src/HOL/Nonstandard_Analysis/NSCA.thy
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```     1 (*  Title:      HOL/Nonstandard_Analysis/NSCA.thy
```
```     2     Author:     Jacques D. Fleuriot
```
```     3     Copyright:  2001, 2002 University of Edinburgh
```
```     4 *)
```
```     5
```
```     6 section\<open>Non-Standard Complex Analysis\<close>
```
```     7
```
```     8 theory NSCA
```
```     9 imports NSComplex HTranscendental
```
```    10 begin
```
```    11
```
```    12 abbreviation
```
```    13    (* standard complex numbers reagarded as an embedded subset of NS complex *)
```
```    14    SComplex  :: "hcomplex set" where
```
```    15    "SComplex \<equiv> Standard"
```
```    16
```
```    17 definition \<comment>\<open>standard part map\<close>
```
```    18   stc :: "hcomplex => hcomplex" where
```
```    19   "stc x = (SOME r. x \<in> HFinite & r:SComplex & r \<approx> x)"
```
```    20
```
```    21
```
```    22 subsection\<open>Closure Laws for SComplex, the Standard Complex Numbers\<close>
```
```    23
```
```    24 lemma SComplex_minus_iff [simp]: "(-x \<in> SComplex) = (x \<in> SComplex)"
```
```    25 by (auto, drule Standard_minus, auto)
```
```    26
```
```    27 lemma SComplex_add_cancel:
```
```    28      "[| x + y \<in> SComplex; y \<in> SComplex |] ==> x \<in> SComplex"
```
```    29 by (drule (1) Standard_diff, simp)
```
```    30
```
```    31 lemma SReal_hcmod_hcomplex_of_complex [simp]:
```
```    32      "hcmod (hcomplex_of_complex r) \<in> \<real>"
```
```    33 by (simp add: Reals_eq_Standard)
```
```    34
```
```    35 lemma SReal_hcmod_numeral [simp]: "hcmod (numeral w ::hcomplex) \<in> \<real>"
```
```    36 by (simp add: Reals_eq_Standard)
```
```    37
```
```    38 lemma SReal_hcmod_SComplex: "x \<in> SComplex ==> hcmod x \<in> \<real>"
```
```    39 by (simp add: Reals_eq_Standard)
```
```    40
```
```    41 lemma SComplex_divide_numeral:
```
```    42      "r \<in> SComplex ==> r/(numeral w::hcomplex) \<in> SComplex"
```
```    43 by simp
```
```    44
```
```    45 lemma SComplex_UNIV_complex:
```
```    46      "{x. hcomplex_of_complex x \<in> SComplex} = (UNIV::complex set)"
```
```    47 by simp
```
```    48
```
```    49 lemma SComplex_iff: "(x \<in> SComplex) = (\<exists>y. x = hcomplex_of_complex y)"
```
```    50 by (simp add: Standard_def image_def)
```
```    51
```
```    52 lemma hcomplex_of_complex_image:
```
```    53      "hcomplex_of_complex `(UNIV::complex set) = SComplex"
```
```    54 by (simp add: Standard_def)
```
```    55
```
```    56 lemma inv_hcomplex_of_complex_image: "inv hcomplex_of_complex `SComplex = UNIV"
```
```    57 by (auto simp add: Standard_def image_def) (metis inj_star_of inv_f_f)
```
```    58
```
```    59 lemma SComplex_hcomplex_of_complex_image:
```
```    60       "[| \<exists>x. x: P; P \<le> SComplex |] ==> \<exists>Q. P = hcomplex_of_complex ` Q"
```
```    61 apply (simp add: Standard_def, blast)
```
```    62 done
```
```    63
```
```    64 lemma SComplex_SReal_dense:
```
```    65      "[| x \<in> SComplex; y \<in> SComplex; hcmod x < hcmod y
```
```    66       |] ==> \<exists>r \<in> Reals. hcmod x< r & r < hcmod y"
```
```    67 apply (auto intro: SReal_dense simp add: SReal_hcmod_SComplex)
