src/HOL/Enum.thy
 author wenzelm Sun Nov 02 18:21:45 2014 +0100 (2014-11-02) changeset 58889 5b7a9633cfa8 parent 58710 7216a10d69ba child 59582 0fbed69ff081 permissions -rw-r--r--
1 (* Author: Florian Haftmann, TU Muenchen *)
3 section {* Finite types as explicit enumerations *}
5 theory Enum
6 imports Map Groups_List
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
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 use in Quickcheck *}
496 datatype (plugins only: code "quickcheck" extraction) finite_1 =
497   a\<^sub>1
499 notation (output) a\<^sub>1  ("a\<^sub>1")
501 lemma UNIV_finite_1:
502   "UNIV = {a\<^sub>1}"
503   by (auto intro: finite_1.exhaust)
505 instantiation finite_1 :: enum
506 begin
508 definition
509   "enum = [a\<^sub>1]"
511 definition
512   "enum_all P = P a\<^sub>1"
514 definition
515   "enum_ex P = P a\<^sub>1"
517 instance proof
518 qed (simp_all only: enum_finite_1_def enum_all_finite_1_def enum_ex_finite_1_def UNIV_finite_1, simp_all)
520 end
522 instantiation finite_1 :: linorder
523 begin
525 definition less_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
526 where
527   "x < (y :: finite_1) \<longleftrightarrow> False"
529 definition less_eq_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool"
530 where
531   "x \<le> (y :: finite_1) \<longleftrightarrow> True"
533 instance
534 apply (intro_classes)
535 apply (auto simp add: less_finite_1_def less_eq_finite_1_def)
536 apply (metis finite_1.exhaust)
537 done
539 end
541 instance finite_1 :: "{dense_linorder, wellorder}"
542 by intro_classes (simp_all add: less_finite_1_def)
544 instantiation finite_1 :: complete_lattice
545 begin
547 definition [simp]: "Inf = (\<lambda>_. a\<^sub>1)"
548 definition [simp]: "Sup = (\<lambda>_. a\<^sub>1)"
549 definition [simp]: "bot = a\<^sub>1"
550 definition [simp]: "top = a\<^sub>1"
551 definition [simp]: "inf = (\<lambda>_ _. a\<^sub>1)"
552 definition [simp]: "sup = (\<lambda>_ _. a\<^sub>1)"
554 instance by intro_classes(simp_all add: less_eq_finite_1_def)
555 end
557 instance finite_1 :: complete_distrib_lattice
558 by intro_classes(simp_all add: INF_def SUP_def)
560 instance finite_1 :: complete_linorder ..
562 lemma finite_1_eq: "x = a\<^sub>1"
563 by(cases x) simp
565 simproc_setup finite_1_eq ("x::finite_1") = {*
566   fn _ => fn _ => fn ct => case term_of ct of
567     Const (@{const_name a\<^sub>1}, _) => NONE
568   | _ => SOME (mk_meta_eq @{thm finite_1_eq})
569 *}
571 instantiation finite_1 :: complete_boolean_algebra begin
572 definition [simp]: "op - = (\<lambda>_ _. a\<^sub>1)"
573 definition [simp]: "uminus = (\<lambda>_. a\<^sub>1)"
574 instance by intro_classes simp_all
575 end
577 instantiation finite_1 ::
578   "{linordered_ring_strict, linordered_comm_semiring_strict, ordered_comm_ring,
579     ordered_cancel_comm_monoid_diff, comm_monoid_mult, ordered_ring_abs,
580     one, Divides.div, sgn_if, inverse}"
581 begin
582 definition [simp]: "Groups.zero = a\<^sub>1"
583 definition [simp]: "Groups.one = a\<^sub>1"
584 definition [simp]: "op + = (\<lambda>_ _. a\<^sub>1)"
585 definition [simp]: "op * = (\<lambda>_ _. a\<^sub>1)"
586 definition [simp]: "op div = (\<lambda>_ _. a\<^sub>1)"
587 definition [simp]: "op mod = (\<lambda>_ _. a\<^sub>1)"
588 definition [simp]: "abs = (\<lambda>_. a\<^sub>1)"
