1(* Title: HOL/Set.thy 2 Author: Tobias Nipkow 3 Author: Lawrence C Paulson 4 Author: Markus Wenzel 5*) 6 7section \<open>Set theory for higher-order logic\<close> 8 9theory Set 10 imports Lattices 11begin 12 13subsection \<open>Sets as predicates\<close> 14 15typedecl 'a set 16 17axiomatization Collect :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set" \<comment> \<open>comprehension\<close> 18 and member :: "'a \<Rightarrow> 'a set \<Rightarrow> bool" \<comment> \<open>membership\<close> 19 where mem_Collect_eq [iff, code_unfold]: "member a (Collect P) = P a" 20 and Collect_mem_eq [simp]: "Collect (\<lambda>x. member x A) = A" 21 22notation 23 member ("'(\<in>')") and 24 member ("(_/ \<in> _)" [51, 51] 50) 25 26abbreviation not_member 27 where "not_member x A \<equiv> \<not> (x \<in> A)" \<comment> \<open>non-membership\<close> 28notation 29 not_member ("'(\<notin>')") and 30 not_member ("(_/ \<notin> _)" [51, 51] 50) 31 32notation (ASCII) 33 member ("'(:')") and 34 member ("(_/ : _)" [51, 51] 50) and 35 not_member ("'(~:')") and 36 not_member ("(_/ ~: _)" [51, 51] 50) 37 38 39text \<open>Set comprehensions\<close> 40 41syntax 42 "_Coll" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a set" ("(1{_./ _})") 43translations 44 "{x. P}" \<rightleftharpoons> "CONST Collect (\<lambda>x. P)" 45 46syntax (ASCII) 47 "_Collect" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> 'a set" ("(1{(_/: _)./ _})") 48syntax 49 "_Collect" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> 'a set" ("(1{(_/ \<in> _)./ _})") 50translations 51 "{p:A. P}" \<rightharpoonup> "CONST Collect (\<lambda>p. p \<in> A \<and> P)" 52 53lemma CollectI: "P a \<Longrightarrow> a \<in> {x. P x}" 54 by simp 55 56lemma CollectD: "a \<in> {x. P x} \<Longrightarrow> P a" 57 by simp 58 59lemma Collect_cong: "(\<And>x. P x = Q x) \<Longrightarrow> {x. P x} = {x. Q x}" 60 by simp 61 62text \<open> 63 Simproc for pulling \<open>x = t\<close> in \<open>{x. \<dots> \<and> x = t \<and> \<dots>}\<close> 64 to the front (and similarly for \<open>t = x\<close>): 65\<close> 66 67simproc_setup defined_Collect ("{x. P x \<and> Q x}") = \<open> 68 fn _ => Quantifier1.rearrange_Collect 69 (fn ctxt => 70 resolve_tac ctxt @{thms Collect_cong} 1 THEN 71 resolve_tac ctxt @{thms iffI} 1 THEN 72 ALLGOALS 73 (EVERY' [REPEAT_DETERM o eresolve_tac ctxt @{thms conjE}, 74 DEPTH_SOLVE_1 o (assume_tac ctxt ORELSE' resolve_tac ctxt @{thms conjI})])) 75\<close> 76 77lemmas CollectE = CollectD [elim_format] 78 79lemma set_eqI: 80 assumes "\<And>x. x \<in> A \<longleftrightarrow> x \<in> B" 81 shows "A = B" 82proof - 83 from assms have "{x. x \<in> A} = {x. x \<in> B}" 84 by simp 85 then show ?thesis by simp 86qed 87 88lemma set_eq_iff: "A = B \<longleftrightarrow> (\<forall>x. x \<in> A \<longleftrightarrow> x \<in> B)" 89 by (auto intro:set_eqI) 90 91lemma Collect_eqI: 92 assumes "\<And>x. P x = Q x" 93 shows "Collect P = Collect Q" 94 using assms by (auto intro: set_eqI) 95 96text \<open>Lifting of predicate class instances\<close> 97 98instantiation set :: (type) boolean_algebra 99begin 100 101definition less_eq_set 102 where "A \<le> B \<longleftrightarrow> (\<lambda>x. member x A) \<le> (\<lambda>x. member x B)" 103 104definition less_set 105 where "A < B \<longleftrightarrow> (\<lambda>x. member x A) < (\<lambda>x. member x B)" 106 107definition inf_set 108 where "A \<sqinter> B = Collect ((\<lambda>x. member x A) \<sqinter> (\<lambda>x. member x B))" 109 110definition sup_set 111 where "A \<squnion> B = Collect ((\<lambda>x. member x A) \<squnion> (\<lambda>x. member x B))" 112 113definition bot_set 114 where "\<bottom> = Collect \<bottom>" 115 116definition top_set 117 where "\<top> = Collect \<top>" 118 119definition uminus_set 120 where "- A = Collect (- (\<lambda>x. member x A))" 121 122definition minus_set 123 where "A - B = Collect ((\<lambda>x. member x A) - (\<lambda>x. member x B))" 124 125instance 126 by standard 127 (simp_all add: less_eq_set_def less_set_def inf_set_def sup_set_def 128 bot_set_def top_set_def uminus_set_def minus_set_def 129 less_le_not_le sup_inf_distrib1 diff_eq set_eqI fun_eq_iff 130 del: inf_apply sup_apply bot_apply top_apply minus_apply uminus_apply) 131 132end 133 134text \<open>Set enumerations\<close> 135 136abbreviation empty :: "'a set" ("{}") 137 where "{} \<equiv> bot" 138 139definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" 140 where insert_compr: "insert a B = {x. x = a \<or> x \<in> B}" 141 142syntax 143 "_Finset" :: "args \<Rightarrow> 'a set" ("{(_)}") 144translations 145 "{x, xs}" \<rightleftharpoons> "CONST insert x {xs}" 146 "{x}" \<rightleftharpoons> "CONST insert x {}" 147 148 149subsection \<open>Subsets and bounded quantifiers\<close> 150 151abbreviation subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" 152 where "subset \<equiv> less" 153 154abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" 155 where "subset_eq \<equiv> less_eq" 156 157notation 158 subset ("'(\<subset>')") and 159 subset ("(_/ \<subset> _)" [51, 51] 50) and 160 subset_eq ("'(\<subseteq>')") and 161 subset_eq ("(_/ \<subseteq> _)" [51, 51] 50) 162 163abbreviation (input) 164 supset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where 165 "supset \<equiv> greater" 166 167abbreviation (input) 168 supset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where 169 "supset_eq \<equiv> greater_eq" 170 171notation 172 supset ("'(\<supset>')") and 173 supset ("(_/ \<supset> _)" [51, 51] 50) and 174 supset_eq ("'(\<supseteq>')") and 175 supset_eq ("(_/ \<supseteq> _)" [51, 51] 50) 176 177notation (ASCII output) 178 subset ("'(<')") and 179 subset ("(_/ < _)" [51, 51] 50) and 180 subset_eq ("'(<=')") and 181 subset_eq ("(_/ <= _)" [51, 51] 50) 182 183definition Ball :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" 184 where "Ball A P \<longleftrightarrow> (\<forall>x. x \<in> A \<longrightarrow> P x)" \<comment> \<open>bounded universal quantifiers\<close> 185 186definition Bex :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" 187 where "Bex A P \<longleftrightarrow> (\<exists>x. x \<in> A \<and> P x)" \<comment> \<open>bounded existential quantifiers\<close> 188 189syntax (ASCII) 190 "_Ball" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3ALL (_/:_)./ _)" [0, 0, 10] 10) 191 "_Bex" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3EX (_/:_)./ _)" [0, 0, 10] 10) 192 "_Bex1" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3EX! (_/:_)./ _)" [0, 0, 10] 10) 193 "_Bleast" :: "id \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> 'a" ("(3LEAST (_/:_)./ _)" [0, 0, 10] 10) 194 195syntax (input) 196 "_Ball" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3! (_/:_)./ _)" [0, 0, 10] 10) 197 "_Bex" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3? (_/:_)./ _)" [0, 0, 10] 10) 198 "_Bex1" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3?! (_/:_)./ _)" [0, 0, 10] 10) 199 200syntax 201 "_Ball" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3\<forall>(_/\<in>_)./ _)" [0, 0, 10] 10) 202 "_Bex" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3\<exists>(_/\<in>_)./ _)" [0, 0, 10] 10) 203 "_Bex1" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3\<exists>!(_/\<in>_)./ _)" [0, 0, 10] 10) 204 "_Bleast" :: "id \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> 'a" ("(3LEAST(_/\<in>_)./ _)" [0, 0, 10] 10) 205 206translations 207 "\<forall>x\<in>A. P" \<rightleftharpoons> "CONST Ball A (\<lambda>x. P)" 208 "\<exists>x\<in>A. P" \<rightleftharpoons> "CONST Bex A (\<lambda>x. P)" 209 "\<exists>!x\<in>A. P" \<rightharpoonup> "\<exists>!x. x \<in> A \<and> P" 210 "LEAST x:A. P" \<rightharpoonup> "LEAST x. x \<in> A \<and> P" 211 212syntax (ASCII output) 213 "_setlessAll" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3ALL _<_./ _)" [0, 0, 10] 10) 214 "_setlessEx" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3EX _<_./ _)" [0, 0, 10] 10) 215 "_setleAll" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10) 216 "_setleEx" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) 217 "_setleEx1" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3EX! _<=_./ _)" [0, 0, 10] 10) 218 219syntax 220 "_setlessAll" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3\<forall>_\<subset>_./ _)" [0, 0, 10] 10) 221 "_setlessEx" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3\<exists>_\<subset>_./ _)" [0, 0, 10] 10) 222 "_setleAll" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10) 223 "_setleEx" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10) 224 "_setleEx1" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10) 225 226translations 227 "\<forall>A\<subset>B. P" \<rightharpoonup> "\<forall>A. A \<subset> B \<longrightarrow> P" 228 "\<exists>A\<subset>B. P" \<rightharpoonup> "\<exists>A. A \<subset> B \<and> P" 229 "\<forall>A\<subseteq>B. P" \<rightharpoonup> "\<forall>A. A \<subseteq> B \<longrightarrow> P" 230 "\<exists>A\<subseteq>B. P" \<rightharpoonup> "\<exists>A. A \<subseteq> B \<and> P" 231 "\<exists>!A\<subseteq>B. P" \<rightharpoonup> "\<exists>!A. A \<subseteq> B \<and> P" 232 233print_translation \<open> 234 let 235 val All_binder = Mixfix.binder_name \<^const_syntax>\<open>All\<close>; 236 val Ex_binder = Mixfix.binder_name \<^const_syntax>\<open>Ex\<close>; 237 val impl = \<^const_syntax>\<open>HOL.implies\<close>; 238 val conj = \<^const_syntax>\<open>HOL.conj\<close>; 239 val sbset = \<^const_syntax>\<open>subset\<close>; 240 val sbset_eq = \<^const_syntax>\<open>subset_eq\<close>; 241 242 val trans = 243 [((All_binder, impl, sbset), \<^syntax_const>\<open>_setlessAll\<close>), 244 ((All_binder, impl, sbset_eq), \<^syntax_const>\<open>_setleAll\<close>), 245 ((Ex_binder, conj, sbset), \<^syntax_const>\<open>_setlessEx\<close>), 246 ((Ex_binder, conj, sbset_eq), \<^syntax_const>\<open>_setleEx\<close>)]; 247 248 fun mk v (v', T) c n P = 249 if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n) 250 then Syntax.const c $ Syntax_Trans.mark_bound_body (v', T) $ n $ P 251 else raise Match; 252 253 fun tr' q = (q, fn _ => 254 (fn [Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (v, Type (\<^type_name>\<open>set\<close>, _)), 255 Const (c, _) $ 256 (Const (d, _) $ (Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (v', T)) $ n) $ P] => 257 (case AList.lookup (=) trans (q, c, d) of 258 NONE => raise Match 259 | SOME l => mk v (v', T) l n P) 260 | _ => raise Match)); 261 in 262 [tr' All_binder, tr' Ex_binder] 263 end 264\<close> 265 266 267text \<open> 268 \<^medskip> 269 Translate between \<open>{e | x1\<dots>xn. P}\<close> and \<open>{u. \<exists>x1\<dots>xn. u = e \<and> P}\<close>; 270 \<open>{y. \<exists>x1\<dots>xn. y = e \<and> P}\<close> is only translated if \<open>[0..n] \<subseteq> bvs e\<close>. 