4 Copyright 1994 TU Muenchen
7 header {* The datatype of finite lists *}
13 | Cons 'a "'a list" (infixr "#" 65)
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"
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"
48 -- {* list Enumeration *}
49 "@list" :: "args => 'a list" ("[(_)]")
51 -- {* Special syntax for filter *}
52 "@filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_:_./ _])")
55 "_lupdbind" :: "['a, 'a] => lupdbind" ("(2_ :=/ _)")
56 "" :: "lupdbind => lupdbinds" ("_")
57 "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds" ("_,/ _")
58 "_LUpdate" :: "['a, lupdbinds] => 'a" ("_/[(_)]" [900,0] 900)
60 upto :: "nat => nat => nat list" ("(1[_../_])")
65 "[x:xs . P]" == "filter (%x. P) xs"
67 "_LUpdate xs (_lupdbinds b bs)" == "_LUpdate (_LUpdate xs b) bs"
68 "xs[i:=x]" == "list_update xs i x"
70 "[i..j]" == "[i..(Suc j)(]"
74 "@filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_\<in>_ ./ _])")
78 Function @{text size} is overloaded for all datatypes. Users may
79 refer to the list version as @{text length}. *}
81 syntax length :: "'a list => nat"
82 translations "length" => "size :: _ list => nat"
84 typed_print_translation {*
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
101 "last(x#xs) = (if xs=[] then x else last xs)"
104 "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
107 "x mem (y#ys) = (if y=x then True else x mem ys)"
110 "set (x#xs) = insert x (set xs)"
112 list_all_Nil: "list_all P [] = True"
113 list_all_Cons: "list_all P (x#xs) = (P(x) \<and> list_all P xs)"
116 "map f (x#xs) = f(x)#map f xs"
118 append_Nil: "[] @ys = ys"
119 append_Cons: "(x#xs)@ys = x#(xs@ys)"
122 "rev(x#xs) = rev(xs) @ [x]"
125 "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
127 foldl_Nil: "foldl f a [] = a"
128 foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
131 "foldr f (x#xs) a = f x (foldr f xs a)"
134 "concat(x#xs) = x @ concat(xs)"
136 drop_Nil: "drop n [] = []"
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"} *}
141 take_Nil: "take n [] = []"
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"} *}
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"} *}
152 (case i of 0 => v # xs
153 | Suc j => x # xs[j:=v])"
155 "takeWhile P [] = []"
156 "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
158 "dropWhile P [] = []"
159 "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
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"} *}
166 upt_0: "[i..0(] = []"
167 upt_Suc: "[i..(Suc j)(] = (if i <= j then [i..j(] @ [j] else [])"
170 "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
173 "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
175 replicate_0: "replicate 0 x = []"
176 replicate_Suc: "replicate (Suc n) x = x # replicate n x"
179 "list_all2 P xs ys == length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)"
182 subsection {* Lexicographic orderings on lists *}
185 lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
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}"
193 lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
194 "lex r == \<Union>n. lexn r n"
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))"
199 sublist :: "'a list => nat set => 'a list"
200 "sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..size xs(]))"
203 lemma not_Cons_self [simp]: "xs \<noteq> x # xs"
206 lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
208 lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
212 "(!!xs. \<forall>ys. length ys < length xs --> P ys ==> P xs) ==> P xs"
213 by (rule measure_induct [of length]) rules
216 subsection {* @{text lists}: the list-forming operator over sets *}
218 consts lists :: "'a set => 'a list set"
221 Nil [intro!]: "[]: lists A"
222 Cons [intro!]: "[| a: A; l: lists A |] ==> a#l : lists A"
224 inductive_cases listsE [elim!]: "x#l : lists A"
226 lemma lists_mono: "A \<subseteq> B ==> lists A \<subseteq> lists B"
227 by (unfold lists.defs) (blast intro!: lfp_mono)
229 lemma lists_IntI [rule_format]:
230 "l: lists A ==> l: lists B --> l: lists (A Int B)"
231 apply (erule lists.induct)
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)
241 lemma append_in_lists_conv [iff]:
242 "(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"
246 subsection {* @{text length} *}
249 Needs to come before @{text "@"} because of theorem @{text
250 append_eq_append_conv}.
