs)"
  3892   by (induct xs) (auto simp:sorted_Cons set_insort)
  3893 
  3894 lemma sorted_insort: "sorted (insort x xs) = sorted xs"
  3895   using sorted_insort_key [where f="\<lambda>x. x"] by simp
  3896 
  3897 theorem sorted_sort_key [simp]: "sorted (map f (sort_key f xs))"
  3898   by (induct xs) (auto simp:sorted_insort_key)
  3899 
  3900 theorem sorted_sort [simp]: "sorted (sort xs)"
  3901   using sorted_sort_key [where f="\<lambda>x. x"] by simp
  3902 
  3903 lemma sorted_butlast:
  3904   assumes "xs \<noteq> []" and "sorted xs"
  3905   shows "sorted (butlast xs)"
  3906 proof -
  3907   from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto
  3908   with `sorted xs` show ?thesis by (simp add: sorted_append)
  3909 qed
  3910   
  3911 lemma insort_not_Nil [simp]:
  3912   "insort_key f a xs \<noteq> []"
  3913   by (induct xs) simp_all
  3914 
  3915 lemma insort_is_Cons: "\<forall>x\<in>set xs. f a \<le> f x \<Longrightarrow> insort_key f a xs = a # xs"
  3916 by (cases xs) auto
  3917 
  3918 lemma sorted_sort_id: "sorted xs \<Longrightarrow> sort xs = xs"
  3919   by (induct xs) (auto simp add: sorted_Cons insort_is_Cons)
  3920 
  3921 lemma sorted_map_remove1:
  3922   "sorted (map f xs) \<Longrightarrow> sorted (map f (remove1 x xs))"
  3923   by (induct xs) (auto simp add: sorted_Cons)
  3924 
  3925 lemma sorted_remove1: "sorted xs \<Longrightarrow> sorted (remove1 a xs)"
  3926   using sorted_map_remove1 [of "\<lambda>x. x"] by simp
  3927 
  3928 lemma insort_key_remove1:
  3929   assumes "a \<in> set xs" and "sorted (map f xs)" and "hd (filter (\<lambda>x. f a = f x) xs) = a"
  3930   shows "insort_key f a (remove1 a xs) = xs"
  3931 using assms proof (induct xs)
  3932   case (Cons x xs)
  3933   then show ?case
  3934   proof (cases "x = a")
  3935     case False
  3936     then have "f x \<noteq> f a" using Cons.prems by auto
  3937     then have "f x < f a" using Cons.prems by (auto simp: sorted_Cons)
  3938     with `f x \<noteq> f a` show ?thesis using Cons by (auto simp: sorted_Cons insort_is_Cons)
  3939   qed (auto simp: sorted_Cons insort_is_Cons)
  3940 qed simp
  3941 
  3942 lemma insort_remove1:
  3943   assumes "a \<in> set xs" and "sorted xs"
  3944   shows "insort a (remove1 a xs) = xs"
  3945 proof (rule insort_key_remove1)
  3946   from `a \<in> set xs` show "a \<in> set xs" .
  3947   from `sorted xs` show "sorted (map (\<lambda>x. x) xs)" by simp
  3948   from `a \<in> set xs` have "a \<in> set (filter (op = a) xs)" by auto
  3949   then have "set (filter (op = a) xs) \<noteq> {}" by auto
  3950   then have "filter (op = a) xs \<noteq> []" by (auto simp only: set_empty)
  3951   then have "length (filter (op = a) xs) > 0" by simp
  3952   then obtain n where n: "Suc n = length (filter (op = a) xs)"
  3953     by (cases "length (filter (op = a) xs)") simp_all
  3954   moreover have "replicate (Suc n) a = a # replicate n a"
  3955     by simp
  3956   ultimately show "hd (filter (op = a) xs) = a" by (simp add: replicate_length_filter)
  3957 qed
  3958 
  3959 lemma sorted_remdups[simp]:
  3960   "sorted l \<Longrightarrow> sorted (remdups l)"
  3961 by (induct l) (auto simp: sorted_Cons)
  3962 
  3963 lemma sorted_distinct_set_unique:
  3964 assumes "sorted xs" "distinct xs" "sorted ys" "distinct ys" "set xs = set ys"
  3965 shows "xs = ys"
  3966 proof -
  3967   from assms have 1: "length xs = length ys" by (auto dest!: distinct_card)
  3968   from assms show ?thesis
  3969   proof(induct rule:list_induct2[OF 1])
  3970     case 1 show ?case by simp
  3971   next
  3972     case 2 thus ?case by (simp add:sorted_Cons)
  3973        (metis Diff_insert_absorb antisym insertE insert_iff)
  3974   qed
  3975 qed
  3976 
  3977 lemma map_sorted_distinct_set_unique:
  3978   assumes "inj_on f (set xs \<union> set ys)"
  3979   assumes "sorted (map f xs)" "distinct (map f xs)"
  3980     "sorted (map f ys)" "distinct (map f ys)"
  3981   assumes "set xs = set ys"
  3982   shows "xs = ys"
  3983 proof -
  3984   from assms have "map f xs = map f ys"
  3985     by (simp add: sorted_distinct_set_unique)
  3986   moreover with `inj_on f (set xs \<union> set ys)` show "xs = ys"
  3987     by (blast intro: map_inj_on)
  3988 qed
  3989 
  3990 lemma finite_sorted_distinct_unique:
  3991 shows "finite A \<Longrightarrow> EX! xs. set xs = A & sorted xs & distinct xs"
  3992 apply(drule finite_distinct_list)
  3993 apply clarify
  3994 apply(rule_tac a="sort xs" in ex1I)
  3995 apply (auto simp: sorted_distinct_set_unique)
  3996 done
  3997 
  3998 lemma
  3999   assumes "sorted xs"
  4000   shows sorted_take: "sorted (take n xs)"
  4001   and sorted_drop: "sorted (drop n xs)"
  4002 proof -
  4003   from assms have "sorted (take n xs @ drop n xs)" by simp
  4004   then show "sorted (take n xs)" and "sorted (drop n xs)"
  4005     unfolding sorted_append by simp_all
  4006 qed
  4007 
  4008 lemma sorted_dropWhile: "sorted xs \<Longrightarrow> sorted (dropWhile P xs)"
  4009   by (auto dest: sorted_drop simp add: dropWhile_eq_drop)
  4010 
  4011 lemma sorted_takeWhile: "sorted xs \<Longrightarrow> sorted (takeWhile P xs)"
  4012   by (subst takeWhile_eq_take) (auto dest: sorted_take)
  4013 
  4014 lemma sorted_filter:
  4015   "sorted (map f xs) \<Longrightarrow> sorted (map f (filter P xs))"
  4016   by (induct xs) (simp_all add: sorted_Cons)
  4017 
  4018 lemma foldr_max_sorted:
  4019   assumes "sorted (rev xs)"
  4020   shows "foldr max xs y = (if xs = [] then y else max (xs ! 0) y)"
  4021 using assms proof (induct xs)
  4022   case (Cons x xs)
  4023   moreover hence "sorted (rev xs)" using sorted_append by auto
  4024   ultimately show ?case
  4025     by (cases xs, auto simp add: sorted_append max_def)
  4026 qed simp
  4027 
  4028 lemma filter_equals_takeWhile_sorted_rev:
  4029   assumes sorted: "sorted (rev (map f xs))"
  4030   shows "[x \<leftarrow> xs. t < f x] = takeWhile (\<lambda> x. t < f x) xs"
  4031     (is "filter ?P xs = ?tW")
  4032 proof (rule takeWhile_eq_filter[symmetric])
  4033   let "?dW" = "dropWhile ?P xs"
  4034   fix x assume "x \<in> set ?dW"
  4035   then obtain i where i: "i < length ?dW" and nth_i: "x = ?dW ! i"
  4036     unfolding in_set_conv_nth by auto
  4037   hence "length ?tW + i < length (?tW @ ?dW)"
  4038     unfolding length_append by simp
  4039   hence i': "length (map f ?tW) + i < length (map f xs)" by simp
  4040   have "(map f ?tW @ map f ?dW) ! (length (map f ?tW) + i) \<le>
  4041         (map f ?tW @ map f ?dW) ! (length (map f ?tW) + 0)"
  4042     using sorted_rev_nth_mono[OF sorted _ i', of "length ?tW"]
  4043     unfolding map_append[symmetric] by simp
  4044   hence "f x \<le> f (?dW ! 0)"
  4045     unfolding nth_append_length_plus nth_i
  4046     using i preorder_class.le_less_trans[OF le0 i] by simp
  4047   also have "... \<le> t"
  4048     using hd_dropWhile[of "?P" xs] le0[THEN preorder_class.le_less_trans, OF i]
  4049     using hd_conv_nth[of "?dW"] by simp
  4050   finally show "\<not> t < f x" by simp
  4051 qed
  4052 
  4053 lemma insort_insert_key_triv:
  4054   "f x \<in> f ` set xs \<Longrightarrow> insort_insert_key f x xs = xs"
  4055   by (simp add: insort_insert_key_def)
  4056 
  4057 lemma insort_insert_triv:
  4058   "x \<in> set xs \<Longrightarrow> insort_insert x xs = xs"
  4059   using insort_insert_key_triv [of "\<lambda>x. x"] by simp
  4060 
  4061 lemma insort_insert_insort_key:
  4062   "f x \<notin> f ` set xs \<Longrightarrow> insort_insert_key f x xs = insort_key f x xs"
  4063   by (simp add: insort_insert_key_def)
  4064 
  4065 lemma insort_insert_insort:
  4066   "x \<notin> set xs \<Longrightarrow> insort_insert x xs = insort x xs"
  4067   using insort_insert_insort_key [of "\<lambda>x. x"] by simp
  4068 
  4069 lemma set_insort_insert:
  4070   "set (insort_insert x xs) = insert x (set xs)"
  4071   by (auto simp add: insort_insert_key_def set_insort)
  4072 
  4073 lemma distinct_insort_insert:
  4074   assumes "distinct xs"
  4075   shows "distinct (insort_insert_key f x xs)"
  4076   using assms by (induct xs) (auto simp add: insort_insert_key_def set_insort)
  4077 
  4078 lemma sorted_insort_insert_key:
  4079   assumes "sorted (map f xs)"
  4080   shows "sorted (map f (insort_insert_key f x xs))"
  4081   using assms by (simp add: insort_insert_key_def sorted_insort_key)
  4082 
  4083 lemma sorted_insort_insert:
  4084   assumes "sorted xs"
  4085   shows "sorted (insort_insert x xs)"
  4086   using assms sorted_insort_insert_key [of "\<lambda>x. x"] by simp
  4087 
  4088 lemma filter_insort_triv:
  4089   "\<not> P x \<Longrightarrow> filter P (insort_key f x xs) = filter P xs"
  4090   by (induct xs) simp_all
  4091 
  4092 lemma filter_insort:
  4093   "sorted (map f xs) \<Longrightarrow> P x \<Longrightarrow> filter P (insort_key f x xs) = insort_key f x (filter P xs)"
  4094   using assms by (induct xs)
  4095     (auto simp add: sorted_Cons, subst insort_is_Cons, auto)
  4096 
  4097 lemma filter_sort:
  4098   "filter P (sort_key f xs) = sort_key f (filter P xs)"
  4099   by (induct xs) (simp_all add: filter_insort_triv filter_insort)
  4100 
  4101 lemma sorted_map_same:
  4102   "sorted (map f [x\<leftarrow>xs. f x = g xs])"
  4103 proof (induct xs arbitrary: g)
  4104   case Nil then show ?case by simp
  4105 next
  4106   case (Cons x xs)
  4107   then have "sorted (map f [y\<leftarrow>xs . f y = (\<lambda>xs. f x) xs])" .
  4108   moreover from Cons have "sorted (map f [y\<leftarrow>xs . f y = (g \<circ> Cons x) xs])" .
  4109   ultimately show ?case by (simp_all add: sorted_Cons)
  4110 qed
  4111 
  4112 lemma sorted_same:
  4113   "sorted [x\<leftarrow>xs. x = g xs]"
  4114   using sorted_map_same [of "\<lambda>x. x"] by simp
  4115 
  4116 lemma remove1_insort [simp]:
  4117   "remove1 x (insort x xs) = xs"
  4118   by (induct xs) simp_all
  4119 
  4120 end
  4121 
  4122 lemma sorted_upt[simp]: "sorted[i..<j]"
  4123 by (induct j) (simp_all add:sorted_append)
  4124 
  4125 lemma sorted_upto[simp]: "sorted[i..j]"
  4126 apply(induct i j rule:upto.induct)
  4127 apply(subst upto.simps)
  4128 apply(simp add:sorted_Cons)
  4129 done
  4130 
  4131 
  4132 subsubsection {* @{const transpose} on sorted lists *}
  4133 
  4134 lemma sorted_transpose[simp]:
  4135   shows "sorted (rev (map length (transpose xs)))"
  4136   by (auto simp: sorted_equals_nth_mono rev_nth nth_transpose
  4137     length_filter_conv_card intro: card_mono)
  4138 
  4139 lemma transpose_max_length:
  4140   "foldr (\<lambda>xs. max (length xs)) (transpose xs) 0 = length [x \<leftarrow> xs. x \<noteq> []]"
  4141   (is "?L = ?R")
  4142 proof (cases "transpose xs = []")
  4143   case False
  4144   have "?L = foldr max (map length (transpose xs)) 0"
  4145     by (simp add: foldr_map comp_def)
  4146   also have "... = length (transpose xs ! 0)"
  4147     using False sorted_transpose by (simp add: foldr_max_sorted)
  4148   finally show ?thesis
  4149     using False by (simp add: nth_transpose)
  4150 next
  4151   case True
  4152   hence "[x \<leftarrow> xs. x \<noteq> []] = []"
  4153     by (auto intro!: filter_False simp: transpose_empty)
  4154   thus ?thesis by (simp add: transpose_empty True)
  4155 qed
  4156 
  4157 lemma length_transpose_sorted:
  4158   fixes xs :: "'a list list"
  4159   assumes sorted: "sorted (rev (map length xs))"
  4160   shows "length (transpose xs) = (if xs = [] then 0 else length (xs ! 0))"
  4161 proof (cases "xs = []")
  4162   case False
  4163   thus ?thesis
  4164     using foldr_max_sorted[OF sorted] False
  4165     unfolding length_transpose foldr_map comp_def
  4166     by simp
  4167 qed simp
  4168 
  4169 lemma nth_nth_transpose_sorted[simp]:
  4170   fixes xs :: "'a list list"
  4171   assumes sorted: "sorted (rev (map length xs))"
  4172   and i: "i < length (transpose xs)"
  4173   and j: "j < length [ys \<leftarrow> xs. i < length ys]"
  4174   shows "transpose xs ! i ! j = xs ! j  ! i"
  4175   using j filter_equals_takeWhile_sorted_rev[OF sorted, of i]
  4176     nth_transpose[OF i] nth_map[OF j]
  4177   by (simp add: takeWhile_nth)
  4178 
  4179 lemma transpose_column_length:
  4180   fixes xs :: "'a list list"
  4181   assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  4182   shows "length (filter (\<lambda>ys. i < length ys) (transpose xs)) = length (xs ! i)"
  4183 proof -
  4184   have "xs \<noteq> []" using `i < length xs` by auto
  4185   note filter_equals_takeWhile_sorted_rev[OF sorted, simp]
  4186   { fix j assume "j \<le> i"
  4187     note sorted_rev_nth_mono[OF sorted, of j i, simplified, OF this `i < length xs`]
  4188   } note sortedE = this[consumes 1]
  4189 
  4190   have "{j. j < length (transpose xs) \<and> i < length (transpose xs ! j)}
  4191     = {..< length (xs ! i)}"
  4192   proof safe
  4193     fix j
  4194     assume "j < length (transpose xs)" and "i < length (transpose xs ! j)"
  4195     with this(2) nth_transpose[OF this(1)]
  4196     have "i < length (takeWhile (\<lambda>ys. j < length ys) xs)" by simp
  4197     from nth_mem[OF this] takeWhile_nth[OF this]
  4198     show "j < length (xs ! i)" by (auto dest: set_takeWhileD)
  4199   next
  4200     fix j assume "j < length (xs ! i)"
  4201     thus "j < length (transpose xs)"
  4202       using foldr_max_sorted[OF sorted] `xs \<noteq> []` sortedE[OF le0]
  4203       by (auto simp: length_transpose comp_def foldr_map)
  4204 
  4205     have "Suc i \<le> length (takeWhile (\<lambda>ys. j < length ys) xs)"
  4206       using `i < length xs` `j < length (xs ! i)` less_Suc_eq_le
  4207       by (auto intro!: length_takeWhile_less_P_nth dest!: sortedE)
  4208     with nth_transpose[OF `j < length (transpose xs)`]
  4209     show "i < length (transpose xs ! j)" by simp
  4210   qed
  4211   thus ?thesis by (simp add: length_filter_conv_card)
  4212 qed
  4213 
  4214 lemma transpose_column:
  4215   fixes xs :: "'a list list"
  4216   assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
  4217   shows "map (\<lambda>ys. ys ! i) (filter (\<lambda>ys. i < length ys) (transpose xs))
  4218     = xs ! i" (is "?R = _")
  4219 proof (rule nth_equalityI, safe)
  4220   show length: "length ?R = length (xs ! i)"
  4221     using transpose_column_length[OF assms] by simp
  4222 
  4223   fix j assume j: "j < length ?R"
  4224   note * = less_le_trans[OF this, unfolded length_map, OF length_filter_le]
  4225   from j have j_less: "j < length (xs ! i)" using length by simp
  4226   have i_less_tW: "Suc i \<le> length (takeWhile (\<lambda>ys. Suc j \<le> length ys) xs)"
  4227   proof (rule length_takeWhile_less_P_nth)
  4228     show "Suc i \<le> length xs" using `i < length xs` by simp
  4229     fix k assume "k < Suc i"
  4230     hence "k \<le> i" by auto
  4231     with sorted_rev_nth_mono[OF sorted this] `i < length xs`
  4232     have "length (xs ! i) \<le> length (xs ! k)" by simp
  4233     thus "Suc j \<le> length (xs ! k)" using j_less by simp
  4234   qed
  4235   have i_less_filter: "i < length [ys\<leftarrow>xs . j < length ys]"
  4236     unfolding filter_equals_takeWhile_sorted_rev[OF sorted, of j]
  4237     using i_less_tW by (simp_all add: Suc_le_eq)
  4238   from j show "?R ! j = xs ! i ! j"
  4239     unfolding filter_equals_takeWhile_sorted_rev[OF sorted_transpose, of i]
  4240     by (simp add: takeWhile_nth nth_nth_transpose_sorted[OF sorted * i_less_filter])
  4241 qed
  4242 
  4243 lemma transpose_transpose:
  4244   fixes xs :: "'a list list"
  4245   assumes sorted: "sorted (rev (map length xs))"
  4246   shows "transpose (transpose xs) = takeWhile (\<lambda>x. x \<noteq> []) xs" (is "?L = ?R")
  4247 proof -
  4248   have len: "length ?L = length ?R"
  4249     unfolding length_transpose transpose_max_length
  4250     using filter_equals_takeWhile_sorted_rev[OF sorted, of 0]
  4251     by simp
  4252 
  4253   { fix i assume "i < length ?R"
  4254     with less_le_trans[OF _ length_takeWhile_le[of _ xs]]
  4255     have "i < length xs" by simp
  4256   } note * = this
  4257   show ?thesis
  4258     by (rule nth_equalityI)
  4259        (simp_all add: len nth_transpose transpose_column[OF sorted] * takeWhile_nth)
  4260 qed
  4261 
  4262 theorem transpose_rectangle:
  4263   assumes "xs = [] \<Longrightarrow> n = 0"
  4264   assumes rect: "\<And> i. i < length xs \<Longrightarrow> length (xs ! i) = n"