271\<close> 272 273syntax 274 "_Setcompr" :: "'a \<Rightarrow> idts \<Rightarrow> bool \<Rightarrow> 'a set" ("(1{_ |/_./ _})") 275 276parse_translation \<open> 277 let 278 val ex_tr = snd (Syntax_Trans.mk_binder_tr ("EX ", \<^const_syntax>\<open>Ex\<close>)); 279 280 fun nvars (Const (\<^syntax_const>\<open>_idts\<close>, _) $ _ $ idts) = nvars idts + 1 281 | nvars _ = 1; 282 283 fun setcompr_tr ctxt [e, idts, b] = 284 let 285 val eq = Syntax.const \<^const_syntax>\<open>HOL.eq\<close> $ Bound (nvars idts) $ e; 286 val P = Syntax.const \<^const_syntax>\<open>HOL.conj\<close> $ eq $ b; 287 val exP = ex_tr ctxt [idts, P]; 288 in Syntax.const \<^const_syntax>\<open>Collect\<close> $ absdummy dummyT exP end; 289 290 in [(\<^syntax_const>\<open>_Setcompr\<close>, setcompr_tr)] end 291\<close> 292 293print_translation \<open> 294 [Syntax_Trans.preserve_binder_abs2_tr' \<^const_syntax>\<open>Ball\<close> \<^syntax_const>\<open>_Ball\<close>, 295 Syntax_Trans.preserve_binder_abs2_tr' \<^const_syntax>\<open>Bex\<close> \<^syntax_const>\<open>_Bex\<close>] 296\<close> \<comment> \<open>to avoid eta-contraction of body\<close> 297 298print_translation \<open> 299let 300 val ex_tr' = snd (Syntax_Trans.mk_binder_tr' (\<^const_syntax>\<open>Ex\<close>, "DUMMY")); 301 302 fun setcompr_tr' ctxt [Abs (abs as (_, _, P))] = 303 let 304 fun check (Const (\<^const_syntax>\<open>Ex\<close>, _) $ Abs (_, _, P), n) = check (P, n + 1) 305 | check (Const (\<^const_syntax>\<open>HOL.conj\<close>, _) $ 306 (Const (\<^const_syntax>\<open>HOL.eq\<close>, _) $ Bound m $ e) $ P, n) = 307 n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso 308 subset (=) (0 upto (n - 1), add_loose_bnos (e, 0, [])) 309 | check _ = false; 310 311 fun tr' (_ $ abs) = 312 let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' ctxt [abs] 313 in Syntax.const \<^syntax_const>\<open>_Setcompr\<close> $ e $ idts $ Q end; 314 in 315 if check (P, 0) then tr' P 316 else 317 let 318 val (x as _ $ Free(xN, _), t) = Syntax_Trans.atomic_abs_tr' abs; 319 val M = Syntax.const \<^syntax_const>\<open>_Coll\<close> $ x $ t; 320 in 321 case t of 322 Const (\<^const_syntax>\<open>HOL.conj\<close>, _) $ 323 (Const (\<^const_syntax>\<open>Set.member\<close>, _) $ 324 (Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (yN, _)) $ A) $ P => 325 if xN = yN then Syntax.const \<^syntax_const>\<open>_Collect\<close> $ x $ A $ P else M 326 | _ => M 327 end 328 end; 329 in [(\<^const_syntax>\<open>Collect\<close>, setcompr_tr')] end 330\<close> 331 332simproc_setup defined_Bex ("\<exists>x\<in>A. P x \<and> Q x") = \<open> 333 fn _ => Quantifier1.rearrange_bex 334 (fn ctxt => 335 unfold_tac ctxt @{thms Bex_def} THEN 336 Quantifier1.prove_one_point_ex_tac ctxt) 337\<close> 338 339simproc_setup defined_All ("\<forall>x\<in>A. P x \<longrightarrow> Q x") = \<open> 340 fn _ => Quantifier1.rearrange_ball 341 (fn ctxt => 342 unfold_tac ctxt @{thms Ball_def} THEN 343 Quantifier1.prove_one_point_all_tac ctxt) 344\<close> 345 346lemma ballI [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> P x) \<Longrightarrow> \<forall>x\<in>A. P x" 347 by (simp add: Ball_def) 348 349lemmas strip = impI allI ballI 350 351lemma bspec [dest?]: "\<forall>x\<in>A. P x \<Longrightarrow> x \<in> A \<Longrightarrow> P x" 352 by (simp add: Ball_def) 353 354text \<open>Gives better instantiation for bound:\<close> 355setup \<open> 356 map_theory_claset (fn ctxt => 357 ctxt addbefore ("bspec", fn ctxt' => dresolve_tac ctxt' @{thms bspec} THEN' assume_tac ctxt')) 358\<close> 359 360ML \<open> 361structure Simpdata = 362struct 363 open Simpdata; 364 val mksimps_pairs = [(\<^const_name>\<open>Ball\<close>, @{thms bspec})] @ mksimps_pairs; 365end; 366 367open Simpdata; 368\<close> 369 370declaration \<open>fn _ => Simplifier.map_ss (Simplifier.set_mksimps (mksimps mksimps_pairs))\<close> 371 372lemma ballE [elim]: "\<forall>x\<in>A. P x \<Longrightarrow> (P x \<Longrightarrow> Q) \<Longrightarrow> (x \<notin> A \<Longrightarrow> Q) \<Longrightarrow> Q" 373 unfolding Ball_def by blast 374 375lemma bexI [intro]: "P x \<Longrightarrow> x \<in> A \<Longrightarrow> \<exists>x\<in>A. P x" 376 \<comment> \<open>Normally the best argument order: \<open>P x\<close> constrains the choice of \<open>x \<in> A\<close>.\<close> 377 unfolding Bex_def by blast 378 379lemma rev_bexI [intro?]: "x \<in> A \<Longrightarrow> P x \<Longrightarrow> \<exists>x\<in>A. P x" 380 \<comment> \<open>The best argument order when there is only one \<open>x \<in> A\<close>.\<close> 381 unfolding Bex_def by blast 382 383lemma bexCI: "(\<forall>x\<in>A. \<not> P x \<Longrightarrow> P a) \<Longrightarrow> a \<in> A \<Longrightarrow> \<exists>x\<in>A. P x" 384 unfolding Bex_def by blast 385 386lemma bexE [elim!]: "\<exists>x\<in>A. P x \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P x \<Longrightarrow> Q) \<Longrightarrow> Q" 387 unfolding Bex_def by blast 388 389lemma ball_triv [simp]: "(\<forall>x\<in>A. P) \<longleftrightarrow> ((\<exists>x. x \<in> A) \<longrightarrow> P)" 390 \<comment> \<open>Trival rewrite rule.\<close> 391 by (simp add: Ball_def) 392 393lemma bex_triv [simp]: "(\<exists>x\<in>A. P) \<longleftrightarrow> ((\<exists>x. x \<in> A) \<and> P)" 394 \<comment> \<open>Dual form for existentials.\<close> 395 by (simp add: Bex_def) 396 397lemma bex_triv_one_point1 [simp]: "(\<exists>x\<in>A. x = a) \<longleftrightarrow> a \<in> A" 398 by blast 399 400lemma bex_triv_one_point2 [simp]: "(\<exists>x\<in>A. a = x) \<longleftrightarrow> a \<in> A" 401 by blast 402 403lemma bex_one_point1 [simp]: "(\<exists>x\<in>A. x = a \<and> P x) \<longleftrightarrow> a \<in> A \<and> P a" 404 by blast 405 406lemma bex_one_point2 [simp]: "(\<exists>x\<in>A. a = x \<and> P x) \<longleftrightarrow> a \<in> A \<and> P a" 407 by blast 408 409lemma ball_one_point1 [simp]: "(\<forall>x\<in>A. x = a \<longrightarrow> P x) \<longleftrightarrow> (a \<in> A \<longrightarrow> P a)" 410 by blast 411 412lemma ball_one_point2 [simp]: "(\<forall>x\<in>A. a = x \<longrightarrow> P x) \<longleftrightarrow> (a \<in> A \<longrightarrow> P a)" 413 by blast 414 415lemma ball_conj_distrib: "(\<forall>x\<in>A. P x \<and> Q x) \<longleftrightarrow> (\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>A. Q x)" 416 by blast 417 418lemma bex_disj_distrib: "(\<exists>x\<in>A. P x \<or> Q x) \<longleftrightarrow> (\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>A. Q x)" 419 by blast 420 421 422text \<open>Congruence rules\<close> 423 424lemma ball_cong: 425 "\<lbrakk> A = B; \<And>x. x \<in> B \<Longrightarrow> P x \<longleftrightarrow> Q x \<rbrakk> \<Longrightarrow> 426 (\<forall>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>B. Q x)" 427by (simp add: Ball_def) 428 429lemma ball_cong_simp [cong]: 430 "\<lbrakk> A = B; \<And>x. x \<in> B =simp=> P x \<longleftrightarrow> Q x \<rbrakk> \<Longrightarrow> 431 (\<forall>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>B. Q x)" 432by (simp add: simp_implies_def Ball_def) 433 434lemma bex_cong: 435 "\<lbrakk> A = B; \<And>x. x \<in> B \<Longrightarrow> P x \<longleftrightarrow> Q x \<rbrakk> \<Longrightarrow> 436 (\<exists>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>B. Q x)" 437by (simp add: Bex_def cong: conj_cong) 438 439lemma bex_cong_simp [cong]: 440 "\<lbrakk> A = B; \<And>x. x \<in> B =simp=> P x \<longleftrightarrow> Q x \<rbrakk> \<Longrightarrow> 441 (\<exists>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>B. Q x)" 442by (simp add: simp_implies_def Bex_def cong: conj_cong) 443 444lemma bex1_def: "(\<exists>!x\<in>X. P x) \<longleftrightarrow> (\<exists>x\<in>X. P x) \<and> (\<forall>x\<in>X. \<forall>y\<in>X. P x \<longrightarrow> P y \<longrightarrow> x = y)" 445 by auto 446 447 448subsection \<open>Basic operations\<close> 449 450subsubsection \<open>Subsets\<close> 451 452lemma subsetI [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> x \<in> B) \<Longrightarrow> A \<subseteq> B" 453 by (simp add: less_eq_set_def le_fun_def) 454 455text \<open> 456 \<^medskip> 457 Map the type \<open>'a set \<Rightarrow> anything\<close> to just \<open>'a\<close>; for overloading constants 458 whose first argument has type \<open>'a set\<close>. 459\<close> 460 461lemma subsetD [elim, intro?]: "A \<subseteq> B \<Longrightarrow> c \<in> A \<Longrightarrow> c \<in> B" 462 by (simp add: less_eq_set_def le_fun_def) 463 \<comment> \<open>Rule in Modus Ponens style.\<close> 464 465lemma rev_subsetD [intro?,no_atp]: "c \<in> A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> c \<in> B" 466 \<comment> \<open>The same, with reversed premises for use with @{method erule} -- cf. @{thm rev_mp}.