author  wenzelm 
Mon, 13 May 2002 11:05:27 +0200  
changeset 13142  1ebd8ed5a1a0 
parent 13124  6e1decd8a7a9 
child 13145  59bc43b51aa2 
permissions  rwrr 
923  1 
(* Title: HOL/List.thy 
2 
ID: $Id$ 

3 
Author: Tobias Nipkow 

4 
Copyright 1994 TU Muenchen 

5 
*) 

6 

13114  7 
header {* The datatype of finite lists *} 
13122  8 

9 
theory List = PreList: 

923  10 

13142  11 
datatype 'a list = 
12 
Nil ("[]") 

13 
 Cons 'a "'a list" (infixr "#" 65) 

923  14 

15 
consts 

13142  16 
"@" :: "'a list => 'a list => 'a list" (infixr 65) 
17 
filter :: "('a => bool) => 'a list => 'a list" 

18 
concat :: "'a list list => 'a list" 

19 
foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b" 

20 
foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b" 

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

22 
tl :: "'a list => 'a list" 

23 
last :: "'a list => 'a" 

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

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

26 
list_all :: "('a => bool) => ('a list => bool)" 

27 
list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" 

28 
map :: "('a=>'b) => ('a list => 'b list)" 

29 
mem :: "'a => 'a list => bool" (infixl 55) 

30 
nth :: "'a list => nat => 'a" (infixl "!" 100) 

31 
list_update :: "'a list => nat => 'a => 'a list" 

32 
take :: "nat => 'a list => 'a list" 

33 
drop :: "nat => 'a list => 'a list" 

34 
takeWhile :: "('a => bool) => 'a list => 'a list" 

35 
dropWhile :: "('a => bool) => 'a list => 'a list" 

36 
rev :: "'a list => 'a list" 

37 
zip :: "'a list => 'b list => ('a * 'b) list" 

38 
upt :: "nat => nat => nat list" ("(1[_../_'(])") 

39 
remdups :: "'a list => 'a list" 

40 
null :: "'a list => bool" 

41 
"distinct" :: "'a list => bool" 

42 
replicate :: "nat => 'a => 'a list" 

923  43 

5077
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the ith
nipkow
parents:
4643
diff
changeset

44 
nonterminals 
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the ith
nipkow
parents:
4643
diff
changeset

45 
lupdbinds lupdbind 
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the ith
nipkow
parents:
4643
diff
changeset

46 

923  47 
syntax 
13142  48 
 {* list Enumeration *} 
49 
"@list" :: "args => 'a list" ("[(_)]") 

923  50 

13142  51 
 {* Special syntax for filter *} 
52 
"@filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_:_./ _])") 

923  53 

13142  54 
 {* list update *} 
55 
"_lupdbind" :: "['a, 'a] => lupdbind" ("(2_ :=/ _)") 

56 
"" :: "lupdbind => lupdbinds" ("_") 

57 
"_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds" ("_,/ _") 

58 
"_LUpdate" :: "['a, lupdbinds] => 'a" ("_/[(_)]" [900,0] 900) 

5077
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the ith
nipkow
parents:
4643
diff
changeset

59 

13142  60 
upto :: "nat => nat => nat list" ("(1[_../_])") 
5427  61 

923  62 
translations 
63 
"[x, xs]" == "x#[xs]" 

64 
"[x]" == "x#[]" 

3842  65 
"[x:xs . P]" == "filter (%x. P) xs" 
923  66 

5077
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the ith
nipkow
parents:
4643
diff
changeset

67 
"_LUpdate xs (_lupdbinds b bs)" == "_LUpdate (_LUpdate xs b) bs" 
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the ith
nipkow
parents:
4643
diff
changeset

68 
"xs[i:=x]" == "list_update xs i x" 
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the ith
nipkow
parents:
4643
diff
changeset

69 

5427  70 
"[i..j]" == "[i..(Suc j)(]" 
71 

72 

12114
a8e860c86252
eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents:
10832
diff
changeset

73 
syntax (xsymbols) 
13142  74 
"@filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_\<in>_ ./ _])") 
3342
ec3b55fcb165
New operator "lists" for formalizing sets of lists
paulson
parents:
3320
diff
changeset

75 

ec3b55fcb165
New operator "lists" for formalizing sets of lists
paulson
parents:
3320
diff
changeset

76 

13142  77 
text {* 
78 
Function @{text size} is overloaded for all datatypes. Users may 

79 
refer to the list version as @{text length}. *} 

80 

81 
syntax length :: "'a list => nat" 

82 
translations "length" => "size :: _ list => nat" 

13114  83 

13142  84 
typed_print_translation {* 
85 
let 

86 
fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] = 

87 
Syntax.const "length" $ t 

88 
 size_tr' _ _ _ = raise Match; 

89 
in [("size", size_tr')] end 

13114  90 
*} 
3437
bea2faf1641d
Replacing the primrec definition of "length" by a translation to the builtin
paulson
parents:
3401
diff
changeset

91 

5183  92 
primrec 
1898  93 
"hd(x#xs) = x" 
5183  94 
primrec 
8972  95 
"tl([]) = []" 
1898  96 
"tl(x#xs) = xs" 
5183  97 
primrec 
8972  98 
"null([]) = True" 
99 
"null(x#xs) = False" 

100 
primrec 

3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3842
diff
changeset

101 
"last(x#xs) = (if xs=[] then x else last xs)" 
5183  102 
primrec 
8972  103 
"butlast [] = []" 
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3842
diff
changeset

104 
"butlast(x#xs) = (if xs=[] then [] else x#butlast xs)" 
5183  105 
primrec 
8972  106 
"x mem [] = False" 
5518  107 
"x mem (y#ys) = (if y=x then True else x mem ys)" 
108 
primrec 

3465  109 
"set [] = {}" 
110 
"set (x#xs) = insert x (set xs)" 

5183  111 
primrec 
13114  112 
list_all_Nil: "list_all P [] = True" 
13142  113 
list_all_Cons: "list_all P (x#xs) = (P(x) \<and> list_all P xs)" 
5518  114 
primrec 
8972  115 
"map f [] = []" 
1898  116 
"map f (x#xs) = f(x)#map f xs" 
5183  117 
primrec 
13114  118 
append_Nil: "[] @ys = ys" 
119 
append_Cons: "(x#xs)@ys = x#(xs@ys)" 