```
```    68 done
```
```    69
```
```    70
```
```    71 subsection\<open>The Finite Elements form a Subring\<close>
```
```    72
```
```    73 lemma HFinite_hcmod_hcomplex_of_complex [simp]:
```
```    74      "hcmod (hcomplex_of_complex r) \<in> HFinite"
```
```    75 by (auto intro!: SReal_subset_HFinite [THEN subsetD])
```
```    76
```
```    77 lemma HFinite_hcmod_iff: "(x \<in> HFinite) = (hcmod x \<in> HFinite)"
```
```    78 by (simp add: HFinite_def)
```
```    79
```
```    80 lemma HFinite_bounded_hcmod:
```
```    81   "[|x \<in> HFinite; y \<le> hcmod x; 0 \<le> y |] ==> y: HFinite"
```
```    82 by (auto intro: HFinite_bounded simp add: HFinite_hcmod_iff)
```
```    83
```
```    84
```
```    85 subsection\<open>The Complex Infinitesimals form a Subring\<close>
```
```    86
```
```    87 lemma hcomplex_sum_of_halves: "x/(2::hcomplex) + x/(2::hcomplex) = x"
```
```    88 by auto
```
```    89
```
```    90 lemma Infinitesimal_hcmod_iff:
```
```    91    "(z \<in> Infinitesimal) = (hcmod z \<in> Infinitesimal)"
```
```    92 by (simp add: Infinitesimal_def)
```
```    93
```
```    94 lemma HInfinite_hcmod_iff: "(z \<in> HInfinite) = (hcmod z \<in> HInfinite)"
```
```    95 by (simp add: HInfinite_def)
```
```    96
```
```    97 lemma HFinite_diff_Infinitesimal_hcmod:
```
```    98      "x \<in> HFinite - Infinitesimal ==> hcmod x \<in> HFinite - Infinitesimal"
```
```    99 by (simp add: HFinite_hcmod_iff Infinitesimal_hcmod_iff)
```
```   100
```
```   101 lemma hcmod_less_Infinitesimal:
```
```   102      "[| e \<in> Infinitesimal; hcmod x < hcmod e |] ==> x \<in> Infinitesimal"
```
```   103 by (auto elim: hrabs_less_Infinitesimal simp add: Infinitesimal_hcmod_iff)
```
```   104
```
```   105 lemma hcmod_le_Infinitesimal:
```
```   106      "[| e \<in> Infinitesimal; hcmod x \<le> hcmod e |] ==> x \<in> Infinitesimal"
```
```   107 by (auto elim: hrabs_le_Infinitesimal simp add: Infinitesimal_hcmod_iff)
```
```   108
```
```   109 lemma Infinitesimal_interval_hcmod:
```
```   110      "[| e \<in> Infinitesimal;
```
```   111           e' \<in> Infinitesimal;
```
```   112           hcmod e' < hcmod x ; hcmod x < hcmod e
```
```   113        |] ==> x \<in> Infinitesimal"
```
```   114 by (auto intro: Infinitesimal_interval simp add: Infinitesimal_hcmod_iff)
```
```   115
```
```   116 lemma Infinitesimal_interval2_hcmod:
```
```   117      "[| e \<in> Infinitesimal;
```
```   118          e' \<in> Infinitesimal;
```
```   119          hcmod e' \<le> hcmod x ; hcmod x \<le> hcmod e
```
```   120       |] ==> x \<in> Infinitesimal"
```
```   121 by (auto intro: Infinitesimal_interval2 simp add: Infinitesimal_hcmod_iff)
```
```   122
```
```   123
```
```   124 subsection\<open>The ``Infinitely Close'' Relation\<close>
```
```   125
```
```   126 (*
```
```   127 Goalw [capprox_def,approx_def] "(z @c= w) = (hcmod z \<approx> hcmod w)"
```
```   128 by (auto_tac (claset(),simpset() addsimps [Infinitesimal_hcmod_iff]));
```
```   129 *)
```
```   130
```
```   131 lemma approx_SComplex_mult_cancel_zero:
```
```   132      "[| a \<in> SComplex; a \<noteq> 0; a*x \<approx> 0 |] ==> x \<approx> 0"
```
```   133 apply (drule Standard_inverse [THEN Standard_subset_HFinite [THEN subsetD]])