589 definition [simp]: "sgn = (\<lambda>_. a\<^sub>1)"
590 definition [simp]: "inverse = (\<lambda>_. a\<^sub>1)"
591 definition [simp]: "op / = (\<lambda>_ _. a\<^sub>1)"
593 instance by intro_classes(simp_all add: less_finite_1_def)
594 end
596 declare [[simproc del: finite_1_eq]]
597 hide_const (open) a\<^sub>1
599 datatype (plugins only: code "quickcheck" extraction) finite_2 =
600   a\<^sub>1 | a\<^sub>2
602 notation (output) a\<^sub>1  ("a\<^sub>1")
603 notation (output) a\<^sub>2  ("a\<^sub>2")
605 lemma UNIV_finite_2:
606   "UNIV = {a\<^sub>1, a\<^sub>2}"
607   by (auto intro: finite_2.exhaust)
609 instantiation finite_2 :: enum
610 begin
612 definition
613   "enum = [a\<^sub>1, a\<^sub>2]"
615 definition
616   "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2"
618 definition
619   "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2"
621 instance proof
622 qed (simp_all only: enum_finite_2_def enum_all_finite_2_def enum_ex_finite_2_def UNIV_finite_2, simp_all)
624 end
626 instantiation finite_2 :: linorder
627 begin
629 definition less_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
630 where
631   "x < y \<longleftrightarrow> x = a\<^sub>1 \<and> y = a\<^sub>2"
633 definition less_eq_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool"
634 where
635   "x \<le> y \<longleftrightarrow> x = y \<or> x < (y :: finite_2)"
637 instance
638 apply (intro_classes)
639 apply (auto simp add: less_finite_2_def less_eq_finite_2_def)
640 apply (metis finite_2.nchotomy)+
641 done
643 end
645 instance finite_2 :: wellorder
646 by(rule wf_wellorderI)(simp add: less_finite_2_def, intro_classes)
648 instantiation finite_2 :: complete_lattice
649 begin
651 definition "\<Sqinter>A = (if a\<^sub>1 \<in> A then a\<^sub>1 else a\<^sub>2)"
652 definition "\<Squnion>A = (if a\<^sub>2 \<in> A then a\<^sub>2 else a\<^sub>1)"
653 definition [simp]: "bot = a\<^sub>1"
654 definition [simp]: "top = a\<^sub>2"
655 definition "x \<sqinter> y = (if x = a\<^sub>1 \<or> y = a\<^sub>1 then a\<^sub>1 else a\<^sub>2)"
656 definition "x \<squnion> y = (if x = a\<^sub>2 \<or> y = a\<^sub>2 then a\<^sub>2 else a\<^sub>1)"
658 lemma neq_finite_2_a\<^sub>1_iff [simp]: "x \<noteq> a\<^sub>1 \<longleftrightarrow> x = a\<^sub>2"
659 by(cases x) simp_all
661 lemma neq_finite_2_a\<^sub>1_iff' [simp]: "a\<^sub>1 \<noteq> x \<longleftrightarrow> x = a\<^sub>2"
662 by(cases x) simp_all
664 lemma neq_finite_2_a\<^sub>2_iff [simp]: "x \<noteq> a\<^sub>2 \<longleftrightarrow> x = a\<^sub>1"
665 by(cases x) simp_all
667 lemma neq_finite_2_a\<^sub>2_iff' [simp]: "a\<^sub>2 \<noteq> x \<longleftrightarrow> x = a\<^sub>1"
668 by(cases x) simp_all
670 instance
671 proof
672   fix x :: finite_2 and A
673   assume "x \<in> A"
674   then show "\<Sqinter>A \<le> x" "x \<le> \<Squnion>A"
675     by(case_tac [!] x)(auto simp add: less_eq_finite_2_def less_finite_2_def Inf_finite_2_def Sup_finite_2_def)
676 qed(auto simp add: less_eq_finite_2_def less_finite_2_def inf_finite_2_def sup_finite_2_def Inf_finite_2_def Sup_finite_2_def)
677 end
679 instance finite_2 :: complete_distrib_lattice
680 by(intro_classes)(auto simp add: INF_def SUP_def sup_finite_2_def inf_finite_2_def Inf_finite_2_def Sup_finite_2_def)
682 instance finite_2 :: complete_linorder ..
