1(* Title: HOL/Library/Option_ord.thy 2 Author: Florian Haftmann, TU Muenchen 3*) 4 5section \<open>Canonical order on option type\<close> 6 7theory Option_ord 8imports Main 9begin 10 11notation 12 bot ("\<bottom>") and 13 top ("\<top>") and 14 inf (infixl "\<sqinter>" 70) and 15 sup (infixl "\<squnion>" 65) and 16 Inf ("\<Sqinter>_" [900] 900) and 17 Sup ("\<Squnion>_" [900] 900) 18 19syntax 20 "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10) 21 "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10) 22 "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10) 23 "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10) 24 25 26instantiation option :: (preorder) preorder 27begin 28 29definition less_eq_option where 30 "x \<le> y \<longleftrightarrow> (case x of None \<Rightarrow> True | Some x \<Rightarrow> (case y of None \<Rightarrow> False | Some y \<Rightarrow> x \<le> y))" 31 32definition less_option where 33 "x < y \<longleftrightarrow> (case y of None \<Rightarrow> False | Some y \<Rightarrow> (case x of None \<Rightarrow> True | Some x \<Rightarrow> x < y))" 34 35lemma less_eq_option_None [simp]: "None \<le> x" 36 by (simp add: less_eq_option_def) 37 38lemma less_eq_option_None_code [code]: "None \<le> x \<longleftrightarrow> True" 39 by simp 40 41lemma less_eq_option_None_is_None: "x \<le> None \<Longrightarrow> x = None" 42 by (cases x) (simp_all add: less_eq_option_def) 43 44lemma less_eq_option_Some_None [simp, code]: "Some x \<le> None \<longleftrightarrow> False" 45 by (simp add: less_eq_option_def) 46 47lemma less_eq_option_Some [simp, code]: "Some x \<le> Some y \<longleftrightarrow> x \<le> y" 48 by (simp add: less_eq_option_def) 49 50lemma less_option_None [simp, code]: "x < None \<longleftrightarrow> False" 51 by (simp add: less_option_def) 52 53lemma less_option_None_is_Some: "None < x \<Longrightarrow> \<exists>z. x = Some z" 54 by (cases x) (simp_all add: less_option_def) 55 56lemma less_option_None_Some [simp]: "None < Some x" 57 by (simp add: less_option_def) 58 59lemma less_option_None_Some_code [code]: "None < Some x \<longleftrightarrow> True" 60 by simp 61 62lemma less_option_Some [simp, code]: "Some x < Some y \<longleftrightarrow> x < y" 63 by (simp add: less_option_def) 64 65instance 66 by standard 67 (auto simp add: less_eq_option_def less_option_def less_le_not_le 68 elim: order_trans split: option.splits) 69 70end 71 72instance option :: (order) order 73 by standard (auto simp add: less_eq_option_def less_option_def split: option.splits) 74 75instance option :: (linorder) linorder 76 by standard (auto simp add: less_eq_option_def less_option_def split: option.splits) 77 78instantiation option :: (order) order_bot 79begin 80 81definition bot_option where "\<bottom> = None" 82 83instance 84 by standard (simp add: bot_option_def) 85 86end 87 88instantiation option :: (order_top) order_top 89begin 90 91definition top_option where "\<top> = Some \<top>" 92 93instance 94 by standard (simp add: top_option_def less_eq_option_def split: option.split) 95 96end 97 98instance option :: (wellorder) wellorder 99proof 100 fix P :: "'a option \<Rightarrow> bool" 101 fix z :: "'a option" 102 assume H: "\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x" 