\<close> 467 by (rule subsetD) 468 469lemma subsetCE [elim,no_atp]: "A \<subseteq> B \<Longrightarrow> (c \<notin> A \<Longrightarrow> P) \<Longrightarrow> (c \<in> B \<Longrightarrow> P) \<Longrightarrow> P" 470 \<comment> \<open>Classical elimination rule.\<close> 471 by (auto simp add: less_eq_set_def le_fun_def) 472 473lemma subset_eq: "A \<subseteq> B \<longleftrightarrow> (\<forall>x\<in>A. x \<in> B)" 474 by blast 475 476lemma contra_subsetD [no_atp]: "A \<subseteq> B \<Longrightarrow> c \<notin> B \<Longrightarrow> c \<notin> A" 477 by blast 478 479lemma subset_refl: "A \<subseteq> A" 480 by (fact order_refl) (* already [iff] *) 481 482lemma subset_trans: "A \<subseteq> B \<Longrightarrow> B \<subseteq> C \<Longrightarrow> A \<subseteq> C" 483 by (fact order_trans) 484 485lemma subset_not_subset_eq [code]: "A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> \<not> B \<subseteq> A" 486 by (fact less_le_not_le) 487 488lemma eq_mem_trans: "a = b \<Longrightarrow> b \<in> A \<Longrightarrow> a \<in> A" 489 by simp 490 491lemmas basic_trans_rules [trans] = 492 order_trans_rules rev_subsetD subsetD eq_mem_trans 493 494 495subsubsection \<open>Equality\<close> 496 497lemma subset_antisym [intro!]: "A \<subseteq> B \<Longrightarrow> B \<subseteq> A \<Longrightarrow> A = B" 498 \<comment> \<open>Anti-symmetry of the subset relation.\<close> 499 by (iprover intro: set_eqI subsetD) 500 501text \<open>\<^medskip> Equality rules from ZF set theory -- are they appropriate here?\<close> 502 503lemma equalityD1: "A = B \<Longrightarrow> A \<subseteq> B" 504 by simp 505 506lemma equalityD2: "A = B \<Longrightarrow> B \<subseteq> A" 507 by simp 508 509text \<open> 510 \<^medskip> 511 Be careful when adding this to the claset as \<open>subset_empty\<close> is in the 512 simpset: \<^prop>\<open>A = {}\<close> goes to \<^prop>\<open>{} \<subseteq> A\<close> and \<^prop>\<open>A \<subseteq> {}\<close> 513 and then back to \<^prop>\<open>A = {}\<close>! 514\<close> 515 516lemma equalityE: "A = B \<Longrightarrow> (A \<subseteq> B \<Longrightarrow> B \<subseteq> A \<Longrightarrow> P) \<Longrightarrow> P" 517 by simp 518 519lemma equalityCE [elim]: "A = B \<Longrightarrow> (c \<in> A \<Longrightarrow> c \<in> B \<Longrightarrow> P) \<Longrightarrow> (c \<notin> A \<Longrightarrow> c \<notin> B \<Longrightarrow> P) \<Longrightarrow> P" 520 by blast 521 522lemma eqset_imp_iff: "A = B \<Longrightarrow> x \<in> A \<longleftrightarrow> x \<in> B" 523 by simp 524 525lemma eqelem_imp_iff: "x = y \<Longrightarrow> x \<in> A \<longleftrightarrow> y \<in> A" 526 by simp 527 528 529subsubsection \<open>The empty set\<close> 530 531lemma empty_def: "{} = {x. False}" 532 by (simp add: bot_set_def bot_fun_def) 533 534lemma empty_iff [simp]: "c \<in> {} \<longleftrightarrow> False" 535 by (simp add: empty_def) 536 537lemma emptyE [elim!]: "a \<in> {} \<Longrightarrow> P" 538 by simp 539 540lemma empty_subsetI [iff]: "{} \<subseteq> A" 541 \<comment> \<open>One effect is to delete the ASSUMPTION \<^prop>\<open>{} \<subseteq> A\<close>\<close> 542 by blast 543 544lemma equals0I: "(\<And>y. y \<in> A \<Longrightarrow> False) \<Longrightarrow> A = {}" 545 by blast 546 547lemma equals0D: "A = {} \<Longrightarrow> a \<notin> A" 548 \<comment> \<open>Use for reasoning about disjointness: \<open>A \<inter> B = {}\<close>\<close> 549 by blast 550 551lemma ball_empty [simp]: "Ball {} P \<longleftrightarrow> True" 552 by (simp add: Ball_def) 553 554lemma bex_empty [simp]: "Bex {} P \<longleftrightarrow> False" 555 by (simp add: Bex_def) 556 557 558subsubsection \<open>The universal set -- UNIV\<close> 559 560abbreviation UNIV :: "'a set" 561 where "UNIV \<equiv> top" 562 563lemma UNIV_def: "UNIV = {x. True}" 564 by (simp add: top_set_def top_fun_def) 565 566lemma UNIV_I [simp]: "x \<in> UNIV" 567 by (simp add: UNIV_def) 568 569declare UNIV_I [intro] \<comment> \<open>unsafe makes it less likely to cause problems\<close> 570 571lemma UNIV_witness [intro?]: "\<exists>x. x \<in> UNIV" 572 by simp 573 574lemma subset_UNIV: "A \<subseteq> UNIV" 575 by (fact top_greatest) (* already simp *) 576 577text \<open> 578 \<^medskip> 579 Eta-contracting these two rules (to remove \<open>P\<close>) causes them 580 to be ignored because of their interaction with congruence rules. 581\<close> 582 583lemma ball_UNIV [simp]: "Ball UNIV P \<longleftrightarrow> All P" 584 by (simp add: Ball_def) 585 586lemma bex_UNIV [simp]: "Bex UNIV P \<longleftrightarrow> Ex P" 587 by (simp add: Bex_def) 588 589lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A" 590 by auto 591 592lemma UNIV_not_empty [iff]: "UNIV \<noteq> {}" 593 by (blast elim: equalityE) 594 595lemma empty_not_UNIV[simp]: "{} \<noteq> UNIV" 596 by blast 597 598 599subsubsection \<open>The Powerset operator -- Pow\<close> 600 601definition Pow :: "'a set \<Rightarrow> 'a set set" 602 where Pow_def: "Pow A = {B. B \<subseteq> A}" 603 604lemma Pow_iff [iff]: "A \<in> Pow B \<longleftrightarrow> A \<subseteq> B" 605 by (simp add: Pow_def) 606 607lemma PowI: "A \<subseteq> B \<Longrightarrow> A \<in> Pow B" 608 by (simp add: Pow_def) 609 610lemma PowD: "A \<in> Pow B \<Longrightarrow> A \<subseteq> B" 611 by (simp add: Pow_def) 612 613lemma Pow_bottom: "{} \<in> Pow B" 614 by simp 615 616lemma Pow_top: "A \<in> Pow A" 617 by simp 618 619lemma Pow_not_empty: "Pow A \<noteq> {}" 620 using Pow_top by blast 621 622 623subsubsection \<open>Set complement\<close> 624 625lemma Compl_iff [simp]: "c \<in> - A \<longleftrightarrow> c \<notin> A" 626 by (simp add: fun_Compl_def uminus_set_def) 627 628lemma ComplI [intro!]: "(c \<in> A \<Longrightarrow> False) \<Longrightarrow> c \<in> - A" 629 by (simp add: fun_Compl_def uminus_set_def) blast 630 631text \<open> 632 \<^medskip> 633 This form, with negated conclusion, works well with the Classical prover. 634 Negated assumptions behave like formulae on the right side of the 635 notional turnstile \dots 636\<close> 637 638lemma ComplD [dest!]: "c \<in> - A \<Longrightarrow> c \<notin> A" 639 by simp 640 641lemmas ComplE = ComplD [elim_format] 642 643lemma Compl_eq: "- A = {x. \<not> x \<in> A}" 644 by blast 645 646 647subsubsection \<open>Binary intersection\<close> 648 649abbreviation inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<inter>" 70) 650 where "(\<inter>) \<equiv> inf" 651 652notation (ASCII) 653 inter (infixl "Int" 70) 654 655lemma Int_def: "A \<inter> B = {x. x \<in> A \<and> x \<in> B}" 656 by (simp add: inf_set_def inf_fun_def) 657 658lemma Int_iff [simp]: "c \<in> A \<inter> B \<longleftrightarrow> c \<in> A \<and> c \<in> B" 659 unfolding Int_def by blast 660 661lemma IntI [intro!]: "c \<in> A \<Longrightarrow> c \<in> B \<Longrightarrow> c \<in> A \<inter> B" 662 by simp 663 664lemma IntD1: "c \<in> A \<inter> B \<Longrightarrow> c \<in> A" 665 by simp 666 667lemma IntD2: "c \<in> A \<inter> B \<Longrightarrow> c \<in> B" 668 by simp 669 670lemma IntE [elim!]: "c \<in> A \<inter> B \<Longrightarrow> (c \<in> A \<Longrightarrow> c \<in> B \<Longrightarrow> P) \<Longrightarrow> P" 671 by simp 672 673lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B" 674 by (fact mono_inf) 675 676 677subsubsection \<open>Binary union\<close> 678 679abbreviation union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<union>" 65) 680 where "union \<equiv> sup" 681 682notation (ASCII) 683 union (infixl "Un" 65) 684 685lemma Un_def: "A \<union> B = {x. x \<in> A \<or> x \<in> B}" 686 by (simp add: sup_set_def sup_fun_def) 687 688lemma Un_iff [simp]: "c \<in> A \<union> B \<longleftrightarrow> c \<in> A \<or> c \<in> B" 689 unfolding Un_def by blast 690 691lemma UnI1 [elim?]: "c \<in> A \<Longrightarrow> c \<in> A \<union> B" 692 by simp 693 694lemma UnI2 [elim?]: "c \<in> B \<Longrightarrow> c \<in> A \<union> B" 695 by simp 696 697text \<open>\<^medskip> Classical introduction rule: no commitment to \<open>A\<close> vs. \<open>B\<close>.\<close> 698lemma UnCI [intro!]: "(c \<notin> B \<Longrightarrow> c \<in> A) \<Longrightarrow> c \<in> A \<union> B" 699 by auto 700 701lemma UnE [elim!]: "c \<in> A \<union> B \<Longrightarrow> (c \<in> A \<Longrightarrow> P) \<Longrightarrow> (c \<in> B \<Longrightarrow> P) \<Longrightarrow> P" 702 unfolding Un_def by blast 703 704lemma insert_def: "insert a B = {x. x = a} \<union> B" 705 by (simp add: insert_compr Un_def) 706 707lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)" 708 by (fact mono_sup) 709 710 711subsubsection \<open>Set difference\<close> 712 713lemma Diff_iff [simp]: "c \<in> A - B \<longleftrightarrow> c \<in> A \<and> c \<notin> B" 714 by (simp add: minus_set_def fun_diff_def) 715 716lemma DiffI [intro!]: "c \<in> A \<Longrightarrow> c \<notin> B \<Longrightarrow> c \<in> A - B" 717 by simp 718 719lemma DiffD1: "c \<in> A - B \<Longrightarrow> c \<in> A" 720 by simp 721 722lemma DiffD2: "c \<in> A - B \<Longrightarrow> c \<in> B \<Longrightarrow> P" 723 by simp 724 725lemma DiffE [elim!]: "c \<in> A - B \<Longrightarrow> (c \<in> A \<Longrightarrow> c \<notin> B \<Longrightarrow> P) \<Longrightarrow> P" 726 by simp 727 728lemma set_diff_eq: "A - B = {x. x \<in> A \<and> x \<notin> B}" 729 by blast 730 731lemma Compl_eq_Diff_UNIV: "- A = (UNIV - A)" 732 by blast 733 734 735subsubsection \<open>Augmenting a set -- \<^const>\<open>insert\<close>\<close> 736 737lemma insert_iff [simp]: "a \<in> insert b A \<longleftrightarrow> a = b \<or> a \<in> A" 738 unfolding insert_def by blast 739 740lemma insertI1: "a \<in> insert a B" 741 by simp 742 743lemma insertI2: "a \<in> B \<Longrightarrow> a \<in> insert b B" 744 by simp 745 746lemma insertE [elim!]: "a \<in> insert b A \<Longrightarrow> (a = b \<Longrightarrow> P) \<Longrightarrow> (a \<in> A \<Longrightarrow> P) \<Longrightarrow> P" 747 unfolding insert_def by blast 748 749lemma insertCI [intro!]: "(a \<notin> B \<Longrightarrow> a = b) \<Longrightarrow> a \<in> insert b B" 750 \<comment> \<open>Classical introduction rule.