1(* Author: Florian Haftmann, TU Muenchen *) 2 3section \<open>Finite types as explicit enumerations\<close> 4 5theory Enum 6imports Map Groups_List 7begin 8 9subsection \<open>Class \<open>enum\<close>\<close> 10 11class 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 \<comment> \<open>tailored towards simple instantiation\<close> 20begin 21 22subclass finite proof 23qed (simp add: UNIV_enum) 24 25lemma enum_UNIV: 26 "set enum = UNIV" 27 by (simp only: UNIV_enum) 28 29lemma in_enum: "x \<in> set enum" 30 by (simp add: enum_UNIV) 31 32lemma enum_eq_I: 33 assumes "\<And>x. x \<in> set xs" 34 shows "set enum = set xs" 35proof - 36 from assms UNIV_eq_I have "UNIV = set xs" by auto 37 with enum_UNIV show ?thesis by simp 38qed 39 40lemma card_UNIV_length_enum: 41 "card (UNIV :: 'a set) = length enum" 42 by (simp add: UNIV_enum distinct_card enum_distinct) 43 44lemma enum_all [simp]: 45 "enum_all = HOL.All" 46 by (simp add: fun_eq_iff enum_all_UNIV) 47 48lemma enum_ex [simp]: 49 "enum_ex = HOL.Ex" 50 by (simp add: fun_eq_iff enum_ex_UNIV) 51 52end 53 54 55subsection \<open>Implementations using @{class enum}\<close> 56 57subsubsection \<open>Unbounded operations and quantifiers\<close> 58 59lemma Collect_code [code]: 60 "Collect P = set (filter P enum)" 61 by (simp add: enum_UNIV) 62 63lemma vimage_code [code]: 64 "f -` B = set (filter (\<lambda>x. f x \<in> B) enum_class.enum)" 65 unfolding vimage_def Collect_code .. 66 67definition card_UNIV :: "'a itself \<Rightarrow> nat" 68where 69 [code del]: "card_UNIV TYPE('a) = card (UNIV :: 'a set)" 70 71lemma [code]: 72 "card_UNIV TYPE('a :: enum) = card (set (Enum.enum :: 'a list))" 73 by (simp only: card_UNIV_def enum_UNIV) 74 75lemma all_code [code]: "(\<forall>x. P x) \<longleftrightarrow> enum_all P" 76 by simp 77 78lemma exists_code [code]: "(\<exists>x. P x) \<longleftrightarrow> enum_ex P" 79 by simp 80 81lemma exists1_code [code]: "(\<exists>!x. P x) \<longleftrightarrow> list_ex1 P enum" 82 by (auto simp add: list_ex1_iff enum_UNIV) 83 84 85subsubsection \<open>An executable choice operator\<close> 86 87definition 88 [code del]: "enum_the = The" 89 90lemma [code]: 91 "The P = (case filter P enum of [x] \<Rightarrow> x | _ \<Rightarrow> enum_the P)" 92proof - 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 \<open>P x\<close> in_enum[of x, unfolded enum_eq] \<open>x \<noteq> a\<close> 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) 117qed 118 119declare [[code abort: enum_the]] 120 121code_printing 122 constant enum_the \<rightharpoonup> (Eval) "(fn '_ => raise Match)" 123 124 125subsubsection \<open>Equality and order on functions\<close> 126 127instantiation "fun" :: (enum, equal) equal 128begin 129 130definition 131 "HOL.equal f g \<longleftrightarrow> (\<forall>x \<in> set enum. f x = g x)" 132 133instance proof 134qed (simp_all add: equal_fun_def fun_eq_iff enum_UNIV) 135 136end 137 138lemma [code]: 139 "HOL.equal f g \<longleftrightarrow> enum_all (%x. f x = g x)" 140 by (auto simp add: equal fun_eq_iff) 141 142lemma [code nbe]: 143 "HOL.equal (f :: _ \<Rightarrow> _) f \<longleftrightarrow> True" 144 by (fact equal_refl) 145 146lemma order_fun [code]: 147 fixes f g :: "'a::enum \<Rightarrow> 'b::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) 151 152 153subsubsection \<open>Operations on relations\<close> 154 155lemma [code]: 156 "Id = image (\<lambda>x. (x, x)) (set Enum.enum)" 157 by (auto intro: imageI in_enum) 158 159lemma tranclp_unfold [code]: 160 "tranclp r a b \<longleftrightarrow> (a, b) \<in> trancl {(x, y). r x y}" 161 by (simp add: trancl_def) 162 163lemma rtranclp_rtrancl_eq [code]: 164 "rtranclp r x y \<longleftrightarrow> (x, y) \<in> rtrancl {(x, y). r x y}" 165 by (simp add: rtrancl_def) 166 167lemma 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) 170 171lemma max_extp_eq [code]: 172 "max_extp r x y \<longleftrightarrow> (x, y) \<in> max_ext {(x, y). r x y}" 173 by (simp add: max_ext_def) 174 175lemma 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) 178 179 180subsubsection \<open>Bounded accessible part\<close> 181 182primrec bacc :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> 'a set" 183where 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})" 186 187lemma bacc_subseteq_acc: 188 "bacc r n \<subseteq> Wellfounded.acc r" 189 by (induct n) (auto intro: acc.intros) 190 191lemma bacc_mono: 192 "n \<le> m \<Longrightarrow> bacc r n \<subseteq> bacc r m" 193 by (induct rule: dec_induct) auto 194 195lemma bacc_upper_bound: 196 "bacc (r :: ('a \<times> 'a) set) (card (UNIV :: 'a::finite set)) = (\<Union>n. bacc r n)" 197proof - 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) 204qed 205 206lemma acc_subseteq_bacc: 207 assumes "finite r" 208 shows "Wellfounded.acc r \<subseteq> (\<Union>n. bacc