103 have "P None" by (rule H) simp 104 then have P_Some [case_names Some]: "P z" if "\<And>x. z = Some x \<Longrightarrow> (P \<circ> Some) x" for z 105 using \<open>P None\<close> that by (cases z) simp_all 106 show "P z" 107 proof (cases z rule: P_Some) 108 case (Some w) 109 show "(P \<circ> Some) w" 110 proof (induct rule: less_induct) 111 case (less x) 112 have "P (Some x)" 113 proof (rule H) 114 fix y :: "'a option" 115 assume "y < Some x" 116 show "P y" 117 proof (cases y rule: P_Some) 118 case (Some v) 119 with \<open>y < Some x\<close> have "v < x" by simp 120 with less show "(P \<circ> Some) v" . 121 qed 122 qed 123 then show ?case by simp 124 qed 125 qed 126qed 127 128instantiation option :: (inf) inf 129begin 130 131definition inf_option where 132 "x \<sqinter> y = (case x of None \<Rightarrow> None | Some x \<Rightarrow> (case y of None \<Rightarrow> None | Some y \<Rightarrow> Some (x \<sqinter> y)))" 133 134lemma inf_None_1 [simp, code]: "None \<sqinter> y = None" 135 by (simp add: inf_option_def) 136 137lemma inf_None_2 [simp, code]: "x \<sqinter> None = None" 138 by (cases x) (simp_all add: inf_option_def) 139 140lemma inf_Some [simp, code]: "Some x \<sqinter> Some y = Some (x \<sqinter> y)" 141 by (simp add: inf_option_def) 142 143instance .. 144 145end 146 147instantiation option :: (sup) sup 148begin 149 150definition sup_option where 151 "x \<squnion> y = (case x of None \<Rightarrow> y | Some x' \<Rightarrow> (case y of None \<Rightarrow> x | Some y \<Rightarrow> Some (x' \<squnion> y)))" 152 153lemma sup_None_1 [simp, code]: "None \<squnion> y = y" 154 by (simp add: sup_option_def) 155 156lemma sup_None_2 [simp, code]: "x \<squnion> None = x" 157 by (cases x) (simp_all add: sup_option_def) 158 159lemma sup_Some [simp, code]: "Some x \<squnion> Some y = Some (x \<squnion> y)" 160 by (simp add: sup_option_def) 161 162instance .. 163 164end 165 166instance option :: (semilattice_inf) semilattice_inf 167proof 168 fix x y z :: "'a option" 169 show "x \<sqinter> y \<le> x" 170 by (cases x, simp_all, cases y, simp_all) 171 show "x \<sqinter> y \<le> y" 172 by (cases x, simp_all, cases y, simp_all) 173 show "x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<sqinter> z" 174 by (cases x, simp_all, cases y, simp_all, cases z, simp_all) 175qed 176 177instance option :: (semilattice_sup) semilattice_sup 178proof 179 fix x y z :: "'a option" 180 show "x \<le> x \<squnion> y" 181 by (cases x, simp_all, cases y, simp_all) 182 show "y \<le> x \<squnion> y" 183 by (cases x, simp_all, cases y, simp_all) 184 fix x y z :: "'a option" 185 show "y \<le> x \<Longrightarrow> z \<le> x \<Longrightarrow> y \<squnion> z \<le> x" 186 by (cases y, simp_all, cases z, simp_all, cases x, simp_all) 187qed 188 189instance option :: (lattice) lattice .. 190 191instance option :: (lattice) bounded_lattice_bot .. 192 193instance option :: (bounded_lattice_top) bounded_lattice_top .. 194 195instance option :: (bounded_lattice_top) bounded_lattice .. 