src/ZF/List_ZF.thy
author wenzelm
Sun Nov 09 17:04:14 2014 +0100 (2014-11-09)
changeset 58957 c9e744ea8a38
parent 58871 c399ae4b836f
child 60770 240563fbf41d
permissions -rw-r--r--
proper context for match_tac etc.;
     1 (*  Title:      ZF/List_ZF.thy
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3     Copyright   1994  University of Cambridge
     4 *)
     5 
     6 section{*Lists in Zermelo-Fraenkel Set Theory*}
     7 
     8 theory List_ZF imports Datatype_ZF ArithSimp begin
     9 
    10 consts
    11   list       :: "i=>i"
    12 
    13 datatype
    14   "list(A)" = Nil | Cons ("a \<in> A", "l \<in> list(A)")
    15 
    16 
    17 syntax
    18  "_Nil" :: i  ("[]")
    19  "_List" :: "is => i"  ("[(_)]")
    20 
    21 translations
    22   "[x, xs]"     == "CONST Cons(x, [xs])"
    23   "[x]"         == "CONST Cons(x, [])"
    24   "[]"          == "CONST Nil"
    25 
    26 
    27 consts
    28   length :: "i=>i"
    29   hd     :: "i=>i"
    30   tl     :: "i=>i"
    31 
    32 primrec
    33   "length([]) = 0"
    34   "length(Cons(a,l)) = succ(length(l))"
    35 
    36 primrec
    37   "hd([]) = 0"
    38   "hd(Cons(a,l)) = a"
    39 
    40 primrec
    41   "tl([]) = []"
    42   "tl(Cons(a,l)) = l"
    43 
    44 
    45 consts
    46   map         :: "[i=>i, i] => i"
    47   set_of_list :: "i=>i"
    48   app         :: "[i,i]=>i"                        (infixr "@" 60)
    49 
    50 (*map is a binding operator -- it applies to meta-level functions, not
    51 object-level functions.  This simplifies the final form of term_rec_conv,
    52 although complicating its derivation.*)
    53 primrec
    54   "map(f,[]) = []"
    55   "map(f,Cons(a,l)) = Cons(f(a), map(f,l))"
    56 
    57 primrec
    58   "set_of_list([]) = 0"
    59   "set_of_list(Cons(a,l)) = cons(a, set_of_list(l))"
    60 
    61 primrec
    62   app_Nil:  "[] @ ys = ys"
    63   app_Cons: "(Cons(a,l)) @ ys = Cons(a, l @ ys)"
    64 
    65 
    66 consts
    67   rev :: "i=>i"
    68   flat       :: "i=>i"
    69   list_add   :: "i=>i"
    70 
    71 primrec
    72   "rev([]) = []"
    73   "rev(Cons(a,l)) = rev(l) @ [a]"
    74 
    75 primrec
    76   "flat([]) = []"
    77   "flat(Cons(l,ls)) = l @ flat(ls)"
    78 
    79 primrec
    80   "list_add([]) = 0"
    81   "list_add(Cons(a,l)) = a #+ list_add(l)"
    82 
    83 consts
    84   drop       :: "[i,i]=>i"
    85 
    86 primrec
    87   drop_0:    "drop(0,l) = l"
    88   drop_succ: "drop(succ(i), l) = tl (drop(i,l))"
    89 
    90 
    91 (*** Thanks to Sidi Ehmety for the following ***)
    92 
    93 definition
    94 (* Function `take' returns the first n elements of a list *)
    95   take     :: "[i,i]=>i"  where
    96   "take(n, as) == list_rec(\<lambda>n\<in>nat. [],
    97                 %a l r. \<lambda>n\<in>nat. nat_case([], %m. Cons(a, r`m), n), as)`n"
    98 
    99 definition
   100   nth :: "[i, i]=>i"  where
   101   --{*returns the (n+1)th element of a list, or 0 if the
   102    list is too short.*}
   103   "nth(n, as) == list_rec(\<lambda>n\<in>nat. 0,
   104                           %a l r. \<lambda>n\<in>nat. nat_case(a, %m. r`m, n), as) ` n"
   105 
   106 definition
   107   list_update :: "[i, i, i]=>i"  where
   108   "list_update(xs, i, v) == list_rec(\<lambda>n\<in>nat. Nil,
   109       %u us vs. \<lambda>n\<in>nat. nat_case(Cons(v, us), %m. Cons(u, vs`m), n), xs)`i"
   110 
   111 consts
   112   filter :: "[i=>o, i] => i"
   113   upt :: "[i, i] =>i"
   114 
   115 primrec
   116   "filter(P, Nil) = Nil"
   117   "filter(P, Cons(x, xs)) =
   118      (if P(x) then Cons(x, filter(P, xs)) else filter(P, xs))"
   119 
   120 primrec
   121   "upt(i, 0) = Nil"
   122   "upt(i, succ(j)) = (if i \<le> j then upt(i, j)@[j] else Nil)"
   123 
   124 definition
   125   min :: "[i,i] =>i"  where
   126     "min(x, y) == (if x \<le> y then x else y)"
   127 
   128 definition
   129   max :: "[i, i] =>i"  where
   130     "max(x, y) == (if x \<le> y then y else x)"
   131 
   132 (*** Aspects of the datatype definition ***)
   133 
   134 declare list.intros [simp,TC]
   135 
   136 (*An elimination rule, for type-checking*)
   137 inductive_cases ConsE: "Cons(a,l) \<in> list(A)"
   138 
   139 lemma Cons_type_iff [simp]: "Cons(a,l) \<in> list(A) \<longleftrightarrow> a \<in> A & l \<in> list(A)"
   140 by (blast elim: ConsE)
   141 
   142 (*Proving freeness results*)
   143 lemma Cons_iff: "Cons(a,l)=Cons(a',l') \<longleftrightarrow> a=a' & l=l'"
   144 by auto
   145 
   146 lemma Nil_Cons_iff: "~ Nil=Cons(a,l)"
   147 by auto
   148 
   149 lemma list_unfold: "list(A) = {0} + (A * list(A))"
   150 by (blast intro!: list.intros [unfolded list.con_defs]
   151           elim: list.cases [unfolded list.con_defs])
   152 
   153 
   154 (**  Lemmas to justify using "list" in other recursive type definitions **)
   155 
   156 lemma list_mono: "A<=B ==> list(A) \<subseteq> list(B)"
   157 apply (unfold list.defs )
   158 apply (rule lfp_mono)
   159 apply (simp_all add: list.bnd_mono)
   160 apply (assumption | rule univ_mono basic_monos)+
   161 done
   162 
   163 (*There is a similar proof by list induction.*)
   164 lemma list_univ: "list(univ(A)) \<subseteq> univ(A)"
   165 apply (unfold list.defs list.con_defs)
   166 apply (rule lfp_lowerbound)
   167 apply (rule_tac [2] A_subset_univ [THEN univ_mono])
   168 apply (blast intro!: zero_in_univ Inl_in_univ Inr_in_univ Pair_in_univ)
   169 done
   170 
   171 (*These two theorems justify datatypes involving list(nat), list(A), ...*)
   172 lemmas list_subset_univ = subset_trans [OF list_mono list_univ]
   173 
   174 lemma list_into_univ: "[| l \<in> list(A);  A \<subseteq> univ(B) |] ==> l \<in> univ(B)"
   175 by (blast intro: list_subset_univ [THEN subsetD])
   176 
   177 lemma list_case_type:
   178     "[| l \<in> list(A);
   179         c \<in> C(Nil);
   180         !!x y. [| x \<in> A;  y \<in> list(A) |] ==> h(x,y): C(Cons(x,y))
   181      |] ==> list_case(c,h,l) \<in> C(l)"
   182 by (erule list.induct, auto)
   183 
   184 lemma list_0_triv: "list(0) = {Nil}"
   185 apply (rule equalityI, auto)
   186 apply (induct_tac x, auto)
   187 done
   188 
   189 
   190 (*** List functions ***)
   191 
   192 lemma tl_type: "l \<in> list(A) ==> tl(l) \<in> list(A)"
   193 apply (induct_tac "l")
   194 apply (simp_all (no_asm_simp) add: list.intros)
   195 done
   196 
   197 (** drop **)
   198 
   199 lemma drop_Nil [simp]: "i \<in> nat ==> drop(i, Nil) = Nil"
