src/HOL/Hyperreal/HyperNat.thy
author wenzelm
Sun Apr 09 18:51:13 2006 +0200 (2006-04-09)
changeset 19380 b808efaa5828
parent 17433 4cf2e7980529
child 19765 dfe940911617
permissions -rw-r--r--
tuned syntax/abbreviations;
     1 (*  Title       : HyperNat.thy
     2     Author      : Jacques D. Fleuriot
     3     Copyright   : 1998  University of Cambridge
     4 
     5 Converted to Isar and polished by lcp    
     6 *)
     7 
     8 header{*Hypernatural numbers*}
     9 
    10 theory HyperNat
    11 imports Star
    12 begin
    13 
    14 types hypnat = "nat star"
    15 
    16 abbreviation
    17   hypnat_of_nat :: "nat => nat star"
    18   "hypnat_of_nat == star_of"
    19 
    20 subsection{*Properties Transferred from Naturals*}
    21 
    22 lemma hypnat_minus_zero [simp]: "!!z. z - z = (0::hypnat)"
    23 by transfer (rule diff_self_eq_0)
    24 
    25 lemma hypnat_diff_0_eq_0 [simp]: "!!n. (0::hypnat) - n = 0"
    26 by transfer (rule diff_0_eq_0)
    27 
    28 lemma hypnat_add_is_0 [iff]: "!!m n. (m+n = (0::hypnat)) = (m=0 & n=0)"
    29 by transfer (rule add_is_0)
    30 
    31 lemma hypnat_diff_diff_left: "!!i j k. (i::hypnat) - j - k = i - (j+k)"
    32 by transfer (rule diff_diff_left)
    33 
    34 lemma hypnat_diff_commute: "!!i j k. (i::hypnat) - j - k = i-k-j"
    35 by transfer (rule diff_commute)
    36 
    37 lemma hypnat_diff_add_inverse [simp]: "!!m n. ((n::hypnat) + m) - n = m"
    38 by transfer (rule diff_add_inverse)
    39 
    40 lemma hypnat_diff_add_inverse2 [simp]:  "!!m n. ((m::hypnat) + n) - n = m"
    41 by transfer (rule diff_add_inverse2)
    42 
    43 lemma hypnat_diff_cancel [simp]: "!!k m n. ((k::hypnat) + m) - (k+n) = m - n"
    44 by transfer (rule diff_cancel)
    45 
    46 lemma hypnat_diff_cancel2 [simp]: "!!k m n. ((m::hypnat) + k) - (n+k) = m - n"
    47 by transfer (rule diff_cancel2)
    48 
    49 lemma hypnat_diff_add_0 [simp]: "!!m n. (n::hypnat) - (n+m) = (0::hypnat)"
    50 by transfer (rule diff_add_0)
    51 
    52 lemma hypnat_diff_mult_distrib: "!!k m n. ((m::hypnat) - n) * k = (m * k) - (n * k)"
    53 by transfer (rule diff_mult_distrib)
    54 
    55 lemma hypnat_diff_mult_distrib2: "!!k m n. (k::hypnat) * (m - n) = (k * m) - (k * n)"
    56 by transfer (rule diff_mult_distrib2)
    57 
    58 lemma hypnat_le_zero_cancel [iff]: "!!n. (n \<le> (0::hypnat)) = (n = 0)"
    59 by transfer (rule le_0_eq)
    60 
    61 lemma hypnat_mult_is_0 [simp]: "!!m n. (m*n = (0::hypnat)) = (m=0 | n=0)"
    62 by transfer (rule mult_is_0)
    63 
    64 lemma hypnat_diff_is_0_eq [simp]: "!!m n. ((m::hypnat) - n = 0) = (m \<le> n)"
    65 by transfer (rule diff_is_0_eq)
    66 
    67 lemma hypnat_not_less0 [iff]: "!!n. ~ n < (0::hypnat)"
    68 by transfer (rule not_less0)
    69 
    70 lemma hypnat_less_one [iff]:
    71       "!!n. (n < (1::hypnat)) = (n=0)"
    72 by transfer (rule less_one)
    73 
    74 lemma hypnat_add_diff_inverse: "!!m n. ~ m<n ==> n+(m-n) = (m::hypnat)"
    75 by transfer (rule add_diff_inverse)
    76 
    77 lemma hypnat_le_add_diff_inverse [simp]: "!!m n. n \<le> m ==> n+(m-n) = (m::hypnat)"
    78 by transfer (rule le_add_diff_inverse)
    79 