\<close> 751 by auto 752 753lemma subset_insert_iff: "A \<subseteq> insert x B \<longleftrightarrow> (if x \<in> A then A - {x} \<subseteq> B else A \<subseteq> B)" 754 by auto 755 756lemma set_insert: 757 assumes "x \<in> A" 758 obtains B where "A = insert x B" and "x \<notin> B" 759proof 760 show "A = insert x (A - {x})" using assms by blast 761 show "x \<notin> A - {x}" by blast 762qed 763 764lemma insert_ident: "x \<notin> A \<Longrightarrow> x \<notin> B \<Longrightarrow> insert x A = insert x B \<longleftrightarrow> A = B" 765 by auto 766 767lemma insert_eq_iff: 768 assumes "a \<notin> A" "b \<notin> B" 769 shows "insert a A = insert b B \<longleftrightarrow> 770 (if a = b then A = B else \<exists>C. A = insert b C \<and> b \<notin> C \<and> B = insert a C \<and> a \<notin> C)" 771 (is "?L \<longleftrightarrow> ?R") 772proof 773 show ?R if ?L 774 proof (cases "a = b") 775 case True 776 with assms \<open>?L\<close> show ?R 777 by (simp add: insert_ident) 778 next 779 case False 780 let ?C = "A - {b}" 781 have "A = insert b ?C \<and> b \<notin> ?C \<and> B = insert a ?C \<and> a \<notin> ?C" 782 using assms \<open>?L\<close> \<open>a \<noteq> b\<close> by auto 783 then show ?R using \<open>a \<noteq> b\<close> by auto 784 qed 785 show ?L if ?R 786 using that by (auto split: if_splits) 787qed 788 789lemma insert_UNIV: "insert x UNIV = UNIV" 790 by auto 791 792 793subsubsection \<open>Singletons, using insert\<close> 794 795lemma singletonI [intro!]: "a \<in> {a}" 796 \<comment> \<open>Redundant? But unlike \<open>insertCI\<close>, it proves the subgoal immediately!\<close> 797 by (rule insertI1) 798 799lemma singletonD [dest!]: "b \<in> {a} \<Longrightarrow> b = a" 800 by blast 801 802lemmas singletonE = singletonD [elim_format] 803 804lemma singleton_iff: "b \<in> {a} \<longleftrightarrow> b = a" 805 by blast 806 807lemma singleton_inject [dest!]: "{a} = {b} \<Longrightarrow> a = b" 808 by blast 809 810lemma singleton_insert_inj_eq [iff]: "{b} = insert a A \<longleftrightarrow> a = b \<and> A \<subseteq> {b}" 811 by blast 812 813lemma singleton_insert_inj_eq' [iff]: "insert a A = {b} \<longleftrightarrow> a = b \<and> A \<subseteq> {b}" 814 by blast 815 816lemma subset_singletonD: "A \<subseteq> {x} \<Longrightarrow> A = {} \<or> A = {x}" 817 by fast 818 819lemma subset_singleton_iff: "X \<subseteq> {a} \<longleftrightarrow> X = {} \<or> X = {a}" 820 by blast 821 822lemma singleton_conv [simp]: "{x. x = a} = {a}" 823 by blast 824 825lemma singleton_conv2 [simp]: "{x. a = x} = {a}" 826 by blast 827 828lemma Diff_single_insert: "A - {x} \<subseteq> B \<Longrightarrow> A \<subseteq> insert x B" 829 by blast 830 831lemma subset_Diff_insert: "A \<subseteq> B - insert x C \<longleftrightarrow> A \<subseteq> B - C \<and> x \<notin> A" 832 by blast 833 834lemma doubleton_eq_iff: "{a, b} = {c, d} \<longleftrightarrow> a = c \<and> b = d \<or> a = d \<and> b = c" 835 by (blast elim: equalityE) 836 837lemma Un_singleton_iff: "A \<union> B = {x} \<longleftrightarrow> A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x}" 838 by auto 839 840lemma singleton_Un_iff: "{x} = A \<union> B \<longleftrightarrow> A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x}" 841 by auto 842 843 844subsubsection \<open>Image of a set under a function\<close> 845 846text \<open>Frequently \<open>b\<close> does not have the syntactic form of \<open>f x\<close>.\<close> 847 848definition image :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixr "`" 90) 849 where "f ` A = {y. \<exists>x\<in>A. y = f x}" 850 851lemma image_eqI [simp, intro]: "b = f x \<Longrightarrow> x \<in> A \<Longrightarrow> b \<in> f ` A" 852 unfolding image_def by blast 853 854lemma imageI: "x \<in> A \<Longrightarrow> f x \<in> f ` A" 855 by (rule image_eqI) (rule refl) 856 857lemma rev_image_eqI: "x \<in> A \<Longrightarrow> b = f x \<Longrightarrow> b \<in> f ` A" 858 \<comment> \<open>This version's more effective when we already have the required \<open>x\<close>.\<close> 859 by (rule image_eqI) 860 861lemma imageE [elim!]: 862 assumes "b \<in> (\<lambda>x. f x) ` A" \<comment> \<open>The eta-expansion gives variable-name preservation.\<close> 863 obtains x where "b = f x" and "x \<in> A" 864 using assms unfolding image_def by blast 865 866lemma Compr_image_eq: "{x \<in> f ` A. P x} = f ` {x \<in> A. P (f x)}" 867 by auto 868 869lemma image_Un: "f ` (A \<union> B) = f ` A \<union> f ` B" 870 by blast 871 872lemma image_iff: "z \<in> f ` A \<longleftrightarrow> (\<exists>x\<in>A. z = f x)" 873 by blast 874 875lemma image_subsetI: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> f ` A \<subseteq> B" 876 \<comment> \<open>Replaces the three steps \<open>subsetI\<close>, \<open>imageE\<close>, 877 \<open>hypsubst\<close>, but breaks too many existing proofs.\<close> 878 by blast 879 880lemma image_subset_iff: "f ` A \<subseteq> B \<longleftrightarrow> (\<forall>x\<in>A. f x \<in> B)" 881 \<comment> \<open>This rewrite rule would confuse users if made default.\<close> 882 by blast 883 884lemma subset_imageE: 885 assumes "B \<subseteq> f ` A" 886 obtains C where "C \<subseteq> A" and "B = f ` C" 887proof - 888 from assms have "B = f ` {a \<in> A. f a \<in> B}" by fast 889 moreover have "{a \<in> A. f a \<in> B} \<subseteq> A" by blast 890 ultimately show thesis by (blast intro: that) 891qed 892 893lemma subset_image_iff: "B \<subseteq> f ` A \<longleftrightarrow> (\<exists>AA\<subseteq>A. B = f ` AA)" 894 by (blast elim: subset_imageE) 895 896lemma image_ident [simp]: "(\<lambda>x. x) ` Y = Y" 897 by blast 898 899lemma image_empty [simp]: "f ` {} = {}" 900 by blast 901 902lemma image_insert [simp]: "f ` insert a B = insert (f a) (f ` B)" 903 by blast 904 905lemma image_constant: "x \<in> A \<Longrightarrow> (\<lambda>x. c) ` A = {c}" 906 by auto 907 908lemma image_constant_conv: "(\<lambda>x. c) ` A = (if A = {} then {} else {c})" 909 by auto 910 911lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A" 912 by blast 913 914lemma insert_image [simp]: "x \<in> A \<Longrightarrow> insert (f x) (f ` A) = f ` A" 915 by blast 916 917lemma image_is_empty [iff]: "f ` A = {} \<longleftrightarrow> A = {}" 918 by blast 919 920lemma empty_is_image [iff]: "{} = f ` A \<longleftrightarrow> A = {}" 921 by blast 922 923lemma image_Collect: "f ` {x. P x} = {f x | x. P x}" 924 \<comment> \<open>NOT suitable as a default simp rule: the RHS isn't simpler than the LHS, 925 with its implicit quantifier and conjunction. Also image enjoys better 926 equational properties than does the RHS.\<close> 927 by blast 928 929lemma if_image_distrib [simp]: 930 "(\<lambda>x. if P x then f x else g x) ` S = f ` (S \<inter> {x. P x}) \<union> g ` (S \<inter> {x. \<not> P x})" 931 by auto 932 933lemma image_cong: 934 "f ` M = g ` N" if "M = N" "\<And>x. x \<in> N \<Longrightarrow> f x = g x" 935 using that by (simp add: image_def) 936 937lemma image_cong_simp [cong]: 938 "f ` M = g ` N" if "M = N" "\<And>x. x \<in> N =simp=> f x = g x" 939 using that image_cong [of M N f g] by (simp add: simp_implies_def) 940 941lemma image_Int_subset: "f ` (A \<inter> B) \<subseteq> f ` A \<inter> f ` B" 942 by blast 943 944lemma image_diff_subset: "f ` A - f ` B \<subseteq> f ` (A - B)" 945 by blast 946 947lemma Setcompr_eq_image: "{f x |x. x \<in> A} = f ` A" 948 by blast 949 950lemma setcompr_eq_image: "{f x |x. P x} = f ` {x. P x}" 951 by auto 952 953lemma ball_imageD: "\<forall>x\<in>f ` A. P x \<Longrightarrow> \<forall>x\<in>A. P (f x)" 954 by simp 955 956lemma bex_imageD: "\<exists>x\<in>f ` A. P x \<Longrightarrow> \<exists>x\<in>A. P (f x)" 957 by auto 958 959lemma image_add_0 [simp]: "(+) (0::'a::comm_monoid_add) ` S = S" 960 by auto 961 962 963text \<open>\<^medskip> Range of a function -- just an abbreviation for image!\<close> 964 965abbreviation range :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set" \<comment> \<open>of function\<close> 966 where "range f \<equiv> f ` UNIV" 967 968lemma range_eqI: "b = f x \<Longrightarrow> b \<in> range f" 969 by simp 970 971lemma rangeI: "f x \<in> range f" 972 by simp 973 974lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) \<Longrightarrow> (\<And>x. b = f x \<Longrightarrow> P) \<Longrightarrow> P" 975 by (rule imageE) 976 977lemma full_SetCompr_eq: "{u. \<exists>x. u = f x} = range f" 978 by auto 979 980lemma range_composition: "range (\<lambda>x. f (g x)) = f ` range g" 981 by auto 982 983lemma range_constant [simp]: "range (\<lambda>_. x) = {x}" 984 by (simp add: image_constant) 985 986lemma range_eq_singletonD: "range f = {a} \<Longrightarrow> f x = a" 987 by auto 988 989 990subsubsection \<open>Some rules with \<open>if\<close>\<close> 991 992text \<open>Elimination of \<open>{x. \<dots> \<and> x = t \<and> \<dots>}\<close>.\<close> 993 994lemma Collect_conv_if: "{x. x = a \<and> P x} = (if P a then {a} else {})" 995 by auto 996 997lemma Collect_conv_if2: "{x. a = x \<and> P x} = (if P a then {a} else {})" 998 by auto 999 1000text \<open> 1001 Rewrite rules for boolean case-splitting: faster than \<open>if_split [split]\<close>. 1002\<close> 1003 1004lemma if_split_eq1: "(if Q then x else y) = b \<longleftrightarrow> (Q \<longrightarrow> x = b) \<and> (\<not> Q \<longrightarrow> y = b)" 1005 by (rule if_split) 1006 1007lemma if_split_eq2: "a = (if Q then x else y) \<longleftrightarrow> (Q \<longrightarrow> a = x) \<and> (\<not> Q \<longrightarrow> a = y)" 1008 by (rule if_split) 1009 1010text \<open> 1011 Split ifs on either side of the membership relation. 