```
```   134 apply (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
```
```   135 done
```
```   136
```
```   137 lemma approx_mult_SComplex1: "[| a \<in> SComplex; x \<approx> 0 |] ==> x*a \<approx> 0"
```
```   138 by (auto dest: Standard_subset_HFinite [THEN subsetD] approx_mult1)
```
```   139
```
```   140 lemma approx_mult_SComplex2: "[| a \<in> SComplex; x \<approx> 0 |] ==> a*x \<approx> 0"
```
```   141 by (auto dest: Standard_subset_HFinite [THEN subsetD] approx_mult2)
```
```   142
```
```   143 lemma approx_mult_SComplex_zero_cancel_iff [simp]:
```
```   144      "[|a \<in> SComplex; a \<noteq> 0 |] ==> (a*x \<approx> 0) = (x \<approx> 0)"
```
```   145 by (blast intro: approx_SComplex_mult_cancel_zero approx_mult_SComplex2)
```
```   146
```
```   147 lemma approx_SComplex_mult_cancel:
```
```   148      "[| a \<in> SComplex; a \<noteq> 0; a* w \<approx> a*z |] ==> w \<approx> z"
```
```   149 apply (drule Standard_inverse [THEN Standard_subset_HFinite [THEN subsetD]])
```
```   150 apply (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
```
```   151 done
```
```   152
```
```   153 lemma approx_SComplex_mult_cancel_iff1 [simp]:
```
```   154      "[| a \<in> SComplex; a \<noteq> 0|] ==> (a* w \<approx> a*z) = (w \<approx> z)"
```
```   155 by (auto intro!: approx_mult2 Standard_subset_HFinite [THEN subsetD]
```
```   156             intro: approx_SComplex_mult_cancel)
```
```   157
```
```   158 (* TODO: generalize following theorems: hcmod -> hnorm *)
```
```   159
```
```   160 lemma approx_hcmod_approx_zero: "(x \<approx> y) = (hcmod (y - x) \<approx> 0)"
```
```   161 apply (subst hnorm_minus_commute)
```
```   162 apply (simp add: approx_def Infinitesimal_hcmod_iff)
```
```   163 done
```
```   164
```
```   165 lemma approx_approx_zero_iff: "(x \<approx> 0) = (hcmod x \<approx> 0)"
```
```   166 by (simp add: approx_hcmod_approx_zero)
```
```   167
```
```   168 lemma approx_minus_zero_cancel_iff [simp]: "(-x \<approx> 0) = (x \<approx> 0)"
```
```   169 by (simp add: approx_def)
```
```   170
```
```   171 lemma Infinitesimal_hcmod_add_diff:
```
```   172      "u \<approx> 0 ==> hcmod(x + u) - hcmod x \<in> Infinitesimal"
```
```   173 apply (drule approx_approx_zero_iff [THEN iffD1])
```
```   174 apply (rule_tac e = "hcmod u" and e' = "- hcmod u" in Infinitesimal_interval2)
```
```   175 apply (auto simp add: mem_infmal_iff [symmetric])
```
```   176 apply (rule_tac c1 = "hcmod x" in add_le_cancel_left [THEN iffD1])
```
```   177 apply auto
```
```   178 done
```
```   179
```
```   180 lemma approx_hcmod_add_hcmod: "u \<approx> 0 ==> hcmod(x + u) \<approx> hcmod x"
```
```   181 apply (rule approx_minus_iff [THEN iffD2])
```
```   182 apply (auto intro: Infinitesimal_hcmod_add_diff simp add: mem_infmal_iff [symmetric])
```
```   183 done
```
```   184
```
```   185
```
```   186 subsection\<open>Zero is the Only Infinitesimal Complex Number\<close>
```
```   187
```
```   188 lemma Infinitesimal_less_SComplex:
```
```   189    "[| x \<in> SComplex; y \<in> Infinitesimal; 0 < hcmod x |] ==> hcmod y < hcmod x"
```
```   190 by (auto intro: Infinitesimal_less_SReal SReal_hcmod_SComplex simp add: Infinitesimal_hcmod_iff)