   200 apply (induct_tac "i")
   201 apply (simp_all (no_asm_simp))
   202 done
   203 
   204 lemma drop_succ_Cons [simp]: "i \<in> nat ==> drop(succ(i), Cons(a,l)) = drop(i,l)"
   205 apply (rule sym)
   206 apply (induct_tac "i")
   207 apply (simp (no_asm))
   208 apply (simp (no_asm_simp))
   209 done
   210 
   211 lemma drop_type [simp,TC]: "[| i \<in> nat; l \<in> list(A) |] ==> drop(i,l) \<in> list(A)"
   212 apply (induct_tac "i")
   213 apply (simp_all (no_asm_simp) add: tl_type)
   214 done
   215 
   216 declare drop_succ [simp del]
   217 
   218 
   219 (** Type checking -- proved by induction, as usual **)
   220 
   221 lemma list_rec_type [TC]:
   222     "[| l \<in> list(A);
   223         c \<in> C(Nil);
   224         !!x y r. [| x \<in> A;  y \<in> list(A);  r \<in> C(y) |] ==> h(x,y,r): C(Cons(x,y))
   225      |] ==> list_rec(c,h,l) \<in> C(l)"
   226 by (induct_tac "l", auto)
   227 
   228 (** map **)
   229 
   230 lemma map_type [TC]:
   231     "[| l \<in> list(A);  !!x. x \<in> A ==> h(x): B |] ==> map(h,l) \<in> list(B)"
   232 apply (simp add: map_list_def)
   233 apply (typecheck add: list.intros list_rec_type, blast)
   234 done
   235 
   236 lemma map_type2 [TC]: "l \<in> list(A) ==> map(h,l) \<in> list({h(u). u \<in> A})"
   237 apply (erule map_type)
   238 apply (erule RepFunI)
   239 done
   240 
   241 (** length **)
   242 
   243 lemma length_type [TC]: "l \<in> list(A) ==> length(l) \<in> nat"
   244 by (simp add: length_list_def)
   245 
   246 lemma lt_length_in_nat:
   247    "[|x < length(xs); xs \<in> list(A)|] ==> x \<in> nat"
   248 by (frule lt_nat_in_nat, typecheck)
   249 
   250 (** app **)
   251 
   252 lemma app_type [TC]: "[| xs: list(A);  ys: list(A) |] ==> xs@ys \<in> list(A)"
   253 by (simp add: app_list_def)
   254 
   255 (** rev **)
   256 
   257 lemma rev_type [TC]: "xs: list(A) ==> rev(xs) \<in> list(A)"
   258 by (simp add: rev_list_def)
   259 
   260 
   261 (** flat **)
   262 
   263 lemma flat_type [TC]: "ls: list(list(A)) ==> flat(ls) \<in> list(A)"
   264 by (simp add: flat_list_def)
   265 
   266 
   267 (** set_of_list **)
   268 
   269 lemma set_of_list_type [TC]: "l \<in> list(A) ==> set_of_list(l) \<in> Pow(A)"
   270 apply (unfold set_of_list_list_def)
   271 apply (erule list_rec_type, auto)
   272 done
   273 
   274 lemma set_of_list_append:
   275      "xs: list(A) ==> set_of_list (xs@ys) = set_of_list(xs) \<union> set_of_list(ys)"
   276 apply (erule list.induct)
   277 apply (simp_all (no_asm_simp) add: Un_cons)
   278 done
   279 
   280 
   281 (** list_add **)
   282 
   283 lemma list_add_type [TC]: "xs: list(nat) ==> list_add(xs) \<in> nat"
   284 by (simp add: list_add_list_def)
   285 
   286 
   287 (*** theorems about map ***)
   288 
   289 lemma map_ident [simp]: "l \<in> list(A) ==> map(%u. u, l) = l"
   290 apply (induct_tac "l")
   291 apply (simp_all (no_asm_simp))
   292 done
   293 
   294 lemma map_compose: "l \<in> list(A) ==> map(h, map(j,l)) = map(%u. h(j(u)), l)"
   295 apply (induct_tac "l")
   296 apply (simp_all (no_asm_simp))
   297 done
   298 
   299 lemma map_app_distrib: "xs: list(A) ==> map(h, xs@ys) = map(h,xs) @ map(h,ys)"
   300 apply (induct_tac "xs")
   301 apply (simp_all (no_asm_simp))
   302 done
   303 
   304 lemma map_flat: "ls: list(list(A)) ==> map(h, flat(ls)) = flat(map(map(h),ls))"
   305 apply (induct_tac "ls")
   306 apply (simp_all (no_asm_simp) add: map_app_distrib)
   307 done
   308 
   309 lemma list_rec_map:
   310      "l \<in> list(A) ==>
   311       list_rec(c, d, map(h,l)) =
   312       list_rec(c, %x xs r. d(h(x), map(h,xs), r), l)"
   313 apply (induct_tac "l")
   314 apply (simp_all (no_asm_simp))
   315 done
   316 
   317 (** theorems about list(Collect(A,P)) -- used in Induct/Term.thy **)
   318 
   319 (* @{term"c \<in> list(Collect(B,P)) ==> c \<in> list"} *)
   320 lemmas list_CollectD = Collect_subset [THEN list_mono, THEN subsetD]
   321 
   322 lemma map_list_Collect: "l \<in> list({x \<in> A. h(x)=j(x)}) ==> map(h,l) = map(j,l)"
   323 apply (induct_tac "l")
   324 apply (simp_all (no_asm_simp))
   325 done
   326 
   327 (*** theorems about length ***)
   328 
   329 lemma length_map [simp]: "xs: list(A) ==> length(map(h,xs)) = length(xs)"
   330 by (induct_tac "xs", simp_all)
   331 
   332 lemma length_app [simp]:
   333      "[| xs: list(A); ys: list(A) |]
   334       ==> length(xs@ys) = length(xs) #+ length(ys)"
   335 by (induct_tac "xs", simp_all)
   336 
   337 lemma length_rev [simp]: "xs: list(A) ==> length(rev(xs)) = length(xs)"
   338 apply (induct_tac "xs")
   339 apply (simp_all (no_asm_simp) add: length_app)
   340 done
   341 
   342 lemma length_flat:
   343      "ls: list(list(A)) ==> length(flat(ls)) = list_add(map(length,ls))"
   344 apply (induct_tac "ls")
   345 apply (simp_all (no_asm_simp) add: length_app)
   346 done
   347 
   348 (** Length and drop **)
   349 
   350 (*Lemma for the inductive step of drop_length*)
   351 lemma drop_length_Cons [rule_format]:
   352      "xs: list(A) ==>
   353            \<forall>x.  \<exists>z zs. drop(length(xs), Cons(x,xs)) = Cons(z,zs)"
   354 by (erule list.induct, simp_all)
   355 
   356 lemma drop_length [rule_format]:
   357      "l \<in> list(A) ==> \<forall>i \<in> length(l). (\<exists>z zs. drop(i,l) = Cons(z,zs))"
   358 apply (erule list.induct, simp_all, safe)
   359 apply (erule drop_length_Cons)
   360 apply (rule natE)
   361 apply (erule Ord_trans [OF asm_rl length_type Ord_nat], assumption, simp_all)
   362 apply (blast intro: succ_in_naturalD length_type)
   363 done
   364 
   365 
   366 (*** theorems about app ***)
   367 
   368 lemma app_right_Nil [simp]: "xs: list(A) ==> xs@Nil=xs"
   369 by (erule list.induct, simp_all)
   370 
   371 lemma app_assoc: "xs: list(A) ==> (xs@ys)@zs = xs@(ys@zs)"
   372 by (induct_tac "xs", simp_all)
   373 
   374 lemma flat_app_distrib: "ls: list(list(A)) ==> flat(ls@ms) = flat(ls)@flat(ms)"
   375 apply (induct_tac "ls")
   376 apply (simp_all (no_asm_simp) add: app_assoc)
   377 done
   378 
   379 (*** theorems about rev ***)
   380 
   381 lemma rev_map_distrib: "l \<in> list(A) ==> rev(map(h,l)) = map(h,rev(l))"
   382 apply (induct_tac "l")
   383 apply (simp_all (no_asm_simp) add: map_app_distrib)
   384 done
   385 
   386 (*Simplifier needs the premises as assumptions because rewriting will not
   387   instantiate the variable ?A in the rules' typing conditions; note that
   388   rev_type does not instantiate ?A.  Only the premises do.
