src/HOL/Hyperreal/HTranscendental.thy
 author paulson Thu Jul 01 12:29:53 2004 +0200 (2004-07-01) changeset 15013 34264f5e4691 parent 14641 79b7bd936264 child 15077 89840837108e permissions -rw-r--r--
new treatment of binary numerals
```     1 (*  Title       : HTranscendental.thy
```
```     2     Author      : Jacques D. Fleuriot
```
```     3     Copyright   : 2001 University of Edinburgh
```
```     4
```
```     5 Converted to Isar and polished by lcp
```
```     6 *)
```
```     7
```
```     8 header{*Nonstandard Extensions of Transcendental Functions*}
```
```     9
```
```    10 theory HTranscendental = Transcendental + Integration:
```
```    11
```
```    12 text{*really belongs in Transcendental*}
```
```    13 lemma sqrt_divide_self_eq:
```
```    14   assumes nneg: "0 \<le> x"
```
```    15   shows "sqrt x / x = inverse (sqrt x)"
```
```    16 proof cases
```
```    17   assume "x=0" thus ?thesis by simp
```
```    18 next
```
```    19   assume nz: "x\<noteq>0"
```
```    20   hence pos: "0<x" using nneg by arith
```
```    21   show ?thesis
```
```    22   proof (rule right_inverse_eq [THEN iffD1, THEN sym])
```
```    23     show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz)
```
```    24     show "inverse (sqrt x) / (sqrt x / x) = 1"
```
```    25       by (simp add: divide_inverse mult_assoc [symmetric]
```
```    26                   power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz)
```
```    27   qed
```
```    28 qed
```
```    29
```
```    30
```
```    31 constdefs
```
```    32
```
```    33   exphr :: "real => hypreal"
```
```    34     --{*define exponential function using standard part *}
```
```    35     "exphr x ==  st(sumhr (0, whn, %n. inverse(real (fact n)) * (x ^ n)))"
```
```    36
```
```    37   sinhr :: "real => hypreal"
```
```    38     "sinhr x == st(sumhr (0, whn, %n. (if even(n) then 0 else
```
```    39              ((-1) ^ ((n - 1) div 2))/(real (fact n))) * (x ^ n)))"
```
```    40
```
```    41   coshr :: "real => hypreal"
```
```    42     "coshr x == st(sumhr (0, whn, %n. (if even(n) then
```
```    43             ((-1) ^ (n div 2))/(real (fact n)) else 0) * x ^ n))"
```
```    44
```
```    45
```
```    46 subsection{*Nonstandard Extension of Square Root Function*}
```
```    47
```
```    48 lemma STAR_sqrt_zero [simp]: "( *f* sqrt) 0 = 0"
```
```    49 by (simp add: starfun hypreal_zero_num)
```
```    50
```
```    51 lemma STAR_sqrt_one [simp]: "( *f* sqrt) 1 = 1"
```
```    52 by (simp add: starfun hypreal_one_num)
```
```    53
```
```    54 lemma hypreal_sqrt_pow2_iff: "(( *f* sqrt)(x) ^ 2 = x) = (0 \<le> x)"
```
```    55 apply (cases x)
```
```    56 apply (auto simp add: hypreal_le hypreal_zero_num starfun hrealpow
```
```    57                       real_sqrt_pow2_iff
```
```    58             simp del: hpowr_Suc realpow_Suc)
```
```    59 done
```
```    60
```
```    61 lemma hypreal_sqrt_gt_zero_pow2: "0 < x ==> ( *f* sqrt) (x) ^ 2 = x"
```
```    62 apply (cases x)
```
```    63 apply (auto intro: FreeUltrafilterNat_subset real_sqrt_gt_zero_pow2
```
```    64             simp add: hypreal_less starfun hrealpow hypreal_zero_num
```
```    65             simp del: hpowr_Suc realpow_Suc)
```
```    66 done
```
```    67
```
```    68 lemma hypreal_sqrt_pow2_gt_zero: "0 < x ==> 0 < ( *f* sqrt) (x) ^ 2"
```
```    69 by (frule hypreal_sqrt_gt_zero_pow2, auto)
```
```    70
```
```    71 lemma hypreal_sqrt_not_zero: "0 < x ==> ( *f* sqrt) (x) \<noteq> 0"
```
```    72 apply (frule hypreal_sqrt_pow2_gt_zero)
```
```    73 apply (auto simp add: numeral_2_eq_2)
```
```    74 done
```
```    75
```
```    76 lemma hypreal_inverse_sqrt_pow2:
```
```    77      "0 < x ==> inverse (( *f* sqrt)(x)) ^ 2 = inverse x"
```
```    78 apply (cut_tac n1 = 2 and a1 = "( *f* sqrt) x" in power_inverse [symmetric])
```
```    79 apply (auto dest: hypreal_sqrt_gt_zero_pow2)
```
```    80 done
```
```    81
```
```    82 lemma hypreal_sqrt_mult_distrib:
```
```    83     "[|0 < x; 0 <y |] ==> ( *f* sqrt)(x*y) =  ( *f* sqrt)(x) * ( *f* sqrt)(y)"
```
```    84 apply (cases x, cases y)
```
```    85 apply (simp add: hypreal_zero_def starfun hypreal_mult hypreal_less hypreal_zero_num, ultra)
```
```    86 apply (auto intro: real_sqrt_mult_distrib)
```
```    87 done
```
```    88
```
```    89 lemma hypreal_sqrt_mult_distrib2:
```
```    90      "[|0\<le>x; 0\<le>y |] ==>
```
```    91      ( *f* sqrt)(x*y) =  ( *f* sqrt)(x) * ( *f* sqrt)(y)"
```
```    92 by (auto intro: hypreal_sqrt_mult_distrib simp add: order_le_less)
```
```    93
```
```    94 lemma hypreal_sqrt_approx_zero [simp]:
```
```    95      "0 < x ==> (( *f* sqrt)(x) @= 0) = (x @= 0)"
```
```    96 apply (auto simp add: mem_infmal_iff [symmetric])
```
```    97 apply (rule hypreal_sqrt_gt_zero_pow2 [THEN subst])
```
```    98 apply (auto intro: Infinitesimal_mult
```
```    99             dest!: hypreal_sqrt_gt_zero_pow2 [THEN ssubst]
```
```   100             simp add: numeral_2_eq_2)
```
```   101 done
```
```   102
```
```   103 lemma hypreal_sqrt_approx_zero2 [simp]:
```