```
```   191
```
```   192 lemma SComplex_Int_Infinitesimal_zero: "SComplex Int Infinitesimal = {0}"
```
```   193 by (auto simp add: Standard_def Infinitesimal_hcmod_iff)
```
```   194
```
```   195 lemma SComplex_Infinitesimal_zero:
```
```   196      "[| x \<in> SComplex; x \<in> Infinitesimal|] ==> x = 0"
```
```   197 by (cut_tac SComplex_Int_Infinitesimal_zero, blast)
```
```   198
```
```   199 lemma SComplex_HFinite_diff_Infinitesimal:
```
```   200      "[| x \<in> SComplex; x \<noteq> 0 |] ==> x \<in> HFinite - Infinitesimal"
```
```   201 by (auto dest: SComplex_Infinitesimal_zero Standard_subset_HFinite [THEN subsetD])
```
```   202
```
```   203 lemma hcomplex_of_complex_HFinite_diff_Infinitesimal:
```
```   204      "hcomplex_of_complex x \<noteq> 0
```
```   205       ==> hcomplex_of_complex x \<in> HFinite - Infinitesimal"
```
```   206 by (rule SComplex_HFinite_diff_Infinitesimal, auto)
```
```   207
```
```   208 lemma numeral_not_Infinitesimal [simp]:
```
```   209      "numeral w \<noteq> (0::hcomplex) ==> (numeral w::hcomplex) \<notin> Infinitesimal"
```
```   210 by (fast dest: Standard_numeral [THEN SComplex_Infinitesimal_zero])
```
```   211
```
```   212 lemma approx_SComplex_not_zero:
```
```   213      "[| y \<in> SComplex; x \<approx> y; y\<noteq> 0 |] ==> x \<noteq> 0"
```
```   214 by (auto dest: SComplex_Infinitesimal_zero approx_sym [THEN mem_infmal_iff [THEN iffD2]])
```
```   215
```
```   216 lemma SComplex_approx_iff:
```
```   217      "[|x \<in> SComplex; y \<in> SComplex|] ==> (x \<approx> y) = (x = y)"
```
```   218 by (auto simp add: Standard_def)
```
```   219
```
```   220 lemma numeral_Infinitesimal_iff [simp]:
```
```   221      "((numeral w :: hcomplex) \<in> Infinitesimal) =
```
```   222       (numeral w = (0::hcomplex))"
```
```   223 apply (rule iffI)
```
```   224 apply (fast dest: Standard_numeral [THEN SComplex_Infinitesimal_zero])
```
```   225 apply (simp (no_asm_simp))
```
```   226 done
```
```   227
```
```   228 lemma approx_unique_complex:
```
```   229      "[| r \<in> SComplex; s \<in> SComplex; r \<approx> x; s \<approx> x|] ==> r = s"
```
```   230 by (blast intro: SComplex_approx_iff [THEN iffD1] approx_trans2)
```
```   231
```
```   232 subsection \<open>Properties of @{term hRe}, @{term hIm} and @{term HComplex}\<close>
```
```   233
```
```   234
```
```   235 lemma abs_hRe_le_hcmod: "\<And>x. \<bar>hRe x\<bar> \<le> hcmod x"
```
```   236 by transfer (rule abs_Re_le_cmod)
```
```   237
```
```   238 lemma abs_hIm_le_hcmod: "\<And>x. \<bar>hIm x\<bar> \<le> hcmod x"
```
```   239 by transfer (rule abs_Im_le_cmod)
```
```   240
```
```   241 lemma Infinitesimal_hRe: "x \<in> Infinitesimal \<Longrightarrow> hRe x \<in> Infinitesimal"
```
```   242 apply (rule InfinitesimalI2, simp)
```
```   243 apply (rule order_le_less_trans [OF abs_hRe_le_hcmod])
```
```   244 apply (erule (1) InfinitesimalD2)
```
```   245 done
```
```   246
```
```   247 lemma Infinitesimal_hIm: "x \<in> Infinitesimal \<Longrightarrow> hIm x \<in> Infinitesimal"
```
```   248 apply (rule InfinitesimalI2, simp)
```
```   249 apply (rule order_le_less_trans [OF abs_hIm_le_hcmod])
```
```   250 apply (erule (1) InfinitesimalD2)
```
```   251 done