    80 lemma hypnat_le_add_diff_inverse2 [simp]: "!!m n. n\<le>m ==> (m-n)+n = (m::hypnat)"
    81 by transfer (rule le_add_diff_inverse2)
    82 
    83 declare hypnat_le_add_diff_inverse2 [OF order_less_imp_le]
    84 
    85 lemma hypnat_le0 [iff]: "!!n. (0::hypnat) \<le> n"
    86 by transfer (rule le0)
    87 
    88 lemma hypnat_add_self_le [simp]: "!!x n. (x::hypnat) \<le> n + x"
    89 by transfer (rule le_add2)
    90 
    91 lemma hypnat_add_one_self_less [simp]: "(x::hypnat) < x + (1::hypnat)"
    92 by (insert add_strict_left_mono [OF zero_less_one], auto)
    93 
    94 lemma hypnat_neq0_conv [iff]: "!!n. (n \<noteq> 0) = (0 < (n::hypnat))"
    95 by transfer (rule neq0_conv)
    96 
    97 lemma hypnat_gt_zero_iff: "((0::hypnat) < n) = ((1::hypnat) \<le> n)"
    98 by (auto simp add: linorder_not_less [symmetric])
    99 
   100 lemma hypnat_gt_zero_iff2: "(0 < n) = (\<exists>m. n = m + (1::hypnat))"
   101 apply safe
   102  apply (rule_tac x = "n - (1::hypnat) " in exI)
   103  apply (simp add: hypnat_gt_zero_iff) 
   104 apply (insert add_le_less_mono [OF _ zero_less_one, of 0], auto) 
   105 done
   106 
   107 lemma hypnat_add_self_not_less: "~ (x + y < (x::hypnat))"
   108 by (simp add: linorder_not_le [symmetric] add_commute [of x]) 
   109 
   110 lemma hypnat_diff_split:
   111     "P(a - b::hypnat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
   112     -- {* elimination of @{text -} on @{text hypnat} *}
   113 proof (cases "a<b" rule: case_split)
   114   case True
   115     thus ?thesis
   116       by (auto simp add: hypnat_add_self_not_less order_less_imp_le 
   117                          hypnat_diff_is_0_eq [THEN iffD2])
   118 next
   119   case False
   120     thus ?thesis
   121       by (auto simp add: linorder_not_less dest: order_le_less_trans) 
   122 qed
   123 
   124 subsection{*Properties of the set of embedded natural numbers*}
   125 
   126 lemma hypnat_of_nat_def: "hypnat_of_nat m == of_nat m"
   127 by (transfer, simp)
   128 
   129 lemma hypnat_of_nat_one [simp]: "hypnat_of_nat (Suc 0) = (1::hypnat)"
   130 by simp
   131 
   132 lemma hypnat_of_nat_Suc [simp]:
   133      "hypnat_of_nat (Suc n) = hypnat_of_nat n + (1::hypnat)"
   134 by (simp add: hypnat_of_nat_def)
   135 
   136 lemma of_nat_eq_add [rule_format]:
   137      "\<forall>d::hypnat. of_nat m = of_nat n + d --> d \<in> range of_nat"
   138 apply (induct n) 
   139 apply (auto simp add: add_assoc) 
   140 apply (case_tac x) 
   141 apply (auto simp add: add_commute [of 1]) 
   142 done
   143 
   144 lemma Nats_diff [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> (a-b :: hypnat) \<in> Nats"
   145 by (auto simp add: of_nat_eq_add Nats_def split: hypnat_diff_split)
   146 
   147 
   148 
   149 subsection{*Existence of an infinite hypernatural number*}
   150 
   151 consts whn :: hypnat
   152 
   153 defs
   154   (* omega is in fact an infinite hypernatural number = [<1,2,3,...>] *)
   155   hypnat_omega_def:  "whn == star_n (%n::nat. n)"
   156 
   157 text{*Existence of infinite number not corresponding to any natural number
   158 follows because member @{term FreeUltrafilterNat} is not finite.