```   104      "0 \<le> x ==> (( *f* sqrt)(x) @= 0) = (x @= 0)"
```
```   105 by (auto simp add: order_le_less)
```
```   106
```
```   107 lemma hypreal_sqrt_sum_squares [simp]:
```
```   108      "(( *f* sqrt)(x*x + y*y + z*z) @= 0) = (x*x + y*y + z*z @= 0)"
```
```   109 apply (rule hypreal_sqrt_approx_zero2)
```
```   110 apply (rule hypreal_le_add_order)+
```
```   111 apply (auto simp add: zero_le_square)
```
```   112 done
```
```   113
```
```   114 lemma hypreal_sqrt_sum_squares2 [simp]:
```
```   115      "(( *f* sqrt)(x*x + y*y) @= 0) = (x*x + y*y @= 0)"
```
```   116 apply (rule hypreal_sqrt_approx_zero2)
```
```   117 apply (rule hypreal_le_add_order)
```
```   118 apply (auto simp add: zero_le_square)
```
```   119 done
```
```   120
```
```   121 lemma hypreal_sqrt_gt_zero: "0 < x ==> 0 < ( *f* sqrt)(x)"
```
```   122 apply (cases x)
```
```   123 apply (auto simp add: starfun hypreal_zero_def hypreal_less hypreal_zero_num, ultra)
```
```   124 apply (auto intro: real_sqrt_gt_zero)
```
```   125 done
```
```   126
```
```   127 lemma hypreal_sqrt_ge_zero: "0 \<le> x ==> 0 \<le> ( *f* sqrt)(x)"
```
```   128 by (auto intro: hypreal_sqrt_gt_zero simp add: order_le_less)
```
```   129
```
```   130 lemma hypreal_sqrt_hrabs [simp]: "( *f* sqrt)(x ^ 2) = abs(x)"
```
```   131 apply (cases x)
```
```   132 apply (simp add: starfun hypreal_hrabs hypreal_mult numeral_2_eq_2)
```
```   133 done
```
```   134
```
```   135 lemma hypreal_sqrt_hrabs2 [simp]: "( *f* sqrt)(x*x) = abs(x)"
```
```   136 apply (rule hrealpow_two [THEN subst])
```
```   137 apply (rule numeral_2_eq_2 [THEN subst])
```
```   138 apply (rule hypreal_sqrt_hrabs)
```
```   139 done
```
```   140
```
```   141 lemma hypreal_sqrt_hyperpow_hrabs [simp]:
```
```   142      "( *f* sqrt)(x pow (hypnat_of_nat 2)) = abs(x)"
```
```   143 apply (cases x)
```
```   144 apply (simp add: starfun hypreal_hrabs hypnat_of_nat_eq hyperpow)
```
```   145 done
```
```   146
```
```   147 lemma star_sqrt_HFinite: "\<lbrakk>x \<in> HFinite; 0 \<le> x\<rbrakk> \<Longrightarrow> ( *f* sqrt) x \<in> HFinite"
```
```   148 apply (rule HFinite_square_iff [THEN iffD1])
```
```   149 apply (simp only: hypreal_sqrt_mult_distrib2 [symmetric], simp)
```
```   150 done
```
```   151
```
```   152 lemma st_hypreal_sqrt:
```
```   153      "[| x \<in> HFinite; 0 \<le> x |] ==> st(( *f* sqrt) x) = ( *f* sqrt)(st x)"
```
```   154 apply (rule power_inject_base [where n=1])
```
```   155 apply (auto intro!: st_zero_le hypreal_sqrt_ge_zero)
```
```   156 apply (rule st_mult [THEN subst])
```
```   157 apply (rule_tac [3] hypreal_sqrt_mult_distrib2 [THEN subst])
```
```   158 apply (rule_tac [5] hypreal_sqrt_mult_distrib2 [THEN subst])
```
```   159 apply (auto simp add: st_hrabs st_zero_le star_sqrt_HFinite)
```
```   160 done
```
```   161
```
```   162 lemma hypreal_sqrt_sum_squares_ge1 [simp]: "x \<le> ( *f* sqrt)(x ^ 2 + y ^ 2)"
```
```   163 apply (cases x, cases y)
```
```   164 apply (simp add: starfun hypreal_add hrealpow hypreal_le
```
```   165             del: hpowr_Suc realpow_Suc)
```
```   166 done
```
```   167
```
```   168 lemma HFinite_hypreal_sqrt:
```
```   169      "[| 0 \<le> x; x \<in> HFinite |] ==> ( *f* sqrt) x \<in> HFinite"
```
```   170 apply (auto simp add: order_le_less)
```
```   171 apply (rule HFinite_square_iff [THEN iffD1])
```
```   172 apply (drule hypreal_sqrt_gt_zero_pow2)
```
```   173 apply (simp add: numeral_2_eq_2)
```
```   174 done
```
```   175
```
```   176 lemma HFinite_hypreal_sqrt_imp_HFinite:
```
```   177      "[| 0 \<le> x; ( *f* sqrt) x \<in> HFinite |] ==> x \<in> HFinite"
```
```   178 apply (auto simp add: order_le_less)
```
```   179 apply (drule HFinite_square_iff [THEN iffD2])
```
```   180 apply (drule hypreal_sqrt_gt_zero_pow2)
```
```   181 apply (simp add: numeral_2_eq_2 del: HFinite_square_iff)
```
```   182 done
```
```   183
```
```   184 lemma HFinite_hypreal_sqrt_iff [simp]:
```
```   185      "0 \<le> x ==> (( *f* sqrt) x \<in> HFinite) = (x \<in> HFinite)"
```
```   186 by (blast intro: HFinite_hypreal_sqrt HFinite_hypreal_sqrt_imp_HFinite)
```
```   187
```
```   188 lemma HFinite_sqrt_sum_squares [simp]:
```
```   189      "(( *f* sqrt)(x*x + y*y) \<in> HFinite) = (x*x + y*y \<in> HFinite)"
```
```   190 apply (rule HFinite_hypreal_sqrt_iff)
```
```   191 apply (rule hypreal_le_add_order)
```
```   192 apply (auto simp add: zero_le_square)
```
```   193 done
```
```   194
```
```   195 lemma Infinitesimal_hypreal_sqrt:
```
```   196      "[| 0 \<le> x; x \<in> Infinitesimal |] ==> ( *f* sqrt) x \<in> Infinitesimal"
```
```   197 apply (auto simp add: order_le_less)
```
```   198 apply (rule Infinitesimal_square_iff [THEN iffD2])
```
```   199 apply (drule hypreal_sqrt_gt_zero_pow2)
```
```   200 apply (simp add: numeral_2_eq_2)
```
```   201 done
```
```   202
```
```   203 lemma Infinitesimal_hypreal_sqrt_imp_Infinitesimal:
```
```   204      "[| 0 \<le> x; ( *f* sqrt) x \<in> Infinitesimal |] ==> x \<in> Infinitesimal"
```
```   205 apply (auto simp add: order_le_less)
```
```   206 apply (drule Infinitesimal_square_iff [THEN iffD1])
