src/HOL/List.thy
 author paulson Thu Feb 19 10:40:28 2004 +0100 (2004-02-19) changeset 14395 cc96cc06abf9 parent 14388 04f04408b99b child 14402 4201e1916482 permissions -rw-r--r--
new theorem
1 (*  Title:      HOL/List.thy
2     ID:         \$Id\$
3     Author:     Tobias Nipkow
4     License:    GPL (GNU GENERAL PUBLIC LICENSE)
5 *)
7 header {* The datatype of finite lists *}
9 theory List = PreList:
11 datatype 'a list =
12     Nil    ("[]")
13   | Cons 'a  "'a list"    (infixr "#" 65)
15 consts
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   fold_rel :: "('a * 'c * 'a) set => ('a * 'c list * 'a) set"
22   hd:: "'a list => 'a"
23   tl:: "'a list => 'a list"
24   last:: "'a list => 'a"
25   butlast :: "'a list => 'a list"
26   set :: "'a list => 'a set"
27   o2l :: "'a option => 'a list"
28   list_all:: "('a => bool) => ('a list => bool)"
29   list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool"
30   map :: "('a=>'b) => ('a list => 'b list)"
31   mem :: "'a => 'a list => bool"    (infixl 55)
32   nth :: "'a list => nat => 'a"    (infixl "!" 100)
33   list_update :: "'a list => nat => 'a => 'a list"
34   take:: "nat => 'a list => 'a list"
35   drop:: "nat => 'a list => 'a list"
36   takeWhile :: "('a => bool) => 'a list => 'a list"
37   dropWhile :: "('a => bool) => 'a list => 'a list"
38   rev :: "'a list => 'a list"
39   zip :: "'a list => 'b list => ('a * 'b) list"
40   upt :: "nat => nat => nat list" ("(1[_../_'(])")
41   remdups :: "'a list => 'a list"
42   null:: "'a list => bool"
43   "distinct":: "'a list => bool"
44   replicate :: "nat => 'a => 'a list"
45   postfix :: "'a list => 'a list => bool"
47 syntax (xsymbols)
48   postfix :: "'a list => 'a list => bool"             ("(_/ \<sqsupseteq> _)" [51, 51] 50)
50 nonterminals lupdbinds lupdbind
52 syntax
53   -- {* list Enumeration *}
54   "@list" :: "args => 'a list"    ("[(_)]")
56   -- {* Special syntax for filter *}
57   "@filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_:_./ _])")
59   -- {* list update *}
60   "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
61   "" :: "lupdbind => lupdbinds"    ("_")
62   "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
63   "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
65   upto:: "nat => nat => nat list"    ("(1[_../_])")
67 translations
68   "[x, xs]" == "x#[xs]"
69   "[x]" == "x#[]"
70   "[x:xs . P]"== "filter (%x. P) xs"
72   "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"
73   "xs[i:=x]" == "list_update xs i x"
75   "[i..j]" == "[i..(Suc j)(]"
78 syntax (xsymbols)
79   "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
82 text {*
83   Function @{text size} is overloaded for all datatypes.Users may
84   refer to the list version as @{text length}. *}
86 syntax length :: "'a list => nat"
87 translations "length" => "size :: _ list => nat"
89 typed_print_translation {*
90   let
91     fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] =
92           Syntax.const "length" \$ t
93       | size_tr' _ _ _ = raise Match;
94   in [("size", size_tr')] end
95 *}
97 primrec
98 "hd(x#xs) = x"
99 primrec
100 "tl([]) = []"
101 "tl(x#xs) = xs"
102 primrec
103 "null([]) = True"
104 "null(x#xs) = False"
105 primrec
106 "last(x#xs) = (if xs=[] then x else last xs)"
107 primrec
108 "butlast []= []"
109 "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
110 primrec
111 "x mem [] = False"
112 "x mem (y#ys) = (if y=x then True else x mem ys)"
113 primrec
114 "set [] = {}"
115 "set (x#xs) = insert x (set xs)"
116 primrec
117  "o2l  None    = []"
118  "o2l (Some x) = [x]"
119 primrec
120 list_all_Nil:"list_all P [] = True"
121 list_all_Cons: "list_all P (x#xs) = (P(x) \<and> list_all P xs)"
122 primrec
123 "map f [] = []"
124 "map f (x#xs) = f(x)#map f xs"
125 primrec
126 append_Nil:"[]@ys = ys"
127 append_Cons: "(x#xs)@ys = x#(xs@ys)"
128 primrec
129 "rev([]) = []"
130 "rev(x#xs) = rev(xs) @ [x]"
131 primrec
132 "filter P [] = []"
133 "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
134 primrec
135 foldl_Nil:"foldl f a [] = a"
136 foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
137 primrec
138 "foldr f [] a = a"
139 "foldr f (x#xs) a = f x (foldr f xs a)"
140 primrec
141 "concat([]) = []"
142 "concat(x#xs) = x @ concat(xs)"
143 primrec
144 drop_Nil:"drop n [] = []"
145 drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
146 -- {* Warning: simpset does not contain this definition *}
147 -- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
148 primrec
149 take_Nil:"take n [] = []"
150 take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
151 -- {* Warning: simpset does not contain this definition *}
152 -- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
153 primrec
154 nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
155 -- {* Warning: simpset does not contain this definition *}
156 -- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
157 primrec
158 "[][i:=v] = []"
159 "(x#xs)[i:=v] =
160 (case i of 0 => v # xs
161 | Suc j => x # xs[j:=v])"
162 primrec
163 "takeWhile P [] = []"
164 "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
165 primrec
166 "dropWhile P [] = []"
167 "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
168 primrec
169 "zip xs [] = []"
170 zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
171 -- {* Warning: simpset does not contain this definition *}
172 -- {* but separate theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
173 primrec
174 upt_0: "[i..0(] = []"
175 upt_Suc: "[i..(Suc j)(] = (if i <= j then [i..j(] @ [j] else [])"
176 primrec
177 "distinct [] = True"
178 "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
179 primrec
180 "remdups [] = []"
181 "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
182 primrec
183 replicate_0: "replicate 0 x = []"