684 instantiation finite_2 :: "{field_inverse_zero, abs_if, ring_div, sgn_if, semiring_div}" begin
685 definition [simp]: "0 = a\<^sub>1"
686 definition [simp]: "1 = a\<^sub>2"
687 definition "x + y = (case (x, y) of (a\<^sub>1, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>1 | _ \<Rightarrow> a\<^sub>2)"
688 definition "uminus = (\<lambda>x :: finite_2. x)"
689 definition "op - = (op + :: finite_2 \<Rightarrow> _)"
690 definition "x * y = (case (x, y) of (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> a\<^sub>1)"
691 definition "inverse = (\<lambda>x :: finite_2. x)"
692 definition "op / = (op * :: finite_2 \<Rightarrow> _)"
693 definition "abs = (\<lambda>x :: finite_2. x)"
694 definition "op div = (op / :: finite_2 \<Rightarrow> _)"
695 definition "x mod y = (case (x, y) of (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> a\<^sub>1)"
696 definition "sgn = (\<lambda>x :: finite_2. x)"
697 instance
698 by intro_classes
699   (simp_all add: plus_finite_2_def uminus_finite_2_def minus_finite_2_def times_finite_2_def
700        inverse_finite_2_def divide_finite_2_def abs_finite_2_def div_finite_2_def mod_finite_2_def sgn_finite_2_def
701      split: finite_2.splits)
702 end
704 lemma two_finite_2 [simp]:
705   "2 = a\<^sub>1"
706   by (simp add: numeral.simps plus_finite_2_def)
708 hide_const (open) a\<^sub>1 a\<^sub>2
710 datatype (plugins only: code "quickcheck" extraction) finite_3 =
711   a\<^sub>1 | a\<^sub>2 | a\<^sub>3
713 notation (output) a\<^sub>1  ("a\<^sub>1")
714 notation (output) a\<^sub>2  ("a\<^sub>2")
715 notation (output) a\<^sub>3  ("a\<^sub>3")
717 lemma UNIV_finite_3:
718   "UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3}"
719   by (auto intro: finite_3.exhaust)
721 instantiation finite_3 :: enum
722 begin
724 definition
725   "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3]"
727 definition
728   "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3"
730 definition
731   "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3"
733 instance proof
734 qed (simp_all only: enum_finite_3_def enum_all_finite_3_def enum_ex_finite_3_def UNIV_finite_3, simp_all)
736 end
738 instantiation finite_3 :: linorder
739 begin
741 definition less_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
742 where
743   "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)"
745 definition less_eq_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool"
746 where
747   "x \<le> y \<longleftrightarrow> x = y \<or> x < (y :: finite_3)"
749 instance proof (intro_classes)
750 qed (auto simp add: less_finite_3_def less_eq_finite_3_def split: finite_3.split_asm)
752 end
754 instance finite_3 :: wellorder
755 proof(rule wf_wellorderI)
756   have "inv_image less_than (case_finite_3 0 1 2) = {(x, y). x < y}"
757     by(auto simp add: less_finite_3_def split: finite_3.splits)
758   from this[symmetric] show "wf \<dots>" by simp
759 qed intro_classes
761 instantiation finite_3 :: complete_lattice
762 begin
764 definition "\<Sqinter>A = (if a\<^sub>1 \<in> A then a\<^sub>1 else if a\<^sub>2 \<in> A then a\<^sub>2 else a\<^sub>3)"
765 definition "\<Squnion>A = (if a\<^sub>3 \<in> A then a\<^sub>3 else if a\<^sub>2 \<in> A then a\<^sub>2 else a\<^sub>1)"
766 definition [simp]: "bot = a\<^sub>1"
767 definition [simp]: "top = a\<^sub>3"
