src/HOL/List.thy
 author berghofe Tue Jul 12 11:51:31 2005 +0200 (2005-07-12) changeset 16770 1f1b1fae30e4 parent 16634 f19d58cfb47a child 16965 46697fa536ab permissions -rw-r--r--
Auxiliary functions to be used in generated code are now defined using "attach".
     1 (*  Title:      HOL/List.thy

     2     ID:         $Id$

     3     Author:     Tobias Nipkow

     4 *)

     5

     6 header {* The datatype of finite lists *}

     7

     8 theory List

     9 imports PreList

    10 begin

    11

    12 datatype 'a list =

    13     Nil    ("[]")

    14   | Cons 'a  "'a list"    (infixr "#" 65)

    15

    16 subsection{*Basic list processing functions*}

    17

    18 consts

    19   "@" :: "'a list => 'a list => 'a list"    (infixr 65)

    20   filter:: "('a => bool) => 'a list => 'a list"

    21   concat:: "'a list list => 'a list"

    22   foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"

    23   foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"

    24   hd:: "'a list => 'a"

    25   tl:: "'a list => 'a list"

    26   last:: "'a list => 'a"

    27   butlast :: "'a list => 'a list"

    28   set :: "'a list => 'a set"

    29   list_all:: "('a => bool) => ('a list => bool)"

    30   list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool"

    31   list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"

    32   map :: "('a=>'b) => ('a list => 'b list)"

    33   mem :: "'a => 'a list => bool"    (infixl 55)

    34   nth :: "'a list => nat => 'a"    (infixl "!" 100)

    35   list_update :: "'a list => nat => 'a => 'a list"

    36   take:: "nat => 'a list => 'a list"

    37   drop:: "nat => 'a list => 'a list"

    38   takeWhile :: "('a => bool) => 'a list => 'a list"

    39   dropWhile :: "('a => bool) => 'a list => 'a list"

    40   rev :: "'a list => 'a list"

    41   zip :: "'a list => 'b list => ('a * 'b) list"

    42   upt :: "nat => nat => nat list" ("(1[_..</_'])")

    43   remdups :: "'a list => 'a list"

    44   remove1 :: "'a => 'a list => 'a list"

    45   null:: "'a list => bool"

    46   "distinct":: "'a list => bool"

    47   replicate :: "nat => 'a => 'a list"

    48   rotate1 :: "'a list \<Rightarrow> 'a list"

    49   rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"

    50   sublist :: "'a list => nat set => 'a list"

    51

    52

    53 nonterminals lupdbinds lupdbind

    54

    55 syntax

    56   -- {* list Enumeration *}

    57   "@list" :: "args => 'a list"    ("[(_)]")

    58

    59   -- {* Special syntax for filter *}

    60   "@filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_:_./ _])")

    61

    62   -- {* list update *}

    63   "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")

    64   "" :: "lupdbind => lupdbinds"    ("_")

    65   "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")

    66   "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)

    67

    68   upto:: "nat => nat => nat list"    ("(1[_../_])")

    69

    70 translations

    71   "[x, xs]" == "x#[xs]"

    72   "[x]" == "x#[]"

    73   "[x:xs . P]"== "filter (%x. P) xs"

    74

    75   "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"

    76   "xs[i:=x]" == "list_update xs i x"

    77

    78   "[i..j]" == "[i..<(Suc j)]"

    79

    80

    81 syntax (xsymbols)

    82   "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")

    83 syntax (HTML output)

    84   "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")

    85

    86

    87 text {*

    88   Function @{text size} is overloaded for all datatypes. Users may

    89   refer to the list version as @{text length}. *}

    90

    91 syntax length :: "'a list => nat"

    92 translations "length" => "size :: _ list => nat"

    93

    94 typed_print_translation {*

    95   let

    96     fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] =

    97           Syntax.const "length" $t   98 | size_tr' _ _ _ = raise Match;   99 in [("size", size_tr')] end   100 *}   101   102   103 primrec   104 "hd(x#xs) = x"   105   106 primrec   107 "tl([]) = []"   108 "tl(x#xs) = xs"   109   110 primrec   111 "null([]) = True"   112 "null(x#xs) = False"   113   114 primrec   115 "last(x#xs) = (if xs=[] then x else last xs)"   116   117 primrec   118 "butlast []= []"   119 "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"   120   121 primrec   122 "x mem [] = False"   123 "x mem (y#ys) = (if y=x then True else x mem ys)"   124   125 primrec   126 "set [] = {}"   127 "set (x#xs) = insert x (set xs)"   128   129 primrec   130 list_all_Nil:"list_all P [] = True"   131 list_all_Cons: "list_all P (x#xs) = (P(x) \<and> list_all P xs)"   132   133 primrec   134 "list_ex P [] = False"   135 "list_ex P (x#xs) = (P x \<or> list_ex P xs)"   136   137 primrec   138 "map f [] = []"   139 "map f (x#xs) = f(x)#map f xs"   140   141 primrec   142 append_Nil:"[]@ys = ys"   143 append_Cons: "(x#xs)@ys = x#(xs@ys)"   144   145 primrec   146 "rev([]) = []"   147 "rev(x#xs) = rev(xs) @ [x]"   148   149 primrec   150 "filter P [] = []"   151 "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"   152   153 primrec   154 foldl_Nil:"foldl f a [] = a"   155 foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"   156   157 primrec   158 "foldr f [] a = a"   159 "foldr f (x#xs) a = f x (foldr f xs a)"   160   161 primrec   162 "concat([]) = []"   163 "concat(x#xs) = x @ concat(xs)"   164   165 primrec   166 drop_Nil:"drop n [] = []"   167 drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"   168 -- {*Warning: simpset does not contain this definition, but separate   169 theorems for @{text "n = 0"} and @{text "n = Suc k"} *}   170   171 primrec   172 take_Nil:"take n [] = []"   173 take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"   174 -- {*Warning: simpset does not contain this definition, but separate   175 theorems for @{text "n = 0"} and @{text "n = Suc k"} *}   176   177 primrec   178 nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"   179 -- {*Warning: simpset does not contain this definition, but separate   180 theorems for @{text "n = 0"} and @{text "n = Suc k"} *}   181   182 primrec   183 "[][i:=v] = []"   184 "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])"   185   186 primrec   187 "takeWhile P [] = []"   188 "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"   189   190 primrec   191 "dropWhile P [] = []"   192 "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"   193   194 primrec   195 "zip xs [] = []"   196 zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"   197 -- {*Warning: simpset does not contain this definition, but separate   198 theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}   199   200 primrec   201 upt_0: "[i..<0] = []"   202 upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"   203   204 primrec   205 "distinct [] = True"   206 "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"   207   208 primrec   209 "remdups [] = []"   210 "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"   211   212 primrec   213 "remove1 x [] = []"   214 "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)"   215   216 primrec   217 replicate_0: "replicate 0 x = []"   218 replicate_Suc: "replicate (Suc n) x = x # replicate n x"   219   220 defs   221 rotate1_def: "rotate1 xs == (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"   222 rotate_def: "rotate n == rotate1 ^ n"   223   224 list_all2_def:   225 "list_all2 P xs ys ==   226 length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)"   227   228 sublist_def:   229 "sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..<size xs]))"   230   231   232 lemma not_Cons_self [simp]: "xs \<noteq> x # xs"   233 by (induct xs) auto   234   235 lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]   236   237 lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"   238 by (induct xs) auto   239   240 lemma length_induct:   241 "(!!xs. \<forall>ys. length ys < length xs --> P ys ==> P xs) ==> P xs"   242 by (rule measure_induct [of length]) rules   243   244   245 subsubsection {* @{text length} *}   246   247 text {*   248 Needs to come before @{text "@"} because of theorem @{text   249 append_eq_append_conv}.   250 *}   251   252 lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"   253 by (induct xs) auto   254   255 lemma length_map [simp]: "length (map f xs) = length xs"   256 by (induct xs) auto   257   258 lemma length_rev [simp]: "length (rev xs) = length xs"   259 by (induct xs) auto   260   261 lemma length_tl [simp]: "length (tl xs) = length xs - 1"   262 by (cases xs) auto   263   264 lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"   265 by (induct xs) auto   266   267 lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"   268 by (induct xs) auto   269   270 lemma length_Suc_conv:   271 "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"   272 by (induct xs) auto   273   274 lemma Suc_length_conv:   275 "(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"   276 apply (induct xs, simp, simp)   277 apply blast   278 done   279   280 lemma impossible_Cons [rule_format]:   281 "length xs <= length ys --> xs = x # ys = False"   282 apply (induct xs, auto)   283 done   284   285 lemma list_induct2[consumes 1]: "\<And>ys.   286 \<lbrakk> length xs = length ys;   287 P [] [];   288 \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>   289 \<Longrightarrow> P xs ys"   290 apply(induct xs)   291 apply simp   292 apply(case_tac ys)   293 apply simp   294 apply(simp)   295 done   296   297 subsubsection {* @{text "@"} -- append *}   298   299 lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"   300 by (induct xs) auto   301   302 lemma append_Nil2 [simp]: "xs @ [] = xs"   303 by (induct xs) auto   304   305 lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"   306 by (induct xs) auto   307   308 lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"   309 by (induct xs) auto   310   311 lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"   312 by (induct xs) auto   313   314 lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"   315 by (induct xs) auto   316   317 lemma append_eq_append_conv [simp]:   318 "!!ys. length xs = length ys \<or> length us = length vs   319 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"   320 apply (induct xs)   321 apply (case_tac ys, simp, force)   322 apply (case_tac ys, force, simp)   323 done   324   325 lemma append_eq_append_conv2: "!!ys zs ts.   326 (xs @ ys = zs @ ts) =   327 (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"   328 apply (induct xs)   329 apply fastsimp   330 apply(case_tac zs)   331 apply simp   332 apply fastsimp   333 done   334   335 lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"   336 by simp   337   338 lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"   339 by simp   340   341 lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"   342 by simp   343   344 lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"   345 using append_same_eq [of _ _ "[]"] by auto   346   347 lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"   348 using append_same_eq [of "[]"] by auto   349   350 lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"   351 by (induct xs) auto   352   353 lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"   354 by (induct xs) auto   355   356 lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"   357 by (simp add: hd_append split: list.split)   358   359 lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"   360 by (simp split: list.split)   361   362 lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"   363 by (simp add: tl_append split: list.split)   364   365   366 lemma Cons_eq_append_conv: "x#xs = ys@zs =   367 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"   368 by(cases ys) auto   369   370 lemma append_eq_Cons_conv: "(ys@zs = x#xs) =   371 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"   372 by(cases ys) auto   373   374   375 text {* Trivial rules for solving @{text "@"}-equations automatically. *}   376   377 lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"   378 by simp   379   380 lemma Cons_eq_appendI:   381 "[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"   382 by (drule sym) simp   383   384 lemma append_eq_appendI:   385 "[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"   386 by (drule sym) simp   387   388   389 text {*   390 Simplification procedure for all list equalities.   391 Currently only tries to rearrange @{text "@"} to see if   392 - both lists end in a singleton list,   393 - or both lists end in the same list.   394 *}   395   396 ML_setup {*   397 local   398   399 val append_assoc = thm "append_assoc";   400 val append_Nil = thm "append_Nil";   401 val append_Cons = thm "append_Cons";   402 val append1_eq_conv = thm "append1_eq_conv";   403 val append_same_eq = thm "append_same_eq";   404   405 fun last (cons as Const("List.list.Cons",_)$ _ $xs) =   406 (case xs of Const("List.list.Nil",_) => cons | _ => last xs)   407 | last (Const("List.op @",_)$ _ $ys) = last ys   408 | last t = t;   409   410 fun list1 (Const("List.list.Cons",_)$ _ $Const("List.list.Nil",_)) = true   411 | list1 _ = false;   412   413 fun butlast ((cons as Const("List.list.Cons",_)$ x) $xs) =   414 (case xs of Const("List.list.Nil",_) => xs | _ => cons$ butlast xs)