5183  120 
primrec 
8972  121 
"rev([]) = []" 
1898  122 
"rev(x#xs) = rev(xs) @ [x]" 
5183  123 
primrec 
8972  124 
"filter P [] = []" 
1898  125 
"filter P (x#xs) = (if P x then x#filter P xs else filter P xs)" 
5183  126 
primrec 
13114  127 
foldl_Nil: "foldl f a [] = a" 
128 
foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs" 

5183  129 
primrec 
8972  130 
"foldr f [] a = a" 
8000  131 
"foldr f (x#xs) a = f x (foldr f xs a)" 
132 
primrec 

8972  133 
"concat([]) = []" 
2608  134 
"concat(x#xs) = x @ concat(xs)" 
5183  135 
primrec 
13114  136 
drop_Nil: "drop n [] = []" 
13142  137 
drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs  Suc(m) => drop m xs)" 
138 
 {* Warning: simpset does not contain this definition *} 

139 
 {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} 

5183  140 
primrec 
13114  141 
take_Nil: "take n [] = []" 
13142  142 
take_Cons: "take n (x#xs) = (case n of 0 => []  Suc(m) => x # take m xs)" 
143 
 {* Warning: simpset does not contain this definition *} 

144 
 {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} 

5183  145 
primrec 
13142  146 
nth_Cons: "(x#xs)!n = (case n of 0 => x  (Suc k) => xs!k)" 
147 
 {* Warning: simpset does not contain this definition *} 

148 
 {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} 

149 
primrec 

150 
"[][i:=v] = []" 

151 
"(x#xs)[i:=v] = 

152 
(case i of 0 => v # xs 

153 
 Suc j => x # xs[j:=v])" 

5183  154 
primrec 
8972  155 
"takeWhile P [] = []" 
2608  156 
"takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])" 
5183  157 
primrec 
8972  158 
"dropWhile P [] = []" 
3584
8f9ee0f79d9a
Corected bug in def of dropWhile (also present in Haskell lib!)
nipkow
parents:
3507
diff
changeset

159 
"dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)" 
5183  160 
primrec 
4132  161 
"zip xs [] = []" 
13142  162 
zip_Cons: "zip xs (y#ys) = (case xs of [] => []  z#zs => (z,y)#zip zs ys)" 
163 
 {* Warning: simpset does not contain this definition *} 

164 
 {* but separate theorems for @{text "xs = []"} and @{text "xs = z # zs"} *} 

5427  165 
primrec 
13114  166 
upt_0: "[i..0(] = []" 
167 
upt_Suc: "[i..(Suc j)(] = (if i <= j then [i..j(] @ [j] else [])" 

5183  168 
primrec 
12887  169 
"distinct [] = True" 
13142  170 
"distinct (x#xs) = (x ~: set xs \<and> distinct xs)" 
5183  171 
primrec 
4605  172 
"remdups [] = []" 
173 
"remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)" 

5183  174 
primrec 
13114  175 
replicate_0: "replicate 0 x = []" 
176 
replicate_Suc: "replicate (Suc n) x = x # replicate n x" 

8115  177 
defs 
13114  178 
list_all2_def: 
13142  179 
"list_all2 P xs ys == length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)" 
8115  180 

3196  181 

13142  182 
subsection {* Lexicographic orderings on lists *} 
5281  183 

184 
consts 

13142  185 
lexn :: "('a * 'a)set => nat => ('a list * 'a list)set" 
5281  186 
primrec 
13142  187 
"lexn r 0 = {}" 
188 
"lexn r (Suc n) = 

189 
(prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int 

190 
{(xs,ys). length xs = Suc n \<and> length ys = Suc n}" 

5281  191 

192 
constdefs 

13142  193 
lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" 
194 
"lex r == \<Union>n. lexn r n" 

5281  195 

13142  196 
lexico :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" 
197 
"lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))" 

9336  198 

13142  199 
sublist :: "'a list => nat set => 'a list" 
200 
"sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..size xs(]))" 

5281  201 

13114  202 

13142  203 
lemma not_Cons_self [simp]: "xs \<noteq> x # xs" 
204 
by (induct xs) auto 

13114  205 

13142  206 
lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric] 
13114  207 

13142  208 
lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)" 
209 
by (induct xs) auto 

13114  210 

13142  211 
lemma length_induct: 
212 
"(!!xs. \<forall>ys. length ys < length xs > P ys ==> P xs) ==> P xs" 

213 
by (rule measure_induct [of length]) rules 

13114  214 

215 

13142  216 
subsection {* @{text lists}: the listforming operator over sets *} 
13114  217 

13142  218 
consts lists :: "'a set => 'a list set" 
219 
inductive "lists A" 

220 
intros 

221 
Nil [intro!]: "[]: lists A" 

222 
Cons [intro!]: "[ a: A; l: lists A ] ==> a#l : lists A" 

13114  223 

13142  224 
inductive_cases listsE [elim!]: "x#l : lists A" 
13114  225 

13142  226 
lemma lists_mono: "A \<subseteq> B ==> lists A \<subseteq> lists B" 
227 
by (unfold lists.defs) (blast intro!: lfp_mono) 

13114  228 

13142  229 
lemma lists_IntI [rule_format]: 
230 
"l: lists A ==> l: lists B > l: lists (A Int B)" 

231 
apply (erule lists.induct) 

232 
apply blast+ 

233 
done 

234 

235 
lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B" 

236 
apply (rule mono_Int [THEN equalityI]) 

237 
apply (simp add: mono_def lists_mono) 

238 
apply (blast intro!: lists_IntI) 

239 
done 

13114  240 

13142  241 
lemma append_in_lists_conv [iff]: 
242 
"(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)" 

243 
by (induct xs) auto 

244 

245 

246 
subsection {* @{text length} *} 

13114  247 

13142  248 
text {* 
249 
Needs to come before @{text "@"} because of theorem @{text 

250 
append_eq_append_conv}. 