  4265   shows "transpose xs = map (\<lambda> i. map (\<lambda> j. xs ! j ! i) [0..<length xs]) [0..<n]"
  4266     (is "?trans = ?map")
  4267 proof (rule nth_equalityI)
  4268   have "sorted (rev (map length xs))"
  4269     by (auto simp: rev_nth rect intro!: sorted_nth_monoI)
  4270   from foldr_max_sorted[OF this] assms
  4271   show len: "length ?trans = length ?map"
  4272     by (simp_all add: length_transpose foldr_map comp_def)
  4273   moreover
  4274   { fix i assume "i < n" hence "[ys\<leftarrow>xs . i < length ys] = xs"
  4275       using rect by (auto simp: in_set_conv_nth intro!: filter_True) }
  4276   ultimately show "\<forall>i < length ?trans. ?trans ! i = ?map ! i"
  4277     by (auto simp: nth_transpose intro: nth_equalityI)
  4278 qed
  4279 
  4280 
  4281 subsubsection {* @{text sorted_list_of_set} *}
  4282 
  4283 text{* This function maps (finite) linearly ordered sets to sorted
  4284 lists. Warning: in most cases it is not a good idea to convert from
  4285 sets to lists but one should convert in the other direction (via
  4286 @{const set}). *}
  4287 
  4288 context linorder
  4289 begin
  4290 
  4291 definition sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where
  4292   "sorted_list_of_set = Finite_Set.fold insort []"
  4293 
  4294 lemma sorted_list_of_set_empty [simp]:
  4295   "sorted_list_of_set {} = []"
  4296   by (simp add: sorted_list_of_set_def)
  4297 
  4298 lemma sorted_list_of_set_insert [simp]:
  4299   assumes "finite A"
  4300   shows "sorted_list_of_set (insert x A) = insort x (sorted_list_of_set (A - {x}))"
  4301 proof -
  4302   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  4303   with assms show ?thesis by (simp add: sorted_list_of_set_def fold_insert_remove)
  4304 qed
  4305 
  4306 lemma sorted_list_of_set [simp]:
  4307   "finite A \<Longrightarrow> set (sorted_list_of_set A) = A \<and> sorted (sorted_list_of_set A) 
  4308     \<and> distinct (sorted_list_of_set A)"
  4309   by (induct A rule: finite_induct) (simp_all add: set_insort sorted_insort distinct_insort)
  4310 
  4311 lemma sorted_list_of_set_sort_remdups:
  4312   "sorted_list_of_set (set xs) = sort (remdups xs)"
  4313 proof -
  4314   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  4315   show ?thesis by (simp add: sort_foldl_insort sorted_list_of_set_def fold_set_remdups)
  4316 qed
  4317 
  4318 lemma sorted_list_of_set_remove:
  4319   assumes "finite A"
  4320   shows "sorted_list_of_set (A - {x}) = remove1 x (sorted_list_of_set A)"
  4321 proof (cases "x \<in> A")
  4322   case False with assms have "x \<notin> set (sorted_list_of_set A)" by simp
  4323   with False show ?thesis by (simp add: remove1_idem)
  4324 next
  4325   case True then obtain B where A: "A = insert x B" by (rule Set.set_insert)
  4326   with assms show ?thesis by simp
  4327 qed
  4328 
  4329 end
  4330 
  4331 lemma sorted_list_of_set_range [simp]:
  4332   "sorted_list_of_set {m..<n} = [m..<n]"
  4333   by (rule sorted_distinct_set_unique) simp_all
  4334 
  4335 
  4336 subsubsection {* @{text lists}: the list-forming operator over sets *}
  4337 
  4338 inductive_set
  4339   lists :: "'a set => 'a list set"
  4340   for A :: "'a set"
  4341 where
  4342     Nil [intro!, simp]: "[]: lists A"
  4343   | Cons [intro!, simp, no_atp]: "[| a: A; l: lists A|] ==> a#l : lists A"
  4344 
  4345 inductive_cases listsE [elim!,no_atp]: "x#l : lists A"
  4346 inductive_cases listspE [elim!,no_atp]: "listsp A (x # l)"
  4347 
  4348 lemma listsp_mono [mono]: "A \<le> B ==> listsp A \<le> listsp B"
  4349 by (rule predicate1I, erule listsp.induct, (blast dest: predicate1D)+)
  4350 
  4351 lemmas lists_mono = listsp_mono [to_set pred_subset_eq]
  4352 
  4353 lemma listsp_infI:
  4354   assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
  4355 by induct blast+
  4356 
  4357 lemmas lists_IntI = listsp_infI [to_set]
  4358 
  4359 lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
  4360 proof (rule mono_inf [where f=listsp, THEN order_antisym])
  4361   show "mono listsp" by (simp add: mono_def listsp_mono)
  4362   show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro!: listsp_infI predicate1I)
  4363 qed
  4364 
  4365 lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_def inf_bool_def]
  4366 
  4367 lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set pred_equals_eq]
  4368 
  4369 lemma Cons_in_lists_iff[simp]: "x#xs : lists A \<longleftrightarrow> x:A \<and> xs : lists A"
  4370 by auto
  4371 
  4372 lemma append_in_listsp_conv [iff]:
  4373      "(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
  4374 by (induct xs) auto
  4375 
  4376 lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]
  4377 
  4378 lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
  4379 -- {* eliminate @{text listsp} in favour of @{text set} *}
  4380 by (induct xs) auto
  4381 
  4382 lemmas in_lists_conv_set = in_listsp_conv_set [to_set]
  4383 
  4384 lemma in_listspD [dest!,no_atp]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
  4385 by (rule in_listsp_conv_set [THEN iffD1])
  4386 
  4387 lemmas in_listsD [dest!,no_atp] = in_listspD [to_set]
  4388 
  4389 lemma in_listspI [intro!,no_atp]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
  4390 by (rule in_listsp_conv_set [THEN iffD2])
  4391 
  4392 lemmas in_listsI [intro!,no_atp] = in_listspI [to_set]
  4393 
  4394 lemma lists_eq_set: "lists A = {xs. set xs <= A}"
  4395 by auto
  4396 
  4397 lemma lists_empty [simp]: "lists {} = {[]}"
  4398 by auto
  4399 
  4400 lemma lists_UNIV [simp]: "lists UNIV = UNIV"
  4401 by auto
  4402 
  4403 
  4404 subsubsection {* Inductive definition for membership *}
  4405 
  4406 inductive ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
  4407 where
  4408     elem:  "ListMem x (x # xs)"
  4409   | insert:  "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"
  4410 
  4411 lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
  4412 apply (rule iffI)
  4413  apply (induct set: ListMem)
  4414   apply auto
  4415 apply (induct xs)
  4416  apply (auto intro: ListMem.intros)
  4417 done
  4418 
  4419 
  4420 subsubsection {* Lists as Cartesian products *}
  4421 
  4422 text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
  4423 @{term A} and tail drawn from @{term Xs}.*}
  4424 
  4425 definition
  4426   set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set" where
  4427   "set_Cons A XS = {z. \<exists>x xs. z = x # xs \<and> x \<in> A \<and> xs \<in> XS}"
  4428 
  4429 lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
  4430 by (auto simp add: set_Cons_def)
  4431 
  4432 text{*Yields the set of lists, all of the same length as the argument and
  4433 with elements drawn from the corresponding element of the argument.*}
  4434 
  4435 primrec
  4436   listset :: "'a set list \<Rightarrow> 'a list set" where
  4437      "listset [] = {[]}"
  4438   |  "listset (A # As) = set_Cons A (listset As)"
  4439 
  4440 
  4441 subsection {* Relations on Lists *}
  4442 
  4443 subsubsection {* Length Lexicographic Ordering *}
  4444 
  4445 text{*These orderings preserve well-foundedness: shorter lists 
  4446   precede longer lists. These ordering are not used in dictionaries.*}
  4447         
  4448 primrec -- {*The lexicographic ordering for lists of the specified length*}
  4449   lexn :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> ('a list \<times> 'a list) set" where
  4450     "lexn r 0 = {}"
  4451   | "lexn r (Suc n) = (map_pair (%(x, xs). x#xs) (%(x, xs). x#xs) ` (r <*lex*> lexn r n)) Int
  4452       {(xs, ys). length xs = Suc n \<and> length ys = Suc n}"
  4453 
  4454 definition
  4455   lex :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4456   "lex r = (\<Union>n. lexn r n)" -- {*Holds only between lists of the same length*}
  4457 
  4458 definition
  4459   lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" where
  4460   "lenlex r = inv_image (less_than <*lex*> lex r) (\<lambda>xs. (length xs, xs))"
  4461         -- {*Compares lists by their length and then lexicographically*}
  4462 
  4463 lemma wf_lexn: "wf r ==> wf (lexn r n)"