   159 See @{text HyperDef.thy} for similar argument.*}
   160 
   161 
   162 lemma lemma_unbounded_set [simp]: "{n::nat. m < n} \<in> FreeUltrafilterNat"
   163 apply (insert finite_atMost [of m]) 
   164 apply (simp add: atMost_def) 
   165 apply (drule FreeUltrafilterNat_finite) 
   166 apply (drule FreeUltrafilterNat_Compl_mem, ultra)
   167 done
   168 
   169 lemma Compl_Collect_le: "- {n::nat. N \<le> n} = {n. n < N}"
   170 by (simp add: Collect_neg_eq [symmetric] linorder_not_le) 
   171 
   172 lemma hypnat_of_nat_eq:
   173      "hypnat_of_nat m  = star_n (%n::nat. m)"
   174 by (simp add: star_of_def)
   175 
   176 lemma SHNat_eq: "Nats = {n. \<exists>N. n = hypnat_of_nat N}"
   177 by (force simp add: hypnat_of_nat_def Nats_def) 
   178 
   179 lemma hypnat_omega_gt_SHNat:
   180      "n \<in> Nats ==> n < whn"
   181 by (auto simp add: hypnat_of_nat_eq star_n_less hypnat_omega_def SHNat_eq)
   182 
   183 (* Infinite hypernatural not in embedded Nats *)
   184 lemma SHNAT_omega_not_mem [simp]: "whn \<notin> Nats"
   185 by (blast dest: hypnat_omega_gt_SHNat)
   186 
   187 lemma hypnat_of_nat_less_whn [simp]: "hypnat_of_nat n < whn"
   188 apply (insert hypnat_omega_gt_SHNat [of "hypnat_of_nat n"])
   189 apply (simp add: hypnat_of_nat_def) 
   190 done
   191 
   192 lemma hypnat_of_nat_le_whn [simp]: "hypnat_of_nat n \<le> whn"
   193 by (rule hypnat_of_nat_less_whn [THEN order_less_imp_le])
   194 
   195 lemma hypnat_zero_less_hypnat_omega [simp]: "0 < whn"
   196 by (simp add: hypnat_omega_gt_SHNat)
   197 
   198 lemma hypnat_one_less_hypnat_omega [simp]: "(1::hypnat) < whn"
   199 by (simp add: hypnat_omega_gt_SHNat)
   200 
   201 
   202 subsection{*Infinite Hypernatural Numbers -- @{term HNatInfinite}*}
   203 
   204 constdefs
   205 
   206   (* the set of infinite hypernatural numbers *)
   207   HNatInfinite :: "hypnat set"
   208   "HNatInfinite == {n. n \<notin> Nats}"
   209 
   210 lemma HNatInfinite_whn [simp]: "whn \<in> HNatInfinite"
   211 by (simp add: HNatInfinite_def)
   212 
   213 lemma Nats_not_HNatInfinite_iff: "(x \<in> Nats) = (x \<notin> HNatInfinite)"
   214 by (simp add: HNatInfinite_def)
   215 
   216 lemma HNatInfinite_not_Nats_iff: "(x \<in> HNatInfinite) = (x \<notin> Nats)"
   217 by (simp add: HNatInfinite_def)
   218 
   219 
   220 subsubsection{*Alternative characterization of the set of infinite hypernaturals*}
   221 
   222 text{* @{term "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}"}*}
   223 
   224 (*??delete? similar reasoning in hypnat_omega_gt_SHNat above*)
   225 lemma HNatInfinite_FreeUltrafilterNat_lemma:
   226      "\<forall>N::nat. {n. f n \<noteq> N} \<in> FreeUltrafilterNat
   227       ==> {n. N < f n} \<in> FreeUltrafilterNat"
   228 apply (induct_tac N)
   229 apply (drule_tac x = 0 in spec)
   230 apply (rule ccontr, drule FreeUltrafilterNat_Compl_mem, drule FreeUltrafilterNat_Int, assumption, simp)
   231 apply (drule_tac x = "Suc n" in spec, ultra)
   232 done
   233 
   234 lemma HNatInfinite_iff: "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}"
   235 apply (auto simp add: HNatInfinite_def SHNat_eq hypnat_of_nat_eq)
   236 apply (rule_tac x = x in star_cases)
   237 apply (auto elim: HNatInfinite_FreeUltrafilterNat_lemma 
   238             simp add: star_n_less FreeUltrafilterNat_Compl_iff1 