251 
*} 

13114  252 

13142  253 
lemma length_append [simp]: "length (xs @ ys) = length xs + length ys" 
254 
by (induct xs) auto 

13114  255 

13142  256 
lemma length_map [simp]: "length (map f xs) = length xs" 
257 
by (induct xs) auto 

13114  258 

13142  259 
lemma length_rev [simp]: "length (rev xs) = length xs" 
260 
by (induct xs) auto 

13114  261 

13142  262 
lemma length_tl [simp]: "length (tl xs) = length xs  1" 
263 
by (cases xs) auto 

13114  264 

13142  265 
lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])" 
266 
by (induct xs) auto 

13114  267 

13142  268 
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])" 
269 
by (induct xs) auto 

13114  270 

271 
lemma length_Suc_conv: 

13142  272 
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)" 
273 
by (induct xs) auto 

274 

13114  275 

13142  276 
subsection {* @{text "@"}  append *} 
13114  277 

13142  278 
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)" 
279 
by (induct xs) auto 

13114  280 

13142  281 
lemma append_Nil2 [simp]: "xs @ [] = xs" 
282 
by (induct xs) auto 

3507  283 

13142  284 
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])" 
285 
by (induct xs) auto 

13114  286 

13142  287 
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])" 
288 
by (induct xs) auto 

13114  289 

13142  290 
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])" 
291 
by (induct xs) auto 

13114  292 

13142  293 
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])" 
294 
by (induct xs) auto 

13114  295 

13142  296 
lemma append_eq_append_conv [rule_format, simp]: 
297 
"\<forall>ys. length xs = length ys \<or> length us = length vs 

298 
> (xs@us = ys@vs) = (xs=ys \<and> us=vs)" 

299 
apply (induct_tac xs) 

300 
apply(rule allI) 

301 
apply (case_tac ys) 

302 
apply simp 

303 
apply force 

304 
apply (rule allI) 

305 
apply (case_tac ys) 

306 
apply force 

13114  307 
apply simp 
13142  308 
done 
309 

310 
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)" 

311 
by simp 

312 

313 
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)" 

314 
by simp 

13114  315 

13142  316 
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)" 
317 
by simp 

13114  318 

13142  319 
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])" 
320 
using append_same_eq [of _ _ "[]"] by auto 

3507  321 

13142  322 
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])" 
323 
using append_same_eq [of "[]"] by auto 

13114  324 

13142  325 
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs" 
326 
by (induct xs) auto 

13114  327 

13142  328 
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)" 
329 
by (induct xs) auto 

13114  330 

13142  331 
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs" 
332 
by (simp add: hd_append split: list.split) 

13114  333 

13142  334 
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys  z#zs => zs @ ys)" 
335 
by (simp split: list.split) 

13114  336 

13142  337 
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys" 
338 
by (simp add: tl_append split: list.split) 

13114  339 

340 

13142  341 
text {* Trivial rules for solving @{text "@"}equations automatically. *} 
13114  342 

343 
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys" 

13142  344 
by simp 
13114  345 

13142  346 
lemma Cons_eq_appendI: 
347 
"[ x # xs1 = ys; xs = xs1 @ zs ] ==> x # xs = ys @ zs" 

348 
by (drule sym) simp 

13114  349 

13142  350 
lemma append_eq_appendI: 
351 
"[ xs @ xs1 = zs; ys = xs1 @ us ] ==> xs @ ys = zs @ us" 

352 
by (drule sym) simp 

13114  353 

354 

13142  355 
text {* 
356 
Simplification procedure for all list equalities. 

357 
Currently only tries to rearrange @{text "@"} to see if 

358 
 both lists end in a singleton list, 

359 
 or both lists end in the same list. 

360 
*} 

361 

362 
ML_setup {* 

3507  363 
local 
364 

13122  365 
val append_assoc = thm "append_assoc"; 
366 
val append_Nil = thm "append_Nil"; 

367 
val append_Cons = thm "append_Cons"; 

368 
val append1_eq_conv = thm "append1_eq_conv"; 

369 
val append_same_eq = thm "append_same_eq"; 

370 

13114  371 
val list_eq_pattern = 
372 
Thm.read_cterm (Theory.sign_of (the_context ())) ("(xs::'a list) = ys",HOLogic.boolT) 

373 

374 
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) = 

375 
(case xs of Const("List.list.Nil",_) => cons  _ => last xs) 

376 
 last (Const("List.op @",_) $ _ $ ys) = last ys 

377 
 last t = t 

378 

379 
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true 

380 
 list1 _ = false 

381 

382 
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) = 

383 
(case xs of Const("List.list.Nil",_) => xs  _ => cons $ butlast xs) 

384 
 butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys 

385 
 butlast xs = Const("List.list.Nil",fastype_of xs) 

386 

387 
val rearr_tac = 

388 
simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons]) 

389 

390 
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) = 

391 
let 

392 
val lastl = last lhs and lastr = last rhs 

393 
fun rearr conv = 

394 
let val lhs1 = butlast lhs and rhs1 = butlast rhs 

395 
val Type(_,listT::_) = eqT 

396 
val appT = [listT,listT] > listT 

397 
val app = Const("List.op @",appT) 

398 
val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr) 

399 
val ct = cterm_of sg (HOLogic.mk_Trueprop(HOLogic.mk_eq(F,F2))) 

400 
val thm = prove_goalw_cterm [] ct (K [rearr_tac 1]) 

401 
handle ERROR => 

402 
error("The error(s) above occurred while trying to prove " ^ 

403 
string_of_cterm ct) 

404 
in Some((conv RS (thm RS trans)) RS eq_reflection) end 

405 

406 
in if list1 lastl andalso list1 lastr 

407 
then rearr append1_eq_conv 

408 
else 

409 
if lastl aconv lastr 

410 
then rearr append_same_eq 

411 
else None 

412 
end 

413 
in 

414 
val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq 

415 
end; 

416 

417 
Addsimprocs [list_eq_simproc]; 

418 
*} 

419 

420 

13142  421 
subsection {* @{text map} *} 
13114  422 

13142  423 
lemma map_ext: "(!!x. x : set xs > f x = g x) ==> map f xs = map g xs" 
424 
by (induct xs) simp_all 

13114  425 

13142  426 
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)" 
427 
by (rule ext, induct_tac xs) auto 

13114  428 

13142  429 
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys" 
430 
by (induct xs) auto 

13114  431 

13142  432 
lemma map_compose: "map (f o g) xs = map f (map g xs)" 
433 
by (induct xs) (auto simp add: o_def) 

13114  434 

13142  435 
lemma rev_map: "rev (map f xs) = map f (rev xs)" 
436 
by (induct xs) auto 

13114  437 

438 
lemma map_cong: 

13142  439 
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys" 
440 
 {* a congruence rule for @{text map} *} 

441 
by (clarify, induct ys) auto 

13114  442 

13142  443 
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])" 
444 
by (cases xs) auto 

13114  445 

13142  446 
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])" 
447 
by (cases xs) auto 

13114  448 

449 
lemma map_eq_Cons: 

13142  450 
"(map f xs = y # ys) = (\<exists>x xs'. xs = x # xs' \<and> f x = y \<and> map f xs' = ys)" 
451 
by (cases xs) auto 

13114  452 

453 
lemma map_injective: 

13142  454 
"!!xs. map f xs = map f ys ==> (\<forall>x y. f x = f y > x = y) ==> xs = ys" 
455 
by (induct ys) (auto simp add: map_eq_Cons) 

