1(* Title: HOL/Nonstandard_Analysis/HyperNat.thy 2 Author: Jacques D. Fleuriot 3 Copyright: 1998 University of Cambridge 4 5Converted to Isar and polished by lcp 6*) 7 8section \<open>Hypernatural numbers\<close> 9 10theory HyperNat 11 imports StarDef 12begin 13 14type_synonym hypnat = "nat star" 15 16abbreviation hypnat_of_nat :: "nat \<Rightarrow> nat star" 17 where "hypnat_of_nat \<equiv> star_of" 18 19definition hSuc :: "hypnat \<Rightarrow> hypnat" 20 where hSuc_def [transfer_unfold]: "hSuc = *f* Suc" 21 22 23subsection \<open>Properties Transferred from Naturals\<close> 24 25lemma hSuc_not_zero [iff]: "\<And>m. hSuc m \<noteq> 0" 26 by transfer (rule Suc_not_Zero) 27 28lemma zero_not_hSuc [iff]: "\<And>m. 0 \<noteq> hSuc m" 29 by transfer (rule Zero_not_Suc) 30 31lemma hSuc_hSuc_eq [iff]: "\<And>m n. hSuc m = hSuc n \<longleftrightarrow> m = n" 32 by transfer (rule nat.inject) 33 34lemma zero_less_hSuc [iff]: "\<And>n. 0 < hSuc n" 35 by transfer (rule zero_less_Suc) 36 37lemma hypnat_minus_zero [simp]: "\<And>z::hypnat. z - z = 0" 38 by transfer (rule diff_self_eq_0) 39 40lemma hypnat_diff_0_eq_0 [simp]: "\<And>n::hypnat. 0 - n = 0" 41 by transfer (rule diff_0_eq_0) 42 43lemma hypnat_add_is_0 [iff]: "\<And>m n::hypnat. m + n = 0 \<longleftrightarrow> m = 0 \<and> n = 0" 44 by transfer (rule add_is_0) 45 46lemma hypnat_diff_diff_left: "\<And>i j k::hypnat. i - j - k = i - (j + k)" 47 by transfer (rule diff_diff_left) 48 49lemma hypnat_diff_commute: "\<And>i j k::hypnat. i - j - k = i - k - j" 50 by transfer (rule diff_commute) 51 52lemma hypnat_diff_add_inverse [simp]: "\<And>m n::hypnat. n + m - n = m" 53 by transfer (rule diff_add_inverse) 54 55lemma hypnat_diff_add_inverse2 [simp]: "\<And>m n::hypnat. m + n - n = m" 56 by transfer (rule diff_add_inverse2) 57 58lemma hypnat_diff_cancel [simp]: "\<And>k m n::hypnat. (k + m) - (k + n) = m - n" 59 by transfer (rule diff_cancel) 60 61lemma hypnat_diff_cancel2 [simp]: "\<And>k m n::hypnat. (m + k) - (n + k) = m - n" 62 by transfer (rule diff_cancel2) 63 64lemma hypnat_diff_add_0 [simp]: "\<And>m n::hypnat. n - (n + m) = 0" 65 by transfer (rule diff_add_0) 66 67lemma hypnat_diff_mult_distrib: "\<And>k m n::hypnat. (m - n) * k = (m * k) - (n * k)" 68 by transfer (rule diff_mult_distrib) 69 70lemma hypnat_diff_mult_distrib2: "\<And>k m n::hypnat. k * (m - n) = (k * m) - (k * n)" 71 by transfer (rule diff_mult_distrib2) 72 73lemma hypnat_le_zero_cancel [iff]: "\<And>n::hypnat. n \<le> 0 \<longleftrightarrow> n = 0" 74 by transfer (rule le_0_eq) 75 76lemma hypnat_mult_is_0 [simp]: "\<And>m n::hypnat. m * n = 0 \<longleftrightarrow> m = 0 \<or> n = 0" 77 by transfer (rule mult_is_0) 78 79lemma hypnat_diff_is_0_eq [simp]: "\<And>m n::hypnat. m - n = 0 \<longleftrightarrow> m \<le> n" 80 by transfer (rule diff_is_0_eq) 81 82lemma hypnat_not_less0 [iff]: "\<And>n::hypnat. \<not> n < 0" 83 by transfer (rule not_less0) 84 85lemma hypnat_less_one [iff]: "\<And>n::hypnat. n < 1 \<longleftrightarrow> n = 0" 86 by transfer (rule less_one) 87 88lemma hypnat_add_diff_inverse: "\<And>m n::hypnat. \<not> m < n \<Longrightarrow> n + (m - n) = m" 89 by transfer (rule add_diff_inverse) 90 91lemma hypnat_le_add_diff_inverse [simp]: "\<And>m n::hypnat. n \<le> m \<Longrightarrow> n + (m - n) = m" 92 by transfer (rule le_add_diff_inverse) 93 94lemma hypnat_le_add_diff_inverse2 [simp]: "\<And>m n::hypnat. n \<le> m \<Longrightarrow> (m - n) + n = m" 95 by transfer (rule le_add_diff_inverse2) 96 97declare hypnat_le_add_diff_inverse2 [OF order_less_imp_le] 98 99lemma hypnat_le0 [iff]: "\<And>n::hypnat. 0 \<le> n" 100 by transfer (rule le0) 101 102lemma hypnat_le_add1 [simp]: "\<And>x n::hypnat. x \<le> x + n" 103 by transfer (rule le_add1) 104 105lemma hypnat_add_self_le [simp]: "\<And>x n::hypnat. x \<le> n + x" 106 by transfer (rule le_add2) 107 108lemma hypnat_add_one_self_less [simp]: "x < x + 1" for x :: hypnat 109 by (fact less_add_one) 110 111lemma hypnat_neq0_conv [iff]: "\<And>n::hypnat. n \<noteq> 0 \<longleftrightarrow> 0 < n" 112 by transfer (rule neq0_conv) 113 114lemma hypnat_gt_zero_iff: "0 < n \<longleftrightarrow> 1 \<le> n" for n :: hypnat 115 by (auto simp add: linorder_not_less [symmetric]) 116 117lemma hypnat_gt_zero_iff2: "0 < n \<longleftrightarrow> (\<exists>m. n = m + 1)" for n :: hypnat 118 by (auto intro!: add_nonneg_pos exI[of _ "n - 1"] simp: hypnat_gt_zero_iff) 119 120lemma hypnat_add_self_not_less: "\<not> x + y < x" for x y :: hypnat 121 by (simp add: linorder_not_le [symmetric] add.commute [of x]) 122 123lemma hypnat_diff_split: "P (a - b) \<longleftrightarrow> (a < b \<longrightarrow> P 0) \<and> (\<forall>d. a = b + d \<longrightarrow> P d)" 124 for a b :: hypnat 125 \<comment> \<open>elimination of \<open>-\<close> on \<open>hypnat\<close>\<close> 126proof (cases "a < b" rule: case_split) 127 case True 128 then show ?thesis 129 by (auto simp add: hypnat_add_self_not_less order_less_imp_le hypnat_diff_is_0_eq [THEN iffD2]) 130next 131 case False 132 then show ?thesis 133 by (auto simp add: linorder_not_less dest: order_le_less_trans) 134qed 135 136 137subsection \<open>Properties of the set of embedded natural numbers\<close> 138 139lemma of_nat_eq_star_of [simp]: "of_nat = star_of" 140proof 141 show "of_nat n = star_of n" for n 142 by transfer simp 143qed 144 145lemma Nats_eq_Standard: "(Nats :: nat star set) = Standard" 146 by (auto simp: Nats_def Standard_def) 147 148lemma hypnat_of_nat_mem_Nats [simp]: "hypnat_of_nat n \<in> Nats" 149 by (simp add: Nats_eq_Standard) 150 151lemma hypnat_of_nat_one [simp]: "hypnat_of_nat (Suc 0) = 1" 152 by transfer simp 153 154lemma hypnat_of_nat_Suc [simp]: "hypnat_of_nat (Suc n) = hypnat_of_nat n + 1" 155 by transfer simp 156 157lemma of_nat_eq_add: 158 fixes d::hypnat 159 shows "of_nat m = of_nat n + d \<Longrightarrow> d \<in> range of_nat" 160proof (induct n arbitrary: d) 161 case (Suc n) 162 then show ?case 163 by (metis Nats_def Nats_eq_Standard Standard_simps(4) hypnat_diff_add_inverse of_nat_in_Nats) 164qed auto 165 166lemma Nats_diff [simp]: "a \<in> Nats \<Longrightarrow> b \<in> Nats \<Longrightarrow> a - b \<in> Nats" for a b :: hypnat 167 by (simp add: Nats_eq_Standard) 168 169 170subsection \<open>Infinite Hypernatural Numbers -- \<^term>\<open>HNatInfinite\<close>\<close> 171 172text \<open>The set of infinite hypernatural numbers.