  4464 apply (induct n, simp, simp)
  4465 apply(rule wf_subset)
  4466  prefer 2 apply (rule Int_lower1)
  4467 apply(rule wf_map_pair_image)
  4468  prefer 2 apply (rule inj_onI, auto)
  4469 done
  4470 
  4471 lemma lexn_length:
  4472   "(xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
  4473 by (induct n arbitrary: xs ys) auto
  4474 
  4475 lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
  4476 apply (unfold lex_def)
  4477 apply (rule wf_UN)
  4478 apply (blast intro: wf_lexn, clarify)
  4479 apply (rename_tac m n)
  4480 apply (subgoal_tac "m \<noteq> n")
  4481  prefer 2 apply blast
  4482 apply (blast dest: lexn_length not_sym)
  4483 done
  4484 
  4485 lemma lexn_conv:
  4486   "lexn r n =
  4487     {(xs,ys). length xs = n \<and> length ys = n \<and>
  4488     (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
  4489 apply (induct n, simp)
  4490 apply (simp add: image_Collect lex_prod_def, safe, blast)
  4491  apply (rule_tac x = "ab # xys" in exI, simp)
  4492 apply (case_tac xys, simp_all, blast)
  4493 done
  4494 
  4495 lemma lex_conv:
  4496   "lex r =
  4497     {(xs,ys). length xs = length ys \<and>
  4498     (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
  4499 by (force simp add: lex_def lexn_conv)
  4500 
  4501 lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
  4502 by (unfold lenlex_def) blast
  4503 
  4504 lemma lenlex_conv:
  4505     "lenlex r = {(xs,ys). length xs < length ys |
  4506                  length xs = length ys \<and> (xs, ys) : lex r}"
  4507 by (simp add: lenlex_def Id_on_def lex_prod_def inv_image_def)
  4508 
  4509 lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
  4510 by (simp add: lex_conv)
  4511 
  4512 lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
  4513 by (simp add:lex_conv)
  4514 
  4515 lemma Cons_in_lex [simp]:
  4516     "((x # xs, y # ys) : lex r) =
  4517       ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
  4518 apply (simp add: lex_conv)
  4519 apply (rule iffI)
  4520  prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
  4521 apply (case_tac xys, simp, simp)
  4522 apply blast
  4523 done
  4524 
  4525 
  4526 subsubsection {* Lexicographic Ordering *}
  4527 
  4528 text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
  4529     This ordering does \emph{not} preserve well-foundedness.
  4530      Author: N. Voelker, March 2005. *} 
  4531 
  4532 definition
  4533   lexord :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4534   "lexord r = {(x,y ). \<exists> a v. y = x @ a # v \<or>
  4535             (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
  4536 
  4537 lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
  4538 by (unfold lexord_def, induct_tac y, auto) 
  4539 
  4540 lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
  4541 by (unfold lexord_def, induct_tac x, auto)
  4542 
  4543 lemma lexord_cons_cons[simp]:
  4544      "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
  4545   apply (unfold lexord_def, safe, simp_all)
  4546   apply (case_tac u, simp, simp)
  4547   apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
  4548   apply (erule_tac x="b # u" in allE)
  4549   by force
  4550 
  4551 lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
  4552 
  4553 lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
  4554 by (induct_tac x, auto)  
  4555 
  4556 lemma lexord_append_left_rightI:
  4557      "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
  4558 by (induct_tac u, auto)
  4559 
  4560 lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
  4561 by (induct x, auto)
  4562 
  4563 lemma lexord_append_leftD:
  4564      "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
  4565 by (erule rev_mp, induct_tac x, auto)
  4566 
  4567 lemma lexord_take_index_conv: 
  4568    "((x,y) : lexord r) = 
  4569     ((length x < length y \<and> take (length x) y = x) \<or> 
  4570      (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
  4571   apply (unfold lexord_def Let_def, clarsimp) 
  4572   apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
  4573   apply auto 
  4574   apply (rule_tac x="hd (drop (length x) y)" in exI)
  4575   apply (rule_tac x="tl (drop (length x) y)" in exI)
  4576   apply (erule subst, simp add: min_def) 
  4577   apply (rule_tac x ="length u" in exI, simp) 
  4578   apply (rule_tac x ="take i x" in exI) 
  4579   apply (rule_tac x ="x ! i" in exI) 
  4580   apply (rule_tac x ="y ! i" in exI, safe) 
  4581   apply (rule_tac x="drop (Suc i) x" in exI)
  4582   apply (drule sym, simp add: drop_Suc_conv_tl) 
  4583   apply (rule_tac x="drop (Suc i) y" in exI)
  4584   by (simp add: drop_Suc_conv_tl) 
  4585 
  4586 -- {* lexord is extension of partial ordering List.lex *} 
  4587 lemma lexord_lex: "(x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
  4588   apply (rule_tac x = y in spec) 
  4589   apply (induct_tac x, clarsimp) 
  4590   by (clarify, case_tac x, simp, force)
  4591 
  4592 lemma lexord_irreflexive: "ALL x. (x,x) \<notin> r \<Longrightarrow> (xs,xs) \<notin> lexord r"
  4593 by (induct xs) auto
  4594 
  4595 text{* By Ren\'e Thiemann: *}
  4596 lemma lexord_partial_trans: 
  4597   "(\<And>x y z. x \<in> set xs \<Longrightarrow> (x,y) \<in> r \<Longrightarrow> (y,z) \<in> r \<Longrightarrow> (x,z) \<in> r)
  4598    \<Longrightarrow>  (xs,ys) \<in> lexord r  \<Longrightarrow>  (ys,zs) \<in> lexord r \<Longrightarrow>  (xs,zs) \<in> lexord r"
  4599 proof (induct xs arbitrary: ys zs)
  4600   case Nil
  4601   from Nil(3) show ?case unfolding lexord_def by (cases zs, auto)
  4602 next
  4603   case (Cons x xs yys zzs)
  4604   from Cons(3) obtain y ys where yys: "yys = y # ys" unfolding lexord_def
  4605     by (cases yys, auto)
  4606   note Cons = Cons[unfolded yys]
  4607   from Cons(3) have one: "(x,y) \<in> r \<or> x = y \<and> (xs,ys) \<in> lexord r" by auto
  4608   from Cons(4) obtain z zs where zzs: "zzs = z # zs" unfolding lexord_def
  4609     by (cases zzs, auto)
  4610   note Cons = Cons[unfolded zzs]
  4611   from Cons(4) have two: "(y,z) \<in> r \<or> y = z \<and> (ys,zs) \<in> lexord r" by auto
  4612   {
  4613     assume "(xs,ys) \<in> lexord r" and "(ys,zs) \<in> lexord r"
  4614     from Cons(1)[OF _ this] Cons(2)
  4615     have "(xs,zs) \<in> lexord r" by auto
  4616   } note ind1 = this
  4617   {
  4618     assume "(x,y) \<in> r" and "(y,z) \<in> r"
  4619     from Cons(2)[OF _ this] have "(x,z) \<in> r" by auto
  4620   } note ind2 = this
  4621   from one two ind1 ind2
  4622   have "(x,z) \<in> r \<or> x = z \<and> (xs,zs) \<in> lexord r" by blast
  4623   thus ?case unfolding zzs by auto
  4624 qed
  4625 
  4626 lemma lexord_trans: 
  4627     "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
  4628 by(auto simp: trans_def intro:lexord_partial_trans)
  4629 
  4630 lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
  4631 by (rule transI, drule lexord_trans, blast) 
  4632 
  4633 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"
  4634   apply (rule_tac x = y in spec) 
  4635   apply (induct_tac x, rule allI) 
  4636   apply (case_tac x, simp, simp) 
  4637   apply (rule allI, case_tac x, simp, simp) 
  4638   by blast
  4639 
  4640 
  4641 subsubsection {* Lexicographic combination of measure functions *}
  4642 
  4643 text {* These are useful for termination proofs *}
  4644 
  4645 definition
  4646   "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"
  4647 
  4648 lemma wf_measures[simp]: "wf (measures fs)"
  4649 unfolding measures_def
  4650 by blast
  4651 
  4652 lemma in_measures[simp]: 
  4653   "(x, y) \<in> measures [] = False"
  4654   "(x, y) \<in> measures (f # fs)
  4655          = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"  
  4656 unfolding measures_def
  4657 by auto
  4658 
  4659 lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
  4660 by simp
  4661 
  4662 lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
  4663 by auto
  4664 
  4665 
  4666 subsubsection {* Lifting Relations to Lists: one element *}
  4667 
  4668 definition listrel1 :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where
  4669 "listrel1 r = {(xs,ys).