r n)" 209proof 210 fix x 211 assume "x \<in> Wellfounded.acc r" 212 then have "\<exists>n. x \<in> 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 \<longrightarrow> y \<in> 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) \<in> r \<Longrightarrow> y \<in> bacc r n" 218 proof 219 fix y assume y: "(y, x) \<in> r" 220 with n have "y \<in> bacc r (n y)" by auto 221 moreover have "n y <= Max ((\<lambda>(y, x). n y) ` r)" 222 using y \<open>finite r\<close> by (auto intro!: Max_ge) 223 note bacc_mono[OF this, of r] 224 ultimately show "y \<in> bacc r (Max ((\<lambda>(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 \<in> (\<Union>n. bacc r n)" by auto 230qed 231 232lemma 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) 237 238lemma [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) 242 243 244subsection \<open>Default instances for @{class enum}\<close> 245 246lemma map_of_zip_enum_is_Some: 247 assumes "length ys = length (enum :: 'a::enum list)" 248 shows "\<exists>y. map_of (zip (enum :: 'a::enum list) ys) x = Some y" 249proof - 250 from assms have "x \<in> set (enum :: 'a::enum list) \<longleftrightarrow> 251 (\<exists>y. map_of (zip (enum :: 'a::enum list) ys) x = Some y)" 252 by (auto intro!: map_of_zip_is_Some) 253 then show ?thesis using enum_UNIV by auto 254qed 255 256lemma map_of_zip_enum_inject: 257 fixes xs ys :: "'b::enum list" 258 assumes length: "length xs = length (enum :: 'a::enum list)" 259 "length ys = length (enum :: 'a::enum list)" 260 and map_of: "the \<circ> map_of (zip (enum :: 'a::enum list) xs) = the \<circ> map_of (zip (enum :: 'a::enum list) ys)" 261 shows "xs = ys" 262proof - 263 have "map_of (zip (enum :: 'a list) xs) = map_of (zip (enum :: '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 :: 'a list) xs) x = Some y1" 268 and "map_of (zip (enum :: 'a list) ys) x = Some y2" by blast 269 moreover from map_of 270 have "the (map_of (zip (enum :: 'a::enum list) xs) x) = the (map_of (zip (enum :: 'a::enum list) ys) x)" 271 by (auto dest: fun_cong) 272 ultimately show "map_of (zip (enum :: 'a::enum list) xs) x = map_of (zip (enum :: 'a::enum list) ys) x" 273 by simp 274 qed 275 with length enum_distinct show "xs = ys" by (rule map_of_zip_inject) 276qed 277 278definition all_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool" 279where 280 "all_n_lists P n \<longleftrightarrow> (\<forall>xs \<in> set (List.n_lists n enum). P xs)" 281 282lemma [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) 286 287definition ex_n_lists :: "(('a :: enum) list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool" 288where 289 "ex_n_lists P n \<longleftrightarrow> (\<exists>xs \<in> set (List.n_lists n enum). P xs)" 290 291lemma [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) 295 296instantiation "fun" :: (enum, enum) enum 297begin 298 299definition 300 "enum = map (\<lambda>ys. the \<circ> map_of (zip (enum::'a list) ys)) (List.n_lists (length (enum::'a::enum list)) enum)" 301 302definition 303 "enum_all P = all_n_lists (\<lambda>bs. P (the \<circ> map_of (zip enum bs))) (length (enum :: 'a list))" 304 305definition 306 "enum_ex P = ex_n_lists (\<lambda>bs. P (the \<circ> map_of (zip enum bs))) (length (enum :: 'a list))" 307 308instance proof 309 show "UNIV = set (enum :: ('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 :: 'a::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 317next 318 from map_of_zip_enum_inject 319 show "distinct (enum :: ('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) 322next 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 :: 'a::enum list) (map f enum))" 331 by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum) 332 from \<open>enum_all P\<close> have "P (the \<circ> map_of (zip enum (map f enum)))" 333 unfolding enum_all_fun_def all_n_lists_def 334 apply (simp add: set_n_lists) 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 345next 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 :: 'a::enum list) (map f enum))" 356 by (auto simp add: map_of_zip_map fun_eq_iff intro: in_enum) 357 from \<open>P f\<close> this have "P (the \<circ> map_of (zip (enum :: 'a::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 366qed 367 368end 369 370lemma enum_fun_code [code]: "enum = (let enum_a = (enum :: 'a::{enum, equal} list) 371 in map (\<lambda>ys. the \<circ> map_of (zip enum_a ys)) (List.n_lists (length enum_a) enum))" 372 by (simp add: enum_fun_def Let_def) 373 374lemma 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 \<circ> map_of (zip enum_a bs))) (length enum_a))" 377 by (simp only: enum_all_fun_def Let_def) 378 379lemma 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 \<circ> map_of (zip enum_a bs))) (length enum_a))" 382 by (simp only: enum_ex_fun_def Let_def) 383 384instantiation