src/HOL/Enum.thy
 author blanchet Mon Jan 20 23:07:23 2014 +0100 (2014-01-20) changeset 55088 57c82e01022b parent 54890 cb892d835803 child 57247 8191ccf6a1bd permissions -rw-r--r--
moved 'bacc' back to 'Enum' (cf. 744934b818c7) -- reduces baggage loaded by 'Hilbert_Choice'
1 (* Author: Florian Haftmann, TU Muenchen *)
3 header {* Finite types as explicit enumerations *}
5 theory Enum
6 imports Map
7 begin
9 subsection {* Class @{text enum} *}
11 class enum =
12   fixes enum :: "'a list"
13   fixes enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
14   fixes enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
15   assumes UNIV_enum: "UNIV = set enum"
16     and enum_distinct: "distinct enum"
17   assumes enum_all_UNIV: "enum_all P \<longleftrightarrow> Ball UNIV P"
18   assumes enum_ex_UNIV: "enum_ex P \<longleftrightarrow> Bex UNIV P"
19    -- {* tailored towards simple instantiation *}
20 begin
22 subclass finite proof
25 lemma enum_UNIV:
26   "set enum = UNIV"
27   by (simp only: UNIV_enum)
29 lemma in_enum: "x \<in> set enum"
32 lemma enum_eq_I:
33   assumes "\<And>x. x \<in> set xs"
34   shows "set enum = set xs"
35 proof -
36   from assms UNIV_eq_I have "UNIV = set xs" by auto
37   with enum_UNIV show ?thesis by simp
38 qed
40 lemma card_UNIV_length_enum:
41   "card (UNIV :: 'a set) = length enum"
42   by (simp add: UNIV_enum distinct_card enum_distinct)
44 lemma enum_all [simp]:
45   "enum_all = HOL.All"
46   by (simp add: fun_eq_iff enum_all_UNIV)
48 lemma enum_ex [simp]:
49   "enum_ex = HOL.Ex"
50   by (simp add: fun_eq_iff enum_ex_UNIV)
52 end
55 subsection {* Implementations using @{class enum} *}
57 subsubsection {* Unbounded operations and quantifiers *}
59 lemma Collect_code [code]:
60   "Collect P = set (filter P enum)"
63 lemma vimage_code [code]:
64   "f -` B = set (filter (%x. f x : B) enum_class.enum)"
65   unfolding vimage_def Collect_code ..
67 definition card_UNIV :: "'a itself \<Rightarrow> nat"
68 where
69   [code del]: "card_UNIV TYPE('a) = card (UNIV :: 'a set)"
71 lemma [code]:
72   "card_UNIV TYPE('a :: enum) = card (set (Enum.enum :: 'a list))"
73   by (simp only: card_UNIV_def enum_UNIV)
75 lemma all_code [code]: "(\<forall>x. P x) \<longleftrightarrow> enum_all P"
76   by simp
78 lemma exists_code [code]: "(\<exists>x. P x) \<longleftrightarrow> enum_ex P"
79   by simp
81 lemma exists1_code [code]: "(\<exists>!x. P x) \<longleftrightarrow> list_ex1 P enum"
82   by (auto simp add: list_ex1_iff enum_UNIV)
85 subsubsection {* An executable choice operator *}
87 definition
88   [code del]: "enum_the = The"
90 lemma [code]:
91   "The P = (case filter P enum of [x] => x | _ => enum_the P)"
92 proof -
93   {
94     fix a
95     assume filter_enum: "filter P enum = [a]"
96     have "The P = a"
97     proof (rule the_equality)
98       fix x
99       assume "P x"
100       show "x = a"
101       proof (rule ccontr)
102         assume "x \<noteq> a"
103         from filter_enum obtain us vs
104           where enum_eq: "enum = us @ [a] @ vs"
105           and "\<forall> x \<in> set us. \<not> P x"
106           and "\<forall> x \<in> set vs. \<not> P x"
107           and "P a"
108           by (auto simp add: filter_eq_Cons_iff) (simp only: filter_empty_conv[symmetric])
109         with `P x` in_enum[of x, unfolded enum_eq] `x \<noteq> a` show "False" by auto
110       qed
111     next
112       from filter_enum show "P a" by (auto simp add: filter_eq_Cons_iff)
113     qed
114   }
115   from this show ?thesis
116     unfolding enum_the_def by (auto split: list.split)