13114  456 

457 
lemma inj_mapI: "inj f ==> inj (map f)" 

13142  458 
by (rules dest: map_injective injD intro: injI) 
13114  459 

460 
lemma inj_mapD: "inj (map f) ==> inj f" 

13142  461 
apply (unfold inj_on_def) 
462 
apply clarify 

463 
apply (erule_tac x = "[x]" in ballE) 

464 
apply (erule_tac x = "[y]" in ballE) 

465 
apply simp 

466 
apply blast 

467 
apply blast 

468 
done 

13114  469 

470 
lemma inj_map: "inj (map f) = inj f" 

13142  471 
by (blast dest: inj_mapD intro: inj_mapI) 
13114  472 

473 

13142  474 
subsection {* @{text rev} *} 
13114  475 

13142  476 
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs" 
477 
by (induct xs) auto 

13114  478 

13142  479 
lemma rev_rev_ident [simp]: "rev (rev xs) = xs" 
480 
by (induct xs) auto 

13114  481 

13142  482 
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])" 
483 
by (induct xs) auto 

13114  484 

13142  485 
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])" 
486 
by (induct xs) auto 

13114  487 

13142  488 
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)" 
489 
apply (induct xs) 

490 
apply force 

491 
apply (case_tac ys) 

492 
apply simp 

493 
apply force 

494 
done 

13114  495 

13142  496 
lemma rev_induct: "[ P []; !!x xs. P xs ==> P (xs @ [x]) ] ==> P xs" 
497 
apply(subst rev_rev_ident[symmetric]) 

498 
apply(rule_tac list = "rev xs" in list.induct, simp_all) 

499 
done 

13114  500 

13142  501 
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}  "compatibility" 
13114  502 

13142  503 
lemma rev_exhaust: "(xs = [] ==> P) ==> (!!ys y. xs = ys @ [y] ==> P) ==> P" 
504 
by (induct xs rule: rev_induct) auto 

13114  505 

506 

13142  507 
subsection {* @{text set} *} 
13114  508 

13142  509 
lemma finite_set [iff]: "finite (set xs)" 
510 
by (induct xs) auto 

13114  511 

13142  512 
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)" 
513 
by (induct xs) auto 

13114  514 

13142  515 
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)" 
516 
by auto 

13114  517 

13142  518 
lemma set_empty [iff]: "(set xs = {}) = (xs = [])" 
519 
by (induct xs) auto 

13114  520 

13142  521 
lemma set_rev [simp]: "set (rev xs) = set xs" 
522 
by (induct xs) auto 

13114  523 

13142  524 
lemma set_map [simp]: "set (map f xs) = f`(set xs)" 
525 
by (induct xs) auto 

13114  526 

13142  527 
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}" 
528 
by (induct xs) auto 

13114  529 

13142  530 
lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> k < j}" 
531 
apply (induct j) 

532 
apply simp_all 

533 
apply(erule ssubst) 

534 
apply auto 

535 
apply arith 

536 
done 

13114  537 

13142  538 
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)" 
539 
apply (induct xs) 

540 
apply simp 

541 
apply simp 

542 
apply (rule iffI) 

543 
apply (blast intro: eq_Nil_appendI Cons_eq_appendI) 

544 
apply (erule exE)+ 

545 
apply (case_tac ys) 

546 
apply auto 

547 
done 

548 

549 
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)" 

550 
 {* eliminate @{text lists} in favour of @{text set} *} 

551 
by (induct xs) auto 

552 

553 
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A" 

554 
by (rule in_lists_conv_set [THEN iffD1]) 

555 

556 
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A" 

557 
by (rule in_lists_conv_set [THEN iffD2]) 

13114  558 

559 

13142  560 
subsection {* @{text mem} *} 
13114  561 

562 
lemma set_mem_eq: "(x mem xs) = (x : set xs)" 

13142  563 
by (induct xs) auto 
13114  564 

565 

13142  566 
subsection {* @{text list_all} *} 
13114  567 

13142  568 
lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)" 
569 
by (induct xs) auto 

13114  570 

13142  571 
lemma list_all_append [simp]: 
572 
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)" 

573 
by (induct xs) auto 

13114  574 

575 

13142  576 
subsection {* @{text filter} *} 
13114  577 

13142  578 
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys" 
579 
by (induct xs) auto 

13114  580 

13142  581 
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs" 
582 
by (induct xs) auto 

13114  583 

13142  584 
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs" 
585 
by (induct xs) auto 

13114  586 

13142  587 
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []" 
588 
by (induct xs) auto 

13114  589 

13142  590 
lemma length_filter [simp]: "length (filter P xs) \<le> length xs" 
591 
by (induct xs) (auto simp add: le_SucI) 

13114  592 

13142  593 
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs" 
594 
by auto 

13114  595 

596 

13142  597 
subsection {* @{text concat} *} 
13114  598 

13142  599 
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys" 
600 
by (induct xs) auto 

13114  601 

13142  602 
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])" 
603 
by (induct xss) auto 

13114  604 

13142  605 
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])" 
606 
by (induct xss) auto 

13114  607 

13142  608 
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)" 
609 
by (induct xs) auto 

13114  610 

13142  611 
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)" 
612 
by (induct xs) auto 

13114  613 

13142  614 
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)" 
615 
by (induct xs) auto 

13114  616 

13142  617 
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))" 
618 
by (induct xs) auto 

13114  619 

620 

13142  621 
subsection {* @{text nth} *} 
13114  622 

13142  623 
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x" 
624 
by auto 

13114  625 

13142  626 
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n" 
627 
by auto 

13114  628 

13142  629 
declare nth.simps [simp del] 
13114  630 

631 
lemma nth_append: 

13142  632 
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n  length xs))" 
633 
apply(induct "xs") 

634 
apply simp 

635 
apply (case_tac n) 

636 
apply auto 

637 
done 

13114  638 

13142  639 
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)" 
640 
apply(induct xs) 

641 
apply simp 

642 
apply (case_tac n) 

643 
apply auto 

644 
done 

13114  645 

13142  646 
lemma set_conv_nth: "set xs = {xs!i  i. i < length xs}" 
647 
apply (induct_tac xs) 

648 
apply simp 

13114  649 
apply simp 
13142  650 
apply safe 
651 
apply (rule_tac x = 0 in exI) 

652 
apply simp 

653 
apply (rule_tac x = "Suc i" in exI) 

654 
apply simp 

655 
apply (case_tac i) 

656 
apply simp 

657 
apply (rename_tac j) 