set :: (enum) enum 385begin 386 387definition 388 "enum = map set (subseqs enum)" 389 390definition 391 "enum_all P \<longleftrightarrow> (\<forall>A\<in>set enum. P (A::'a set))" 392 393definition 394 "enum_ex P \<longleftrightarrow> (\<exists>A\<in>set enum. P (A::'a set))" 395 396instance proof 397qed (simp_all add: enum_set_def enum_all_set_def enum_ex_set_def subseqs_powset distinct_set_subseqs 398 enum_distinct enum_UNIV) 399 400end 401 402instantiation unit :: enum 403begin 404 405definition 406 "enum = [()]" 407 408definition 409 "enum_all P = P ()" 410 411definition 412 "enum_ex P = P ()" 413 414instance proof 415qed (auto simp add: enum_unit_def enum_all_unit_def enum_ex_unit_def) 416 417end 418 419instantiation bool :: enum 420begin 421 422definition 423 "enum = [False, True]" 424 425definition 426 "enum_all P \<longleftrightarrow> P False \<and> P True" 427 428definition 429 "enum_ex P \<longleftrightarrow> P False \<or> P True" 430 431instance proof 432qed (simp_all only: enum_bool_def enum_all_bool_def enum_ex_bool_def UNIV_bool, simp_all) 433 434end 435 436instantiation prod :: (enum, enum) enum 437begin 438 439definition 440 "enum = List.product enum enum" 441 442definition 443 "enum_all P = enum_all (%x. enum_all (%y. P (x, y)))" 444 445definition 446 "enum_ex P = enum_ex (%x. enum_ex (%y. P (x, y)))" 447 448 449instance 450 by standard 451 (simp_all add: enum_prod_def distinct_product 452 enum_UNIV enum_distinct enum_all_prod_def enum_ex_prod_def) 453 454end 455 456instantiation sum :: (enum, enum) enum 457begin 458 459definition 460 "enum = map Inl enum @ map Inr enum" 461 462definition 463 "enum_all P \<longleftrightarrow> enum_all (\<lambda>x. P (Inl x)) \<and> enum_all (\<lambda>x. P (Inr x))" 464 465definition 466 "enum_ex P \<longleftrightarrow> enum_ex (\<lambda>x. P (Inl x)) \<or> enum_ex (\<lambda>x. P (Inr x))" 467 468instance proof 469qed (simp_all only: enum_sum_def enum_all_sum_def enum_ex_sum_def UNIV_sum, 470 auto simp add: enum_UNIV distinct_map enum_distinct) 471 472end 473 474instantiation option :: (enum) enum 475begin 476 477definition 478 "enum = None # map Some enum" 479 480definition 481 "enum_all P \<longleftrightarrow> P None \<and> enum_all (\<lambda>x. P (Some x))" 482 483definition 484 "enum_ex P \<longleftrightarrow> P None \<or> enum_ex (\<lambda>x. P (Some x))" 485 486instance proof 487qed (simp_all only: enum_option_def enum_all_option_def enum_ex_option_def UNIV_option_conv, 488 auto simp add: distinct_map enum_UNIV enum_distinct) 489 490end 491 492 493subsection \<open>Small finite types\<close> 494 495text \<open>We define small finite types for use in Quickcheck\<close> 496 497datatype (plugins only: code "quickcheck" extraction) finite_1 = 498 a\<^sub>1 499 500notation (output) a\<^sub>1 ("a\<^sub>1") 501 502lemma UNIV_finite_1: 503 "UNIV = {a\<^sub>1}" 504 by (auto intro: finite_1.exhaust) 505 506instantiation finite_1 :: enum 507begin 508 509definition 510 "enum = [a\<^sub>1]" 511 512definition 513 "enum_all P = P a\<^sub>1" 514 515definition 516 "enum_ex P = P a\<^sub>1" 517 518instance proof 519qed (simp_all only: enum_finite_1_def enum_all_finite_1_def enum_ex_finite_1_def UNIV_finite_1, simp_all) 520 521end 522 523instantiation finite_1 :: linorder 524begin 525 526definition less_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool" 527where 528 "x < (y :: finite_1) \<longleftrightarrow> False" 529 530definition less_eq_finite_1 :: "finite_1 \<Rightarrow> finite_1 \<Rightarrow> bool" 531where 532 "x \<le> (y :: finite_1) \<longleftrightarrow> True" 533 534instance 535apply (intro_classes) 536apply (auto simp add: less_finite_1_def less_eq_finite_1_def) 537apply (metis finite_1.exhaust) 538done 539 540end 541 542instance finite_1 :: "{dense_linorder, wellorder}" 543by intro_classes (simp_all add: less_finite_1_def) 544 545instantiation finite_1 :: complete_lattice 546begin 547 548definition [simp]: "Inf = (\<lambda>_. a\<^sub>1)" 549definition [simp]: "Sup = (\<lambda>_. a\<^sub>1)" 550definition [simp]: "bot = a\<^sub>1" 551definition [simp]: "top = a\<^sub>1" 552definition [simp]: "inf = (\<lambda>_ _. a\<^sub>1)" 553definition [simp]: "sup = (\<lambda>_ _. a\<^sub>1)" 554 555instance by intro_classes(simp_all add: less_eq_finite_1_def) 556end 557 558instance finite_1 :: complete_distrib_lattice 559 by standard simp_all 560 561instance finite_1 :: complete_linorder .. 