```
```   252
```
```   253 lemma real_sqrt_lessI: "\<lbrakk>0 < u; x < u\<^sup>2\<rbrakk> \<Longrightarrow> sqrt x < u"
```
```   254 (* TODO: this belongs somewhere else *)
```
```   255 by (frule real_sqrt_less_mono) simp
```
```   256
```
```   257 lemma hypreal_sqrt_lessI:
```
```   258   "\<And>x u. \<lbrakk>0 < u; x < u\<^sup>2\<rbrakk> \<Longrightarrow> ( *f* sqrt) x < u"
```
```   259 by transfer (rule real_sqrt_lessI)
```
```   260
```
```   261 lemma hypreal_sqrt_ge_zero: "\<And>x. 0 \<le> x \<Longrightarrow> 0 \<le> ( *f* sqrt) x"
```
```   262 by transfer (rule real_sqrt_ge_zero)
```
```   263
```
```   264 lemma Infinitesimal_sqrt:
```
```   265   "\<lbrakk>x \<in> Infinitesimal; 0 \<le> x\<rbrakk> \<Longrightarrow> ( *f* sqrt) x \<in> Infinitesimal"
```
```   266 apply (rule InfinitesimalI2)
```
```   267 apply (drule_tac r="r\<^sup>2" in InfinitesimalD2, simp)
```
```   268 apply (simp add: hypreal_sqrt_ge_zero)
```
```   269 apply (rule hypreal_sqrt_lessI, simp_all)
```
```   270 done
```
```   271
```
```   272 lemma Infinitesimal_HComplex:
```
```   273   "\<lbrakk>x \<in> Infinitesimal; y \<in> Infinitesimal\<rbrakk> \<Longrightarrow> HComplex x y \<in> Infinitesimal"
```
```   274 apply (rule Infinitesimal_hcmod_iff [THEN iffD2])
```
```   275 apply (simp add: hcmod_i)
```
```   276 apply (rule Infinitesimal_add)
```
```   277 apply (erule Infinitesimal_hrealpow, simp)
```
```   278 apply (erule Infinitesimal_hrealpow, simp)
```
```   279 done
```
```   280
```
```   281 lemma hcomplex_Infinitesimal_iff:
```
```   282   "(x \<in> Infinitesimal) = (hRe x \<in> Infinitesimal \<and> hIm x \<in> Infinitesimal)"
```
```   283 apply (safe intro!: Infinitesimal_hRe Infinitesimal_hIm)
```
```   284 apply (drule (1) Infinitesimal_HComplex, simp)
```
```   285 done
```
```   286
```
```   287 lemma hRe_diff [simp]: "\<And>x y. hRe (x - y) = hRe x - hRe y"
```
```   288 by transfer simp
```
```   289
```
```   290 lemma hIm_diff [simp]: "\<And>x y. hIm (x - y) = hIm x - hIm y"
```
```   291 by transfer simp
```
```   292
```
```   293 lemma approx_hRe: "x \<approx> y \<Longrightarrow> hRe x \<approx> hRe y"
```
```   294 unfolding approx_def by (drule Infinitesimal_hRe) simp
```
```   295
```
```   296 lemma approx_hIm: "x \<approx> y \<Longrightarrow> hIm x \<approx> hIm y"
```
```   297 unfolding approx_def by (drule Infinitesimal_hIm) simp
```
```   298
```
```   299 lemma approx_HComplex:
```
```   300   "\<lbrakk>a \<approx> b; c \<approx> d\<rbrakk> \<Longrightarrow> HComplex a c \<approx> HComplex b d"
```
```   301 unfolding approx_def by (simp add: Infinitesimal_HComplex)
```
```   302
```
```   303 lemma hcomplex_approx_iff:
```
```   304   "(x \<approx> y) = (hRe x \<approx> hRe y \<and> hIm x \<approx> hIm y)"
```
```   305 unfolding approx_def by (simp add: hcomplex_Infinitesimal_iff)
```
```   306
```
```   307 lemma HFinite_hRe: "x \<in> HFinite \<Longrightarrow> hRe x \<in> HFinite"
```
```   308 apply (auto simp add: HFinite_def SReal_def)
```
```   309 apply (rule_tac x="star_of r" in exI, simp)
```
```   310 apply (erule order_le_less_trans [OF abs_hRe_le_hcmod])
```
```   311 done
```
```   312
```
```   313 lemma HFinite_hIm: "x \<in> HFinite \<Longrightarrow> hIm x \<in> HFinite"