  4670    \<exists>us z z' vs. xs = us @ z # vs \<and> (z,z') \<in> r \<and> ys = us @ z' # vs}"
  4671 
  4672 lemma listrel1I:
  4673   "\<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow>
  4674   (xs, ys) \<in> listrel1 r"
  4675 unfolding listrel1_def by auto
  4676 
  4677 lemma listrel1E:
  4678   "\<lbrakk> (xs, ys) \<in> listrel1 r;
  4679      !!x y us vs. \<lbrakk> (x, y) \<in> r;  xs = us @ x # vs;  ys = us @ y # vs \<rbrakk> \<Longrightarrow> P
  4680    \<rbrakk> \<Longrightarrow> P"
  4681 unfolding listrel1_def by auto
  4682 
  4683 lemma not_Nil_listrel1 [iff]: "([], xs) \<notin> listrel1 r"
  4684 unfolding listrel1_def by blast
  4685 
  4686 lemma not_listrel1_Nil [iff]: "(xs, []) \<notin> listrel1 r"
  4687 unfolding listrel1_def by blast
  4688 
  4689 lemma Cons_listrel1_Cons [iff]:
  4690   "(x # xs, y # ys) \<in> listrel1 r \<longleftrightarrow>
  4691    (x,y) \<in> r \<and> xs = ys \<or> x = y \<and> (xs, ys) \<in> listrel1 r"
  4692 by (simp add: listrel1_def Cons_eq_append_conv) (blast)
  4693 
  4694 lemma listrel1I1: "(x,y) \<in> r \<Longrightarrow> (x # xs, y # xs) \<in> listrel1 r"
  4695 by (metis Cons_listrel1_Cons)
  4696 
  4697 lemma listrel1I2: "(xs, ys) \<in> listrel1 r \<Longrightarrow> (x # xs, x # ys) \<in> listrel1 r"
  4698 by (metis Cons_listrel1_Cons)
  4699 
  4700 lemma append_listrel1I:
  4701   "(xs, ys) \<in> listrel1 r \<and> us = vs \<or> xs = ys \<and> (us, vs) \<in> listrel1 r
  4702     \<Longrightarrow> (xs @ us, ys @ vs) \<in> listrel1 r"
  4703 unfolding listrel1_def
  4704 by auto (blast intro: append_eq_appendI)+
  4705 
  4706 lemma Cons_listrel1E1[elim!]:
  4707   assumes "(x # xs, ys) \<in> listrel1 r"
  4708     and "\<And>y. ys = y # xs \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
  4709     and "\<And>zs. ys = x # zs \<Longrightarrow> (xs, zs) \<in> listrel1 r \<Longrightarrow> R"
  4710   shows R
  4711 using assms by (cases ys) blast+
  4712 
  4713 lemma Cons_listrel1E2[elim!]:
  4714   assumes "(xs, y # ys) \<in> listrel1 r"
  4715     and "\<And>x. xs = x # ys \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R"
  4716     and "\<And>zs. xs = y # zs \<Longrightarrow> (zs, ys) \<in> listrel1 r \<Longrightarrow> R"
  4717   shows R
  4718 using assms by (cases xs) blast+
  4719 
  4720 lemma snoc_listrel1_snoc_iff:
  4721   "(xs @ [x], ys @ [y]) \<in> listrel1 r
  4722     \<longleftrightarrow> (xs, ys) \<in> listrel1 r \<and> x = y \<or> xs = ys \<and> (x,y) \<in> r" (is "?L \<longleftrightarrow> ?R")
  4723 proof
  4724   assume ?L thus ?R
  4725     by (fastforce simp: listrel1_def snoc_eq_iff_butlast butlast_append)
  4726 next
  4727   assume ?R then show ?L unfolding listrel1_def by force
  4728 qed
  4729 
  4730 lemma listrel1_eq_len: "(xs,ys) \<in> listrel1 r \<Longrightarrow> length xs = length ys"
  4731 unfolding listrel1_def by auto
  4732 
  4733 lemma listrel1_mono:
  4734   "r \<subseteq> s \<Longrightarrow> listrel1 r \<subseteq> listrel1 s"
  4735 unfolding listrel1_def by blast
  4736 
  4737 
  4738 lemma listrel1_converse: "listrel1 (r^-1) = (listrel1 r)^-1"
  4739 unfolding listrel1_def by blast
  4740 
  4741 lemma in_listrel1_converse:
  4742   "(x,y) : listrel1 (r^-1) \<longleftrightarrow> (x,y) : (listrel1 r)^-1"
  4743 unfolding listrel1_def by blast
  4744 
  4745 lemma listrel1_iff_update:
  4746   "(xs,ys) \<in> (listrel1 r)
  4747    \<longleftrightarrow> (\<exists>y n. (xs ! n, y) \<in> r \<and> n < length xs \<and> ys = xs[n:=y])" (is "?L \<longleftrightarrow> ?R")
  4748 proof
  4749   assume "?L"
  4750   then obtain x y u v where "xs = u @ x # v"  "ys = u @ y # v"  "(x,y) \<in> r"
  4751     unfolding listrel1_def by auto
  4752   then have "ys = xs[length u := y]" and "length u < length xs"
  4753     and "(xs ! length u, y) \<in> r" by auto
  4754   then show "?R" by auto
  4755 next
  4756   assume "?R"
  4757   then obtain x y n where "(xs!n, y) \<in> r" "n < size xs" "ys = xs[n:=y]" "x = xs!n"
  4758     by auto
  4759   then obtain u v where "xs = u @ x # v" and "ys = u @ y # v" and "(x, y) \<in> r"
  4760     by (auto intro: upd_conv_take_nth_drop id_take_nth_drop)
  4761   then show "?L" by (auto simp: listrel1_def)
  4762 qed
  4763 
  4764 
  4765 text{* Accessible part and wellfoundedness: *}
  4766 
  4767 lemma Cons_acc_listrel1I [intro!]:
  4768   "x \<in> acc r \<Longrightarrow> xs \<in> acc (listrel1 r) \<Longrightarrow> (x # xs) \<in> acc (listrel1 r)"
  4769 apply (induct arbitrary: xs set: acc)
  4770 apply (erule thin_rl)
  4771 apply (erule acc_induct)
  4772 apply (rule accI)
  4773 apply (blast)
  4774 done
  4775 
  4776 lemma lists_accD: "xs \<in> lists (acc r) \<Longrightarrow> xs \<in> acc (listrel1 r)"
  4777 apply (induct set: lists)
  4778  apply (rule accI)
  4779  apply simp
  4780 apply (rule accI)
  4781 apply (fast dest: acc_downward)
  4782 done
  4783 
  4784 lemma lists_accI: "xs \<in> acc (listrel1 r) \<Longrightarrow> xs \<in> lists (acc r)"
  4785 apply (induct set: acc)
  4786 apply clarify
  4787 apply (rule accI)
  4788 apply (fastforce dest!: in_set_conv_decomp[THEN iffD1] simp: listrel1_def)
  4789 done
  4790 
  4791 lemma wf_listrel1_iff[simp]: "wf(listrel1 r) = wf r"
  4792 by(metis wf_acc_iff in_lists_conv_set lists_accI lists_accD Cons_in_lists_iff)
  4793 
  4794 
  4795 subsubsection {* Lifting Relations to Lists: all elements *}
  4796 
  4797 inductive_set
  4798   listrel :: "('a * 'a)set => ('a list * 'a list)set"
  4799   for r :: "('a * 'a)set"
  4800 where
  4801     Nil:  "([],[]) \<in> listrel r"
  4802   | Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
  4803 
  4804 inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
  4805 inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
  4806 inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
  4807 inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
  4808 
  4809 
  4810 lemma listrel_eq_len:  "(xs, ys) \<in> listrel r \<Longrightarrow> length xs = length ys"
  4811 by(induct rule: listrel.induct) auto
  4812 
  4813 lemma listrel_iff_zip: "(xs,ys) : listrel r \<longleftrightarrow>
  4814   length xs = length ys & (\<forall>(x,y) \<in> set(zip xs ys). (x,y) \<in> r)" (is "?L \<longleftrightarrow> ?R")
  4815 proof
  4816   assume ?L thus ?R by induct (auto intro: listrel_eq_len)
  4817 next
  4818   assume ?R thus ?L
  4819     apply (clarify)
  4820     by (induct rule: list_induct2) (auto intro: listrel.intros)
  4821 qed
  4822 
  4823 lemma listrel_iff_nth: "(xs,ys) : listrel r \<longleftrightarrow>
  4824   length xs = length ys & (\<forall>n < length xs. (xs!n, ys!n) \<in> r)" (is "?L \<longleftrightarrow> ?R")
  4825 by (auto simp add: all_set_conv_all_nth listrel_iff_zip)
  4826 
  4827 
  4828 lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