196 197instance option :: (distrib_lattice) distrib_lattice 198proof 199 fix x y z :: "'a option" 200 show "x \<squnion> y \<sqinter> z = (x \<squnion> y) \<sqinter> (x \<squnion> z)" 201 by (cases x, simp_all, cases y, simp_all, cases z, simp_all add: sup_inf_distrib1 inf_commute) 202qed 203 204instantiation option :: (complete_lattice) complete_lattice 205begin 206 207definition Inf_option :: "'a option set \<Rightarrow> 'a option" where 208 "\<Sqinter>A = (if None \<in> A then None else Some (\<Sqinter>Option.these A))" 209 210lemma None_in_Inf [simp]: "None \<in> A \<Longrightarrow> \<Sqinter>A = None" 211 by (simp add: Inf_option_def) 212 213definition Sup_option :: "'a option set \<Rightarrow> 'a option" where 214 "\<Squnion>A = (if A = {} \<or> A = {None} then None else Some (\<Squnion>Option.these A))" 215 216lemma empty_Sup [simp]: "\<Squnion>{} = None" 217 by (simp add: Sup_option_def) 218 219lemma singleton_None_Sup [simp]: "\<Squnion>{None} = None" 220 by (simp add: Sup_option_def) 221 222instance 223proof 224 fix x :: "'a option" and A 225 assume "x \<in> A" 226 then show "\<Sqinter>A \<le> x" 227 by (cases x) (auto simp add: Inf_option_def in_these_eq intro: Inf_lower) 228next 229 fix z :: "'a option" and A 230 assume *: "\<And>x. x \<in> A \<Longrightarrow> z \<le> x" 231 show "z \<le> \<Sqinter>A" 232 proof (cases z) 233 case None then show ?thesis by simp 234 next 235 case (Some y) 236 show ?thesis 237 by (auto simp add: Inf_option_def in_these_eq Some intro!: Inf_greatest dest!: *) 238 qed 239next 240 fix x :: "'a option" and A 241 assume "x \<in> A" 242 then show "x \<le> \<Squnion>A" 243 by (cases x) (auto simp add: Sup_option_def in_these_eq intro: Sup_upper) 244next 245 fix z :: "'a option" and A 246 assume *: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z" 247 show "\<Squnion>A \<le> z " 248 proof (cases z) 249 case None 250 with * have "\<And>x. x \<in> A \<Longrightarrow> x = None" by (auto dest: less_eq_option_None_is_None) 251 then have "A = {} \<or> A = {None}" by blast 252 then show ?thesis by (simp add: Sup_option_def) 253 next 254 case (Some y) 255 from * have "\<And>w. Some w \<in> A \<Longrightarrow> Some w \<le> z" . 256 with Some have "\<And>w. w \<in> Option.these A \<Longrightarrow> w \<le> y" 257 by (simp add: in_these_eq) 258 then have "\<Squnion>Option.these A \<le> y" by (rule Sup_least) 259 with Some show ?thesis by (simp add: Sup_option_def) 260 qed 261next 262 show "\<Squnion>{} = (\<bottom>::'a option)" 263 by (auto simp: bot_option_def) 264 show "\<Sqinter>{} = (\<top>::'a option)" 265 by (auto simp: top_option_def Inf_option_def) 266qed 267 268end 269 270lemma Some_Inf: 271 "Some (\<Sqinter>A) = \<Sqinter>(Some ` A)" 272 by (auto simp add: Inf_option_def) 273 274lemma Some_Sup: 275 "A \<noteq> {} \<Longrightarrow> Some (\<Squnion>A) = \<Squnion>(Some ` A)" 276 by (auto simp add: Sup_option_def) 277 278lemma Some_INF: 279 "Some (\<Sqinter>x\<in>A. f x) = (\<Sqinter>x\<in>A. Some (f x))" 280 using Some_Inf [of "f ` A"] by (simp add: comp_def) 281 282lemma Some_SUP: 283 "A \<noteq> {} \<Longrightarrow> Some (\<Squnion>x\<in>A. f x) = (\<Squnion>x\<in>A. Some (f x))" 284 using Some_Sup [of "f ` A"] by (simp add: comp_def) 285 286lemma option_Inf_Sup: "INFIMUM (A::('a::complete_distrib_lattice option) set set) Sup \<le> SUPREMUM {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y} Inf" 287proof (cases "{} \<in> A") 288 case True 