```
```   207 apply (drule hypreal_sqrt_gt_zero_pow2)
```
```   208 apply (simp add: numeral_2_eq_2 del: Infinitesimal_square_iff [symmetric])
```
```   209 done
```
```   210
```
```   211 lemma Infinitesimal_hypreal_sqrt_iff [simp]:
```
```   212      "0 \<le> x ==> (( *f* sqrt) x \<in> Infinitesimal) = (x \<in> Infinitesimal)"
```
```   213 by (blast intro: Infinitesimal_hypreal_sqrt_imp_Infinitesimal Infinitesimal_hypreal_sqrt)
```
```   214
```
```   215 lemma Infinitesimal_sqrt_sum_squares [simp]:
```
```   216      "(( *f* sqrt)(x*x + y*y) \<in> Infinitesimal) = (x*x + y*y \<in> Infinitesimal)"
```
```   217 apply (rule Infinitesimal_hypreal_sqrt_iff)
```
```   218 apply (rule hypreal_le_add_order)
```
```   219 apply (auto simp add: zero_le_square)
```
```   220 done
```
```   221
```
```   222 lemma HInfinite_hypreal_sqrt:
```
```   223      "[| 0 \<le> x; x \<in> HInfinite |] ==> ( *f* sqrt) x \<in> HInfinite"
```
```   224 apply (auto simp add: order_le_less)
```
```   225 apply (rule HInfinite_square_iff [THEN iffD1])
```
```   226 apply (drule hypreal_sqrt_gt_zero_pow2)
```
```   227 apply (simp add: numeral_2_eq_2)
```
```   228 done
```
```   229
```
```   230 lemma HInfinite_hypreal_sqrt_imp_HInfinite:
```
```   231      "[| 0 \<le> x; ( *f* sqrt) x \<in> HInfinite |] ==> x \<in> HInfinite"
```
```   232 apply (auto simp add: order_le_less)
```
```   233 apply (drule HInfinite_square_iff [THEN iffD2])
```
```   234 apply (drule hypreal_sqrt_gt_zero_pow2)
```
```   235 apply (simp add: numeral_2_eq_2 del: HInfinite_square_iff)
```
```   236 done
```
```   237
```
```   238 lemma HInfinite_hypreal_sqrt_iff [simp]:
```
```   239      "0 \<le> x ==> (( *f* sqrt) x \<in> HInfinite) = (x \<in> HInfinite)"
```
```   240 by (blast intro: HInfinite_hypreal_sqrt HInfinite_hypreal_sqrt_imp_HInfinite)
```
```   241
```
```   242 lemma HInfinite_sqrt_sum_squares [simp]:
```
```   243      "(( *f* sqrt)(x*x + y*y) \<in> HInfinite) = (x*x + y*y \<in> HInfinite)"
```
```   244 apply (rule HInfinite_hypreal_sqrt_iff)
```
```   245 apply (rule hypreal_le_add_order)
```
```   246 apply (auto simp add: zero_le_square)
```
```   247 done
```
```   248
```
```   249 lemma HFinite_exp [simp]:
```
```   250      "sumhr (0, whn, %n. inverse (real (fact n)) * x ^ n) \<in> HFinite"
```
```   251 by (auto intro!: NSBseq_HFinite_hypreal NSconvergent_NSBseq
```
```   252          simp add: starfunNat_sumr [symmetric] starfunNat hypnat_omega_def
```
```   253                    convergent_NSconvergent_iff [symmetric]
```
```   254                    summable_convergent_sumr_iff [symmetric] summable_exp)
```
```   255
```
```   256 lemma exphr_zero [simp]: "exphr 0 = 1"
```
```   257 apply (simp add: exphr_def sumhr_split_add
```
```   258                    [OF hypnat_one_less_hypnat_omega, symmetric])
```
```   259 apply (simp add: sumhr hypnat_zero_def starfunNat hypnat_one_def hypnat_add
```
```   260                  hypnat_omega_def hypreal_add
```
```   261             del: hypnat_add_zero_left)
```
```   262 apply (simp add: hypreal_one_num [symmetric])
```
```   263 done
```
```   264
```
```   265 lemma coshr_zero [simp]: "coshr 0 = 1"
```
```   266 apply (simp add: coshr_def sumhr_split_add
```
```   267                    [OF hypnat_one_less_hypnat_omega, symmetric])
```
```   268 apply (simp add: sumhr hypnat_zero_def starfunNat hypnat_one_def
```
```   269          hypnat_add hypnat_omega_def st_add [symmetric]
```
```   270          hypreal_one_def [symmetric] hypreal_zero_def [symmetric]
```
```   271        del: hypnat_add_zero_left)
```
```   272 done
```
```   273
```
```   274 lemma STAR_exp_zero_approx_one [simp]: "( *f* exp) 0 @= 1"
```
```   275 by (simp add: hypreal_zero_def hypreal_one_def starfun hypreal_one_num)
```
```   276
```
```   277 lemma STAR_exp_Infinitesimal: "x \<in> Infinitesimal ==> ( *f* exp) x @= 1"
```
```   278 apply (case_tac "x = 0")
```
```   279 apply (cut_tac [2] x = 0 in DERIV_exp)
```
```   280 apply (auto simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def)
```
```   281 apply (drule_tac x = x in bspec, auto)
```
```   282 apply (drule_tac c = x in approx_mult1)
```
```   283 apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
```
```   284             simp add: mult_assoc)
```
```   285 apply (rule approx_add_right_cancel [where d="-1"])
```
```   286 apply (rule approx_sym [THEN [2] approx_trans2])
```
```   287 apply (auto simp add: mem_infmal_iff)
```
```   288 done
```
```   289
```
```   290 lemma STAR_exp_epsilon [simp]: "( *f* exp) epsilon @= 1"
```
```   291 by (auto intro: STAR_exp_Infinitesimal)
```
```   292
```
```   293 lemma STAR_exp_add: "( *f* exp)(x + y) = ( *f* exp) x * ( *f* exp) y"
```
```   294 apply (cases x, cases y)
```
```   295 apply (simp add: starfun hypreal_add hypreal_mult exp_add)
```
```   296 done
```
```   297
```
```   298 lemma exphr_hypreal_of_real_exp_eq: "exphr x = hypreal_of_real (exp x)"
```
```   299 apply (simp add: exphr_def)
```
```   300 apply (rule st_hypreal_of_real [THEN subst])
```