184 replicate_Suc: "replicate (Suc n) x = x # replicate n x"
185 defs
186  postfix_def: "postfix xs ys == \<exists>zs. xs = zs @ ys"
187 defs
188  list_all2_def:
189  "list_all2 P xs ys == length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)"
192 subsection {* Lexicographic orderings on lists *}
194 consts
195 lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
196 primrec
197 "lexn r 0 = {}"
198 "lexn r (Suc n) =
199 (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
200 {(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
202 constdefs
203 lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
204 "lex r == \<Union>n. lexn r n"
206 lexico :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
207 "lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
209 sublist :: "'a list => nat set => 'a list"
210 "sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..size xs(]))"
213 lemma not_Cons_self [simp]: "xs \<noteq> x # xs"
214 by (induct xs) auto
216 lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
218 lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
219 by (induct xs) auto
221 lemma length_induct:
222 "(!!xs. \<forall>ys. length ys < length xs --> P ys ==> P xs) ==> P xs"
223 by (rule measure_induct [of length]) rules
226 subsection {* @{text lists}: the list-forming operator over sets *}
228 consts lists :: "'a set => 'a list set"
229 inductive "lists A"
230 intros
231 Nil [intro!]: "[]: lists A"
232 Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A"
234 inductive_cases listsE [elim!]: "x#l : lists A"
236 lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B"
237 by (unfold lists.defs) (blast intro!: lfp_mono)
239 lemma lists_IntI:
240   assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l
241   by induct blast+
243 lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B"
244 apply (rule mono_Int [THEN equalityI])
245 apply (simp add: mono_def lists_mono)
246 apply (blast intro!: lists_IntI)
247 done
249 lemma append_in_lists_conv [iff]:
250 "(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"
251 by (induct xs) auto
254 subsection {* @{text length} *}
256 text {*
257 Needs to come before @{text "@"} because of theorem @{text
258 append_eq_append_conv}.
259 *}
261 lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
262 by (induct xs) auto
264 lemma length_map [simp]: "length (map f xs) = length xs"
265 by (induct xs) auto
267 lemma length_rev [simp]: "length (rev xs) = length xs"
268 by (induct xs) auto
270 lemma length_tl [simp]: "length (tl xs) = length xs - 1"
271 by (cases xs) auto
273 lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
274 by (induct xs) auto
276 lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
277 by (induct xs) auto
279 lemma length_Suc_conv:
280 "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
281 by (induct xs) auto
283 lemma Suc_length_conv:
284 "(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
285 apply (induct xs, simp, simp)
286 apply blast
287 done
289 lemma impossible_Cons [rule_format]:
290   "length xs <= length ys --> xs = x # ys = False"
291 apply (induct xs, auto)
292 done
294 lemma list_induct2[consumes 1]: "\<And>ys.
295  \<lbrakk> length xs = length ys;
296    P [] [];
297    \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
298  \<Longrightarrow> P xs ys"
299 apply(induct xs)
300  apply simp
301 apply(case_tac ys)
302  apply simp
303 apply(simp)
304 done
306 subsection {* @{text "@"} -- append *}
308 lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
309 by (induct xs) auto
311 lemma append_Nil2 [simp]: "xs @ [] = xs"
312 by (induct xs) auto
314 lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
315 by (induct xs) auto
317 lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
318 by (induct xs) auto
320 lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
321 by (induct xs) auto
323 lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
324 by (induct xs) auto
326 lemma append_eq_append_conv [simp]:
327  "!!ys. length xs = length ys \<or> length us = length vs
328  ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
329 apply (induct xs)
330  apply (case_tac ys, simp, force)
331 apply (case_tac ys, force, simp)
332 done
334 lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
335 by simp
337 lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
338 by simp
340 lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
341 by simp
343 lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
344 using append_same_eq [of _ _ "[]"] by auto
346 lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
347 using append_same_eq [of "[]"] by auto
349 lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
350 by (induct xs) auto
352 lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
353 by (induct xs) auto
355 lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
356 by (simp add: hd_append split: list.split)
358 lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
359 by (simp split: list.split)
361 lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
362 by (simp add: tl_append split: list.split)
365 lemma Cons_eq_append_conv: "x#xs = ys@zs =
366  (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
367 by(cases ys) auto
370 text {* Trivial rules for solving @{text "@"}-equations automatically. *}
372 lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
373 by simp
375 lemma Cons_eq_appendI:
376 "[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
377 by (drule sym) simp
379 lemma append_eq_appendI:
380 "[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
381 by (drule sym) simp
384 text {*
385 Simplification procedure for all list equalities.
386 Currently only tries to rearrange @{text "@"} to see if
387 - both lists end in a singleton list,
388 - or both lists end in the same list.