  4829 apply clarify  
  4830 apply (erule listrel.induct)
  4831 apply (blast intro: listrel.intros)+
  4832 done
  4833 
  4834 lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
  4835 apply clarify 
  4836 apply (erule listrel.induct, auto) 
  4837 done
  4838 
  4839 lemma listrel_refl_on: "refl_on A r \<Longrightarrow> refl_on (lists A) (listrel r)" 
  4840 apply (simp add: refl_on_def listrel_subset Ball_def)
  4841 apply (rule allI) 
  4842 apply (induct_tac x) 
  4843 apply (auto intro: listrel.intros)
  4844 done
  4845 
  4846 lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
  4847 apply (auto simp add: sym_def)
  4848 apply (erule listrel.induct) 
  4849 apply (blast intro: listrel.intros)+
  4850 done
  4851 
  4852 lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
  4853 apply (simp add: trans_def)
  4854 apply (intro allI) 
  4855 apply (rule impI) 
  4856 apply (erule listrel.induct) 
  4857 apply (blast intro: listrel.intros)+
  4858 done
  4859 
  4860 theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
  4861 by (simp add: equiv_def listrel_refl_on listrel_sym listrel_trans) 
  4862 
  4863 lemma listrel_rtrancl_refl[iff]: "(xs,xs) : listrel(r^*)"
  4864 using listrel_refl_on[of UNIV, OF refl_rtrancl]
  4865 by(auto simp: refl_on_def)
  4866 
  4867 lemma listrel_rtrancl_trans:
  4868   "\<lbrakk> (xs,ys) : listrel(r^*);  (ys,zs) : listrel(r^*) \<rbrakk>
  4869   \<Longrightarrow> (xs,zs) : listrel(r^*)"
  4870 by (metis listrel_trans trans_def trans_rtrancl)
  4871 
  4872 
  4873 lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
  4874 by (blast intro: listrel.intros)
  4875 
  4876 lemma listrel_Cons:
  4877      "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})"
  4878 by (auto simp add: set_Cons_def intro: listrel.intros)
  4879 
  4880 text {* Relating @{term listrel1}, @{term listrel} and closures: *}
  4881 
  4882 lemma listrel1_rtrancl_subset_rtrancl_listrel1:
  4883   "listrel1 (r^*) \<subseteq> (listrel1 r)^*"
  4884 proof (rule subrelI)
  4885   fix xs ys assume 1: "(xs,ys) \<in> listrel1 (r^*)"
  4886   { fix x y us vs
  4887     have "(x,y) : r^* \<Longrightarrow> (us @ x # vs, us @ y # vs) : (listrel1 r)^*"
  4888     proof(induct rule: rtrancl.induct)
  4889       case rtrancl_refl show ?case by simp
  4890     next
  4891       case rtrancl_into_rtrancl thus ?case
  4892         by (metis listrel1I rtrancl.rtrancl_into_rtrancl)
  4893     qed }
  4894   thus "(xs,ys) \<in> (listrel1 r)^*" using 1 by(blast elim: listrel1E)
  4895 qed
  4896 
  4897 lemma rtrancl_listrel1_eq_len: "(x,y) \<in> (listrel1 r)^* \<Longrightarrow> length x = length y"
  4898 by (induct rule: rtrancl.induct) (auto intro: listrel1_eq_len)
  4899 
  4900 lemma rtrancl_listrel1_ConsI1:
  4901   "(xs,ys) : (listrel1 r)^* \<Longrightarrow> (x#xs,x#ys) : (listrel1 r)^*"
  4902 apply(induct rule: rtrancl.induct)
  4903  apply simp
  4904 by (metis listrel1I2 rtrancl.rtrancl_into_rtrancl)
  4905 
  4906 lemma rtrancl_listrel1_ConsI2:
  4907   "(x,y) \<in> r^* \<Longrightarrow> (xs, ys) \<in> (listrel1 r)^*
  4908   \<Longrightarrow> (x # xs, y # ys) \<in> (listrel1 r)^*"
  4909   by (blast intro: rtrancl_trans rtrancl_listrel1_ConsI1 
  4910     subsetD[OF listrel1_rtrancl_subset_rtrancl_listrel1 listrel1I1])
  4911 
  4912 lemma listrel1_subset_listrel:
  4913   "r \<subseteq> r' \<Longrightarrow> refl r' \<Longrightarrow> listrel1 r \<subseteq> listrel(r')"
  4914 by(auto elim!: listrel1E simp add: listrel_iff_zip set_zip refl_on_def)
  4915 
  4916 lemma listrel_reflcl_if_listrel1:
  4917   "(xs,ys) : listrel1 r \<Longrightarrow> (xs,ys) : listrel(r^*)"
  4918 by(erule listrel1E)(auto simp add: listrel_iff_zip set_zip)
  4919 
  4920 lemma listrel_rtrancl_eq_rtrancl_listrel1: "listrel (r^*) = (listrel1 r)^*"
  4921 proof
  4922   { fix x y assume "(x,y) \<in> listrel (r^*)"
  4923     then have "(x,y) \<in> (listrel1 r)^*"
  4924     by induct (auto intro: rtrancl_listrel1_ConsI2) }
  4925   then show "listrel (r^*) \<subseteq> (listrel1 r)^*"
  4926     by (rule subrelI)
  4927 next
  4928   show "listrel (r^*) \<supseteq> (listrel1 r)^*"
  4929   proof(rule subrelI)
  4930     fix xs ys assume "(xs,ys) \<in> (listrel1 r)^*"
  4931     then show "(xs,ys) \<in> listrel (r^*)"
  4932     proof induct
  4933       case base show ?case by(auto simp add: listrel_iff_zip set_zip)
  4934     next
  4935       case (step ys zs)
  4936       thus ?case  by (metis listrel_reflcl_if_listrel1 listrel_rtrancl_trans)
  4937     qed
  4938   qed
  4939 qed
  4940 
  4941 lemma rtrancl_listrel1_if_listrel:
  4942   "(xs,ys) : listrel r \<Longrightarrow> (xs,ys) : (listrel1 r)^*"
  4943 by(metis listrel_rtrancl_eq_rtrancl_listrel1 subsetD[OF listrel_mono] r_into_rtrancl subsetI)
  4944 
  4945 lemma listrel_subset_rtrancl_listrel1: "listrel r \<subseteq> (listrel1 r)^*"
  4946 by(fast intro:rtrancl_listrel1_if_listrel)
  4947 
  4948 
  4949 subsection {* Size function *}
  4950 
  4951 lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (list_size f)"
  4952 by (rule is_measure_trivial)
  4953 
  4954 lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (option_size f)"
  4955 by (rule is_measure_trivial)
  4956 
  4957 lemma list_size_estimation[termination_simp]: 
  4958   "x \<in> set xs \<Longrightarrow> y < f x \<Longrightarrow> y < list_size f xs"
  4959 by (induct xs) auto
  4960 
  4961 lemma list_size_estimation'[termination_simp]: 
  4962   "x \<in> set xs \<Longrightarrow> y \<le> f x \<Longrightarrow> y \<le> list_size f xs"
  4963 by (induct xs) auto
  4964 
  4965 lemma list_size_map[simp]: "list_size f (map g xs) = list_size (f o g) xs"
  4966 by (induct xs) auto
  4967 
  4968 lemma list_size_append[simp]: "list_size f (xs @ ys) = list_size f xs + list_size f ys"
  4969 by (induct xs, auto)
  4970 
  4971 lemma list_size_pointwise[termination_simp]: 
  4972   "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> list_size f xs \<le> list_size g xs"
  4973 by (induct xs) force+
  4974 
  4975 
  4976 subsection {* Transfer *}
  4977 
  4978 definition
  4979   embed_list :: "nat list \<Rightarrow> int list"
  4980 where
  4981   "embed_list l = map int l"
  4982 
  4983 definition
  4984   nat_list :: "int list \<Rightarrow> bool"
  4985 where
  4986   "nat_list l = nat_set (set l)"
  4987 
  4988 definition
  4989   return_list :: "int list \<Rightarrow> nat list"
  4990 where
  4991   "return_list l = map nat l"
  4992 
  4993 lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow>
  4994     embed_list (return_list l) = l"
  4995   unfolding embed_list_def return_list_def nat_list_def nat_set_def
  4996   apply (induct l)
  4997   apply auto
  4998 done
  4999 
  5000 lemma transfer_nat_int_list_functions:
  5001   "l @ m = return_list (embed_list l @ embed_list m)"
  5002   "[] = return_list []"
  5003   unfolding return_list_def embed_list_def
  5004   apply auto
  5005   apply (induct l, auto)
  5006   apply (induct m, auto)
  5007 done
  5008 
  5009 (*
  5010 lemma transfer_nat_int_fold1: "fold f l x =
  5011     fold (%x. f (nat x)) (embed_list l) x";
  5012 *)
  5013 
  5014 
  5015 subsection {* Code generation *}
  5016 
  5017 subsubsection {* Counterparts for set-related operations *}
  5018 
  5019 definition member :: "'a list \<Rightarrow> 'a \<Rightarrow> bool" where
  5020   [code_abbrev]: "member xs x \<longleftrightarrow> x \<in> set xs"
  5021 
  5022 text {*
  5023   Use @{text member} only for generating executable code.  Otherwise use
  5024   @{prop "x \<in> set xs"} instead --- it is much easier to reason about.
  5025 *}
  5026 
  5027 lemma member_rec [code]:
  5028   "member (x # xs) y \<longleftrightarrow> x = y \<or> member xs y"
  5029   "member [] y \<longleftrightarrow> False"
  5030   by (auto simp add: member_def)
  5031 
  5032 lemma in_set_member (* FIXME delete candidate *):
  5033   "x \<in> set xs \<longleftrightarrow> member xs x"
  5034   by (simp add: member_def)
  5035 
  5036 declare INF_def [code_unfold]
  5037 declare SUP_def [code_unfold]
  5038 
  5039 declare set_map [symmetric, code_unfold]
  5040 
  5041 definition list_all :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5042   list_all_iff [code_abbrev]: "list_all P xs \<longleftrightarrow> Ball (set xs) P"
  5043 
  5044 definition list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5045   list_ex_iff [code_abbrev]: "list_ex P xs \<longleftrightarrow> Bex (set xs) P"
  5046 
  5047 definition list_ex1 :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  5048   list_ex1_iff [code_abbrev]: "list_ex1 P xs \<longleftrightarrow> (\<exists>! x. x \<in> set xs \<and> P x)"
  5049 
  5050 text {*
  5051   Usually you should prefer @{text "\<forall>x\<in>set xs"}, @{text "\<exists>x\<in>set xs"}
  5052   and @{text "\<exists>!x. x\<in>set xs \<and> _"} over @{const list_all}, @{const list_ex}
  5053   and @{const list_ex1} in specifications.
  5054 *}
  5055 
  5056 lemma list_all_simps [simp, code]:
  5057   "list_all P (x # xs) \<longleftrightarrow> P x \<and> list_all P xs"
  5058   "list_all P [] \<longleftrightarrow> True"
  5059   by (simp_all add: list_all_iff)
  5060 
  5061 lemma list_ex_simps [simp, code]:
  5062   "list_ex P (x # xs) \<longleftrightarrow> P x \<or> list_ex P xs"
  5063   "list_ex P [] \<longleftrightarrow> False"
  5064   by (simp_all add: list_ex_iff)
  5065 
  5066 lemma list_ex1_simps [simp, code]:
  5067   "list_ex1 P [] = False"
  5068   "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)"
  5069   by (auto simp add: list_ex1_iff list_all_iff)
  5070 
  5071 lemma Ball_set_list_all: (* FIXME delete candidate *)
  5072   "Ball (set xs) P \<longleftrightarrow> list_all P xs"
  5073   by (simp add: list_all_iff)
  5074 
  5075 lemma Bex_set_list_ex: (* FIXME delete candidate *)
  5076   "Bex (set xs) P \<longleftrightarrow> list_ex P xs"
  5077   by (simp add: list_ex_iff)
  5078 
  5079 lemma list_all_append [simp]:
  5080   "list_all P (xs @ ys) \<longleftrightarrow> list_all P xs \<and> list_all P ys"
  5081   by (auto simp add: list_all_iff)
  5082 
  5083 lemma list_ex_append [simp]:
  5084   "list_ex P (xs @ ys) \<longleftrightarrow> list_ex P xs \<or> list_ex P ys"
  5085   by (auto simp add: list_ex_iff)
  5086 
  5087 lemma list_all_rev [simp]:
  5088   "list_all P (rev xs) \<longleftrightarrow> list_all P xs"
  5089   by (simp add: list_all_iff)
  5090 
  5091 lemma list_ex_rev [simp]:
  5092   "list_ex P (rev xs) \<longleftrightarrow> list_ex P xs"
  5093   by (simp add: list_ex_iff)
  5094 
  5095 lemma list_all_length:
  5096   "list_all P xs \<longleftrightarrow> (\<forall>n < length xs. P (xs ! n))"
  5097   by (auto simp add: list_all_iff set_conv_nth)
  5098 
  5099 lemma list_ex_length:
  5100   "list_ex P xs \<longleftrightarrow> (\<exists>n < length xs. P (xs ! n))"
  5101   by (auto simp add: list_ex_iff set_conv_nth)
  5102 
  5103 lemma list_all_cong [fundef_cong]:
  5104   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_all f xs = list_all g ys"
  5105   by (simp add: list_all_iff)
  5106 
  5107 lemma list_any_cong [fundef_cong]:
  5108   "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_ex f xs = list_ex g ys"
  5109   by (simp add: list_ex_iff)
  5110 
  5111 text {* Bounded quantification and summation over nats. *}
  5112 
  5113 lemma atMost_upto [code_unfold]:
  5114   "{..n} = set [0..<Suc n]"
  5115   by auto
  5116 
  5117 lemma atLeast_upt [code_unfold]:
  5118   "{..<n} = set [0..<n]"
  5119   by auto
  5120 
  5121 lemma greaterThanLessThan_upt [code_unfold]:
  5122   "{n<..<m} = set [Suc n..<m]"
  5123   by auto
  5124 
  5125 lemmas atLeastLessThan_upt [code_unfold] = set_upt [symmetric]
  5126 
  5127 lemma greaterThanAtMost_upt [code_unfold]:
  5128   "{n<..m} = set [Suc n..<Suc m]"
  5129   by auto
  5130 
  5131 lemma atLeastAtMost_upt [code_unfold]:
  5132   "{n..m} = set [n..<Suc m]"
  5133   by auto
  5134 
  5135 lemma all_nat_less_eq [code_unfold]:
  5136   "(\<forall>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..<n}. P m)"
  5137   by auto
  5138 
  5139 lemma ex_nat_less_eq [code_unfold]:
  5140   "(\<exists>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..<n}. P m)"
  5141   by auto
  5142 
  5143 lemma all_nat_less [code_unfold]:
  5144   "(\<forall>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..n}. P m)"
  5145   by auto
  5146 
  5147 lemma ex_nat_less [code_unfold]:
  5148   "(\<exists>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..n}. P m)"
  5149   by auto
  5150 
  5151 lemma setsum_set_upt_conv_listsum_nat [code_unfold]:
  5152   "setsum f (set [m..<n]) = listsum (map f [m..<n])"
  5153   by (simp add: interv_listsum_conv_setsum_set_nat)
  5154 
  5155 text {* Summation over ints. *}
  5156 
  5157 lemma greaterThanLessThan_upto [code_unfold]:
  5158   "{i<..<j::int} = set [i+1..j - 1]"
  5159 by auto
  5160 
  5161 lemma atLeastLessThan_upto [code_unfold]:
  5162   "{i..<j::int} = set [i..j - 1]"
  5163 by auto
  5164 
  5165 lemma greaterThanAtMost_upto [code_unfold]:
  5166   "{i<..j::int} = set [i+1..j]"
  5167 by auto
  5168 
  5169 lemmas atLeastAtMost_upto [code_unfold] = set_upto [symmetric]
  5170 
  5171 lemma setsum_set_upto_conv_listsum_int [code_unfold]:
  5172   "setsum f (set [i..j::int]) = listsum (map f [i..j])"
  5173   by (simp add: interv_listsum_conv_setsum_set_int)
  5174 
  5175 
  5176 subsubsection {* Optimizing by rewriting *}
  5177 
  5178 definition null :: "'a list \<Rightarrow> bool" where
  5179   [code_abbrev]: "null xs \<longleftrightarrow> xs = []"