```   301 apply (rule approx_st_eq, auto)
```
```   302 apply (rule approx_minus_iff [THEN iffD2])
```
```   303 apply (auto simp add: mem_infmal_iff [symmetric] hypreal_of_real_def hypnat_zero_def hypnat_omega_def sumhr hypreal_minus hypreal_add)
```
```   304 apply (rule NSLIMSEQ_zero_Infinitesimal_hypreal)
```
```   305 apply (insert exp_converges [of x])
```
```   306 apply (simp add: sums_def)
```
```   307 apply (drule LIMSEQ_const [THEN [2] LIMSEQ_add, where b = "- exp x"])
```
```   308 apply (simp add: LIMSEQ_NSLIMSEQ_iff)
```
```   309 done
```
```   310
```
```   311 lemma starfun_exp_ge_add_one_self [simp]: "0 \<le> x ==> (1 + x) \<le> ( *f* exp) x"
```
```   312 apply (cases x)
```
```   313 apply (simp add: starfun hypreal_add hypreal_le hypreal_zero_num hypreal_one_num, ultra)
```
```   314 done
```
```   315
```
```   316 (* exp (oo) is infinite *)
```
```   317 lemma starfun_exp_HInfinite:
```
```   318      "[| x \<in> HInfinite; 0 \<le> x |] ==> ( *f* exp) x \<in> HInfinite"
```
```   319 apply (frule starfun_exp_ge_add_one_self)
```
```   320 apply (rule HInfinite_ge_HInfinite, assumption)
```
```   321 apply (rule order_trans [of _ "1+x"], auto)
```
```   322 done
```
```   323
```
```   324 lemma starfun_exp_minus: "( *f* exp) (-x) = inverse(( *f* exp) x)"
```
```   325 apply (cases x)
```
```   326 apply (simp add: starfun hypreal_inverse hypreal_minus exp_minus)
```
```   327 done
```
```   328
```
```   329 (* exp (-oo) is infinitesimal *)
```
```   330 lemma starfun_exp_Infinitesimal:
```
```   331      "[| x \<in> HInfinite; x \<le> 0 |] ==> ( *f* exp) x \<in> Infinitesimal"
```
```   332 apply (subgoal_tac "\<exists>y. x = - y")
```
```   333 apply (rule_tac [2] x = "- x" in exI)
```
```   334 apply (auto intro!: HInfinite_inverse_Infinitesimal starfun_exp_HInfinite
```
```   335             simp add: starfun_exp_minus HInfinite_minus_iff)
```
```   336 done
```
```   337
```
```   338 lemma starfun_exp_gt_one [simp]: "0 < x ==> 1 < ( *f* exp) x"
```
```   339 apply (cases x)
```
```   340 apply (simp add: starfun hypreal_one_num hypreal_zero_num hypreal_less, ultra)
```
```   341 done
```
```   342
```
```   343 (* needs derivative of inverse function
```
```   344    TRY a NS one today!!!
```
```   345
```
```   346 Goal "x @= 1 ==> ( *f* ln) x @= 1"
```
```   347 by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
```
```   348 by (auto_tac (claset(),simpset() addsimps [hypreal_one_def]));
```
```   349
```
```   350
```
```   351 Goalw [nsderiv_def] "0r < x ==> NSDERIV ln x :> inverse x";
```
```   352
```
```   353 *)
```
```   354
```
```   355
```
```   356 lemma starfun_ln_exp [simp]: "( *f* ln) (( *f* exp) x) = x"
```
```   357 apply (cases x)
```
```   358 apply (simp add: starfun)
```
```   359 done
```
```   360
```
```   361 lemma starfun_exp_ln_iff [simp]: "(( *f* exp)(( *f* ln) x) = x) = (0 < x)"
```
```   362 apply (cases x)
```
```   363 apply (simp add: starfun hypreal_zero_num hypreal_less)
```
```   364 done
```
```   365
```
```   366 lemma starfun_exp_ln_eq: "( *f* exp) u = x ==> ( *f* ln) x = u"
```
```   367 by (auto simp add: starfun)
```
```   368
```
```   369 lemma starfun_ln_less_self [simp]: "0 < x ==> ( *f* ln) x < x"
```
```   370 apply (cases x)
```
```   371 apply (simp add: starfun hypreal_zero_num hypreal_less, ultra)
```
```   372 done
```
```   373
```
```   374 lemma starfun_ln_ge_zero [simp]: "1 \<le> x ==> 0 \<le> ( *f* ln) x"
```
```   375 apply (cases x)
```
```   376 apply (simp add: starfun hypreal_zero_num hypreal_le hypreal_one_num, ultra)
```
```   377 done
```
```   378
```
```   379 lemma starfun_ln_gt_zero [simp]: "1 < x ==> 0 < ( *f* ln) x"
```
```   380 apply (cases x)
```
```   381 apply (simp add: starfun hypreal_zero_num hypreal_less hypreal_one_num, ultra)
```
```   382 done
```
```   383
```
```   384 lemma starfun_ln_not_eq_zero [simp]: "[| 0 < x; x \<noteq> 1 |] ==> ( *f* ln) x \<noteq> 0"
```
```   385 apply (cases x)
```
```   386 apply (auto simp add: starfun hypreal_zero_num hypreal_less hypreal_one_num, ultra)
```
```   387 apply (auto dest: ln_not_eq_zero)
```
```   388 done
```
```   389
```
```   390 lemma starfun_ln_HFinite: "[| x \<in> HFinite; 1 \<le> x |] ==> ( *f* ln) x \<in> HFinite"
```
```   391 apply (rule HFinite_bounded)
```
```   392 apply (rule_tac [2] order_less_imp_le)
```
```   393 apply (rule_tac [2] starfun_ln_less_self)
```
```   394 apply (rule_tac [2] order_less_le_trans, auto)
```
```   395 done
```
```   396
```
```   397 lemma starfun_ln_inverse: "0 < x ==> ( *f* ln) (inverse x) = -( *f* ln) x"
```
```   398 apply (cases x)
```
```   399 apply (simp add: starfun hypreal_zero_num hypreal_minus hypreal_inverse hypreal_less, ultra)
```
```   400 apply (simp add: ln_inverse)
```
```   401 done
```
```   402
```
```   403 lemma starfun_exp_HFinite: "x \<in> HFinite ==> ( *f* exp) x \<in> HFinite"
```
```   404 apply (cases x)
```
```   405 apply (auto simp add: starfun HFinite_FreeUltrafilterNat_iff)
```
```   406 apply (rule bexI, rule_tac [2] lemma_hyprel_refl, auto)
```