562 563lemma finite_1_eq: "x = a\<^sub>1" 564by(cases x) simp 565 566simproc_setup finite_1_eq ("x::finite_1") = \<open> 567 fn _ => fn _ => fn ct => 568 (case Thm.term_of ct of 569 Const (@{const_name a\<^sub>1}, _) => NONE 570 | _ => SOME (mk_meta_eq @{thm finite_1_eq})) 571\<close> 572 573instantiation finite_1 :: complete_boolean_algebra 574begin 575definition [simp]: "(-) = (\<lambda>_ _. a\<^sub>1)" 576definition [simp]: "uminus = (\<lambda>_. a\<^sub>1)" 577instance by intro_classes simp_all 578end 579 580instantiation finite_1 :: 581 "{linordered_ring_strict, linordered_comm_semiring_strict, ordered_comm_ring, 582 ordered_cancel_comm_monoid_diff, comm_monoid_mult, ordered_ring_abs, 583 one, modulo, sgn, inverse}" 584begin 585definition [simp]: "Groups.zero = a\<^sub>1" 586definition [simp]: "Groups.one = a\<^sub>1" 587definition [simp]: "(+) = (\<lambda>_ _. a\<^sub>1)" 588definition [simp]: "( * ) = (\<lambda>_ _. a\<^sub>1)" 589definition [simp]: "(mod) = (\<lambda>_ _. a\<^sub>1)" 590definition [simp]: "abs = (\<lambda>_. a\<^sub>1)" 591definition [simp]: "sgn = (\<lambda>_. a\<^sub>1)" 592definition [simp]: "inverse = (\<lambda>_. a\<^sub>1)" 593definition [simp]: "divide = (\<lambda>_ _. a\<^sub>1)" 594 595instance by intro_classes(simp_all add: less_finite_1_def) 596end 597 598declare [[simproc del: finite_1_eq]] 599hide_const (open) a\<^sub>1 600 601datatype (plugins only: code "quickcheck" extraction) finite_2 = 602 a\<^sub>1 | a\<^sub>2 603 604notation (output) a\<^sub>1 ("a\<^sub>1") 605notation (output) a\<^sub>2 ("a\<^sub>2") 606 607lemma UNIV_finite_2: 608 "UNIV = {a\<^sub>1, a\<^sub>2}" 609 by (auto intro: finite_2.exhaust) 610 611instantiation finite_2 :: enum 612begin 613 614definition 615 "enum = [a\<^sub>1, a\<^sub>2]" 616 617definition 618 "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2" 619 620definition 621 "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2" 622 623instance proof 624qed (simp_all only: enum_finite_2_def enum_all_finite_2_def enum_ex_finite_2_def UNIV_finite_2, simp_all) 625 626end 627 628instantiation finite_2 :: linorder 629begin 630 631definition less_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool" 632where 633 "x < y \<longleftrightarrow> x = a\<^sub>1 \<and> y = a\<^sub>2" 634 635definition less_eq_finite_2 :: "finite_2 \<Rightarrow> finite_2 \<Rightarrow> bool" 636where 637 "x \<le> y \<longleftrightarrow> x = y \<or> x < (y :: finite_2)" 638 639instance 640apply (intro_classes) 641apply (auto simp add: less_finite_2_def less_eq_finite_2_def) 642apply (metis finite_2.nchotomy)+ 643done 644 645end 646 647instance finite_2 :: wellorder 648by(rule wf_wellorderI)(simp add: less_finite_2_def, intro_classes) 649 650instantiation finite_2 :: complete_lattice 651begin 652 653definition "\<Sqinter>A = (if a\<^sub>1 \<in> A then a\<^sub>1 else a\<^sub>2)" 654definition "\<Squnion>A = (if a\<^sub>2 \<in> A then a\<^sub>2 else a\<^sub>1)" 655definition [simp]: "bot = a\<^sub>1" 656definition [simp]: "top = a\<^sub>2" 657definition "x \<sqinter> y = (if x = a\<^sub>1 \<or> y = a\<^sub>1 then a\<^sub>1 else a\<^sub>2)" 658definition "x \<squnion> y = (if x = a\<^sub>2 \<or> y = a\<^sub>2 then a\<^sub>2 else a\<^sub>1)" 659 660lemma neq_finite_2_a\<^sub>1_iff [simp]: "x \<noteq> a\<^sub>1 \<longleftrightarrow> x = a\<^sub>2" 661by(cases x) simp_all 662 663lemma neq_finite_2_a\<^sub>1_iff' [simp]: "a\<^sub>1 \<noteq> x \<longleftrightarrow> x = a\<^sub>2" 664by(cases x) simp_all 665 666lemma neq_finite_2_a\<^sub>2_iff [simp]: "x \<noteq> a\<^sub>2 \<longleftrightarrow> x = a\<^sub>1" 667by(cases x) simp_all 668 669lemma neq_finite_2_a\<^sub>2_iff' [simp]: "a\<^sub>2 \<noteq> x \<longleftrightarrow> x = a\<^sub>1" 670by(cases x) simp_all 671 672instance 673proof 674 fix x :: finite_2 and A 675 assume "x \<in> A" 676 then show "\<Sqinter>A \<le> x" "x \<le> \<Squnion>A" 677 by(case_tac [!] x)(auto simp add: less_eq_finite_2_def less_finite_2_def Inf_finite_2_def Sup_finite_2_def) 678qed(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) 679end 680 681instance finite_2 :: complete_linorder .. 682 683instance finite_2 :: complete_distrib_lattice .. 684 685instantiation finite_2 :: "{field, idom_abs_sgn, idom_modulo}" begin 686definition [simp]: "0 = a\<^sub>1" 687definition [simp]: "1 = a\<^sub>2" 688definition "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)" 689definition "uminus = (\<lambda>x :: finite_2. x)" 690definition "(-) = ((+) :: finite_2 \<Rightarrow> _)" 691definition "x * y = (case (x, y) of (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> a\<^sub>1)" 692definition "inverse = (\<lambda>x :: finite_2. x)" 693definition "divide = (( * ) :: finite_2 \<Rightarrow> _)" 694definition "x mod y = (case (x, y) of (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> a\<^sub>1)" 695definition "abs = (\<lambda>x :: finite_2. x)" 696definition "sgn = (\<lambda>x :: finite_2. x)" 697instance 698 by standard 699 (simp_all add: plus_finite_2_def uminus_finite_2_def minus_finite_2_def 700 times_finite_2_def 701 inverse_finite_2_def divide_finite_2_def modulo_finite_2_def 702 abs_finite_2_def sgn_finite_2_def 703 