768 definition [simp]: "inf = (min :: finite_3 \<Rightarrow> _)"
769 definition [simp]: "sup = (max :: finite_3 \<Rightarrow> _)"
771 instance
772 proof
773   fix x :: finite_3 and A
774   assume "x \<in> A"
775   then show "\<Sqinter>A \<le> x" "x \<le> \<Squnion>A"
776     by(case_tac [!] x)(auto simp add: Inf_finite_3_def Sup_finite_3_def less_eq_finite_3_def less_finite_3_def)
777 next
778   fix A and z :: finite_3
779   assume "\<And>x. x \<in> A \<Longrightarrow> z \<le> x"
780   then show "z \<le> \<Sqinter>A"
781     by(cases z)(auto simp add: Inf_finite_3_def less_eq_finite_3_def less_finite_3_def)
782 next
783   fix A and z :: finite_3
784   assume *: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z"
785   show "\<Squnion>A \<le> z"
786     by(auto simp add: Sup_finite_3_def less_eq_finite_3_def less_finite_3_def dest: *)
787 qed(auto simp add: Inf_finite_3_def Sup_finite_3_def)
788 end
790 instance finite_3 :: complete_distrib_lattice
791 proof
792   fix a :: finite_3 and B
793   show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
794   proof(cases a "\<Sqinter>B" rule: finite_3.exhaust[case_product finite_3.exhaust])
795     case a\<^sub>2_a\<^sub>3
796     then have "\<And>x. x \<in> B \<Longrightarrow> x = a\<^sub>3"
797       by(case_tac x)(auto simp add: Inf_finite_3_def split: split_if_asm)
798     then show ?thesis using a\<^sub>2_a\<^sub>3
799       by(auto simp add: INF_def Inf_finite_3_def max_def less_eq_finite_3_def less_finite_3_def split: split_if_asm)
800   qed(auto simp add: INF_def Inf_finite_3_def max_def less_finite_3_def less_eq_finite_3_def split: split_if_asm)
801   show "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
802     by(cases a "\<Squnion>B" rule: finite_3.exhaust[case_product finite_3.exhaust])
803       (auto simp add: SUP_def Sup_finite_3_def min_def less_finite_3_def less_eq_finite_3_def split: split_if_asm)
804 qed
806 instance finite_3 :: complete_linorder ..
808 instantiation finite_3 :: "{field_inverse_zero, abs_if, ring_div, semiring_div, sgn_if}" begin
809 definition [simp]: "0 = a\<^sub>1"
810 definition [simp]: "1 = a\<^sub>2"
811 definition
812   "x + y = (case (x, y) of
813      (a\<^sub>1, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>1
814    | (a\<^sub>1, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>3) \<Rightarrow> a\<^sub>2
815    | _ \<Rightarrow> a\<^sub>3)"
816 definition "- x = (case x of a\<^sub>1 \<Rightarrow> a\<^sub>1 | a\<^sub>2 \<Rightarrow> a\<^sub>3 | a\<^sub>3 \<Rightarrow> a\<^sub>2)"
817 definition "x - y = x + (- y :: finite_3)"
818 definition "x * y = (case (x, y) of (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>3) \<Rightarrow> a\<^sub>2 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>3 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>3 | _ \<Rightarrow> a\<^sub>1)"
819 definition "inverse = (\<lambda>x :: finite_3. x)"
820 definition "x / y = x * inverse (y :: finite_3)"
821 definition "abs = (\<lambda>x :: finite_3. x)"
822 definition "op div = (op / :: finite_3 \<Rightarrow> _)"
823 definition "x mod y = (case (x, y) of (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>1) \<Rightarrow> a\<^sub>3 | _ \<Rightarrow> a\<^sub>1)"
824 definition "sgn = (\<lambda>x. case x of a\<^sub>1 \<Rightarrow> a\<^sub>1 | _ \<Rightarrow> a\<^sub>2)"
825 instance
826 by intro_classes