289 then show ?thesis 290 by (rule INF_lower2, simp_all) 291next 292 case False 293 from this have X: "{} \<notin> A" 294 by simp 295 then show ?thesis 296 proof (cases "{None} \<in> A") 297 case True 298 then show ?thesis 299 by (rule INF_lower2, simp_all) 300 next 301 case False 302 303 {fix y 304 assume A: "y \<in> A" 305 have "Sup (y - {None}) = Sup y" 306 by (metis (no_types, lifting) Sup_option_def insert_Diff_single these_insert_None these_not_empty_eq) 307 from A and this have "(\<exists>z. y - {None} = z - {None} \<and> z \<in> A) \<and> \<Squnion>y = \<Squnion>(y - {None})" 308 by auto 309 } 310 from this have A: "Sup ` A = (Sup ` {y - {None} | y. y\<in>A})" 311 by (auto simp add: image_def) 312 313 have [simp]: "\<And>y. y \<in> A \<Longrightarrow> \<exists>ya. {ya. \<exists>x. x \<in> y \<and> (\<exists>y. x = Some y) \<and> ya = the x} 314 = {y. \<exists>x\<in>ya - {None}. y = the x} \<and> ya \<in> A" 315 by (rule exI, auto) 316 317 have [simp]: "\<And>y. y \<in> A \<Longrightarrow> 318 (\<exists>ya. y - {None} = ya - {None} \<and> ya \<in> A) \<and> \<Squnion>{ya. \<exists>x\<in>y - {None}. ya = the x} 319 = \<Squnion>{ya. \<exists>x. x \<in> y \<and> (\<exists>y. x = Some y) \<and> ya = the x}" 320 apply (safe, blast) 321 by (rule arg_cong [of _ _ Sup], auto) 322 {fix y 323 assume [simp]: "y \<in> A" 324 have "\<exists>x. (\<exists>y. x = {ya. \<exists>x\<in>y - {None}. ya = the x} \<and> y \<in> A) \<and> \<Squnion>{ya. \<exists>x. x \<in> y \<and> (\<exists>y. x = Some y) \<and> ya = the x} = \<Squnion>x" 325 and "\<exists>x. (\<exists>y. x = y - {None} \<and> y \<in> A) \<and> \<Squnion>{ya. \<exists>x\<in>y - {None}. ya = the x} = \<Squnion>{y. \<exists>xa. xa \<in> x \<and> (\<exists>y. xa = Some y) \<and> y = the xa}" 326 apply (rule exI [of _ "{ya. \<exists>x. x \<in> y \<and> (\<exists>y. x = Some y) \<and> ya = the x}"], simp) 327 by (rule exI [of _ "y - {None}"], simp) 328 } 329 from this have C: "(\<lambda>x. (\<Squnion>Option.these x)) ` {y - {None} |y. y \<in> A} = (Sup ` {the ` (y - {None}) |y. y \<in> A})" 330 by (simp add: image_def Option.these_def, safe, simp_all) 331 332 have D: "\<forall> f . \<exists>Y\<in>A. f Y \<notin> Y \<Longrightarrow> False" 333 by (drule spec [of _ "\<lambda> Y . SOME x . x \<in> Y"], simp add: X some_in_eq) 334 335 define F where "F = (\<lambda> Y . SOME x::'a option . x \<in> (Y - {None}))" 336 337 have G: "\<And> Y . Y \<in> A \<Longrightarrow> \<exists> x . x \<in> Y - {None}" 338 by (metis False X all_not_in_conv insert_Diff_single these_insert_None these_not_empty_eq) 339 340 have F: "\<And> Y . Y \<in> A \<Longrightarrow> F Y \<in> (Y - {None})" 341 by (metis F_def G empty_iff some_in_eq) 342 343 have "Some \<bottom> \<le> Inf (F ` A)" 344 by (metis (no_types, lifting) Diff_iff F Inf_option_def bot.extremum image_iff 345 less_eq_option_Some singletonI) 346 347 from this have "Inf (F ` A) \<noteq> None" 348 by (cases "\<Sqinter>x\<in>A. F x", simp_all) 349 350 from this have "Inf (F ` A) \<noteq> None \<and> Inf (F ` A) \<in> Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}" 351 using F by auto 352 353 from this have "\<exists> x . x \<noteq> None \<and> x \<in> Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}" 