split: finite_2.splits) 704end 705 706lemma two_finite_2 [simp]: 707 "2 = a\<^sub>1" 708 by (simp add: numeral.simps plus_finite_2_def) 709 710lemma dvd_finite_2_unfold: 711 "x dvd y \<longleftrightarrow> x = a\<^sub>2 \<or> y = a\<^sub>1" 712 by (auto simp add: dvd_def times_finite_2_def split: finite_2.splits) 713 714instantiation finite_2 :: "{normalization_semidom, unique_euclidean_semiring}" begin 715definition [simp]: "normalize = (id :: finite_2 \<Rightarrow> _)" 716definition [simp]: "unit_factor = (id :: finite_2 \<Rightarrow> _)" 717definition [simp]: "euclidean_size x = (case x of a\<^sub>1 \<Rightarrow> 0 | a\<^sub>2 \<Rightarrow> 1)" 718definition [simp]: "division_segment (x :: finite_2) = 1" 719instance 720 by standard 721 (auto simp add: divide_finite_2_def times_finite_2_def dvd_finite_2_unfold 722 split: finite_2.splits) 723end 724 725 726hide_const (open) a\<^sub>1 a\<^sub>2 727 728datatype (plugins only: code "quickcheck" extraction) finite_3 = 729 a\<^sub>1 | a\<^sub>2 | a\<^sub>3 730 731notation (output) a\<^sub>1 ("a\<^sub>1") 732notation (output) a\<^sub>2 ("a\<^sub>2") 733notation (output) a\<^sub>3 ("a\<^sub>3") 734 735lemma UNIV_finite_3: 736 "UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3}" 737 by (auto intro: finite_3.exhaust) 738 739instantiation finite_3 :: enum 740begin 741 742definition 743 "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3]" 744 745definition 746 "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3" 747 748definition 749 "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3" 750 751instance proof 752qed (simp_all only: enum_finite_3_def enum_all_finite_3_def enum_ex_finite_3_def UNIV_finite_3, simp_all) 753 754end 755 756lemma finite_3_not_eq_unfold: 757 "x \<noteq> a\<^sub>1 \<longleftrightarrow> x \<in> {a\<^sub>2, a\<^sub>3}" 758 "x \<noteq> a\<^sub>2 \<longleftrightarrow> x \<in> {a\<^sub>1, a\<^sub>3}" 759 "x \<noteq> a\<^sub>3 \<longleftrightarrow> x \<in> {a\<^sub>1, a\<^sub>2}" 760 by (cases x; simp)+ 761 762instantiation finite_3 :: linorder 763begin 764 765definition less_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool" 766where 767 "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)" 768 769definition less_eq_finite_3 :: "finite_3 \<Rightarrow> finite_3 \<Rightarrow> bool" 770where 771 "x \<le> y \<longleftrightarrow> x = y \<or> x < (y :: finite_3)" 772 773instance proof (intro_classes) 774qed (auto simp add: less_finite_3_def less_eq_finite_3_def split: finite_3.split_asm) 775 776end 777 778instance finite_3 :: wellorder 779proof(rule wf_wellorderI) 780 have "inv_image less_than (case_finite_3 0 1 2) = {(x, y). x < y}" 781 by(auto simp add: less_finite_3_def split: finite_3.splits) 782 from this[symmetric] show "wf \<dots>" by simp 783qed intro_classes 784 785class finite_lattice = finite + lattice + Inf + Sup + bot + top + 786 assumes Inf_finite_empty: "Inf {} = Sup UNIV" 787 assumes Inf_finite_insert: "Inf (insert a A) = a \<sqinter> Inf A" 788 assumes Sup_finite_empty: "Sup {} = Inf UNIV" 789 assumes Sup_finite_insert: "Sup (insert a A) = a \<squnion> Sup A" 790 assumes bot_finite_def: "bot = Inf UNIV" 791 assumes top_finite_def: "top = Sup UNIV" 792begin 793 794subclass complete_lattice 795proof 796 fix x A 797 show "x \<in> A \<Longrightarrow> \<Sqinter>A \<le> x" 798 by (metis Set.set_insert abel_semigroup.commute local.Inf_finite_insert local.inf.abel_semigroup_axioms local.inf.left_idem local.inf.orderI) 799 show "x \<in> A \<Longrightarrow> x \<le> \<Squnion>A" 800 by (metis Set.set_insert insert_absorb2 local.Sup_finite_insert local.sup.absorb_iff2) 801next 802 fix A z 803 have "\<Squnion> UNIV = z \<squnion> \<Squnion>UNIV" 804 by (subst Sup_finite_insert [symmetric], simp add: insert_UNIV) 805 from this have [simp]: "z \<le> \<Squnion>UNIV" 806 using local.le_iff_sup by auto 807 have "(\<forall> x. x \<in> A \<longrightarrow> z \<le> x) \<longrightarrow> z \<le> \<Sqinter>A" 808 apply (rule finite_induct [of A "\<lambda> A . (\<forall> x. x \<in> A \<longrightarrow> z \<le> x) \<longrightarrow> z \<le> \<Sqinter>A"]) 809 by (simp_all add: Inf_finite_empty Inf_finite_insert) 810 from this show "(\<And>x. x \<in> A \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> \<Sqinter>A" 811 by simp 812 813 have "\<Sqinter> UNIV = z \<sqinter> \<Sqinter>UNIV" 814 by (subst Inf_finite_insert [symmetric], simp add: insert_UNIV) 815 from this have [simp]: "\<Sqinter>UNIV \<le> z" 816 by (simp add: local.inf.absorb_iff2) 817 have "(\<forall> x. x \<in> A \<longrightarrow> x \<le> z) \<longrightarrow> \<Squnion>A \<le> z" 818 by (rule finite_induct [of A "\<lambda> A . (\<forall> x. x \<in> A \<longrightarrow> x \<le> z) \<longrightarrow> \<Squnion>A \<le> z" ], simp_all add: Sup_finite_empty Sup_finite_insert) 819 from