```   407 apply (rule_tac x = "exp u" in exI)
```
```   408 apply (ultra, arith)
```
```   409 done
```
```   410
```
```   411 lemma starfun_exp_add_HFinite_Infinitesimal_approx:
```
```   412      "[|x \<in> Infinitesimal; z \<in> HFinite |] ==> ( *f* exp) (z + x) @= ( *f* exp) z"
```
```   413 apply (simp add: STAR_exp_add)
```
```   414 apply (frule STAR_exp_Infinitesimal)
```
```   415 apply (drule approx_mult2)
```
```   416 apply (auto intro: starfun_exp_HFinite)
```
```   417 done
```
```   418
```
```   419 (* using previous result to get to result *)
```
```   420 lemma starfun_ln_HInfinite:
```
```   421      "[| x \<in> HInfinite; 0 < x |] ==> ( *f* ln) x \<in> HInfinite"
```
```   422 apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
```
```   423 apply (drule starfun_exp_HFinite)
```
```   424 apply (simp add: starfun_exp_ln_iff [THEN iffD2] HFinite_HInfinite_iff)
```
```   425 done
```
```   426
```
```   427 lemma starfun_exp_HInfinite_Infinitesimal_disj:
```
```   428  "x \<in> HInfinite ==> ( *f* exp) x \<in> HInfinite | ( *f* exp) x \<in> Infinitesimal"
```
```   429 apply (insert linorder_linear [of x 0])
```
```   430 apply (auto intro: starfun_exp_HInfinite starfun_exp_Infinitesimal)
```
```   431 done
```
```   432
```
```   433 (* check out this proof!!! *)
```
```   434 lemma starfun_ln_HFinite_not_Infinitesimal:
```
```   435      "[| x \<in> HFinite - Infinitesimal; 0 < x |] ==> ( *f* ln) x \<in> HFinite"
```
```   436 apply (rule ccontr, drule HInfinite_HFinite_iff [THEN iffD2])
```
```   437 apply (drule starfun_exp_HInfinite_Infinitesimal_disj)
```
```   438 apply (simp add: starfun_exp_ln_iff [symmetric] HInfinite_HFinite_iff
```
```   439             del: starfun_exp_ln_iff)
```
```   440 done
```
```   441
```
```   442 (* we do proof by considering ln of 1/x *)
```
```   443 lemma starfun_ln_Infinitesimal_HInfinite:
```
```   444      "[| x \<in> Infinitesimal; 0 < x |] ==> ( *f* ln) x \<in> HInfinite"
```
```   445 apply (drule Infinitesimal_inverse_HInfinite)
```
```   446 apply (frule positive_imp_inverse_positive)
```
```   447 apply (drule_tac [2] starfun_ln_HInfinite)
```
```   448 apply (auto simp add: starfun_ln_inverse HInfinite_minus_iff)
```
```   449 done
```
```   450
```
```   451 lemma starfun_ln_less_zero: "[| 0 < x; x < 1 |] ==> ( *f* ln) x < 0"
```
```   452 apply (cases x)
```
```   453 apply (simp add: hypreal_zero_num hypreal_one_num hypreal_less starfun, ultra)
```
```   454 apply (auto intro: ln_less_zero)
```
```   455 done
```
```   456
```
```   457 lemma starfun_ln_Infinitesimal_less_zero:
```
```   458      "[| x \<in> Infinitesimal; 0 < x |] ==> ( *f* ln) x < 0"
```
```   459 apply (auto intro!: starfun_ln_less_zero simp add: Infinitesimal_def)
```
```   460 apply (drule bspec [where x = 1])
```
```   461 apply (auto simp add: abs_if)
```
```   462 done
```
```   463
```
```   464 lemma starfun_ln_HInfinite_gt_zero:
```
```   465      "[| x \<in> HInfinite; 0 < x |] ==> 0 < ( *f* ln) x"
```
```   466 apply (auto intro!: starfun_ln_gt_zero simp add: HInfinite_def)
```
```   467 apply (drule bspec [where x = 1])
```
```   468 apply (auto simp add: abs_if)
```
```   469 done
```
```   470
```
```   471 (*
```
```   472 Goalw [NSLIM_def] "(%h. ((x powr h) - 1) / h) -- 0 --NS> ln x"
```
```   473 *)
```
```   474
```
```   475 lemma HFinite_sin [simp]:
```
```   476      "sumhr (0, whn, %n. (if even(n) then 0 else
```
```   477               ((- 1) ^ ((n - 1) div 2))/(real (fact n))) * x ^ n)
```
```   478               \<in> HFinite"
```
```   479 apply (auto intro!: NSBseq_HFinite_hypreal NSconvergent_NSBseq
```
```   480             simp add: starfunNat_sumr [symmetric] starfunNat hypnat_omega_def
```
```   481                       convergent_NSconvergent_iff [symmetric]
```
```   482                       summable_convergent_sumr_iff [symmetric])
```
```   483 apply (simp only: One_nat_def summable_sin)
```
```   484 done
```
```   485
```
```   486 lemma STAR_sin_zero [simp]: "( *f* sin) 0 = 0"
```
```   487 by (simp add: starfun hypreal_zero_num)
```
```   488
```
```   489 lemma STAR_sin_Infinitesimal [simp]: "x \<in> Infinitesimal ==> ( *f* sin) x @= x"
```
```   490 apply (case_tac "x = 0")
```
```   491 apply (cut_tac [2] x = 0 in DERIV_sin)
```
```   492 apply (auto simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def hypreal_of_real_zero)
```
```   493 apply (drule bspec [where x = x], auto)
```
```   494 apply (drule approx_mult1 [where c = x])
```
```   495 apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
```
```   496            simp add: mult_assoc)
```
```   497 done
```
```   498
```
```   499 lemma HFinite_cos [simp]:
```
```   500      "sumhr (0, whn, %n. (if even(n) then
```
```   501             ((- 1) ^ (n div 2))/(real (fact n)) else
```
```   502             0) * x ^ n) \<in> HFinite"
```
```   503 by (auto intro!: NSBseq_HFinite_hypreal NSconvergent_NSBseq
```
```   504          simp add: starfunNat_sumr [symmetric] starfunNat hypnat_omega_def
```
```   505                    convergent_NSconvergent_iff [symmetric]
```