1012 Not for \<open>[simp]\<close> -- can cause goals to blow up! 1013\<close> 1014 1015lemma if_split_mem1: "(if Q then x else y) \<in> b \<longleftrightarrow> (Q \<longrightarrow> x \<in> b) \<and> (\<not> Q \<longrightarrow> y \<in> b)" 1016 by (rule if_split) 1017 1018lemma if_split_mem2: "(a \<in> (if Q then x else y)) \<longleftrightarrow> (Q \<longrightarrow> a \<in> x) \<and> (\<not> Q \<longrightarrow> a \<in> y)" 1019 by (rule if_split [where P = "\<lambda>S. a \<in> S"]) 1020 1021lemmas split_ifs = if_bool_eq_conj if_split_eq1 if_split_eq2 if_split_mem1 if_split_mem2 1022 1023(*Would like to add these, but the existing code only searches for the 1024 outer-level constant, which in this case is just Set.member; we instead need 1025 to use term-nets to associate patterns with rules. Also, if a rule fails to 1026 apply, then the formula should be kept. 1027 [("uminus", Compl_iff RS iffD1), ("minus", [Diff_iff RS iffD1]), 1028 ("Int", [IntD1,IntD2]), 1029 ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])] 1030 *) 1031 1032 1033subsection \<open>Further operations and lemmas\<close> 1034 1035subsubsection \<open>The ``proper subset'' relation\<close> 1036 1037lemma psubsetI [intro!]: "A \<subseteq> B \<Longrightarrow> A \<noteq> B \<Longrightarrow> A \<subset> B" 1038 unfolding less_le by blast 1039 1040lemma psubsetE [elim!]: "A \<subset> B \<Longrightarrow> (A \<subseteq> B \<Longrightarrow> \<not> B \<subseteq> A \<Longrightarrow> R) \<Longrightarrow> R" 1041 unfolding less_le by blast 1042 1043lemma psubset_insert_iff: 1044 "A \<subset> insert x B \<longleftrightarrow> (if x \<in> B then A \<subset> B else if x \<in> A then A - {x} \<subset> B else A \<subseteq> B)" 1045 by (auto simp add: less_le subset_insert_iff) 1046 1047lemma psubset_eq: "A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> A \<noteq> B" 1048 by (simp only: less_le) 1049 1050lemma psubset_imp_subset: "A \<subset> B \<Longrightarrow> A \<subseteq> B" 1051 by (simp add: psubset_eq) 1052 1053lemma psubset_trans: "A \<subset> B \<Longrightarrow> B \<subset> C \<Longrightarrow> A \<subset> C" 1054 unfolding less_le by (auto dest: subset_antisym) 1055 1056lemma psubsetD: "A \<subset> B \<Longrightarrow> c \<in> A \<Longrightarrow> c \<in> B" 1057 unfolding less_le by (auto dest: subsetD) 1058 1059lemma psubset_subset_trans: "A \<subset> B \<Longrightarrow> B \<subseteq> C \<Longrightarrow> A \<subset> C" 1060 by (auto simp add: psubset_eq) 1061 1062lemma subset_psubset_trans: "A \<subseteq> B \<Longrightarrow> B \<subset> C \<Longrightarrow> A \<subset> C" 1063 by (auto simp add: psubset_eq) 1064 1065lemma psubset_imp_ex_mem: "A \<subset> B \<Longrightarrow> \<exists>b. b \<in> B - A" 1066 unfolding less_le by blast 1067 1068lemma atomize_ball: "(\<And>x. x \<in> A \<Longrightarrow> P x) \<equiv> Trueprop (\<forall>x\<in>A. P x)" 1069 by (simp only: Ball_def atomize_all atomize_imp) 1070 1071lemmas [symmetric, rulify] = atomize_ball 1072 and [symmetric, defn] = atomize_ball 1073 1074lemma image_Pow_mono: "f ` A \<subseteq> B \<Longrightarrow> image f ` Pow A \<subseteq> Pow B" 1075 by blast 1076 1077lemma image_Pow_surj: "f ` A = B \<Longrightarrow> image f ` Pow A = Pow B" 1078 by (blast elim: subset_imageE) 1079 1080 1081subsubsection \<open>Derived rules involving subsets.\<close> 1082 1083text \<open>\<open>insert\<close>.\<close> 1084 1085lemma subset_insertI: "B \<subseteq> insert a B" 1086 by (rule subsetI) (erule insertI2) 1087 1088lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B" 1089 by blast 1090 1091lemma subset_insert: "x \<notin> A \<Longrightarrow> A \<subseteq> insert x B \<longleftrightarrow> A \<subseteq> B" 1092 by blast 1093 1094 1095text \<open>\<^medskip> Finite Union -- the least upper bound of two sets.\<close> 1096 1097lemma Un_upper1: "A \<subseteq> A \<union> B" 1098 by (fact sup_ge1) 1099 1100lemma Un_upper2: "B \<subseteq> A \<union> B" 1101 by (fact sup_ge2) 1102 1103lemma Un_least: "A \<subseteq> C \<Longrightarrow> B \<subseteq> C \<Longrightarrow> A \<union> B \<subseteq> C" 1104 by (fact sup_least) 1105 1106 1107text \<open>\<^medskip> Finite Intersection -- the greatest lower bound of two sets.\<close> 1108 1109lemma Int_lower1: "A \<inter> B \<subseteq> A" 1110 by (fact inf_le1) 1111 1112lemma Int_lower2: "A \<inter> B \<subseteq> B" 1113 by (fact inf_le2) 1114 1115lemma Int_greatest: "C \<subseteq> A \<Longrightarrow> C \<subseteq> B \<Longrightarrow> C \<subseteq> A \<inter> B" 1116 by (fact inf_greatest) 1117 1118 1119text \<open>\<^medskip> Set difference.\<close> 1120 1121lemma Diff_subset[simp]: "A - B \<subseteq> A" 1122 by blast 1123 1124lemma Diff_subset_conv: "A - B \<subseteq> C \<longleftrightarrow> A \<subseteq> B \<union> C" 1125 by blast 1126 1127 1128subsubsection \<open>Equalities involving union, intersection, inclusion, etc.\<close> 1129 1130text \<open>\<open>{}\<close>.\<close> 1131 1132lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})" 1133 \<comment> \<open>supersedes \<open>Collect_False_empty\<close>\<close> 1134 by auto 1135 1136lemma subset_empty [simp]: "A \<subseteq> {} \<longleftrightarrow> A = {}" 1137 by (fact bot_unique) 1138 1139lemma not_psubset_empty [iff]: "\<not> (A < {})" 1140 by (fact not_less_bot) (* FIXME: already simp *) 1141 1142lemma Collect_subset [simp]: "{x\<in>A. P x} \<subseteq> A" by auto 1143 1144lemma Collect_empty_eq [simp]: "Collect P = {} \<longleftrightarrow> (\<forall>x. \<not> P x)" 1145 by blast 1146 1147lemma empty_Collect_eq [simp]: "{} = Collect P \<longleftrightarrow> (\<forall>x. \<not> P x)" 1148 by blast 1149 1150lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}" 1151 by blast 1152 1153lemma Collect_disj_eq: "{x. P x \<or> Q x} = {x. P x} \<union> {x. Q x}" 1154 by blast 1155 1156lemma Collect_imp_eq: "{x. P x \<longrightarrow> Q x} = - {x. P x} \<union> {x. Q x}" 1157 by blast 1158 1159lemma Collect_conj_eq: "{x. P x \<and> Q x} = {x. P x} \<inter> {x. Q x}" 1160 by blast 1161 1162lemma Collect_mono_iff: "Collect P \<subseteq> Collect Q \<longleftrightarrow> (\<forall>x. P x \<longrightarrow> Q x)" 1163 by blast 1164 1165 1166text \<open>\<^medskip> \<open>insert\<close>.\<close> 1167 1168lemma insert_is_Un: "insert a A = {a} \<union> A" 1169 \<comment> \<open>NOT SUITABLE FOR REWRITING since \<open>{a} \<equiv> insert a {}\<close>\<close> 1170 by blast 1171 1172lemma insert_not_empty [simp]: "insert a A \<noteq> {}" 1173 and empty_not_insert [simp]: "{} \<noteq> insert a A" 1174 by blast+ 1175 1176lemma insert_absorb: "a \<in> A \<Longrightarrow> insert a A = A" 1177 \<comment> \<open>\<open>[simp]\<close> causes recursive calls when there are nested inserts\<close> 1178 \<comment> \<open>with \<^emph>\<open>quadratic\<close> running time\<close> 1179 by blast 1180 1181lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A" 1182 by blast 1183 1184lemma insert_commute: "insert x (insert y A) = insert y (insert x A)" 1185 by blast 1186 1187lemma insert_subset [simp]: "insert x A \<subseteq> B \<longleftrightarrow> x \<in> B \<and> A \<subseteq> B" 1188 by blast 1189 1190lemma mk_disjoint_insert: "a \<in> A \<Longrightarrow> \<exists>B. A = insert a B \<and> a \<notin> B" 1191 \<comment> \<open>use new \<open>B\<close> rather than \<open>A - {a}\<close> to avoid infinite unfolding\<close> 1192 by (rule exI [where x = "A - {a}"]) blast 1193 1194lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a \<longrightarrow> P u}" 1195 by auto 1196 1197lemma insert_inter_insert [simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)" 1198 by blast 1199 1200lemma insert_disjoint [simp]: 1201 "insert a A \<inter> B = {} \<longleftrightarrow> a \<notin> B \<and> A \<inter> B = {}" 1202 "{} = insert a A \<inter> B \<longleftrightarrow> a \<notin> B \<and> {} = A \<inter> B" 1203 by auto 1204 1205lemma disjoint_insert [simp]: 1206 "B \<inter> insert a A = {} \<longleftrightarrow> a \<notin> B \<and> B \<inter> A = {}" 1207 "{} = A \<inter> insert b B \<longleftrightarrow> b \<notin> A \<and> {} = A \<inter> B" 1208 by auto 1209 1210 1211text \<open>\<^medskip> \<open>Int\<close>\<close> 1212 1213lemma Int_absorb: "A \<inter> A = A" 1214 by (fact inf_idem) (* already simp *) 1215 1216lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B" 1217 by (fact inf_left_idem) 1218 1219lemma Int_commute: "A \<inter> B = B \<inter> A" 1220 by (fact inf_commute) 1221 1222lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)" 1223 by (fact inf_left_commute) 1224 1225lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)" 1226 by (fact inf_assoc) 1227 1228lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute 1229 \<comment> \<open>Intersection is an AC-operator\<close> 1230 1231lemma Int_absorb1: "B \<subseteq> A \<Longrightarrow> A \<inter> B = B" 1232 by (fact inf_absorb2) 1233 1234lemma Int_absorb2: "A \<subseteq> B \<Longrightarrow> A \<inter> B = A" 1235 by (fact inf_absorb1) 1236 1237lemma Int_empty_left: "{} \<inter> B = {}" 1238 by (fact inf_bot_left) (* already simp *) 1239 1240lemma Int_empty_right: "A \<inter> {} = {}" 1241 by (fact inf_bot_right) (* already simp *) 1242 1243lemma disjoint_eq_subset_Compl: "A \<inter> B = {} \<longleftrightarrow> A \<subseteq> - B" 1244 by blast 1245 1246lemma disjoint_iff_not_equal: "A \<inter> B = {} \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)" 1247 by blast 1248 1249lemma Int_UNIV_left: "UNIV \<inter> B = B" 1250 by (fact inf_top_left) (* already simp *) 1251 1252lemma Int_UNIV_right: "A \<inter> UNIV = A" 1253 by (fact inf_top_right) (* already simp *) 1254 1255lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)" 1256 by (fact inf_sup_distrib1) 1257 1258lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)" 1259 by (fact inf_sup_distrib2) 1260 1261lemma Int_UNIV [simp]: "A \<inter> B = UNIV \<longleftrightarrow> A = UNIV \<and> B = UNIV" 1262 by (fact inf_eq_top_iff) (* already simp *) 1263 1264lemma Int_subset_iff [simp]: "C \<subseteq> A \<inter> B \<longleftrightarrow> C \<subseteq> A \<and> C \<subseteq> B" 1265 by (fact le_inf_iff) 1266 1267lemma Int_Collect: "x \<in> A \<inter> {x. P x} \<longleftrightarrow> x \<in> A \<and> P x" 1268 by blast 1269 1270 1271text \<open>\<^medskip> \<open>Un\<close>.