658 
apply (rule_tac x = j in exI) 

659 
apply simp 

660 
done 

13114  661 

13142  662 
lemma list_ball_nth: "[ n < length xs; !x : set xs. P x ] ==> P(xs!n)" 
663 
by (auto simp add: set_conv_nth) 

13114  664 

13142  665 
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs" 
666 
by (auto simp add: set_conv_nth) 

13114  667 

668 
lemma all_nth_imp_all_set: 

13142  669 
"[ !i < length xs. P(xs!i); x : set xs ] ==> P x" 
670 
by (auto simp add: set_conv_nth) 

13114  671 

672 
lemma all_set_conv_all_nth: 

13142  673 
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs > P (xs ! i))" 
674 
by (auto simp add: set_conv_nth) 

13114  675 

676 

13142  677 
subsection {* @{text list_update} *} 
13114  678 

13142  679 
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs" 
680 
by (induct xs) (auto split: nat.split) 

13114  681 

682 
lemma nth_list_update: 

13142  683 
"!!i j. i < length xs ==> (xs[i:=x])!j = (if i = j then x else xs!j)" 
684 
by (induct xs) (auto simp add: nth_Cons split: nat.split) 

13114  685 

13142  686 
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x" 
687 
by (simp add: nth_list_update) 

13114  688 

13142  689 
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j" 
690 
by (induct xs) (auto simp add: nth_Cons split: nat.split) 

13114  691 

13142  692 
lemma list_update_overwrite [simp]: 
693 
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]" 

694 
by (induct xs) (auto split: nat.split) 

13114  695 

696 
lemma list_update_same_conv: 

13142  697 
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)" 
698 
by (induct xs) (auto split: nat.split) 

13114  699 

700 
lemma update_zip: 

13142  701 
"!!i xy xs. length xs = length ys ==> 
13114  702 
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])" 
13142  703 
by (induct ys) (auto, case_tac xs, auto split: nat.split) 
13114  704 

705 
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)" 

13142  706 
by (induct xs) (auto split: nat.split) 
13114  707 

708 
lemma set_update_subsetI: "[ set xs <= A; x:A ] ==> set(xs[i := x]) <= A" 

13142  709 
by (blast dest!: set_update_subset_insert [THEN subsetD]) 
13114  710 

711 

13142  712 
subsection {* @{text last} and @{text butlast} *} 
13114  713 

13142  714 
lemma last_snoc [simp]: "last (xs @ [x]) = x" 
715 
by (induct xs) auto 

13114  716 

13142  717 
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs" 
718 
by (induct xs) auto 

13114  719 

13142  720 
lemma length_butlast [simp]: "length (butlast xs) = length xs  1" 
721 
by (induct xs rule: rev_induct) auto 

13114  722 

723 
lemma butlast_append: 

13142  724 
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)" 
725 
by (induct xs) auto 

13114  726 

13142  727 
lemma append_butlast_last_id [simp]: 
728 
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs" 

729 
by (induct xs) auto 

13114  730 

13142  731 
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs" 
732 
by (induct xs) (auto split: split_if_asm) 

13114  733 

734 
lemma in_set_butlast_appendI: 

13142  735 
"x : set (butlast xs)  x : set (butlast ys) ==> x : set (butlast (xs @ ys))" 
736 
by (auto dest: in_set_butlastD simp add: butlast_append) 

13114  737 

13142  738 

739 
subsection {* @{text take} and @{text drop} *} 

13114  740 

13142  741 
lemma take_0 [simp]: "take 0 xs = []" 
742 
by (induct xs) auto 

13114  743 

13142  744 
lemma drop_0 [simp]: "drop 0 xs = xs" 
745 
by (induct xs) auto 

13114  746 

13142  747 
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs" 
748 
by simp 

13114  749 

13142  750 
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs" 
751 
by simp 

13114  752 

13142  753 
declare take_Cons [simp del] and drop_Cons [simp del] 
13114  754 

13142  755 
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n" 
756 
by (induct n) (auto, case_tac xs, auto) 

13114  757 

13142  758 
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs  n)" 
759 
by (induct n) (auto, case_tac xs, auto) 

13114  760 

13142  761 
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs" 
762 
by (induct n) (auto, case_tac xs, auto) 

13114  763 

13142  764 
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []" 
765 
by (induct n) (auto, case_tac xs, auto) 

13114  766 

13142  767 
lemma take_append [simp]: 
768 
"!!xs. take n (xs @ ys) = (take n xs @ take (n  length xs) ys)" 

769 
by (induct n) (auto, case_tac xs, auto) 

13114  770 

13142  771 
lemma drop_append [simp]: 
772 
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n  length xs) ys" 

773 
by (induct n) (auto, case_tac xs, auto) 

13114  774 

13142  775 
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs" 
776 
apply (induct m) 

777 
apply auto 

778 
apply (case_tac xs) 

779 
apply auto 

780 
apply (case_tac na) 

781 
apply auto 

782 
done 

13114  783 

13142  784 
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs" 
785 
apply (induct m) 

786 
apply auto 

787 
apply (case_tac xs) 

788 
apply auto 

789 
done 

13114  790 

791 
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)" 

13142  792 
apply (induct m) 
793 
apply auto 

794 
apply (case_tac xs) 

795 
apply auto 

796 
done 

13114  797 

13142  798 
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs" 
799 
apply (induct n) 

800 
apply auto 

801 
apply (case_tac xs) 

802 
apply auto 

803 
done 

13114  804 

805 
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)" 

13142  806 
apply (induct n) 
807 
apply auto 

808 
apply (case_tac xs) 

809 
apply auto 

810 
done 

13114  811 

13142  812 
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)" 
813 
apply (induct n) 

814 
apply auto 

815 
apply (case_tac xs) 

816 
apply auto 

817 
done 

13114  818 

819 
lemma rev_take: "!!i. rev (take i xs) = drop (length xs  i) (rev xs)" 

13142  820 
apply (induct xs) 
821 
apply auto 

822 
apply (case_tac i) 

823 
apply auto 

824 
done 

13114  825 

826 
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs  i) (rev xs)" 

13142  827 
apply (induct xs) 
828 
apply auto 

829 
apply (case_tac i) 

830 
apply auto 

831 
done 

13114  832 

13142  833 
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i" 
834 
apply (induct xs) 

835 
apply auto 

836 
apply (case_tac n) 

837 
apply(blast ) 

838 
apply (case_tac i) 

839 
apply auto 

840 
done 

13114  841 

13142  842 
lemma nth_drop [simp]: 
843 
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)" 