   389 *)
   390 lemma rev_app_distrib:
   391      "[| xs: list(A);  ys: list(A) |] ==> rev(xs@ys) = rev(ys)@rev(xs)"
   392 apply (erule list.induct)
   393 apply (simp_all add: app_assoc)
   394 done
   395 
   396 lemma rev_rev_ident [simp]: "l \<in> list(A) ==> rev(rev(l))=l"
   397 apply (induct_tac "l")
   398 apply (simp_all (no_asm_simp) add: rev_app_distrib)
   399 done
   400 
   401 lemma rev_flat: "ls: list(list(A)) ==> rev(flat(ls)) = flat(map(rev,rev(ls)))"
   402 apply (induct_tac "ls")
   403 apply (simp_all add: map_app_distrib flat_app_distrib rev_app_distrib)
   404 done
   405 
   406 
   407 (*** theorems about list_add ***)
   408 
   409 lemma list_add_app:
   410      "[| xs: list(nat);  ys: list(nat) |]
   411       ==> list_add(xs@ys) = list_add(ys) #+ list_add(xs)"
   412 apply (induct_tac "xs", simp_all)
   413 done
   414 
   415 lemma list_add_rev: "l \<in> list(nat) ==> list_add(rev(l)) = list_add(l)"
   416 apply (induct_tac "l")
   417 apply (simp_all (no_asm_simp) add: list_add_app)
   418 done
   419 
   420 lemma list_add_flat:
   421      "ls: list(list(nat)) ==> list_add(flat(ls)) = list_add(map(list_add,ls))"
   422 apply (induct_tac "ls")
   423 apply (simp_all (no_asm_simp) add: list_add_app)
   424 done
   425 
   426 (** New induction rules **)
   427 
   428 lemma list_append_induct [case_names Nil snoc, consumes 1]:
   429     "[| l \<in> list(A);
   430         P(Nil);
   431         !!x y. [| x \<in> A;  y \<in> list(A);  P(y) |] ==> P(y @ [x])
   432      |] ==> P(l)"
   433 apply (subgoal_tac "P(rev(rev(l)))", simp)
   434 apply (erule rev_type [THEN list.induct], simp_all)
   435 done
   436 
   437 lemma list_complete_induct_lemma [rule_format]:
   438  assumes ih:
   439     "\<And>l. [| l \<in> list(A);
   440              \<forall>l' \<in> list(A). length(l') < length(l) \<longrightarrow> P(l')|]
   441           ==> P(l)"
   442   shows "n \<in> nat ==> \<forall>l \<in> list(A). length(l) < n \<longrightarrow> P(l)"
   443 apply (induct_tac n, simp)
   444 apply (blast intro: ih elim!: leE)
   445 done
   446 
   447 theorem list_complete_induct:
   448       "[| l \<in> list(A);
   449           \<And>l. [| l \<in> list(A);
   450                   \<forall>l' \<in> list(A). length(l') < length(l) \<longrightarrow> P(l')|]
   451                ==> P(l)
   452        |] ==> P(l)"
   453 apply (rule list_complete_induct_lemma [of A])
   454    prefer 4 apply (rule le_refl, simp)
   455   apply blast
   456  apply simp
   457 apply assumption
   458 done
   459 
   460 
   461 (*** Thanks to Sidi Ehmety for these results about min, take, etc. ***)
   462 
   463 (** min FIXME: replace by Int! **)
   464 (* Min theorems are also true for i, j ordinals *)
   465 lemma min_sym: "[| i \<in> nat; j \<in> nat |] ==> min(i,j)=min(j,i)"
   466 apply (unfold min_def)
   467 apply (auto dest!: not_lt_imp_le dest: lt_not_sym intro: le_anti_sym)
   468 done
   469 
   470 lemma min_type [simp,TC]: "[| i \<in> nat; j \<in> nat |] ==> min(i,j):nat"
   471 by (unfold min_def, auto)
   472 
   473 lemma min_0 [simp]: "i \<in> nat ==> min(0,i) = 0"
   474 apply (unfold min_def)
   475 apply (auto dest: not_lt_imp_le)
   476 done
   477 
   478 lemma min_02 [simp]: "i \<in> nat ==> min(i, 0) = 0"
   479 apply (unfold min_def)
   480 apply (auto dest: not_lt_imp_le)
   481 done
   482 
   483 lemma lt_min_iff: "[| i \<in> nat; j \<in> nat; k \<in> nat |] ==> i<min(j,k) \<longleftrightarrow> i<j & i<k"
   484 apply (unfold min_def)
   485 apply (auto dest!: not_lt_imp_le intro: lt_trans2 lt_trans)
   486 done
   487 
   488 lemma min_succ_succ [simp]:
   489      "[| i \<in> nat; j \<in> nat |] ==>  min(succ(i), succ(j))= succ(min(i, j))"
   490 apply (unfold min_def, auto)
   491 done
   492 
   493 (*** more theorems about lists ***)
   494 
   495 (** filter **)
   496 
   497 lemma filter_append [simp]:
   498      "xs:list(A) ==> filter(P, xs@ys) = filter(P, xs) @ filter(P, ys)"
   499 by (induct_tac "xs", auto)
   500 
   501 lemma filter_type [simp,TC]: "xs:list(A) ==> filter(P, xs):list(A)"
   502 by (induct_tac "xs", auto)
   503 
   504 lemma length_filter: "xs:list(A) ==> length(filter(P, xs)) \<le> length(xs)"
   505 apply (induct_tac "xs", auto)
   506 apply (rule_tac j = "length (l) " in le_trans)
   507 apply (auto simp add: le_iff)
   508 done
   509 
   510 lemma filter_is_subset: "xs:list(A) ==> set_of_list(filter(P,xs)) \<subseteq> set_of_list(xs)"
   511 by (induct_tac "xs", auto)
   512 
   513 lemma filter_False [simp]: "xs:list(A) ==> filter(%p. False, xs) = Nil"
   514 by (induct_tac "xs", auto)
   515 
   516 lemma filter_True [simp]: "xs:list(A) ==> filter(%p. True, xs) = xs"
   517 by (induct_tac "xs", auto)
   518 
   519 (** length **)
   520 
   521 lemma length_is_0_iff [simp]: "xs:list(A) ==> length(xs)=0 \<longleftrightarrow> xs=Nil"
   522 by (erule list.induct, auto)
   523 
   524 lemma length_is_0_iff2 [simp]: "xs:list(A) ==> 0 = length(xs) \<longleftrightarrow> xs=Nil"
   525 by (erule list.induct, auto)
   526 
   527 lemma length_tl [simp]: "xs:list(A) ==> length(tl(xs)) = length(xs) #- 1"
   528 by (erule list.induct, auto)
   529 
   530 lemma length_greater_0_iff: "xs:list(A) ==> 0<length(xs) \<longleftrightarrow> xs \<noteq> Nil"
   531 by (erule list.induct, auto)
   532 
   533 lemma length_succ_iff: "xs:list(A) ==> length(xs)=succ(n) \<longleftrightarrow> (\<exists>y ys. xs=Cons(y, ys) & length(ys)=n)"
   534 by (erule list.induct, auto)
   535 
   536 (** more theorems about append **)
   537 
   538 lemma append_is_Nil_iff [simp]:
   539      "xs:list(A) ==> (xs@ys = Nil) \<longleftrightarrow> (xs=Nil & ys = Nil)"
   540 by (erule list.induct, auto)
   541 
   542 lemma append_is_Nil_iff2 [simp]:
   543      "xs:list(A) ==> (Nil = xs@ys) \<longleftrightarrow> (xs=Nil & ys = Nil)"
   544 by (erule list.induct, auto)
   545 
   546 lemma append_left_is_self_iff [simp]:
   547      "xs:list(A) ==> (xs@ys = xs) \<longleftrightarrow> (ys = Nil)"
   548 by (erule list.induct, auto)
   549 