\<close> 1272 1273lemma Un_absorb: "A \<union> A = A" 1274 by (fact sup_idem) (* already simp *) 1275 1276lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B" 1277 by (fact sup_left_idem) 1278 1279lemma Un_commute: "A \<union> B = B \<union> A" 1280 by (fact sup_commute) 1281 1282lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)" 1283 by (fact sup_left_commute) 1284 1285lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)" 1286 by (fact sup_assoc) 1287 1288lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute 1289 \<comment> \<open>Union is an AC-operator\<close> 1290 1291lemma Un_absorb1: "A \<subseteq> B \<Longrightarrow> A \<union> B = B" 1292 by (fact sup_absorb2) 1293 1294lemma Un_absorb2: "B \<subseteq> A \<Longrightarrow> A \<union> B = A" 1295 by (fact sup_absorb1) 1296 1297lemma Un_empty_left: "{} \<union> B = B" 1298 by (fact sup_bot_left) (* already simp *) 1299 1300lemma Un_empty_right: "A \<union> {} = A" 1301 by (fact sup_bot_right) (* already simp *) 1302 1303lemma Un_UNIV_left: "UNIV \<union> B = UNIV" 1304 by (fact sup_top_left) (* already simp *) 1305 1306lemma Un_UNIV_right: "A \<union> UNIV = UNIV" 1307 by (fact sup_top_right) (* already simp *) 1308 1309lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)" 1310 by blast 1311 1312lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)" 1313 by blast 1314 1315lemma Int_insert_left: "(insert a B) \<inter> C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)" 1316 by auto 1317 1318lemma Int_insert_left_if0 [simp]: "a \<notin> C \<Longrightarrow> (insert a B) \<inter> C = B \<inter> C" 1319 by auto 1320 1321lemma Int_insert_left_if1 [simp]: "a \<in> C \<Longrightarrow> (insert a B) \<inter> C = insert a (B \<inter> C)" 1322 by auto 1323 1324lemma Int_insert_right: "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)" 1325 by auto 1326 1327lemma Int_insert_right_if0 [simp]: "a \<notin> A \<Longrightarrow> A \<inter> (insert a B) = A \<inter> B" 1328 by auto 1329 1330lemma Int_insert_right_if1 [simp]: "a \<in> A \<Longrightarrow> A \<inter> (insert a B) = insert a (A \<inter> B)" 1331 by auto 1332 1333lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)" 1334 by (fact sup_inf_distrib1) 1335 1336lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)" 1337 by (fact sup_inf_distrib2) 1338 1339lemma Un_Int_crazy: "(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)" 1340 by blast 1341 1342lemma subset_Un_eq: "A \<subseteq> B \<longleftrightarrow> A \<union> B = B" 1343 by (fact le_iff_sup) 1344 1345lemma Un_empty [iff]: "A \<union> B = {} \<longleftrightarrow> A = {} \<and> B = {}" 1346 by (fact sup_eq_bot_iff) (* FIXME: already simp *) 1347 1348lemma Un_subset_iff [simp]: "A \<union> B \<subseteq> C \<longleftrightarrow> A \<subseteq> C \<and> B \<subseteq> C" 1349 by (fact le_sup_iff) 1350 1351lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A" 1352 by blast 1353 1354lemma Diff_Int2: "A \<inter> C - B \<inter> C = A \<inter> C - B" 1355 by blast 1356 1357lemma subset_UnE: 1358 assumes "C \<subseteq> A \<union> B" 1359 obtains A' B' where "A' \<subseteq> A" "B' \<subseteq> B" "C = A' \<union> B'" 1360proof 1361 show "C \<inter> A \<subseteq> A" "C \<inter> B \<subseteq> B" "C = (C \<inter> A) \<union> (C \<inter> B)" 1362 using assms by blast+ 1363qed 1364 1365text \<open>\<^medskip> Set complement\<close> 1366 1367lemma Compl_disjoint [simp]: "A \<inter> - A = {}" 1368 by (fact inf_compl_bot) 1369 1370lemma Compl_disjoint2 [simp]: "- A \<inter> A = {}" 1371 by (fact compl_inf_bot) 1372 1373lemma Compl_partition: "A \<union> - A = UNIV" 1374 by (fact sup_compl_top) 1375 1376lemma Compl_partition2: "- A \<union> A = UNIV" 1377 by (fact compl_sup_top) 1378 1379lemma double_complement: "- (-A) = A" for A :: "'a set" 1380 by (fact double_compl) (* already simp *) 1381 1382lemma Compl_Un: "- (A \<union> B) = (- A) \<inter> (- B)" 1383 by (fact compl_sup) (* already simp *) 1384 1385lemma Compl_Int: "- (A \<inter> B) = (- A) \<union> (- B)" 1386 by (fact compl_inf) (* already simp *) 1387 1388lemma subset_Compl_self_eq: "A \<subseteq> - A \<longleftrightarrow> A = {}" 1389 by blast 1390 1391lemma Un_Int_assoc_eq: "(A \<inter> B) \<union> C = A \<inter> (B \<union> C) \<longleftrightarrow> C \<subseteq> A" 1392 \<comment> \<open>Halmos, Naive Set Theory, page 16.\<close> 1393 by blast 1394 1395lemma Compl_UNIV_eq: "- UNIV = {}" 1396 by (fact compl_top_eq) (* already simp *) 1397 1398lemma Compl_empty_eq: "- {} = UNIV" 1399 by (fact compl_bot_eq) (* already simp *) 1400 1401lemma Compl_subset_Compl_iff [iff]: "- A \<subseteq> - B \<longleftrightarrow> B \<subseteq> A" 1402 by (fact compl_le_compl_iff) (* FIXME: already simp *) 1403 1404lemma Compl_eq_Compl_iff [iff]: "- A = - B \<longleftrightarrow> A = B" 1405 for A B :: "'a set" 1406 by (fact compl_eq_compl_iff) (* FIXME: already simp *) 1407 1408lemma Compl_insert: "- insert x A = (- A) - {x}" 1409 by blast 1410 1411text \<open>\<^medskip> Bounded quantifiers. 1412 1413 The following are not added to the default simpset because 1414 (a) they duplicate the body and (b) there are no similar rules for \<open>Int\<close>. 1415\<close> 1416 1417lemma ball_Un: "(\<forall>x \<in> A \<union> B. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>B. P x)" 1418 by blast 1419 1420lemma bex_Un: "(\<exists>x \<in> A \<union> B. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>B. P x)" 1421 by blast 1422 1423 1424text \<open>\<^medskip> Set difference.\<close> 1425 1426lemma Diff_eq: "A - B = A \<inter> (- B)" 1427 by blast 1428 1429lemma Diff_eq_empty_iff [simp]: "A - B = {} \<longleftrightarrow> A \<subseteq> B" 1430 by blast 1431 1432lemma Diff_cancel [simp]: "A - A = {}" 1433 by blast 1434 1435lemma Diff_idemp [simp]: "(A - B) - B = A - B" 1436 for A B :: "'a set" 1437 by blast 1438 1439lemma Diff_triv: "A \<inter> B = {} \<Longrightarrow> A - B = A" 1440 by (blast elim: equalityE) 1441 1442lemma empty_Diff [simp]: "{} - A = {}" 1443 by blast 1444 1445lemma Diff_empty [simp]: "A - {} = A" 1446 by blast 1447 1448lemma Diff_UNIV [simp]: "A - UNIV = {}" 1449 by blast 1450 1451lemma Diff_insert0 [simp]: "x \<notin> A \<Longrightarrow> A - insert x B = A - B" 1452 by blast 1453 1454lemma Diff_insert: "A - insert a B = A - B - {a}" 1455 \<comment> \<open>NOT SUITABLE FOR REWRITING since \<open>{a} \<equiv> insert a 0\<close>\<close> 1456 by blast 1457 1458lemma Diff_insert2: "A - insert a B = A - {a} - B" 1459 \<comment> \<open>NOT SUITABLE FOR REWRITING since \<open>{a} \<equiv> insert a 0\<close>\<close> 1460 by blast 1461 1462lemma insert_Diff_if: "insert x A - B = (if x \<in> B then A - B else insert x (A - B))" 1463 by auto 1464 1465lemma insert_Diff1 [simp]: "x \<in> B \<Longrightarrow> insert x A - B = A - B" 1466 by blast 1467 1468lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert a A" 1469 by blast 1470 1471lemma insert_Diff: "a \<in> A \<Longrightarrow> insert a (A - {a}) = A" 1472 by blast 1473 1474lemma Diff_insert_absorb: "x \<notin> A \<Longrightarrow> (insert x A) - {x} = A" 1475 by auto 1476 1477lemma Diff_disjoint [simp]: "A \<inter> (B - A) = {}" 1478 by blast 1479 1480lemma Diff_partition: "A \<subseteq> B \<Longrightarrow> A \<union> (B - A) = B" 1481 by blast 1482 1483lemma double_diff: "A \<subseteq> B \<Longrightarrow> B \<subseteq> C \<Longrightarrow> B - (C - A) = A" 1484 by blast 1485 1486lemma Un_Diff_cancel [simp]: "A \<union> (B - A) = A \<union> B" 1487 by blast 1488 1489lemma Un_Diff_cancel2 [simp]: "(B - A) \<union> A = B \<union> A" 1490 by blast 1491 1492lemma Diff_Un: "A - (B \<union> C) = (A - B) \<inter> (A - C)" 1493 by blast 1494 1495lemma Diff_Int: "A - (B \<inter> C) = (A - B) \<union> (A - C)" 1496 by blast 1497 1498lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B" 1499 by blast 1500 1501lemma Un_Diff: "(A \<union> B) - C = (A - C) \<union> (B - C)" 1502 by blast 1503 1504lemma Int_Diff: "(A \<inter> B) - C = A \<inter> (B - C)" 1505 by blast 1506 1507lemma Diff_Int_distrib: "C \<inter> (A - B) = (C \<inter> A) - (C \<inter> B)" 1508 by blast 1509 1510lemma Diff_Int_distrib2: "(A - B) \<inter> C = (A \<inter> C) - (B \<inter> C)" 1511 by blast 1512 1513lemma Diff_Compl [simp]: "A - (- B) = A \<inter> B" 1514 by auto 1515 1516lemma Compl_Diff_eq [simp]: "- (A - B) = - A \<union> B" 1517 by blast 1518 1519lemma subset_Compl_singleton [simp]: "A \<subseteq> - {b} \<longleftrightarrow> b \<notin> A" 1520 by blast 1521 1522text \<open>\<^medskip> Quantification over type \<^typ>\<open>bool\<close>.\<close> 1523 1524lemma bool_induct: "P True \<Longrightarrow> P False \<Longrightarrow> P x" 1525 by (cases x) auto 1526 1527lemma all_bool_eq: "(\<forall>b. P b) \<longleftrightarrow> P True \<and> P False" 1528 by (auto intro: bool_induct) 1529 1530lemma bool_contrapos: "P x \<Longrightarrow> \<not> P False \<Longrightarrow> P True" 1531 by (cases x) auto 1532 1533lemma ex_bool_eq: "(\<exists>b. P b) \<longleftrightarrow> P True \<or> P False" 1534 by (auto intro: bool_contrapos) 1535 1536lemma UNIV_bool: "UNIV = {False, True}" 1537 by (auto intro: bool_induct) 1538 1539text \<open>\<^medskip> \<open>Pow\<close>\<close> 1540 1541lemma Pow_empty [simp]: "Pow {} = {{}}" 1542 by (auto simp add: Pow_def) 1543 1544lemma Pow_singleton_iff [simp]: "Pow X = {Y} \<longleftrightarrow> X = {} \<and> Y = {}" 1545 by blast (* somewhat slow *) 1546 1547lemma Pow_insert: "Pow (insert a A) = Pow A \<union> (insert a ` Pow A)" 1548 by (blast intro: image_eqI [where ?x = "u - {a}" for u]) 1549 1550lemma Pow_Compl: "Pow (- A) = {- B | B. A \<in> Pow B}" 1551 by (blast intro: exI [where ?x = "- u" for u]) 1552 1553lemma Pow_UNIV [simp]: "Pow UNIV = UNIV" 1554 by blast 1555 1556lemma Un_Pow_subset: "Pow A \<union> Pow B \<subseteq> Pow (A \<union> B)" 1557 by blast 1558 1559lemma Pow_Int_eq [simp]: "Pow (A \<inter> B) = Pow A \<inter> Pow B" 1560 by blast 1561 1562 1563text \<open>\<^medskip> Miscellany.