389 *}
391 ML_setup {*
392 local
394 val append_assoc = thm "append_assoc";
395 val append_Nil = thm "append_Nil";
396 val append_Cons = thm "append_Cons";
397 val append1_eq_conv = thm "append1_eq_conv";
398 val append_same_eq = thm "append_same_eq";
400 fun last (cons as Const("List.list.Cons",_) \$ _ \$ xs) =
401   (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
402   | last (Const("List.op @",_) \$ _ \$ ys) = last ys
403   | last t = t;
405 fun list1 (Const("List.list.Cons",_) \$ _ \$ Const("List.list.Nil",_)) = true
406   | list1 _ = false;
408 fun butlast ((cons as Const("List.list.Cons",_) \$ x) \$ xs) =
409   (case xs of Const("List.list.Nil",_) => xs | _ => cons \$ butlast xs)
410   | butlast ((app as Const("List.op @",_) \$ xs) \$ ys) = app \$ butlast ys
411   | butlast xs = Const("List.list.Nil",fastype_of xs);
413 val rearr_tac =
414   simp_tac (HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]);
416 fun list_eq sg _ (F as (eq as Const(_,eqT)) \$ lhs \$ rhs) =
417   let
418     val lastl = last lhs and lastr = last rhs;
419     fun rearr conv =
420       let
421         val lhs1 = butlast lhs and rhs1 = butlast rhs;
422         val Type(_,listT::_) = eqT
423         val appT = [listT,listT] ---> listT
424         val app = Const("List.op @",appT)
425         val F2 = eq \$ (app\$lhs1\$lastl) \$ (app\$rhs1\$lastr)
426         val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
427         val thm = Tactic.prove sg [] [] eq (K (rearr_tac 1));
428       in Some ((conv RS (thm RS trans)) RS eq_reflection) end;
430   in
431     if list1 lastl andalso list1 lastr then rearr append1_eq_conv
432     else if lastl aconv lastr then rearr append_same_eq
433     else None
434   end;
436 in
438 val list_eq_simproc =
439   Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq;
441 end;
444 *}
447 subsection {* @{text map} *}
449 lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
450 by (induct xs) simp_all
452 lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
453 by (rule ext, induct_tac xs) auto
455 lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
456 by (induct xs) auto
458 lemma map_compose: "map (f o g) xs = map f (map g xs)"
459 by (induct xs) (auto simp add: o_def)
461 lemma rev_map: "rev (map f xs) = map f (rev xs)"
462 by (induct xs) auto
464 lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
465 by (induct xs) auto
467 lemma map_cong [recdef_cong]:
468 "xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
469 -- {* a congruence rule for @{text map} *}
470 by simp
472 lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
473 by (cases xs) auto
475 lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
476 by (cases xs) auto
478 lemma map_eq_Cons_conv[iff]:
479  "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
480 by (cases xs) auto
482 lemma Cons_eq_map_conv[iff]:
483  "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
484 by (cases ys) auto
486 lemma ex_map_conv:
487   "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
488 by(induct ys, auto)
490 lemma map_injective:
491  "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
492 by (induct ys) (auto dest!:injD)
494 lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
495 by(blast dest:map_injective)
497 lemma inj_mapI: "inj f ==> inj (map f)"
498 by (rules dest: map_injective injD intro: inj_onI)
500 lemma inj_mapD: "inj (map f) ==> inj f"
501 apply (unfold inj_on_def, clarify)
502 apply (erule_tac x = "[x]" in ballE)
503  apply (erule_tac x = "[y]" in ballE, simp, blast)
504 apply blast
505 done
507 lemma inj_map[iff]: "inj (map f) = inj f"
508 by (blast dest: inj_mapD intro: inj_mapI)
510 lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
511 by (induct xs, auto)
513 subsection {* @{text rev} *}
515 lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
516 by (induct xs) auto
518 lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
519 by (induct xs) auto
521 lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
522 by (induct xs) auto
524 lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
525 by (induct xs) auto
527 lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"
528 apply (induct xs, force)
529 apply (case_tac ys, simp, force)
530 done
532 lemma rev_induct [case_names Nil snoc]:
533   "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
534 apply(subst rev_rev_ident[symmetric])
535 apply(rule_tac list = "rev xs" in list.induct, simp_all)
536 done
538 ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
540 lemma rev_exhaust [case_names Nil snoc]:
541   "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
542 by (induct xs rule: rev_induct) auto
544 lemmas rev_cases = rev_exhaust
547 subsection {* @{text set} *}
549 lemma finite_set [iff]: "finite (set xs)"
550 by (induct xs) auto
552 lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
553 by (induct xs) auto
555 lemma hd_in_set: "l = x#xs \<Longrightarrow> x\<in>set l"
556 by (case_tac l, auto)
558 lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
559 by auto
561 lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs"
562 by auto
564 lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
565 by (induct xs) auto
567 lemma set_rev [simp]: "set (rev xs) = set xs"
568 by (induct xs) auto
570 lemma set_map [simp]: "set (map f xs) = f`(set xs)"
571 by (induct xs) auto
573 lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
574 by (induct xs) auto
576 lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> k < j}"
577 apply (induct j, simp_all)
578 apply (erule ssubst, auto)
579 done
581 lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
582 apply (induct xs, simp, simp)
583 apply (rule iffI)
584  apply (blast intro: eq_Nil_appendI Cons_eq_appendI)
585 apply (erule exE)+
586 apply (case_tac ys, auto)
587 done
589 lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"
590 -- {* eliminate @{text lists} in favour of @{text set} *}
591 by (induct xs) auto
593 lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"
594 by (rule in_lists_conv_set [THEN iffD1])
596 lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"
597 by (rule in_lists_conv_set [THEN iffD2])
599 lemma finite_list: "finite A ==> EX l. set l = A"
600 apply (erule finite_induct, auto)
601 apply (rule_tac x="x#l" in exI, auto)
602 done
604 lemma card_length: "card (set xs) \<le> length xs"
605 by (induct xs) (auto simp add: card_insert_if)
607 subsection {* @{text mem} *}
609 lemma set_mem_eq: "(x mem xs) = (x : set xs)"
610 by (induct xs) auto
613 subsection {* @{text list_all} *}
615 lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)"
616 by (induct xs) auto
618 lemma list_all_append [simp]:
619 "list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
620 by (induct xs) auto
623 subsection {* @{text filter} *}
625 lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
626 by (induct xs) auto
628 lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
629 by (induct xs) auto
631 lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
632 by (induct xs) auto
634 lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
635 by (induct xs) auto
637 lemma length_filter [simp]: "length (filter P xs) \<le> length xs"
638 by (induct xs) (auto simp add: le_SucI)
640 lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
641 by auto
644 subsection {* @{text concat} *}
646 lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
647 by (induct xs) auto
649 lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
650 by (induct xss) auto
652 lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
653 by (induct xss) auto
655 lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
656 by (induct xs) auto
658 lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
659 by (induct xs) auto
661 lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
662 by (induct xs) auto
664 lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
665 by (induct xs) auto
668 subsection {* @{text nth} *}
670 lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
671 by auto