253 lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
256 lemma length_map [simp]: "length (map f xs) = length xs"
259 lemma length_rev [simp]: "length (rev xs) = length xs"
262 lemma length_tl [simp]: "length (tl xs) = length xs - 1"
265 lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
268 lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
271 lemma length_Suc_conv:
272 "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
276 subsection {* @{text "@"} -- append *}
278 lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
281 lemma append_Nil2 [simp]: "xs @ [] = xs"
284 lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
287 lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
290 lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
293 lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
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)
310 lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
313 lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
316 lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
319 lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
320 using append_same_eq [of _ _ "[]"] by auto
322 lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
323 using append_same_eq [of "[]"] by auto
325 lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
328 lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
331 lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
332 by (simp add: hd_append split: list.split)
334 lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
335 by (simp split: list.split)
337 lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
338 by (simp add: tl_append split: list.split)
341 text {* Trivial rules for solving @{text "@"}-equations automatically. *}
343 lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
346 lemma Cons_eq_appendI:
347 "[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
350 lemma append_eq_appendI:
351 "[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
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.
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";
371 val list_eq_pattern =
372 Thm.read_cterm (Theory.sign_of (the_context ())) ("(xs::'a list) = ys",HOLogic.boolT)
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
379 fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
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)
388 simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons])
390 fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
392 val lastl = last lhs and lastr = last rhs
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])
402 error("The error(s) above occurred while trying to prove " ^
404 in Some((conv RS (thm RS trans)) RS eq_reflection) end
406 in if list1 lastl andalso list1 lastr
407 then rearr append1_eq_conv
410 then rearr append_same_eq
414 val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq
417 Addsimprocs [list_eq_simproc];
421 subsection {* @{text map} *}
423 lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
424 by (induct xs) simp_all
426 lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
427 by (rule ext, induct_tac xs) auto
429 lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
432 lemma map_compose: "map (f o g) xs = map f (map g xs)"
433 by (induct xs) (auto simp add: o_def)
435 lemma rev_map: "rev (map f xs) = map f (rev xs)"
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
443 lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
446 lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
450 "(map f xs = y # ys) = (\<exists>x xs'. xs = x # xs' \<and> f x = y \<and> map f xs' = ys)"
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)
457 lemma inj_mapI: "inj f ==> inj (map f)"
458 by (rules dest: map_injective injD intro: injI)
460 lemma inj_mapD: "inj (map f) ==> inj f"
461 apply (unfold inj_on_def)
463 apply (erule_tac x = "[x]" in ballE)
464 apply (erule_tac x = "[y]" in ballE)
470 lemma inj_map: "inj (map f) = inj f"
471 by (blast dest: inj_mapD intro: inj_mapI)
474 subsection {* @{text rev} *}
476 lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
479 lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
482 lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
485 lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
488 lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"
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)
501 ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *} -- "compatibility"
503 lemma rev_exhaust: "(xs = [] ==> P) ==> (!!ys y. xs = ys @ [y] ==> P) ==> P"
504 by (induct xs rule: rev_induct) auto
507 subsection {* @{text set} *}
509 lemma finite_set [iff]: "finite (set xs)"
512 lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
515 lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
518 lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
521 lemma set_rev [simp]: "set (rev xs) = set xs"
524 lemma set_map [simp]: "set (map f xs) = f`(set xs)"
527 lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
530 lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> k < j}"
538 lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
543 apply (blast intro: eq_Nil_appendI Cons_eq_appendI)
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} *}
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])
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])
560 subsection {* @{text mem} *}
562 lemma set_mem_eq: "(x mem xs) = (x : set xs)"
566 subsection {* @{text list_all} *}
568 lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)"
571 lemma list_all_append [simp]:
572 "list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
576 subsection {* @{text filter} *}
578 lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
581 lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
584 lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
587 lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
590 lemma length_filter [simp]: "length (filter P xs) \<le> length xs"
591 by (induct xs) (auto simp add: le_SucI)
593 lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
597 subsection {* @{text concat} *}
599 lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
602 lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
605 lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
608 lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
611 lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
614 lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
617 lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
621 subsection {* @{text nth} *}
623 lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
626 lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
629 declare nth.simps [simp del]
632 "!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