\<close> 1564 1565lemma set_eq_subset: "A = B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A" 1566 by blast 1567 1568lemma subset_iff: "A \<subseteq> B \<longleftrightarrow> (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)" 1569 by blast 1570 1571lemma subset_iff_psubset_eq: "A \<subseteq> B \<longleftrightarrow> A \<subset> B \<or> A = B" 1572 unfolding less_le by blast 1573 1574lemma all_not_in_conv [simp]: "(\<forall>x. x \<notin> A) \<longleftrightarrow> A = {}" 1575 by blast 1576 1577lemma ex_in_conv: "(\<exists>x. x \<in> A) \<longleftrightarrow> A \<noteq> {}" 1578 by blast 1579 1580lemma ball_simps [simp, no_atp]: 1581 "\<And>A P Q. (\<forall>x\<in>A. P x \<or> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<or> Q)" 1582 "\<And>A P Q. (\<forall>x\<in>A. P \<or> Q x) \<longleftrightarrow> (P \<or> (\<forall>x\<in>A. Q x))" 1583 "\<And>A P Q. (\<forall>x\<in>A. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q x))" 1584 "\<And>A P Q. (\<forall>x\<in>A. P x \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<longrightarrow> Q)" 1585 "\<And>P. (\<forall>x\<in>{}. P x) \<longleftrightarrow> True" 1586 "\<And>P. (\<forall>x\<in>UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)" 1587 "\<And>a B P. (\<forall>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<and> (\<forall>x\<in>B. P x))" 1588 "\<And>P Q. (\<forall>x\<in>Collect Q. P x) \<longleftrightarrow> (\<forall>x. Q x \<longrightarrow> P x)" 1589 "\<And>A P f. (\<forall>x\<in>f`A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P (f x))" 1590 "\<And>A P. (\<not> (\<forall>x\<in>A. P x)) \<longleftrightarrow> (\<exists>x\<in>A. \<not> P x)" 1591 by auto 1592 1593lemma bex_simps [simp, no_atp]: 1594 "\<And>A P Q. (\<exists>x\<in>A. P x \<and> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<and> Q)" 1595 "\<And>A P Q. (\<exists>x\<in>A. P \<and> Q x) \<longleftrightarrow> (P \<and> (\<exists>x\<in>A. Q x))" 1596 "\<And>P. (\<exists>x\<in>{}. P x) \<longleftrightarrow> False" 1597 "\<And>P. (\<exists>x\<in>UNIV. P x) \<longleftrightarrow> (\<exists>x. P x)" 1598 "\<And>a B P. (\<exists>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<or> (\<exists>x\<in>B. P x))" 1599 "\<And>P Q. (\<exists>x\<in>Collect Q. P x) \<longleftrightarrow> (\<exists>x. Q x \<and> P x)" 1600 "\<And>A P f. (\<exists>x\<in>f`A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))" 1601 "\<And>A P. (\<not>(\<exists>x\<in>A. P x)) \<longleftrightarrow> (\<forall>x\<in>A. \<not> P x)" 1602 by auto 1603 1604lemma ex_image_cong_iff [simp, no_atp]: 1605 "(\<exists>x. x\<in>f`A) \<longleftrightarrow> A \<noteq> {}" "(\<exists>x. x\<in>f`A \<and> P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))" 1606 by auto 1607 1608subsubsection \<open>Monotonicity of various operations\<close> 1609 1610lemma image_mono: "A \<subseteq> B \<Longrightarrow> f ` A \<subseteq> f ` B" 1611 by blast 1612 1613lemma Pow_mono: "A \<subseteq> B \<Longrightarrow> Pow A \<subseteq> Pow B" 1614 by blast 1615 1616lemma insert_mono: "C \<subseteq> D \<Longrightarrow> insert a C \<subseteq> insert a D" 1617 by blast 1618 1619lemma Un_mono: "A \<subseteq> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> A \<union> B \<subseteq> C \<union> D" 1620 by (fact sup_mono) 1621 1622lemma Int_mono: "A \<subseteq> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> A \<inter> B \<subseteq> C \<inter> D" 1623 by (fact inf_mono) 1624 1625lemma Diff_mono: "A \<subseteq> C \<Longrightarrow> D \<subseteq> B \<Longrightarrow> A - B \<subseteq> C - D" 1626 by blast 1627 1628lemma Compl_anti_mono: "A \<subseteq> B \<Longrightarrow> - B \<subseteq> - A" 1629 by (fact compl_mono) 1630 1631text \<open>\<^medskip> Monotonicity of implications.\<close> 1632 1633lemma in_mono: "A \<subseteq> B \<Longrightarrow> x \<in> A \<longrightarrow> x \<in> B" 1634 by (rule impI) (erule subsetD) 1635 1636lemma conj_mono: "P1 \<longrightarrow> Q1 \<Longrightarrow> P2 \<longrightarrow> Q2 \<Longrightarrow> (P1 \<and> P2) \<longrightarrow> (Q1 \<and> Q2)" 1637 by iprover 1638 1639lemma disj_mono: "P1 \<longrightarrow> Q1 \<Longrightarrow> P2 \<longrightarrow> Q2 \<Longrightarrow> (P1 \<or> P2) \<longrightarrow> (Q1 \<or> Q2)" 1640 by iprover 1641 1642lemma imp_mono: "Q1 \<longrightarrow> P1 \<Longrightarrow> P2 \<longrightarrow> Q2 \<Longrightarrow> (P1 \<longrightarrow> P2) \<longrightarrow> (Q1 \<longrightarrow> Q2)" 1643 by iprover 1644 1645lemma imp_refl: "P \<longrightarrow> P" .. 1646 1647lemma not_mono: "Q \<longrightarrow> P \<Longrightarrow> \<not> P \<longrightarrow> \<not> Q" 1648 by iprover 1649 1650lemma ex_mono: "(\<And>x. P x \<longrightarrow> Q x) \<Longrightarrow> (\<exists>x. P x) \<longrightarrow> (\<exists>x. Q x)" 1651 by iprover 1652 1653lemma all_mono: "(\<And>x. P x \<longrightarrow> Q x) \<Longrightarrow> (\<forall>x. P x) \<longrightarrow> (\<forall>x. Q x)" 1654 by iprover 1655 1656lemma Collect_mono: "(\<And>x. P x \<longrightarrow> Q x) \<Longrightarrow> Collect P \<subseteq> Collect Q" 1657 by blast 1658 1659lemma Int_Collect_mono: "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P x \<longrightarrow> Q x) \<Longrightarrow> A \<inter> Collect P \<subseteq> B \<inter> Collect Q" 1660 by blast 1661 1662lemmas basic_monos = 1663 subset_refl imp_refl disj_mono conj_mono ex_mono Collect_mono in_mono 1664 1665lemma eq_to_mono: "a = b \<Longrightarrow> c = d \<Longrightarrow> b \<longrightarrow> d \<Longrightarrow> a \<longrightarrow> c" 1666 by iprover 1667 1668 1669subsubsection \<open>Inverse image of a function\<close> 1670 1671definition vimage :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set \<Rightarrow> 'a set" (infixr "-`" 90) 1672 where "f -` B \<equiv> {x. f x \<in> B}" 1673 1674lemma vimage_eq [simp]: "a \<in> f -` B \<longleftrightarrow> f a \<in> B" 1675 unfolding vimage_def by blast 1676 1677lemma vimage_singleton_eq: "a \<in> f -` {b} \<longleftrightarrow> f a = b" 1678 by simp 1679 1680lemma vimageI [intro]: "f a = b \<Longrightarrow> b \<in> B \<Longrightarrow> a \<in> f -` B" 1681 unfolding vimage_def by blast 1682 1683lemma vimageI2: "f a \<in> A \<Longrightarrow> a \<in> f -` A" 1684 unfolding vimage_def by fast 1685 1686lemma vimageE [elim!]: "a \<in> f -` B \<Longrightarrow> (\<And>x. f a = x \<Longrightarrow> x \<in> B \<Longrightarrow> P) \<Longrightarrow> P" 1687 unfolding vimage_def by blast 1688 1689lemma vimageD: "a \<in> f -` A \<Longrightarrow> f a \<in> A" 1690 unfolding vimage_def by fast 1691 1692lemma vimage_empty [simp]: "f -` {} = {}" 1693 by blast 1694 1695lemma vimage_Compl: "f -` (- A) = - (f -` A)" 1696 by blast 1697 1698lemma vimage_Un [simp]: "f -` (A \<union> B) = (f -` A) \<union> (f -` B)" 1699 by blast 1700 1701lemma vimage_Int [simp]: "f -` (A \<inter> B) = (f -` A) \<inter> (f -` B)" 1702 by fast 1703 1704lemma vimage_Collect_eq [simp]: "f -` Collect P = {y. P (f y)}" 1705 by blast 1706 1707lemma vimage_Collect: "(\<And>x. P (f x) = Q x) \<Longrightarrow> f -` (Collect P) = Collect Q" 1708 by blast 1709 1710lemma vimage_insert: "f -` (insert a B) = (f -` {a}) \<union> (f -` B)" 1711 \<comment> \<open>NOT suitable for rewriting because of the recurrence of \<open>{a}\<close>.\<close> 1712 by blast 1713 1714lemma vimage_Diff: "f -` (A - B) = (f -` A) - (f -` B)" 1715 by blast 1716 1717lemma vimage_UNIV [simp]: "f -` UNIV = UNIV" 1718 by blast 1719 1720lemma vimage_mono: "A \<subseteq> B \<Longrightarrow> f -` A \<subseteq> f -` B" 1721 \<comment> \<open>monotonicity\<close> 1722 by blast 1723 1724lemma vimage_image_eq: "f -` (f ` A) = {y. \<exists>x\<in>A. f x = f y}" 1725 by (blast intro: sym) 1726 1727lemma image_vimage_subset: "f ` (f -` A) \<subseteq> A" 1728 by blast 1729 1730lemma image_vimage_eq [simp]: "f ` (f -` A) = A \<inter> range f" 1731 by blast 1732 1733lemma image_subset_iff_subset_vimage: "f ` A \<subseteq> B \<longleftrightarrow> A \<subseteq> f -` B" 1734 by blast 1735 1736lemma vimage_const [simp]: "((\<lambda>x. c) -` A) = (if c \<in> A then UNIV else {})" 1737 by auto 1738 1739lemma vimage_if [simp]: "((\<lambda>x. if x \<in> B then c else d) -` A) = 1740 (if c \<in> A then (if d \<in> A then UNIV else B) 1741 else if d \<in> A then - B else {})" 1742 by (auto simp add: vimage_def) 1743 1744lemma vimage_inter_cong: "(\<And> w. w \<in> S \<Longrightarrow> f w = g w) \<Longrightarrow> f -` y \<inter> S = g -` y \<inter> S" 1745 by auto 1746 1747lemma vimage_ident [simp]: "(\<lambda>x. x) -` Y = Y" 1748 by blast 1749 1750 1751subsubsection \<open>Singleton sets\<close> 1752 1753definition is_singleton :: "'a set \<Rightarrow> bool" 1754 where "is_singleton A \<longleftrightarrow> (\<exists>x. A = {x})" 1755 1756lemma is_singletonI [simp, intro!]: "is_singleton {x}" 1757 unfolding is_singleton_def by simp 1758 1759lemma is_singletonI': "A \<noteq> {} \<Longrightarrow> (\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x = y) \<Longrightarrow> is_singleton A" 1760 unfolding is_singleton_def by blast 1761 1762lemma is_singletonE: "is_singleton A \<Longrightarrow> (\<And>x. A = {x} \<Longrightarrow> P) \<Longrightarrow> P" 1763 unfolding is_singleton_def by blast 1764 1765 1766subsubsection \<open>Getting the