354 by blast 355 356 from this have E:" Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y} = {None} \<Longrightarrow> False" 357 by blast 358 359 have [simp]: "((\<Squnion>x\<in>{f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}. \<Sqinter>x) = None) = False" 360 by (metis (no_types, lifting) E Sup_option_def \<open>\<exists>x. x \<noteq> None \<and> x \<in> Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}\<close> 361 ex_in_conv option.simps(3)) 362 363 have B: "Option.these ((\<lambda>x. Some (\<Squnion>Option.these x)) ` {y - {None} |y. y \<in> A}) 364 = ((\<lambda>x. (\<Squnion> Option.these x)) ` {y - {None} |y. y \<in> A})" 365 by (metis image_image these_image_Some_eq) 366 { 367 fix f 368 assume A: "\<And> Y . (\<exists>y. Y = the ` (y - {None}) \<and> y \<in> A) \<Longrightarrow> f Y \<in> Y" 369 370 have "\<And>xa. xa \<in> A \<Longrightarrow> f {y. \<exists>a\<in>xa - {None}. y = the a} = f (the ` (xa - {None}))" 371 by (simp add: image_def) 372 from this have [simp]: "\<And>xa. xa \<in> A \<Longrightarrow> \<exists>x\<in>A. f {y. \<exists>a\<in>xa - {None}. y = the a} = f (the ` (x - {None}))" 373 by blast 374 have "\<And>xa. xa \<in> A \<Longrightarrow> f (the ` (xa - {None})) = f {y. \<exists>a \<in> xa - {None}. y = the a} \<and> xa \<in> A" 375 by (simp add: image_def) 376 from this have [simp]: "\<And>xa. xa \<in> A \<Longrightarrow> \<exists>x. f (the ` (xa - {None})) = f {y. \<exists>a\<in>x - {None}. y = the a} \<and> x \<in> A" 377 by blast 378 379 { 380 fix Y 381 have "Y \<in> A \<Longrightarrow> Some (f (the ` (Y - {None}))) \<in> Y" 382 using A [of "the ` (Y - {None})"] apply (simp add: image_def) 383 using option.collapse by fastforce 384 } 385 from this have [simp]: "\<And> Y . Y \<in> A \<Longrightarrow> Some (f (the ` (Y - {None}))) \<in> Y" 386 by blast 387 have [simp]: "(\<Sqinter>x\<in>A. Some (f {y. \<exists>x\<in>x - {None}. y = the x})) = \<Sqinter>{Some (f {y. \<exists>a\<in>x - {None}. y = the a}) |x. x \<in> A}" 388 by (simp add: Setcompr_eq_image) 389 390 have [simp]: "\<exists>x. (\<exists>f. x = {y. \<exists>x\<in>A. y = f x} \<and> (\<forall>Y\<in>A. f Y \<in> Y)) \<and> \<Sqinter>{Some (f {y. \<exists>a\<in>x - {None}. y = the a}) |x. x \<in> A} = \<Sqinter>x" 391 apply (rule exI [of _ "{Some (f {y. \<exists>a\<in>x - {None}. y = the a}) | x . x\<in> A}"], safe) 392 by (rule exI [of _ "(\<lambda> Y . Some (f (the ` (Y - {None})))) "], safe, simp_all) 393 394 { 395 fix xb 396 have "xb \<in> A \<Longrightarrow> (\<Sqinter>x\<in>{{ya. \<exists>x\<in>y - {None}. ya = the x} |y. y \<in> A}. f x) \<le> f {y. \<exists>x\<in>xb - {None}. y = the x}" 397 apply (rule INF_lower2 [of "{y. \<exists>x\<in>xb - {None}. y = the x}"]) 398 by blast+ 399 } 400 from this have [simp]: "(\<Sqinter>x\<in>{the ` (y - {None}) |y. y \<in> A}. f x) \<le> the (\<Sqinter>Y\<in>A. Some (f (the ` (Y - {None}))))" 401 apply (simp add: Inf_option_def image_def Option.these_def) 402 by (rule Inf_greatest, clarsimp) 403 404 have [simp]: "the (\<Sqinter>Y\<in>A. Some (f (the ` (Y - {None})))) \<in> Option.these (Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})" 405 apply (simp add: Option.these_def image_def) 406 apply (rule