this show " (\<And>x. x \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> \<Squnion>A \<le> z" 820 by blast 821next 822 show "\<Sqinter>{} = \<top>" 823 by (simp add: Inf_finite_empty top_finite_def) 824 show " \<Squnion>{} = \<bottom>" 825 by (simp add: Sup_finite_empty bot_finite_def) 826qed 827end 828 829class finite_distrib_lattice = finite_lattice + distrib_lattice 830begin 831lemma finite_inf_Sup: "a \<sqinter> (Sup A) = Sup {a \<sqinter> b | b . b \<in> A}" 832proof (rule finite_induct [of A "\<lambda> A . a \<sqinter> (Sup A) = Sup {a \<sqinter> b | b . b \<in> A}"], simp_all) 833 fix x::"'a" 834 fix F 835 assume "x \<notin> F" 836 assume [simp]: "a \<sqinter> \<Squnion>F = \<Squnion>{a \<sqinter> b |b. b \<in> F}" 837 have [simp]: " insert (a \<sqinter> x) {a \<sqinter> b |b. b \<in> F} = {a \<sqinter> b |b. b = x \<or> b \<in> F}" 838 by blast 839 have "a \<sqinter> (x \<squnion> \<Squnion>F) = a \<sqinter> x \<squnion> a \<sqinter> \<Squnion>F" 840 by (simp add: inf_sup_distrib1) 841 also have "... = a \<sqinter> x \<squnion> \<Squnion>{a \<sqinter> b |b. b \<in> F}" 842 by simp 843 also have "... = \<Squnion>{a \<sqinter> b |b. b = x \<or> b \<in> F}" 844 by (unfold Sup_insert[THEN sym], simp) 845 finally show "a \<sqinter> (x \<squnion> \<Squnion>F) = \<Squnion>{a \<sqinter> b |b. b = x \<or> b \<in> F}" 846 by simp 847qed 848 849lemma finite_Inf_Sup: "INFIMUM A Sup \<le> SUPREMUM {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y} Inf" 850proof (rule finite_induct [of A "\<lambda> A .INFIMUM A Sup \<le> SUPREMUM {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y} Inf"], simp_all add: finite_UnionD) 851 fix x::"'a set" 852 fix F 853 assume "x \<notin> F" 854 have [simp]: "{\<Squnion>x \<sqinter> b |b . b \<in> Inf ` {f ` F |f. \<forall>Y\<in>F. f Y \<in> Y} } = {\<Squnion>x \<sqinter> (Inf (f ` F)) |f . (\<forall>Y\<in>F. f Y \<in> Y)}" 855 by auto 856 define fa where "fa = (\<lambda> (b::'a) f Y . (if Y = x then b else f Y))" 857 have "\<And>f b. \<forall>Y\<in>F. f Y \<in> Y \<Longrightarrow> b \<in> x \<Longrightarrow> insert b (f ` (F \<inter> {Y. Y \<noteq> x})) = insert (fa b f x) (fa b f ` F) \<and> fa b f x \<in> x \<and> (\<forall>Y\<in>F. fa b f Y \<in> Y)" 858 by (auto simp add: fa_def) 859 from this have B: "\<And>f b. \<forall>Y\<in>F. f Y \<in> Y \<Longrightarrow> b \<in> x \<Longrightarrow> fa b f ` ({x} \<union> F) \<in> {insert (f x) (f ` F) |f. f x \<in> x \<and> (\<forall>Y\<in>F. f Y \<in> Y)}" 860 by blast 861 have [simp]: "\<And>f b. \<forall>Y\<in>F. f Y \<in> Y \<Longrightarrow> b \<in> x \<Longrightarrow> b \<sqinter> (\<Sqinter>x\<in>F. f x) \<le> SUPREMUM {insert (f x) (f ` F) |f. f x \<in> x \<and> (\<forall>Y\<in>F. f Y \<in> Y)} Inf" 862 using B apply (rule SUP_upper2, simp_all) 863 by (simp_all add: INF_greatest Inf_lower inf.coboundedI2 fa_def) 864 865 assume "INFIMUM F Sup \<le> SUPREMUM {f ` F |f. \<forall>Y\<in>F. f Y \<in> Y} Inf" 866 867 from this have "\<Squnion>x \<sqinter> INFIMUM F Sup \<le> \<Squnion>x \<sqinter> SUPREMUM {f ` F |f. \<forall>Y\<in>F. f Y \<in> Y} Inf" 868 apply simp 869 using inf.coboundedI2 by blast 870 also have "... = Sup {\<Squnion>x \<sqinter> (Inf (f ` F)) |f . (\<forall>Y\<in>F. f Y \<in> Y)}" 871 by (simp add: finite_inf_Sup) 872 873 also have "... = Sup {Sup {Inf (f ` F) \<sqinter> b | b . b \<in> x} |f . (\<forall>Y\<in>F. f Y \<in> Y)}" 874 apply (subst inf_commute) 875 by (simp add: finite_inf_Sup) 876 877 also have "... \<le> SUPREMUM {insert (f x) (f ` F) |f. f x \<in> x \<and> (\<forall>Y\<in>F. f Y \<in> Y)} Inf" 878 apply (rule Sup_least, clarsimp)+ 879 by (subst inf_commute, simp) 880 881 finally show "\<Squnion>x \<sqinter> INFIMUM F Sup \<le> SUPREMUM {insert (f x) (f ` F) |f. f x \<in> x \<and> (\<forall>Y\<in>F. f Y \<in> Y)} Inf" 882 by simp 883qed 884 885subclass complete_distrib_lattice 886 by (standard, rule finite_Inf_Sup) 887end 888 889instantiation finite_3 :: finite_lattice 890begin 891 892definition "\<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)" 893definition "\<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)" 894definition [simp]: "bot = a\<^sub>1" 895definition [simp]: "top = a\<^sub>3" 896definition [simp]: "inf = (min :: finite_3 \<Rightarrow> _)" 897definition [simp]: "sup = (max :: finite_3 \<Rightarrow> _)" 898 899instance 900proof 901qed (auto simp add: Inf_finite_3_def Sup_finite_3_def max_def min_def less_eq_finite_3_def less_finite_3_def split: finite_3.split) 902end 903 904instance finite_3 :: complete_lattice .. 905 906instance finite_3 :: finite_distrib_lattice 907proof 908qed (auto simp add: min_def max_def) 909 910instance finite_3 :: complete_distrib_lattice .. 911 912instance finite_3 :: complete_linorder .. 