827   (simp_all add: plus_finite_3_def uminus_finite_3_def minus_finite_3_def times_finite_3_def
828        inverse_finite_3_def divide_finite_3_def abs_finite_3_def div_finite_3_def mod_finite_3_def sgn_finite_3_def
829        less_finite_3_def
830      split: finite_3.splits)
831 end
835 hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3
837 datatype (plugins only: code "quickcheck" extraction) finite_4 =
838   a\<^sub>1 | a\<^sub>2 | a\<^sub>3 | a\<^sub>4
840 notation (output) a\<^sub>1  ("a\<^sub>1")
841 notation (output) a\<^sub>2  ("a\<^sub>2")
842 notation (output) a\<^sub>3  ("a\<^sub>3")
843 notation (output) a\<^sub>4  ("a\<^sub>4")
845 lemma UNIV_finite_4:
846   "UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4}"
847   by (auto intro: finite_4.exhaust)
849 instantiation finite_4 :: enum
850 begin
852 definition
853   "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4]"
855 definition
856   "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3 \<and> P a\<^sub>4"
858 definition
859   "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3 \<or> P a\<^sub>4"
861 instance proof
862 qed (simp_all only: enum_finite_4_def enum_all_finite_4_def enum_ex_finite_4_def UNIV_finite_4, simp_all)
864 end
866 instantiation finite_4 :: complete_lattice begin
868 text {* @{term a\<^sub>1} \$<\$ @{term a\<^sub>2},@{term a\<^sub>3} \$<\$ @{term a\<^sub>4},
869   but @{term a\<^sub>2} and @{term a\<^sub>3} are incomparable. *}
871 definition
872   "x < y \<longleftrightarrow> (case (x, y) of
873      (a\<^sub>1, a\<^sub>1) \<Rightarrow> False | (a\<^sub>1, _) \<Rightarrow> True
874    |  (a\<^sub>2, a\<^sub>4) \<Rightarrow> True
875    |  (a\<^sub>3, a\<^sub>4) \<Rightarrow> True  | _ \<Rightarrow> False)"
877 definition
878   "x \<le> y \<longleftrightarrow> (case (x, y) of
879      (a\<^sub>1, _) \<Rightarrow> True
880    | (a\<^sub>2, a\<^sub>2) \<Rightarrow> True | (a\<^sub>2, a\<^sub>4) \<Rightarrow> True
881    | (a\<^sub>3, a\<^sub>3) \<Rightarrow> True | (a\<^sub>3, a\<^sub>4) \<Rightarrow> True
882    | (a\<^sub>4, a\<^sub>4) \<Rightarrow> True | _ \<Rightarrow> False)"
884 definition
885   "\<Sqinter>A = (if a\<^sub>1 \<in> A \<or> a\<^sub>2 \<in> A \<and> a\<^sub>3 \<in> A then a\<^sub>1 else if a\<^sub>2 \<in> A then a\<^sub>2 else if a\<^sub>3 \<in> A then a\<^sub>3 else a\<^sub>4)"
886 definition
887   "\<Squnion>A = (if a\<^sub>4 \<in> A \<or> a\<^sub>2 \<in> A \<and> a\<^sub>3 \<in> A then a\<^sub>4 else if a\<^sub>2 \<in> A then a\<^sub>2 else if a\<^sub>3 \<in> A then a\<^sub>3 else a\<^sub>1)"
888 definition [simp]: "bot = a\<^sub>1"
889 definition [simp]: "top = a\<^sub>4"
890 definition
891   "x \<sqinter> y = (case (x, y) of
892      (a\<^sub>1, _) \<Rightarrow> a\<^sub>1 | (_, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>1
893    | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
894    | (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
895    | _ \<Rightarrow> a\<^sub>4)"
896 definition
897   "x \<squnion> y = (case (x, y) of
898      (a\<^sub>4, _) \<Rightarrow> a\<^sub>4 | (_, a\<^sub>4) \<Rightarrow> a\<^sub>4 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>4 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>4
899   | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
900   | (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