   550 lemma append_left_is_self_iff2 [simp]:
   551      "xs:list(A) ==> (xs = xs@ys) \<longleftrightarrow> (ys = Nil)"
   552 by (erule list.induct, auto)
   553 
   554 (*TOO SLOW as a default simprule!*)
   555 lemma append_left_is_Nil_iff [rule_format]:
   556      "[| xs:list(A); ys:list(A); zs:list(A) |] ==>
   557    length(ys)=length(zs) \<longrightarrow> (xs@ys=zs \<longleftrightarrow> (xs=Nil & ys=zs))"
   558 apply (erule list.induct)
   559 apply (auto simp add: length_app)
   560 done
   561 
   562 (*TOO SLOW as a default simprule!*)
   563 lemma append_left_is_Nil_iff2 [rule_format]:
   564      "[| xs:list(A); ys:list(A); zs:list(A) |] ==>
   565    length(ys)=length(zs) \<longrightarrow> (zs=ys@xs \<longleftrightarrow> (xs=Nil & ys=zs))"
   566 apply (erule list.induct)
   567 apply (auto simp add: length_app)
   568 done
   569 
   570 lemma append_eq_append_iff [rule_format,simp]:
   571      "xs:list(A) ==> \<forall>ys \<in> list(A).
   572       length(xs)=length(ys) \<longrightarrow> (xs@us = ys@vs) \<longleftrightarrow> (xs=ys & us=vs)"
   573 apply (erule list.induct)
   574 apply (simp (no_asm_simp))
   575 apply clarify
   576 apply (erule_tac a = ys in list.cases, auto)
   577 done
   578 
   579 lemma append_eq_append [rule_format]:
   580   "xs:list(A) ==>
   581    \<forall>ys \<in> list(A). \<forall>us \<in> list(A). \<forall>vs \<in> list(A).
   582    length(us) = length(vs) \<longrightarrow> (xs@us = ys@vs) \<longrightarrow> (xs=ys & us=vs)"
   583 apply (induct_tac "xs")
   584 apply (force simp add: length_app, clarify)
   585 apply (erule_tac a = ys in list.cases, simp)
   586 apply (subgoal_tac "Cons (a, l) @ us =vs")
   587  apply (drule rev_iffD1 [OF _ append_left_is_Nil_iff], simp_all, blast)
   588 done
   589 
   590 lemma append_eq_append_iff2 [simp]:
   591  "[| xs:list(A); ys:list(A); us:list(A); vs:list(A); length(us)=length(vs) |]
   592   ==>  xs@us = ys@vs \<longleftrightarrow> (xs=ys & us=vs)"
   593 apply (rule iffI)
   594 apply (rule append_eq_append, auto)
   595 done
   596 
   597 lemma append_self_iff [simp]:
   598      "[| xs:list(A); ys:list(A); zs:list(A) |] ==> xs@ys=xs@zs \<longleftrightarrow> ys=zs"
   599 by simp
   600 
   601 lemma append_self_iff2 [simp]:
   602      "[| xs:list(A); ys:list(A); zs:list(A) |] ==> ys@xs=zs@xs \<longleftrightarrow> ys=zs"
   603 by simp
   604 
   605 (* Can also be proved from append_eq_append_iff2,
   606 but the proof requires two more hypotheses: x \<in> A and y \<in> A *)
   607 lemma append1_eq_iff [rule_format,simp]:
   608      "xs:list(A) ==> \<forall>ys \<in> list(A). xs@[x] = ys@[y] \<longleftrightarrow> (xs = ys & x=y)"
   609 apply (erule list.induct)
   610  apply clarify
   611  apply (erule list.cases)
   612  apply simp_all
   613 txt{*Inductive step*}
   614 apply clarify
   615 apply (erule_tac a=ys in list.cases, simp_all)
   616 done
   617 
   618 
   619 lemma append_right_is_self_iff [simp]:
   620      "[| xs:list(A); ys:list(A) |] ==> (xs@ys = ys) \<longleftrightarrow> (xs=Nil)"
   621 by (simp (no_asm_simp) add: append_left_is_Nil_iff)
   622 
   623 lemma append_right_is_self_iff2 [simp]:
   624      "[| xs:list(A); ys:list(A) |] ==> (ys = xs@ys) \<longleftrightarrow> (xs=Nil)"
   625 apply (rule iffI)
   626 apply (drule sym, auto)
   627 done
   628 
   629 lemma hd_append [rule_format,simp]:
   630      "xs:list(A) ==> xs \<noteq> Nil \<longrightarrow> hd(xs @ ys) = hd(xs)"
   631 by (induct_tac "xs", auto)
   632 
   633 lemma tl_append [rule_format,simp]:
   634      "xs:list(A) ==> xs\<noteq>Nil \<longrightarrow> tl(xs @ ys) = tl(xs)@ys"
   635 by (induct_tac "xs", auto)
   636 
   637 (** rev **)
   638 lemma rev_is_Nil_iff [simp]: "xs:list(A) ==> (rev(xs) = Nil \<longleftrightarrow> xs = Nil)"
   639 by (erule list.induct, auto)
   640 
   641 lemma Nil_is_rev_iff [simp]: "xs:list(A) ==> (Nil = rev(xs) \<longleftrightarrow> xs = Nil)"
   642 by (erule list.induct, auto)
   643 
   644 lemma rev_is_rev_iff [rule_format,simp]:
   645      "xs:list(A) ==> \<forall>ys \<in> list(A). rev(xs)=rev(ys) \<longleftrightarrow> xs=ys"
   646 apply (erule list.induct, force, clarify)
   647 apply (erule_tac a = ys in list.cases, auto)
   648 done
   649 
   650 lemma rev_list_elim [rule_format]:
   651      "xs:list(A) ==>
   652       (xs=Nil \<longrightarrow> P) \<longrightarrow> (\<forall>ys \<in> list(A). \<forall>y \<in> A. xs =ys@[y] \<longrightarrow>P)\<longrightarrow>P"
   653 by (erule list_append_induct, auto)
   654 
   655 
   656 (** more theorems about drop **)
   657 
   658 lemma length_drop [rule_format,simp]:
   659      "n \<in> nat ==> \<forall>xs \<in> list(A). length(drop(n, xs)) = length(xs) #- n"
   660 apply (erule nat_induct)
   661 apply (auto elim: list.cases)
   662 done
   663 
   664 lemma drop_all [rule_format,simp]:
   665      "n \<in> nat ==> \<forall>xs \<in> list(A). length(xs) \<le> n \<longrightarrow> drop(n, xs)=Nil"
   666 apply (erule nat_induct)
   667 apply (auto elim: list.cases)
   668 done
   669 
   670 lemma drop_append [rule_format]:
   671      "n \<in> nat ==>
   672       \<forall>xs \<in> list(A). drop(n, xs@ys) = drop(n,xs) @ drop(n #- length(xs), ys)"
   673 apply (induct_tac "n")
   674 apply (auto elim: list.cases)
   675 done
   676 
   677 lemma drop_drop:
   678     "m \<in> nat ==> \<forall>xs \<in> list(A). \<forall>n \<in> nat. drop(n, drop(m, xs))=drop(n #+ m, xs)"
   679 apply (induct_tac "m")
   680 apply (auto elim: list.cases)
   681 done
   682 
   683 (** take **)
   684 
   685 lemma take_0 [simp]: "xs:list(A) ==> take(0, xs) =  Nil"
   686 apply (unfold take_def)
   687 apply (erule list.induct, auto)
   688 done
   689 
   690 lemma take_succ_Cons [simp]:
   691     "n \<in> nat ==> take(succ(n), Cons(a, xs)) = Cons(a, take(n, xs))"
   692 by (simp add: take_def)
   693 
   694 (* Needed for proving take_all *)
   695 lemma take_Nil [simp]: "n \<in> nat ==> take(n, Nil) = Nil"
   696 by (unfold take_def, auto)
   697 
   698 lemma take_all [rule_format,simp]:
   699      "n \<in> nat ==> \<forall>xs \<in> list(A). length(xs) \<le> n  \<longrightarrow> take(n, xs) = xs"
   700 apply (erule nat_induct)
   701 apply (auto elim: list.cases)
   702 done
   703 
   704 lemma take_type [rule_format,simp,TC]:
   705      "xs:list(A) ==> \<forall>n \<in> nat. take(n, xs):list(A)"
   706 apply (erule list.induct, simp, clarify)
   707 apply (erule natE, auto)
   708 done
   709 
   710 lemma take_append [rule_format,simp]:
   711  "xs:list(A) ==>
   712   \<forall>ys \<in> list(A). \<forall>n \<in> nat. take(n, xs @ ys) =
   713                             take(n, xs) @ take(n #- length(xs), ys)"
   714 apply (erule list.induct, simp, clarify)
   715 apply (erule natE, auto)
   716 done
   717 
   718 lemma take_take [rule_format]:
   719    "m \<in> nat ==>
   720     \<forall>xs \<in> list(A). \<forall>n \<in> nat. take(n, take(m,xs))= take(min(n, m), xs)"
   721 apply (induct_tac "m", auto)
   722 apply (erule_tac a = xs in list.cases)
   723 apply (auto simp add: take_Nil)
   724 apply (erule_tac n=n in natE)
   725 apply (auto intro: take_0 take_type)
   726 done
   727 
   728 (** nth **)
   729 
   730 lemma nth_0 [simp]: "nth(0, Cons(a, l)) = a"
   731 by (simp add: nth_def)
   732 
   733 lemma nth_Cons [simp]: "n \<in> nat ==> nth(succ(n), Cons(a,l)) = nth(n,l)"
   734 by (simp add: nth_def)
   735 
   736 lemma nth_empty [simp]: "nth(n, Nil) = 0"
   737 by (simp add: nth_def)
   738 
   739 lemma nth_type [rule_format,simp,TC]:
   740      "xs:list(A) ==> \<forall>n. n < length(xs) \<longrightarrow> nth(n,xs) \<in> A"
   741 apply (erule list.induct, simp, clarify)
   742 apply (subgoal_tac "n \<in> nat")
   743  apply (erule natE, auto dest!: le_in_nat)
   744 done
   745 
   746 lemma nth_eq_0 [rule_format]:
   747      "xs:list(A) ==> \<forall>n \<in> nat. length(xs) \<le> n \<longrightarrow> nth(n,xs) = 0"
   748 apply (erule list.induct, simp, clarify)
   749 apply (erule natE, auto)
   750 done
   751 
   752 lemma nth_append [rule_format]:
   753   "xs:list(A) ==>
   754    \<forall>n \<in> nat. nth(n, xs @ ys) = (if n < length(xs) then nth(n,xs)
   755                                 else nth(n #- length(xs), ys))"
   756 apply (induct_tac "xs", simp, clarify)
   757 apply (erule natE, auto)
   758 done
   759 
   760 lemma set_of_list_conv_nth:
   761     "xs:list(A)
   762      ==> set_of_list(xs) = {x \<in> A. \<exists>i\<in>nat. i<length(xs) & x = nth(i,xs)}"
   763 apply (induct_tac "xs", simp_all)
   764 apply (rule equalityI, auto)
   765 apply (rule_tac x = 0 in bexI, auto)
   766 apply (erule natE, auto)
   767 done
   768 
   769 (* Other theorems about lists *)
   770 
   771 lemma nth_take_lemma [rule_format]:
   772  "k \<in> nat ==>
   773   \<forall>xs \<in> list(A). (\<forall>ys \<in> list(A). k \<le> length(xs) \<longrightarrow> k \<le> length(ys) \<longrightarrow>
   774       (\<forall>i \<in> nat. i<k \<longrightarrow> nth(i,xs) = nth(i,ys))\<longrightarrow> take(k,xs) = take(k,ys))"
   775 apply (induct_tac "k")
   776 apply (simp_all (no_asm_simp) add: lt_succ_eq_0_disj all_conj_distrib)
   777 apply clarify
   778 (*Both lists are non-empty*)
   779 apply (erule_tac a=xs in list.cases, simp)
   780 apply (erule_tac a=ys in list.cases, clarify)
   781 apply (simp (no_asm_use) )
   782 apply clarify
   783 apply (simp (no_asm_simp))
   784 apply (rule conjI, force)
   785 apply (rename_tac y ys z zs)
   786 apply (drule_tac x = zs and x1 = ys in bspec [THEN bspec], auto)
   787 done
   788 
   789 lemma nth_equalityI [rule_format]:
   790      "[| xs:list(A); ys:list(A); length(xs) = length(ys);
   791          \<forall>i \<in> nat. i < length(xs) \<longrightarrow> nth(i,xs) = nth(i,ys) |]
   792       ==> xs = ys"
   793 apply (subgoal_tac "length (xs) \<le> length (ys) ")
   794 apply (cut_tac k="length(xs)" and xs=xs and ys=ys in nth_take_lemma)
   795 apply (simp_all add: take_all)
   796 done
   797 
   798 (*The famous take-lemma*)
   799 
   800 lemma take_equalityI [rule_format]:
   801     "[| xs:list(A); ys:list(A); (\<forall>i \<in> nat. take(i, xs) = take(i,ys)) |]
   802      ==> xs = ys"
   803 apply (case_tac "length (xs) \<le> length (ys) ")
   804 apply (drule_tac x = "length (ys) " in bspec)
   805 apply (drule_tac [3] not_lt_imp_le)
   806 apply (subgoal_tac [5] "length (ys) \<le> length (xs) ")
   807 apply (rule_tac [6] j = "succ (length (ys))" in le_trans)
   808 apply (rule_tac [6] leI)
   809 apply (drule_tac [5] x = "length (xs) " in bspec)
   810 apply (simp_all add: take_all)
   811 done
   812 
   813 lemma nth_drop [rule_format]:
   814   "n \<in> nat ==> \<forall>i \<in> nat. \<forall>xs \<in> list(A). nth(i, drop(n, xs)) = nth(n #+ i, xs)"
   815 apply (induct_tac "n", simp_all, clarify)
   816 apply (erule list.cases, auto)
   817 done
   818 
   819 lemma take_succ [rule_format]:
   820   "xs\<in>list(A)
   821    ==> \<forall>i. i < length(xs) \<longrightarrow> take(succ(i), xs) = take(i,xs) @ [nth(i, xs)]"
   822 apply (induct_tac "xs", auto)
   823 apply (subgoal_tac "i\<in>nat")
   824 apply (erule natE)
   825 apply (auto simp add: le_in_nat)
   826 done
   827 
   828 lemma take_add [rule_format]:
   829      "[|xs\<in>list(A); j\<in>nat|]
   830       ==> \<forall>i\<in>nat. take(i #+ j, xs) = take(i,xs) @ take(j, drop(i,xs))"
   831 apply (induct_tac "xs", simp_all, clarify)
   832 apply (erule_tac n = i in natE, simp_all)
   833 done
   834 
   835 lemma length_take:
   836      "l\<in>list(A) ==> \<forall>n\<in>nat. length(take(n,l)) = min(n, length(l))"
   837 apply (induct_tac "l", safe, simp_all)
   838 apply (erule natE, simp_all)
   839 done
   840 
   841 subsection{*The function zip*}
   842 
   843 text{*Crafty definition to eliminate a type argument*}
   844 
   845 consts
   846   zip_aux        :: "[i,i]=>i"
   847 
   848 primrec (*explicit lambda is required because both arguments of "un" vary*)
   849   "zip_aux(B,[]) =
   850      (\<lambda>ys \<in> list(B). list_case([], %y l. [], ys))"
   851 
   852   "zip_aux(B,Cons(x,l)) =
   853      (\<lambda>ys \<in> list(B).