exI [of _ "(\<Sqinter>x\<in>A. Some (f {y. \<exists>x\<in>x - {None}. y = the x}))"], simp) 407 by (simp add: Inf_option_def) 408 409 have "(\<Sqinter>x\<in>{the ` (y - {None}) |y. y \<in> A}. f x) \<le> \<Squnion>Option.these (Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})" 410 by (rule Sup_upper2 [of "the (Inf ((\<lambda> Y . Some (f (the ` (Y - {None})) )) ` A))"], simp_all) 411 } 412 from this have X: "\<And> f . \<forall>Y. (\<exists>y. Y = the ` (y - {None}) \<and> y \<in> A) \<longrightarrow> f Y \<in> Y \<Longrightarrow> 413 (\<Sqinter>x\<in>{the ` (y - {None}) |y. y \<in> A}. f x) \<le> \<Squnion>Option.these (Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})" 414 by blast 415 416 417 have [simp]: "\<And> x . x\<in>{y - {None} |y. y \<in> A} \<Longrightarrow> x \<noteq> {} \<and> x \<noteq> {None}" 418 using F by fastforce 419 420 have "(Inf (Sup `A)) = (Inf (Sup ` {y - {None} | y. y\<in>A}))" 421 by (subst A, simp) 422 423 also have "... = (\<Sqinter>x\<in>{y - {None} |y. y \<in> A}. if x = {} \<or> x = {None} then None else Some (\<Squnion>Option.these x))" 424 by (simp add: Sup_option_def) 425 426 also have "... = (\<Sqinter>x\<in>{y - {None} |y. y \<in> A}. Some (\<Squnion>Option.these x))" 427 using G by fastforce 428 429 also have "... = Some (\<Sqinter>Option.these ((\<lambda>x. Some (\<Squnion>Option.these x)) ` {y - {None} |y. y \<in> A}))" 430 by (simp add: Inf_option_def, safe) 431 432 also have "... = Some (\<Sqinter> ((\<lambda>x. (\<Squnion>Option.these x)) ` {y - {None} |y. y \<in> A}))" 433 by (simp add: B) 434 435 also have "... = Some (Inf (Sup ` {the ` (y - {None}) |y. y \<in> A}))" 436 by (unfold C, simp) 437 thm Inf_Sup 438 also have "... = Some (\<Squnion>x\<in>{f ` {the ` (y - {None}) |y. y \<in> A} |f. \<forall>Y. (\<exists>y. Y = the ` (y - {None}) \<and> y \<in> A) \<longrightarrow> f Y \<in> Y}. \<Sqinter>x) " 439 by (simp add: Inf_Sup) 440 441 also have "... \<le> SUPREMUM {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y} Inf" 442 proof (cases "SUPREMUM {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y} Inf") 443 case None 444 then show ?thesis by (simp add: less_eq_option_def) 445 next 446 case (Some a) 447 then show ?thesis 448 apply simp 449 apply (rule Sup_least, safe) 450 apply (simp add: Sup_option_def) 451 apply (cases "(\<forall>f. \<exists>Y\<in>A. f Y \<notin> Y) \<or> Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y} = {None}", simp_all) 452 by (drule X, simp) 453 qed 454 finally show ?thesis by simp 455 qed 456qed 457 458instance option :: (complete_distrib_lattice) complete_distrib_lattice 459 by (standard, simp add: option_Inf_Sup) 460 461instance option :: (complete_linorder) complete_linorder .. 462 463 464no_notation 465 bot ("\<bottom>") and 466 top ("\<top>") and 467 inf (infixl "\<sqinter>" 70) and 468 sup (infixl "\<squnion>" 65) and 469 Inf ("\<Sqinter>_" [900] 900) and 470 Sup ("\<Squnion>_" [900] 900) 471 472no_syntax 473 "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10) 474 "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10) 475 "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10) 476 "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10) 477 478end 479