901   | _ \<Rightarrow> a\<^sub>1)"
903 instance
904 proof
905   fix A and z :: finite_4
906   assume *: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z"
907   show "\<Squnion>A \<le> z"
908     by(auto simp add: Sup_finite_4_def less_eq_finite_4_def dest!: * split: finite_4.splits)
909 next
910   fix A and z :: finite_4
911   assume *: "\<And>x. x \<in> A \<Longrightarrow> z \<le> x"
912   show "z \<le> \<Sqinter>A"
913     by(auto simp add: Inf_finite_4_def less_eq_finite_4_def dest!: * split: finite_4.splits)
914 qed(auto simp add: less_finite_4_def less_eq_finite_4_def Inf_finite_4_def Sup_finite_4_def inf_finite_4_def sup_finite_4_def split: finite_4.splits)
916 end
918 instance finite_4 :: complete_distrib_lattice
919 proof
920   fix a :: finite_4 and B
921   show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
922     by(cases a "\<Sqinter>B" rule: finite_4.exhaust[case_product finite_4.exhaust])
923       (auto simp add: sup_finite_4_def Inf_finite_4_def INF_def split: finite_4.splits split_if_asm)
924   show "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
925     by(cases a "\<Squnion>B" rule: finite_4.exhaust[case_product finite_4.exhaust])
926       (auto simp add: inf_finite_4_def Sup_finite_4_def SUP_def split: finite_4.splits split_if_asm)
927 qed
929 instantiation finite_4 :: complete_boolean_algebra begin
930 definition "- x = (case x of a\<^sub>1 \<Rightarrow> a\<^sub>4 | a\<^sub>2 \<Rightarrow> a\<^sub>3 | a\<^sub>3 \<Rightarrow> a\<^sub>2 | a\<^sub>4 \<Rightarrow> a\<^sub>1)"
931 definition "x - y = x \<sqinter> - (y :: finite_4)"
932 instance
933 by intro_classes
934   (simp_all add: inf_finite_4_def sup_finite_4_def uminus_finite_4_def minus_finite_4_def split: finite_4.splits)
935 end
937 hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4
939 datatype (plugins only: code "quickcheck" extraction) finite_5 =
940   a\<^sub>1 | a\<^sub>2 | a\<^sub>3 | a\<^sub>4 | a\<^sub>5
942 notation (output) a\<^sub>1  ("a\<^sub>1")
943 notation (output) a\<^sub>2  ("a\<^sub>2")
944 notation (output) a\<^sub>3  ("a\<^sub>3")
945 notation (output) a\<^sub>4  ("a\<^sub>4")
946 notation (output) a\<^sub>5  ("a\<^sub>5")
948 lemma UNIV_finite_5:
949   "UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5}"
950   by (auto intro: finite_5.exhaust)
952 instantiation finite_5 :: enum
953 begin
955 definition
956   "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5]"
958 definition
959   "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"
961 definition
962   "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"
964 instance proof
965 qed (simp_all only: enum_finite_5_def enum_all_finite_5_def enum_ex_finite_5_def UNIV_finite_5, simp_all)
967 end
969 instantiation finite_5 :: complete_lattice
970 begin
972 text {* The non-distributive pentagon lattice \$N_5\$ *}
974 definition
975   "x < y \<longleftrightarrow> (case (x, y) of
976      (a\<^sub>1, a\<^sub>1) \<Rightarrow> False | (a\<^sub>1, _) \<Rightarrow> True
977    | (a\<^sub>2, a\<^sub>3) \<Rightarrow> True  | (a\<^sub>2, a\<^sub>5) \<Rightarrow> True
978    | (a\<^sub>3, a\<^sub>5) \<Rightarrow> True
979    | (a\<^sub>4, a\<^sub>5) \<Rightarrow> True  | _ \<Rightarrow> False)"
981 definition
982   "x \<le> y \<longleftrightarrow> (case (x, y) of