```   506                    summable_convergent_sumr_iff [symmetric] summable_cos)
```
```   507
```
```   508 lemma STAR_cos_zero [simp]: "( *f* cos) 0 = 1"
```
```   509 by (simp add: starfun hypreal_zero_num hypreal_one_num)
```
```   510
```
```   511 lemma STAR_cos_Infinitesimal [simp]: "x \<in> Infinitesimal ==> ( *f* cos) x @= 1"
```
```   512 apply (case_tac "x = 0")
```
```   513 apply (cut_tac [2] x = 0 in DERIV_cos)
```
```   514 apply (auto simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def hypreal_of_real_zero)
```
```   515 apply (drule bspec [where x = x])
```
```   516 apply (auto simp add: hypreal_of_real_zero hypreal_of_real_one)
```
```   517 apply (drule approx_mult1 [where c = x])
```
```   518 apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
```
```   519             simp add: mult_assoc hypreal_of_real_one)
```
```   520 apply (rule approx_add_right_cancel [where d = "-1"], auto)
```
```   521 done
```
```   522
```
```   523 lemma STAR_tan_zero [simp]: "( *f* tan) 0 = 0"
```
```   524 by (simp add: starfun hypreal_zero_num)
```
```   525
```
```   526 lemma STAR_tan_Infinitesimal: "x \<in> Infinitesimal ==> ( *f* tan) x @= x"
```
```   527 apply (case_tac "x = 0")
```
```   528 apply (cut_tac [2] x = 0 in DERIV_tan)
```
```   529 apply (auto simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def hypreal_of_real_zero)
```
```   530 apply (drule bspec [where x = x], auto)
```
```   531 apply (drule approx_mult1 [where c = x])
```
```   532 apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
```
```   533              simp add: mult_assoc)
```
```   534 done
```
```   535
```
```   536 lemma STAR_sin_cos_Infinitesimal_mult:
```
```   537      "x \<in> Infinitesimal ==> ( *f* sin) x * ( *f* cos) x @= x"
```
```   538 apply (insert approx_mult_HFinite [of "( *f* sin) x" _ "( *f* cos) x" 1])
```
```   539 apply (simp add: Infinitesimal_subset_HFinite [THEN subsetD])
```
```   540 done
```
```   541
```
```   542 lemma HFinite_pi: "hypreal_of_real pi \<in> HFinite"
```
```   543 by simp
```
```   544
```
```   545 (* lemmas *)
```
```   546
```
```   547 lemma lemma_split_hypreal_of_real:
```
```   548      "N \<in> HNatInfinite
```
```   549       ==> hypreal_of_real a =
```
```   550           hypreal_of_hypnat N * (inverse(hypreal_of_hypnat N) * hypreal_of_real a)"
```
```   551 by (simp add: mult_assoc [symmetric] HNatInfinite_not_eq_zero)
```
```   552
```
```   553 lemma STAR_sin_Infinitesimal_divide:
```
```   554      "[|x \<in> Infinitesimal; x \<noteq> 0 |] ==> ( *f* sin) x/x @= 1"
```
```   555 apply (cut_tac x = 0 in DERIV_sin)
```
```   556 apply (simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def hypreal_of_real_zero hypreal_of_real_one)
```
```   557 done
```
```   558
```
```   559 (*------------------------------------------------------------------------*)
```
```   560 (* sin* (1/n) * 1/(1/n) @= 1 for n = oo                                   *)
```
```   561 (*------------------------------------------------------------------------*)
```
```   562
```
```   563 lemma lemma_sin_pi:
```
```   564      "n \<in> HNatInfinite
```
```   565       ==> ( *f* sin) (inverse (hypreal_of_hypnat n))/(inverse (hypreal_of_hypnat n)) @= 1"
```
```   566 apply (rule STAR_sin_Infinitesimal_divide)
```
```   567 apply (auto simp add: HNatInfinite_not_eq_zero)
```
```   568 done
```
```   569
```
```   570 lemma STAR_sin_inverse_HNatInfinite:
```
```   571      "n \<in> HNatInfinite
```
```   572       ==> ( *f* sin) (inverse (hypreal_of_hypnat n)) * hypreal_of_hypnat n @= 1"
```
```   573 apply (frule lemma_sin_pi)
```
```   574 apply (simp add: divide_inverse)
```
```   575 done
```
```   576
```
```   577 lemma Infinitesimal_pi_divide_HNatInfinite:
```
```   578      "N \<in> HNatInfinite
```
```   579       ==> hypreal_of_real pi/(hypreal_of_hypnat N) \<in> Infinitesimal"
```
```   580 apply (simp add: divide_inverse)
```
```   581 apply (auto intro: Infinitesimal_HFinite_mult2)
```
```   582 done
```
```   583
```
```   584 lemma pi_divide_HNatInfinite_not_zero [simp]:
```
```   585      "N \<in> HNatInfinite ==> hypreal_of_real pi/(hypreal_of_hypnat N) \<noteq> 0"
```
```   586 by (simp add: HNatInfinite_not_eq_zero)
```
```   587
```
```   588 lemma STAR_sin_pi_divide_HNatInfinite_approx_pi:
```
```   589      "n \<in> HNatInfinite
```
```   590       ==> ( *f* sin) (hypreal_of_real pi/(hypreal_of_hypnat n)) * hypreal_of_hypnat n
```
```   591           @= hypreal_of_real pi"
```
```   592 apply (frule STAR_sin_Infinitesimal_divide
```
```   593                [OF Infinitesimal_pi_divide_HNatInfinite
```
```   594                    pi_divide_HNatInfinite_not_zero])
```
```   595 apply (auto simp add: inverse_mult_distrib)
```
```   596 apply (rule approx_SReal_mult_cancel [of "inverse (hypreal_of_real pi)"])
```
```   597 apply (auto intro: SReal_inverse simp add: divide_inverse mult_ac)
```
```   598 done
```
```   599
```
```   600 lemma STAR_sin_pi_divide_HNatInfinite_approx_pi2:
```
```   601      "n \<in> HNatInfinite
```
```   602       ==> hypreal_of_hypnat n *