673 lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
674 by auto
676 declare nth.simps [simp del]
678 lemma nth_append:
679 "!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
680 apply (induct "xs", simp)
681 apply (case_tac n, auto)
682 done
684 lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
685 apply (induct xs, simp)
686 apply (case_tac n, auto)
687 done
689 lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
690 apply (induct_tac xs, simp, simp)
691 apply safe
692 apply (rule_tac x = 0 in exI, simp)
693  apply (rule_tac x = "Suc i" in exI, simp)
694 apply (case_tac i, simp)
695 apply (rename_tac j)
696 apply (rule_tac x = j in exI, simp)
697 done
699 lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
700 by (auto simp add: set_conv_nth)
702 lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
703 by (auto simp add: set_conv_nth)
705 lemma all_nth_imp_all_set:
706 "[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
707 by (auto simp add: set_conv_nth)
709 lemma all_set_conv_all_nth:
710 "(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
711 by (auto simp add: set_conv_nth)
714 subsection {* @{text list_update} *}
716 lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
717 by (induct xs) (auto split: nat.split)
719 lemma nth_list_update:
720 "!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
721 by (induct xs) (auto simp add: nth_Cons split: nat.split)
723 lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
724 by (simp add: nth_list_update)
726 lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
727 by (induct xs) (auto simp add: nth_Cons split: nat.split)
729 lemma list_update_overwrite [simp]:
730 "!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
731 by (induct xs) (auto split: nat.split)
733 lemma list_update_id[simp]: "!!i. i < length xs \<Longrightarrow> xs[i := xs!i] = xs"
734 apply (induct xs, simp)
735 apply(simp split:nat.splits)
736 done
738 lemma list_update_same_conv:
739 "!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
740 by (induct xs) (auto split: nat.split)
742 lemma list_update_append1:
743  "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
744 apply (induct xs, simp)
745 apply(simp split:nat.split)
746 done
748 lemma update_zip:
749 "!!i xy xs. length xs = length ys ==>
750 (zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
751 by (induct ys) (auto, case_tac xs, auto split: nat.split)
753 lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
754 by (induct xs) (auto split: nat.split)
756 lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
757 by (blast dest!: set_update_subset_insert [THEN subsetD])
760 subsection {* @{text last} and @{text butlast} *}
762 lemma last_snoc [simp]: "last (xs @ [x]) = x"
763 by (induct xs) auto
765 lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
766 by (induct xs) auto
768 lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
771 lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
774 lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
775 by (induct xs) (auto)
777 lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
780 lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
785 lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
786 by (induct xs rule: rev_induct) auto
788 lemma butlast_append:
789 "!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
790 by (induct xs) auto
792 lemma append_butlast_last_id [simp]:
793 "xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
794 by (induct xs) auto
796 lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
797 by (induct xs) (auto split: split_if_asm)
799 lemma in_set_butlast_appendI:
800 "x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
801 by (auto dest: in_set_butlastD simp add: butlast_append)
804 subsection {* @{text take} and @{text drop} *}
806 lemma take_0 [simp]: "take 0 xs = []"
807 by (induct xs) auto
809 lemma drop_0 [simp]: "drop 0 xs = xs"
810 by (induct xs) auto
812 lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
813 by simp
815 lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
816 by simp
818 declare take_Cons [simp del] and drop_Cons [simp del]
820 lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
821 by(cases xs, simp_all)
823 lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
824 by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
826 lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
827 apply (induct xs, simp)
828 apply(simp add:drop_Cons nth_Cons split:nat.splits)
829 done
831 lemma take_Suc_conv_app_nth:
832  "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
833 apply (induct xs, simp)
834 apply (case_tac i, auto)
835 done
837 lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
838 by (induct n) (auto, case_tac xs, auto)
840 lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
841 by (induct n) (auto, case_tac xs, auto)
843 lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
844 by (induct n) (auto, case_tac xs, auto)
846 lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
847 by (induct n) (auto, case_tac xs, auto)
849 lemma take_append [simp]:
850 "!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
851 by (induct n) (auto, case_tac xs, auto)
853 lemma drop_append [simp]:
854 "!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
855 by (induct n) (auto, case_tac xs, auto)
857 lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
858 apply (induct m, auto)
859 apply (case_tac xs, auto)
860 apply (case_tac na, auto)
861 done
863 lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
864 apply (induct m, auto)
865 apply (case_tac xs, auto)
866 done
868 lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
869 apply (induct m, auto)
870 apply (case_tac xs, auto)
871 done
873 lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
874 apply (induct n, auto)
875 apply (case_tac xs, auto)
876 done
878 lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
879 apply (induct n, auto)
880 apply (case_tac xs, auto)
881 done
883 lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
884 apply (induct n, auto)
885 apply (case_tac xs, auto)
886 done
888 lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
889 apply (induct xs, auto)
890 apply (case_tac i, auto)
891 done
893 lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
894 apply (induct xs, auto)
895 apply (case_tac i, auto)
896 done
898 lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
899 apply (induct xs, auto)
900 apply (case_tac n, blast)
901 apply (case_tac i, auto)
902 done
904 lemma nth_drop [simp]:
905 "!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
906 apply (induct n, auto)
907 apply (case_tac xs, auto)
908 done
910 lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
911 by(induct xs)(auto simp:take_Cons split:nat.split)
913 lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
914 by(induct xs)(auto simp:drop_Cons split:nat.split)
916 lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
917 using set_take_subset by fast
919 lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
920 using set_drop_subset by fast
922 lemma append_eq_conv_conj:
923 "!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
924 apply (induct xs, simp, clarsimp)
925 apply (case_tac zs, auto)
926 done
928 lemma take_add [rule_format]:
929     "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
930 apply (induct xs, auto)
931 apply (case_tac i, simp_all)
932 done
934 lemma append_eq_append_conv_if:
935  "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
936   (if size xs\<^isub>1 \<le> size ys\<^isub>1
937    then xs\<^isub>1 = take (size xs\<^isub>1) ys\<^isub>1 \<and> xs\<^isub>2 = drop (size xs\<^isub>1) ys\<^isub>1 @ ys\<^isub>2
938    else take (size ys\<^isub>1) xs\<^isub>1 = ys\<^isub>1 \<and> drop (size ys\<^isub>1) xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>2)"
939 apply(induct xs\<^isub>1)
940  apply simp
941 apply(case_tac ys\<^isub>1)
942 apply simp_all
943 done
946 subsection {* @{text takeWhile} and @{text dropWhile} *}
948 lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