   239                       star_n_eq_iff Collect_neg_eq [symmetric])
   240 done
   241 
   242 
   243 subsubsection{*Alternative Characterization of @{term HNatInfinite} using 
   244 Free Ultrafilter*}
   245 
   246 lemma HNatInfinite_FreeUltrafilterNat:
   247      "x \<in> HNatInfinite 
   248       ==> \<exists>X \<in> Rep_star x. \<forall>u. {n. u < X n}:  FreeUltrafilterNat"
   249 apply (cases x)
   250 apply (auto simp add: HNatInfinite_iff SHNat_eq hypnat_of_nat_eq)
   251 apply (rule bexI [OF _ Rep_star_star_n], clarify) 
   252 apply (auto simp add: hypnat_of_nat_def star_n_less)
   253 done
   254 
   255 lemma FreeUltrafilterNat_HNatInfinite:
   256      "\<exists>X \<in> Rep_star x. \<forall>u. {n. u < X n}:  FreeUltrafilterNat
   257       ==> x \<in> HNatInfinite"
   258 apply (cases x)
   259 apply (auto simp add: star_n_less HNatInfinite_iff SHNat_eq hypnat_of_nat_eq)
   260 apply (drule spec, ultra, auto) 
   261 done
   262 
   263 lemma HNatInfinite_FreeUltrafilterNat_iff:
   264      "(x \<in> HNatInfinite) = 
   265       (\<exists>X \<in> Rep_star x. \<forall>u. {n. u < X n}:  FreeUltrafilterNat)"
   266 by (blast intro: HNatInfinite_FreeUltrafilterNat 
   267                  FreeUltrafilterNat_HNatInfinite)
   268 
   269 lemma HNatInfinite_gt_one [simp]: "x \<in> HNatInfinite ==> (1::hypnat) < x"
   270 by (auto simp add: HNatInfinite_iff)
   271 
   272 lemma zero_not_mem_HNatInfinite [simp]: "0 \<notin> HNatInfinite"
   273 apply (auto simp add: HNatInfinite_iff)
   274 apply (drule_tac a = " (1::hypnat) " in equals0D)
   275 apply simp
   276 done
   277 
   278 lemma HNatInfinite_not_eq_zero: "x \<in> HNatInfinite ==> 0 < x"
   279 apply (drule HNatInfinite_gt_one) 
   280 apply (auto simp add: order_less_trans [OF zero_less_one])
   281 done
   282 
   283 lemma HNatInfinite_ge_one [simp]: "x \<in> HNatInfinite ==> (1::hypnat) \<le> x"
   284 by (blast intro: order_less_imp_le HNatInfinite_gt_one)
   285 
   286 
   287 subsubsection{*Closure Rules*}
   288 
   289 lemma HNatInfinite_add:
   290      "[| x \<in> HNatInfinite; y \<in> HNatInfinite |] ==> x + y \<in> HNatInfinite"
   291 apply (auto simp add: HNatInfinite_iff)
   292 apply (drule bspec, assumption)
   293 apply (drule bspec [OF _ Nats_0])
   294 apply (drule add_strict_mono, assumption, simp)
   295 done
   296 
   297 lemma HNatInfinite_SHNat_add:
   298      "[| x \<in> HNatInfinite; y \<in> Nats |] ==> x + y \<in> HNatInfinite"
   299 apply (auto simp add: HNatInfinite_not_Nats_iff) 
   300 apply (drule_tac a = "x + y" in Nats_diff, auto) 
   301 done
   302 
   303 lemma HNatInfinite_Nats_imp_less: "[| x \<in> HNatInfinite; y \<in> Nats |] ==> y < x"
   304 by (simp add: HNatInfinite_iff) 
   305 
   306 lemma HNatInfinite_SHNat_diff:
   307   assumes x: "x \<in> HNatInfinite" and y: "y \<in> Nats" 
   308   shows "x - y \<in> HNatInfinite"
   309 proof -
   310   have "y < x" by (simp add: HNatInfinite_Nats_imp_less prems)
   311   hence "x - y + y = x" by (simp add: order_less_imp_le)
   312   with x show ?thesis
   313     by (force simp add: HNatInfinite_not_Nats_iff 
   314               dest: Nats_add [of "x-y", OF _ y]) 
   315 qed
   316 
   317 lemma HNatInfinite_add_one:
   318      "x \<in> HNatInfinite ==> x + (1::hypnat) \<in> HNatInfinite"
   319 by (auto intro: HNatInfinite_SHNat_add)
   320 