\<close> 173definition HNatInfinite :: "hypnat set" 174 where "HNatInfinite = {n. n \<notin> Nats}" 175 176lemma Nats_not_HNatInfinite_iff: "x \<in> Nats \<longleftrightarrow> x \<notin> HNatInfinite" 177 by (simp add: HNatInfinite_def) 178 179lemma HNatInfinite_not_Nats_iff: "x \<in> HNatInfinite \<longleftrightarrow> x \<notin> Nats" 180 by (simp add: HNatInfinite_def) 181 182lemma star_of_neq_HNatInfinite: "N \<in> HNatInfinite \<Longrightarrow> star_of n \<noteq> N" 183 by (auto simp add: HNatInfinite_def Nats_eq_Standard) 184 185lemma star_of_Suc_lessI: "\<And>N. star_of n < N \<Longrightarrow> star_of (Suc n) \<noteq> N \<Longrightarrow> star_of (Suc n) < N" 186 by transfer (rule Suc_lessI) 187 188lemma star_of_less_HNatInfinite: 189 assumes N: "N \<in> HNatInfinite" 190 shows "star_of n < N" 191proof (induct n) 192 case 0 193 from N have "star_of 0 \<noteq> N" 194 by (rule star_of_neq_HNatInfinite) 195 then show ?case by simp 196next 197 case (Suc n) 198 from N have "star_of (Suc n) \<noteq> N" 199 by (rule star_of_neq_HNatInfinite) 200 with Suc show ?case 201 by (rule star_of_Suc_lessI) 202qed 203 204lemma star_of_le_HNatInfinite: "N \<in> HNatInfinite \<Longrightarrow> star_of n \<le> N" 205 by (rule star_of_less_HNatInfinite [THEN order_less_imp_le]) 206 207 208subsubsection \<open>Closure Rules\<close> 209 210lemma Nats_less_HNatInfinite: "x \<in> Nats \<Longrightarrow> y \<in> HNatInfinite \<Longrightarrow> x < y" 211 by (auto simp add: Nats_def star_of_less_HNatInfinite) 212 213lemma Nats_le_HNatInfinite: "x \<in> Nats \<Longrightarrow> y \<in> HNatInfinite \<Longrightarrow> x \<le> y" 214 by (rule Nats_less_HNatInfinite [THEN order_less_imp_le]) 215 216lemma zero_less_HNatInfinite: "x \<in> HNatInfinite \<Longrightarrow> 0 < x" 217 by (simp add: Nats_less_HNatInfinite) 218 219lemma one_less_HNatInfinite: "x \<in> HNatInfinite \<Longrightarrow> 1 < x" 220 by (simp add: Nats_less_HNatInfinite) 221 222lemma one_le_HNatInfinite: "x \<in> HNatInfinite \<Longrightarrow> 1 \<le> x" 223 by (simp add: Nats_le_HNatInfinite) 224 225lemma zero_not_mem_HNatInfinite [simp]: "0 \<notin> HNatInfinite" 226 by (simp add: HNatInfinite_def) 227 228lemma Nats_downward_closed: "x \<in> Nats \<Longrightarrow> y \<le> x \<Longrightarrow> y \<in> Nats" for x y :: hypnat 229 using HNatInfinite_not_Nats_iff Nats_le_HNatInfinite by fastforce 230 231lemma HNatInfinite_upward_closed: "x \<in> HNatInfinite \<Longrightarrow> x \<le> y \<Longrightarrow> y \<in> HNatInfinite" 232 using HNatInfinite_not_Nats_iff Nats_downward_closed by blast 233 234lemma HNatInfinite_add: "x \<in> HNatInfinite \<Longrightarrow> x + y \<in> HNatInfinite" 235 using HNatInfinite_upward_closed hypnat_le_add1 by blast 236 237lemma HNatInfinite_diff: "\<lbrakk>x \<in> HNatInfinite; y \<in> Nats\<rbrakk> \<Longrightarrow> x - y \<in> HNatInfinite" 238 by (metis HNatInfinite_not_Nats_iff Nats_add Nats_le_HNatInfinite le_add_diff_inverse) 239 240lemma HNatInfinite_is_Suc: "x \<in> HNatInfinite \<Longrightarrow> \<exists>y. x = y + 1" for x :: hypnat 241 using hypnat_gt_zero_iff2 zero_less_HNatInfinite by blast 242 243 244subsection \<open>Existence of an infinite hypernatural number\<close> 245 246text \<open>\<open>\<omega>\<close> is in fact an infinite hypernatural number = \<open>[<1, 2, 3, \<dots>>]\<close>\<close> 247definition whn :: hypnat 248 where hypnat_omega_def: "whn = star_n (\<lambda>n::nat. n)" 249 250lemma hypnat_of_nat_neq_whn: "hypnat_of_nat n \<noteq> whn" 251 by (simp add: FreeUltrafilterNat.singleton' hypnat_omega_def star_of_def star_n_eq_iff) 252 253lemma whn_neq_hypnat_of_nat: "whn \<noteq> hypnat_of_nat n" 254 by (simp add: FreeUltrafilterNat.singleton hypnat_omega_def star_of_def star_n_eq_iff) 255 256lemma whn_not_Nats [simp]: "whn \<notin> Nats" 257 by (simp add: Nats_def image_def whn_neq_hypnat_of_nat) 258 259lemma HNatInfinite_whn [simp]: "whn \<in> HNatInfinite" 260 by (simp add: HNatInfinite_def) 261 262lemma lemma_unbounded_set [simp]: "eventually (\<lambda>n::nat. m < n) \<U>" 263 by (rule filter_leD[OF FreeUltrafilterNat.le_cofinite]) 264 (auto simp add: cofinite_eq_sequentially eventually_at_top_dense) 265 266lemma hypnat_of_nat_eq: "hypnat_of_nat m = star_n (\<lambda>n::nat. m)" 267 by (simp add: star_of_def) 268 269lemma SHNat_eq: "Nats = {n. \<exists>N. n = hypnat_of_nat N}" 270 by (simp add: Nats_def image_def) 271 272lemma Nats_less_whn: "n \<in> Nats \<Longrightarrow> n < whn" 273 by (simp add: Nats_less_HNatInfinite) 274 275lemma Nats_le_whn: "n \<in> Nats \<Longrightarrow> n \<le> whn" 276 by (simp add: Nats_le_HNatInfinite) 277 278lemma hypnat_of_nat_less_whn [simp]: "hypnat_of_nat n < whn" 279 by (simp add: Nats_less_whn) 280 281lemma hypnat_of_nat_le_whn [simp]: "hypnat_of_nat n \<le> whn" 282 by (simp add: Nats_le_whn) 283 284lemma hypnat_zero_less_hypnat_omega [simp]: "0 < whn" 285 by (simp add: Nats_less_whn) 286 287lemma hypnat_one_less_hypnat_omega [simp]: "1 < whn" 288 by (simp add: Nats_less_whn) 289 290 291subsubsection \<open>Alternative characterization of the set of infinite hypernaturals\<close> 292 293text \<open>\<^term>\<open>HNatInfinite = {N. \<forall>n \<in> Nats. n < N}\<close>\<close> 294 295text\<open>unused, but possibly interesting\<close> 296lemma HNatInfinite_FreeUltrafilterNat_eventually: 297 assumes "\<And>k::nat. eventually (\<lambda>n. f n \<noteq> k) \<U>" 298 shows "eventually (\<lambda>n. m < f n) \<U>" 299proof (induct m) 300 case 0 301 then show ?case 302 using assms eventually_mono by fastforce 303next 304 case (Suc m) 305 then show ?case 306 using assms [of "Suc m"] eventually_elim2 by fastforce 307qed 308 309lemma HNatInfinite_iff: "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}" 310 using HNatInfinite_def Nats_less_HNatInfinite by auto 311 312 313subsubsection \<open>Alternative Characterization of \<^term>\<open>HNatInfinite\<close> using