117 qed
119 declare [[code abort: enum_the]]
121 code_printing
122   constant enum_the \<rightharpoonup> (Eval) "(fn '_ => raise Match)"
125 subsubsection {* Equality and order on functions *}
127 instantiation "fun" :: (enum, equal) equal
128 begin
130 definition
131   "HOL.equal f g \<longleftrightarrow> (\<forall>x \<in> set enum. f x = g x)"
133 instance proof
134 qed (simp_all add: equal_fun_def fun_eq_iff enum_UNIV)
136 end
138 lemma [code]:
139   "HOL.equal f g \<longleftrightarrow> enum_all (%x. f x = g x)"
140   by (auto simp add: equal fun_eq_iff)
142 lemma [code nbe]:
143   "HOL.equal (f :: _ \<Rightarrow> _) f \<longleftrightarrow> True"
144   by (fact equal_refl)
146 lemma order_fun [code]:
147   fixes f g :: "'a\<Colon>enum \<Rightarrow> 'b\<Colon>order"
148   shows "f \<le> g \<longleftrightarrow> enum_all (\<lambda>x. f x \<le> g x)"
149     and "f < g \<longleftrightarrow> f \<le> g \<and> enum_ex (\<lambda>x. f x \<noteq> g x)"
150   by (simp_all add: fun_eq_iff le_fun_def order_less_le)
153 subsubsection {* Operations on relations *}
155 lemma [code]:
156   "Id = image (\<lambda>x. (x, x)) (set Enum.enum)"
157   by (auto intro: imageI in_enum)
159 lemma tranclp_unfold [code]:
160   "tranclp r a b \<longleftrightarrow> (a, b) \<in> trancl {(x, y). r x y}"
163 lemma rtranclp_rtrancl_eq [code]:
164   "rtranclp r x y \<longleftrightarrow> (x, y) \<in> rtrancl {(x, y). r x y}"
167 lemma max_ext_eq [code]:
168   "max_ext R = {(X, Y). finite X \<and> finite Y \<and> Y \<noteq> {} \<and> (\<forall>x. x \<in> X \<longrightarrow> (\<exists>xa \<in> Y. (x, xa) \<in> R))}"
169   by (auto simp add: max_ext.simps)
171 lemma max_extp_eq [code]:
172   "max_extp r x y \<longleftrightarrow> (x, y) \<in> max_ext {(x, y). r x y}"
175 lemma mlex_eq [code]:
176   "f <*mlex*> R = {(x, y). f x < f y \<or> (f x \<le> f y \<and> (x, y) \<in> R)}"
177   by (auto simp add: mlex_prod_def)
180 subsubsection {* Bounded accessible part *}
182 primrec bacc :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> 'a set"
183 where
184   "bacc r 0 = {x. \<forall> y. (y, x) \<notin> r}"
185 | "bacc r (Suc n) = (bacc r n \<union> {x. \<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r n})"
187 lemma bacc_subseteq_acc:
188   "bacc r n \<subseteq> Wellfounded.acc r"
189   by (induct n) (auto intro: acc.intros)
191 lemma bacc_mono:
192   "n \<le> m \<Longrightarrow> bacc r n \<subseteq> bacc r m"
193   by (induct rule: dec_induct) auto
195 lemma bacc_upper_bound:
196   "bacc (r :: ('a \<times> 'a) set)  (card (UNIV :: 'a::finite set)) = (\<Union>n. bacc r n)"
197 proof -
198   have "mono (bacc r)" unfolding mono_def by (simp add: bacc_mono)
199   moreover have "\<forall>n. bacc r n = bacc r (Suc n) \<longrightarrow> bacc r (Suc n) = bacc r (Suc (Suc n))" by auto
200   moreover have "finite (range (bacc r))" by auto
201   ultimately show ?thesis
202    by (intro finite_mono_strict_prefix_implies_finite_fixpoint)
203      (auto intro: finite_mono_remains_stable_implies_strict_prefix)
204 qed
206 lemma acc_subseteq_bacc:
207   assumes "finite r"
208   shows "Wellfounded.acc r \<subseteq> (\<Union>n. bacc r n)"
209 proof
210   fix x
211   assume "x : Wellfounded.acc r"
212   then have "\<exists> n. x : bacc r n"
213   proof (induct x arbitrary: rule: acc.induct)
214     case (accI x)
215     then have "\<forall>y. \<exists> n. (y, x) \<in> r --> y : bacc r n" by simp
216     from choice[OF this] obtain n where n: "\<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r (n y)" ..