844 
apply (induct n) 

845 
apply auto 

846 
apply (case_tac xs) 

847 
apply auto 

848 
done 

3507  849 

13114  850 
lemma append_eq_conv_conj: 
13142  851 
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)" 
852 
apply(induct xs) 

853 
apply simp 

854 
apply clarsimp 

855 
apply (case_tac zs) 

856 
apply auto 

857 
done 

858 

13114  859 

13142  860 
subsection {* @{text takeWhile} and @{text dropWhile} *} 
13114  861 

13142  862 
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs" 
863 
by (induct xs) auto 

13114  864 

13142  865 
lemma takeWhile_append1 [simp]: 
866 
"[ x:set xs; ~P(x) ] ==> takeWhile P (xs @ ys) = takeWhile P xs" 

867 
by (induct xs) auto 

13114  868 

13142  869 
lemma takeWhile_append2 [simp]: 
870 
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys" 

871 
by (induct xs) auto 

13114  872 

13142  873 
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs" 
874 
by (induct xs) auto 

13114  875 

13142  876 
lemma dropWhile_append1 [simp]: 
877 
"[ x : set xs; ~P(x) ] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys" 

878 
by (induct xs) auto 

13114  879 

13142  880 
lemma dropWhile_append2 [simp]: 
881 
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys" 

882 
by (induct xs) auto 

13114  883 

13142  884 
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x" 
885 
by (induct xs) (auto split: split_if_asm) 

13114  886 

887 

13142  888 
subsection {* @{text zip} *} 
13114  889 

13142  890 
lemma zip_Nil [simp]: "zip [] ys = []" 
891 
by (induct ys) auto 

13114  892 

13142  893 
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys" 
894 
by simp 

13114  895 

13142  896 
declare zip_Cons [simp del] 
13114  897 

13142  898 
lemma length_zip [simp]: 
899 
"!!xs. length (zip xs ys) = min (length xs) (length ys)" 

900 
apply(induct ys) 

901 
apply simp 

902 
apply (case_tac xs) 

903 
apply auto 

904 
done 

13114  905 

906 
lemma zip_append1: 

13142  907 
"!!xs. zip (xs @ ys) zs = 
908 
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)" 

909 
apply (induct zs) 

910 
apply simp 

911 
apply (case_tac xs) 

912 
apply simp_all 

913 
done 

13114  914 

915 
lemma zip_append2: 

13142  916 
"!!ys. zip xs (ys @ zs) = 
917 
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs" 

918 
apply (induct xs) 

919 
apply simp 

920 
apply (case_tac ys) 

921 
apply simp_all 

922 
done 

13114  923 

13142  924 
lemma zip_append [simp]: 
925 
"[ length xs = length us; length ys = length vs ] ==> 

926 
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs" 

927 
by (simp add: zip_append1) 

13114  928 

929 
lemma zip_rev: 

13142  930 
"!!xs. length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)" 
931 
apply(induct ys) 

932 
apply simp 

933 
apply (case_tac xs) 

934 
apply simp_all 

935 
done 

13114  936 

13142  937 
lemma nth_zip [simp]: 
938 
"!!i xs. [ i < length xs; i < length ys ] ==> (zip xs ys)!i = (xs!i, ys!i)" 

939 
apply (induct ys) 

940 
apply simp 

941 
apply (case_tac xs) 

942 
apply (simp_all add: nth.simps split: nat.split) 

943 
done 

13114  944 

945 
lemma set_zip: 

13142  946 
"set (zip xs ys) = {(xs!i, ys!i)  i. i < min (length xs) (length ys)}" 
947 
by (simp add: set_conv_nth cong: rev_conj_cong) 

13114  948 

949 
lemma zip_update: 

13142  950 
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]" 
951 
by (rule sym, simp add: update_zip) 

13114  952 

13142  953 
lemma zip_replicate [simp]: 
954 
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)" 

955 
apply (induct i) 

956 
apply auto 

957 
apply (case_tac j) 

958 
apply auto 

959 
done 

13114  960 

13142  961 

962 
subsection {* @{text list_all2} *} 

13114  963 

964 
lemma list_all2_lengthD: "list_all2 P xs ys ==> length xs = length ys" 

13142  965 
by (simp add: list_all2_def) 
13114  966 

13142  967 
lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])" 
968 
by (simp add: list_all2_def) 

13114  969 

13142  970 
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])" 
971 
by (simp add: list_all2_def) 

13114  972 

13142  973 
lemma list_all2_Cons [iff]: 
974 
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)" 

975 
by (auto simp add: list_all2_def) 

13114  976 

977 
lemma list_all2_Cons1: 

13142  978 
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)" 
979 
by (cases ys) auto 

13114  980 

981 
lemma list_all2_Cons2: 

13142  982 
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)" 
983 
by (cases xs) auto 

13114  984 

13142  985 
lemma list_all2_rev [iff]: 
986 
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys" 

987 
by (simp add: list_all2_def zip_rev cong: conj_cong) 

13114  988 

989 
lemma list_all2_append1: 

13142  990 
"list_all2 P (xs @ ys) zs = 
991 
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and> 

992 
list_all2 P xs us \<and> list_all2 P ys vs)" 

993 
apply (simp add: list_all2_def zip_append1) 

994 
apply (rule iffI) 

995 
apply (rule_tac x = "take (length xs) zs" in exI) 

996 
apply (rule_tac x = "drop (length xs) zs" in exI) 

997 
apply (force split: nat_diff_split simp add: min_def) 

998 
apply clarify 

999 
apply (simp add: ball_Un) 

1000 
done 

13114  1001 

1002 
lemma list_all2_append2: 

13142  1003 
"list_all2 P xs (ys @ zs) = 
1004 
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and> 

1005 
list_all2 P us ys \<and> list_all2 P vs zs)" 

1006 
apply (simp add: list_all2_def zip_append2) 

1007 
apply (rule iffI) 

1008 
apply (rule_tac x = "take (length ys) xs" in exI) 

1009 
apply (rule_tac x = "drop (length ys) xs" in exI) 

1010 
apply (force split: nat_diff_split simp add: min_def) 

1011 
apply clarify 

1012 
apply (simp add: ball_Un) 

1013 
done 

13114  1014 

1015 
lemma list_all2_conv_all_nth: 

1016 
"list_all2 P xs ys = 

13142  1017 
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))" 
1018 
by (force simp add: list_all2_def set_zip) 

13114  1019 

1020 
lemma list_all2_trans[rule_format]: 

13142  1021 
"\<forall>a b c. P1 a b > P2 b c > P3 a c ==> 
1022 
\<forall>bs cs. list_all2 P1 as bs > list_all2 P2 bs cs > list_all2 P3 as cs" 

1023 
apply(induct_tac as) 

1024 
apply simp 

1025 
apply(rule allI) 

1026 
apply(induct_tac bs) 

1027 
apply simp 

1028 
apply(rule allI) 

1029 
apply(induct_tac cs) 

1030 
apply auto 

1031 
done 

1032 

1033 

1034 
subsection {* @{text foldl} *} 

1035 

1036 
lemma foldl_append [simp]: 

1037 
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys" 

1038 
by (induct xs) auto 

1039 

1040 
text {* 

1041 
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more 

1042 
difficult to use because it requires an additional transitivity step. 