639 lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
646 lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
647 apply (induct_tac xs)
651 apply (rule_tac x = 0 in exI)
653 apply (rule_tac x = "Suc i" in exI)
658 apply (rule_tac x = j in exI)
662 lemma list_ball_nth: "[| n < length xs; !x : set xs. P x |] ==> P(xs!n)"
663 by (auto simp add: set_conv_nth)
665 lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
666 by (auto simp add: set_conv_nth)
668 lemma all_nth_imp_all_set:
669 "[| !i < length xs. P(xs!i); x : set xs |] ==> P x"
670 by (auto simp add: set_conv_nth)
672 lemma all_set_conv_all_nth:
673 "(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
674 by (auto simp add: set_conv_nth)
677 subsection {* @{text list_update} *}
679 lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
680 by (induct xs) (auto split: nat.split)
682 lemma nth_list_update:
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)
686 lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
687 by (simp add: nth_list_update)
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)
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)
696 lemma list_update_same_conv:
697 "!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
698 by (induct xs) (auto split: nat.split)
701 "!!i xy xs. length xs = length ys ==>
702 (zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
703 by (induct ys) (auto, case_tac xs, auto split: nat.split)
705 lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
706 by (induct xs) (auto split: nat.split)
708 lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
709 by (blast dest!: set_update_subset_insert [THEN subsetD])
712 subsection {* @{text last} and @{text butlast} *}
714 lemma last_snoc [simp]: "last (xs @ [x]) = x"
717 lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
720 lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
721 by (induct xs rule: rev_induct) auto
723 lemma butlast_append:
724 "!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
727 lemma append_butlast_last_id [simp]:
728 "xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
731 lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
732 by (induct xs) (auto split: split_if_asm)
734 lemma in_set_butlast_appendI:
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)
739 subsection {* @{text take} and @{text drop} *}
741 lemma take_0 [simp]: "take 0 xs = []"
744 lemma drop_0 [simp]: "drop 0 xs = xs"
747 lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
750 lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
753 declare take_Cons [simp del] and drop_Cons [simp del]
755 lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
756 by (induct n) (auto, case_tac xs, auto)
758 lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
759 by (induct n) (auto, case_tac xs, auto)
761 lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
762 by (induct n) (auto, case_tac xs, auto)
764 lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
765 by (induct n) (auto, case_tac xs, auto)
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)
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)
775 lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
784 lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
791 lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
798 lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
805 lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
812 lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
819 lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
826 lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
833 lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
842 lemma nth_drop [simp]:
843 "!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
850 lemma append_eq_conv_conj:
851 "!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
860 subsection {* @{text takeWhile} and @{text dropWhile} *}
862 lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
865 lemma takeWhile_append1 [simp]:
866 "[| x:set xs; ~P(x) |] ==> takeWhile P (xs @ ys) = takeWhile P xs"
869 lemma takeWhile_append2 [simp]:
870 "(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
873 lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
876 lemma dropWhile_append1 [simp]:
877 "[| x : set xs; ~P(x) |] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
880 lemma dropWhile_append2 [simp]:
881 "(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
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)
888 subsection {* @{text zip} *}
890 lemma zip_Nil [simp]: "zip [] ys = []"
893 lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
896 declare zip_Cons [simp del]
898 lemma length_zip [simp]:
899 "!!xs. length (zip xs ys) = min (length xs) (length ys)"
907 "!!xs. zip (xs @ ys) zs =
908 zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
916 "!!ys. zip xs (ys @ zs) =
917 zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
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)
930 "!!xs. length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
937 lemma nth_zip [simp]:
938 "!!i xs. [| i < length xs; i < length ys |] ==> (zip xs ys)!i = (xs!i, ys!i)"
942 apply (simp_all add: nth.simps split: nat.split)
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)
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)
953 lemma zip_replicate [simp]:
954 "!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
962 subsection {* @{text list_all2} *}
964 lemma list_all2_lengthD: "list_all2 P xs ys ==> length xs = length ys"
965 by (simp add: list_all2_def)
967 lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])"
968 by (simp add: list_all2_def)
970 lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"
971 by (simp add: list_all2_def)
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)
977 lemma list_all2_Cons1:
978 "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
981 lemma list_all2_Cons2:
982 "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
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)
989 lemma list_all2_append1:
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)
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)
999 apply (simp add: ball_Un)
1002 lemma list_all2_append2:
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)
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)
1012 apply (simp add: ball_Un)
1015 lemma list_all2_conv_all_nth:
1016 "list_all2 P xs ys =
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)
1020 lemma list_all2_trans[rule_format]:
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)
1026 apply(induct_tac bs)
1029 apply(induct_tac cs)
1034 subsection {* @{text foldl} *}
1036 lemma foldl_append [simp]:
1037 "!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
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.