  5180 
  5181 text {*
  5182   Efficient emptyness check is implemented by @{const null}.
  5183 *}
  5184 
  5185 lemma null_rec [code]:
  5186   "null (x # xs) \<longleftrightarrow> False"
  5187   "null [] \<longleftrightarrow> True"
  5188   by (simp_all add: null_def)
  5189 
  5190 lemma eq_Nil_null: (* FIXME delete candidate *)
  5191   "xs = [] \<longleftrightarrow> null xs"
  5192   by (simp add: null_def)
  5193 
  5194 lemma equal_Nil_null [code_unfold]:
  5195   "HOL.equal xs [] \<longleftrightarrow> null xs"
  5196   by (simp add: equal eq_Nil_null)
  5197 
  5198 definition maps :: "('a \<Rightarrow> 'b list) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  5199   [code_abbrev]: "maps f xs = concat (map f xs)"
  5200 
  5201 definition map_filter :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
  5202   [code_post]: "map_filter f xs = map (the \<circ> f) (filter (\<lambda>x. f x \<noteq> None) xs)"
  5203 
  5204 text {*
  5205   Operations @{const maps} and @{const map_filter} avoid
  5206   intermediate lists on execution -- do not use for proving.
  5207 *}
  5208 
  5209 lemma maps_simps [code]:
  5210   "maps f (x # xs) = f x @ maps f xs"
  5211   "maps f [] = []"
  5212   by (simp_all add: maps_def)
  5213 
  5214 lemma map_filter_simps [code]:
  5215   "map_filter f (x # xs) = (case f x of None \<Rightarrow> map_filter f xs | Some y \<Rightarrow> y # map_filter f xs)"
  5216   "map_filter f [] = []"
  5217   by (simp_all add: map_filter_def split: option.split)
  5218 
  5219 lemma concat_map_maps: (* FIXME delete candidate *)
  5220   "concat (map f xs) = maps f xs"
  5221   by (simp add: maps_def)
  5222 
  5223 lemma map_filter_map_filter [code_unfold]:
  5224   "map f (filter P xs) = map_filter (\<lambda>x. if P x then Some (f x) else None) xs"
  5225   by (simp add: map_filter_def)
  5226 
  5227 text {* Optimized code for @{text"\<forall>i\<in>{a..b::int}"} and @{text"\<forall>n:{a..<b::nat}"}
  5228 and similiarly for @{text"\<exists>"}. *}
  5229 
  5230 definition all_interval_nat :: "(nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
  5231   "all_interval_nat P i j \<longleftrightarrow> (\<forall>n \<in> {i..<j}. P n)"
  5232 
  5233 lemma [code]:
  5234   "all_interval_nat P i j \<longleftrightarrow> i \<ge> j \<or> P i \<and> all_interval_nat P (Suc i) j"
  5235 proof -
  5236   have *: "\<And>n. P i \<Longrightarrow> \<forall>n\<in>{Suc i..<j}. P n \<Longrightarrow> i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n"
  5237   proof -
  5238     fix n
  5239     assume "P i" "\<forall>n\<in>{Suc i..<j}. P n" "i \<le> n" "n < j"
  5240     then show "P n" by (cases "n = i") simp_all
  5241   qed
  5242   show ?thesis by (auto simp add: all_interval_nat_def intro: *)
  5243 qed
  5244 
  5245 lemma list_all_iff_all_interval_nat [code_unfold]:
  5246   "list_all P [i..<j] \<longleftrightarrow> all_interval_nat P i j"
  5247   by (simp add: list_all_iff all_interval_nat_def)
  5248 
  5249 lemma list_ex_iff_not_all_inverval_nat [code_unfold]:
  5250   "list_ex P [i..<j] \<longleftrightarrow> \<not> (all_interval_nat (Not \<circ> P) i j)"
  5251   by (simp add: list_ex_iff all_interval_nat_def)
  5252 
  5253 definition all_interval_int :: "(int \<Rightarrow> bool) \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool" where
  5254   "all_interval_int P i j \<longleftrightarrow> (\<forall>k \<in> {i..j}. P k)"
  5255 
  5256 lemma [code]:
  5257   "all_interval_int P i j \<longleftrightarrow> i > j \<or> P i \<and> all_interval_int P (i + 1) j"
  5258 proof -
  5259   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"
  5260   proof -
  5261     fix k
  5262     assume "P i" "\<forall>k\<in>{i+1..j}. P k" "i \<le> k" "k \<le> j"
  5263     then show "P k" by (cases "k = i") simp_all
  5264   qed
  5265   show ?thesis by (auto simp add: all_interval_int_def intro: *)
  5266 qed
  5267 
  5268 lemma list_all_iff_all_interval_int [code_unfold]:
  5269   "list_all P [i..j] \<longleftrightarrow> all_interval_int P i j"
  5270   by (simp add: list_all_iff all_interval_int_def)
  5271 
  5272 lemma list_ex_iff_not_all_inverval_int [code_unfold]:
  5273   "list_ex P [i..j] \<longleftrightarrow> \<not> (all_interval_int (Not \<circ> P) i j)"
  5274   by (simp add: list_ex_iff all_interval_int_def)
  5275 
  5276 hide_const (open) member null maps map_filter all_interval_nat all_interval_int
  5277 
  5278 
  5279 subsubsection {* Pretty lists *}
  5280 
  5281 use "Tools/list_code.ML"
  5282 
  5283 code_type list
  5284   (SML "_ list")
  5285   (OCaml "_ list")
  5286   (Haskell "![(_)]")
  5287   (Scala "List[(_)]")
  5288 
  5289 code_const Nil
  5290   (SML "[]")
  5291   (OCaml "[]")
  5292   (Haskell "[]")
  5293   (Scala "!Nil")
  5294 
  5295 code_instance list :: equal
  5296   (Haskell -)
  5297 
  5298 code_const "HOL.equal \<Colon> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
  5299   (Haskell infix 4 "==")
  5300 
  5301 code_reserved SML
  5302   list
  5303 
  5304 code_reserved OCaml
  5305   list
  5306 
  5307 setup {* fold (List_Code.add_literal_list) ["SML", "OCaml", "Haskell", "Scala"] *}
  5308 
  5309 
  5310 subsubsection {* Use convenient predefined operations *}
  5311 
  5312 code_const "op @"
  5313   (SML infixr 7 "@")
  5314   (OCaml infixr 6 "@")
  5315   (Haskell infixr 5 "++")
  5316   (Scala infixl 7 "++")
  5317 
  5318 code_const map
  5319   (Haskell "map")
  5320 
  5321 code_const filter
  5322   (Haskell "filter")
  5323 
  5324 code_const concat
  5325   (Haskell "concat")
  5326 
  5327 code_const List.maps
  5328   (Haskell "concatMap")
  5329 
  5330 code_const rev
  5331   (Haskell "reverse")
  5332 
  5333 code_const zip
  5334   (Haskell "zip")
  5335 
  5336 code_const List.null
  5337   (Haskell "null")
  5338 
  5339 code_const takeWhile
  5340   (Haskell "takeWhile")
  5341 
  5342 code_const dropWhile
  5343   (Haskell "dropWhile")
  5344 
  5345 code_const list_all
  5346   (Haskell "all")
  5347 
  5348 code_const list_ex
  5349   (Haskell "any")
  5350 
  5351 end