913 914instantiation finite_3 :: "{field, idom_abs_sgn, idom_modulo}" begin 915definition [simp]: "0 = a\<^sub>1" 916definition [simp]: "1 = a\<^sub>2" 917definition 918 "x + y = (case (x, y) of 919 (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>1 920 | (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>2 921 | _ \<Rightarrow> a\<^sub>3)" 922definition "- 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)" 923definition "x - y = x + (- y :: finite_3)" 924definition "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)" 925definition "inverse = (\<lambda>x :: finite_3. x)" 926definition "x div y = x * inverse (y :: finite_3)" 927definition "x mod y = (case y of a\<^sub>1 \<Rightarrow> x | _ \<Rightarrow> a\<^sub>1)" 928definition "abs = (\<lambda>x. case x of a\<^sub>3 \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> x)" 929definition "sgn = (\<lambda>x :: finite_3. x)" 930instance 931 by standard 932 (simp_all add: plus_finite_3_def uminus_finite_3_def minus_finite_3_def 933 times_finite_3_def 934 inverse_finite_3_def divide_finite_3_def modulo_finite_3_def 935 abs_finite_3_def sgn_finite_3_def 936 less_finite_3_def 937 split: finite_3.splits) 938end 939 940lemma two_finite_3 [simp]: 941 "2 = a\<^sub>3" 942 by (simp add: numeral.simps plus_finite_3_def) 943 944lemma dvd_finite_3_unfold: 945 "x dvd y \<longleftrightarrow> x = a\<^sub>2 \<or> x = a\<^sub>3 \<or> y = a\<^sub>1" 946 by (cases x) (auto simp add: dvd_def times_finite_3_def split: finite_3.splits) 947 948instantiation finite_3 :: "{normalization_semidom, unique_euclidean_semiring}" begin 949definition [simp]: "normalize x = (case x of a\<^sub>3 \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> x)" 950definition [simp]: "unit_factor = (id :: finite_3 \<Rightarrow> _)" 951definition [simp]: "euclidean_size x = (case x of a\<^sub>1 \<Rightarrow> 0 | _ \<Rightarrow> 1)" 952definition [simp]: "division_segment (x :: finite_3) = 1" 953instance proof 954 fix x :: finite_3 955 assume "x \<noteq> 0" 956 then show "is_unit (unit_factor x)" 957 by (cases x) (simp_all add: dvd_finite_3_unfold) 958qed (auto simp add: divide_finite_3_def times_finite_3_def 959 dvd_finite_3_unfold inverse_finite_3_def plus_finite_3_def 960 split: finite_3.splits) 961end 962 963hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 964 965datatype (plugins only: code "quickcheck" extraction) finite_4 = 966 a\<^sub>1 | a\<^sub>2 | a\<^sub>3 | a\<^sub>4 967 968notation (output) a\<^sub>1 ("a\<^sub>1") 969notation (output) a\<^sub>2 ("a\<^sub>2") 970notation (output) a\<^sub>3 ("a\<^sub>3") 971notation (output) a\<^sub>4 ("a\<^sub>4") 972 973lemma UNIV_finite_4: 974 "UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4}" 975 by (auto intro: finite_4.exhaust) 976 977instantiation finite_4 :: enum 978begin 979 980definition 981 "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4]" 982 983definition 984 "enum_all P \<longleftrightarrow> P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3 \<and> P a\<^sub>4" 985 986definition 987 "enum_ex P \<longleftrightarrow> P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3 \<or> P a\<^sub>4" 988 989instance proof 990qed (simp_all only: enum_finite_4_def enum_all_finite_4_def enum_ex_finite_4_def UNIV_finite_4, simp_all) 991 992end 993 994instantiation finite_4 :: finite_distrib_lattice begin 995 996text \<open>@{term a\<^sub>1} $<$ @{term a\<^sub>2},@{term a\<^sub>3} $<$ @{term a\<^sub>4}, 997 but @{term a\<^sub>2} and @{term a\<^sub>3} are incomparable.\<close> 998 999definition 1000 "x < y \<longleftrightarrow> (case (x, y) of 1001 (a\<^sub>1, a\<^sub>1) \<Rightarrow> False | (a\<^sub>1, _) \<Rightarrow> True 1002 | (a\<^sub>2, a\<^sub>4) \<Rightarrow> True 1003 | (a\<^sub>3, a\<^sub>4) \<Rightarrow> True | _ \<Rightarrow> False)" 1004 1005definition 1006 "x \<le> y \<longleftrightarrow> (case (x, y) of 1007 (a\<^sub>1, _) \<Rightarrow> True 1008 | (a\<^sub>2, a\<^sub>2) \<Rightarrow> True | (a\<^sub>2, a\<^sub>4) \<Rightarrow> True 1009 | (a\<^sub>3, a\<^sub>3) \<Rightarrow> True | (a\<^sub>3, a\<^sub>4) \<Rightarrow> True 1010 | (a\<^sub>4, a\<^sub>4) \<Rightarrow> True | _ \<Rightarrow> False)" 1011 1012definition 1013 "\<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)" 1014definition 1015 "\<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)" 1016definition [simp]: "bot = a\<^sub>1" 1017definition [simp]: "top = a\<^sub>4" 1018definition 1019 "x \<sqinter> y = (case (x, y) of 1020 (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 1021 | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2 1022 | (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3 1023 | _ \<Rightarrow> a\<^sub>4)" 1024definition 1025 "x \<squnion> y = (case (x, y) of 1026 (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 1027 | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2 1028 | (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3 1029 | _ \<Rightarrow> a\<^sub>1)" 1030 1031instance proof 1032qed(auto simp add: less_finite_4_def less_eq_finite_4_def Inf_finite_4_def Sup_finite_4_def 1033 inf_finite_4_def sup_finite_4_def split: finite_4.splits) 1034end 1035 1036instance finite_4 :: complete_lattice .. 