   854         list_case(Nil, %y zs. Cons(<x,y>, zip_aux(B,l)`zs), ys))"
   855 
   856 definition
   857   zip :: "[i, i]=>i"  where
   858    "zip(xs, ys) == zip_aux(set_of_list(ys),xs)`ys"
   859 
   860 
   861 (* zip equations *)
   862 
   863 lemma list_on_set_of_list: "xs \<in> list(A) ==> xs \<in> list(set_of_list(xs))"
   864 apply (induct_tac xs, simp_all)
   865 apply (blast intro: list_mono [THEN subsetD])
   866 done
   867 
   868 lemma zip_Nil [simp]: "ys:list(A) ==> zip(Nil, ys)=Nil"
   869 apply (simp add: zip_def list_on_set_of_list [of _ A])
   870 apply (erule list.cases, simp_all)
   871 done
   872 
   873 lemma zip_Nil2 [simp]: "xs:list(A) ==> zip(xs, Nil)=Nil"
   874 apply (simp add: zip_def list_on_set_of_list [of _ A])
   875 apply (erule list.cases, simp_all)
   876 done
   877 
   878 lemma zip_aux_unique [rule_format]:
   879      "[|B<=C;  xs \<in> list(A)|]
   880       ==> \<forall>ys \<in> list(B). zip_aux(C,xs) ` ys = zip_aux(B,xs) ` ys"
   881 apply (induct_tac xs)
   882  apply simp_all
   883  apply (blast intro: list_mono [THEN subsetD], clarify)
   884 apply (erule_tac a=ys in list.cases, auto)
   885 apply (blast intro: list_mono [THEN subsetD])
   886 done
   887 
   888 lemma zip_Cons_Cons [simp]:
   889      "[| xs:list(A); ys:list(B); x \<in> A; y \<in> B |] ==>
   890       zip(Cons(x,xs), Cons(y, ys)) = Cons(<x,y>, zip(xs, ys))"
   891 apply (simp add: zip_def, auto)
   892 apply (rule zip_aux_unique, auto)
   893 apply (simp add: list_on_set_of_list [of _ B])
   894 apply (blast intro: list_on_set_of_list list_mono [THEN subsetD])
   895 done
   896 
   897 lemma zip_type [rule_format,simp,TC]:
   898      "xs:list(A) ==> \<forall>ys \<in> list(B). zip(xs, ys):list(A*B)"
   899 apply (induct_tac "xs")
   900 apply (simp (no_asm))
   901 apply clarify
   902 apply (erule_tac a = ys in list.cases, auto)
   903 done
   904 
   905 (* zip length *)
   906 lemma length_zip [rule_format,simp]:
   907      "xs:list(A) ==> \<forall>ys \<in> list(B). length(zip(xs,ys)) =
   908                                      min(length(xs), length(ys))"
   909 apply (unfold min_def)
   910 apply (induct_tac "xs", simp_all, clarify)
   911 apply (erule_tac a = ys in list.cases, auto)
   912 done
   913 
   914 lemma zip_append1 [rule_format]:
   915  "[| ys:list(A); zs:list(B) |] ==>
   916   \<forall>xs \<in> list(A). zip(xs @ ys, zs) =
   917                  zip(xs, take(length(xs), zs)) @ zip(ys, drop(length(xs),zs))"
   918 apply (induct_tac "zs", force, clarify)
   919 apply (erule_tac a = xs in list.cases, simp_all)
   920 done
   921 
   922 lemma zip_append2 [rule_format]:
   923  "[| xs:list(A); zs:list(B) |] ==> \<forall>ys \<in> list(B). zip(xs, ys@zs) =
   924        zip(take(length(ys), xs), ys) @ zip(drop(length(ys), xs), zs)"
   925 apply (induct_tac "xs", force, clarify)
   926 apply (erule_tac a = ys in list.cases, auto)
   927 done
   928 
   929 lemma zip_append [simp]:
   930  "[| length(xs) = length(us); length(ys) = length(vs);
   931      xs:list(A); us:list(B); ys:list(A); vs:list(B) |]
   932   ==> zip(xs@ys,us@vs) = zip(xs, us) @ zip(ys, vs)"
   933 by (simp (no_asm_simp) add: zip_append1 drop_append diff_self_eq_0)
   934 
   935 
   936 lemma zip_rev [rule_format,simp]:
   937  "ys:list(B) ==> \<forall>xs \<in> list(A).
   938     length(xs) = length(ys) \<longrightarrow> zip(rev(xs), rev(ys)) = rev(zip(xs, ys))"
   939 apply (induct_tac "ys", force, clarify)
   940 apply (erule_tac a = xs in list.cases)
   941 apply (auto simp add: length_rev)
   942 done
   943 
   944 lemma nth_zip [rule_format,simp]:
   945    "ys:list(B) ==> \<forall>i \<in> nat. \<forall>xs \<in> list(A).