217     obtain n where "\<And>y. (y, x) : r \<Longrightarrow> y : bacc r n"
218     proof
219       fix y assume y: "(y, x) : r"
220       with n have "y : bacc r (n y)" by auto
221       moreover have "n y <= Max ((%(y, x). n y) ` r)"
222         using y `finite r` by (auto intro!: Max_ge)
223       note bacc_mono[OF this, of r]
224       ultimately show "y : bacc r (Max ((%(y, x). n y) ` r))" by auto
225     qed
226     then show ?case
227       by (auto simp add: Let_def intro!: exI[of _ "Suc n"])
228   qed
229   then show "x : (UN n. bacc r n)" by auto
230 qed
232 lemma acc_bacc_eq:
233   fixes A :: "('a :: finite \<times> 'a) set"
234   assumes "finite A"
235   shows "Wellfounded.acc A = bacc A (card (UNIV :: 'a set))"
236   using assms by (metis acc_subseteq_bacc bacc_subseteq_acc bacc_upper_bound order_eq_iff)
238 lemma [code]:
239   fixes xs :: "('a::finite \<times> 'a) list"
240   shows "Wellfounded.acc (set xs) = bacc (set xs) (card_UNIV TYPE('a))"
241   by (simp add: card_UNIV_def acc_bacc_eq)
244 subsection {* Default instances for @{class enum} *}
246 lemma map_of_zip_enum_is_Some:
247   assumes "length ys = length (enum \<Colon> 'a\<Colon>enum list)"
248   shows "\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y"
249 proof -
250   from assms have "x \<in> set (enum \<Colon> 'a\<Colon>enum list) \<longleftrightarrow>
251     (\<exists>y. map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x = Some y)"
252     by (auto intro!: map_of_zip_is_Some)
253   then show ?thesis using enum_UNIV by auto
254 qed
256 lemma map_of_zip_enum_inject:
257   fixes xs ys :: "'b\<Colon>enum list"
258   assumes length: "length xs = length (enum \<Colon> 'a\<Colon>enum list)"
259       "length ys = length (enum \<Colon> 'a\<Colon>enum list)"
260     and map_of: "the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) xs) = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys)"
261   shows "xs = ys"
262 proof -
263   have "map_of (zip (enum \<Colon> 'a list) xs) = map_of (zip (enum \<Colon> 'a list) ys)"
264   proof
265     fix x :: 'a
266     from length map_of_zip_enum_is_Some obtain y1 y2
267       where "map_of (zip (enum \<Colon> 'a list) xs) x = Some y1"
268         and "map_of (zip (enum \<Colon> 'a list) ys) x = Some y2" by blast
269     moreover from map_of
270       have "the (map_of (zip (enum \<Colon> 'a\<Colon>enum list) xs) x) = the (map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x)"
271       by (auto dest: fun_cong)
272     ultimately show "map_of (zip (enum \<Colon> 'a\<Colon>enum list) xs) x = map_of (zip (enum \<Colon> 'a\<Colon>enum list) ys) x"
273       by simp
274   qed
275   with length enum_distinct show "xs = ys" by (rule map_of_zip_inject)
276 qed
278 definition all_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"
279 where
280   "all_n_lists P n \<longleftrightarrow> (\<forall>xs \<in> set (List.n_lists n enum). P xs)"
282 lemma [code]:
283   "all_n_lists P n \<longleftrightarrow> (if n = 0 then P [] else enum_all (%x. all_n_lists (%xs. P (x # xs)) (n - 1)))"
284   unfolding all_n_lists_def enum_all
285   by (cases n) (auto simp add: enum_UNIV)
287 definition ex_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"
288 where
289   "ex_n_lists P n \<longleftrightarrow> (\<exists>xs \<in> set (List.n_lists n enum). P xs)"
291 lemma [code]:
292   "ex_n_lists P n \<longleftrightarrow> (if n = 0 then P [] else enum_ex (%x. ex_n_lists (%xs. P (x # xs)) (n - 1)))"
293   unfolding ex_n_lists_def enum_ex
294   by (cases n) (auto simp add: enum_UNIV)
296 instantiation "fun" :: (enum, enum) enum
297 begin
299 definition
300   "enum = map (\<lambda>ys. the o map_of (zip (enum\<Colon>'a list) ys)) (List.n_lists (length (enum\<Colon>'a\<Colon>enum list)) enum)"
302 definition
303   "enum_all P = all_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))"