   321 lemma HNatInfinite_is_Suc: "x \<in> HNatInfinite ==> \<exists>y. x = y + (1::hypnat)"
   322 apply (rule_tac x = "x - (1::hypnat) " in exI)
   323 apply auto
   324 done
   325 
   326 
   327 subsection{*Embedding of the Hypernaturals into the Hyperreals*}
   328 text{*Obtained using the nonstandard extension of the naturals*}
   329 
   330 constdefs
   331   hypreal_of_hypnat :: "hypnat => hypreal"
   332    "hypreal_of_hypnat == *f* real"
   333 
   334 declare hypreal_of_hypnat_def [transfer_unfold]
   335 
   336 lemma HNat_hypreal_of_nat [simp]: "hypreal_of_nat N \<in> Nats"
   337 by (simp add: hypreal_of_nat_def) 
   338 
   339 lemma hypreal_of_hypnat:
   340       "hypreal_of_hypnat (star_n X) = star_n (%n. real (X n))"
   341 by (simp add: hypreal_of_hypnat_def starfun)
   342 
   343 lemma hypreal_of_hypnat_inject [simp]:
   344      "!!m n. (hypreal_of_hypnat m = hypreal_of_hypnat n) = (m=n)"
   345 by (transfer, simp)
   346 
   347 lemma hypreal_of_hypnat_zero: "hypreal_of_hypnat 0 = 0"
   348 by (simp add: star_n_zero_num hypreal_of_hypnat)
   349 
   350 lemma hypreal_of_hypnat_one: "hypreal_of_hypnat (1::hypnat) = 1"
   351 by (simp add: star_n_one_num hypreal_of_hypnat)
   352 
   353 lemma hypreal_of_hypnat_add [simp]:
   354      "!!m n. hypreal_of_hypnat (m + n) = hypreal_of_hypnat m + hypreal_of_hypnat n"
   355 by (transfer, rule real_of_nat_add)
   356 
   357 lemma hypreal_of_hypnat_mult [simp]:
   358      "!!m n. hypreal_of_hypnat (m * n) = hypreal_of_hypnat m * hypreal_of_hypnat n"
   359 by (transfer, rule real_of_nat_mult)
   360 
   361 lemma hypreal_of_hypnat_less_iff [simp]:
   362      "!!m n. (hypreal_of_hypnat n < hypreal_of_hypnat m) = (n < m)"
   363 by (transfer, simp)
   364 
   365 lemma hypreal_of_hypnat_eq_zero_iff: "(hypreal_of_hypnat N = 0) = (N = 0)"
   366 by (simp add: hypreal_of_hypnat_zero [symmetric])
   367 declare hypreal_of_hypnat_eq_zero_iff [simp]
   368 
   369 lemma hypreal_of_hypnat_ge_zero [simp]: "!!n. 0 \<le> hypreal_of_hypnat n"
   370 by (transfer, simp)
   371 
   372 lemma HNatInfinite_inverse_Infinitesimal [simp]:
   373      "n \<in> HNatInfinite ==> inverse (hypreal_of_hypnat n) \<in> Infinitesimal"
   374 apply (cases n)
   375 apply (auto simp add: hypreal_of_hypnat star_n_inverse 
   376       HNatInfinite_FreeUltrafilterNat_iff Infinitesimal_FreeUltrafilterNat_iff2)
   377 apply (rule bexI [OF _ Rep_star_star_n], auto)
   378 apply (drule_tac x = "m + 1" in spec, ultra)
   379 done
   380 
   381 lemma HNatInfinite_hypreal_of_hypnat_gt_zero:
   382      "N \<in> HNatInfinite ==> 0 < hypreal_of_hypnat N"
   383 apply (rule ccontr)
   384 apply (simp add: hypreal_of_hypnat_zero [symmetric] linorder_not_less)
   385 done
   386 
   387 
   388 ML
   389 {*
   390 val hypnat_of_nat_def = thm"hypnat_of_nat_def";
   391 val HNatInfinite_def = thm"HNatInfinite_def";
   392 val hypreal_of_hypnat_def = thm"hypreal_of_hypnat_def";
   393 val hypnat_omega_def = thm"hypnat_omega_def";
   394 
   395 val starrel_iff = thm "starrel_iff";
   396 val lemma_starrel_refl = thm "lemma_starrel_refl";
   397 val hypnat_minus_zero = thm "hypnat_minus_zero";
   398 val hypnat_diff_0_eq_0 = thm "hypnat_diff_0_eq_0";
   399 val hypnat_add_is_0 = thm "hypnat_add_is_0";
   400 val hypnat_diff_diff_left = thm "hypnat_diff_diff_left";
   401 val hypnat_diff_commute = thm "hypnat_diff_commute";