1045 lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
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)
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))"
1056 subsection {* @{text upto} *}
1058 lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])"
1059 -- {* Does not terminate! *}
1062 lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []"
1063 by (subst upt_rec) simp
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. *}
1069 lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]"
1071 apply(subst upt_rec)
1072 prefer 2 apply(rule refl)
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"}. *}
1080 lemma length_upt [simp]: "length [i..j(] = j - i"
1081 by (induct j) (auto simp add: Suc_diff_le)
1083 lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k"
1085 apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
1088 lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]"
1091 apply (subst upt_rec)
1093 apply (subst upt_rec)
1094 apply (simp del: upt.simps)
1097 lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]"
1100 lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..n(]) ! i = f(m+i)"
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)
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"
1111 apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib)
1113 txt {* Both lists must be non-empty *}
1118 apply (simp (no_asm_use))
1120 txt {* prenexing's needed, not miniscoping *}
1121 apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
1125 lemma nth_equalityI:
1126 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
1127 apply (frule nth_take_lemma [OF le_refl eq_imp_le])
1128 apply (simp_all add: take_all)
1131 lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
1132 -- {* The famous take-lemma. *}
1133 apply (drule_tac x = "max (length xs) (length ys)" in spec)
1134 apply (simp add: le_max_iff_disj take_all)
1138 subsection {* @{text "distinct"} and @{text remdups} *}
1140 lemma distinct_append [simp]:
1141 "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
1144 lemma set_remdups [simp]: "set (remdups xs) = set xs"
1145 by (induct xs) (auto simp add: insert_absorb)
1147 lemma distinct_remdups [iff]: "distinct (remdups xs)"
1150 lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
1154 It is best to avoid this indexed version of distinct, but sometimes
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)
1166 apply (simp add: set_conv_nth)
1168 apply (clarsimp simp add: set_conv_nth)
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)
1176 apply (erule_tac x = "Suc i" in allE)
1177 apply (erule_tac x = "Suc j" in allE)
1182 subsection {* @{text replicate} *}
1184 lemma length_replicate [simp]: "length (replicate n x) = n"
1187 lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
1190 lemma replicate_app_Cons_same:
1191 "(replicate n x) @ (x # xs) = x # replicate n x @ xs"
1194 lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
1197 apply (simp add: replicate_app_Cons_same)
1200 lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
1203 lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
1206 lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
1209 lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
1210 by (atomize (full), induct n) auto
1212 lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
1215 apply (simp add: nth_Cons split: nat.split)
1218 lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
1221 lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
1222 by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
1224 lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
1227 lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
1228 by (simp add: set_replicate_conv_if split: split_if_asm)
1231 subsection {* Lexcicographic orderings on lists *}
1233 lemma wf_lexn: "wf r ==> wf (lexn r n)"
1234 apply (induct_tac n)
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)
1245 "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
1248 lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
1249 apply (unfold lex_def)
1251 apply (blast intro: wf_lexn)
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)
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)
1266 apply (simp add: image_Collect lex_prod_def)
1269 apply (rename_tac a xys x xs' y ys')
1270 apply (rule_tac x = "a # xys" in exI)
1272 apply (case_tac xys)
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)
1283 lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)"
1284 by (unfold lexico_def) blast
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)
1291 lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
1292 by (simp add: lex_conv)
1294 lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
1295 by (simp add:lex_conv)
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)
1302 prefer 2 apply (blast intro: Cons_eq_appendI)
1304 apply (case_tac xys)
1311 subsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
1313 lemma sublist_empty [simp]: "sublist xs {} = []"
1314 by (auto simp add: sublist_def)
1316 lemma sublist_nil [simp]: "sublist [] A = []"
1317 by (auto simp add: sublist_def)
1319 lemma sublist_shift_lemma:
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)
1324 lemma sublist_append:
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)
1329 apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
1330 apply (simp add: add_commute)
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)
1340 lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
1341 by (simp add: sublist_Cons)
1343 lemma sublist_upt_eq_take [simp]: "sublist l {..n(} = take n l"
1344 apply (induct l rule: rev_induct)
1346 apply (simp split: nat_diff_split add: sublist_append)
1351 "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
1352 by (cases n) simp_all
1355 "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
1356 by (cases n) simp_all
1358 lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
1359 by (cases n) simp_all
1361 lemmas [of "number_of v", standard, simp] =
1362 take_Cons' drop_Cons' nth_Cons'