   415   | butlast ((app as Const("List.op @",_) $xs)$ ys) = app $butlast ys   416 | butlast xs = Const("List.list.Nil",fastype_of xs);   417   418 val rearr_tac =   419 simp_tac (HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]);   420   421 fun list_eq sg _ (F as (eq as Const(_,eqT))$ lhs $rhs) =   422 let   423 val lastl = last lhs and lastr = last rhs;   424 fun rearr conv =   425 let   426 val lhs1 = butlast lhs and rhs1 = butlast rhs;   427 val Type(_,listT::_) = eqT   428 val appT = [listT,listT] ---> listT   429 val app = Const("List.op @",appT)   430 val F2 = eq$ (app$lhs1$lastl) $(app$rhs1$lastr)   431 val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));   432 val thm = Tactic.prove sg [] [] eq (K (rearr_tac 1));   433 in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;   434   435 in   436 if list1 lastl andalso list1 lastr then rearr append1_eq_conv   437 else if lastl aconv lastr then rearr append_same_eq   438 else NONE   439 end;   440   441 in   442   443 val list_eq_simproc =   444 Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq;   445   446 end;   447   448 Addsimprocs [list_eq_simproc];   449 *}   450   451   452 subsubsection {* @{text map} *}   453   454 lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"   455 by (induct xs) simp_all   456   457 lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"   458 by (rule ext, induct_tac xs) auto   459   460 lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"   461 by (induct xs) auto   462   463 lemma map_compose: "map (f o g) xs = map f (map g xs)"   464 by (induct xs) (auto simp add: o_def)   465   466 lemma rev_map: "rev (map f xs) = map f (rev xs)"   467 by (induct xs) auto   468   469 lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"   470 by (induct xs) auto   471   472 lemma map_cong [recdef_cong]:   473 "xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"   474 -- {* a congruence rule for @{text map} *}   475 by simp   476   477 lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"   478 by (cases xs) auto   479   480 lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"   481 by (cases xs) auto   482   483 lemma map_eq_Cons_conv[iff]:   484 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"   485 by (cases xs) auto   486   487 lemma Cons_eq_map_conv[iff]:   488 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"   489 by (cases ys) auto   490   491 lemma ex_map_conv:   492 "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"   493 by(induct ys, auto)   494   495 lemma map_eq_imp_length_eq:   496 "!!xs. map f xs = map f ys ==> length xs = length ys"   497 apply (induct ys)   498 apply simp   499 apply(simp (no_asm_use))   500 apply clarify   501 apply(simp (no_asm_use))   502 apply fast   503 done   504   505 lemma map_inj_on:   506 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]   507 ==> xs = ys"   508 apply(frule map_eq_imp_length_eq)   509 apply(rotate_tac -1)   510 apply(induct rule:list_induct2)   511 apply simp   512 apply(simp)   513 apply (blast intro:sym)   514 done   515   516 lemma inj_on_map_eq_map:   517 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"   518 by(blast dest:map_inj_on)   519   520 lemma map_injective:   521 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"   522 by (induct ys) (auto dest!:injD)   523   524 lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"   525 by(blast dest:map_injective)   526   527 lemma inj_mapI: "inj f ==> inj (map f)"   528 by (rules dest: map_injective injD intro: inj_onI)   529   530 lemma inj_mapD: "inj (map f) ==> inj f"   531 apply (unfold inj_on_def, clarify)   532 apply (erule_tac x = "[x]" in ballE)   533 apply (erule_tac x = "[y]" in ballE, simp, blast)   534 apply blast   535 done   536   537 lemma inj_map[iff]: "inj (map f) = inj f"   538 by (blast dest: inj_mapD intro: inj_mapI)   539   540 lemma inj_on_mapI: "inj_on f (\<Union>(set  A)) \<Longrightarrow> inj_on (map f) A"   541 apply(rule inj_onI)   542 apply(erule map_inj_on)   543 apply(blast intro:inj_onI dest:inj_onD)   544 done   545   546 lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"   547 by (induct xs, auto)   548   549 lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"   550 by (induct xs) auto   551   552 lemma map_fst_zip[simp]:   553 "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"   554 by (induct rule:list_induct2, simp_all)   555   556 lemma map_snd_zip[simp]:   557 "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"   558 by (induct rule:list_induct2, simp_all)   559   560   561 subsubsection {* @{text rev} *}   562   563 lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"   564 by (induct xs) auto   565   566 lemma rev_rev_ident [simp]: "rev (rev xs) = xs"   567 by (induct xs) auto   568   569 lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"   570 by auto   571   572 lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"   573 by (induct xs) auto   574   575 lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"   576 by (induct xs) auto   577   578 lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"   579 by (cases xs) auto   580   581 lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"   582 by (cases xs) auto   583   584 lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"   585 apply (induct xs, force)   586 apply (case_tac ys, simp, force)   587 done   588   589 lemma inj_on_rev[iff]: "inj_on rev A"   590 by(simp add:inj_on_def)   591   592 lemma rev_induct [case_names Nil snoc]:   593 "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"   594 apply(simplesubst rev_rev_ident[symmetric])   595 apply(rule_tac list = "rev xs" in list.induct, simp_all)   596 done   597   598 ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"   599   600 lemma rev_exhaust [case_names Nil snoc]:   601 "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"   602 by (induct xs rule: rev_induct) auto   603   604 lemmas rev_cases = rev_exhaust   605   606   607 subsubsection {* @{text set} *}   608   609 lemma finite_set [iff]: "finite (set xs)"   610 by (induct xs) auto   611   612 lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"   613 by (induct xs) auto   614   615 lemma hd_in_set: "l = x#xs \<Longrightarrow> x\<in>set l"   616 by (case_tac l, auto)   617   618 lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"   619 by auto   620   621 lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs"   622 by auto   623   624 lemma set_empty [iff]: "(set xs = {}) = (xs = [])"   625 by (induct xs) auto   626   627 lemma set_empty2[iff]: "({} = set xs) = (xs = [])"   628 by(induct xs) auto   629   630 lemma set_rev [simp]: "set (rev xs) = set xs"   631 by (induct xs) auto   632   633 lemma set_map [simp]: "set (map f xs) = f(set xs)"   634 by (induct xs) auto   635   636 lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"   637 by (induct xs) auto   638   639 lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"   640 apply (induct j, simp_all)   641 apply (erule ssubst, auto)   642 done   643   644 lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"   645 proof (induct xs)   646 case Nil show ?case by simp   647 case (Cons a xs)   648 show ?case   649 proof   650 assume "x \<in> set (a # xs)"   651 with prems show "\<exists>ys zs. a # xs = ys @ x # zs"   652 by (simp, blast intro: Cons_eq_appendI)   653 next   654 assume "\<exists>ys zs. a # xs = ys @ x # zs"   655 then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast   656 show "x \<in> set (a # xs)"   657 by (cases ys, auto simp add: eq)   658 qed   659 qed   660   661 lemma finite_list: "finite A ==> EX l. set l = A"   662 apply (erule finite_induct, auto)   663 apply (rule_tac x="x#l" in exI, auto)   664 done   665   666 lemma card_length: "card (set xs) \<le> length xs"   667 by (induct xs) (auto simp add: card_insert_if)   668   669   670 subsubsection {* @{text mem}, @{text list_all} and @{text list_ex} *}   671   672 text{* Only use @{text mem} for generating executable code. Otherwise   673 use @{prop"x : set xs"} instead --- it is much easier to reason about.   