```
```   314 apply (auto simp add: HFinite_def SReal_def)
```
```   315 apply (rule_tac x="star_of r" in exI, simp)
```
```   316 apply (erule order_le_less_trans [OF abs_hIm_le_hcmod])
```
```   317 done
```
```   318
```
```   319 lemma HFinite_HComplex:
```
```   320   "\<lbrakk>x \<in> HFinite; y \<in> HFinite\<rbrakk> \<Longrightarrow> HComplex x y \<in> HFinite"
```
```   321 apply (subgoal_tac "HComplex x 0 + HComplex 0 y \<in> HFinite", simp)
```
```   322 apply (rule HFinite_add)
```
```   323 apply (simp add: HFinite_hcmod_iff hcmod_i)
```
```   324 apply (simp add: HFinite_hcmod_iff hcmod_i)
```
```   325 done
```
```   326
```
```   327 lemma hcomplex_HFinite_iff:
```
```   328   "(x \<in> HFinite) = (hRe x \<in> HFinite \<and> hIm x \<in> HFinite)"
```
```   329 apply (safe intro!: HFinite_hRe HFinite_hIm)
```
```   330 apply (drule (1) HFinite_HComplex, simp)
```
```   331 done
```
```   332
```
```   333 lemma hcomplex_HInfinite_iff:
```
```   334   "(x \<in> HInfinite) = (hRe x \<in> HInfinite \<or> hIm x \<in> HInfinite)"
```
```   335 by (simp add: HInfinite_HFinite_iff hcomplex_HFinite_iff)
```
```   336
```
```   337 lemma hcomplex_of_hypreal_approx_iff [simp]:
```
```   338      "(hcomplex_of_hypreal x \<approx> hcomplex_of_hypreal z) = (x \<approx> z)"
```
```   339 by (simp add: hcomplex_approx_iff)
```
```   340
```
```   341 lemma Standard_HComplex:
```
```   342   "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> HComplex x y \<in> Standard"
```
```   343 by (simp add: HComplex_def)
```
```   344
```
```   345 (* Here we go - easy proof now!! *)
```
```   346 lemma stc_part_Ex: "x:HFinite ==> \<exists>t \<in> SComplex. x \<approx> t"
```
```   347 apply (simp add: hcomplex_HFinite_iff hcomplex_approx_iff)
```
```   348 apply (rule_tac x="HComplex (st (hRe x)) (st (hIm x))" in bexI)
```
```   349 apply (simp add: st_approx_self [THEN approx_sym])
```
```   350 apply (simp add: Standard_HComplex st_SReal [unfolded Reals_eq_Standard])
```
```   351 done
```
```   352
```
```   353 lemma stc_part_Ex1: "x:HFinite ==> \<exists>!t. t \<in> SComplex &  x \<approx> t"
```
```   354 apply (drule stc_part_Ex, safe)
```
```   355 apply (drule_tac [2] approx_sym, drule_tac [2] approx_sym, drule_tac [2] approx_sym)
```
```   356 apply (auto intro!: approx_unique_complex)
```
```   357 done
```
```   358
```
```   359 lemmas hcomplex_of_complex_approx_inverse =
```
```   360   hcomplex_of_complex_HFinite_diff_Infinitesimal [THEN [2] approx_inverse]
```
```   361
```
```   362
```
```   363 subsection\<open>Theorems About Monads\<close>
```
```   364
```
```   365 lemma monad_zero_hcmod_iff: "(x \<in> monad 0) = (hcmod x:monad 0)"
```
```   366 by (simp add: Infinitesimal_monad_zero_iff [symmetric] Infinitesimal_hcmod_iff)
```
```   367
```
```   368
```
```   369 subsection\<open>Theorems About Standard Part\<close>
```
```   370
```
```   371 lemma stc_approx_self: "x \<in> HFinite ==> stc x \<approx> x"
```
```   372 apply (simp add: stc_def)
```
```   373 apply (frule stc_part_Ex, safe)
```
```   374 apply (rule someI2)
```
```   375 apply (auto intro: approx_sym)
```
```   376 done
```
```   377
```
```   378 lemma stc_SComplex: "x \<in> HFinite ==> stc x \<in> SComplex"