   946                     i < length(xs) \<longrightarrow> i < length(ys) \<longrightarrow>
   947                     nth(i,zip(xs, ys)) = <nth(i,xs),nth(i, ys)>"
   948 apply (induct_tac "ys", force, clarify)
   949 apply (erule_tac a = xs in list.cases, simp)
   950 apply (auto elim: natE)
   951 done
   952 
   953 lemma set_of_list_zip [rule_format]:
   954      "[| xs:list(A); ys:list(B); i \<in> nat |]
   955       ==> set_of_list(zip(xs, ys)) =
   956           {<x, y>:A*B. \<exists>i\<in>nat. i < min(length(xs), length(ys))
   957           & x = nth(i, xs) & y = nth(i, ys)}"
   958 by (force intro!: Collect_cong simp add: lt_min_iff set_of_list_conv_nth)
   959 
   960 (** list_update **)
   961 
   962 lemma list_update_Nil [simp]: "i \<in> nat ==>list_update(Nil, i, v) = Nil"
   963 by (unfold list_update_def, auto)
   964 
   965 lemma list_update_Cons_0 [simp]: "list_update(Cons(x, xs), 0, v)= Cons(v, xs)"
   966 by (unfold list_update_def, auto)
   967 
   968 lemma list_update_Cons_succ [simp]:
   969   "n \<in> nat ==>
   970     list_update(Cons(x, xs), succ(n), v)= Cons(x, list_update(xs, n, v))"
   971 apply (unfold list_update_def, auto)
   972 done
   973 
   974 lemma list_update_type [rule_format,simp,TC]:
   975      "[| xs:list(A); v \<in> A |] ==> \<forall>n \<in> nat. list_update(xs, n, v):list(A)"
   976 apply (induct_tac "xs")
   977 apply (simp (no_asm))
   978 apply clarify
   979 apply (erule natE, auto)
   980 done
   981 
   982 lemma length_list_update [rule_format,simp]:
   983      "xs:list(A) ==> \<forall>i \<in> nat. length(list_update(xs, i, v))=length(xs)"
   984 apply (induct_tac "xs")
   985 apply (simp (no_asm))
   986 apply clarify
   987 apply (erule natE, auto)
   988 done
   989 
   990 lemma nth_list_update [rule_format]:
   991      "[| xs:list(A) |] ==> \<forall>i \<in> nat. \<forall>j \<in> nat. i < length(xs)  \<longrightarrow>
   992          nth(j, list_update(xs, i, x)) = (if i=j then x else nth(j, xs))"
   993 apply (induct_tac "xs")
   994  apply simp_all
   995 apply clarify
   996 apply (rename_tac i j)
   997 apply (erule_tac n=i in natE)
   998 apply (erule_tac [2] n=j in natE)
   999 apply (erule_tac n=j in natE, simp_all, force)
  1000 done
  1001 
  1002 lemma nth_list_update_eq [simp]:
  1003      "[| i < length(xs); xs:list(A) |] ==> nth(i, list_update(xs, i,x)) = x"
  1004 by (simp (no_asm_simp) add: lt_length_in_nat nth_list_update)
  1005 
  1006 
  1007 lemma nth_list_update_neq [rule_format,simp]:
  1008   "xs:list(A) ==>
  1009      \<forall>i \<in> nat. \<forall>j \<in> nat. i \<noteq> j \<longrightarrow> nth(j, list_update(xs,i,x)) = nth(j,xs)"
  1010 apply (induct_tac "xs")
  1011  apply (simp (no_asm))
  1012 apply clarify
  1013 apply (erule natE)
  1014 apply (erule_tac [2] natE, simp_all)
  1015 apply (erule natE, simp_all)
  1016 done
  1017 
  1018 lemma list_update_overwrite [rule_format,simp]:
  1019      "xs:list(A) ==> \<forall>i \<in> nat. i < length(xs)
  1020    \<longrightarrow> list_update(list_update(xs, i, x), i, y) = list_update(xs, i,y)"
  1021 apply (induct_tac "xs")
  1022  apply (simp (no_asm))
  1023 apply clarify
  1024 apply (erule natE, auto)
  1025 done
  1026 
  1027 lemma list_update_same_conv [rule_format]:
  1028      "xs:list(A) ==>
  1029       \<forall>i \<in> nat. i < length(xs) \<longrightarrow>
  1030                  (list_update(xs, i, x) = xs) \<longleftrightarrow> (nth(i, xs) = x)"
  1031 apply (induct_tac "xs")
  1032  apply (simp (no_asm))
  1033 apply clarify
  1034 apply (erule natE, auto)
  1035 done
  1036 
  1037 lemma update_zip [rule_format]:
  1038      "ys:list(B) ==>
  1039       \<forall>i \<in> nat. \<forall>xy \<in> A*B. \<forall>xs \<in> list(A).
  1040         length(xs) = length(ys) \<longrightarrow>
  1041         list_update(zip(xs, ys), i, xy) = zip(list_update(xs, i, fst(xy)),
  1042                                               list_update(ys, i, snd(xy)))"
  1043 apply (induct_tac "ys")
  1044  apply auto
  1045 apply (erule_tac a = xs in list.cases)
  1046 apply (auto elim: natE)
  1047 done
  1048 
  1049 lemma set_update_subset_cons [rule_format]:
  1050   "xs:list(A) ==>
  1051    \<forall>i \<in> nat. set_of_list(list_update(xs, i, x)) \<subseteq> cons(x, set_of_list(xs))"
  1052 apply (induct_tac "xs")
  1053  apply simp
  1054 apply (rule ballI)
  1055 apply (erule natE, simp_all, auto)
  1056 done
  1057 
  1058 lemma set_of_list_update_subsetI:
  1059      "[| set_of_list(xs) \<subseteq> A; xs:list(A); x \<in> A; i \<in> nat|]
  1060    ==> set_of_list(list_update(xs, i,x)) \<subseteq> A"
  1061 apply (rule subset_trans)
  1062 apply (rule set_update_subset_cons, auto)
  1063 done
  1064 
  1065 (** upt **)
  1066 
  1067 lemma upt_rec:
  1068      "j \<in> nat ==> upt(i,j) = (if i<j then Cons(i, upt(succ(i), j)) else Nil)"
  1069 apply (induct_tac "j", auto)
  1070 apply (drule not_lt_imp_le)
  1071 apply (auto simp: lt_Ord intro: le_anti_sym)
  1072 done
  1073 
  1074 lemma upt_conv_Nil [simp]: "[| j \<le> i; j \<in> nat |] ==> upt(i,j) = Nil"
  1075 apply (subst upt_rec, auto)
  1076 apply (auto simp add: le_iff)
  1077 apply (drule lt_asym [THEN notE], auto)
  1078 done
  1079 
  1080 (*Only needed if upt_Suc is deleted from the simpset*)
  1081 lemma upt_succ_append:
  1082      "[| i \<le> j; j \<in> nat |] ==> upt(i,succ(j)) = upt(i, j)@[j]"
  1083 by simp
  1084 
  1085 lemma upt_conv_Cons:
  1086      "[| i<j; j \<in> nat |]  ==> upt(i,j) = Cons(i,upt(succ(i),j))"
  1087 apply (rule trans)
  1088 apply (rule upt_rec, auto)
  1089 done
  1090 
  1091 lemma upt_type [simp,TC]: "j \<in> nat ==> upt(i,j):list(nat)"
  1092 by (induct_tac "j", auto)
  1093 
  1094 (*LOOPS as a simprule, since j<=j*)
  1095 lemma upt_add_eq_append:
  1096      "[| i \<le> j; j \<in> nat; k \<in> nat |] ==> upt(i, j #+k) = upt(i,j)@upt(j,j#+k)"
  1097 apply (induct_tac "k")