305 definition
306   "enum_ex P = ex_n_lists (\<lambda>bs. P (the o map_of (zip enum bs))) (length (enum :: 'a list))"
308 instance proof
309   show "UNIV = set (enum \<Colon> ('a \<Rightarrow> 'b) list)"
310   proof (rule UNIV_eq_I)
311     fix f :: "'a \<Rightarrow> 'b"
312     have "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
313       by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
314     then show "f \<in> set enum"
315       by (auto simp add: enum_fun_def set_n_lists intro: in_enum)
316   qed
317 next
318   from map_of_zip_enum_inject
319   show "distinct (enum \<Colon> ('a \<Rightarrow> 'b) list)"
320     by (auto intro!: inj_onI simp add: enum_fun_def
321       distinct_map distinct_n_lists enum_distinct set_n_lists)
322 next
323   fix P
324   show "enum_all (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = Ball UNIV P"
325   proof
326     assume "enum_all P"
327     show "Ball UNIV P"
328     proof
329       fix f :: "'a \<Rightarrow> 'b"
330       have f: "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
331         by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
332       from `enum_all P` have "P (the \<circ> map_of (zip enum (map f enum)))"
333         unfolding enum_all_fun_def all_n_lists_def
335         apply (erule_tac x="map f enum" in allE)
336         apply (auto intro!: in_enum)
337         done
338       from this f show "P f" by auto
339     qed
340   next
341     assume "Ball UNIV P"
342     from this show "enum_all P"
343       unfolding enum_all_fun_def all_n_lists_def by auto
344   qed
345 next
346   fix P
347   show "enum_ex (P :: ('a \<Rightarrow> 'b) \<Rightarrow> bool) = Bex UNIV P"
348   proof
349     assume "enum_ex P"
350     from this show "Bex UNIV P"
351       unfolding enum_ex_fun_def ex_n_lists_def by auto
352   next
353     assume "Bex UNIV P"
354     from this obtain f where "P f" ..
355     have f: "f = the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum))"
356       by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum)
357     from `P f` this have "P (the \<circ> map_of (zip (enum \<Colon> 'a\<Colon>enum list) (map f enum)))"
358       by auto
359     from  this show "enum_ex P"
360       unfolding enum_ex_fun_def ex_n_lists_def
361       apply (auto simp add: set_n_lists)
362       apply (rule_tac x="map f enum" in exI)
363       apply (auto intro!: in_enum)
364       done
365   qed
366 qed
368 end
370 lemma enum_fun_code [code]: "enum = (let enum_a = (enum \<Colon> 'a\<Colon>{enum, equal} list)
371   in map (\<lambda>ys. the o map_of (zip enum_a ys)) (List.n_lists (length enum_a) enum))"
372   by (simp add: enum_fun_def Let_def)
374 lemma enum_all_fun_code [code]:
375   "enum_all P = (let enum_a = (enum :: 'a::{enum, equal} list)
376    in all_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))"
377   by (simp only: enum_all_fun_def Let_def)
379 lemma enum_ex_fun_code [code]:
380   "enum_ex P = (let enum_a = (enum :: 'a::{enum, equal} list)
381    in ex_n_lists (\<lambda>bs. P (the o map_of (zip enum_a bs))) (length enum_a))"
382   by (simp only: enum_ex_fun_def Let_def)
384 instantiation set :: (enum) enum
385 begin
387 definition
388   "enum = map set (sublists enum)"
390 definition
391   "enum_all P \<longleftrightarrow> (\<forall>A\<in>set enum. P (A::'a set))"
393 definition
394   "enum_ex P \<longleftrightarrow> (\<exists>A\<in>set enum. P (A::'a set))"
396 instance proof
397 qed (simp_all add: enum_set_def enum_all_set_def enum_ex_set_def sublists_powset distinct_set_sublists
398   enum_distinct enum_UNIV)
400 end
402 instantiation unit :: enum
403 begin
405 definition
406   "enum = [()]"
408 definition
409   "enum_all P = P ()"
411 definition
412   "enum_ex P = P ()"
414 instance proof