1037 1038instance finite_4 :: complete_distrib_lattice .. 1039 1040instantiation finite_4 :: complete_boolean_algebra begin 1041definition "- 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)" 1042definition "x - y = x \<sqinter> - (y :: finite_4)" 1043instance 1044by intro_classes 1045 (simp_all add: inf_finite_4_def sup_finite_4_def uminus_finite_4_def minus_finite_4_def 1046 split: finite_4.splits) 1047end 1048 1049hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4 1050 1051datatype (plugins only: code "quickcheck" extraction) finite_5 = 1052 a\<^sub>1 | a\<^sub>2 | a\<^sub>3 | a\<^sub>4 | a\<^sub>5 1053 1054notation (output) a\<^sub>1 ("a\<^sub>1") 1055notation (output) a\<^sub>2 ("a\<^sub>2") 1056notation (output) a\<^sub>3 ("a\<^sub>3") 1057notation (output) a\<^sub>4 ("a\<^sub>4") 1058notation (output) a\<^sub>5 ("a\<^sub>5") 1059 1060lemma UNIV_finite_5: 1061 "UNIV = {a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5}" 1062 by (auto intro: finite_5.exhaust) 1063 1064instantiation finite_5 :: enum 1065begin 1066 1067definition 1068 "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4, a\<^sub>5]" 1069 1070definition 1071 "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" 1072 1073definition 1074 "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" 1075 1076instance proof 1077qed (simp_all only: enum_finite_5_def enum_all_finite_5_def enum_ex_finite_5_def UNIV_finite_5, simp_all) 1078 1079end 1080 1081instantiation finite_5 :: finite_lattice 1082begin 1083 1084text \<open>The non-distributive pentagon lattice $N_5$\<close> 1085 1086definition 1087 "x < y \<longleftrightarrow> (case (x, y) of 1088 (a\<^sub>1, a\<^sub>1) \<Rightarrow> False | (a\<^sub>1, _) \<Rightarrow> True 1089 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> True | (a\<^sub>2, a\<^sub>5) \<Rightarrow> True 1090 | (a\<^sub>3, a\<^sub>5) \<Rightarrow> True 1091 | (a\<^sub>4, a\<^sub>5) \<Rightarrow> True | _ \<Rightarrow> False)" 1092 1093definition 1094 "x \<le> y \<longleftrightarrow> (case (x, y) of 1095 (a\<^sub>1, _) \<Rightarrow> True 1096 | (a\<^sub>2, a\<^sub>2) \<Rightarrow> True | (a\<^sub>2, a\<^sub>3) \<Rightarrow> True | (a\<^sub>2, a\<^sub>5) \<Rightarrow> True 1097 | (a\<^sub>3, a\<^sub>3) \<Rightarrow> True | (a\<^sub>3, a\<^sub>5) \<Rightarrow> True 1098 | (a\<^sub>4, a\<^sub>4) \<Rightarrow> True | (a\<^sub>4, a\<^sub>5) \<Rightarrow> True 1099 | (a\<^sub>5, a\<^sub>5) \<Rightarrow> True | _ \<Rightarrow> False)" 1100 1101definition 1102 "\<Sqinter>A = 1103 (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 1104 else if a\<^sub>2 \<in> A then a\<^sub>2 1105 else if a\<^sub>3 \<in> A then a\<^sub>3 1106 else if a\<^sub>4 \<in> A then a\<^sub>4 1107 else a\<^sub>5)" 1108definition 1109 "\<Squnion>A = 1110 (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 1111 else if a\<^sub>3 \<in> A then a\<^sub>3 1112 else if a\<^sub>2 \<in> A then a\<^sub>2 1113 else if a\<^sub>4 \<in> A then a\<^sub>4 1114 else a\<^sub>1)" 1115definition [simp]: "bot = a\<^sub>1" 1116definition [simp]: "top = a\<^sub>5" 1117definition 1118 "x \<sqinter> y = (case (x, y) of 1119 (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 1120 | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2 1121 | (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3 1122 | (a\<^sub>4, _) \<Rightarrow> a\<^sub>4 | (_, a\<^sub>4) \<Rightarrow> a\<^sub>4 1123 | _ \<Rightarrow> a\<^sub>5)" 1124definition 1125 "x \<squnion> y = (case (x, y) of 1126 (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 1127 | (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (_, a\<^sub>3) \<Rightarrow> a\<^sub>3 1128 | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (_, a\<^sub>2) \<Rightarrow> a\<^sub>2 1129 | (a\<^sub>4, _) \<Rightarrow> a\<^sub>4 | (_, a\<^sub>4) \<Rightarrow> a\<^sub>4 1130 | _ \<Rightarrow> a\<^sub>1)" 1131 1132instance 1133proof 1134qed (auto simp add: less_eq_finite_5_def less_finite_5_def inf_finite_5_def sup_finite_5_def 1135 Inf_finite_5_def Sup_finite_5_def split: finite_5.splits if_split_asm) 1136end 1137 1138 1139instance finite_5 :: complete_lattice .. 1140 1141 1142hide_const (open) a\<^sub>1 a\<^sub>2 a\<^sub>3 a\<^sub>4 a\<^sub>5 1143 1144 1145subsection \<open>Closing up\<close> 1146 1147hide_type (open) finite_1 finite_2 finite_3 finite_4 finite_5 1148hide_const (open) enum enum_all enum_ex all_n_lists ex_n_lists ntrancl 1149 1150end 1151