949 by (induct xs) auto
951 lemma takeWhile_append1 [simp]:
952 "[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
953 by (induct xs) auto
955 lemma takeWhile_append2 [simp]:
956 "(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
957 by (induct xs) auto
959 lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
960 by (induct xs) auto
962 lemma dropWhile_append1 [simp]:
963 "[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
964 by (induct xs) auto
966 lemma dropWhile_append2 [simp]:
967 "(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
968 by (induct xs) auto
970 lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
971 by (induct xs) (auto split: split_if_asm)
973 lemma takeWhile_eq_all_conv[simp]:
974  "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
975 by(induct xs, auto)
977 lemma dropWhile_eq_Nil_conv[simp]:
978  "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
979 by(induct xs, auto)
981 lemma dropWhile_eq_Cons_conv:
982  "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
983 by(induct xs, auto)
986 subsection {* @{text zip} *}
988 lemma zip_Nil [simp]: "zip [] ys = []"
989 by (induct ys) auto
991 lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
992 by simp
994 declare zip_Cons [simp del]
996 lemma length_zip [simp]:
997 "!!xs. length (zip xs ys) = min (length xs) (length ys)"
998 apply (induct ys, simp)
999 apply (case_tac xs, auto)
1000 done
1002 lemma zip_append1:
1003 "!!xs. zip (xs @ ys) zs =
1004 zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
1005 apply (induct zs, simp)
1006 apply (case_tac xs, simp_all)
1007 done
1009 lemma zip_append2:
1010 "!!ys. zip xs (ys @ zs) =
1011 zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
1012 apply (induct xs, simp)
1013 apply (case_tac ys, simp_all)
1014 done
1016 lemma zip_append [simp]:
1017  "[| length xs = length us; length ys = length vs |] ==>
1018 zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
1019 by (simp add: zip_append1)
1021 lemma zip_rev:
1022 "length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
1023 by (induct rule:list_induct2, simp_all)
1025 lemma nth_zip [simp]:
1026 "!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
1027 apply (induct ys, simp)
1028 apply (case_tac xs)
1029  apply (simp_all add: nth.simps split: nat.split)
1030 done
1032 lemma set_zip:
1033 "set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
1034 by (simp add: set_conv_nth cong: rev_conj_cong)
1036 lemma zip_update:
1037 "length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
1038 by (rule sym, simp add: update_zip)
1040 lemma zip_replicate [simp]:
1041 "!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
1042 apply (induct i, auto)
1043 apply (case_tac j, auto)
1044 done
1047 subsection {* @{text list_all2} *}
1049 lemma list_all2_lengthD [intro?]:
1050   "list_all2 P xs ys ==> length xs = length ys"
1051 by (simp add: list_all2_def)
1053 lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])"
1054 by (simp add: list_all2_def)
1056 lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"
1057 by (simp add: list_all2_def)
1059 lemma list_all2_Cons [iff]:
1060 "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
1061 by (auto simp add: list_all2_def)
1063 lemma list_all2_Cons1:
1064 "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
1065 by (cases ys) auto
1067 lemma list_all2_Cons2:
1068 "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
1069 by (cases xs) auto
1071 lemma list_all2_rev [iff]:
1072 "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
1073 by (simp add: list_all2_def zip_rev cong: conj_cong)
1075 lemma list_all2_rev1:
1076 "list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
1077 by (subst list_all2_rev [symmetric]) simp
1079 lemma list_all2_append1:
1080 "list_all2 P (xs @ ys) zs =
1081 (EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
1082 list_all2 P xs us \<and> list_all2 P ys vs)"
1083 apply (simp add: list_all2_def zip_append1)
1084 apply (rule iffI)
1085  apply (rule_tac x = "take (length xs) zs" in exI)
1086  apply (rule_tac x = "drop (length xs) zs" in exI)
1087  apply (force split: nat_diff_split simp add: min_def, clarify)
1088 apply (simp add: ball_Un)
1089 done
1091 lemma list_all2_append2:
1092 "list_all2 P xs (ys @ zs) =
1093 (EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
1094 list_all2 P us ys \<and> list_all2 P vs zs)"
1095 apply (simp add: list_all2_def zip_append2)
1096 apply (rule iffI)
1097  apply (rule_tac x = "take (length ys) xs" in exI)
1098  apply (rule_tac x = "drop (length ys) xs" in exI)
1099  apply (force split: nat_diff_split simp add: min_def, clarify)
1100 apply (simp add: ball_Un)
1101 done
1103 lemma list_all2_append:
1104   "length xs = length ys \<Longrightarrow>
1105   list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
1106 by (induct rule:list_induct2, simp_all)
1108 lemma list_all2_appendI [intro?, trans]:
1109   "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
1110   by (simp add: list_all2_append list_all2_lengthD)
1112 lemma list_all2_conv_all_nth:
1113 "list_all2 P xs ys =
1114 (length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
1115 by (force simp add: list_all2_def set_zip)
1117 lemma list_all2_trans:
1118   assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
1119   shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
1120         (is "!!bs cs. PROP ?Q as bs cs")
1121 proof (induct as)
1122   fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
1123   show "!!cs. PROP ?Q (x # xs) bs cs"
1124   proof (induct bs)
1125     fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
1126     show "PROP ?Q (x # xs) (y # ys) cs"
1127       by (induct cs) (auto intro: tr I1 I2)
1128   qed simp
1129 qed simp
1131 lemma list_all2_all_nthI [intro?]:
1132   "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
1133   by (simp add: list_all2_conv_all_nth)
1135 lemma list_all2I:
1136   "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
1137   by (simp add: list_all2_def)
1139 lemma list_all2_nthD:
1140   "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
1141   by (simp add: list_all2_conv_all_nth)
1143 lemma list_all2_nthD2:
1144   "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
1145   by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
1147 lemma list_all2_map1:
1148   "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
1149   by (simp add: list_all2_conv_all_nth)
1151 lemma list_all2_map2:
1152   "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
1153   by (auto simp add: list_all2_conv_all_nth)
1155 lemma list_all2_refl [intro?]:
1156   "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
1157   by (simp add: list_all2_conv_all_nth)
1159 lemma list_all2_update_cong:
1160   "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
1161   by (simp add: list_all2_conv_all_nth nth_list_update)
1163 lemma list_all2_update_cong2:
1164   "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
1165   by (simp add: list_all2_lengthD list_all2_update_cong)
1167 lemma list_all2_takeI [simp,intro?]:
1168   "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
1169   apply (induct xs)
1170    apply simp
1171   apply (clarsimp simp add: list_all2_Cons1)
1172   apply (case_tac n)
1173   apply auto
1174   done
1176 lemma list_all2_dropI [simp,intro?]:
1177   "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
1178   apply (induct as, simp)
1179   apply (clarsimp simp add: list_all2_Cons1)
1180   apply (case_tac n, simp, simp)
1181   done
1183 lemma list_all2_mono [intro?]:
1184   "\<And>y. list_all2 P x y \<Longrightarrow> (\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> list_all2 Q x y"
1185   apply (induct x, simp)
1186   apply (case_tac y, auto)
1187   done
1190 subsection {* @{text foldl} *}
1192 lemma foldl_append [simp]:
1193 "!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
1194 by (induct xs) auto
1196 text {*
1197 Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
1198 difficult to use because it requires an additional transitivity step.