```
```   603           ( *f* sin) (hypreal_of_real pi/(hypreal_of_hypnat n))
```
```   604           @= hypreal_of_real pi"
```
```   605 apply (rule mult_commute [THEN subst])
```
```   606 apply (erule STAR_sin_pi_divide_HNatInfinite_approx_pi)
```
```   607 done
```
```   608
```
```   609 lemma starfunNat_pi_divide_n_Infinitesimal:
```
```   610      "N \<in> HNatInfinite ==> ( *fNat* (%x. pi / real x)) N \<in> Infinitesimal"
```
```   611 by (auto intro!: Infinitesimal_HFinite_mult2
```
```   612          simp add: starfunNat_mult [symmetric] divide_inverse
```
```   613                    starfunNat_inverse [symmetric] starfunNat_real_of_nat)
```
```   614
```
```   615 lemma STAR_sin_pi_divide_n_approx:
```
```   616      "N \<in> HNatInfinite ==>
```
```   617       ( *f* sin) (( *fNat* (%x. pi / real x)) N) @=
```
```   618       hypreal_of_real pi/(hypreal_of_hypnat N)"
```
```   619 by (auto intro!: STAR_sin_Infinitesimal Infinitesimal_HFinite_mult2
```
```   620          simp add: starfunNat_mult [symmetric] divide_inverse
```
```   621                    starfunNat_inverse_real_of_nat_eq)
```
```   622
```
```   623 lemma NSLIMSEQ_sin_pi: "(%n. real n * sin (pi / real n)) ----NS> pi"
```
```   624 apply (auto simp add: NSLIMSEQ_def starfunNat_mult [symmetric] starfunNat_real_of_nat)
```
```   625 apply (rule_tac f1 = sin in starfun_stafunNat_o2 [THEN subst])
```
```   626 apply (auto simp add: starfunNat_mult [symmetric] starfunNat_real_of_nat divide_inverse)
```
```   627 apply (rule_tac f1 = inverse in starfun_stafunNat_o2 [THEN subst])
```
```   628 apply (auto dest: STAR_sin_pi_divide_HNatInfinite_approx_pi
```
```   629             simp add: starfunNat_real_of_nat mult_commute divide_inverse)
```
```   630 done
```
```   631
```
```   632 lemma NSLIMSEQ_cos_one: "(%n. cos (pi / real n))----NS> 1"
```
```   633 apply (simp add: NSLIMSEQ_def, auto)
```
```   634 apply (rule_tac f1 = cos in starfun_stafunNat_o2 [THEN subst])
```
```   635 apply (rule STAR_cos_Infinitesimal)
```
```   636 apply (auto intro!: Infinitesimal_HFinite_mult2
```
```   637             simp add: starfunNat_mult [symmetric] divide_inverse
```
```   638                       starfunNat_inverse [symmetric] starfunNat_real_of_nat)
```
```   639 done
```
```   640
```
```   641 lemma NSLIMSEQ_sin_cos_pi:
```
```   642      "(%n. real n * sin (pi / real n) * cos (pi / real n)) ----NS> pi"
```
```   643 by (insert NSLIMSEQ_mult [OF NSLIMSEQ_sin_pi NSLIMSEQ_cos_one], simp)
```
```   644
```
```   645
```
```   646 text{*A familiar approximation to @{term "cos x"} when @{term x} is small*}
```
```   647
```
```   648 lemma STAR_cos_Infinitesimal_approx:
```
```   649      "x \<in> Infinitesimal ==> ( *f* cos) x @= 1 - x ^ 2"
```
```   650 apply (rule STAR_cos_Infinitesimal [THEN approx_trans])
```
```   651 apply (auto simp add: Infinitesimal_approx_minus [symmetric]
```
```   652             diff_minus add_assoc [symmetric] numeral_2_eq_2)
```
```   653 done
```
```   654
```
```   655 lemma STAR_cos_Infinitesimal_approx2:
```
```   656      "x \<in> Infinitesimal ==> ( *f* cos) x @= 1 - (x ^ 2)/2"
```
```   657 apply (rule STAR_cos_Infinitesimal [THEN approx_trans])
```
```   658 apply (auto intro: Infinitesimal_SReal_divide
```
```   659             simp add: Infinitesimal_approx_minus [symmetric] numeral_2_eq_2)
```
```   660 done
```
```   661
```
```   662 ML
```
```   663 {*
```
```   664 val STAR_sqrt_zero = thm "STAR_sqrt_zero";
```
```   665 val STAR_sqrt_one = thm "STAR_sqrt_one";
```
```   666 val hypreal_sqrt_pow2_iff = thm "hypreal_sqrt_pow2_iff";
```
```   667 val hypreal_sqrt_gt_zero_pow2 = thm "hypreal_sqrt_gt_zero_pow2";
```
```   668 val hypreal_sqrt_pow2_gt_zero = thm "hypreal_sqrt_pow2_gt_zero";
```
```   669 val hypreal_sqrt_not_zero = thm "hypreal_sqrt_not_zero";
```
```   670 val hypreal_inverse_sqrt_pow2 = thm "hypreal_inverse_sqrt_pow2";
```
```   671 val hypreal_sqrt_mult_distrib = thm "hypreal_sqrt_mult_distrib";
```
```   672 val hypreal_sqrt_mult_distrib2 = thm "hypreal_sqrt_mult_distrib2";
```
```   673 val hypreal_sqrt_approx_zero = thm "hypreal_sqrt_approx_zero";
```
```   674 val hypreal_sqrt_approx_zero2 = thm "hypreal_sqrt_approx_zero2";
```
```   675 val hypreal_sqrt_sum_squares = thm "hypreal_sqrt_sum_squares";
```
```   676 val hypreal_sqrt_sum_squares2 = thm "hypreal_sqrt_sum_squares2";
```
```   677 val hypreal_sqrt_gt_zero = thm "hypreal_sqrt_gt_zero";
```
```   678 val hypreal_sqrt_ge_zero = thm "hypreal_sqrt_ge_zero";
```
```   679 val hypreal_sqrt_hrabs = thm "hypreal_sqrt_hrabs";
```
```   680 val hypreal_sqrt_hrabs2 = thm "hypreal_sqrt_hrabs2";
```
```   681 val hypreal_sqrt_hyperpow_hrabs = thm "hypreal_sqrt_hyperpow_hrabs";
```
```   682 val star_sqrt_HFinite = thm "star_sqrt_HFinite";
```
```   683 val st_hypreal_sqrt = thm "st_hypreal_sqrt";
```
```   684 val hypreal_sqrt_sum_squares_ge1 = thm "hypreal_sqrt_sum_squares_ge1";
```
```   685 val HFinite_hypreal_sqrt = thm "HFinite_hypreal_sqrt";
```
```   686 val HFinite_hypreal_sqrt_imp_HFinite = thm "HFinite_hypreal_sqrt_imp_HFinite";