674 The same is true for @{text list_all} and @{text list_ex}: write   675 @{text"\<forall>x\<in>set xs"} and @{text"\<exists>x\<in>set xs"} instead because the HOL   676 quantifiers are aleady known to the automatic provers. For the purpose   677 of generating executable code use the theorems @{text set_mem_eq},   678 @{text list_all_conv} and @{text list_ex_iff} to get rid off or   679 introduce the combinators. *}   680   681 lemma set_mem_eq: "(x mem xs) = (x : set xs)"   682 by (induct xs) auto   683   684 lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)"   685 by (induct xs) auto   686   687 lemma list_all_append [simp]:   688 "list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"   689 by (induct xs) auto   690   691 lemma list_all_rev [simp]: "list_all P (rev xs) = list_all P xs"   692 by (simp add: list_all_conv)   693   694 lemma list_ex_iff: "list_ex P xs = (\<exists>x \<in> set xs. P x)"   695 by (induct xs) simp_all   696   697   698 subsubsection {* @{text filter} *}   699   700 lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"   701 by (induct xs) auto   702   703 lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"   704 by (induct xs) simp_all   705   706 lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"   707 by (induct xs) auto   708   709 lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"   710 by (induct xs) auto   711   712 lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"   713 by (induct xs) auto   714   715 lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"   716 by (induct xs) (auto simp add: le_SucI)   717   718 lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"   719 by auto   720   721 lemma length_filter_less:   722 "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"   723 proof (induct xs)   724 case Nil thus ?case by simp   725 next   726 case (Cons x xs) thus ?case   727 apply (auto split:split_if_asm)   728 using length_filter_le[of P xs] apply arith   729 done   730 qed   731   732 lemma length_filter_conv_card:   733 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"   734 proof (induct xs)   735 case Nil thus ?case by simp   736 next   737 case (Cons x xs)   738 let ?S = "{i. i < length xs & p(xs!i)}"   739 have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)   740 show ?case (is "?l = card ?S'")   741 proof (cases)   742 assume "p x"   743 hence eq: "?S' = insert 0 (Suc  ?S)"   744 by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)   745 have "length (filter p (x # xs)) = Suc(card ?S)"   746 using Cons by simp   747 also have "\<dots> = Suc(card(Suc  ?S))" using fin   748 by (simp add: card_image inj_Suc)   749 also have "\<dots> = card ?S'" using eq fin   750 by (simp add:card_insert_if) (simp add:image_def)   751 finally show ?thesis .   752 next   753 assume "\<not> p x"   754 hence eq: "?S' = Suc  ?S"   755 by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)   756 have "length (filter p (x # xs)) = card ?S"   757 using Cons by simp   758 also have "\<dots> = card(Suc  ?S)" using fin   759 by (simp add: card_image inj_Suc)   760 also have "\<dots> = card ?S'" using eq fin   761 by (simp add:card_insert_if)   762 finally show ?thesis .   763 qed   764 qed   765   766   767 subsubsection {* @{text concat} *}   768   769 lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"   770 by (induct xs) auto   771   772 lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"   773 by (induct xss) auto   774   775 lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"   776 by (induct xss) auto   777   778 lemma set_concat [simp]: "set (concat xs) = \<Union>(set  set xs)"   779 by (induct xs) auto   780   781 lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"   782 by (induct xs) auto   783   784 lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"   785 by (induct xs) auto   786   787 lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"   788 by (induct xs) auto   789   790   791 subsubsection {* @{text nth} *}   792   793 lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"   794 by auto   795   796 lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"   797 by auto   798   799 declare nth.simps [simp del]   800   801 lemma nth_append:   802 "!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"   803 apply (induct "xs", simp)   804 apply (case_tac n, auto)   805 done   806   807 lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"   808 by (induct "xs") auto   809   810 lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"   811 by (induct "xs") auto   812   813 lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"   814 apply (induct xs, simp)   815 apply (case_tac n, auto)   816 done   817   818 lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"   819 apply (induct xs, simp, simp)   820 apply safe   821 apply (rule_tac x = 0 in exI, simp)   822 apply (rule_tac x = "Suc i" in exI, simp)   823 apply (case_tac i, simp)   824 apply (rename_tac j)   825 apply (rule_tac x = j in exI, simp)   826 done   827   828 lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"   829 by (auto simp add: set_conv_nth)   830   831 lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"   832 by (auto simp add: set_conv_nth)   833   834 lemma all_nth_imp_all_set:   835 "[| !i < length xs. P(xs!i); x : set xs|] ==> P x"   836 by (auto simp add: set_conv_nth)   837   838 lemma all_set_conv_all_nth:   839 "(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"   840 by (auto simp add: set_conv_nth)   841   842   843 subsubsection {* @{text list_update} *}   844   845 lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"   846 by (induct xs) (auto split: nat.split)   847   848 lemma nth_list_update:   849 "!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"   850 by (induct xs) (auto simp add: nth_Cons split: nat.split)   851   852 lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"   853 by (simp add: nth_list_update)   854   855 lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"   856 by (induct xs) (auto simp add: nth_Cons split: nat.split)   857   858 lemma list_update_overwrite [simp]:   859 "!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"   860 by (induct xs) (auto split: nat.split)   861   862 lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"   863 apply (induct xs, simp)   864 apply(simp split:nat.splits)   865 done   866   867 lemma list_update_same_conv:   868 "!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"   869 by (induct xs) (auto split: nat.split)   870   871 lemma list_update_append1:   872 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"   873 apply (induct xs, simp)   874 apply(simp split:nat.split)   875 done   876   877 lemma list_update_append:   878 "!!n. (xs @ ys) [n:= x] =   879 (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"   880 by (induct xs) (auto split:nat.splits)   881   882 lemma list_update_length [simp]:   883 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"   884 by (induct xs, auto)   885   886 lemma update_zip:   887 "!!i xy xs. length xs = length ys ==>   888 (zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"   889 by (induct ys) (auto, case_tac xs, auto split: nat.split)   890   891 lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"   892 by (induct xs) (auto split: nat.split)   893   894 lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"   895 by (blast dest!: set_update_subset_insert [THEN subsetD])   896   897 lemma set_update_memI: "!!n. n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"   898 by (induct xs) (auto split:nat.splits)   899   900   901 subsubsection {* @{text last} and @{text butlast} *}   902   903 lemma last_snoc [simp]: "last (xs @ [x]) = x"   904 by (induct xs) auto   905   906 lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"   907 by (induct xs) auto   908   909 lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"   910 by(simp add:last.simps)   911   912 lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"   913 by(simp add:last.simps)   914   915 lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"   916 by (induct xs) (auto)   917   918 lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"   919 by(simp add:last_append)   920   921 lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"   922 by(simp add:last_append)   923   924   925 lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"   926 by (induct xs rule: rev_induct) auto   927   928 lemma butlast_append:   929 "!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"   930 by (induct xs) auto   931   932 lemma append_butlast_last_id [simp]:   933 "xs \<noteq> [] ==> butlast xs @ [last xs] = xs"   934 by (induct xs) auto   935   936 lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"   937 by (induct xs) (auto split: split_if_asm)   938   939 lemma in_set_butlast_appendI:   940 "x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"   941 by (auto dest: in_set_butlastD simp add: butlast_append)   942   943   944 subsubsection {* @{text take} and @{text drop} *}   945   946 lemma take_0 [simp]: "take 0 xs = []"   947 by (induct xs) auto   948   949 lemma drop_0 [simp]: "drop 0 xs = xs"   950 by (induct xs) auto   951   952 lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"   953 by simp   954   955 lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"   956 by simp   957   958 declare take_Cons [simp del] and drop_Cons [simp del]   959   960 lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"   961 by(clarsimp simp add:neq_Nil_conv)   962   963 lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"   964 by(cases xs, simp_all)   965   966 lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"   967 by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)   968   969 lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"   970 apply (induct xs, simp)   971 apply(simp add:drop_Cons nth_Cons split:nat.splits)   972 done   973   974 lemma take_Suc_conv_app_nth:   975 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"   976 apply (induct xs, simp)   977 apply (case_tac i, auto)   978 done   979   980 lemma drop_Suc_conv_tl:   981 "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"   982 apply (induct xs, simp)   983 apply (case_tac i, auto)   984 done   985   986 lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"   987 by (induct n) (auto, case_tac xs, auto)   988   989 lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"   990 by (induct n) (auto, case_tac xs, auto)   991   992 lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"   993 by (induct n) (auto, case_tac xs, auto)   994   995 lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"   996 by (induct n) (auto, case_tac xs, auto)   997   998 lemma take_append [simp]:   999 "!