Free Ultrafilter\<close> 314 315lemma HNatInfinite_FreeUltrafilterNat: 316 "star_n X \<in> HNatInfinite \<Longrightarrow> \<forall>u. eventually (\<lambda>n. u < X n) \<U>" 317 by (metis (full_types) starP2_star_of starP_star_n star_less_def star_of_less_HNatInfinite) 318 319lemma FreeUltrafilterNat_HNatInfinite: 320 "\<forall>u. eventually (\<lambda>n. u < X n) \<U> \<Longrightarrow> star_n X \<in> HNatInfinite" 321 by (auto simp add: star_less_def starP2_star_n HNatInfinite_iff SHNat_eq hypnat_of_nat_eq) 322 323lemma HNatInfinite_FreeUltrafilterNat_iff: 324 "(star_n X \<in> HNatInfinite) = (\<forall>u. eventually (\<lambda>n. u < X n) \<U>)" 325 by (rule iffI [OF HNatInfinite_FreeUltrafilterNat FreeUltrafilterNat_HNatInfinite]) 326 327 328subsection \<open>Embedding of the Hypernaturals into other types\<close> 329 330definition of_hypnat :: "hypnat \<Rightarrow> 'a::semiring_1_cancel star" 331 where of_hypnat_def [transfer_unfold]: "of_hypnat = *f* of_nat" 332 333lemma of_hypnat_0 [simp]: "of_hypnat 0 = 0" 334 by transfer (rule of_nat_0) 335 336lemma of_hypnat_1 [simp]: "of_hypnat 1 = 1" 337 by transfer (rule of_nat_1) 338 339lemma of_hypnat_hSuc: "\<And>m. of_hypnat (hSuc m) = 1 + of_hypnat m" 340 by transfer (rule of_nat_Suc) 341 342lemma of_hypnat_add [simp]: "\<And>m n. of_hypnat (m + n) = of_hypnat m + of_hypnat n" 343 by transfer (rule of_nat_add) 344 345lemma of_hypnat_mult [simp]: "\<And>m n. of_hypnat (m * n) = of_hypnat m * of_hypnat n" 346 by transfer (rule of_nat_mult) 347 348lemma of_hypnat_less_iff [simp]: 349 "\<And>m n. of_hypnat m < (of_hypnat n::'a::linordered_semidom star) \<longleftrightarrow> m < n" 350 by transfer (rule of_nat_less_iff) 351 352lemma of_hypnat_0_less_iff [simp]: 353 "\<And>n. 0 < (of_hypnat n::'a::linordered_semidom star) \<longleftrightarrow> 0 < n" 354 by transfer (rule of_nat_0_less_iff) 355 356lemma of_hypnat_less_0_iff [simp]: "\<And>m. \<not> (of_hypnat m::'a::linordered_semidom star) < 0" 357 by transfer (rule of_nat_less_0_iff) 358 359lemma of_hypnat_le_iff [simp]: 360 "\<And>m n. of_hypnat m \<le> (of_hypnat n::'a::linordered_semidom star) \<longleftrightarrow> m \<le> n" 361 by transfer (rule of_nat_le_iff) 362 363lemma of_hypnat_0_le_iff [simp]: "\<And>n. 0 \<le> (of_hypnat n::'a::linordered_semidom star)" 364 by transfer (rule of_nat_0_le_iff) 365 366lemma of_hypnat_le_0_iff [simp]: "\<And>m. (of_hypnat m::'a::linordered_semidom star) \<le> 0 \<longleftrightarrow> m = 0" 367 by transfer (rule of_nat_le_0_iff) 368 369lemma of_hypnat_eq_iff [simp]: 370 "\<And>m n. of_hypnat m = (of_hypnat n::'a::linordered_semidom star) \<longleftrightarrow> m = n" 371 by transfer (rule of_nat_eq_iff) 372 373lemma of_hypnat_eq_0_iff [simp]: "\<And>m. (of_hypnat m::'a::linordered_semidom star) = 0 \<longleftrightarrow> m = 0" 374 by transfer (rule of_nat_eq_0_iff) 375 376lemma HNatInfinite_of_hypnat_gt_zero: 377 "N \<in> HNatInfinite \<Longrightarrow> (0::'a::linordered_semidom star) < of_hypnat N" 378 by (rule ccontr) (simp add: linorder_not_less) 379 380end 381