983      (a\<^sub>1, _) \<Rightarrow> True
984    | (a\<^sub>2, a\<^sub>2) \<Rightarrow> True | (a\<^sub>2, a\<^sub>3) \<Rightarrow> True | (a\<^sub>2, a\<^sub>5) \<Rightarrow> True
985    | (a\<^sub>3, a\<^sub>3) \<Rightarrow> True | (a\<^sub>3, a\<^sub>5) \<Rightarrow> True
986    | (a\<^sub>4, a\<^sub>4) \<Rightarrow> True | (a\<^sub>4, a\<^sub>5) \<Rightarrow> True
987    | (a\<^sub>5, a\<^sub>5) \<Rightarrow> True | _ \<Rightarrow> False)"
989 definition
990   "\<Sqinter>A =
991   (if a\<^sub>1 \<in> A \<or> a\<^sub>4 \<in> A \<and> (a\<^sub>2 \<in> A \<or> a\<^sub>3 \<in> A) then a\<^sub>1
992    else if a\<^sub>2 \<in> A then a\<^sub>2
993    else if a\<^sub>3 \<in> A then a\<^sub>3
994    else if a\<^sub>4 \<in> A then a\<^sub>4
995    else a\<^sub>5)"
996 definition
997   "\<Squnion>A =
998   (if a\<^sub>5 \<in> A \<or> a\<^sub>4 \<in> A \<and> (a\<^sub>2 \<in> A \<or> a\<^sub>3 \<in> A) then a\<^sub>5
999    else if a\<^sub>3 \<in> A then a\<^sub>3
1000    else if a\<^sub>2 \<in> A then a\<^sub>2
1001    else if a\<^sub>4 \<in> A then a\<^sub>4
1002    else a\<^sub>1)"
1003 definition [simp]: "bot = a\<^sub>1"
1004 definition [simp]: "top = a\<^sub>5"
1005 definition
1006   "x \<sqinter> y = (case (x, y) of
1007      (a\<^sub>1, _) \<Rightarrow> a\<^sub>1 | (_, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>4) \<Rightarrow> a\<^sub>1 | (a\<^sub>4, a\<^sub>2) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>4) \<Rightarrow> a\<^sub>1 | (a\<^sub>4, a\<^sub>3) \<Rightarrow> a\<^sub>1
1008    | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
1009    | (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
1010    | (a\<^sub>4, _) \<Rightarrow> a\<^sub>4 | (_, a\<^sub>4) \<Rightarrow> a\<^sub>4
1011    | _ \<Rightarrow> a\<^sub>5)"
1012 definition
1013   "x \<squnion> y = (case (x, y) of
1014      (a\<^sub>5, _) \<Rightarrow> a\<^sub>5 | (_, a\<^sub>5) \<Rightarrow> a\<^sub>5 | (a\<^sub>2, a\<^sub>4) \<Rightarrow> a\<^sub>5 | (a\<^sub>4, a\<^sub>2) \<Rightarrow> a\<^sub>5 | (a\<^sub>3, a\<^sub>4) \<Rightarrow> a\<^sub>5 | (a\<^sub>4, a\<^sub>3) \<Rightarrow> a\<^sub>5
1015    | (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3
1016    | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2
1017    | (a\<^sub>4, _) \<Rightarrow> a\<^sub>4 | (_, a\<^sub>4) \<Rightarrow> a\<^sub>4
1018    | _ \<Rightarrow> a\<^sub>1)"
1020 instance
1021 proof intro_classes
1022   fix A and z :: finite_5
1023   assume *: "\<And>x. x \<in> A \<Longrightarrow> z \<le> x"
1024   show "z \<le> \<Sqinter>A"
1025     by(auto simp add: less_eq_finite_5_def Inf_finite_5_def split: finite_5.splits split_if_asm dest!: *)
1026 next
1027   fix A and z :: finite_5
1028   assume *: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z"
1029   show "\<Squnion>A \<le> z"
1030     by(auto simp add: less_eq_finite_5_def Sup_finite_5_def split: finite_5.splits split_if_asm dest!: *)
1031 qed(auto simp add: less_eq_finite_5_def less_finite_5_def inf_finite_5_def sup_finite_5_def Inf_finite_5_def Sup_finite_5_def split: finite_5.splits split_if_asm)
1033 end
1035 hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4 a\<^sub>5
1038 subsection {* Closing up *}
1040 hide_type (open) finite_1 finite_2 finite_3 finite_4 finite_5
1041 hide_const (open) enum enum_all enum_ex all_n_lists ex_n_lists ntrancl
1043 end