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"   1000 by (induct n) (auto, case_tac xs, auto)   1001   1002 lemma drop_append [simp]:   1003 "!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"   1004 by (induct n) (auto, case_tac xs, auto)   1005   1006 lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"   1007 apply (induct m, auto)   1008 apply (case_tac xs, auto)   1009 apply (case_tac n, auto)   1010 done   1011   1012 lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"   1013 apply (induct m, auto)   1014 apply (case_tac xs, auto)   1015 done   1016   1017 lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"   1018 apply (induct m, auto)   1019 apply (case_tac xs, auto)   1020 done   1021   1022 lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"   1023 apply(induct xs)   1024 apply simp   1025 apply(simp add: take_Cons drop_Cons split:nat.split)   1026 done   1027   1028 lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"   1029 apply (induct n, auto)   1030 apply (case_tac xs, auto)   1031 done   1032   1033 lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"   1034 apply(induct xs)   1035 apply simp   1036 apply(simp add:take_Cons split:nat.split)   1037 done   1038   1039 lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"   1040 apply(induct xs)   1041 apply simp   1042 apply(simp add:drop_Cons split:nat.split)   1043 done   1044   1045 lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"   1046 apply (induct n, auto)   1047 apply (case_tac xs, auto)   1048 done   1049   1050 lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"   1051 apply (induct n, auto)   1052 apply (case_tac xs, auto)   1053 done   1054   1055 lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"   1056 apply (induct xs, auto)   1057 apply (case_tac i, auto)   1058 done   1059   1060 lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"   1061 apply (induct xs, auto)   1062 apply (case_tac i, auto)   1063 done   1064   1065 lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"   1066 apply (induct xs, auto)   1067 apply (case_tac n, blast)   1068 apply (case_tac i, auto)   1069 done   1070   1071 lemma nth_drop [simp]:   1072 "!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"   1073 apply (induct n, auto)   1074 apply (case_tac xs, auto)   1075 done   1076   1077 lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"   1078 by(induct xs)(auto simp:take_Cons split:nat.split)   1079   1080 lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"   1081 by(induct xs)(auto simp:drop_Cons split:nat.split)   1082   1083 lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"   1084 using set_take_subset by fast   1085   1086 lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"   1087 using set_drop_subset by fast   1088   1089 lemma append_eq_conv_conj:   1090 "!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"   1091 apply (induct xs, simp, clarsimp)   1092 apply (case_tac zs, auto)   1093 done   1094   1095 lemma take_add [rule_format]:   1096 "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"   1097 apply (induct xs, auto)   1098 apply (case_tac i, simp_all)   1099 done   1100   1101 lemma append_eq_append_conv_if:   1102 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =   1103 (if size xs\<^isub>1 \<le> size ys\<^isub>1   1104 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   1105 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)"   1106 apply(induct xs\<^isub>1)   1107 apply simp   1108 apply(case_tac ys\<^isub>1)   1109 apply simp_all   1110 done   1111   1112 lemma take_hd_drop:   1113 "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"   1114 apply(induct xs)   1115 apply simp   1116 apply(simp add:drop_Cons split:nat.split)   1117 done   1118   1119   1120 subsubsection {* @{text takeWhile} and @{text dropWhile} *}   1121   1122 lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"   1123 by (induct xs) auto   1124   1125 lemma takeWhile_append1 [simp]:   1126 "[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"   1127 by (induct xs) auto   1128   1129 lemma takeWhile_append2 [simp]:   1130 "(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"   1131 by (induct xs) auto   1132   1133 lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"   1134 by (induct xs) auto   1135   1136 lemma dropWhile_append1 [simp]:   1137 "[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"   1138 by (induct xs) auto   1139   1140 lemma dropWhile_append2 [simp]:   1141 "(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"   1142 by (induct xs) auto   1143   1144 lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"   1145 by (induct xs) (auto split: split_if_asm)   1146   1147 lemma takeWhile_eq_all_conv[simp]:   1148 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"   1149 by(induct xs, auto)   1150   1151 lemma dropWhile_eq_Nil_conv[simp]:   1152 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"   1153 by(induct xs, auto)   1154   1155 lemma dropWhile_eq_Cons_conv:   1156 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"   1157 by(induct xs, auto)   1158   1159   1160 subsubsection {* @{text zip} *}   1161   1162 lemma zip_Nil [simp]: "zip [] ys = []"   1163 by (induct ys) auto   1164   1165 lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"   1166 by simp   1167   1168 declare zip_Cons [simp del]   1169   1170 lemma zip_Cons1:   1171 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"   1172 by(auto split:list.split)   1173   1174 lemma length_zip [simp]:   1175 "!!xs. length (zip xs ys) = min (length xs) (length ys)"   1176 apply (induct ys, simp)   1177 apply (case_tac xs, auto)   1178 done   1179   1180 lemma zip_append1:   1181 "!!xs. zip (xs @ ys) zs =   1182 zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"   1183 apply (induct zs, simp)   1184 apply (case_tac xs, simp_all)   1185 done   1186   1187 lemma zip_append2:   1188 "!!ys. zip xs (ys @ zs) =   1189 zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"   1190 apply (induct xs, simp)   1191 apply (case_tac ys, simp_all)   1192 done   1193   1194 lemma zip_append [simp]:   1195 "[| length xs = length us; length ys = length vs |] ==>   1196 zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"   1197 by (simp add: zip_append1)   1198   1199 lemma zip_rev:   1200 "length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"   1201 by (induct rule:list_induct2, simp_all)   1202   1203 lemma nth_zip [simp]:   1204 "!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"   1205 apply (induct ys, simp)   1206 apply (case_tac xs)   1207 apply (simp_all add: nth.simps split: nat.split)   1208 done   1209   1210 lemma set_zip:   1211 "set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"   1212 by (simp add: set_conv_nth cong: rev_conj_cong)   1213   1214 lemma zip_update:   1215 "length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"   1216 by (rule sym, simp add: update_zip)   1217   1218 lemma zip_replicate [simp]:   1219 "!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"   1220 apply (induct i, auto)   1221 apply (case_tac j, auto)   1222 done   1223   1224   1225 subsubsection {* @{text list_all2} *}   1226   1227 lemma list_all2_lengthD [intro?]:   1228 "list_all2 P xs ys ==> length xs = length ys"   1229 by (simp add: list_all2_def)   1230   1231 lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])"   1232 by (simp add: list_all2_def)   1233   1234 lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"   1235 by (simp add: list_all2_def)   1236   1237 lemma list_all2_Cons [iff]:   1238 "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"   1239 by (auto simp add: list_all2_def)   1240   1241 lemma list_all2_Cons1:   1242 "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"   1243 by (cases ys) auto   1244   1245 lemma list_all2_Cons2:   1246 "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"   1247 by (cases xs) auto   1248   1249 lemma list_all2_rev [iff]:   1250 "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"   1251 by (simp add: list_all2_def zip_rev cong: conj_cong)   1252   1253 lemma list_all2_rev1:   1254 "list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"   1255 by (subst list_all2_rev [symmetric]) simp   1256   1257 lemma list_all2_append1:   1258 "list_all2 P (xs @ ys) zs =   1259 (EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>   1260 list_all2 P xs us \<and> list_all2 P ys vs)"   1261 apply (simp add: list_all2_def zip_append1)   1262 apply (rule iffI)   1263 apply (rule_tac x = "take (length xs) zs" in exI)   1264 apply (rule_tac x = "drop (length xs) zs" in exI)   1265 apply (force split: nat_diff_split simp add: min_def, clarify)   1266 apply (simp add: ball_Un)   1267 done   1268   1269 lemma list_all2_append2:   1270 "list_all2 P xs (ys @ zs) =   1271 (EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>   1272 list_all2 P us ys \<and> list_all2 P vs zs)"   1273 apply (simp add: list_all2_def zip_append2)   1274 apply (rule iffI)   1275 apply (rule_tac x = "take (length ys) xs" in exI)   1276 apply (rule_tac x = "drop (length ys) xs" in exI)   1277 apply (force split: nat_diff_split simp add: min_def, clarify)   1278 apply (simp add: ball_Un)   1279 done   1280   1281 lemma list_all2_append:   1282 "length xs = length ys \<Longrightarrow>   1283 list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"   1284 by (induct rule:list_induct2, simp_all)   1285   1286 lemma list_all2_appendI [intro?, trans]:   1287 "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"   1288 by (simp add: list_all2_append list_all2_lengthD)   1289   1290 lemma list_all2_conv_all_nth:   1291 "list_all2 P xs ys =   1292 (length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"   1293 by (force simp add: list_all2_def set_zip)   1294   1295 lemma list_all2_trans:   1296 assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"   1297 shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"   1298 (is "!!bs cs. PROP ?Q as bs cs")   1299 proof (induct as)   1300 fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"   1301 show "!!cs. PROP ?Q (x # xs) bs cs"   1302 proof (induct bs)   1303 fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"   1304 show "PROP ?Q (x # xs) (y # ys) cs"   1305 by (induct cs) (auto intro: tr I1 I2)   1306 qed simp   1307 qed simp   1308   1309 lemma list_all2_all_nthI [intro?]:   1310 "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"   1311 by (simp add: list_all2_conv_all_nth)   1312   1313 lemma list_all2I:   1314 "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"   1315 by (simp add: list_all2_def)   1316   1317 lemma list_all2_nthD:   1318 "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"   1319 by (simp add: list_all2_conv_all_nth)   1320   1321 lemma list_all2_nthD2:   1322 "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"   1323 by (frule list_all2_lengthD) (auto intro: list_all2_nthD)   1324   1325 lemma list_all2_map1:   1326 "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"   1327 by (simp add: list_all2_conv_all_nth)   1328   1329 lemma list_all2_map2:   1330 "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"   1331 by (auto simp add: list_all2_conv_all_nth)   1332   1333 lemma list_all2_refl [intro?]:   1334 "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"   1335 by (simp add: list_all2_conv_all_nth)   1336   1337 lemma list_all2_update_cong:   1338 "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"   1339 by (simp add: list_all2_conv_all_nth nth_list_update)   1340   1341 lemma list_all2_update_cong2:   1342 "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"   1343 by (simp add: list_all2_lengthD list_all2_update_cong)   1344   1345 lemma list_all2_takeI [simp,intro?]:   1346 "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"   1347 apply (induct xs)   1348 apply simp   1349 apply (clarsimp simp add: list_all2_Cons1)   1350 apply (case_tac n)   1351 apply auto   1352 done   1353   1354 lemma list_all2_dropI [simp,intro?]:   1355 "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"   1356 apply (induct as, simp)   1357 apply (clarsimp simp add: list_all2_Cons1)   1358 apply (case_tac n, simp, simp)   1359 done   1360   1361 lemma list_all2_mono [intro?]:   1362 "\<And>y. list_all2 P x y \<Longrightarrow> (\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> list_all2 Q x y"   1363 apply (induct x, simp)   1364 apply (case_tac y, auto)   1365 done   1366   1367   1368 subsubsection {* @{text foldl} and @{text foldr} *}   1369   1370 lemma foldl_append [simp]:   1371 "!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"   1372 by (induct xs) auto   1373   1374 lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"   1375 by (induct xs) auto   1376   1377 lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"   1378 by (induct xs) auto   1379   1380 lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"   1381 by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])   1382   1383 text {*   1384 Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more   1385 difficult to use because it requires an additional transitivity step.   1386 *}   1387   1388 lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"   1389 by (induct ns) auto   1390   1391 lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"   1392 by (force intro: start_le_sum simp add: in_set_conv_decomp)   1393   1394 lemma sum_eq_0_conv [iff]:   1395 "!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"   1396 by (induct ns) auto   1397   1398   1399 subsubsection {* @{text upto} *}   1400   1401 lemma upt_rec: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"   1402 -- {* Does not terminate! *}   1403 by (induct j) auto   1404   1405 lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"   1406 by (subst upt_rec) simp   1407   1408 lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"   1409 by(induct j)simp_all   1410   1411 lemma upt_eq_Cons_conv:   1412 "!!x xs. ([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"   1413 apply(induct j)   1414 apply simp   1415 apply(clarsimp simp add: append_eq_Cons_conv)   1416 apply arith   1417 done   1418   1419 lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"   1420 -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}   1421 by simp   1422   1423 lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"   1424 apply(rule trans)   1425 apply(subst upt_rec)   1426 prefer 2 apply (rule refl, simp)   1427 done   1428   1429 lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"   1430 -- {* LOOPS as a simprule, since @{text "j <= j"}. *}   1431 by (induct k) auto   1432   1433 lemma length_upt [simp]: "length [i..<j] = j - i"   1434 by (induct j) (auto simp add: Suc_diff_le)   1435   1436 lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"   1437 apply (induct j)   1438 apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)   1439 done   1440   1441 lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..<n] = [i..<i+m]"   1442 apply (induct m, simp)   1443 apply (subst upt_rec)   1444 apply (rule sym)   1445 apply (subst upt_rec)   1446 apply (simp del: upt.simps)   1447 done   1448   1449 lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..n]"   1450 by (induct n) auto   1451   1452 lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..<n]) ! i = f(m+i)"   1453 apply (induct n m rule: diff_induct)   1454 prefer 3 apply (subst map_Suc_upt[symmetric])   1455 apply (auto simp add: less_diff_conv nth_upt)   1456 done   1457   1458 lemma nth_take_lemma:   1459 "!!xs ys. k <= length xs ==> k <= length ys ==>   1460 (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"   1461 apply (atomize, induct k)   1462 apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)   1463 txt {* Both lists must be non-empty *}   1464 apply (case_tac xs, simp)   1465 apply (case_tac ys, clarify)   1466 apply (simp (no_asm_use))   1467 apply clarify   1468 txt {* prenexing's needed, not miniscoping *}   1469 apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)   1470 apply blast   1471 done   1472   1473 lemma nth_equalityI:   1474 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"   1475 apply (frule nth_take_lemma [OF le_refl eq_imp_le])   1476 apply (simp_all add: take_all)   1477 done   1478   1479 (* needs nth_equalityI *)   1480 lemma list_all2_antisym:   1481 "\<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>   1482 \<Longrightarrow> xs = ys"   1483 apply (simp add: list_all2_conv_all_nth)   1484 apply (rule nth_equalityI, blast, simp)   1485 done   1486   1487 lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"   1488 -- {* The famous take-lemma. *}   1489 apply (drule_tac x = "max (length xs) (length ys)" in spec)   1490 apply (simp add: le_max_iff_disj take_all)   1491 done   1492   1493   1494 lemma take_Cons':   1495 "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"   1496 by (cases n) simp_all   1497   1498 lemma drop_Cons':   1499 "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"   1500 by (cases n) simp_all   1501   1502 lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"   1503 by (cases n) simp_all   1504   1505 lemmas [simp] = take_Cons'[of "number_of v",standard]   1506 drop_Cons'[of "number_of v",standard]   1507 nth_Cons'[of _ _ "number_of v",standard]   1508   1509   1510 subsubsection {* @{text "distinct"} and @{text remdups} *}   1511   1512 lemma distinct_append [simp]:   1513 "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"   1514 by (induct xs) auto   1515   1516 lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"   1517 by(induct xs) auto   1518   1519 lemma set_remdups [simp]: "set (remdups xs) = set xs"   1520 by (induct xs) (auto simp add: insert_absorb)   1521   1522 lemma distinct_remdups [iff]: "distinct (remdups xs)"   1523 by (induct