```
```   379 apply (simp add: stc_def)
```
```   380 apply (frule stc_part_Ex, safe)
```
```   381 apply (rule someI2)
```
```   382 apply (auto intro: approx_sym)
```
```   383 done
```
```   384
```
```   385 lemma stc_HFinite: "x \<in> HFinite ==> stc x \<in> HFinite"
```
```   386 by (erule stc_SComplex [THEN Standard_subset_HFinite [THEN subsetD]])
```
```   387
```
```   388 lemma stc_unique: "\<lbrakk>y \<in> SComplex; y \<approx> x\<rbrakk> \<Longrightarrow> stc x = y"
```
```   389 apply (frule Standard_subset_HFinite [THEN subsetD])
```
```   390 apply (drule (1) approx_HFinite)
```
```   391 apply (unfold stc_def)
```
```   392 apply (rule some_equality)
```
```   393 apply (auto intro: approx_unique_complex)
```
```   394 done
```
```   395
```
```   396 lemma stc_SComplex_eq [simp]: "x \<in> SComplex ==> stc x = x"
```
```   397 apply (erule stc_unique)
```
```   398 apply (rule approx_refl)
```
```   399 done
```
```   400
```
```   401 lemma stc_hcomplex_of_complex:
```
```   402      "stc (hcomplex_of_complex x) = hcomplex_of_complex x"
```
```   403 by auto
```
```   404
```
```   405 lemma stc_eq_approx:
```
```   406      "[| x \<in> HFinite; y \<in> HFinite; stc x = stc y |] ==> x \<approx> y"
```
```   407 by (auto dest!: stc_approx_self elim!: approx_trans3)
```
```   408
```
```   409 lemma approx_stc_eq:
```
```   410      "[| x \<in> HFinite; y \<in> HFinite; x \<approx> y |] ==> stc x = stc y"
```
```   411 by (blast intro: approx_trans approx_trans2 SComplex_approx_iff [THEN iffD1]
```
```   412           dest: stc_approx_self stc_SComplex)
```
```   413
```
```   414 lemma stc_eq_approx_iff:
```
```   415      "[| x \<in> HFinite; y \<in> HFinite|] ==> (x \<approx> y) = (stc x = stc y)"
```
```   416 by (blast intro: approx_stc_eq stc_eq_approx)
```
```   417
```
```   418 lemma stc_Infinitesimal_add_SComplex:
```
```   419      "[| x \<in> SComplex; e \<in> Infinitesimal |] ==> stc(x + e) = x"
```
```   420 apply (erule stc_unique)
```
```   421 apply (erule Infinitesimal_add_approx_self)
```
```   422 done
```
```   423
```
```   424 lemma stc_Infinitesimal_add_SComplex2:
```
```   425      "[| x \<in> SComplex; e \<in> Infinitesimal |] ==> stc(e + x) = x"
```
```   426 apply (erule stc_unique)
```
```   427 apply (erule Infinitesimal_add_approx_self2)
```
```   428 done
```
```   429
```
```   430 lemma HFinite_stc_Infinitesimal_add:
```
```   431      "x \<in> HFinite ==> \<exists>e \<in> Infinitesimal. x = stc(x) + e"
```
```   432 by (blast dest!: stc_approx_self [THEN approx_sym] bex_Infinitesimal_iff2 [THEN iffD2])
```
```   433
```
```   434 lemma stc_add:
```
```   435      "[| x \<in> HFinite; y \<in> HFinite |] ==> stc (x + y) = stc(x) + stc(y)"
```
```   436 by (simp add: stc_unique stc_SComplex stc_approx_self approx_add)
```
```   437
```
```   438 lemma stc_numeral [simp]: "stc (numeral w) = numeral w"
```
```   439 by (rule Standard_numeral [THEN stc_SComplex_eq])
```
```   440
```
```   441 lemma stc_zero [simp]: "stc 0 = 0"
```
```   442 by simp
```
```   443
```
```   444 lemma stc_one [simp]: "stc 1 = 1"
```
```   445 by simp
```
```   446
```
```   447 lemma stc_minus: "y \<in> HFinite ==> stc(-y) = -stc(y)"