```
```   687 val HFinite_hypreal_sqrt_iff = thm "HFinite_hypreal_sqrt_iff";
```
```   688 val HFinite_sqrt_sum_squares = thm "HFinite_sqrt_sum_squares";
```
```   689 val Infinitesimal_hypreal_sqrt = thm "Infinitesimal_hypreal_sqrt";
```
```   690 val Infinitesimal_hypreal_sqrt_imp_Infinitesimal = thm "Infinitesimal_hypreal_sqrt_imp_Infinitesimal";
```
```   691 val Infinitesimal_hypreal_sqrt_iff = thm "Infinitesimal_hypreal_sqrt_iff";
```
```   692 val Infinitesimal_sqrt_sum_squares = thm "Infinitesimal_sqrt_sum_squares";
```
```   693 val HInfinite_hypreal_sqrt = thm "HInfinite_hypreal_sqrt";
```
```   694 val HInfinite_hypreal_sqrt_imp_HInfinite = thm "HInfinite_hypreal_sqrt_imp_HInfinite";
```
```   695 val HInfinite_hypreal_sqrt_iff = thm "HInfinite_hypreal_sqrt_iff";
```
```   696 val HInfinite_sqrt_sum_squares = thm "HInfinite_sqrt_sum_squares";
```
```   697 val HFinite_exp = thm "HFinite_exp";
```
```   698 val exphr_zero = thm "exphr_zero";
```
```   699 val coshr_zero = thm "coshr_zero";
```
```   700 val STAR_exp_zero_approx_one = thm "STAR_exp_zero_approx_one";
```
```   701 val STAR_exp_Infinitesimal = thm "STAR_exp_Infinitesimal";
```
```   702 val STAR_exp_epsilon = thm "STAR_exp_epsilon";
```
```   703 val STAR_exp_add = thm "STAR_exp_add";
```
```   704 val exphr_hypreal_of_real_exp_eq = thm "exphr_hypreal_of_real_exp_eq";
```
```   705 val starfun_exp_ge_add_one_self = thm "starfun_exp_ge_add_one_self";
```
```   706 val starfun_exp_HInfinite = thm "starfun_exp_HInfinite";
```
```   707 val starfun_exp_minus = thm "starfun_exp_minus";
```
```   708 val starfun_exp_Infinitesimal = thm "starfun_exp_Infinitesimal";
```
```   709 val starfun_exp_gt_one = thm "starfun_exp_gt_one";
```
```   710 val starfun_ln_exp = thm "starfun_ln_exp";
```
```   711 val starfun_exp_ln_iff = thm "starfun_exp_ln_iff";
```
```   712 val starfun_exp_ln_eq = thm "starfun_exp_ln_eq";
```
```   713 val starfun_ln_less_self = thm "starfun_ln_less_self";
```
```   714 val starfun_ln_ge_zero = thm "starfun_ln_ge_zero";
```
```   715 val starfun_ln_gt_zero = thm "starfun_ln_gt_zero";
```
```   716 val starfun_ln_not_eq_zero = thm "starfun_ln_not_eq_zero";
```
```   717 val starfun_ln_HFinite = thm "starfun_ln_HFinite";
```
```   718 val starfun_ln_inverse = thm "starfun_ln_inverse";
```
```   719 val starfun_exp_HFinite = thm "starfun_exp_HFinite";
```
```   720 val starfun_exp_add_HFinite_Infinitesimal_approx = thm "starfun_exp_add_HFinite_Infinitesimal_approx";
```
```   721 val starfun_ln_HInfinite = thm "starfun_ln_HInfinite";
```
```   722 val starfun_exp_HInfinite_Infinitesimal_disj = thm "starfun_exp_HInfinite_Infinitesimal_disj";
```
```   723 val starfun_ln_HFinite_not_Infinitesimal = thm "starfun_ln_HFinite_not_Infinitesimal";
```
```   724 val starfun_ln_Infinitesimal_HInfinite = thm "starfun_ln_Infinitesimal_HInfinite";
```
```   725 val starfun_ln_less_zero = thm "starfun_ln_less_zero";
```
```   726 val starfun_ln_Infinitesimal_less_zero = thm "starfun_ln_Infinitesimal_less_zero";
```
```   727 val starfun_ln_HInfinite_gt_zero = thm "starfun_ln_HInfinite_gt_zero";
```
```   728 val HFinite_sin = thm "HFinite_sin";
```
```   729 val STAR_sin_zero = thm "STAR_sin_zero";
```
```   730 val STAR_sin_Infinitesimal = thm "STAR_sin_Infinitesimal";
```
```   731 val HFinite_cos = thm "HFinite_cos";
```
```   732 val STAR_cos_zero = thm "STAR_cos_zero";
```
```   733 val STAR_cos_Infinitesimal = thm "STAR_cos_Infinitesimal";
```
```   734 val STAR_tan_zero = thm "STAR_tan_zero";
```
```   735 val STAR_tan_Infinitesimal = thm "STAR_tan_Infinitesimal";
```
```   736 val STAR_sin_cos_Infinitesimal_mult = thm "STAR_sin_cos_Infinitesimal_mult";
```
```   737 val HFinite_pi = thm "HFinite_pi";
```
```   738 val lemma_split_hypreal_of_real = thm "lemma_split_hypreal_of_real";
```
```   739 val STAR_sin_Infinitesimal_divide = thm "STAR_sin_Infinitesimal_divide";
```
```   740 val lemma_sin_pi = thm "lemma_sin_pi";
```
```   741 val STAR_sin_inverse_HNatInfinite = thm "STAR_sin_inverse_HNatInfinite";
```
```   742 val Infinitesimal_pi_divide_HNatInfinite = thm "Infinitesimal_pi_divide_HNatInfinite";
```
```   743 val pi_divide_HNatInfinite_not_zero = thm "pi_divide_HNatInfinite_not_zero";
```
```   744 val STAR_sin_pi_divide_HNatInfinite_approx_pi = thm "STAR_sin_pi_divide_HNatInfinite_approx_pi";
```
```   745 val STAR_sin_pi_divide_HNatInfinite_approx_pi2 = thm "STAR_sin_pi_divide_HNatInfinite_approx_pi2";
```
```   746 val starfunNat_pi_divide_n_Infinitesimal = thm "starfunNat_pi_divide_n_Infinitesimal";
```
```   747 val STAR_sin_pi_divide_n_approx = thm "STAR_sin_pi_divide_n_approx";
```
```   748 val NSLIMSEQ_sin_pi = thm "NSLIMSEQ_sin_pi";
```
```   749 val NSLIMSEQ_cos_one = thm "NSLIMSEQ_cos_one";
```
```   750 val NSLIMSEQ_sin_cos_pi = thm "NSLIMSEQ_sin_cos_pi";
```
```   751 val STAR_cos_Infinitesimal_approx = thm "STAR_cos_Infinitesimal_approx";
```
```   752 val STAR_cos_Infinitesimal_approx2 = thm "STAR_cos_Infinitesimal_approx2";
```
```   753 *}
```
```   754
```
```   755
```
```   756 end
```