415 qed (auto simp add: enum_unit_def enum_all_unit_def enum_ex_unit_def)
417 end
419 instantiation bool :: enum
420 begin
422 definition
423   "enum = [False, True]"
425 definition
426   "enum_all P \<longleftrightarrow> P False \<and> P True"
428 definition
429   "enum_ex P \<longleftrightarrow> P False \<or> P True"
431 instance proof
432 qed (simp_all only: enum_bool_def enum_all_bool_def enum_ex_bool_def UNIV_bool, simp_all)
434 end
436 instantiation prod :: (enum, enum) enum
437 begin
439 definition
440   "enum = List.product enum enum"
442 definition
443   "enum_all P = enum_all (%x. enum_all (%y. P (x, y)))"
445 definition
446   "enum_ex P = enum_ex (%x. enum_ex (%y. P (x, y)))"
449 instance by default
450   (simp_all add: enum_prod_def product_list_set distinct_product
451     enum_UNIV enum_distinct enum_all_prod_def enum_ex_prod_def)
453 end
455 instantiation sum :: (enum, enum) enum
456 begin
458 definition
459   "enum = map Inl enum @ map Inr enum"
461 definition
462   "enum_all P \<longleftrightarrow> enum_all (\<lambda>x. P (Inl x)) \<and> enum_all (\<lambda>x. P (Inr x))"
464 definition
465   "enum_ex P \<longleftrightarrow> enum_ex (\<lambda>x. P (Inl x)) \<or> enum_ex (\<lambda>x. P (Inr x))"
467 instance proof
468 qed (simp_all only: enum_sum_def enum_all_sum_def enum_ex_sum_def UNIV_sum,
469   auto simp add: enum_UNIV distinct_map enum_distinct)
471 end
473 instantiation option :: (enum) enum
474 begin
476 definition
477   "enum = None # map Some enum"
479 definition
480   "enum_all P \<longleftrightarrow> P None \<and> enum_all (\<lambda>x. P (Some x))"
482 definition
483   "enum_ex P \<longleftrightarrow> P None \<or> enum_ex (\<lambda>x. P (Some x))"
485 instance proof
486 qed (simp_all only: enum_option_def enum_all_option_def enum_ex_option_def UNIV_option_conv,
487   auto simp add: distinct_map enum_UNIV enum_distinct)
489 end
492 subsection {* Small finite types *}
494 text {* We define small finite types for the use in Quickcheck *}
496 datatype finite_1 = a\<^sub>1
498 notation (output) a\<^sub>1  ("a\<^sub>1")
500 lemma UNIV_finite_1:
501   "UNIV = {a\<^sub>1}"
502   by (auto intro: finite_1.exhaust)
504 instantiation finite_1 :: enum
505 begin
507 definition
508   "enum = [a\<^sub>1]"
510 definition
511   "enum_all P = P a\<^sub>1"
513 definition
514   "enum_ex P = P a\<^sub>1"
516 instance proof
517 qed (simp_all only: enum_finite_1_def enum_all_finite_1_def enum_ex_finite_1_def UNIV_finite_1, simp_all)
519 end
521 instantiation finite_1 :: linorder
522 begin
524 definition less_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
525 where
526   "x < (y :: finite_1) \<longleftrightarrow> False"
528 definition less_eq_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
529 where
530   "x \<le> (y :: finite_1) \<longleftrightarrow> True"
532 instance
533 apply (intro_classes)
534 apply (auto simp add: less_finite_1_def less_eq_finite_1_def)
535 apply (metis finite_1.exhaust)
536 done
538 end
540 hide_const (open) a\<^sub>1
542 datatype finite_2 = a\<^sub>1 | a\<^sub>2
544 notation (output) a\<^sub>1  ("a\<^sub>1")
545 notation (output) a\<^sub>2  ("a\<^sub>2")
547 lemma UNIV_finite_2:
548   "UNIV = {a\<^sub>1, a\<^sub>2}"
549   by (auto intro: finite_2.exhaust)
551 instantiation finite_2 :: enum
552 begin
554 definition
555   "enum = [a\<^sub>1, a\<^sub>2]"
557 definition
558   "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2"
560 definition
561   "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2"
563 instance proof
564 qed (simp_all only: enum_finite_2_def enum_all_finite_2_def enum_ex_finite_2_def UNIV_finite_2, simp_all)
566 end
568 instantiation finite_2 :: linorder
569 begin
571 definition less_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
572 where
573   "x < y \<longleftrightarrow> x = a\<^sub>1 \<and> y = a\<^sub>2"