1199 *}
1201 lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
1202 by (induct ns) auto
1204 lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
1205 by (force intro: start_le_sum simp add: in_set_conv_decomp)
1207 lemma sum_eq_0_conv [iff]:
1208 "!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
1209 by (induct ns) auto
1212 subsection {* folding a relation over a list *}
1214 (*"fold_rel R cs \<equiv> foldl (%r c. r O {(x,y). (c,x,y):R}) Id cs"*)
1215 inductive "fold_rel R" intros
1216   Nil:  "(a, [],a) : fold_rel R"
1217   Cons: "[|(a,x,b) : R; (b,xs,c) : fold_rel R|] ==> (a,x#xs,c) : fold_rel R"
1218 inductive_cases fold_rel_elim_case [elim!]:
1219    "(a, [] , b) : fold_rel R"
1220    "(a, x#xs, b) : fold_rel R"
1222 lemma fold_rel_Nil [intro!]: "a = b ==> (a, [], b) : fold_rel R"
1223 by (simp add: fold_rel.Nil)
1224 declare fold_rel.Cons [intro!]
1227 subsection {* @{text upto} *}
1229 lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])"
1230 -- {* Does not terminate! *}
1231 by (induct j) auto
1233 lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []"
1234 by (subst upt_rec) simp
1236 lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]"
1237 -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
1238 by simp
1240 lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]"
1241 apply(rule trans)
1242 apply(subst upt_rec)
1243  prefer 2 apply (rule refl, simp)
1244 done
1246 lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]"
1247 -- {* LOOPS as a simprule, since @{text "j <= j"}. *}
1248 by (induct k) auto
1250 lemma length_upt [simp]: "length [i..j(] = j - i"
1251 by (induct j) (auto simp add: Suc_diff_le)
1253 lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k"
1254 apply (induct j)
1255 apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
1256 done
1258 lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]"
1259 apply (induct m, simp)
1260 apply (subst upt_rec)
1261 apply (rule sym)
1262 apply (subst upt_rec)
1263 apply (simp del: upt.simps)
1264 done
1266 lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]"
1267 by (induct n) auto
1269 lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..n(]) ! i = f(m+i)"
1270 apply (induct n m rule: diff_induct)
1271 prefer 3 apply (subst map_Suc_upt[symmetric])
1272 apply (auto simp add: less_diff_conv nth_upt)
1273 done
1275 lemma nth_take_lemma:
1276   "!!xs ys. k <= length xs ==> k <= length ys ==>
1277      (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
1278 apply (atomize, induct k)
1279 apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
1280 txt {* Both lists must be non-empty *}
1281 apply (case_tac xs, simp)
1282 apply (case_tac ys, clarify)
1283  apply (simp (no_asm_use))
1284 apply clarify
1285 txt {* prenexing's needed, not miniscoping *}
1286 apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
1287 apply blast
1288 done
1290 lemma nth_equalityI:
1291  "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
1292 apply (frule nth_take_lemma [OF le_refl eq_imp_le])
1293 apply (simp_all add: take_all)
1294 done
1296 (* needs nth_equalityI *)
1297 lemma list_all2_antisym:
1298   "\<lbrakk> (\<And>x y. \<lbrakk>P x y; Q y x\<rbrakk> \<Longrightarrow> x = y); list_all2 P xs ys; list_all2 Q ys xs \<rbrakk>
1299   \<Longrightarrow> xs = ys"
1300   apply (simp add: list_all2_conv_all_nth)
1301   apply (rule nth_equalityI, blast, simp)
1302   done
1304 lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
1305 -- {* The famous take-lemma. *}
1306 apply (drule_tac x = "max (length xs) (length ys)" in spec)
1307 apply (simp add: le_max_iff_disj take_all)
1308 done
1311 subsection {* @{text "distinct"} and @{text remdups} *}
1313 lemma distinct_append [simp]:
1314 "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
1315 by (induct xs) auto
1317 lemma set_remdups [simp]: "set (remdups xs) = set xs"
1318 by (induct xs) (auto simp add: insert_absorb)
1320 lemma distinct_remdups [iff]: "distinct (remdups xs)"
1321 by (induct xs) auto
1323 lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
1324 by (induct xs) auto
1326 text {*
1327 It is best to avoid this indexed version of distinct, but sometimes
1328 it is useful. *}
1329 lemma distinct_conv_nth:
1330 "distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)"
1331 apply (induct_tac xs, simp, simp)
1332 apply (rule iffI, clarsimp)
1333  apply (case_tac i)
1334 apply (case_tac j, simp)
1335 apply (simp add: set_conv_nth)
1336  apply (case_tac j)
1337 apply (clarsimp simp add: set_conv_nth, simp)
1338 apply (rule conjI)
1339  apply (clarsimp simp add: set_conv_nth)
1340  apply (erule_tac x = 0 in allE)
1341  apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
1342 apply (erule_tac x = "Suc i" in allE)
1343 apply (erule_tac x = "Suc j" in allE, simp)
1344 done
1346 lemma distinct_card: "distinct xs \<Longrightarrow> card (set xs) = size xs"
1347   by (induct xs) auto
1349 lemma card_distinct: "card (set xs) = size xs \<Longrightarrow> distinct xs"
1350 proof (induct xs)
1351   case Nil thus ?case by simp
1352 next
1353   case (Cons x xs)
1354   show ?case
1355   proof (cases "x \<in> set xs")
1356     case False with Cons show ?thesis by simp
1357   next
1358     case True with Cons.prems
1359     have "card (set xs) = Suc (length xs)"
1360       by (simp add: card_insert_if split: split_if_asm)
1361     moreover have "card (set xs) \<le> length xs" by (rule card_length)
1362     ultimately have False by simp
1363     thus ?thesis ..