  1098 apply (auto simp add: app_assoc app_type)
  1099 apply (rule_tac j = j in le_trans, auto)
  1100 done
  1101 
  1102 lemma length_upt [simp]: "[| i \<in> nat; j \<in> nat |] ==>length(upt(i,j)) = j #- i"
  1103 apply (induct_tac "j")
  1104 apply (rule_tac [2] sym)
  1105 apply (auto dest!: not_lt_imp_le simp add: diff_succ diff_is_0_iff)
  1106 done
  1107 
  1108 lemma nth_upt [rule_format,simp]:
  1109      "[| i \<in> nat; j \<in> nat; k \<in> nat |] ==> i #+ k < j \<longrightarrow> nth(k, upt(i,j)) = i #+ k"
  1110 apply (induct_tac "j", simp)
  1111 apply (simp add: nth_append le_iff)
  1112 apply (auto dest!: not_lt_imp_le
  1113             simp add: nth_append less_diff_conv add_commute)
  1114 done
  1115 
  1116 lemma take_upt [rule_format,simp]:
  1117      "[| m \<in> nat; n \<in> nat |] ==>
  1118          \<forall>i \<in> nat. i #+ m \<le> n \<longrightarrow> take(m, upt(i,n)) = upt(i,i#+m)"
  1119 apply (induct_tac "m")
  1120 apply (simp (no_asm_simp) add: take_0)
  1121 apply clarify
  1122 apply (subst upt_rec, simp)
  1123 apply (rule sym)
  1124 apply (subst upt_rec, simp)
  1125 apply (simp_all del: upt.simps)
  1126 apply (rule_tac j = "succ (i #+ x) " in lt_trans2)
  1127 apply auto
  1128 done
  1129 
  1130 lemma map_succ_upt:
  1131      "[| m \<in> nat; n \<in> nat |] ==> map(succ, upt(m,n))= upt(succ(m), succ(n))"
  1132 apply (induct_tac "n")
  1133 apply (auto simp add: map_app_distrib)
  1134 done
  1135 
  1136 lemma nth_map [rule_format,simp]:
  1137      "xs:list(A) ==>
  1138       \<forall>n \<in> nat. n < length(xs) \<longrightarrow> nth(n, map(f, xs)) = f(nth(n, xs))"
  1139 apply (induct_tac "xs", simp)
  1140 apply (rule ballI)
  1141 apply (induct_tac "n", auto)
  1142 done
  1143 
  1144 lemma nth_map_upt [rule_format]:
  1145      "[| m \<in> nat; n \<in> nat |] ==>
  1146       \<forall>i \<in> nat. i < n #- m \<longrightarrow> nth(i, map(f, upt(m,n))) = f(m #+ i)"
  1147 apply (rule_tac n = m and m = n in diff_induct, typecheck, simp, simp)
  1148 apply (subst map_succ_upt [symmetric], simp_all, clarify)
  1149 apply (subgoal_tac "i < length (upt (0, x))")
  1150  prefer 2
  1151  apply (simp add: less_diff_conv)
  1152  apply (rule_tac j = "succ (i #+ y) " in lt_trans2)
  1153   apply simp
  1154  apply simp
  1155 apply (subgoal_tac "i < length (upt (y, x))")
  1156  apply (simp_all add: add_commute less_diff_conv)
  1157 done
  1158 
  1159 (** sublist (a generalization of nth to sets) **)
  1160 
  1161 definition
  1162   sublist :: "[i, i] => i"  where
  1163     "sublist(xs, A)==
  1164      map(fst, (filter(%p. snd(p): A, zip(xs, upt(0,length(xs))))))"
  1165 
  1166 lemma sublist_0 [simp]: "xs:list(A) ==>sublist(xs, 0) =Nil"
  1167 by (unfold sublist_def, auto)
  1168 
  1169 lemma sublist_Nil [simp]: "sublist(Nil, A) = Nil"
  1170 by (unfold sublist_def, auto)
  1171 
  1172 lemma sublist_shift_lemma:
  1173  "[| xs:list(B); i \<in> nat |] ==>
  1174   map(fst, filter(%p. snd(p):A, zip(xs, upt(i,i #+ length(xs))))) =
  1175   map(fst, filter(%p. snd(p):nat & snd(p) #+ i \<in> A, zip(xs,upt(0,length(xs)))))"
  1176 apply (erule list_append_induct)
  1177 apply (simp (no_asm_simp))
  1178 apply (auto simp add: add_commute length_app filter_append map_app_distrib)
  1179 done
  1180 
  1181 lemma sublist_type [simp,TC]:
  1182      "xs:list(B) ==> sublist(xs, A):list(B)"
  1183 apply (unfold sublist_def)
  1184 apply (induct_tac "xs")
  1185 apply (auto simp add: filter_append map_app_distrib)
  1186 done
  1187 
  1188 lemma upt_add_eq_append2:
  1189      "[| i \<in> nat; j \<in> nat |] ==> upt(0, i #+ j) = upt(0, i) @ upt(i, i #+ j)"
  1190 by (simp add: upt_add_eq_append [of 0] nat_0_le)
  1191 
  1192 lemma sublist_append:
  1193  "[| xs:list(B); ys:list(B)  |] ==>
  1194   sublist(xs@ys, A) = sublist(xs, A) @ sublist(ys, {j \<in> nat. j #+ length(xs): A})"
  1195 apply (unfold sublist_def)
  1196 apply (erule_tac l = ys in list_append_induct, simp)
  1197 apply (simp (no_asm_simp) add: upt_add_eq_append2 app_assoc [symmetric])
  1198 apply (auto simp add: sublist_shift_lemma length_type map_app_distrib app_assoc)
  1199 apply (simp_all add: add_commute)
  1200 done
  1201 
  1202 
  1203 lemma sublist_Cons:
  1204      "[| xs:list(B); x \<in> B |] ==>
  1205       sublist(Cons(x, xs), A) =
  1206       (if 0 \<in> A then [x] else []) @ sublist(xs, {j \<in> nat. succ(j) \<in> A})"
  1207 apply (erule_tac l = xs in list_append_induct)
  1208 apply (simp (no_asm_simp) add: sublist_def)
  1209 apply (simp del: app_Cons add: app_Cons [symmetric] sublist_append, simp)
  1210 done
  1211 
  1212 lemma sublist_singleton [simp]:
  1213      "sublist([x], A) = (if 0 \<in> A then [x] else [])"
  1214 by (simp add: sublist_Cons)
  1215 
  1216 lemma sublist_upt_eq_take [rule_format, simp]:
  1217     "xs:list(A) ==> \<forall>n\<in>nat. sublist(xs,n) = take(n,xs)"
  1218 apply (erule list.induct, simp)
  1219 apply (clarify )
  1220 apply (erule natE)
  1221 apply (simp_all add: nat_eq_Collect_lt Ord_mem_iff_lt sublist_Cons)
  1222 done
  1223 
  1224 lemma sublist_Int_eq:
  1225      "xs \<in> list(B) ==> sublist(xs, A \<inter> nat) = sublist(xs, A)"
  1226 apply (erule list.induct)
  1227 apply (simp_all add: sublist_Cons)
  1228 done
  1229 
  1230 text{*Repetition of a List Element*}
  1231 
  1232 consts   repeat :: "[i,i]=>i"
  1233 primrec
  1234   "repeat(a,0) = []"
  1235 
  1236   "repeat(a,succ(n)) = Cons(a,repeat(a,n))"
  1237 
  1238 lemma length_repeat: "n \<in> nat ==> length(repeat(a,n)) = n"
  1239 by (induct_tac n, auto)
  1240 
  1241 lemma repeat_succ_app: "n \<in> nat ==> repeat(a,succ(n)) = repeat(a,n) @ [a]"
  1242 apply (induct_tac n)
  1243 apply (simp_all del: app_Cons add: app_Cons [symmetric])
  1244 done
  1245 
  1246 lemma repeat_type [TC]: "[|a \<in> A; n \<in> nat|] ==> repeat(a,n) \<in> list(A)"
  1247 by (induct_tac n, auto)
  1248 
  1249 end