```
```   448 by (simp add: stc_unique stc_SComplex stc_approx_self approx_minus)
```
```   449
```
```   450 lemma stc_diff:
```
```   451      "[| x \<in> HFinite; y \<in> HFinite |] ==> stc (x-y) = stc(x) - stc(y)"
```
```   452 by (simp add: stc_unique stc_SComplex stc_approx_self approx_diff)
```
```   453
```
```   454 lemma stc_mult:
```
```   455      "[| x \<in> HFinite; y \<in> HFinite |]
```
```   456                ==> stc (x * y) = stc(x) * stc(y)"
```
```   457 by (simp add: stc_unique stc_SComplex stc_approx_self approx_mult_HFinite)
```
```   458
```
```   459 lemma stc_Infinitesimal: "x \<in> Infinitesimal ==> stc x = 0"
```
```   460 by (simp add: stc_unique mem_infmal_iff)
```
```   461
```
```   462 lemma stc_not_Infinitesimal: "stc(x) \<noteq> 0 ==> x \<notin> Infinitesimal"
```
```   463 by (fast intro: stc_Infinitesimal)
```
```   464
```
```   465 lemma stc_inverse:
```
```   466      "[| x \<in> HFinite; stc x \<noteq> 0 |]
```
```   467       ==> stc(inverse x) = inverse (stc x)"
```
```   468 apply (drule stc_not_Infinitesimal)
```
```   469 apply (simp add: stc_unique stc_SComplex stc_approx_self approx_inverse)
```
```   470 done
```
```   471
```
```   472 lemma stc_divide [simp]:
```
```   473      "[| x \<in> HFinite; y \<in> HFinite; stc y \<noteq> 0 |]
```
```   474       ==> stc(x/y) = (stc x) / (stc y)"
```
```   475 by (simp add: divide_inverse stc_mult stc_not_Infinitesimal HFinite_inverse stc_inverse)
```
```   476
```
```   477 lemma stc_idempotent [simp]: "x \<in> HFinite ==> stc(stc(x)) = stc(x)"
```
```   478 by (blast intro: stc_HFinite stc_approx_self approx_stc_eq)
```
```   479
```
```   480 lemma HFinite_HFinite_hcomplex_of_hypreal:
```
```   481      "z \<in> HFinite ==> hcomplex_of_hypreal z \<in> HFinite"
```
```   482 by (simp add: hcomplex_HFinite_iff)
```
```   483
```
```   484 lemma SComplex_SReal_hcomplex_of_hypreal:
```
```   485      "x \<in> \<real> ==>  hcomplex_of_hypreal x \<in> SComplex"
```
```   486 apply (rule Standard_of_hypreal)
```
```   487 apply (simp add: Reals_eq_Standard)
```
```   488 done
```
```   489
```
```   490 lemma stc_hcomplex_of_hypreal:
```
```   491  "z \<in> HFinite ==> stc(hcomplex_of_hypreal z) = hcomplex_of_hypreal (st z)"
```
```   492 apply (rule stc_unique)
```
```   493 apply (rule SComplex_SReal_hcomplex_of_hypreal)
```
```   494 apply (erule st_SReal)
```
```   495 apply (simp add: hcomplex_of_hypreal_approx_iff st_approx_self)
```
```   496 done
```
```   497
```
```   498 (*
```
```   499 Goal "x \<in> HFinite ==> hcmod(stc x) = st(hcmod x)"
```
```   500 by (dtac stc_approx_self 1)
```
```   501 by (auto_tac (claset(),simpset() addsimps [bex_Infinitesimal_iff2 RS sym]));
```
```   502
```
```   503
```
```   504 approx_hcmod_add_hcmod
```
```   505 *)
```
```   506
```
```   507 lemma Infinitesimal_hcnj_iff [simp]:
```
```   508      "(hcnj z \<in> Infinitesimal) = (z \<in> Infinitesimal)"
```
```   509 by (simp add: Infinitesimal_hcmod_iff)
```
```   510
```
```   511 lemma Infinitesimal_hcomplex_of_hypreal_epsilon [simp]:
```
```   512      "hcomplex_of_hypreal \<epsilon> \<in> Infinitesimal"
```
```   513 by (simp add: Infinitesimal_hcmod_iff)
```
```   514
```
```   515 end
```