xs) auto   1524   1525 lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"   1526 by (induct x, auto)   1527   1528 lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"   1529 by (induct x, auto)   1530   1531 lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"   1532 by (induct xs) auto   1533   1534 lemma length_remdups_eq[iff]:   1535 "(length (remdups xs) = length xs) = (remdups xs = xs)"   1536 apply(induct xs)   1537 apply auto   1538 apply(subgoal_tac "length (remdups xs) <= length xs")   1539 apply arith   1540 apply(rule length_remdups_leq)   1541 done   1542   1543 lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"   1544 by (induct xs) auto   1545   1546 lemma distinct_map_filterI:   1547 "distinct(map f xs) \<Longrightarrow> distinct(map f (filter P xs))"   1548 apply(induct xs)   1549 apply simp   1550 apply force   1551 done   1552   1553 text {*   1554 It is best to avoid this indexed version of distinct, but sometimes   1555 it is useful. *}   1556 lemma distinct_conv_nth:   1557 "distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)"   1558 apply (induct xs, simp, simp)   1559 apply (rule iffI, clarsimp)   1560 apply (case_tac i)   1561 apply (case_tac j, simp)   1562 apply (simp add: set_conv_nth)   1563 apply (case_tac j)   1564 apply (clarsimp simp add: set_conv_nth, simp)   1565 apply (rule conjI)   1566 apply (clarsimp simp add: set_conv_nth)   1567 apply (erule_tac x = 0 in allE)   1568 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)   1569 apply (erule_tac x = "Suc i" in allE)   1570 apply (erule_tac x = "Suc j" in allE, simp)   1571 done   1572   1573 lemma distinct_card: "distinct xs ==> card (set xs) = size xs"   1574 by (induct xs) auto   1575   1576 lemma card_distinct: "card (set xs) = size xs ==> distinct xs"   1577 proof (induct xs)   1578 case Nil thus ?case by simp   1579 next   1580 case (Cons x xs)   1581 show ?case   1582 proof (cases "x \<in> set xs")   1583 case False with Cons show ?thesis by simp   1584 next   1585 case True with Cons.prems   1586 have "card (set xs) = Suc (length xs)"   1587 by (simp add: card_insert_if split: split_if_asm)   1588 moreover have "card (set xs) \<le> length xs" by (rule card_length)   1589 ultimately have False by simp   1590 thus ?thesis ..   1591 qed   1592 qed   1593   1594 lemma inj_on_setI: "distinct(map f xs) ==> inj_on f (set xs)"   1595 apply(induct xs)   1596 apply simp   1597 apply fastsimp   1598 done   1599   1600 lemma inj_on_set_conv:   1601 "distinct xs \<Longrightarrow> inj_on f (set xs) = distinct(map f xs)"   1602 apply(induct xs)   1603 apply simp   1604 apply fastsimp   1605 done   1606   1607   1608 subsubsection {* @{text remove1} *}   1609   1610 lemma set_remove1_subset: "set(remove1 x xs) <= set xs"   1611 apply(induct xs)   1612 apply simp   1613 apply simp   1614 apply blast   1615 done   1616   1617 lemma [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"   1618 apply(induct xs)   1619 apply simp   1620 apply simp   1621 apply blast   1622 done   1623   1624 lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"   1625 apply(insert set_remove1_subset)   1626 apply fast   1627 done   1628   1629 lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"   1630 by (induct xs) simp_all   1631   1632   1633 subsubsection {* @{text replicate} *}   1634   1635 lemma length_replicate [simp]: "length (replicate n x) = n"   1636 by (induct n) auto   1637   1638 lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"   1639 by (induct n) auto   1640   1641 lemma replicate_app_Cons_same:   1642 "(replicate n x) @ (x # xs) = x # replicate n x @ xs"   1643 by (induct n) auto   1644   1645 lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"   1646 apply (induct n, simp)   1647 apply (simp add: replicate_app_Cons_same)   1648 done   1649   1650 lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"   1651 by (induct n) auto   1652   1653 text{* Courtesy of Matthias Daum: *}   1654 lemma append_replicate_commute:   1655 "replicate n x @ replicate k x = replicate k x @ replicate n x"   1656 apply (simp add: replicate_add [THEN sym])   1657 apply (simp add: add_commute)   1658 done   1659   1660 lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"   1661 by (induct n) auto   1662   1663 lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"   1664 by (induct n) auto   1665   1666 lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"   1667 by (atomize (full), induct n) auto   1668   1669 lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"   1670 apply (induct n, simp)   1671 apply (simp add: nth_Cons split: nat.split)   1672 done   1673   1674 text{* Courtesy of Matthias Daum (2 lemmas): *}   1675 lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"   1676 apply (case_tac "k \<le> i")   1677 apply (simp add: min_def)   1678 apply (drule not_leE)   1679 apply (simp add: min_def)   1680 apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")   1681 apply simp   1682 apply (simp add: replicate_add [symmetric])   1683 done   1684   1685 lemma drop_replicate[simp]: "!!i. drop i (replicate k x) = replicate (k-i) x"   1686 apply (induct k)   1687 apply simp   1688 apply clarsimp   1689 apply (case_tac i)   1690 apply simp   1691 apply clarsimp   1692 done   1693   1694   1695 lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"   1696 by (induct n) auto   1697   1698 lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"   1699 by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)   1700   1701 lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"   1702 by auto   1703   1704 lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"   1705 by (simp add: set_replicate_conv_if split: split_if_asm)   1706   1707   1708 subsubsection{*@{text rotate1} and @{text rotate}*}   1709   1710 lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"   1711 by(simp add:rotate1_def)   1712   1713 lemma rotate0[simp]: "rotate 0 = id"   1714 by(simp add:rotate_def)   1715   1716 lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"   1717 by(simp add:rotate_def)   1718   1719 lemma rotate_add:   1720 "rotate (m+n) = rotate m o rotate n"   1721 by(simp add:rotate_def funpow_add)   1722   1723 lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"   1724 by(simp add:rotate_add)   1725   1726 lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"   1727 by(cases xs) simp_all   1728   1729 lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"   1730 apply(induct n)   1731 apply simp   1732 apply (simp add:rotate_def)   1733 done   1734   1735 lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"   1736 by(simp add:rotate1_def split:list.split)   1737   1738 lemma rotate_drop_take:   1739 "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"   1740 apply(induct n)   1741 apply simp   1742 apply(simp add:rotate_def)   1743 apply(cases "xs = []")   1744 apply (simp)   1745 apply(case_tac "n mod length xs = 0")   1746 apply(simp add:mod_Suc)   1747 apply(simp add: rotate1_hd_tl drop_Suc take_Suc)   1748 apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]   1749 take_hd_drop linorder_not_le)   1750 done   1751   1752 lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"   1753 by(simp add:rotate_drop_take)   1754   1755 lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"   1756 by(simp add:rotate_drop_take)   1757   1758 lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"   1759 by(simp add:rotate1_def split:list.split)   1760   1761 lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs"   1762 by (induct n) (simp_all add:rotate_def)   1763   1764 lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"   1765 by(simp add:rotate1_def split:list.split) blast   1766   1767 lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"   1768 by (induct n) (simp_all add:rotate_def)   1769   1770 lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"   1771 by(simp add:rotate_drop_take take_map drop_map)   1772   1773 lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"   1774 by(simp add:rotate1_def split:list.split)   1775   1776 lemma set_rotate[simp]: "set(rotate n xs) = set xs"   1777 by (induct n) (simp_all add:rotate_def)   1778   1779 lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"   1780 by(simp add:rotate1_def split:list.split)   1781   1782 lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"   1783 by (induct n) (simp_all add:rotate_def)   1784   1785 lemma rotate_rev:   1786 "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"   1787 apply(simp add:rotate_drop_take rev_drop rev_take)   1788 apply(cases "length xs = 0")   1789 apply simp   1790 apply(cases "n mod length xs = 0")   1791 apply simp   1792 apply(simp add:rotate_drop_take rev_drop rev_take)   1793 done   1794   1795   1796 subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}   1797   1798 lemma sublist_empty [simp]: "sublist xs {} = []"   1799 by (auto simp add: sublist_def)   1800   1801 lemma sublist_nil [simp]: "sublist [] A = []"   1802 by (auto simp add: sublist_def)   1803   1804 lemma length_sublist:   1805 "length(sublist xs I) = card{i. i < length xs \<and> i : I}"   1806 by(simp add: sublist_def length_filter_conv_card cong:conj_cong)   1807   1808 lemma sublist_shift_lemma_Suc:   1809 "!