1364   qed
1365 qed
1368 subsection {* @{text replicate} *}
1370 lemma length_replicate [simp]: "length (replicate n x) = n"
1371 by (induct n) auto
1373 lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
1374 by (induct n) auto
1376 lemma replicate_app_Cons_same:
1377 "(replicate n x) @ (x # xs) = x # replicate n x @ xs"
1378 by (induct n) auto
1380 lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
1381 apply (induct n, simp)
1382 apply (simp add: replicate_app_Cons_same)
1383 done
1385 lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
1386 by (induct n) auto
1388 lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
1389 by (induct n) auto
1391 lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
1392 by (induct n) auto
1394 lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
1395 by (atomize (full), induct n) auto
1397 lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
1398 apply (induct n, simp)
1399 apply (simp add: nth_Cons split: nat.split)
1400 done
1402 lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
1403 by (induct n) auto
1405 lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
1406 by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
1408 lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
1409 by auto
1411 lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
1412 by (simp add: set_replicate_conv_if split: split_if_asm)
1415 subsection {* @{text postfix} *}
1417 lemma postfix_refl [simp, intro!]: "xs \<sqsupseteq> xs" by (auto simp add: postfix_def)
1418 lemma postfix_trans: "\<lbrakk>xs \<sqsupseteq> ys; ys \<sqsupseteq> zs\<rbrakk> \<Longrightarrow> xs \<sqsupseteq> zs"
1419          by (auto simp add: postfix_def)
1420 lemma postfix_antisym: "\<lbrakk>xs \<sqsupseteq> ys; ys \<sqsupseteq> xs\<rbrakk> \<Longrightarrow> xs = ys"
1421          by (auto simp add: postfix_def)
1423 lemma postfix_emptyI [simp, intro!]: "xs \<sqsupseteq> []" by (auto simp add: postfix_def)
1424 lemma postfix_emptyD [dest!]: "[] \<sqsupseteq> xs \<Longrightarrow> xs = []"by(auto simp add:postfix_def)
1425 lemma postfix_ConsI: "xs \<sqsupseteq> ys \<Longrightarrow> x#xs \<sqsupseteq> ys" by (auto simp add: postfix_def)
1426 lemma postfix_ConsD: "xs \<sqsupseteq> y#ys \<Longrightarrow> xs \<sqsupseteq> ys" by (auto simp add: postfix_def)
1427 lemma postfix_appendI: "xs \<sqsupseteq> ys \<Longrightarrow> zs@xs \<sqsupseteq> ys" by (auto simp add: postfix_def)
1428 lemma postfix_appendD: "xs \<sqsupseteq> zs@ys \<Longrightarrow> xs \<sqsupseteq> ys" by (auto simp add: postfix_def)
1430 lemma postfix_is_subset_lemma: "xs = zs @ ys \<Longrightarrow> set ys \<subseteq> set xs"
1431 by (induct zs, auto)
1432 lemma postfix_is_subset: "xs \<sqsupseteq> ys \<Longrightarrow> set ys \<subseteq> set xs"
1433 by (unfold postfix_def, erule exE, erule postfix_is_subset_lemma)
1435 lemma postfix_ConsD2_lemma [rule_format]: "x#xs = zs @ y#ys \<longrightarrow> xs \<sqsupseteq> ys"
1436 by (induct zs, auto intro!: postfix_appendI postfix_ConsI)
1437 lemma postfix_ConsD2: "x#xs \<sqsupseteq> y#ys \<Longrightarrow> xs \<sqsupseteq> ys"
1438 by (auto simp add: postfix_def dest!: postfix_ConsD2_lemma)
1440 subsection {* Lexicographic orderings on lists *}
1442 lemma wf_lexn: "wf r ==> wf (lexn r n)"
1443 apply (induct_tac n, simp, simp)
1444 apply(rule wf_subset)
1445  prefer 2 apply (rule Int_lower1)
1446 apply(rule wf_prod_fun_image)
1447  prefer 2 apply (rule inj_onI, auto)
1448 done
1450 lemma lexn_length:
1451 "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
1452 by (induct n) auto
1454 lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
1455 apply (unfold lex_def)
1456 apply (rule wf_UN)
1457 apply (blast intro: wf_lexn, clarify)
1458 apply (rename_tac m n)
1459 apply (subgoal_tac "m \<noteq> n")
1460  prefer 2 apply blast
1461 apply (blast dest: lexn_length not_sym)
1462 done
1464 lemma lexn_conv:
1465 "lexn r n =
1466 {(xs,ys). length xs = n \<and> length ys = n \<and>
1467 (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
1468 apply (induct_tac n, simp, blast)
1469 apply (simp add: image_Collect lex_prod_def, safe, blast)
1470  apply (rule_tac x = "ab # xys" in exI, simp)
1471 apply (case_tac xys, simp_all, blast)
1472 done
1474 lemma lex_conv:
1475 "lex r =
1476 {(xs,ys). length xs = length ys \<and>
1477 (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
1478 by (force simp add: lex_def lexn_conv)
1480 lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)"
1481 by (unfold lexico_def) blast
1483 lemma lexico_conv:
1484 "lexico r = {(xs,ys). length xs < length ys |
1485 length xs = length ys \<and> (xs, ys) : lex r}"
1486 by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def)
1488 lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
1489 by (simp add: lex_conv)
1491 lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
1492 by (simp add:lex_conv)
1494 lemma Cons_in_lex [iff]:
1495 "((x # xs, y # ys) : lex r) =
1496 ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
1497 apply (simp add: lex_conv)
1498 apply (rule iffI)
1499  prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
1500 apply (case_tac xys, simp, simp)
1501 apply blast
1502 done