575 definition less_eq_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
576 where
577   "x \<le> y \<longleftrightarrow> x = y \<or> x < (y :: finite_2)"
579 instance
580 apply (intro_classes)
581 apply (auto simp add: less_finite_2_def less_eq_finite_2_def)
582 apply (metis finite_2.nchotomy)+
583 done
585 end
587 hide_const (open) a\<^sub>1 a\<^sub>2
589 datatype finite_3 = a\<^sub>1 | a\<^sub>2 | a\<^sub>3
591 notation (output) a\<^sub>1  ("a\<^sub>1")
592 notation (output) a\<^sub>2  ("a\<^sub>2")
593 notation (output) a\<^sub>3  ("a\<^sub>3")
595 lemma UNIV_finite_3:
596   "UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3}"
597   by (auto intro: finite_3.exhaust)
599 instantiation finite_3 :: enum
600 begin
602 definition
603   "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3]"
605 definition
606   "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3"
608 definition
609   "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3"
611 instance proof
612 qed (simp_all only: enum_finite_3_def enum_all_finite_3_def enum_ex_finite_3_def UNIV_finite_3, simp_all)
614 end
616 instantiation finite_3 :: linorder
617 begin
619 definition less_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
620 where
621   "x < y = (case x of a\<^sub>1 \<Rightarrow> y \<noteq> a\<^sub>1 | a\<^sub>2 \<Rightarrow> y = a\<^sub>3 | a\<^sub>3 \<Rightarrow> False)"
623 definition less_eq_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
624 where
625   "x \<le> y \<longleftrightarrow> x = y \<or> x < (y :: finite_3)"
627 instance proof (intro_classes)
628 qed (auto simp add: less_finite_3_def less_eq_finite_3_def split: finite_3.split_asm)
630 end
632 hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3
634 datatype finite_4 = a\<^sub>1 | a\<^sub>2 | a\<^sub>3 | a\<^sub>4
636 notation (output) a\<^sub>1  ("a\<^sub>1")
637 notation (output) a\<^sub>2  ("a\<^sub>2")
638 notation (output) a\<^sub>3  ("a\<^sub>3")
639 notation (output) a\<^sub>4  ("a\<^sub>4")
641 lemma UNIV_finite_4:
642   "UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4}"
643   by (auto intro: finite_4.exhaust)
645 instantiation finite_4 :: enum
646 begin
648 definition
649   "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4]"
651 definition
652   "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3 \<and> P a\<^sub>4"
654 definition
655   "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3 \<or> P a\<^sub>4"
657 instance proof
658 qed (simp_all only: enum_finite_4_def enum_all_finite_4_def enum_ex_finite_4_def UNIV_finite_4, simp_all)
660 end
662 hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4
665 datatype finite_5 = a\<^sub>1 | a\<^sub>2 | a\<^sub>3 | a\<^sub>4 | a\<^sub>5
667 notation (output) a\<^sub>1  ("a\<^sub>1")
668 notation (output) a\<^sub>2  ("a\<^sub>2")
669 notation (output) a\<^sub>3  ("a\<^sub>3")
670 notation (output) a\<^sub>4  ("a\<^sub>4")
671 notation (output) a\<^sub>5  ("a\<^sub>5")
673 lemma UNIV_finite_5:
674   "UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5}"
675   by (auto intro: finite_5.exhaust)
677 instantiation finite_5 :: enum
678 begin
680 definition
681   "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5]"
683 definition
684   "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3 \<and> P a\<^sub>4 \<and> P a\<^sub>5"
686 definition
687   "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3 \<or> P a\<^sub>4 \<or> P a\<^sub>5"
689 instance proof
690 qed (simp_all only: enum_finite_5_def enum_all_finite_5_def enum_ex_finite_5_def UNIV_finite_5, simp_all)
692 end
694 hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4 a\<^sub>5
697 subsection {* Closing up *}
699 hide_type (open) finite_1 finite_2 finite_3 finite_4 finite_5
700 hide_const (open) enum enum_all enum_ex all_n_lists ex_n_lists ntrancl
702 end