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =   1810 map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"   1811 apply(induct xs)   1812 apply simp   1813 apply (case_tac "is")   1814 apply simp   1815 apply simp   1816 done   1817   1818 lemma sublist_shift_lemma:   1819 "map fst [p:zip xs [i..<i + length xs] . snd p : A] =   1820 map fst [p:zip xs [0..<length xs] . snd p + i : A]"   1821 by (induct xs rule: rev_induct) (simp_all add: add_commute)   1822   1823 lemma sublist_append:   1824 "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"   1825 apply (unfold sublist_def)   1826 apply (induct l' rule: rev_induct, simp)   1827 apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)   1828 apply (simp add: add_commute)   1829 done   1830   1831 lemma sublist_Cons:   1832 "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"   1833 apply (induct l rule: rev_induct)   1834 apply (simp add: sublist_def)   1835 apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)   1836 done   1837   1838 lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"   1839 apply(induct xs)   1840 apply simp   1841 apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE)   1842 apply(erule lessE)   1843 apply auto   1844 apply(erule lessE)   1845 apply auto   1846 done   1847   1848 lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"   1849 by(auto simp add:set_sublist)   1850   1851 lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"   1852 by(auto simp add:set_sublist)   1853   1854 lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"   1855 by(auto simp add:set_sublist)   1856   1857 lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"   1858 by (simp add: sublist_Cons)   1859   1860   1861 lemma distinct_sublistI[simp]: "!!I. distinct xs \<Longrightarrow> distinct(sublist xs I)"   1862 apply(induct xs)   1863 apply simp   1864 apply(auto simp add:sublist_Cons)   1865 done   1866   1867   1868 lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"   1869 apply (induct l rule: rev_induct, simp)   1870 apply (simp split: nat_diff_split add: sublist_append)   1871 done   1872   1873   1874 subsubsection{*Sets of Lists*}   1875   1876 subsubsection {* @{text lists}: the list-forming operator over sets *}   1877   1878 consts lists :: "'a set => 'a list set"   1879 inductive "lists A"   1880 intros   1881 Nil [intro!]: "[]: lists A"   1882 Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A"   1883   1884 inductive_cases listsE [elim!]: "x#l : lists A"   1885   1886 lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B"   1887 by (unfold lists.defs) (blast intro!: lfp_mono)   1888   1889 lemma lists_IntI:   1890 assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l   1891 by induct blast+   1892   1893 lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B"   1894 proof (rule mono_Int [THEN equalityI])   1895 show "mono lists" by (simp add: mono_def lists_mono)   1896 show "lists A \<inter> lists B \<subseteq> lists (A \<inter> B)" by (blast intro: lists_IntI)   1897 qed   1898   1899 lemma append_in_lists_conv [iff]:   1900 "(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"   1901 by (induct xs) auto   1902   1903 lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"   1904 -- {* eliminate @{text lists} in favour of @{text set} *}   1905 by (induct xs) auto   1906   1907 lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"   1908 by (rule in_lists_conv_set [THEN iffD1])   1909   1910 lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"   1911 by (rule in_lists_conv_set [THEN iffD2])   1912   1913 lemma lists_UNIV [simp]: "lists UNIV = UNIV"   1914 by auto   1915   1916 subsubsection{*Lists as Cartesian products*}   1917   1918 text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from   1919 @{term A} and tail drawn from @{term Xs}.*}   1920   1921 constdefs   1922 set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"   1923 "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"   1924   1925 lemma [simp]: "set_Cons A {[]} = (%x. [x])A"   1926 by (auto simp add: set_Cons_def)   1927   1928 text{*Yields the set of lists, all of the same length as the argument and   1929 with elements drawn from the corresponding element of the argument.*}   1930   1931 consts listset :: "'a set list \<Rightarrow> 'a list set"   1932 primrec   1933 "listset [] = {[]}"   1934 "listset(A#As) = set_Cons A (listset As)"   1935   1936   1937 subsection{*Relations on Lists*}   1938   1939 subsubsection {* Length Lexicographic Ordering *}   1940   1941 text{*These orderings preserve well-foundedness: shorter lists   1942 precede longer lists. These ordering are not used in dictionaries.*}   1943   1944 consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"   1945 --{*The lexicographic ordering for lists of the specified length*}   1946 primrec   1947 "lexn r 0 = {}"   1948 "lexn r (Suc n) =   1949 (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs)  (r <*lex*> lexn r n)) Int   1950 {(xs,ys). length xs = Suc n \<and> length ys = Suc n}"   1951   1952 constdefs   1953 lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"   1954 "lex r == \<Union>n. lexn r n"   1955 --{*Holds only between lists of the same length*}   1956   1957 lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"   1958 "lenlex r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"   1959 --{*Compares lists by their length and then lexicographically*}   1960   1961   1962 lemma wf_lexn: "wf r ==> wf (lexn r n)"   1963 apply (induct n, simp, simp)   1964 apply(rule wf_subset)   1965 prefer 2 apply (rule Int_lower1)   1966 apply(rule wf_prod_fun_image)   1967 prefer 2 apply (rule inj_onI, auto)   1968 done   1969   1970 lemma lexn_length:   1971 "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"   1972 by (induct n) auto   1973   1974 lemma wf_lex [intro!]: "wf r ==> wf (lex r)"   1975 apply (unfold lex_def)   1976 apply (rule wf_UN)   1977 apply (blast intro: wf_lexn, clarify)   1978 apply (rename_tac m n)   1979 apply (subgoal_tac "m \<noteq> n")   1980 prefer 2 apply blast   1981 apply (blast dest: lexn_length not_sym)   1982 done   1983   1984 lemma lexn_conv:   1985 "lexn r n =   1986 {(xs,ys). length xs = n \<and> length ys = n \<and>   1987 (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"   1988 apply (induct n, simp, blast)   1989 apply (simp add: image_Collect lex_prod_def, safe, blast)   1990 apply (rule_tac x = "ab # xys" in exI, simp)   1991 apply (case_tac xys, simp_all, blast)   1992 done   1993   1994 lemma lex_conv:   1995 "lex r =   1996 {(xs,ys). length xs = length ys \<and>   1997 (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"   1998 by (force simp add: lex_def lexn_conv)   1999   2000 lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"   2001 by (unfold lenlex_def) blast   2002   2003 lemma lenlex_conv:   2004 "lenlex r = {(xs,ys). length xs < length ys |   2005 length xs = length ys \<and> (xs, ys) : lex r}"   2006 by (simp add: lenlex_def diag_def lex_prod_def measure_def inv_image_def)   2007   2008 lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"   2009 by (simp add: lex_conv)   2010   2011 lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"   2012 by (simp add:lex_conv)   2013   2014 lemma Cons_in_lex [iff]:   2015 "((x # xs, y # ys) : lex r) =   2016 ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"   2017 apply (simp add: lex_conv)   2018 apply (rule iffI)   2019 prefer 2 apply (blast intro: Cons_eq_appendI, clarify)   2020 apply (case_tac xys, simp, simp)   2021 apply blast   2022 done   2023   2024   2025 subsubsection {* Lexicographic Ordering *}   2026   2027 text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".   2028 This ordering does \emph{not} preserve well-foundedness.   