1043 
*} 

1044 

1045 
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns" 

1046 
by (induct ns) auto 

1047 

1048 
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns" 

1049 
by (force intro: start_le_sum simp add: in_set_conv_decomp) 

1050 

1051 
lemma sum_eq_0_conv [iff]: 

1052 
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))" 

1053 
by (induct ns) auto 

13114  1054 

1055 

13142  1056 
subsection {* @{text upto} *} 
13114  1057 

13142  1058 
lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])" 
1059 
 {* Does not terminate! *} 

1060 
by (induct j) auto 

1061 

1062 
lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []" 

1063 
by (subst upt_rec) simp 

13114  1064 

13142  1065 
lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]" 
1066 
 {* Only needed if @{text upt_Suc} is deleted from the simpset. *} 

1067 
by simp 

13114  1068 

13142  1069 
lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]" 
1070 
apply(rule trans) 

1071 
apply(subst upt_rec) 

1072 
prefer 2 apply(rule refl) 

1073 
apply simp 

1074 
done 

13114  1075 

13142  1076 
lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]" 
1077 
 {* LOOPS as a simprule, since @{text "j <= j"}. *} 

1078 
by (induct k) auto 

13114  1079 

13142  1080 
lemma length_upt [simp]: "length [i..j(] = j  i" 
1081 
by (induct j) (auto simp add: Suc_diff_le) 

13114  1082 

13142  1083 
lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k" 
1084 
apply (induct j) 

1085 
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split) 

1086 
done 

13114  1087 

13142  1088 
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]" 
1089 
apply (induct m) 

1090 
apply simp 

1091 
apply (subst upt_rec) 

1092 
apply (rule sym) 

1093 
apply (subst upt_rec) 

1094 
apply (simp del: upt.simps) 

1095 
done 

3507  1096 

13114  1097 
lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]" 
13142  1098 
by (induct n) auto 
13114  1099 

1100 
lemma nth_map_upt: "!!i. i < nm ==> (map f [m..n(]) ! i = f(m+i)" 

13142  1101 
apply (induct n m rule: diff_induct) 
1102 
prefer 3 apply (subst map_Suc_upt[symmetric]) 

1103 
apply (auto simp add: less_diff_conv nth_upt) 

1104 
done 

13114  1105 

13142  1106 
lemma nth_take_lemma [rule_format]: 
1107 
"ALL xs ys. k <= length xs > k <= length ys 

1108 
> (ALL i. i < k > xs!i = ys!i) 

1109 
> take k xs = take k ys" 

1110 
apply (induct k) 

1111 
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib) 

1112 
apply clarify 

1113 
txt {* Both lists must be nonempty *} 

1114 
apply (case_tac xs) 

1115 
apply simp 

1116 
apply (case_tac ys) 

1117 
apply clarify 

1118 
apply (simp (no_asm_use)) 

1119 
apply clarify 

1120 
txt {* prenexing's needed, not miniscoping *} 

1121 
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps) 

1122 
apply blast 

1123 
done 

13114  1124 

1125 
lemma nth_equalityI: 

1126 
"[ length xs = length ys; ALL i < length xs. xs!i = ys!i ] ==> xs = ys" 

13142  1127 
apply (frule nth_take_lemma [OF le_refl eq_imp_le]) 
1128 
apply (simp_all add: take_all) 

1129 
done 

1130 

1131 
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys" 

1132 
 {* The famous takelemma. *} 

1133 
apply (drule_tac x = "max (length xs) (length ys)" in spec) 

1134 
apply (simp add: le_max_iff_disj take_all) 

1135 
done 

1136 

1137 

1138 
subsection {* @{text "distinct"} and @{text remdups} *} 

1139 

1140 
lemma distinct_append [simp]: 

1141 
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})" 

1142 
by (induct xs) auto 

1143 

1144 
lemma set_remdups [simp]: "set (remdups xs) = set xs" 

1145 
by (induct xs) (auto simp add: insert_absorb) 

1146 

1147 
lemma distinct_remdups [iff]: "distinct (remdups xs)" 

1148 
by (induct xs) auto 

1149 

1150 
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)" 

1151 
by (induct xs) auto 

13114  1152 

13142  1153 
text {* 
1154 
It is best to avoid this indexed version of distinct, but sometimes 

1155 
it is useful. *} 

1156 
lemma distinct_conv_nth: 

1157 
"distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j > xs!i \<noteq> xs!j)" 

1158 
apply (induct_tac xs) 

1159 
apply simp 

1160 
apply simp 

1161 
apply (rule iffI) 

1162 
apply clarsimp 

1163 
apply (case_tac i) 

1164 
apply (case_tac j) 

1165 
apply simp 

1166 
apply (simp add: set_conv_nth) 

1167 
apply (case_tac j) 

1168 
apply (clarsimp simp add: set_conv_nth) 

1169 
apply simp 

1170 
apply (rule conjI) 

1171 
apply (clarsimp simp add: set_conv_nth) 

1172 
apply (erule_tac x = 0 in allE) 

1173 
apply (erule_tac x = "Suc i" in allE) 

1174 
apply simp 

1175 
apply clarsimp 

1176 
apply (erule_tac x = "Suc i" in allE) 

1177 
apply (erule_tac x = "Suc j" in allE) 

1178 
apply simp 

1179 
done 

13114  1180 

1181 

13142  1182 
subsection {* @{text replicate} *} 
13114  1183 

13142  1184 
lemma length_replicate [simp]: "length (replicate n x) = n" 
1185 
by (induct n) auto 

13124  1186 

13142  1187 
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)" 
1188 
by (induct n) auto 

13114  1189 

1190 
lemma replicate_app_Cons_same: 

13142  1191 
"(replicate n x) @ (x # xs) = x # replicate n x @ xs" 
1192 
by (induct n) auto 

13114  1193 

13142  1194 
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x" 
1195 
apply(induct n) 