   402 val hypnat_diff_add_inverse = thm "hypnat_diff_add_inverse";
   403 val hypnat_diff_add_inverse2 = thm "hypnat_diff_add_inverse2";
   404 val hypnat_diff_cancel = thm "hypnat_diff_cancel";
   405 val hypnat_diff_cancel2 = thm "hypnat_diff_cancel2";
   406 val hypnat_diff_add_0 = thm "hypnat_diff_add_0";
   407 val hypnat_diff_mult_distrib = thm "hypnat_diff_mult_distrib";
   408 val hypnat_diff_mult_distrib2 = thm "hypnat_diff_mult_distrib2";
   409 val hypnat_mult_is_0 = thm "hypnat_mult_is_0";
   410 val hypnat_not_less0 = thm "hypnat_not_less0";
   411 val hypnat_less_one = thm "hypnat_less_one";
   412 val hypnat_add_diff_inverse = thm "hypnat_add_diff_inverse";
   413 val hypnat_le_add_diff_inverse = thm "hypnat_le_add_diff_inverse";
   414 val hypnat_le_add_diff_inverse2 = thm "hypnat_le_add_diff_inverse2";
   415 val hypnat_le0 = thm "hypnat_le0";
   416 val hypnat_add_self_le = thm "hypnat_add_self_le";
   417 val hypnat_add_one_self_less = thm "hypnat_add_one_self_less";
   418 val hypnat_neq0_conv = thm "hypnat_neq0_conv";
   419 val hypnat_gt_zero_iff = thm "hypnat_gt_zero_iff";
   420 val hypnat_gt_zero_iff2 = thm "hypnat_gt_zero_iff2";
   421 val SHNat_eq = thm"SHNat_eq"
   422 val hypnat_of_nat_one = thm "hypnat_of_nat_one";
   423 val hypnat_of_nat_Suc = thm "hypnat_of_nat_Suc";
   424 val SHNAT_omega_not_mem = thm "SHNAT_omega_not_mem";
   425 val cofinite_mem_FreeUltrafilterNat = thm "cofinite_mem_FreeUltrafilterNat";
   426 val hypnat_omega_gt_SHNat = thm "hypnat_omega_gt_SHNat";
   427 val hypnat_of_nat_less_whn = thm "hypnat_of_nat_less_whn";
   428 val hypnat_of_nat_le_whn = thm "hypnat_of_nat_le_whn";
   429 val hypnat_zero_less_hypnat_omega = thm "hypnat_zero_less_hypnat_omega";
   430 val hypnat_one_less_hypnat_omega = thm "hypnat_one_less_hypnat_omega";
   431 val HNatInfinite_whn = thm "HNatInfinite_whn";
   432 val HNatInfinite_iff = thm "HNatInfinite_iff";
   433 val HNatInfinite_FreeUltrafilterNat = thm "HNatInfinite_FreeUltrafilterNat";
   434 val FreeUltrafilterNat_HNatInfinite = thm "FreeUltrafilterNat_HNatInfinite";
   435 val HNatInfinite_FreeUltrafilterNat_iff = thm "HNatInfinite_FreeUltrafilterNat_iff";
   436 val HNatInfinite_gt_one = thm "HNatInfinite_gt_one";
   437 val zero_not_mem_HNatInfinite = thm "zero_not_mem_HNatInfinite";
   438 val HNatInfinite_not_eq_zero = thm "HNatInfinite_not_eq_zero";
   439 val HNatInfinite_ge_one = thm "HNatInfinite_ge_one";
   440 val HNatInfinite_add = thm "HNatInfinite_add";
   441 val HNatInfinite_SHNat_add = thm "HNatInfinite_SHNat_add";
   442 val HNatInfinite_SHNat_diff = thm "HNatInfinite_SHNat_diff";
   443 val HNatInfinite_add_one = thm "HNatInfinite_add_one";
   444 val HNatInfinite_is_Suc = thm "HNatInfinite_is_Suc";
   445 val HNat_hypreal_of_nat = thm "HNat_hypreal_of_nat";
   446 val hypreal_of_hypnat = thm "hypreal_of_hypnat";
   447 val hypreal_of_hypnat_zero = thm "hypreal_of_hypnat_zero";
   448 val hypreal_of_hypnat_one = thm "hypreal_of_hypnat_one";
   449 val hypreal_of_hypnat_add = thm "hypreal_of_hypnat_add";
   450 val hypreal_of_hypnat_mult = thm "hypreal_of_hypnat_mult";
   451 val hypreal_of_hypnat_less_iff = thm "hypreal_of_hypnat_less_iff";
   452 val hypreal_of_hypnat_ge_zero = thm "hypreal_of_hypnat_ge_zero";
   453 val HNatInfinite_inverse_Infinitesimal = thm "HNatInfinite_inverse_Infinitesimal";
   454 *}
   455 
   456 end