1505 subsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
1507 lemma sublist_empty [simp]: "sublist xs {} = []"
1508 by (auto simp add: sublist_def)
1510 lemma sublist_nil [simp]: "sublist [] A = []"
1511 by (auto simp add: sublist_def)
1513 lemma sublist_shift_lemma:
1514 "map fst [p:zip xs [i..i + length xs(] . snd p : A] =
1515 map fst [p:zip xs [0..length xs(] . snd p + i : A]"
1516 by (induct xs rule: rev_induct) (simp_all add: add_commute)
1518 lemma sublist_append:
1519 "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
1520 apply (unfold sublist_def)
1521 apply (induct l' rule: rev_induct, simp)
1522 apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
1524 done
1526 lemma sublist_Cons:
1527 "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
1528 apply (induct l rule: rev_induct)
1529  apply (simp add: sublist_def)
1530 apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
1531 done
1533 lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
1534 by (simp add: sublist_Cons)
1536 lemma sublist_upt_eq_take [simp]: "sublist l {..n(} = take n l"
1537 apply (induct l rule: rev_induct, simp)
1538 apply (simp split: nat_diff_split add: sublist_append)
1539 done
1542 lemma take_Cons':
1543 "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
1544 by (cases n) simp_all
1546 lemma drop_Cons':
1547 "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
1548 by (cases n) simp_all
1550 lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
1551 by (cases n) simp_all
1553 lemmas [simp] = take_Cons'[of "number_of v",standard]
1554                 drop_Cons'[of "number_of v",standard]
1555                 nth_Cons'[of _ _ "number_of v",standard]
1558 lemma distinct_card: "distinct xs \<Longrightarrow> card (set xs) = size xs"
1559   by (induct xs) auto
1561 lemma card_length: "card (set xs) \<le> length xs"
1562   by (induct xs) (auto simp add: card_insert_if)
1564 lemma "card (set xs) = size xs \<Longrightarrow> distinct xs"
1565 proof (induct xs)
1566   case Nil thus ?case by simp
1567 next
1568   case (Cons x xs)
1569   show ?case
1570   proof (cases "x \<in> set xs")
1571     case False with Cons show ?thesis by simp
1572   next
1573     case True with Cons.prems
1574     have "card (set xs) = Suc (length xs)"
1575       by (simp add: card_insert_if split: split_if_asm)
1576     moreover have "card (set xs) \<le> length xs" by (rule card_length)
1577     ultimately have False by simp
1578     thus ?thesis ..
1579   qed
1580 qed
1582 subsection {* Characters and strings *}
1584 datatype nibble =
1585     Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
1586   | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
1588 datatype char = Char nibble nibble
1589   -- "Note: canonical order of character encoding coincides with standard term ordering"
1591 types string = "char list"
1593 syntax
1594   "_Char" :: "xstr => char"    ("CHR _")
1595   "_String" :: "xstr => string"    ("_")
1597 parse_ast_translation {*
1598   let
1599     val constants = Syntax.Appl o map Syntax.Constant;
1601     fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10));
1602     fun mk_char c =
1603       if Symbol.is_ascii c andalso Symbol.is_printable c then
1604         constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)]
1605       else error ("Printable ASCII character expected: " ^ quote c);
1607     fun mk_string [] = Syntax.Constant "Nil"
1608       | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs];
1610     fun char_ast_tr [Syntax.Variable xstr] =
1611         (case Syntax.explode_xstr xstr of
1612           [c] => mk_char c
1613         | _ => error ("Single character expected: " ^ xstr))
1614       | char_ast_tr asts = raise AST ("char_ast_tr", asts);
1616     fun string_ast_tr [Syntax.Variable xstr] =
1617         (case Syntax.explode_xstr xstr of
1618           [] => constants [Syntax.constrainC, "Nil", "string"]
1619         | cs => mk_string cs)
1620       | string_ast_tr asts = raise AST ("string_tr", asts);
1621   in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end;
1622 *}
1624 print_ast_translation {*
1625   let
1626     fun dest_nib (Syntax.Constant c) =
1627         (case explode c of
1628           ["N", "i", "b", "b", "l", "e", h] =>
1629             if "0" <= h andalso h <= "9" then ord h - ord "0"
1630             else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10
1631             else raise Match
1632         | _ => raise Match)
1633       | dest_nib _ = raise Match;
1635     fun dest_chr c1 c2 =
1636       let val c = chr (dest_nib c1 * 16 + dest_nib c2)
1637       in if Symbol.is_printable c then c else raise Match end;
1639     fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2
1640       | dest_char _ = raise Match;
1642     fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)];
1644     fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]]
1645       | char_ast_tr' _ = raise Match;
1647     fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String",
1648             xstr (map dest_char (Syntax.unfold_ast "_args" args))]
1649       | list_ast_tr' ts = raise Match;
1650   in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end;
1651 *}
1653 end