1196 
apply simp 

1197 
apply (simp add: replicate_app_Cons_same) 

1198 
done 

13114  1199 

13142  1200 
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x" 
1201 
by (induct n) auto 

13114  1202 

13142  1203 
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x" 
1204 
by (induct n) auto 

13114  1205 

13142  1206 
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n  1) x" 
1207 
by (induct n) auto 

13114  1208 

13142  1209 
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x" 
1210 
by (atomize (full), induct n) auto 

13114  1211 

13142  1212 
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x" 
1213 
apply(induct n) 

1214 
apply simp 

1215 
apply (simp add: nth_Cons split: nat.split) 

1216 
done 

13114  1217 

13142  1218 
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}" 
1219 
by (induct n) auto 

13114  1220 

13142  1221 
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}" 
1222 
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc) 

13114  1223 

13142  1224 
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})" 
1225 
by auto 

13114  1226 

13142  1227 
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y" 
1228 
by (simp add: set_replicate_conv_if split: split_if_asm) 

13114  1229 

1230 

13142  1231 
subsection {* Lexcicographic orderings on lists *} 
3507  1232 

13142  1233 
lemma wf_lexn: "wf r ==> wf (lexn r n)" 
1234 
apply (induct_tac n) 

1235 
apply simp 

1236 
apply simp 

1237 
apply(rule wf_subset) 

1238 
prefer 2 apply (rule Int_lower1) 

1239 
apply(rule wf_prod_fun_image) 

1240 
prefer 2 apply (rule injI) 

1241 
apply auto 

1242 
done 

13114  1243 

1244 
lemma lexn_length: 

13142  1245 
"!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n" 
1246 
by (induct n) auto 

13114  1247 

13142  1248 
lemma wf_lex [intro!]: "wf r ==> wf (lex r)" 
1249 
apply (unfold lex_def) 

1250 
apply (rule wf_UN) 

1251 
apply (blast intro: wf_lexn) 

1252 
apply clarify 

1253 
apply (rename_tac m n) 

1254 
apply (subgoal_tac "m \<noteq> n") 

1255 
prefer 2 apply blast 

1256 
apply (blast dest: lexn_length not_sym) 

1257 
done 

13114  1258 

1259 
lemma lexn_conv: 

13142  1260 
"lexn r n = 
1261 
{(xs,ys). length xs = n \<and> length ys = n \<and> 

1262 
(\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}" 

1263 
apply (induct_tac n) 

1264 
apply simp 

1265 
apply blast 

1266 
apply (simp add: image_Collect lex_prod_def) 

1267 
apply auto 

1268 
apply blast 

1269 
apply (rename_tac a xys x xs' y ys') 

1270 
apply (rule_tac x = "a # xys" in exI) 

1271 
apply simp 

1272 
apply (case_tac xys) 

1273 
apply simp_all 

13114  1274 
apply blast 
13142  1275 
done 
13114  1276 

1277 
lemma lex_conv: 

13142  1278 
"lex r = 
1279 
{(xs,ys). length xs = length ys \<and> 

1280 
(\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}" 

1281 
by (force simp add: lex_def lexn_conv) 

13114  1282 

13142  1283 
lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)" 
1284 
by (unfold lexico_def) blast 

13114  1285 

1286 
lemma lexico_conv: 

13142  1287 
"lexico r = {(xs,ys). length xs < length ys  
1288 
length xs = length ys \<and> (xs, ys) : lex r}" 

1289 
by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def) 

13114  1290 

13142  1291 
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r" 
1292 
by (simp add: lex_conv) 

13114  1293 

13142  1294 
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r" 
1295 
by (simp add:lex_conv) 

13114  1296 

13142  1297 
lemma Cons_in_lex [iff]: 
1298 
"((x # xs, y # ys) : lex r) = 

1299 
((x, y) : r \<and> length xs = length ys  x = y \<and> (xs, ys) : lex r)" 

1300 
apply (simp add: lex_conv) 

1301 
apply (rule iffI) 

1302 
prefer 2 apply (blast intro: Cons_eq_appendI) 

1303 
apply clarify 

1304 
apply (case_tac xys) 

1305 
apply simp 

1306 
apply simp 

1307 
apply blast 

1308 
done 

13114  1309 

1310 

13142  1311 
subsection {* @{text sublist}  a generalization of @{text nth} to sets *} 
13114  1312 

13142  1313 
lemma sublist_empty [simp]: "sublist xs {} = []" 
1314 
by (auto simp add: sublist_def) 

13114  1315 

13142  1316 
lemma sublist_nil [simp]: "sublist [] A = []" 
1317 
by (auto simp add: sublist_def) 

13114  1318 

1319 
lemma sublist_shift_lemma: 

13142  1320 
"map fst [p:zip xs [i..i + length xs(] . snd p : A] = 
1321 
map fst [p:zip xs [0..length xs(] . snd p + i : A]" 

1322 
by (induct xs rule: rev_induct) (simp_all add: add_commute) 

13114  1323 

1324 
lemma sublist_append: 

13142  1325 
"sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}" 
1326 
apply (unfold sublist_def) 

1327 
apply (induct l' rule: rev_induct) 

1328 
apply simp 

1329 
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma) 

1330 
apply (simp add: add_commute) 

1331 
done 

13114  1332 

1333 
lemma sublist_Cons: 

13142  1334 
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}" 
1335 
apply (induct l rule: rev_induct) 

1336 
apply (simp add: sublist_def) 

1337 
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append) 

1338 
done 

13114  1339 

13142  1340 
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])" 
1341 
by (simp add: sublist_Cons) 

13114  1342 

13142  1343 
lemma sublist_upt_eq_take [simp]: "sublist l {..n(} = take n l" 
1344 
apply (induct l rule: rev_induct) 

1345 
apply simp 

1346 
apply (simp split: nat_diff_split add: sublist_append) 

1347 
done 

13114  1348 

1349 

13142  1350 
lemma take_Cons': 
1351 
"take n (x # xs) = (if n = 0 then [] else x # take (n  1) xs)" 

1352 
by (cases n) simp_all 

13114  1353 

13142  1354 
lemma drop_Cons': 
1355 
"drop n (x # xs) = (if n = 0 then x # xs else drop (n  1) xs)" 

1356 
by (cases n) simp_all 

13114  1357 

13142  1358 
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n  1))" 
1359 
by (cases n) simp_all 

1360 

1361 
lemmas [of "number_of v", standard, simp] = 

1362 
take_Cons' drop_Cons' nth_Cons' 

3507  1363 

13122  1364 
end 