contents of a singleton set\<close> 1767 1768definition the_elem :: "'a set \<Rightarrow> 'a" 1769 where "the_elem X = (THE x. X = {x})" 1770 1771lemma the_elem_eq [simp]: "the_elem {x} = x" 1772 by (simp add: the_elem_def) 1773 1774lemma is_singleton_the_elem: "is_singleton A \<longleftrightarrow> A = {the_elem A}" 1775 by (auto simp: is_singleton_def) 1776 1777lemma the_elem_image_unique: 1778 assumes "A \<noteq> {}" 1779 and *: "\<And>y. y \<in> A \<Longrightarrow> f y = f x" 1780 shows "the_elem (f ` A) = f x" 1781 unfolding the_elem_def 1782proof (rule the1_equality) 1783 from \<open>A \<noteq> {}\<close> obtain y where "y \<in> A" by auto 1784 with * have "f x = f y" by simp 1785 with \<open>y \<in> A\<close> have "f x \<in> f ` A" by blast 1786 with * show "f ` A = {f x}" by auto 1787 then show "\<exists>!x. f ` A = {x}" by auto 1788qed 1789 1790 1791subsubsection \<open>Least value operator\<close> 1792 1793lemma Least_mono: "mono f \<Longrightarrow> \<exists>x\<in>S. \<forall>y\<in>S. x \<le> y \<Longrightarrow> (LEAST y. y \<in> f ` S) = f (LEAST x. x \<in> S)" 1794 for f :: "'a::order \<Rightarrow> 'b::order" 1795 \<comment> \<open>Courtesy of Stephan Merz\<close> 1796 apply clarify 1797 apply (erule_tac P = "\<lambda>x. x \<in> S" in LeastI2_order) 1798 apply fast 1799 apply (rule LeastI2_order) 1800 apply (auto elim: monoD intro!: order_antisym) 1801 done 1802 1803 1804subsubsection \<open>Monad operation\<close> 1805 1806definition bind :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" 1807 where "bind A f = {x. \<exists>B \<in> f`A. x \<in> B}" 1808 1809hide_const (open) bind 1810 1811lemma bind_bind: "Set.bind (Set.bind A B) C = Set.bind A (\<lambda>x. Set.bind (B x) C)" 1812 for A :: "'a set" 1813 by (auto simp: bind_def) 1814 1815lemma empty_bind [simp]: "Set.bind {} f = {}" 1816 by (simp add: bind_def) 1817 1818lemma nonempty_bind_const: "A \<noteq> {} \<Longrightarrow> Set.bind A (\<lambda>_. B) = B" 1819 by (auto simp: bind_def) 1820 1821lemma bind_const: "Set.bind A (\<lambda>_. B) = (if A = {} then {} else B)" 1822 by (auto simp: bind_def) 1823 1824lemma bind_singleton_conv_image: "Set.bind A (\<lambda>x. {f x}) = f ` A" 1825 by (auto simp: bind_def) 1826 1827 1828subsubsection \<open>Operations for execution\<close> 1829 1830definition is_empty :: "'a set \<Rightarrow> bool" 1831 where [code_abbrev]: "is_empty A \<longleftrightarrow> A = {}" 1832 1833hide_const (open) is_empty 1834 1835definition remove :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" 1836 where [code_abbrev]: "remove x A = A - {x}" 1837 1838hide_const (open) remove 1839 1840lemma member_remove [simp]: "x \<in> Set.remove y A \<longleftrightarrow> x \<in> A \<and> x \<noteq> y" 1841 by (simp add: remove_def) 1842 1843definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" 1844 where [code_abbrev]: "filter P A = {a \<in> A. P a}" 1845 1846hide_const (open) filter 1847 1848lemma member_filter [simp]: "x \<in> Set.filter P A \<longleftrightarrow> x \<in> A \<and> P x" 1849 by (simp add: filter_def) 1850 1851instantiation set :: (equal) equal 1852begin 1853 1854definition "HOL.equal A B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A" 1855 1856instance by standard (auto simp add: equal_set_def) 1857 1858end 1859 1860 1861text \<open>Misc\<close> 1862 1863definition pairwise :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" 1864 where "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y \<in> S. x \<noteq> y \<longrightarrow> R x y)" 1865 1866lemma pairwise_alt: "pairwise R S \<longleftrightarrow> (\<forall>x\<in>S. \<forall>y\<in>S-{x}. R x y)" 1867by (auto simp add: pairwise_def) 1868 1869lemma pairwise_trivial [simp]: "pairwise (\<lambda>i j. j \<noteq> i) I" 1870 by (auto simp: pairwise_def) 1871 1872lemma pairwiseI [intro?]: 1873 "pairwise R S" if "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> R x y" 1874 using that by (simp add: pairwise_def) 1875 1876lemma pairwiseD: 1877 "R x y" and "R y x" 1878 if "pairwise R S" "x \<in> S" and "y \<in> S" and "x \<noteq> y" 1879 using that by (simp_all add: pairwise_def) 1880 1881lemma pairwise_empty [simp]: "pairwise P {}" 1882 by (simp add: pairwise_def) 1883 1884lemma pairwise_singleton [simp]: "pairwise P {A}" 1885 by (simp add: pairwise_def) 1886 1887lemma pairwise_insert: 1888 "pairwise r (insert x s) \<longleftrightarrow> (\<forall>y. y \<in> s \<and> y \<noteq> x \<longrightarrow> r x y \<and> r y x) \<and> pairwise r s" 1889 by (force simp: pairwise_def) 1890 1891lemma pairwise_subset: "pairwise P S \<Longrightarrow> T \<subseteq> S \<Longrightarrow> pairwise P T" 1892 by (force simp: pairwise_def) 1893 1894lemma pairwise_mono: "\<lbrakk>pairwise P A; \<And>x y. P x y \<Longrightarrow> Q x y; B \<subseteq> A\<rbrakk> \<Longrightarrow> pairwise Q B" 1895 by (fastforce simp: pairwise_def) 1896 1897lemma pairwise_imageI: 1898 "pairwise P (f ` A)" 1899 if "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x \<noteq> f y \<Longrightarrow> P (f x) (f y)" 1900 using that by (auto intro: pairwiseI) 1901 1902lemma pairwise_image: "pairwise r (f ` s) \<longleftrightarrow> pairwise (\<lambda>x y. (f x \<noteq> f y) \<longrightarrow> r (f x) (f y)) s" 1903 by (force simp: pairwise_def) 1904 1905definition disjnt :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" 1906 where "disjnt A B \<longleftrightarrow> A \<inter> B = {}" 1907 1908lemma disjnt_self_iff_empty [simp]: "disjnt S S \<longleftrightarrow> S = {}" 1909 by (auto simp: disjnt_def) 1910 1911lemma disjnt_iff: "disjnt A B \<longleftrightarrow> (\<forall>x. \<not> (x \<in> A \<and> x \<in> B))" 1912 by (force simp: disjnt_def) 1913 1914lemma disjnt_sym: "disjnt A B \<Longrightarrow> disjnt B A" 1915 using disjnt_iff by blast 1916 1917lemma disjnt_empty1 [simp]: "disjnt {} A" and disjnt_empty2 [simp]: "disjnt A {}" 1918 by (auto simp: disjnt_def) 1919 1920lemma disjnt_insert1 [simp]: "disjnt (insert a X) Y \<longleftrightarrow> a \<notin> Y \<and> disjnt X Y" 1921 by (simp add: disjnt_def) 1922 1923lemma disjnt_insert2 [simp]: "disjnt Y (insert a X) \<longleftrightarrow> a \<notin> Y \<and> disjnt Y X" 1924 by (simp add: disjnt_def) 1925 1926lemma disjnt_subset1 : "\<lbrakk>disjnt X Y; Z \<subseteq> X\<rbrakk> \<Longrightarrow> disjnt Z Y" 1927 by (auto simp: disjnt_def) 1928 1929lemma disjnt_subset2 : "\<lbrakk>disjnt X Y; Z \<subseteq> Y\<rbrakk> \<Longrightarrow> disjnt X Z" 1930 by (auto simp: disjnt_def) 1931 1932lemma disjnt_Un1 [simp]: "disjnt (A \<union> B) C \<longleftrightarrow> disjnt A C \<and> disjnt B C" 1933 by (auto simp: disjnt_def) 1934 1935lemma disjnt_Un2 [simp]: "disjnt C (A \<union> B) \<longleftrightarrow> disjnt C A \<and> disjnt C B" 1936 by (auto simp: disjnt_def) 1937 1938lemma disjoint_image_subset: "\<lbrakk>pairwise disjnt \<A>; \<And>X. X \<in> \<A> \<Longrightarrow> f X \<subseteq> X\<rbrakk> \<Longrightarrow> pairwise disjnt (f `\<A>)" 1939 unfolding disjnt_def pairwise_def by fast 1940 1941lemma Int_emptyI: "(\<And>x. x \<in> A \<Longrightarrow> x \<in> B \<Longrightarrow> False) \<Longrightarrow> A \<inter> B = {}" 1942 by blast 1943 1944lemma in_image_insert_iff: 1945 assumes "\<And>C. C \<in> B \<Longrightarrow> x \<notin> C" 1946 shows "A \<in> insert x ` B \<longleftrightarrow> x \<in> A \<and> A - {x} \<in> B" (is "?P \<longleftrightarrow> ?Q") 1947proof 1948 assume ?P then show ?Q 1949 using assms by auto 1950next 1951 assume ?Q 1952 then have "x \<in> A" and "A - {x} \<in> B" 1953 by simp_all 1954 from \<open>A - {x} \<in> B\<close> have "insert x (A - {x}) \<in> insert x ` B" 1955 by (rule imageI) 1956 also from \<open>x \<in> A\<close> 1957 have "insert x (A - {x}) = A" 1958 by auto 1959 finally show ?P . 1960qed 1961 1962hide_const (open) member not_member 1963 1964lemmas equalityI = subset_antisym 1965lemmas set_mp = subsetD 1966lemmas set_rev_mp = rev_subsetD 1967 1968ML \<open> 1969val Ball_def = @{thm Ball_def} 1970val Bex_def = @{thm Bex_def} 1971val CollectD = @{thm CollectD} 1972val CollectE = @{thm CollectE} 1973val CollectI = @{thm CollectI} 1974val Collect_conj_eq = @{thm Collect_conj_eq} 1975val Collect_mem_eq = @{thm Collect_mem_eq} 1976val IntD1 = @{thm IntD1} 1977val IntD2 = @{thm IntD2} 1978val IntE = @{thm IntE} 1979val IntI = @{thm IntI} 1980val Int_Collect = @{thm Int_Collect} 1981val UNIV_I = @{thm UNIV_I} 1982val UNIV_witness = @{thm UNIV_witness} 1983val UnE = @{thm UnE} 1984val UnI1 = @{thm UnI1} 1985val UnI2 = @{thm UnI2} 1986val ballE = @{thm ballE} 1987val ballI = @{thm ballI} 1988val bexCI = @{thm bexCI} 1989val bexE = @{thm bexE} 1990val bexI = @{thm bexI} 1991val bex_triv = @{thm bex_triv} 1992val bspec = @{thm bspec} 1993val contra_subsetD = @{thm contra_subsetD} 1994val equalityCE = @{thm equalityCE} 1995val equalityD1 = @{thm equalityD1} 1996val equalityD2 = @{thm equalityD2} 1997val equalityE = @{thm equalityE} 1998val equalityI = @{thm equalityI} 1999val imageE = @{thm imageE} 2000val imageI = @{thm imageI} 2001val image_Un = @{thm image_Un} 2002val image_insert = @{thm image_insert} 2003val insert_commute = @{thm insert_commute} 2004val insert_iff = @{thm insert_iff} 2005val mem_Collect_eq = @{thm mem_Collect_eq} 2006val rangeE = @{thm rangeE} 2007val rangeI = @{thm rangeI} 2008val range_eqI = @{thm range_eqI} 2009val subsetCE = @{thm subsetCE} 2010val subsetD = @{thm subsetD} 2011val subsetI = @{thm subsetI} 2012val subset_refl = @{thm subset_refl} 2013val subset_trans = @{thm subset_trans} 2014val vimageD = @{thm vimageD} 2015val vimageE = @{thm vimageE} 2016val vimageI = @{thm vimageI} 2017val vimageI2 = @{thm vimageI2} 2018val vimage_Collect = @{thm vimage_Collect} 2019val vimage_Int = @{thm vimage_Int} 2020val vimage_Un = @{thm vimage_Un} 2021\<close> 2022 2023end 2024