2029 Author: N. Voelker, March 2005 *}   2030   2031 constdefs   2032 lexord :: "('a * 'a)set \<Rightarrow> ('a list * 'a list) set"   2033 "lexord r == {(x,y). \<exists> a v. y = x @ a # v \<or>   2034 (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"   2035   2036 lemma lexord_Nil_left[simp]: "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"   2037 by (unfold lexord_def, induct_tac y, auto)   2038   2039 lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"   2040 by (unfold lexord_def, induct_tac x, auto)   2041   2042 lemma lexord_cons_cons[simp]:   2043 "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"   2044 apply (unfold lexord_def, safe, simp_all)   2045 apply (case_tac u, simp, simp)   2046 apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)   2047 apply (erule_tac x="b # u" in allE)   2048 by force   2049   2050 lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons   2051   2052 lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"   2053 by (induct_tac x, auto)   2054   2055 lemma lexord_append_left_rightI:   2056 "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"   2057 by (induct_tac u, auto)   2058   2059 lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"   2060 by (induct x, auto)   2061   2062 lemma lexord_append_leftD:   2063 "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"   2064 by (erule rev_mp, induct_tac x, auto)   2065   2066 lemma lexord_take_index_conv:   2067 "((x,y) : lexord r) =   2068 ((length x < length y \<and> take (length x) y = x) \<or>   2069 (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"   2070 apply (unfold lexord_def Let_def, clarsimp)   2071 apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)   2072 apply auto   2073 apply (rule_tac x="hd (drop (length x) y)" in exI)   2074 apply (rule_tac x="tl (drop (length x) y)" in exI)   2075 apply (erule subst, simp add: min_def)   2076 apply (rule_tac x ="length u" in exI, simp)   2077 apply (rule_tac x ="take i x" in exI)   2078 apply (rule_tac x ="x ! i" in exI)   2079 apply (rule_tac x ="y ! i" in exI, safe)   2080 apply (rule_tac x="drop (Suc i) x" in exI)   2081 apply (drule sym, simp add: drop_Suc_conv_tl)   2082 apply (rule_tac x="drop (Suc i) y" in exI)   2083 by (simp add: drop_Suc_conv_tl)   2084   2085 -- {* lexord is extension of partial ordering List.lex *}   2086 lemma lexord_lex: " (x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"   2087 apply (rule_tac x = y in spec)   2088 apply (induct_tac x, clarsimp)   2089 by (clarify, case_tac x, simp, force)   2090   2091 lemma lexord_irreflexive: "(! x. (x,x) \<notin> r) \<Longrightarrow> (y,y) \<notin> lexord r"   2092 by (induct y, auto)   2093   2094 lemma lexord_trans:   2095 "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"   2096 apply (erule rev_mp)+   2097 apply (rule_tac x = x in spec)   2098 apply (rule_tac x = z in spec)   2099 apply ( induct_tac y, simp, clarify)   2100 apply (case_tac xa, erule ssubst)   2101 apply (erule allE, erule allE) -- {* avoid simp recursion *}   2102 apply (case_tac x, simp, simp)   2103 apply (case_tac x, erule allE, erule allE, simp)   2104 apply (erule_tac x = listb in allE)   2105 apply (erule_tac x = lista in allE, simp)   2106 apply (unfold trans_def)   2107 by blast   2108   2109 lemma lexord_transI: "trans r \<Longrightarrow> trans (lexord r)"   2110 by (rule transI, drule lexord_trans, blast)   2111   2112 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"   2113 apply (rule_tac x = y in spec)   2114 apply (induct_tac x, rule allI)   2115 apply (case_tac x, simp, simp)   2116 apply (rule allI, case_tac x, simp, simp)   2117 by blast   2118   2119   2120 subsubsection{*Lifting a Relation on List Elements to the Lists*}   2121   2122 consts listrel :: "('a * 'a)set => ('a list * 'a list)set"   2123   2124 inductive "listrel(r)"   2125 intros   2126 Nil: "([],[]) \<in> listrel r"   2127 Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"   2128   2129 inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"   2130 inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"   2131 inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"   2132 inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"   2133   2134   2135 lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"   2136 apply clarify   2137 apply (erule listrel.induct)   2138 apply (blast intro: listrel.intros)+   2139 done   2140   2141 lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"   2142 apply clarify   2143 apply (erule listrel.induct, auto)   2144 done   2145   2146 lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)"   2147 apply (simp add: refl_def listrel_subset Ball_def)   2148 apply (rule allI)   2149 apply (induct_tac x)   2150 apply (auto intro: listrel.intros)   2151 done   2152   2153 lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)"   2154 apply (auto simp add: sym_def)   2155 apply (erule listrel.induct)   2156 apply (blast intro: listrel.intros)+   2157 done   2158   2159 lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)"   2160 apply (simp add: trans_def)   2161 apply (intro allI)   2162 apply (rule impI)   2163 apply (erule listrel.induct)   2164 apply (blast intro: listrel.intros)+   2165 done   2166   2167 theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"   2168 by (simp add: equiv_def listrel_refl listrel_sym listrel_trans)   2169   2170 lemma listrel_Nil [simp]: "listrel r  {[]} = {[]}"   2171 by (blast intro: listrel.intros)   2172   2173 lemma listrel_Cons:   2174 "listrel r  {x#xs} = set_Cons (r{x}) (listrel r  {xs})";   2175 by (auto simp add: set_Cons_def intro: listrel.intros)   2176   2177   2178 subsection{*Miscellany*}   2179   2180 subsubsection {* Characters and strings *}   2181   2182 datatype nibble =   2183 Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7   2184 | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF   2185   2186 datatype char = Char nibble nibble   2187 -- "Note: canonical order of character encoding coincides with standard term ordering"   2188   2189 types string = "char list"   2190   2191 syntax   2192 "_Char" :: "xstr => char" ("CHR _")   2193 "_String" :: "xstr => string" ("_")   2194   2195 parse_ast_translation {*   2196 let   2197 val constants = Syntax.Appl o map Syntax.Constant;   2198   2199 fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10));   2200 fun mk_char c =   2201 if Symbol.is_ascii c andalso Symbol.is_printable c then   2202 constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)]   2203 else error ("Printable ASCII character expected: " ^ quote c);   2204   2205 fun mk_string [] = Syntax.Constant "Nil"   2206 | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs];   2207   2208 fun char_ast_tr [Syntax.Variable xstr] =   2209 (case Syntax.explode_xstr xstr of   2210 [c] => mk_char c   2211 | _ => error ("Single character expected: " ^ xstr))   2212 | char_ast_tr asts = raise AST ("char_ast_tr", asts);   2213   2214 fun string_ast_tr [Syntax.Variable xstr] =   2215 (case Syntax.explode_xstr xstr of   2216 [] => constants [Syntax.constrainC, "Nil", "string"]   2217 | cs => mk_string cs)   2218 | string_ast_tr asts = raise AST ("string_tr", asts);   2219 in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end;   2220 *}   2221   2222 ML {*   2223 fun int_of_nibble h =   2224 if "0" <= h andalso h <= "9" then ord h - ord "0"   2225 else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10   2226 else raise Match;   2227   2228 fun nibble_of_int i =   2229 if i <= 9 then chr (ord "0" + i) else chr (ord "A" + i - 10);   2230 *}   2231   2232 print_ast_translation {*   2233 let   2234 fun dest_nib (Syntax.Constant c) =   2235 (case explode c of   2236 ["N", "i", "b", "b", "l", "e", h] => int_of_nibble h   2237 | _ => raise Match)   2238 | dest_nib _ = raise Match;   2239   2240 fun dest_chr c1 c2 =   2241 let val c = chr (dest_nib c1 * 16 + dest_nib c2)   2242 in if Symbol.is_printable c then c else raise Match end;   2243   2244 fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2   2245 | dest_char _ = raise Match;   2246   2247 fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)];   2248   2249 fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]]   2250 | char_ast_tr' _ = raise Match;   2251   2252 fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String",   2253 xstr (map dest_char (Syntax.unfold_ast "_args" args))]   2254 | list_ast_tr' ts = raise Match;   2255 in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end;   2256 *}   2257   2258 subsubsection {* Code generator setup *}   2259   2260 ML {*   2261 local   2262   2263 fun list_codegen thy defs gr dep thyname b t =   2264 let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy defs dep thyname false)   2265 (gr, HOLogic.dest_list t)   2266 in SOME (gr', Pretty.list "[" "]" ps) end handle TERM _ => NONE;   2267   2268 fun dest_nibble (Const (s, _)) = int_of_nibble (unprefix "List.nibble.Nibble" s)   2269 | dest_nibble _ = raise Match;   2270   2271 fun char_codegen thy defs gr dep thyname b (Const ("List.char.Char", _)$ c1 $c2) =   2272 (let val c = chr (dest_nibble c1 * 16 + dest_nibble c2)   2273 in if Symbol.is_printable c then SOME (gr, Pretty.quote (Pretty.str c))   2274 else NONE   2275 end handle Fail _ => NONE | Match => NONE)   2276 | char_codegen thy defs gr dep thyname b _ = NONE;   2277   2278 in   2279   2280 val list_codegen_setup =   2281 [Codegen.add_codegen "list_codegen" list_codegen,   2282 Codegen.add_codegen "char_codegen" char_codegen];   2283   2284 end;   2285 *}   2286   2287 types_code   2288 "list" ("_ list")   2289 attach (term_of) {*   2290 val term_of_list = HOLogic.mk_list;   2291 *}   2292 attach (test) {*   2293 fun gen_list' aG i j = frequency   2294 [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] ()   2295 and gen_list aG i = gen_list' aG i i;   2296 *}   2297 "char" ("string")   2298 attach (term_of) {*   2299 val nibbleT = Type ("List.nibble", []);   2300   2301 fun term_of_char c =   2302 Const ("List.char.Char", nibbleT --> nibbleT --> Type ("List.char", []))$

  2303     Const ("List.nibble.Nibble" ^ nibble_of_int (ord c div 16), nibbleT) \$

  2304     Const ("List.nibble.Nibble" ^ nibble_of_int (ord c mod 16), nibbleT);

  2305 *}

  2306 attach (test) {*

  2307 fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z")));

  2308 *}

  2309

  2310 consts_code "Cons" ("(_ ::/ _)")

  2311

  2312 setup list_codegen_setup

  2313

  2314 end
`